Enhancemen Humanoid - 大阪大学...Mobilit y Enhancemen t Con trol of Humanoid Rob ot based on...

Mobility Enhancement Control of Humanoid Robot

based on Reaction Force Manipulation

via Whole Body Motion

Tomomichi Sugihara

Mobility Enhancement Control of Humanoid Robotbased on Reaction Force Manipulation via Whole Body Motion

byTomomichi Sugihara

Abstract

This dissertation aims at the mobility enhancement of humanoid robots, especially,agile response to the command and exible absorption of disturbances caused by collisionswith the obstacles and so forth. It fucntions as fundamental re ex and equilibratory sensewhich a hierarchical controlling system for the high-level behavior stands on.

The strong nonlinearity of the dynamics gives no closed solutions to the motor controlof humanoid robots in principle. In addition, much uncertainty in the real environmentaccelerates the di�culty. The key issue is how to manipulate the reaction force from theenvironment through the interaction with it, since legged systems including humanoidsare driven via conversion of the internal joint torques to the external reaction force, dueto the absence of �xed points in the inertia frame. Another key is how to manage a largenumber of degrees-of-freedom. Such a kinematic complexity causes a serious increase ofcomputational cost. Although not a few groups have tackled to the problem, killer solutionis yet to appear, and the humanoid robots at present still lack of the high mobility. Theenhancement of it requires both a theoretical and a technical breakthrough.

This thesis introduces a simple model which approximately represents the dynamicsof humanoids by abstracting a relationship between the center of gravity(COG), the totalexternal force and its point of action called Zero Moment Point(ZMP), and makes it aloadstar to devise controlling strategies. The gap between the simple model and the strictmodel is complemented with COG Jacobian, which maps the whole joint angle velocitiesto COG velocity. Thanks to it, the desired reaction force is converted to the equivalentwhole body motion through COG motion rather in a small computational cost.

And this thesis discusses two strategies, namely, a pattern-based strategy which appliespreplanned motion trajectory, and a non-pattern-based strategy.

The former approach works for advanced tasks under the known environment, andrequires a stabilization controller to absorb disturbances due to the error in the modelsand so forth. The stabilization is achieved by simultaneous satisfaction of both the forcepattern to maintain the contact state and the geometric pattern to accomplish the task,and often su�ers from the con ict of them. In order to solve this problem, Dual TermAbsorption of Disturbance is proposed in accordance with the di�erence of their spans.

The latter approach has such a robustness that it absorbs much uncertainty of theenvironment and enables quick response to the command, though its application to high-level tasks is challenging. The horizontal motion is controlled by ZMP manipulation inaccordance with the similarity between the simple model of the humanoid dynamics andan inverted pendulum. And the vertical motion is controlled by impedance control. Theidea of impedance-switching is also introduced to manipulate the contact state even torealize the motion through aerial phase.

3

Contents

1 Introduction 91.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 History of Humanoid Robots . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Survey of Biped Motion Control . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 General Research Orientation Towards Biped Motion . . . . . . . . 131.3.2 Principle of Legged Motion . . . . . . . . . . . . . . . . . . . . . . . 141.3.3 Systematic Classi�cation of Control Strategies . . . . . . . . . . . . 15

1.4 The Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 The COG Jacobian of Legged Robots 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 The Concept of the COG Jacobian . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 The COG Jacobian of general oating-base multibody . . . . . . . 222.2.2 The Legged Robot Model . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 The COG Jacobian in each contact state . . . . . . . . . . . . . . . . . . . 252.3.1 Classi�cation of Contact State and Angular Momentum . . . . . . . 252.3.2 COG Movement in 0 Point Contact State . . . . . . . . . . . . . . 262.3.3 COG Movement in 1 Point Contact State . . . . . . . . . . . . . . 262.3.4 COG Movement in 2 Points(or Edge) Contact . . . . . . . . . . . . 272.3.5 COG Movement in 3 Points(or Face) Contact . . . . . . . . . . . . 292.3.6 The COG Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 A Motion control using the COG Jacobian . . . . . . . . . . . . . . . . . . 302.4.1 Simpli�cation of Equation of Motion using the COG Jacobian . . . 302.4.2 COG control via reaction force manipulation . . . . . . . . . . . . . 322.4.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Dual Term Absorption of Disturbance for Motion Stabilization 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Stabilization Strategy on Mass-Concentrated Model . . . . . . . . . . . . . 383.3 Dual Term Absorption of Disturbance . . . . . . . . . . . . . . . . . . . . . 41

3.3.1 Short Term Absorption of Disturbance . . . . . . . . . . . . . . . . 413.3.2 Long Term Absorption of Disturbance . . . . . . . . . . . . . . . . 41

5

6 CONTENTS

3.3.3 Simultaneous Realization of Short/Long Term Absorption of Dis-turbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 The Whole Body Cooperative Motion Control . . . . . . . . . . . . . . . . 43

3.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 ZMP Manipulation based on Inverted Pendulum Model 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Inverted Pendulum Model based ZMP Manipulation . . . . . . . . . . . . . 50

4.3 The Whole Body Cooperative External Force Manipulation . . . . . . . . . 53

4.3.1 The Equivalent COG Acceleration to the External Force . . . . . . 53

4.3.2 The Whole Body Motion Generation . . . . . . . . . . . . . . . . . 53

4.3.3 Structure of the Controller and the System . . . . . . . . . . . . . . 53

4.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Impedance Switching for Contact State Transition Control 59

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Inverted Pendulum Model with Impedance . . . . . . . . . . . . . . . . . . 60

5.3 Energy Control by Impedance Switching . . . . . . . . . . . . . . . . . . . 61

5.4 Contact Phase Invariant Whole-body Controller based on ConstraintsSwitching and SRV Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.6 Multiple Inverted Pendulum(MIP) Model Control . . . . . . . . . . . . . . 67

5.6.1 MIP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.6.2 Design of Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.6.3 Indirect Manipulation of Reaction Force . . . . . . . . . . . . . . . 70

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Miniature Humanoid Robot System towards High Mobility 73

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 Humanoid Robot \UT-�: mighty" . . . . . . . . . . . . . . . . . . . . . . . 73

6.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2.2 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2.3 Hardware system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3 Development of Z-DYNAFORM . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3.1 Motivation and Software Design policy . . . . . . . . . . . . . . . . 79

6.3.2 Construction of Z-DYNAFORM . . . . . . . . . . . . . . . . . . . . 80

6.4 pow(A,4); { an Architecture for Animatronic Arti�cial Agents . . . . . . . 82

6.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4.2 Emulation of pow(A,4); on RTLinux in mighty system . . . . . . . 83

CONTENTS 7

7 Conclusion 877.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2 Known problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.3 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.4 The Last Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Acknowledgement 91

Bibliography 93

Appendix 115

A ZMP, Interpretation and Application 117A.1 The Original Idea of ZMP . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

A.2 Spread of the Use of ZMP . . . . . . . . . . . . . . . . . . . . . . . . . . . 118A.2.1 Motion Planning based on ZMP Criterion . . . . . . . . . . . . . . 118A.2.2 Online Motion Modi�cation based on ZMP criterion . . . . . . . . . 119

A.2.3 ZMP Manipulation as the Input . . . . . . . . . . . . . . . . . . . . 119A.3 Extension of ZMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.4 Calculation of ZMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.5 Mass-Concentrated Model Approximation . . . . . . . . . . . . . . . . . . 121

B Micro/Macro Multipoint Contact Model in Forward Dynamics 123B.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

B.2 Model of Rigid Multibody System contacting with the Environment . . . . 125B.2.1 Classi�cation of Contact State . . . . . . . . . . . . . . . . . . . . . 125B.2.2 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 126B.2.3 Collision Detection and Friction . . . . . . . . . . . . . . . . . . . . 126

B.3 Uni�cation of Micro/Macro Contact Model . . . . . . . . . . . . . . . . . . 128

B.3.1 Micro Contact Model based on Spring-Damper Dynamics . . . . . . 128

B.3.2 Macro Contact Model and Equation of Contact . . . . . . . . . . . 129B.3.3 Solution of Equation of Contact with Quadratic Programming . . . 131B.3.4 Modi�cation of Friction Force . . . . . . . . . . . . . . . . . . . . . 132B.3.5 Procedure of Forward Dynamics Computation . . . . . . . . . . . . 132

B.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

C High Precision ZMP Manipulation with O�set Acceleration Estimation137

D Jacobian of Angular Momentum 139

E The Whole Body Motion Planning using the COG Jacobian 141

F The Equation of Motion of Inverted Pendulum 145

G Virtual Horizontal Plane 147

8 CONTENTS

H Generalized Inverse Matrix 149

I Singularity-Robust Inverse Matrix 153

J Complementary Pivoting Method 155

K Convex Hull on a plane 159

List of Publications 161

Chapter 1

Introduction

1.1 Background

This dissertation deals with the motion dynamics and control of humanoid robots { an-thropomorphic mechatronic devices. Greatly advanced modern technology has enabledto take the robots out to the human environment. And, not a few researchers have be-come focusing on the potential of humanoid robots. It is, in fact, partly because theyhave suitable shapes and forms for human-supporting activities as our future utilities,meeting our various needs in the shared infrastructures with the human beings as somementioned[1]. However, it should be mainly due to the promising sound of \humanoid";one would associate it with the robot which can go and act wherever human can, andperform a variety of motion as if they had intelligence.

In spite of such an expectation, the real humanoid robots in the current stage disap-pointingly lack of mobility against the severity of the real environment, which is �lled withmuch uncertainty, so that they are still threats for us human beings as if they are murder-ous weapons. Insu�cient mobility may cause some man-machine disasters as Fig.1.1 tellsin cartoon fashion. Research on the motion control of humanoid is still in the midway,and requires much further development.

The humanoid as a dynamical system has the following three features, namely,

1. A large number of degrees-of-freedom system, consisting of from 20 to 50 joints.Such a multibody system possesses chaotic dynamics intrinsically, so that it needsa sophisticated management of coordinates.

2. An underactuated system because of the absence of �xed points in the inertia frame.It necessarily requires a conversion from internal joint forces to the external reactionforces through the interaction with the environment at each contact point.

3. Structure-varying system, the total degrees of freedom of which varies as the contactstate changes, collocating with the environment.

These facts imply that it is a strongly nonlinear system, so that there are no closed solutionto control it basically. Besides the fact, the requirement of high-level tasks acceleratesthe di�culty. In order to enhance their mobility much more and put them into practicaluses in the severe environment, we have to overcome it.

9

10 1.2. HISTORY OF HUMANOID ROBOTS

Figure 1.1: Possible man-machine disasterous incidents caused by the lack of mobility

1.2 History of Humanoid Robots

\Robot"[2] is originated from a Czechoslovak word \robota", which stands for a labor,and means the automaton which serves us human beings as if it were a real creature,despite it isn't alive.

The history of the robot began with the humanoid robot. At the moment of birthin some myths, they were formed anthropomorphic shapes even before named \robot".Their imaginary ancestors { Talose in Greek myths, robots in \R.U.R."[2], Hadaly in\L'Eve future"[3], Maria in \Metropolis"[4] and so forth { and the robots appeared beforeapproved as a �eld of science { Android desinged by Da Vince, Steam Man by Moore[5],the �rst oriental android Gakutensoku by Nishimura and so forth { are thought to beprimitive humanoid robots. They were the very result of the curiosity towards ourselves.

On the other hand, the history of the robotics[6] began with a control of manipulator[7, 8] at Argonne National Laboratory, which was a trial to realize a human-like device,particularly focusing on the arm function. Since then, robotics have widely concernedabout functional mimic of biotical systems, not only the whole body type humanoidrobots. And in the early stage, it was mainly targeting to the industrial robots whichworked in limited environments such as plants.

After a decade passed since the robotics had begun, Katoh et al.[9] developed WABOT-1, which equipped visual recognition system, verbal communication system and quasi-static walking controller. Then, Katoh et al.[10, 11] also developed WABOT-2. Thoughit didn't have a walking ability, it had visual and auditory sensations and succeeded toplay the erectronic organ. The history of humanoid robot research after Katoh et al's is

CHAPTER 1. INTRODUCTION 11

also the history of an evolution of their body, which are classi�ed into i) the upperbodytype, ii) the clawler type and ii) the whole body type.

i)The Upperbody Type Humanoid Robots

are the extension of industorial manipulators in the sense that they are �xed to thebase. The study theme through them mainly lies on the fusion of intelligence and motorcontrol. Fanuc Ltd.[12] developed a large size humanoid Fanuc man. Tachi et al.[13]developed TELESAR for a study of R3(Realtime Remote Robot), to exploit the kinemat-ically human-like structure for tele-existence. Brooks et al.[14] developed Cog, based onthe idea that human intelligence is emerged through human interaction. Nagashima et al.developed Chiye(H1)[15], Saika(H2)[16, 17], respectively, aiming at the implementationof vision based autokinesis and re ex [18, 19, 20]. Takuma et al.[21] also developed H3,which is portably controlled via internet. Okada et al.[22] developed Cybernetic Shoul-der to reconstruct dynamical human skill on the motion based on active/passive hybridcompliance in the frequency domain[23] and programmable passive compliance[24].

ii)The Clawler Type Humanoid Robots

have wheels or caterpillars on their lower bodies, so that they can locomote in the world.Sugano et al.[11] developed Hadaly-2 with mechanically variable compliance on its arms,and studied the man-machine interaction and cooperation. Morita et al.[25, 26] developedWENDY and realized egg-clacking to show its precise force controllability. Kageyamaet al.[27] developed H4, on which tactile sensors are mounted, and studied the man-machine interaction based on the visual-tactile sensor fusion. tmsuk Co.,Ltd.[28] devel-oped TMSUK-4 and controlled it via PHS, and then developed TMSUK-5 with hydraulicactuators. Imai et al.[29] developed Robovie, which is commercially available for anypurposes[30], and have studied human-humanoid interaction and communication.

iii)The Whole Body Type Humanoid Robots

have legs, and thanks to it, they can move on much rougher terrains than clawler types.Hirai et al.[31, 32] developed P2, which was an epoch-making not only in humanoidrobotics but in the entire �eld of robotics. The latest achievement of them, ASIMO[33],features highly sophisticated control technology called i-Walk. Yamaguchi et al.[34, 35]developed WABIAN and realized a variety of motion, emotional walking, carrying ob-jects, dancing and so forth, and human-guided navigation on the walking [36]. Fujimotoet al.[37] developed a 20-axis humanoid robot MARI-I and examined position/force hy-brid control. Nagasaka et al.[38] developed H5, which has a light-weight body made byaluminum, to realize the dynamic whole body motion and vision-based walking motionusing a stereo vision system. Nishiwaki et al.[39] developed H6, mounting force sensor andtactile sensors in addition to the stereo vision on its body, as a platform of perception-action integration. And, H7[40] has been developed as a later version of H6. Kawadaindustories, Inc.[41] developed isamu on the technology of H6[39]. Konno et al.[42, 43]developed Saika-3 and proposed posture stability estimation on some arm-leg cooperatedtasks. Kuniyoshi et al.[44, 45] developed R.Daneel and experimentally achieved dynamicroll-and-rise motion[46, 47]. Kim et al.[48] developed CENTAUR, which is a centaurustype robot, having four legs and two arms, and did bilateral teleoperation of it. Yamano

12 1.2. HISTORY OF HUMANOID ROBOTS

et al.[49] developed Bonten-Maru as a study platform of the whole body motion con-trol. Kim et al.[50] developed KHR-1. Inaba et al.[51] developed Nagara and showedsome performance at RoboCup[52]. And, robos corp.[53] developed KOZOH based on thetechnology of Nagara. Okada et al.[54, 55] developed UT-�, which consists of CyberneticShoulder, Double Spherical hip joint and Backlash Clutch at its knee joints. Yamaguchiet al.[56] developed TH1 driven by arti�cial wire-tendons. Mizuuchi et al.[57, 58] devel-oped a fully tendon-driven humanoid Kenta with a exible spine, and proposed a motiondesign method buided by multi-sensors [59]. Kawato et al.[60] developed DB, which isdriven by hydraulic actuators, and realized a snare-drumming motion synchronized witha rhythmic beat sound using CPG oscillatory pattern[61]. Ministry of Economy, Tradeand Industry(METI) in Japan and New Energy and Industrial Technology DevelopmentOrganization(NEDO) began and promoted Humanoid Robot Project(HRP)[62] with Na-tional Institute of Advanced Industrial Science and Technology(AIST), ManufacturingScience and Technology Center(MSTC), Honda Co.,Ltd., Kawasaki Heavy IndustoriesLtd., Hitachi Ltd., Fujitsu corp., Kawada industries, Inc., university of Tokyo, Wasedauniversity, Tohoku university, Tsukuba university, Tokyo institute of technology, Osakauniversity, Kyoto university and other organizations. The project resulted remote opera-tion cockpit[63], simulation platform[64], HRP-1S[65], HRP-2P and HRP-2(Promet)[66].Pfei�er et al. [67, 68, 69] developed Johnnie, which hopefully jogs.

Recent advancement of components such as processors, motor drivers and batteriesencouraged the design of the above human-sized robots. On the other hand, Inaba etal.[70] proposed the concept of Remote-brained Robotics(RBR) in early '90s. Small sizerobots, built with DC servo modules and controlled from a remote host computer via radiotransmission, solved the technical issues, excluding heavy processors out of the robot body.They have evolved to humanoid robots by Igarashi et al. [71] and Kanehiro et al.[72].Using those small humanoids, Nagasaka et al.[73] realized visually guided swing, iron barand braquiation motion. Waita et al.[74] structurally reinforced the ordinary Remote-brained and developed Albiss. Ogura et al.[75] developed ToT, which one can also use asan input device for human-sized robots. Yamasaki et al.[76] developed PINO on the similartechnique to RBR. Furuta et al.[77, 78] promoted E-SYS project and developed some smallhumanoid robots named Mk. series. Using those robots, some walking motion controllingmethods were proposed[79, 80, 81, 82, 83, 84]. Furuta et al.[85] also developed morph3, onwhich a number of sensors was mounted, and realized some acrobatic whole body motion.Kuroki et al.[86] developed SDR-4X(or QRIO) as the entertainment robot and showed itshigh performance by robust locomotion on the uneven terrain, and visually guided kickingmotion of a football. Murase et al.[87, 88] developed HOAP-1, a commercially availablesmall size humanoid robot, and then, Kimura et al.[89] re�ned it as HOAP-2.

Astoundingly, the above listed researches and developments are only a part of thehumanoid robots in the world. One can �nd a number of personally created humanoidrobots by hobbyists, or even toys evolved almost to robots, introduced in websites onthe internet. Such recent explosively growing interests in humanoid robots vividly showsever-increasing expectations of society regarding them.


1.3 Survey of Biped Motion Control

1.3.1 General Research Orientation Towards Biped Motion

Robotics

EngineeringScience

MedicineSport Medicine

Rehabilitaion Medicine

Artificial LegBiomechanism

BiomechanicsCybernetics

Figure 1.2: Research orientation towards biped motion

The most fundamental issue on the humanoid robot is about the biped motion. Itvividly shows human's skill in the motion control, and has been a challenging problemof several �elds of researches. Fig.1.2 surveys the �elds which are interested in bipeds,described as follows.

Sport medicine, Rehabilitation medicion :rehabilitation and augmentation of hu-man's motion ability.

Biomechanics :mechanical analysis of biosystems.

Cybernetics :scienti�c analysis of information process inside biosystems.

Biomechanism :functional analysis and synthesis of intelligence and motor controller inbiosystems

Arti�cial leg :design and development of locomotion assistive device

Robotics :design and development of a locomotive machine and its controller.

As contrasted to medical and scienti�c interests which put an emphasis on the discov-ery of humanmechanism [90, 91, 92, 93, 94, 95] from an analytic point of view, engineeringis motivated by the craftsmanship to build up arti�cial biomimetic systems. And, an ex-pectation for the possibility of a pragmatic use of them as locomotors has encouragedsuch a synthetic approach.

14 1.3. SURVEY OF BIPED MOTION CONTROL

(A) Humanoid Robot System (B) Rolling-Contact System

f f

Figure 1.3: A metaphore of legged system for rolling-contact model

1.3.2 Principle of Legged Motion

In order to establish the controlling strategy of bipedal robots, let us generalize the prob-lem as much as possible and perceive its essence.

Summarizing the dynamical features itemized in section 1.1, one can conclude that abipedal, or legged system in general is modeled as a kinematic chain which lacks of the�xed root body and contacts with the environment. Due to the absence of an actuatorat the contact point, the control of this system requires the conversion from the internalforces generated by each joint actuator to the external reaction forces via interaction withthe environment.

From this point of view, it behaves as if it is the rolling-contact system illustrated byFig.1.3, which is strongly dominated by nonholonomic constraints. In this analogy, thecontact point is equivalently located at the point of action of the total external force, whileit can move discontinuously in the surface contact case di�erent from the actual rolling-contact objects. This fact implies that the control of legged necessitates a manipulationof the contact point, or the point of action of the total external force, and the force itselfacting at the point. A hyper redundancy with a number of degrees-of-freedom, however,makes it nontrivial to translate the above manipulation problem into the hole body motionof the humanoid system. Though such enormous DOFs enable the robot to perform avariety of motion, it also can causes a serious increase of computational cost. In otherwords, the control of it is achieved by a skillful manipulation of the reaction force onthe ever-changing contact state under nonholonomic constraints and a management of anumber of bodies without a bankruptcy. The following subsection reviews the previous


studies on bipedal robots from this point of view.

1.3.3 Systematic Classi�cation of Control Strategies

Biped Robotics

Mechanics DesignMotor Control

Tracking Control Passive Dynamics Neuro-systemMimetics

Design of ControllerDesign of Reference

Simple ModelBased

DynamicalConversion

EvolutionalAcquisition

Foot PlacementFeedforward

MomentumFeedback

PhaseDivisional

Figure 1.4: Classi�cation of researches on bipedal robots

Fig.1.4 shows a systematic classi�cation of biped robot researches from the viewpointof how to tackle the above requirements. Note that the actual results are multiple com-binations of these ideas, not necessarily classi�ed into them obviously.

Since Witt[96] reported the dynamic control of stepping motion in lateral plane by amechanical two-leg system, the fact that bipedal motion is the result of well-coupled me-chanical body and motor control has been realized. Biped robotics, then, has approachedeach aspect.

In the studies of mechanical design of biped bodies, various new actuators, gears,structures, springy joint mechanisms, dampers, etc. have been introduced (Katoh etal.[97], Funabashi et al.[98, 99], Yoneda et al.[100], Pratt et al.[101], Koganezawa etal.[102, 103], Demura et al. [104, 105, 106], Kaneko et al.[107], Sugahara et al.[108, 109],Okada et al.[54, 55] ).

Here, we review the previous studies of motor control, grouping them according to thebasic idea to deal with the problem of reaction force manipulation as follows.

The �rst group which is based on a straightforward tracking control began in late '60s.Vukobratovi�c et al. [110, 111] had already showed a basic strategy to devide the probleminto a design of referential motion pattern and of a stabilization controller. Thoughtheir �rst model was a simple connection of shu�ing lower extremities and a lever on itsupperbody, it would be augmented to an anthropomorphic mechanics [112]. They pointedout that one could discuss a planning of the reference in the absence of a model of theenvironment in accordance with an equilibrium between the resultant external force givento the system from the environment and the inertia force generated by a motion of thesystem itself, and proposed the concept of ZMP(Zero Moment Point, see Appendix A) asthe center of equilibrium.

Most of the succeeding researches have focused on the design of reference. Katoh etal.[9] adopted this ZMP as a criterion to check the physical validity of motion trajectories,

16 1.3. SURVEY OF BIPED MOTION CONTROL

despite they gave no solution to the problem to redesign them. After that, some simplemodel based methods have been proposed; they solved the equation of motion of eachmodel of a biped analytically and obtain the reference. Kajita et al. [113, 114, 115]proposed the Linear Inverted Pendulum Mode and designed the trajectory of COG(centerof gravity) by strictly linearized equation of motion. Thanks to it, a fast online trajectoryplanning based of sensory information about the terrain was achieved. Minakata et al.[116,117] proposed Virtual Inverted Pendulum Method which used the dynamics of virtuallyvaried its length and changed the walking speed. Kurematsu et al.[118] proposed to planthe COG trajectory and aquire the whole body motion which realized the desired COGmotion by neural network. The largest drawback of this simple model based is that iteasily su�ers from a modeling error.

Some have tried to design trajectories based on evolutional acquisition. The referentialpatterns are acquired through frequent trials and errors, using genetic algorithm, neuralnetwork and so forth (Arakawa et al.[119, 120], Kanehiro et al.[121], Yamasaki et al.[122],and Wol� et al.[123]). Though it requires less consideration of dynamics which botherssome designers, it does never create reliable motion patterns.

The problem of how to obtain trajectories inversely from the desired ZMP movementwas solved �rst by Yamaguchi et al.[34, 35]. They have proposed a computation of up-perbody trajectory which compensates the moment around the desired ZMP using FastFourier Transformation, and developed WL(Waseda Leg) series and WABIAN series. Na-gasaka et al.[124, 125] proposed a motion-captured trajectory modi�cation using geneticalgorithm, and realized biped walking and jumping motion. DasGupta et al.[126] pro-posed a design of waist joint trajectory by Fourier series using steepest descent methodwith a combination of motion-captured trajectory. Yamane et al.[127] developed Dynam-ics Filter which converts the input trajectory to physically consistent one. Nagasaka[128]developed Dynamics Filter, which consists of some \�lters" implementing partial opti-mization problems decomposed from the original problem, to convert the whole bodymotion trajectory to one with desired characteristics. Kitagawa et al.[129] proposed awhole body motion trajectory designing method based on enhanced ZMP. On the back-ground of the above dynamical conversion methods lies the birth of powerful computers,which can deal with the strict equation of motion.

These approaches that exclude the model of environment can substantially reduce thedi�culty of the problem, so that they are in the main stream of the control of bipedrobots. However, they are vulnerable to uncertainties in the real environment. A designof controller has been the other key issue in this group. Such controllers have been de-veloped based on momentum feedback. Some of them (Miyazaki et al.[130], Furusho etal.[131], Takenaka et al.[32], Nagasaka[128], Huang et al.[132], Park et al.[133]) supposeda combination with detailed reference obtained by the above methods, so that less ro-bustness against disturbances necessarily associated with them. A number of trials todesign e�ective controllers (Chow et al.[134], Gubina et al.[135], Mita et al.[136], Oga-sawara et al.[137], Kumagai et al.[138], Sorao et al.[139], Fujimoto et al.[140, 37], Obata etal.[141], Tamiya et al.[142][143], Takahashi et al.[144], Mitobe et al.[145], Inoue et al.[146],Takeuchi[147] Azevedo et al.[148] ), however, commonly su�ers from the complexity ofthe system and the requirement for a variety of motion.


Some have been focused on the function of foot placement feedforward control (Miuraand Shimoyama[149], Raibert et al.[150, 151, 152, 153], Westervelt et al.[154] ). Theirinterests are in a realization of an asymptotically stable system.

Since each gait ordinarily consists of typical phases in terms of supporting states, somephase divisional methods (Sano et al.[155], Pratt et al.[156, 157, 158], Kanzaki et al.[159],Toda et al.[81] ) have been proposed based on heuristic approaches, so that they are lessversatile.

The second group is based on the passive dynamics of the system itself, introduced anddiscussed �rst by McGeer[160, 161]. They basically try to utilize an inherent dynamicsof a walking system which shows a repetitive cyclic gait on the slope even without actu-ation (Kuo et al.[162], Ruina et al.[163, 164], Goswami et al.[165], Sugimoto et al.[166],Yamakita et al.[167] ). Their main concern is an emergence of periodic gait, not a respon-sive and agile motion control.

Neuro-system mimetics is the third group, based on an idea that a complex biomimeticcoupling of an arti�cial neural network and body dynamics might emerge a behavior whichis robust against disturbances and e�cient in terms of energy consumption. Doya[168] wasthe pioneer who developed a neural network controller for a walking machine { though itwas a simple 3-link robot { to acquire a variety of motion. After Taga [169, 170, 171, 172]applied CPG(central pattern generator [173]) for the control of a musculo-skeletal bipedalmodel and showed that an adaptive walking motion against the variation of terrains couldbe emerged from the interaction between neural oscillatory controller, body dynamicsand the dynamics of environment, not a few researchers (Katayama et al.[174], Hase etal.[175], Fujiwara et al.[176], Miyakoshi et al.[177], Cao et al.[178], Kimura et al.[179], Huet al.[180, 181], Matsuura et al.[182], Gotoh et al.[183], Endo et al.[184] ) have challengedin similar strategies. However, a clear methodology to design such neural controllers isyet to be shown.

1.4 The Goal

The goal of this dissertation is to develop a controlling method of humanoid robots fora mobility enhancement of them to realize robots with so high mobility that can quicklyresponse to commands from operators and can exibly absorb disturbances from uncer-tainties in the real world. Various di�culties, particularly in their dynamical complexityhave prevented the improvement of the mobility, despite a number of groups have tackledto this problem. The mobility enhancement requires a breakthrough to overcome them.

The former studies have mostly focused on a periodic and steady walking, which is atypical motion of legged robots, assuming a �xed assignment of joints, so that they wereended as speci�cs. However, it is though that legged motion in general is based on thefundamental principle as is mentioned in section1.3.2. A common solution to manipulatethe reaction force from the environment through the interaction with it and to manage alarge number of joints should be established.

Our basic policy towards the problem is a straightforward approach in accordance withan understanding of the plant dynamics, which one can conclude is the most promising

18 1.5. OUTLINE OF THIS THESIS

from the above survey of previous works. The �rst thing to do is to reveal the essenceof this quite complicated dynamical system, and set up a reasonable lower-dimensionalmodel, which the designing strategy of controller will be based on. In this case, the largegap between such the simple model and the strict humanoid model has to be comple-mented by any means, and it will naturally be the core technique to convert the internalforces to the equivalent external reaction forces with a cooperation of the whole jointsof the robot body. Once one gets such a manipulation maneuver of reaction forces, theremaining problem is how to decide them as manipulated variables to achieve desiredmotions and tasks.

The main contribution of proposed is to give a fundamental re ex and an equilibratorysense, on which a hierarchical controlling system stands, to the robots.

1.5 Outline of this thesis

This dissertation consists of seven chapters.Chapter 2 will present a concept and a mathematical expression of the COG Jaco-

bian, a map from the whole joint angle velocity to COG velocity, of the legged robotin generalized form with a classi�ed variation of the contact state of legged. It will bequantitatively proved that COG, a kinematic parameter originally, approximately repre-sents the core dynamics of the total system and helps to design the controller of leggedrobots. Thanks to it, the lower-dimensional model is associated with the strict humanoidmodel in substantialy less computational cost, and thus, the desired reaction force will betranslated into the equivalent whole joint movement.

From chapter 3 to 5, some strategies to decide the reaction force to accelerate andstabilize the whole body motion will be proposed.

In chapter 3, Dual Term Absorption of Disturbance for a pattern-based motion controlwill be presented. A pattern-based approach greatly reduces the di�culty on the designof a variety of motion, while it requires a stabilization, which in the case of humanoidsmeans the maintenance of the consistency between the geometric pattern and the forcepattern. Though they are frequently in con ict with each other, a di�erence of the span inconsidering the avoidance of short term crisis and the recovery of posture will be pointedout. Dual Term Absorption of Disturbance is an idea to stabilized the robot in suchmultiply-layered time order.

In chpter 4, an online motion controlling method via ZMP manipulation will be pro-posed based on the dynamical similarity of humanoid robots and inverted pendulums.Manipulation of ZMP enables to handle directly severe physical constraints due to theabsence of any �xation in the inertia frame. The control of dynamic motion on humanoidrobots with a large number of degrees-of-freedom will be achieved as well as of invertedpendulum. It doesn't need detailed motion trajectory patterns as inputs, and thus, quickresponse the commands and robust absorption of large disturbances will be achieved. Thestability margin around the reference will be expanded than a pattern-based approach.

Chapter 5 will augment the ZMPmanipulation method introduced in chapter 4, addingan impedance to the axis of the inverted pendulum model. The vertical energy controlwill be achieved by switching the impedance, and the contact state will be manipulated


rather easily.In chapter 6, a miniature size humanoid robot developed towards the high mobility,

its design, mechanics and hardware will be introduced. And then, a design policy of soft-wares in the �eld of robotics will be discussed, and commonly useful dynamics calculationlibrary Z-DYNAFORM will be presented. A new architecture of robot controlling system,pow(A,4);, will also be shown.

The contribution of this dissertation will be summarized in chapter 7.

Chapter 2

The COG Jacobian of Legged Robots

2.1 Introduction

In this chapter, we discuss the COG Jacobian of legged robots, which helps to denote,understand and even control the dynamics of the system.

Legged robots consist of a large number of links and joints (20{50) in general. And theyare also structure-varying systems [185], namely, the total degrees of freedom varies as thecontact state changes; they behave as oating-base multilinks, or as rolling-contact bodieson the ground in some situations, and moreover, contact state transition is a discontinuousphenomenon. These features give robots non-holonomic constraints, so that the design ofthe controller is still a challanging problem. In order to solve it, it is e�cient to abstractthe essence of system dynamics and to compose a lower-dimensional model.

Miyazaki and Arimoto[130] applied Singular Perturbation method to a multibodybiped robot and classi�ed its modes into the fast mode and the slow mode. Furushoand Masubuchi[131] showed it possible to aproximate the total behavior only by the slowmode with a good precision, servocontrolling all joints except the ankle joint of supportingleg strongly. The slow mode introduced by these two researches is due to the similarityof dynamics with inverted pendulums in the sense that the center of gravity(COG) islocated above the point of action of the external force. Not a few controlling methodsbased on the lower-dimensional model which assumes the total mass is concentrated atCOG have been proposed according to this fact [116] [113] [32] [140] [145] [147]. Thevalidity of focusing on COG as it represents the core dynamics of legged systems has beencon�rmed. However, a gap between the mass-concentrated model presented in terms ofCOG and the precise multibody model is so large that how to complement it is also aproblem; since COG is a many variable nonlinear function de�ned by the whole jointangles, it is hard to solve the inverse problem to substitute the control of COG withthe control of joint angles. Though Kurematsu, Kitamura and Nakai[118] tried to aquirethat inverse function which maps the desired COG motion to the whole body motion byneural network, it has some reservation about if it can surely reconstruct the nonlinearcharacteristics of COG function. And this also has a drawback not to be applied in thecase that mass distribution of the robot varies by picking up some object for instance. Inthe study of Mizuuchi et al.[186], COG is carried to the desired position by controlling

21

22 2.2. THE CONCEPT OF THE COG JACOBIAN

each joint to make the error between the referential and actual torques minimized. Itdoesn't consider the e�ect of inertia force, so that it is only applicable for quasi-staticmotion. Nagasaka et al.[128] proposed Trunk Position Compliance Control based on theidea that COG displacement is proportional to trunk position displacement in most cases.This assumption has a natural limit to the applicable types of motion.

Manipulator Jacobian which relates the joints movement to the endpoint movement isfrequently used for the control of manipulators [187][188]. Based on the same idea, somehas reported that COG is controlled using the COG Jacobian which relates the jointsmovement to COG movement. Hirano et al.[189] modeled the biped robot as a serialthree linkage in the sagittal plane, and derived the COG Jacobian analytically. Tamiyaet al.[142] proposed the numerical calculation of the COG Jacobian, giving �nite tinydisplacements to each joint and computing the corresponding COG displacement. It isless accurate and requires a large amount of computation. Furthermore, it models therobot as a kinematic chain whose root is located at the supporting foot, so that one hasto switch the model with the contact state transient.

This chapter introduces the expression and computation of the COG Jacobian forgeneral legged robots. Umetani and Yoshida[190] proposed the concept of generalizedJacobian for oating-base multibody systems as space robot manipulators for instance,based on the face that they are constrained by the momentum conservation law. In thecase of legged robots, the representation of the COG Jacobian is more complicated thanthat of space robots since they are structure-varying.

We model the legged robot as a oating-base multibody just like the space robotsat �rst. And we classify the contact state of the robot in accordance with the physicallaws which constrains the system. Then, we discuss the relationship between momentumand COG velocity, and derive the COG Jacobian. Revealing the physical meaning of theCOG Jacobian from the equation which relates the joints movement to COG movement,the e�cacy of it for designing of the controller will be explained. An example motioncontrolling method for humanoid robot will also be shown.

The signi�cance of our method is that

1. one doesn't have to switch the model, or modify its structure even when the contactstate changes.

2. the strict COG Jacobian is computed only by doing the whole body kinematics inany cases. It has an advantage in terms of both the computational cost and theaccuracy.

2.2 The Concept of the COG Jacobian

2.2.1 The COG Jacobian of general oating-base multibody

In this section, the concept of the COG Jacobian of the general oating-base multibody isformulated. COG pG of a rigid multibody system is a function of generalized coordinatesq.

pG = pG(q) (2.1)

CHAPTER 2. THE COG JACOBIAN OF LEGGED ROBOTS 23

Thus, there exists Jacobian matrix shown as follows.

dpG =@pG@q

dq (2.2)

In the case of the oating-base multibody, the generalized coordinates q includes un-deractuated coordinates for the base link. We classify q into the underactuated qP andactuated qA. Then Eq.(2.2) becomes as follows.

dpG =@pG@qP

dqP +@pG@qA

dqA (2.3)

If qP is related with qA by Eq.(2.4), Eq.(2.3) turns as Eq.(2.5) and the movement ofactuated coordinates will be related to COG movement, involving the movement of un-deractuated coordnates implicitly.

dqP = QP (qA)dqA (2.4)

dpG =

�@pG@qP

QP (qA) +@pG@qA

�dqA (2.5)

The following equation appearing in Eq.(2.5) is the generalized form of the COG Jacobianof the oating-base multibody.

JG � @pG@qP

QP (qA) +@pG@qA

(2.6)

If the whole joints are actuated, qP represents the position and orientation of the base linkin the inertia frame, while qA represents the joint angles. And when the system doesn'thave any �xed point in the inertia frame, Eq.(2.4) corresponds to the linear/angularmomentum conservation law with zero initial velocity. When it contacts with the envi-ronment, Eq.(2.4) represents the relative movement of the contact point with respect tothe base link, and qP corresponds to the counteraction of the base link.

2.2.2 The Legged Robot Model

Here, we model the legged robot as an open-loop kinematic chain whose base link(link0, hereafter) oats as is shown by Fig.2.1. Let �W the inertia frame and �0 the totalbody frame which is �xed on the base link. Then, an arbitrary point p in the robot body,and the angular velocity of an arbitrary link k is expressed by the following equations,respectively.

p = p0 +R00p (2.7)

!k = !0 +R00!k (2.8)

where p0 is the position of the original point of �0 with respect to �W , R0 is the trans-portation matrix of the attitude from �0 to �W , !0 is the angular velocity of �0 withrespect to �W , 0!k is the relative angular velocity of link k with respect to �0, and

0p is

24 2.2. THE CONCEPT OF THE COG JACOBIAN

Σ

Σ

W

0p R00 ,

Figure 2.1: Multibody Legged Robot Model

the relative position of p with respect to �0. Di�erentiating Eq.(2.7) by time, the velocityof the point p with respect to �W is calculated as follows.

_p = _p0 + !0 �R00p+R0

0 _p

= _p0 � (R00p)� !0 +R0

0 _p (2.9)

where 0 _p is the relative velocity of p with respect to �0. And v� for an arbitrary vectorv = [ vx vy vz ]

T represents the 3 � 3 outer products matrix. Applying the methodproposed by Orin et al.[191], we can rather easily obtain Jacobian matrices 0J!k and

0J

which satisfy the following equations.

0 _p = 0J _� (2.10)0 _!k =

0J!k_� (2.11)

Now, according to Eq.(2.9), COG velocity of the robot _pG is expressed by the followingequation.

_pG = _p0 � (R00pG)� !0 +R0

0 _pG (2.12)

And 0 _pG is calculated as follows.

0 _pG =n�1Xk=0

�k0 _pGk =

n�1Xk=0

�k0JGk

_� (2.13)


where n is the number of links, 0pGk is COG position of link k with respect to �0, and� is the whole joint angles of the robot. 0JGk is available from Eq.(2.10). �k is the massratio of link k to the total mass of the robot m, de�ned by the following equation.

�k � mk

m

m �

n�1Xj=0

mj

!(2.14)

From Eq.(2.13), we de�ne the relative COG Jacobian in �0,0JG as follows.

0JG �n�1Xk=0

�k0JGk (2.15)

Using this, Eq.(2.12) turns as follows.

_pG = _p0 � (R00pG)� !0 +R0

0JG_� (2.16)

Here, we make the following assumptions for the sake of simplicity.

1. The whole joints included in � are actuated.

2. Su�cient friction force prevents the contact points from slipping.

3. The environment is modeled as a rigid body. Deformation of it by the e�ect of robotmotion can be ignored.

If the generalized velocity of underactuated coordinates _p0 and !0 are related with _� inthe form of Eq.(2.4), the COG Jacobian in the form of Eq.(2.6) will be derived. In thenext section, we will discuss how _p0 and !0 are expressed in each contact state, and derivethe COG Jacobian.

2.3 The COG Jacobian in each contact state

2.3.1 Classi�cation of Contact State and Angular Momentum

(A)0 Point Contact (B)1 Point Contact (C)2 Points (edge) Contact (D)3 Points (face) Contact

Figure 2.2: Contact States of Legged

26 2.3. THE COG JACOBIAN IN EACH CONTACT STATE

The contact state of the legged robot, which varies as the robot moves, can be classi�edinto the following four types shown in Fig.2.2 in accordance the physical laws whichconstrains the system, namely, i) 0 point contact, ii) 1 point contact, iii) 2 points(or edge)contact, and iv) more than 3 points(or face) contact. Especially, the system is dominatedby angular momentum conservation law, which is a really strong constraint, in state i)ii)and iii). Then, we introduce the angular momentum of the system in advance.

The angular momentum of the system around the point p, L(p) with respect to theinertia frame �W is calculated by the following equation.

L(p) =n�1Xi=0

�mi(pG;i � p)� _pG;i + I i!i

= R0

n�1Xi=0

�mi(

0pG;i �0p)� (0R _p0 �0pG;i �0R!0 +0 _pG;i) +

0Ii(0R!0 +

0!i)(2.17)

where I i is the inertia tensor around COG of link i, 0R is the inverse matrix of R0

(� R�10 ), and 0I i is the transported I i from �W to �0

�� 0RIiR0

�. Thus, we get the

angular momentum with respect to �0 as follows.

0L(0p) =n�1Xi=0

�mi(

0pG;i �0p)� (0R _p0 �0pG;i �0R!0 +0 _pG;i) +

0I i(0R!0 +

0!i)

(2.18)

Using this, COG movement is derived in the following subsections.

2.3.2 COG Movement in 0 Point Contact State

When the robot doesn't have any contact points with the environment and the reactionforce doesn't work to it, COG velocity is absolutely ruled by momentum conservationlaw. That is to say,

m _pG =M �mgt (2.19)

or,_pG = vG � gt (2.20)

whereM is the initial linear momentum of the system, vG is the initial COG velocity, gis the acceleration of gravity, and t is the time. COG velocity is uncontrollable by jointactuation under this state, and therefore, the COG Jacobian doesn't exist.

2.3.3 COG Movement in 1 Point Contact State

When the robot contacts with the environment at only one point pF1, the velocity of pF1is zero, namely,

_pF1 = 0 , _p0 � (R00pF1)� !0 +R0

0 _pF1 = 0 (2.21)


Thus,

_p0 = R0(0pF1 � !0 �0 _pF1) (2.22)

And, since the angular momentum around pF1 is conserved as is shown in the followingequation.

0L(0pF1) =0LF1

,n�1Xi=0

�mi(

0pG;i �0pF1)��(0pG;i �0 _pF1)�0R!0 +

0 _pG;i �0 _pF1+0Ii(

0R!0 +0!i)

�=

n�1Xi=0

�0I i �mi(

0pG;i �0pF1)20R!0 +

n�1Xi=0

�mi(

0pG;i �0pF1)� (0JG;i �0JF1) +0I i

0J!i

_�

= 0IF10R!0 +

0JAF1_� =0LF1 (2.23)

where 0LF1 is the initial angular momentum around pF1 with respect to �0, and0IF1,

0JAF1 are de�ned respectively as follows.

0IF1 �n�1Xi=0

�0I i �mi(

0pG;i �0pF1)2

(2.24)

0JAF1 �n�1Xi=0

�mi(

0pG;i �0pF1)� (0JG;i �0JF1) +0I i

0J!i

(2.25)

We note that 0IF1 is the inertia tensor of the whole system around 0pF1 with respect to�0, and thus, it is nonsingular matrix. COG velocity is derived from Eq.(2.16)(2.22) and(2.23) as follows.

_pG = �R0

�(0pG �0pF1)�0I�1F1

0LF1

+R0

�0JG �0JF1 + (0pG �0pF1)�0I�1F1

0JAF1

_�

(2.26)

2.3.4 COG Movement in 2 Points(or Edge) Contact

When the robot contacts with the environment at two points pF1, pF2, the velocity ofboth these points are zero.

_pF1 = 0 , _p0 � (R00pF1)� !0 +R0

0 _pF1 = 0 (2.27)

_pF2 = 0 , _p0 � (R00pF2)� !0 +R0

0 _pF2 = 0 (2.28)

Thus, we obtain the following two equations.

_p0 = R0(0pF1 �0R!0 �0 _pF1) (2.29)

28 2.3. THE COG JACOBIAN IN EACH CONTACT STATE

(0pF2 �0pF1)�0R!0 =0 _pF2 �0 _pF1

= (0JF2 �0JF1) _� (2.30)

And, the angular momentum around the edge which is created by connecting pF1 withpF2 as follows.

0dT0L(0pF1) =0LF1

, 0dT0IF10R!0 +

0dT0JAF1_� = 0LF1 (2.31)

where 0d is de�ned as follows.

0d �0pF2 �0pF1k0pF2 �0pF1 k

(2.32)

and 0LF1 is the initial angular momentum around the edge. Now, since we are focusingon the case that the two contact points are di�erent with each other (i.e. 0pF1 6= 0pF2),the rank of the matrix (0pF2 �0pF1)� is 2, namely, Eq.(2.30) has two independent sube-quations. Here, we abstract them as:

0P F1;20R!0 =

0JF1;2_� (2.33)

and combining them with Eq.(2.31).

0eIF10R!0 = �0eJF1_� +0eLF1 (2.34)

where

0eIF1 � � 0dT0IF10P F1;2

�(2.35)

0eJF1 ��0dT0JAF1

�0JF1;2

�(2.36)

0eLF1 ��0LF10

�(2.37)

Therefore, COG velocity is derived from Eq.(2.16)(2.29) and (2.34) as follows.

_pG = �R0

n(0pG �0pF1)�0eI�1F10eLF1

o+R0

n0JG �0JF1 + (0pG �0pF1)�0eI�1F10eJAF1

o_�

(2.38)

Let us consider a particular case that both the two contact points exist on the samelink, including such the case that the robot stands on its toe edge. And, let link F thecontacting link, and 0d?(3� 2 matrix) the orthogonal complementary bases of 0d. Then,link F doesn't rotate around two lines whose direction vectors are the bases of 0d?, whichis expressed by the following equations.

0d?T (0R!0 +0!F ) = 0

, 0d?T0R!0 = �0d?T0J!F_� (2.39)


Thus, we can substitute Eq.(2.35) and (2.36) for the following one in Eq.(2.34) in thiscase.

0eIF1 � � 0dT0IF10d?T

�(2.40)

0eJF1 ��0dT0JAF10d?T0J!F

�(2.41)

2.3.5 COG Movement in 3 Points(or Face) Contact

When the robot contact with the environment at three points pF1, pF2 and pF3, thevelocity of all these points are zero.

_pF1 = 0 , _p0 � (R00pF1)� !0 +R0

0 _pF1 = 0 (2.42)

_pF2 = 0 , _p0 � (R00pF2)� !0 +R0

0 _pF2 = 0 (2.43)

_pF3 = 0 , _p0 � (R00pF3)� !0 +R0

0 _pF3 = 0 (2.44)

Then, we get the following three equations.

_p0 = R0(0pF1 �0R!0 �0 _pF1) (2.45)

(0pF2 �0pF1)�0R!0 =0 _pF2 �0 _pF1

= (0JF2 �0JF1) _� (2.46)

(0pF3 �0pF1)�0R!0 =0 _pF3 �0 _pF1

= (0JF3 �0JF1) _� (2.47)

Now, we are considering the case that the three points0pF1,0pF2 and

0pF3 are di�erent witheach other, and three independent subequations exist in Eq.(2.46) and (2.47). Abstractingthem as:

0P F1;2;30R!0 =

0JF1;2;3_� (2.48)

and COG velocity is derived from Eq.(2.16)(2.45) and (2.48) as follows.

_pG = R0

�0JG �0JF1 � (0pG �0pF1)�0P�1

F1;2;30JF1;2;3

_� (2.49)

Let us consider a particular case that the whole three points exist on the same linkF , including the robot stands on its sole. Here, link F doesn't rotate in any direction asis expressed by the following equation.

0R!0 +0!F = 0

, 0R!0 = �0J!F_� (2.50)

Then, COG velocity in this case is represented by the following simple form.

_pG = R0

�0JG �0JF1 + (0pG �0pF1)�0J!F

_� (2.51)

30 2.4. A MOTION CONTROL USING THE COG JACOBIAN

In the cases that the robot contacts with the environment at more than three points,the same equation with Eq.(2.51) or Eq.(2.49) is derived, choosing arbitrary three con-tact points which don't exist on the same line, though one should take the additionalconstraints which denote the condition that the other points stay contact with the envi-ronment into account.

We additionally remark that, the robot can generate a certain amount of momentagainst the ground in this state, so that the angular momentum is controllable through themovement of actuated joints. Jacobian of angular momentum which maps the movementof joints to the angular momentum is derived by deforming Eq.(2.18) in Appendix D.Kajita et al.[192] also discussed a similar idea.

2.3.6 The COG Jacobian

Consequently, the COG Jacobian in each case is represented as follows fromEq.(2.26)(2.38) (2.49) and (2.51).

JG =

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

unde�ned (non-contact)

R0

�0JG �0JF1 + (0pG �0pF1)�0I�1F1

0JAF1

(1 point contact)

R0

n0JG �0JF1 + (0pG �0pF1)�0eI�1F10eJAF1

o(2 points(or edge) contact)

R0

�0JG �0JF1 � (0pG �0pF1)�0P�1

F1;2;30JF1;2;3

(3 points contact)

R0

�0JG �0JF1 + (0pG �0pF1)�0J!F

(face contact)

(2.52)

And in 1 point contact and 2 points contact cases, the movement around the contactpoint is dominated by angular momentum conservation law, so that the e�ect of the initialangular momentum remains in the subsequent COG motion as is shown by Eq.(2.26) and(2.38).

In addition, one should note that the COG Jacobian only describes the kinematicrelationship between the joint angles and COG under certain contact state, not assurethe physical consistency of the contact state itself, though it involves some mass propertyof the model.

2.4 A Motion control using the COG Jacobian

2.4.1 Simpli�cation of Equation of Motion using the COG Ja-cobian

Although the COG Jacobian originally means the kinematic map from the movement ofjoints to COG movement, it can represent the dynamics of the systems which is largely


a�ected by the external reaction force such as legged robots in nature. In this section, weexplain it and show an example motion control of a humanoid robot based on the idea.

The equation of motion of the model introduced in section 2.2.2 is expressed as follows[140]. �

H11H12

H21H22

��q0��

�+

�b1b2

�=

�0�

�+

NXk=1

�Kk11Kk12

Kk21Kk22

��fknk

�(2.53)

where

�q0 ��p0_!0

�(2.54)

and H ij is the inertia matrix, bi is the nonlinear term involving Coriolis, centrifugal andgravity force, � is the joint torques, N is the number of contact points between the robotand the environment, fk, nk are the force and moment which act at contact point k,respectively, and K ik is the map from the external force to generalized force.

The upper rows in Eq.(2.53) derive from the absence of generalized force which ac-tuates 6 DOFs of the base link, and are nonholonomic constraints including angularmomentum condition. Conclusively, the whole body motion requires a conversion fromthe internal force � to the external force fk and nk via interactions with the environmentunder these comstraints. The dynamics of this conversion mechanics appears in the lowerrows in Eq.(2.53). Since H ij and other vectors and matrices are strong nonlinear, fk,nk are generated not only from internal condition of the legged system but also physicalcharacteristics of the environment, and it often becomes hyperstatic problem, the rela-tionship between � and fk, nk is not trivial. Due to these facts, it's not a practical ideato design the controller directly based on the equation of motion (2.53). Here, we focuson the total linear movement of the system, ignoring the local motion of each link, andthen, derive the following equation.

n�1Xi=0

mi(�pG;i + g) =NXk=1

f k

, m(�pG + g) = f (2.55)

where f = [ fx fy fz ]T is the total external force working to the system. It implies

that movement of COG, which is originally a geometric property, can represent the coredynamics of the system without considering local dynamics of each link.

Suppose the robot contacts with the environment at no fewer than three points asshown in section 2.3.5 for the sake of simplicity. COG acceleration in this case is expressedas follows under the assumption in section 2.2.2.

�pG = JG�� + _JG

_� (2.56)

( _JG_� is obtained by a computation proposed in [193]). Manipulation of COG according to

Eq.(2.56) equivalently means indirect manipulation of the external force from Eq.(2.55).

32 2.4. A MOTION CONTROL USING THE COG JACOBIAN

2.4.2 COG control via reaction force manipulation

Let us control COG position of the robot under double leg supported condition via manip-ulation of the external force using the COG Jacobian. Suppose the desired COG positionis refpG, it could be achieved by PD control as follows.

f = P (refpG � pG)�D _pG +mg (2.57)

where P and D are the proportional gain matrix and the di�erential gain matrix, re-spectively. Legged robot, however, doesn't have any �xed point in the inertia frame, sothat the robot cannot generate arbitrary reaction force. In particular, the following twoconditions must be satis�ed at least.

f � g � 0 (2.58)

pZ 2 S (2.59)

pZ = [ xZ yZ zZ ]T is the point which the horizontal component of the moment workingaround is zero, so that it is called ZMP(Zero Moment Point)[112]. And, S represents thesupporting region of the robot (the convex hull of the contact points). Both Eq.(2.58)and (2.59) are due to the fact that the external force never works in the direction of pullthe robot.

pZ is calculated from the following equation.

(pG � pZ)� f + nG = nZ (2.60)

where nG = [ nGx nGy nGz ]T and nZ = [ nZx nZy nZz ]

T are the moment around COGand ZMP, respectively. Suppose z-axis of the inertia frame coincides with the direction ofgravity, both the x and y components of nZ are zero from the de�nition of ZMP. Thus,the following equations are derived from Eq.(2.55) and (2.60).

xZ = xG � fx(zG � zZ) + nGyfz

(2.61)

yZ = yG � fy(zG � zZ)� nGxfz

(2.62)

Then, we control COG by the following method.

i) Decide a tentative indirect input f according to Eq.(2.57).

ii) If fz becomes less than a certain limit minfz, replace it byminfz in order to satisfy

Eq.(2.58),

iii) In Eq.(2.61) and (2.62), zZ is the height of the ground(known parameter). SupposenGz and nGy are ignorable conparing with the e�ect of linear force. And, if Eq.(2.59)is not satis�ed by f , replace pZ with the nearest point to it in S. Intuitivelyspeaking, it functions as if the robot plant its opposite foot of the desired motiondirection �rmly [145].


iv) Calculate the desired COG acceleration �pG according to Eq.(2.55).

�xG =fz(xG � xZ)

m(zG � zZ)(2.63)

�yG =fz(yG � yZ)

m(zG � zZ)(2.64)

�zG =fzm� g (2.65)

where g is the absolute value of the acceleration of gravity.

v) Aside from this procedure, consider the other constraint to remain double leg sup-porting as follows.

R�qL =RJL�� +R _JL

_� (2.66)R�qL = P R;L(

refRqL �RqL)�DR;LR _qL (2.67)

where RqL is the relative position and orientation of the left foot with respect tothe right foot (6times1), RqL is the desired vector of it, and PR;L, DR;L are theproportional gain matrix and the di�erential gain matrix, respectively. RJL andR _JL are also computed by the same method in [193].

vi) Combine Eq.(2.56) and (2.66).

�q = J �� + _J _� (2.68)

Obtain the joints angle acceleration by solving Eq.(2.68) by weighted generalizedinverse matrix .

�� = J#(�q � _J _�) (2.69)

2.4.3 Simulation

We simulated the COG position control of a bipedal robot using the method in theprevious subsection. Fig.2.3 is the model, joint con�guration and speci�cation of therobot. The simulation is done with Z-DYNAFORM [194]. Firstly, we set the desiredCOG position near the left foot, and when COG comes within 5mm from it, then, re-placed the desired position near the other foot, and repeated it. The gains were de-cided by trial and error to be P = d iagf1000; 1000; 1000g, D = d iagf500; 500; 1000gPR;L = d iagf100; 100; 100; 100; 100; 100g and DR;L = d iagf100; 100; 100; 100; 100; 100g,respectively. Fig.2.4 is a snapshot of the animation of this motion. Fig.2.5 shows the lociof the desired / actual COG, and the desired / actual ZMP. One can see that the actualZMP comparatively well follows the desired. And can see that the larger the gap betweenthem becomes, the farther the desired ZMP is located from COG, which is thought to bebecause of the in uence of ignored moment nG around COG.

Since this simulation was done under the condition that the both feet stay contactwith the ground on their sole, the COG Jacobian is calculated only by Eq.(2.51). Thetotal time of computation (including the computation of the COG Jacobian, Jacobian

34 2.5. SUMMARY

Height 58[cm]Weight 6.5[kg]Number of joints 20

Figure 2.3: Bipedal robot model

for the relative movement of the left foot with respect to the right foot, decision of thedesired ZMP, calculation of equivalent COG acceleration and solution of simultaneouslinear equation by weighted generalized inverse matrix) was shorter than 10 �sec pera cycle by PC/AT compatible machine (CPU Celeron 2GHz, RAM 512MB, RTLinux).One should note that the total amount of computation increases in the case that angularmomentum conservation law a�ects COG movement.

2.5 Summary

In this chapter, the COG Jacobian of general legged robots, the concept, the expressionand the way of computation was introduced. Despite the legged robots are structure-varying system, it doesn't require any replacement or modi�cation of the model even ascontact state varies. And, in any cases, the strict COG Jacobian is obtained by calculationof kinematics, so that it has an advantage in both terms of computational cost and theaccuracy.

The signi�cance of proposed lies on the following points.

1. It is quantitatively promised that COG, which originally is a kinematic parameter,can represent the global dynamics of the total system and can help to design thecontroller of legged robots.

2. Supplement of the gap between the mass-concentrated model and the precise multi-body model is enabled by the COG Jacobian in generalized form. Since it is morph-independent in the sense of not concerning the number of legs, it is thought to beof wide application.


Figure 2.4: COG control in double leg supported posture on the oor

-0.15

-0.1

-0.05

0

0.05

0.1

[m]

0 500 1000 1500 2000 2500 [msec]

x

xx

G

Z

Z

actual

referential xG

actual

referential

-0.15

-0.1

-0.05

0

0.05

0.1

[m]

0 500 1000 1500 2000 2500 [msec]

yG

yG

yZ

yZ

referential

actual

referential

actual

Figure 2.5: Loci of referential and actual COG, ZMP

Chapter 3

Dual Term Absorption ofDisturbance for Motion Stabilization

3.1 Introduction

In this chapter, a new motion stabilization method on pattern-based control for humanoidrobots is introduced.

Since humanoid robots commonly feature a large number of degrees-of-freedom, man-agement of the whole body to make a variety of motion with keeping stability is not aneasy problem. Beside the fact, the higher-level operation robots are required to carry out,the severer constraints on both kinematics and dynamics are imposed on their motion.From this point of view, a so-called pattern-based approach can reduce the di�culty, andthen is applied for not a few robots; operators prepare the whole body motion trajectoryfor achievement of desired performance, taking some constraints { collision avoidance withobstacles, physical consistency and so forth { into account, and let the robot follows thetrajectory by feedback control. This approach consists not only of a motion planner [35][128] [114] [127] [195] but also a stabilization controller which absorbs disturbances due tothe error in the model of both the robot and the environment, dynamic variation of theenvironment, contact with unknown objects and so forth. In order to improve reliabilityof robots, the design of the stabilizer should be focused on. Kajita et al.[196] controlsthe angular momentum around the ankle joint of the supporting leg. It has a drawbackthat loads concentrate to the ankle joint, only controlling by the ankle torque. Further-more, applicable types of motion are disappointingly restricted to single-leg-supportedones. Park et al.[133] proposed an online modi�cation methods of pre-designed trajec-tories. They only consider walking motion on the sagittal plane. And the criterion ofstabilization they adopted is only about the ZMP, which is not enough to discuss thestability. In the control by Huang et al.[132], they focused on upperbody attitude, ZMPand foot landing time, and tried to manage them by modifying certain parts in the robotbody. It is, however, specialized only for walking motion, not suitable for the whole bodymotion in general. Tamiya et al.[143] developed Auto Balancer, which modi�es the orig-inal input trajectories so as to reduce the moment both by the gravity and by the inertiaforce within a certain range. This idea contrary limits the acceleration generated and

37

38 3.2. STABILIZATION STRATEGY ON MASS-CONCENTRATED MODEL

prevents from dynamic legged locomotion. And it also has a problem of computationalcost. Nagasaka[128] proposed Trunk Position Compliance Control. And, Hirai et al.[32]also applied a compliance control strategy to build a stabilizer. Their ideas are to ab-sorb disturbances by modi�cation of the motion of lower extremities. On the other hand,human beings can absorb disturbances by skillful cooperation of the whole body in a con-text. More exible stabilization method for humanoid robots is expected. Yoshino[197]proposed a highly precise playback control of planned trajectories even in a fast walkingmotion by linearizing the equation of motion at each moment around the desired postureand using linear feedback. Humanoids, however, are unstable systems in nature because ofthe absence of any �xed point in the inertia frame, so that a conversion from the internalforce(joint torques) to the external force is required as they make movements. It meansthat, not only an error compensation of geometric pattern, but the interaction with theenvironment should be taken into account for stabilization of them, In this sense, motionpattern should be described by both the geometric pattern and the force pattern. Then,as is �gured in Fig.3.1, the following two schemes should be realized simultaneously

i) Playback of the force pattern to maintain a contact state.

ii) Playback of the geometric pattern with a compensation of the error.

However, these two schemes frequently con ict with each other. It is because the geo-metric pattern and the force pattern depend on each other; the former scheme requiresa modi�cation of the geometric pattern, while the latter requires of the force pattern.In other words, stabilization control of humanoid robots means a maintenance of theconsistency between those two con icting schemes by a mutual modi�cation of both thegeometric and force patterns.

In this chapter, a stabilization control, named \Dual Term Absorption of Disturbance"is proposed. We focus on the fact that a maintenance of the force pattern should beconsidered in a severely shorter span than of the geometric pattern, and place the above i)as a short term absorption of disturbance, and ii) as a long term absorption of disturbance.The idea is to resolve that con icting issues in this multiply-layered time order. Then, awhole-body cooperative motion controlling method will be introduced in accordance withthe idea. Though the control of humanoids often su�ers from the amount of computationsince they commonly consist of a large number of degrees-of-freedom, the method shown inthis chapter enables i) an implementation on comparatively small computational cost, andii) an kinematic-structure-independent application for humanoid robots with a numberof degrees-of-freedom, by abstracting the core dynamics of the robot in terms of COGmovement. In addition, it features a exibility that each o�set of joint from the desiredcan be coordinated by arbitrarily weighting.

3.2 Stabilization Strategy on Mass-Concentrated

Model

As is already explained in section 2.4, the total dynamics of a humanoid robot shownin Fig.3.2(A) is represented by Eq.(2.55), and direct manipulation of COG acceleration

CHAPTER 3. DUAL TERM ABSORPTION OF DISTURBANCE FOR MOTIONSTABILIZATION 39

Absorption of Unpredicted Force Postural Recovery(Short-term Absorption of Disturbance) (Long-term Absorption of Disturbance)

Figure 3.1: Two basic schemes for stabilization

equivalently means indirect manipulation of the external force. Here, the point of actionof the total external force f on the ground, or the center of pressure is ZMP pZ =[ xZ yZ zZ ]T , and we have Eq.(2.60) about ZMP. Suppose the e�ect of nG is ignorableagainst the linear momentum in the same way with section 2.4, we can assume a mass-concentrated model �gured in Fig.3.2(B). Let z-axis of the inertia frame coincide with thedirection of gravity, and we get the following equation from Eq.(2.55) and (2.60), sinceboth x and y components of nZ are zero by the de�nition of ZMP.

�xG = �(xG � xZ) (3.1)

�yG = �(yG � yZ) (3.2)

�zG =fzm� g (2.65)

where � is de�ned as follows.

� � fzm(zG � zZ)

(3.3)

zZ is a known parameter as the height of the ground.As is mentioned in the previous section, motion of the humanoid robot should be

described by a pair of the geometric pattern and the force pattern, and be stabilized byresolving the con ict between them. We note that the geometric pattern is representedby COG pG, while the force pattern is represented by ZMP pZ and the vertical reactionforce fz in Eq.(3.1)(3.2) and (2.65). That is to say, a mass-concentrated model helps oneto understand the relationship between those two types of patterns.

Let us formulate the problem concretely. Suppose a motion pattern of the humanoidrobot is described by the geometric pattern �, pG, and the force pattern pZ , fz. And,

40 3.2. STABILIZATION STRATEGY ON MASS-CONCENTRATED MODEL

pZ

pG

(A)Mass distributed model (B)Mass concentrated model

f n1 1,

f n2 2, f

Figure 3.2: Mass-distributed and mass-concentrated humanoid model

suppose the commanded value of them, cmd�, cmdpG,cmdpZ ,

cmdfz satisfy the physicalconsistency, namely, when the robot performs in the ideal environment without any dis-turbances according to cmd�, actual pG, pZ and fz strictly coincide with

cmdpG,cmdpZ and

cmdfz, respectively. This condition is expressed as follows.

cmd�xG = cmd�(cmdxG �cmdxZ) (3.4)cmd�yG = cmd�(cmdyG �cmdyZ) (3.5)

cmd�zG =cmdfzm

� g (3.6)

where cmd� is de�ned as follows.

cmd� �cmdfz

m(cmdzG �cmdzZ)(3.7)

Then, the problem can be broken into the following two steps, namely,

i) Computation of the referential COG acceleration based on Dual Term Absorptionof Disturbance

ii) Computation of the whole joint angle movement which achieves the referential COGacceleration computed by i)

and, � is calculated so as close to the commanded value cmd� as possible.


3.3 Dual Term Absorption of Disturbance

3.3.1 Short Term Absorption of Disturbance

In this section, the idea of Dual Term Absorption of Disturbance, and based on it, thecomputation method of COG acceleration which works to keep the consistency betweenthe geometric and force pattern.

In the motion of humanoid robots, it is required to remain a surface contact conditionbetween the sole and the ground in order to avoid the short term crisis of upsetting. It isachieved by playback of the force pattern, namely, letting the actual ZMP pZ and verticalreaction force fz get close to the commanded value cmdpZ and cmdfz, respectively. Here,we suppose cmdpZ has an enough margin within the supporting region. An o�set from thetrue COG acceleration

A playback of the force pattern should be considered in severely short span, namely, theorder of acceleration. Thus, we call it \short term absorption of disturbance" hereafter.

3.3.2 Long Term Absorption of Disturbance

In the case of actual robots, modeling error, dynamic variation of environment, or someother factors will cause a gap from the commanded COG motion pattern as follows.

pG = cmdpG +�pG (3.8)

In order to accomplish the planned task, this �pG = [ �xG �xG �xG ]T should becompensated and the robot should robustly follow the commanded. This issue may acceptto be solved in rather long span than the previous short term absorption of disturbance,since it is a problem of a geometric pattern convergence. Thus, we place it as \long termabsorption of disturbance" hereafter.

3.3.3 Simultaneous Realization of Short/Long Term Absorptionof Disturbance

This subsection concretely shows how to compute COG acceleration for the sake of si-multaneous realization of short/long term absorption of disturbance. The absence of any�xed point in the inertia frame poses a constraint about the acceleration to the robot,namely, only the acceleration which is consistent with the following conditions is feasible.

i) the vertical reaction force is not negative.

ii) ZMP exists within the supporting region (convex hull of the contact points).

These conditions are represented by Eq.(3.9) and (2.59), respectively.

fz � 0 (3.9)

pZ 2 S (2.59)

42 3.3. DUAL TERM ABSORPTION OF DISTURBANCE

SupportingRegion

Right Foot Stamp

x

y

Left Foot Stamp

Actual

Calculated

pZ

pZ

Figure 3.3: Feasible area of the referential ZMP

Then, we tentatively modify the force pattern as follows.

xZ = cmdxZ + Px�xG +Dx� _xG (3.10)

yZ = cmdyZ + Py�yG +Dy� _yG (3.11)

fz =cmdfz � Pz�zG �Dz� _zG (3.12)

where P� and D� are the proportional gain and the di�erential gain, respectively, tocompensate the error of �-axis (� for x, y or z) component of COG. Next, we consider theabove conditions i) and ii), or Eq.(2.58) and (2.59). When the vertical reaction force fzdecided by Eq.(3.12) is less than a certain positive valueminfz, we replace it with

minfz.Using it, � is calculated. And after that, decide ZMP pZ according to Eq.(3.10) and(3.11). When it is out of the supporting region, we replace it with the nearest point inthe region as is �gured in Fig.3.3. Putting these pZ and fz into Eq.(3.1)(3.2) and (2.65),the referential COG acceleration is computed.

If pZ andminfz decided just by Eq.(3.10)(3.11) and (3.12) satisfy the above constraintsi) ii), Eq.(3.1)(3.2) and (2.65) are deformed as follows, respectively.

�xG = �(xG �cmdxZ � Px�xG �Dx� _xG) (3.13)

�yG = �(yG �cmdyZ � Py�yG �Dy� _yG) (3.14)

�zG =cmdfz � Pz�zG �Dz� _zG

m� g (3.15)


Namely,

�xG = �(xG �cmdxZ)� �(Px�xG �Dx� _xG) (3.16)

�yG = �(yG �cmdyZ)� �(Py�yG �Dy� _yG) (3.17)

�zG =

�cmdfzm

� g

�� Pz�zG +Dz� _zG

m(3.18)

These equations show that the external force acting to the robot will get close to thecommanded pattern and short term absorption of disturbance will be achieved. And, weget the following equation by di�erentiating Eq.(3.8) twice.

�pG = cmd�pG +��pG (3.19)

From Eq.(3.19)(3.4)(3.5) and (3.6), Eq.(3.16)(3.17) and (3.18) are deformed as follows.

��xG + �Dx� _xG + �(Px � 1)�xG = 0 (3.20)

��yG + �Dy� _yG + �(Py � 1)�yG = 0 (3.21)

��zG +Dz

m� _zG +

Pz � 1

m�zG = 0 (3.22)

Although it is hard to discuss the stability of the system since � varies as the robot movesin z-axis direction, �pG will converge to 0 under the following condition, if the variationof � is a little enough to be ignored against that of xG and yG so that it can be regardedas a constant value.

P� > 1; D� > 0 (for any � = x; y; z) (3.23)

It assures that long term absorption of disturbance is achieved.In practice, acceleration o�set estimation method (see Appendix.C) is used to deal

with a model error between the mass-concentrated model and precise humanoid robotmodel, which causes an o�set of ZMP from desired position.

3.4 The Whole Body Cooperative Motion Control

This section shows how to compute the referential whole joint movementsref� which givesCOG the referential �pG and lets � converge to the commanded cmd�. Firstly, ref� can berelated to �pG by the COG Jacobian, which was introduced in the previous chapter, asfollows.

�pG = JGref �� + _JG

ref _� (3.24)

However, a strict discussion of the dynamics in accordance with Eq.(2.53) doesn't onlynecessitates a quite complicated procedure as is mentioned in section 3.2, but also increasesthe complexity of computation, which discourages the implementation of the controller.Then, we integrate it as follows, and consider it as a condition of velocity.

ref _pG � _pG +

Z�t

�pG dt = JGref _� (3.25)

44 3.4. THE WHOLE BODY COOPERATIVE MOTION CONTROL

where �t is a quantized period. Thanks to this, the amount of computation is substan-tially reduced, despite it surely keeps the core dynamics.

Aside from Eq.(3.25), we represent the motion of extremities, or some other constraintson the task as the following form of velocity condition.

ref _pC = JCref _� (3.26)

Combining Eq.(3.25) and (3.26) as follows.

ref _pU = JUref _� (3.27)

As the result, the problem is translated into the computation of the referential joint anglevelocity which satis�es Eq.(3.27) and gets � as close to cmd� as possible. It is formulatedas follows.

1

2

cmd�(t+�t)��(t) +

Z�t

ref _� dt

� 2W�! min.

subject to JUref _� = ref _pU

(3.28)

where W = d iagfwig is a weighting matrix (regular and positive de�nite), and kvk2Wfor an arbitrary 3times1 vector v means a weighted squared norm de�ned as follows.

kvk2W � vTWv (3.29)

If �t is tiny enough, the problem (3.28) can be replaced as follows.

1

2

cmd _� �ref _� 2W�! min.


(3.30)

where cmd _�� is de�ned as follows.

cmd _� �cmd�(t+�t)� �(t)

�t(3.31)

Problem (3.30) is equivalently solved by the following simultaneous equation.�W JT

U

JU O

� �ref _��

�=

"Wcmd _�ref _pU

#(3.32)

where � is the co-state vector of ref _�. Solving Eq.(3.32), we obtain the referential jointangle velocity as follows.

ref _� =cmd _�

+W�1JTU(JUW

�1JTU)�1refvU (3.33)

vU �JUcmd _� �ref _pU


Contribution of each joint to the stabilization of total system is coordinated by weight-ing matrix W . The o�set of ith joint angle �i from the commanded value will be a littleagainst a large wi, while it is positively modi�ed to stabilize the system against a smallwi. In this way, such types of tasks that carrying glasses with water or going throughnarrow space among obstacles are achieved, choosing positively modi�able properties ineach situation.

3.5 Simulation

DOF: 20 (8 for arm,12 for leg)height: 480 [mm]weight: 6.5 [kg]

Figure 3.4: Kinematic structure, size and mass of the robot

This section shows some results of simulations. The commanded pattern planningmethod for each motion is explained in Appendix E. The humanoid robot was supposedto be HOAP-1 [88](Fujitsu Automation Inc.). Fig.3.4 shows the appearance, kinematicstructure, size and mass of the model. The simulation is done with Z-DYNAFORM [194].

The �rst example is a simple standing motion in double leg supporting posture. Somerandom impulses as pseudo disturbances were given to the robot during the motion.Fig.3.5 is a snapshot of the animation of this motion. (A) is a result with the stabilizationcontrol, which shows the function of the controller. The robot was skillfully able to absorbthe impact. (B) is one without it and shows the robot upset. The weights correspondingto the arm motion were set for little values so that one can easily see the whole bodycooperation. Fig.3.6 is the loci of each component of COG and ZMP in motion (A). Thesecond one is a walking motion. Some impulses were given to the robot as the samemanner with the former example. Fig.3.7 is a snapshot of the motion. (A) is a resultwith the stabilization, while (B) is one without it. Fig.3.8 shows the loci of COG andZMP in motion (A). In these simulations, �t was set for 1[msec]. From these results, wecan verify the function of the stabilizer which works as the robot avoids the short termcrisis immediately after given impacts, and the robot posture gradually converges to thecommanded. Both short/long term absorption of disturbance is realized by the controller.

One can �nd a small oscillation in the loci of ZMP, which is though to depend onthe model of external force in forward dynamics calculation. We applied a perfect plastic

46 3.6. SUMMARY

contact model[140] to it. Though it need some discussion, it is out of the main stream ofthis dissertation.

3.6 Summary

In this chapter, a new stabilizer for a pattern-based motion control of humanoid robotsis presented. Firstly, the meaning of stabilization in the case of humanoids was revealed,namely, it is achieved by maintaining the consistency between the geometric pattern andthe force pattern. Secondly, a di�erence of the span in considering the avoidance of shortterm crisis and the recovery of posture was pointed out, and based on the fact, Dual TermAbsorption of Disturbance was introduced. It is an idea to make use of multiply-layeredtime order to stabilize the system. Thirdly, the relationship between the geometric patternand the force pattern was clari�ed through the mass-concentrated model. And �nally, itwas shown how to compute the referential COG acceleration which realizes Dual TermAbsorption of Disturbance without any con icts, and to compute the equivalent referentialwhole joint movements using the COG Jacobian.

The e�cacy of proposed is listed as follows.

1. It is widely applicable in the sense that it doesn't specify the structure of the robot,not only humanoid but any rigid multibody robots, and that it doesn't limit thetype of motion so long as it goes under continuous contact with the ground.

2. Contribution of each joint to the stabilization, namely, the amount of o�set fromthe commanded value is coordinated by a modi�cation of weighting matrix.

3. Adoption of the mass-concentrated model substantially reduces the computationalcost, so that it is suitable for online implementation. In spite of that, it catches thecore dynamics of the total system and functions as a practical stabilizer.


(A) a success case with stabilization control

(B) a failure case without stabilization control

Figure 3.5: Snapshots of a Simulation of Balance Control on Standing

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0

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0 1000 2000 3000 4000 5000 [msec]

real COG

cmd COG

ZMP

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real COG

cmd COG

ZMP

0.206

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0.21

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0.214

0.216

[m]

0 1000 2000 3000 4000 5000 [msec]

real COG

cmd COG

Figure 3.6: Loci of the COG along x, y, z axes

(A) a success case with stabilization control

(B) a failure case without stabilization control

Figure 3.7: Snapshots of a Simulation of Balance Control on Walking

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0

0.05

0.1

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0.3

[m]

0 1000 2000 3000 4000 [msec]

real COG

cmd COG

ZMP-0.08

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cmd COG

ZMP

0.2

0.205

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[m]

0 1000 2000 3000 4000 [msec]

real COG

cmd COG

Figure 3.8: Loci of the COG along x, y, z axes

Chapter 4

ZMP Manipulation based onInverted Pendulum Model

4.1 Introduction

In this chapter, a new realtime controlling method of humanoid robots is introduced.

A pattern-based approach discussed in the previous chapter solves a large part ofthe di�culty in o�ine stage with concomitant use of inverse dynamics, optimizationand so forth. And thus, it accepts even some fragile stabilizers for not so high leveltasks, so that it is adopted by many research groups [149] [198] [35] [128] [32] [199] [68].However, it only works in the case that the operator has su�cient information aboutthe environment in advance, and disturbances are small enough. The above groups havemainly addressed steady, periodic walking motion at all. Since the real environmentis �lled with variations and unpredictable factors, humanoid robots should have highmobility to cope with some situations which call for instant reaction and re ex. It isunlikely thought that the pattern-based control leads to such the improvement of motionability. Developments of sophisticated controllers would be the breakthrough.

Gubina et al.[135] applied the pole assignment for the dynamic motion of biped sys-tem, which assumes an oversimpli�ed model which consists only of massless legs and asolid torso on the sagittal plane, and was not practical. Miyakoshi et al.[177] constructeda neuro-musculo-skeletal biped system model and simulated its adaptive behavior viaoscillation by CPG and interaction with the environment. It targeted the functional re-construction of the lower neural system in terms of a biological interest. Pratt et al.[156]proposed Virtual Model Control which virtually connects the robot with the environmentby spring-damper, and realized a robust walking motion against disturbances. However,it doesn't take the limitation of applicable external force into account, so that the motionrealized is disappointingly slow. Fujimoto and Kawamura [140] developed a hierarchicalcontrol system which directly controls the reaction force, so that it is robust against avariation of terrain. It only considers periodic walking motion. Moreover, the implemen-tation of its reaction force controller is open to question. Mita and Ikeda[200] proposedVariable Constraint Control and realized running motion control of one-leg, two-leg andfour-leg machines. Though those results showed the e�ectiveness against nonholonomic

49

50 4.2. INVERTED PENDULUM MODEL BASED ZMP MANIPULATION

constraint systems, a strict and explicit representation of equation of motion is needed.Thus, robots with a large number of degrees-of-freedom su�er from the serious problemof computational cost. The strong nonlinearity of legged robots has been a barrier to beovercome.

In the previous chapter, we discussed the core dynamics of humanoids represented bya rather simple model consisting of COG and ZMP, which is the key to open the problem.Since the horizontal components of the moment which works around ZMP are zero, onecan notice that it has a similar dynamics to that of an inverted pendulum, where COGcorresponds to the mass at the tip and the supporting point is equivalently located atZMP. This chapter proposes a realtime motion controller of humanoid robots throughthe indirect manipulation of ZMP based on this analogy. This has several advantages asfollows.

1. Manipulation of ZMP enables to handle physical constraints which is conditionedby contact state of the robot with the environment, directly.

2. Dynamics motion of humanoid robots is achieved on the same principle with thecontrol of well-known inverted pendulum[201].

3. The amount of computation is substantially reduced with the mass-concentratedmodel which ignores local dynamics of each link.

There has been some studies focusing on ZMP as manipulated variable to control COG.Sorao et al.[139] applied fuzzy control for planning the amount of ZMP manipulationon the control of COG. It is a sort of trial to control an inverted pendulum by fuzzycontrol, and is irrelevant solution. In the study by Obata et al.[141], only a positioncontrol in a standing posture is considered by ZMP manipulation, not augmented to thelocomotion. Takeuchi[147] proposed a realtime concurrent decision of COG accelerationand ZMP as input values using Receding Horizon control, which doesn't assure the e�cacyof itself, not explaining the relationship between COG and ZMP. Though Mitobe etal.[145] showed a controller of a massless-legged biped robot for COG movement throughZMP manipulation, an e�ective application for multibody legged robots is yet to be found.

The uniqueness of proposed lies at its strategy that the controller is quantitativelydesigned based upon the metaphor of inverted pendulum, while it has been thought asno more than a qualitative one as Kato[202] and Arimoto et al.[130] mentioned. It is alsoshown that the inverted pendulum model is widely available for general multibody robotmodels by virtue of the COG Jacobian.

4.2 Inverted Pendulum Model based ZMP Manipu-

lation

Let us again introduce Eq.(2.55) and (2.60) to represent the total dynamics of the hu-manoid robot shown in Fig.4.1(A), and approximate it by the mass-concentrated model,ignoring the e�ect of nG. Here, we notice that the horizontal components of nZ are zero,and �nd that the system has quite a similar dynamics to that of inverted pendulum as is

CHAPTER 4. ZMP MANIPULATION BASED ON INVERTED PENDULUM MODEL51

pZ

pG

(A)Precise anthropomorphic model (B)Inverted Pendulum Metaphorized Model

f n1 1,

f n2 2,f

Figure 4.1: An inverted pendulum metaphor for biped robot

shown in Fig.4.1(B), whose supporting point is equivalently located at ZMP. In order tohighlight that dynamical similarity, we rewrite Eq.(3.1) and (3.2) as follows.

�xG = !2(xG � xZ) (4.1)

�yG = !2(yG � yZ) (4.2)

where !2 is de�ned as follows.

!2 � �zG + g

zG � zZ� 0 (4.3)

( .:. �zG � �g; zG > zZ)

Based on Eq.(2.65), which represents the vertical movement of COG, we tentatively decidethe input vertical reaction force fz by PID control as follows.

fz = Pz(refzG � zG) + Iz

Z(refzG � zG) dt +Dz(

ref _zG � _zG) (4.4)

Then, Eq(2.65) becomes as follows by Laplace transform.

ms2ZG =

�Pz +

Izs

+Dzs

��refZG � ZG

�� mg

s

, ZG =Dzs

2 + Pzs+ Izms3 +Dzs2 + Pzs + Iz

refZG � mg

ms3 +Dzs2 + Pzs+ Iz(4.5)

52 4.2. INVERTED PENDULUM MODEL BASED ZMP MANIPULATION

It is stable under the following conditions.

Pz; Iz;Dz > 0; mIz � PzDz > 0 (4.6)

And, if refzG = const:, it is assured that zG will converge to refzG by �nal-value theorem.

limt!1

zG = lims!0

sZG = refzG

�.:. refZG =

refzGs

�(4.7)

When zG converges to refzG so quickly that �zG is small enough to be ignored, !2 isapproximately regarded as a constant value, and both Eq.(4.1) and (4.2) coincide withthe equation of motion of an inverted pendulum derived in Appendix F.

In this research, the horizontal COG movement is commanded by refvGx and refvGy,which are x and y components, respectively, of the referential COG velocity vG �[ vGx vGy vGz ]

T , in accordance with the idea that legged robots should be maneuveredby giving referential velocity just as wheeled vehicles.

Now, we give a thought only to x component for the symmetry of Eq.(4.1) and (4.2).Di�erentiating Eq.(4.1) and ignoring a derivation of !2, the following equation is obtained.

�vGx = !2(vGx � vZx) (4.8)

where vZx � _xZ . Eq.(4.8) means that vGx is controllable with the input vZx, whichis dynamically equivalent to the velocity of the supporting point of inverted pendulum.Therefore, a controlling method of inverted pendulum, PID control, optimal regulator, orH1 control for instance, is available for a decision of manipulated variable.

Here, we adopt PID control for simplicity, and tentatively decide vZx as follows.

vZx = Px(refvGx � vGx) + Ix

Z(refvGx � vGx) dt+Dx(

ref _xG � _xG) (4.9)

where Px, Ix and Dx are the proportional, integral and di�erential gains, respectively.Then, Eq.(4.8) becomes as follow by Laplace transform.

VGx =�!2(Dxs

2 + Pxs+ Ix)

s3 � !2Dxs2 � !2(Px + 1)s� !2IxrefVGx (4.10)

Though it is not simple to discuss its stability since !2 in fact varies as the robot movesin the vertical direction, if the variation is small enough, it is stable under the followingcondition from Hurwitz's stability criterion.

Px; Ix;Dx < 0; !2Dx(Px + 1) + Ix < 0 (4.11)

And, if refvGx = const:, vGx will converge torefvGx by �nal-value theorem.

limt!1

vGx = lims!0

sVGx =refvGx

�.:. refVGx =

refvGxs

�(4.12)

vZy � _yZ is also decided tentatively in the same way. Integrating them, tentative refer-ential ZMP pZ is calculated.

We should also consider the constraints of ZMP and the vertical force represented byEq.(3.9) and (2.59). minfz and the way shown in Fig.3.3 are used again.

The actual robot contacts not only with a level ground, but with slopes, rough terrains,desks or walls 3-dimensionally. In order to deal with such situations, the idea of VHPexplained in Appendix G is introduced.


4.3 The Whole Body Cooperative External Force

Manipulation

4.3.1 The Equivalent COG Acceleration to the External Force

This section shows the computation of the whole body movement which equivalentlyrealizes the desired external force.

Firstly, the equivalent COG acceleration to fz and pZ decided in the previous sectionis calculated by putting them into Eq.(4.1)(4.2) and (2.65).

4.3.2 The Whole Body Motion Generation

The same equation with Eq.(3.27) is considered, which includes the relationship betweenCOG movement and the whole joint angle movement, and the geometric constraints onthe task.

ref _pU = JUref _� (3.27)

Having a large number of degrees-of-freedom, ref _� which satis�es Eq.(3.27) is not alwaysunique in the case of humanoid robots. Then, we solve it by minimizing the weightedsquared norm. It is formulated as follows.

1

2

ref _� 2W�! min.


(4.13)

where W = d iagfwig is a regular and positive de�nite weighting matrix, and kvk2W foran arbitrary 3times 1 vector v is de�ned by Eq.(3.29). Problem (4.13) is equivalentlysolved by the following simultaneous equation.�

W JTU

JU O

� �ref _��

�=

�0

ref _pU

�(4.14)

where � is the co-state vector of ref _�. Solving Eq.(4.14), we obtain the referential jointangle velocity as follows.

ref _� =W�1JTU(JUW

�1JTU)�1ref _pU (4.15)

Contribution of each joint to COG acceleration is coordinated by weighting matrix W .

4.3.3 Structure of the Controller and the System

The control system designed consists of four subsystems as is �gured by a block diagramFig.4.2.

1.Referential ZMP Planner decides the referential ZMP as manipulated variable inaccordance with the analogy of inverted pendulum and physical constraints.

54 4.4. SIMULATION

+

-

vref +

+

+

+

+

-JointActuator

pZ

Robot

pG

.Referential ZMP Planner

ZMPManipulator

Decomposerof the StrictReferentialCOG Velocity

θref.

prefG

.pref

Z

θ.

Figure 4.2: Block diagram of the motion controller

2.ZMP Manipulator computes an equivalent COG acceleration to manipulatedamount of ZMP.

3.Motion Rate Decomposer computes the referential whole joint angle velocity whichrealizes the desired COG acceleration and satis�es the constraints posed on the taskrequired.

4.Joint Angle Velocity Controller controls each joint angle velocity by local feed-back.

The part including ZMP manipulator, motion rate decomposer and joint angle velocitycontroller inside the hatched box has the equivalent dynamics with an inverted pendulumcontrolled by referential ZMP planner. And the actual controller of the system consistsof the parts inside the dotted box.

4.4 Simulation

Figure 4.3: A stepping motion with an impact

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refxZ

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[m]

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GyyZ

refyZ

0.2

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0.28

[m]

0 1000 2000 3000 4000 5000 6000 [ms]

zG

Figure 4.4: Loci of COG and the referential/actual ZMP



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refxZ

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refyZ

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zG


This section shows some results of motion control simulations. The appearance, jointcon�guration and speci�cation of the robot model in the simulations is shown by Fig.2.3in section 2.4.3.

The �rst example is COG displacement motion with both feet contact with the ground.COG was carried repetitively on each foot. The height of COG was neither explicitlycontrolled, nor was given any reference. Fig.4.3 shows a snapshot of the motion, andFig.4.4 is the loci of COG, referential and actual ZMP. x-axis was supposed to be directedahead, while y-axis to be directed heft hand side of the robot. One can see that ZMPmoves round to accelerate COG in horizontal components in accordance with the sameprinciple with inverted pendulum. Although the uctuation of COG height grew as timepassed, apparently smooth motion was generated without any constraints.

The second one is a stepping motion, which was realized by giving an initial verticalvelocity to the left foot at a certain moment when ZMP is on the right foot. Fig.4.5 isa snapshot of the motion, and Fig.4.6 is the loci of COG, referential and actual ZMP.A perturbation of ZMP was occurred at the moment the left foot was lifted up, whichconverged within the supporting region. And the robot continued the motion withoutupsetting. Then, an impact was given to the robot in the same motion during the singleleg supported phase. Giving up controlling the sideway motion of the swinging footagainst an excess impact, the robot voluntarily carried the left foot towards the directionto cushion the shock. Fig.4.7 is a snapshot of the motion, and Fig.4.8 is the loci of COG,referential and actual ZMP. A large movement of ZMP immediately after landing the leftfoot on the ground is observed, which is thought to be in order to accelerate COG tothe desired position. A dotted area in the center graph of Fig.4.8 shows the supportingregion. Though the referential ZMP is saturated within the region, the actual ZMP issometimes out of it. It is guessed to be because of the in uence of the ignored momentaround COG in Eq.(4.1) and (4.2). The supporting region set up was narrower than theactual convex hull of the contact points, so that the robot avoided the upsetting. How

56 4.5. SUMMARY


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0 1000 2000 3000 4000 5000 6000 [ms]

GxxZ

refxZ

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0

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yZ

Gy

yZref

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[m]

0 1000 2000 3000 4000 5000 6000 [ms]

zG


large o�set is accepted should be examined in the future works. In addition, it shouldalso be e�ective to apply acceleration o�set estimation method (see Appendix.C).

Giving the forward and upward initial velocity to the swinging foot, one-step-forwardmotion is generated. Repeating it, a walking motion shown in Fig.4.9 is realized. Fig.4.10is the loci of COG, referential and actual ZMP. The timing of giving initial foot velocityand displacement of COG is decided by a �nite state automaton. This result suggests thepossibility of steady motion without setting the period explicitly by proposed.

4.5 Summary

An online motion controlling method via ZMP manipulation was presented based on thesimilarity of dynamics between humanoid robots and inverted pendulums in accordancewith the mass-concentrated model. The gap between the lower-dimensional model andthe multibody robot model is complemented by the COG Jacobian. Some results ofsimulations showed the e�cacy of proposed. It leads to the following advantages.

1. Manipulation of ZMP enables to directly handle severe physical constraints due tothe absence of any �xation in the inertia frame.

2. The control of dynamic motion on humanoid robots with a large number of degrees-of-freedom is achieved as well as of inverted pendulum. It doesn't require detailedmotion trajectory pattern prepared with considering even acceleration in advance. Avariety of motion is created only by a set of command for COG and other extremities.It means that the stability margin around the reference is more expanded than apattern-based approach, and robust behavior against large disturbances is expected.

3. The computational cost is substantially reduced with the mass-concentrated model,focusing on the total dynamics of the system rather than local dynamics of each



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0

0.1

0.2

0.3

0.4

[m]

0 1000 2000 3000 4000 5000 [ms]

GxxZ

refxZ

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0

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0.1

[m]

0 1000 2000 3000 4000 5000 [ms]

yZ

Gy

yZref

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[m]

0 1000 2000 3000 4000 5000 [ms]

zG


link. Thanks to it, a realtime implementation is encouraged. Moreover, the use ofthe COG Jacobian accepts reduction of such the lower-dimensional model to generallegged model. In this sense, it is structure-invariant.

In this chapter, constraints about the friction force, which is also of importance ondynamics motion, was not taken into consideration. When the robot slips at contactpoints, failing to obtain enough friction, the a�ect on the movement of COG should bemodeled by some means, and the controlling system should be modi�ed adaptively. It isthe future work.

Chapter 5

Impedance Switching for ContactState Transition Control

5.1 Introduction

Inverted pendulum model based ZMP manipulation method introduced in the previouschapter equivalently enables to handle horizontal components of the external force ratherin a small amount of computation. Thanks to it, quick responsive motion and robustabsorption of unpredicted impact in horizontal movement have been achieved. However,it assumes that the robot keeps continuous contact with the ground during the motion. Acontact state transition is required in order to cope with emergency in the case that therobot su�ers from much larger impacts, and to expand the range of their activities andto perform more variety of motion. The vertical movement of COG, which plays an im-portant role in contact state transition, has not been su�ciently taken into considerationso far.

In this chapter, we augment inverted pendulum model based control to realize a re-sponsive transition between contact and aerial phase. The requirement for more positivemanipulation of the external force is accomplished by a control based on an inverted pen-dulum model with impedance. Some previous researches have achieved typical types ofmotion which go through aerial phase such as jumping and running with legged machines.Raibert et al.[153] realized hopping motion and even somersault with simple body robotsby a combination of simple maneuvers. It owes to hydraulic actuator which works peri-odically, and is not promising for humanoid robots. Nagasaka[128], Yamane et al.[127]and Kajita et al.[203] developed pattern planning methods of jumping and running forhumanoids. They are simple extensions of a conventional pattern-based motion control,and aims only at intentional motions. Hirano et al.[189] studied jumping motion of ahumanoid robot in computer simulation using an adaptive impedance control. It is forachievement of repetitive jumping, not responsive motion. Mita et al.[200] proposed Vari-able Constraint Control. Though it is e�ective against nonholonomic constraint, a strictand explicit representation of equation of motion is needed. Thus, the more complicatedthe system is, the more amount of computation it requires. It is necessary for quick mo-tion to make the computational cost less and the control period short. Arikawa et al.[204]

59

60 5.2. INVERTED PENDULUM MODEL WITH IMPEDANCE

developed a Multi-DOF jumping Robot and controlled it according to pre-planned poly-nomial trajectories, which is not robust against disturbance. Pfei�er et al.[67][68] aredeveloping Jogging JOHNNIE, aiming at fast movement through running. It is still un-der development. Our stance is close to that of Mita et al. in the sense that manipulationof the external force is treated just as a part of the dynamical constraints which denote themotion. Consequently, the controller is invariant against motion types, and even againstcontact state. In addition, it has an advantage in small computational cost because of anadoption of the mass-concentrated model.

Then, we present the idea of multiple inverted pendulum model control. Inverted pen-dulum model mainly intends COG control. Responsive legged motion, however, consistsnot only of COG control but of skillful footwork, namely, manipulation of reaction forceand contact state of each foot. Not having any mechanical connection with the envi-ronment, footwork of humanoid robots through manipulation of external reaction forceconverted from internal torques often becomes a tough problem even in the case of asimple motion. Only a few researches have addressed this problem. Nishiwaki et al.[205]tried to generate stepping motion towards any direction composed of elementary patternsin motion database, pre-designed by genetic algorithm. Such a strategy doesn't explicitlytake the conversion from internal force to external into account, so that it's hard to dealwith a various type of terrain and to create a variety of motion with it. Multiple invertedpendulum model is aiming at rather easy manipulation of contact state between the feetand the ground. Impedance control are ordinarily for the purpose of shock absorption atthe landing in the cases of legged robots. Sorao et al.[206] and Park et al.[207] applied itfor such the purpose for walking motion. We focus on the aspect that it gives the systema semi-passive characteristics which works to generate reactive footwork.

5.2 Inverted Pendulum Model with Impedance

When in contact phase, the robot can convert the internal force generated at each jointactuator to the external reaction force through the interaction with the environment.Inverted pendulum model with impedance functions during this phase to control COG ef-fectively. As is already discussed in chapter 4, a dynamical similarity between humanoidrobots and inverted pendulums hints that horizontal components of COG can be con-trolled via manipulation of ZMP. And the controller was designed based on Eq.(4.1) and(4.2).

�xG = !2(xG � xZ) (4.1)

�yG = !2(yG � yZ) (4.2)

Thus, the referential ZMP as manipulated variable to control the horizontal componentsof COG is calculated in accordance with Eq.(4.9) and the dynamical constraint �guredin Fig.3.3.

In this chapter, we focus on Eq.(2.65), which denotes vertical COG movement.

�zG =fzm� g (2.65)

CHAPTER 5. IMPEDANCE SWITCHING FOR CONTACT STATE TRANSITIONCONTROL 61

Since the robot is un�xed to the ground, fz must satisfy Eq.(3.9).

fz � 0 (3.9)

When fz equals to zero, the robot is in aerial phase, while fz is positive, the robot is incontact with the ground. It means that manipulation of fz plays an important role forthe transition of contact phase. Detachment o� the ground requires large fz to accelerateCOG enough against the gravity. And, at the moment of touchdown, fz should be givencompliant characteristic in order to absorb the impact. Inverted pendulum model withimpedance shown in Fig.5.1 is that to realize such responsive and exible motion in theuni�ed way.

Based on the model, fz as an input is decided as

fz = mfKPz(refzG � zG) +KDz(

ref _zG � _zG) + gg (5.1)

whererefzG is the referential COG in z-axis, whose meaning varies depending on the casesas is mentioned later. One can note that Eq.(5.1) includes a compensation of the gravity.From Eq.(2.65) and (5.1), we get the following equation.

�zG = KPz(refzG � zG) +KDz(

ref _zG � _zG) (5.2)

5.3 Energy Control by Impedance Switching

KPz and KDz in Eq.(5.2) is decided in accordance with the contact state and the motionscheme as is shown in Fig.5.2. I, II and III in Fig.5.2 are described as follows.

I)Impedance for lift-o�

In order to reach the desired height, COG has to be accelerated to enough velocityagainst gravity. And the initial velocity at the lift o� determines the maximum height inaerial phase. A simple spring model helps to meet this requirement. Suppose the robotlifts o� the ground when zG equals to refzG, the planned maximum jumping height fromrefzG in aerial phase is zH , and the stooping depth from refzG is zd as is shown in Fig.5.3.In order to give the maximum vertical speed to the robot at the moment of detachmento�, KDz should be set for zero.

KDz = 0 (5.3)

Then, we get the following equation from physical energy conservation law.

1

2mKPzz

2d = mgzH () KPz =

2gzHz2d

(5.4)

In other words, the potential energy of the springy model is virtually accumulated by anintentional switching of the parameters KPz, KDz, zH and zd. It yields a passive charac-teristic of the system and the acceleration against the gravity implicitly by dischargingthe virtual potential energy in restitution phase.

62 5.3. ENERGY CONTROL BY IMPEDANCE SWITCHING

pZ

pG

Figure 5.1: Legged system and inverted pendulum model with impedance

II)Impedance for touchdown

The spring model is also applicable for shock absorption at the touchdown. Supposethe robot lands onto the ground at the height refzG with falling speed immediately beforecontact _zG�, and the desired maximum stooping depth from refzG is zd, as is shown inFig.5.4. The impact at the touchdown is ideally eliminated if KDz is equal to zero. Then,we get the following equation which is also derived from physical energy conservation law.

1

2m _z2G� =

1

2mKPzz

2d () KPz =

�_zG�zd

�2(5.5)

The springy model in compression phase contributes to absorb the kinematic energyincreased as the robot falls down, which means that the shock is diverged along the timeaxis.

III)Impedance in standing phase

Since Eq.(5.2) represents a second-order-lag system if both KPz and KDz are positive.Characteristic frequency and damping coe�cient are

! =pKPz; � =

KDz

2pKPz

(5.6)


AerialPhase

ImpedanceControl

InvertedPendulumControl

RestitutionPhase

CompressionPhase

StandingPhase

StandingPhase

III I IIIII

z G

z G

refz H

z d

Figure 5.2: Contact phase, impedance and inverted pendulum control

For instance, zG stably converges to refzG when KPz and KDz satisfy the following condi-tion as is shown in Fig.5.5.

� > 1 () K2Dz � 4KPz > 0 (5.7)

In order to remain contact with the ground, fz must be positive, so that it should belimited by a certain minimum valueminfz(> 0).

5.4 Contact Phase Invariant Whole-body Controller

based on Constraints Switching and SRV Reso-

lution

In this section, the construction of a contact phase invariant controller is shown.Firstly, the equivalent COG acceleration �pG to fz and pZ decided is calculated by

putting them into Eq.(4.1)(4.2) and (2.65). And, it is related to the whole joint anglemovement by the same idea with Eq.(3.25). Here, we call ref _pG in Eq.(3.25) the strictreferential velocity(SRV) of COG, hereafter, since it is expected to be realized instanta-neously by the whole body movement. Fig.5.6 shows a block diagram of SRV planner forCOG.

645.4. CONTACT PHASE INVARIANT WHOLE-BODY CONTROLLER BASED ON

CONSTRAINTS SWITCHING AND SRV RESOLUTION

AerialPhase

RestitutionPhase

z Gref

z H

z d

Detachoff!

Figure 5.3: Lift-o� motion from contact phase to aerial phase

And, the tasks are described by a combination of some geometric and dynamicalconstraints. All the constraints are classi�ed into those originated either in physical lawor in controlling scheme. Suppose motion commands such as motion of arm, foot step andso forth are given by a set of their strict referential velocity, represented in the followingform.

J iref _� = refvi (5.8)

where i is an index of the equation. Eq.(5.8) can be regarded as a sort of constraint forcontrol. Eq.(3.25) is also included in this type of constraints. Let us consider anothercase that the robot is in aerial phase. The movement of robot is dominated by the linearand angular momentum conservation law due to the absence of external input from theenvironment. Since both the linear and angular momentum are mapped from the velocityof the system, they have the same dimension with the velocity and are also represented bythe same form with Eq.(5.8). Then, Eq.(5.8) in this case is regarded as a sort of naturaldynamical constraint. Moreover, COG is uncontrollable and Eq.(3.25) is invalidated inthis phase. On the other hand, the attitude and posture should be controlled instead, sincethey largely a�ects on the stability after landing as some former studies[153] revealed. Asthis example shows, physically acceptable constraints have to be selected in accordancewith the contact condition. This is a similar idea to Variable Constraint Control[200]proposed by Mita et al., except it denotes the constraints in a dimension of accelerationand force, while the proposed in this paper denotes them in a dimension of velocity andmomentum. This di�erence is attributable to that we gave preference to the realtimeimplementation even in the case of humanoid robots as highly redundant system over the


AerialPhase

CompressionPhase

z Gref

z H

z d

Touchdown!

Figure 5.4: Touchdown motion from aerial phase to contact phase

exactitude in the sense of dynamics.Now, the problem is how to resolve these constraints into each joint angle velocity.

And, it is translated into the following quadratic programming to expect the minimummovement of each joint angle. It is solve by the same way with that for Problem 4.13,except the condition is substituted by the following.264 J0

...JN

375ref _� =

264refv0...

refvN

375 (5.9)

or,

Jref _� = refv (5.10)

where N is the number of constraints. And, the problem is solved as follows.

ref _� =W�1JT (JW�1JT )�1refv (5.11)

The set of joint torque which makes the joint angles follow ref _� is calculated by localfeedback controller, such as simple PD controller, at each joint.

Fig.5.7 shows the procedure. They are switched in accordance with contact conditionand motion scheme. In other words, the motion both in contact and in aerial phase iscreated only by switching of the constraints and no other extra procedure. Consequently,the controller stands invariable on contact phase.

66 5.5. SIMULATION

RestitutionPhase

z Gref

z d

Figure 5.5: Standing motion in contact phase

Although the humanoid has tens of degrees-of-freedom and thus is apparently compli-cated, the combination of various types of constraints due to contact with the environmentand motion scheme itself determines a large part of the con�guration of the robot. Thisidea has a similar aspect with what is so-called synergetics in biological �eld. Biologicalsystem in general consists of an extremely large number of muscles as active elements,so that the management of them seems quite challenging. However, natural ingeniousmechanism { connection with bones and tendons, internal coupling of joints, contact withthe environment, and so forth { functions as the constraint and reduces the actual degreesof freedom of the whole system. Then, the skillful whole body cooperation is achieved asthe result.

5.5 Simulation

A jumping motion was examined in computer simulation with the controller proposed,using a robot model of HOAP-1 �gured in Fig.3.4.

The planned maximum stooping depth, height at the detachment-o� and the maxi-mum height of jumping were set for 50[mm], 220[mm] and 50[mm], respectively, and thereferential height of COG after the touchdown was 220[mm]. Impedance in each phasewere decided in accordance with values and equations derived in section 5.3. Fig.5.8 is asnapshot of the motion. And the loci of COG and ZMP are shown in Fig.5.9. We can seethat the impedance control works and the stable jumping motion is achieved.


+

-

vref

pG

.

Referential ZMP Planner

ExternalForce

Manipulator

frefz

+

-

pG

Referential COG Velocity Planner

pG

prefG

.pref

G

pZMP

prefZMP

pG

.

Figure 5.6: The external force manipulator

5.6 Multiple Inverted Pendulum(MIP) Model Con-

trol

5.6.1 MIP Model

In this section, we introduce the idea of Multiple Inverted Pendulum(MIP) model shown inFig.5.10, aiming at easy manipulation of contact state between each foot and the ground.

We have mainly been focusing on COG control by the inverted pendulum modelcontrol so far. Responsive legged motion, however, consists not only of COG control butof skillful manipulation of reaction force and contact state of each foot. The absence ofany mechanical connection with the environment puts a di�culty on the footwork of therobot through manipulation of the external reaction force. Let us consider a simple case ofstepping motion. At the moment of kicking against the ground, the reaction force actingto the foot of the kicking leg must be zero, that is to say, ZMP must completely be on thesole of the supporting leg. However, COG should be accelerated enough in the directiontowards the supporting leg in order to stay standing posture or go forward over the peak,which means, ZMP should be contrarily on the sole of the kicking leg at the beginning ofkicking motion. As this fact shows, the implementation of footwork on humanoid robotsrequires a sophisticated manipulation of the external reaction force and ZMP.

MIP model enables an intuitive manipulation of reaction force acting at each foot.ZMP is shifted in accordance with a distribution of them. MIP also gives the system asemi-passive characteristic, which encourages a quick, reactive footwork in addition to acompliant characteristic to absorb the impact as ordinary impedance control works.

5.6.2 Design of Impedance

Here, we model the biped system just as shown in the left side of Fig.5.11. We also havethe following assumptions.

68 5.6. MULTIPLE INVERTED PENDULUM(MIP) MODEL CONTROL

+

-JointActuator

SRVResolver+

-

prefN

τθref.

θref.

vrefJ =N N

pN

+pref1

θref.

vrefJ =1 1

p1

-

+

-

prefG

θref.

prefJ =G G

pG

.

.

.

θ.

.

Figure 5.7: Procedure of Constraints Switching and SRV Resolution

i) The total mass is concentrated at the point pG.

ii) Only telescopic force is applied at each leg.

iii) Each foot contacts with the ground at one point, which is not connected to. Namely,when the force applied at the foot is 0, the foot detaches o� the ground.

iv) While in the noncontact phase, the foot position of the swinging leg is arbitrarilycontrollable.

We distinguish the two legs into the supporting leg S and the kicking leg K, and let fXand pX be the force applied at leg X and the foot position of leg X (X is for S or K),respectively.

Designing fK and fS according to the following Eq.(5.12) and (5.13), the dynamics ofthe system becomes equivalent to one which consists of the rigid supporting leg connectedto the ground by one revolution joint and the kicking leg having a spring-damper, just asshown in the right side of Fig.5.11.

fK =

�k(ref lK � lK) + c(ref _lK � _lK) (lK < ref lK)0 (lK � ref lK)

(5.12)

fS = (mg � fKdK) � dS (5.13)


Figure 5.8: Snapshot of a jumping motion simulation

-100

-50

0

50

[mm]

0 200 400 600 800 [msec]

COG

ZMP

x-axis

-100

-50

0

50

[mm]

0 200 400 600 800 [msec]

COG

ZMP

y-axis

0

50

100

150

200

250

[mm]

0 200 400 600 800 [msec]

COG

z-axis

Figure 5.9: Loci of COG and ZMP in each axis

where

lK = kpG � pK k (5.14)

dK = (pG � pK)=lK (5.15)

dS =pG � pSkpG � pS k

(5.16)

and ref lK is the virtual neutral length, which one can set arbitrarily, of the kicking leg. kand c are designed according to the following rules.

I)Impedance against the gravity to kick the ground

In order to lift COG to the height of h from the bottom of stooping(minlK the length ofleg K at the point), k and c should be designed as follows.

c = 0; k =2mgh

(ref lK �minlK)2(5.17)

In this case, it is expected that leg K detaches o� when its length becomes lK = ref lK . Ifh > lS �minh (lS =kpG�pS k, andminh is the height of COG at the bottom of stooping),COG goes forward with the following speed v over the peak.

v =p2g(h� lS +minh) (5.18)

II)Impedance for shock absorption at the touchdown

In order to absorb the impact at the touchdown of the foot of leg K, k and c should bedesigned as follows.

c = 0; k =mv2�

(ref lK �minlK)2(5.19)

70 5.6. MULTIPLE INVERTED PENDULUM(MIP) MODEL CONTROL

pG

-mg

fSfK

pKpS

Figure 5.10: Legged System and Multiple Inverted Pendulum(MIP) model

where ref lK is the length of leg K at the touchdown, v� is the COG speed in verticaldirection immediately before touchdown andminlK is the minimum length of leg K, whichis at the bottom of stooping.

III)Impedance to stay standing

When k and c satisfy the following condition, COG converges to the referential position.

k > 0; c > 0; c2 � 4k > 0 (5.20)

Since fK must satisfy the following condition to stay contact, fK � 0 it should be limitedto a certain minimum valueminfK(> 0).

5.6.3 Indirect Manipulation of Reaction Force

Although a precise manipulation of the reaction force at each foot is a hard problemsince the humanoid robot is a highly redundant multibody system with underactuatedlinks, one can calculate the referential COG acceleration ref �pG from the summation of thereferential fK ; fS and the equivalent gravity force as follows.

�pG =fK + fS

m� g (5.21)


k

c

-mg

f fS K

p pS K

pG

l

l

K

S

Figure 5.11: Legged system and MIP model

Figure 5.12: Snapshots of a stepping motion realized by MIP model control

Using Eq.(3.25), it turns to a constraint equation about COG.

When one of the feet at least is in face contact with the ground, it becomes a hyper-static problem; the points of action on each sole don't always coincide with the desiredpS;pK even if the acceleration of COG is �pG in Eq.(5.21). Though it didn't matter in thefollowing simulation, it should be solved later, using such as robust force control.

We examined a stepping motion in a simulation based on the proposed, letting theright leg be leg S and the left leg be leg K. Firstly, we set the initial stance betweenboth feet was 10cm and the height of COG was 20cm. When incrementing the virtualneutral length of the supporting leg 2cm instantaneously, a kicking motion immediatelyemerged and the left foot lifted o� the ground. After the left foot again touched down,we modi�ed a damping coe�cient, and COG converged to the position with a constantheight. In this simulation, a PID control of the supporting leg to maintain its length wasapplied simultaneously with Eq.(5.13) to avoid the drifting caused by the integral error.Fig.5.12 shows a snapshot of the motion of both a concentrated mass with massless legs

72 5.7. SUMMARY

and HOAP-1 model created in the simulation.

5.7 Summary

We presented an inverted pendulum model with impedance, augmenting the invertedpendulum model control in chapter 4. Thanks to it, indirect manipulation of the externalreaction force, especially the vertical component of the force is enabled rather easily,which achieves the transition between contact and aerial phase with impedance switchingbasically in accordance with the physical energy conservation law.

Handling of the external force, which commonly requires so a large amount of com-putation that it is time-consuming, is translated into COG velocity control equivalently,so that it has an advantage of less computational cost enough to be applied for realtimeimplementation.

Another advantage is that it is invariant on contact phase since COG velocity controlis represented as no more than a part of the constraints which are switched in accordancewith contact phase and motion scheme. The structure of controller itself doesn't varyin accordance with contact phase, and the motion both in contact and in aerial phase isrealized in the uni�ed way.

Finally, MIP model was introduced in order to handle the contact state and reac-tion force manipulation of each grounding foot. Though it is yet be developed, furthersophisticated legged motion control and even the whole body contact motion with the en-vironment is expected to be achieved. Simultaneous management of COG movement andcontact state manipulation will be realized by robust controls, and the implementation isthe future work.

Chapter 6

Miniature Humanoid Robot Systemtowards High Mobility

6.1 Introduction

We have mainly discussed the dynamics and controlling method of humanoid robots sofar in the previous chapters. High Mobility of humanoid robots, however, is the result ofdeveloped controlling theories, sophisticated softwares, and reliable hardwares.

This chapter introduces a hardware and software system for motion control implemen-tation and experiments towards high mobility of humanoid robots.

6.2 Humanoid Robot \UT-�: mighty"

6.2.1 Motivation

In this section, a mechanics and hardware system of a new miniature humanoid robotdeveloped by the authors will be introduced.

Greatly advanced technology in the robotics �eld have aroused people's expectationthat humanoid robots will work in the human society. Some have begun to discuss thepractical use of humanoids [62]. In spite of that, the real humanoid robots in the currentstage disappointingly lack of mobility against the severity of real environment. Robotsover 1 meter high [33] [35] [39] are still threats as if they are murderous weapons. Researchon the motion control of humanoid is still in the midway, and requires much furtherdevelopment of basic theory. From this point of view, light miniature anthropomorphicrobots turn the experiments to be smoother and safer. The new robot has been developedaccording to this requirement.

Some small humanoid robots has previously been developed. Nakai et al.[208], Waitaet al.[74], Furuta et al.[78, 85], and Yamasaki et al.[76] constructed lightweight and low-cost anthropomorphic robots from small servo DC motor modules originally for radio-controlled toys. However, low precision on the control of those actuator modules fortoys prevents a progress of dynamic motion. Murase et al.[87, 88] produced HOAP-1for the purpose of o�ering an open platform to researchers. It unfortunately restricts

73

74 6.2. HUMANOID ROBOT \UT-�: MIGHTY"

an expansion of the system. Kuroki et al.[86] developed SDR-4X, a large part of whosetechnology is not revealed, so that it is not available for other researchers.

6.2.2 Mechanics

Figure 6.1: External view and joint con�guration of the robot UT-�: mighty

Name Developer Height[cm] Weight[kg] DOF

Kaz[208] Nakai et al. 34 1.6 18Alviss[74] Waita et al. 43 2.1 20Mk.6[78] Furuta et al. 39 2.0(?) 26PINO[76] Yamasaki et al. 70 4.5 26morph3[85] Furuta et al. 38 2.4 30HOAP-1[87] Murase et al. 48 6.0 20SDR-4X[86] Kuroki et al. 58 6.5 38

Table 6.1: Size/weight of representative miniature anthropomorphic robots

Fig.6.1 shows UT-�: mighty. The robot is about 58 cm tall, and about 7kg weightincluding the weigh of battery and processor on the body. Table.6.1 shows some repre-sentative miniature anthropomorphic robots and their speci�cations for reference.

CHAPTER 6. MINIATURE HUMANOID ROBOT SYSTEM TOWARDS HIGHMOBILITY 75

Figure 6.2: Total joint assignment of the robot

The total number of joints is 20; 4 are located at each arm, and 6 at each leg. All thosejoints are actuated by coreless DC motor manufuctured by MAXON, and decelerated withHarmonic Drive gears. The gear ratios are all 100:1, and the output of motors are 4.5W(forelbow rotational joints), 6.5W(for shoulder abductional joints, elbow exional joints andknee rotational joints), and 11W(for the other joints), respectively. The choice of thosehigh performance actuators mounted on the arms and legs encourages active whole bodymotion, and also promises feeding the results of experiments back to as practically tallrobots as human-beings, though it causes a slight increase of weight at each joint. Thethin-shelled exoskeletal structure realizes the light and sti� body. The main structure ofthe robot is made of magnesium alloy casting, and is designed as curved surface. Eachjoint connected with those external skeletons is modularized as a orthogonal double axialunit, which leads to reduction of the number of parts and improves an ease of maintenance.

Fig.6.2 shows unique features on its joint assignment, listed as follows.

i) Hip joint(Fig.6.3(A))

Two axes of hip joint are assigned so as to enlarge the range of movement { especially


(A) Hip joint (B) Flexional leg joints

(C) Shoulder joint

Figure 6.3: Spotlighted unique joint assignment of the robot

of exion, extension and abduction. And, the distance between the two hip joints iscountered by inward o�set of knee rotation joints to prevent the sideward perturbationof the trunk.

ii) Flexional joint at the leg(Fig.6.3(B))

Ankle exional joint is assigned backward to hip and knee exional joints for the sake ofsingularity avoidance. And, knee exional joint has a mechanical stopper.

iii) Shoulder joint(Fig.6.3(C))

The root axis of each shoulder is slanted from the perpendicular axis to 45 degrees, aimingat a natural avor on its mixed motion of extension and abduction.

Pictures in Fig.6.4 are some postures of the robot to show the performance of theabove feature.

6.2.3 Hardware system

The hardware system architecture of the robot is shown in Fig.6.5. It is an independentsystem in the sense that it has the processor board and battery within, excluding anycables from outer resources which disturbs robot's performance. The processor board isCARD-PCI/GX(EPSON) with a special I/O board (Fujitsu Japan Automation), featur-ing Geode GX1(National Semiconductor). Communication with internet is ensured viawireless LAN card(Melco), so that it can be remotely operated. It also has the internalUSB LAN which branches at small USB Hubs(Sanwa supply), and communicates withactuator drivers and A/D converters. The actuators, coreless DC motors, are controlledby the motor controller iMCs01 + motor driver iMDs03(iXs research corp.). The inputsignal from sensors, which includes the gyro sensor MG2(MicroStone), the accelerometerMA3(Microstone) and the 3-axes force sensor PicoForce(NITTA), are processed at A/D


Figure 6.4: Joint assignment of the robot

converter iMCs03(iXs research corp.). 3-axes force sensors are mounted on the soles andwrists, 4 for each sole and 1 for each wrist. Those on the sole are located at four cornersto calculate the resultant linear force and the moment of a couple [209].

6.2.4 Experiment

Fig.6.6 shows a snapshot of a bow motion. The motion trajectory was planned in o�ineaccording to the strategy that the projection of COG onto the level ground had stayedat the initial position during the motion, and the ankle joint angle was automaticallycomputed from a given hip joint angle at each frame. The time step between frames was10[msec], and the total time of the motion was 4[sec].

Fig.6.7 is the resultant motion by inverted pendulum model based ZMP manipulationcontrol introduced in chapter 4. The operator switched command of the referential po-sition of COG to the above of each foot at ad libitum timing. The stance between thetwo feet was 14[cm]. The referential height of COG is about 27[cm]. And the referentialCOG velocity was decided by the following equation.

refvG = P (refpG � pG) (6.1)

P was set for 3.0 in this experiment. The parameters(gains) set are shown by Tab.6.2.All arm joints were �xed to the initial posture, and leg joints were used for the motion


3-axesforce sensor

Motorcontroller A/D

converter

US

B H

ub

CorelessDC motor+Encoder Accelerometer Gyro

Internet

Motordriver

Processor boardCPU:GeodeRAM:64MB

NiH 24V

+-

DC-DC

I/O boardRadio LANCompact FlashUSB

Figure 6.5: Hardware system of the robot

PGx 40 IGx 10 DGx 60PGy 40 IGy 10 DGy 60PGz 6000 IGz 5000 DGz 10000

Table 6.2: Parameters for inverted pendulum model based ZMP manipulation

control. The referential whole body motion was updated every 5[msec]. As the snapshotshows, a steady repetetive rolling motion was generated by the control.

Figure 6.6: A bow motion


Figure 6.7: A rolling motion

6.3 Development of Z-DYNAFORM

6.3.1 Motivation and Software Design policy

Motor ControlSensor Processing

Structural Analysis

Real RobotVirtual RobotCAD/CAM Simulation Control

Operation Interface

Motion PlanningBehavior Planning

Visualization

KinematicsInverse Dynamics

Device DriverNumerical Analysis3D Geometry Data Processing

Forward Dynamics

Figure 6.8: Computer Aided Robotics

Technical use of computer softwares in the robotics �eld typically extends toCAD/CAM, dynamics simulation, control of real robots and so forth. Commercial appli-cations are frequently used for each purpose. Those applications are commonly packagedas all-in-one integrated softwares, so that it has some drawbacks; excess functionality

80 6.3. DEVELOPMENT OF Z-DYNAFORM

often causes overheads on the access and consumption of a large amount of resources,and, rarely provided with means to be interlocked with other softwares, data portabilityis less promised. Kanehiro[210] proposed a composition methodology of humanoid robotswhich positively combines those commercial softwares with the core system written inEusLisp[211], emphasizing the high extensibility of such an approach. However, since thesoftware system of robots becomes broad-scale, including from low-level device drivers tohigh-level intelligent operation managers and working on parallel dependent or indepen-dent operations of them, it tends to be \patched", namely, to lose the consistency and leadine�ciency of maintenance. In order to ensure the consistency of the system with lessoverheads and high portability, Computer Aided Robotics(CAR) requires a distinctivesoftware design policy.

Fig.6.8 shows some typical components of CAR. One can notice that, even thoughthey are independent in application level, they share some basic components internally.Then, we thought it the core issue to provide that basic operations as a library(hatchedpart in Fig.6.8) in order to build the consistent and portable software environment. \Z-DYNAFORM" (which is an acronym for Z's DYNamics Analyser FOr Rigid Multibody) isproduced according to such an idea. It mainly targets dynamical analysis and computationof rigid multibody systems, which are so severely constrained by physical laws that preciseanalyses of dynamics is needed. Especially, it features the computation for legged systems,which have no �xation in the inertia frame.

6.3.2 Construction of Z-DYNAFORM

The library is developed with language C on Linux and GNU tools. It is multiplatform,working on Linux, BSD(including Solaris) and MS-Windows. And, it has an extensionto the Linux kernel, so that it is available even in the development of device drivers forLinux and in realtime process on RTLinux. Fig.6.9 shows the construction of the library.Each unit of modules listed as follows is decided to keep the well-balanced maintainabilityand extensibility.

z3d is for 3-dimensional vector analysis.

zmath provides several numerical vector and matrix calculations.

zmaterial models solid state properties of objects.

zoptic does optical computation for visualization.

zactua models some kinds of actuators for simulations.

zshape manages 3-dimensional objects and their geometry.

zopt provides some optimization tools.

zpoly prepares polynomial expression and its applications.

zctrl models some dynamical systems for automatic control.


Applicationsmotion generator

dynamics simulator

animation viewer etc...

z3d3D

geometry

zrfdcmultibody

forward dynamics

zlinkmultibody kinematics& inverse dynamics

zseqvalue sequencemanagement

zviewview

transformation

zshape3D shape

zoptoptimization

tools

zctrlautomatic

controll

Z-DYNAFORM

ZBL

zgagenetic

algorithm

zlistlist operation

zmisc

zstringstring operation

miscellanious

other libraries

zactuaactuatormodel

zopticcolor & optics

model

zmaterialmaterial

model

zmathmathematic

object

znnneural

network

zarrayarray operation

zipsequence

interpolation

zpolypolynomialexpression

zstreamstream operation

Figure 6.9: Construction of the software environment around Z-DYNAFORM

zga is an implementation of genetic algorithm.

znn is an implementation of neural network.

zlink treats description and kinetics of multibody systems.

zseq manages vector sequences.

zip gives some interpolation methods.

zview is for visualization of 3-dimensional objects.

zrfdc calculates forward dynamics of multibody systems.

Users can easily code motion planner, motion sequencer, dynamics simulator, animationviewer, robot controller, robot model designer and so forth on the library. Actually, it isused in some humanoid systems [38, 39, 40, 88, 54].

826.4. POW(A,4); { AN ARCHITECTURE FOR ANIMATRONIC ARTIFICIAL

AGENTS

6.4 pow(A,4); { an Architecture for Animatronic Ar-

ti�cial Agents

6.4.1 Motivation

The software system of robots, especially of humanoid robots, should be built in line witha peculiar paradigm against ordinary ones for o�ces, plants, arts, personal uses and soforth, due to the following schemes.

1. Since robots are dominated by strict physical laws and their state dynamicallychanges as time passes, frequent collection of sensory information and motor actionare required in order to recognize the real world and even to act towards it. In thissense, the system should be opened to the world via hardwares, including sensorsand actuators. The enhancement of its function often means an exchange or anaddition of hardware devices in the case of robots, and it largely a�ects on theupper parts of the whole system. From this point of view, the architecture shouldencourage an easy extension of devices.

2. The system necessarily consists of a variety of processes from low-level ones whichdirectly control hardwares to high-level ones which manages the motor control, be-havior planning, control interfaces for the operator and so forth. And, it leads toa highly multilayered massive construction. Further more, those processes commu-nicate, share resources, and issue interruptions with each other over the hierarchy,though such a feature likely causes a \spaghetti" construction. In spite of that, theyshould be managed by a strict hard-realtime scheduler.

These schemes require the following speci�cations on the operation system.

I.Micro Kernel Architecture

The system should work on a micro kernel and ensure an extensibility of hardwares,and easy communications between processes. The total function of the operation systemshould be departmentalized and individually managed by daemon processes, which workas servers. Such a client-server-model-based system also encourages an availability ofshared resources and a memory protection for safety.

II.Controllable Hard-realtime Scheduler

A hard-realtime process scheduling should be controlled by the operator explicitly.

II.Usability of Signal

The use of signal interruption accelerates cooperations between upper and lower processeswithout any overheads.

Present operation systems disappointingly lack of these speci�cations. The Linuxsystem, for instance, allows user processes to communicate with device drivers only viadevice �les, and dynamic memory allocation is not available in shared memory. Althoughsome operation systems [212, 213] [214] [215] have realized an user-space hard-realtime


operations, a commercial provision reduces availability of them. The UNIX system pre-pares only a couple of user-de�nable signals. Then, an operation system specialized forthe robot control is projected to be required in the near future.

The realization of OS prospected in the above also facilitates a hardware abstraction,namely, the function of upper parts of the system can easily examined on alternatedhardware emulators, which reduces danger and cost on experiments of real robots. Mostof commercial simulators lack of portability to be directly fed back to real robot controllers,and thus, is hard to be embedded into the system. Kanehiro[210] proposed a hierarchicalconstruction of robot system which features HAL(Hardware Abstraction Layer) and plug-in implementation of re ex modules, and developed a software platform called OpenHRP[216] which enables to handle both the real robot and the virtual robot in the simulatorcompatibly. Though it encourages easy expansion of the system without changing the corepart, plug-in architecture causes serious overheads and loose connectivities of processes.Fujimoto et al.[189] developed ROCOS(an acronym of RObot COntrol Simulator), whichmainly aims at constructing a virtual realistic robot controlling system as a simulator.

This system can also contribute to a CG animation. An utility of knowledge of roboticsabout the expression, composition and presentation of motion on CG characters has beendiscussed []. It is expected that, if such a character is controlled by the robot system asa virtual robot with hardware abstraction, arti�cial intelligences and control techniquescan be returned to it.

Then, we name this framework \pow(A,4);", which is an acronym for an Architecturefor Animatronic Arti�cial Agents.

6.4.2 Emulation of pow(A,4); on RTLinux in mighty system

We've functionally emulated pow(A,4); on RTLinux [212, 213]. The total construction isshown in Fig.6.10 It consists of several daemons to emulate micro kernel architecture. Anordinary shell such as bash or tcsh is available as an user interface to call client programsas co-processes. Each daemon provides some interfaces to communicate with the clientprocesses.

Some of the daemons function as wrappers of the device drivers for the hardwaresmounted on the robot in order to remove a barrier that user processes can communicatewith the drivers only via Linux device �les. In the case of virtual robots in the simulator,they are completely substitutable with virtual hardware daemons, so that the systemensures portability of the upper parts.

These daemons are managed by a hard-realtime process scheduler with PSC mech-anism, which enables the user space realtime programming on RTLinux. Thanks to it,the system is protected from invalid memory access in the kernel space. And, usinguser-de�nable signals prepared, the processes can interrupt immediately with each other.

Fig.6.11 shows the software system of mighty designed based on the emulatedpow(A,4);.

846.4. POW(A,4); { AN ARCHITECTURE FOR ANIMATRONIC ARTIFICIAL

AGENTS

devfs

FIFO

sharedmemory

daemon

daemon

daemon

daemon

daemon

pow(A,4); kernel

client

client

client

devicedriver

devicedriver

devicedriver

kernel space

user space

Linux kernel

Figure 6.10: Structure of pow(A,4); emulated on RTLinux


modeld

imcs01d

pow(A,4); kernel

monitor

imcs03_mod imcs01_mod

kernel space

user space

Linux kernel

robotmodeldaemon

motorcontrollerdaemon

seqdmotionsequencedaemon

mmdmotionmanagementdaemon

imcs03dsensordaemon

seq_load

seq_ctrl

set_gain

set_angle

servo

etc.

Figure 6.11: Software system of mighty based on pow(A,4);

Chapter 7

Conclusion

7.1 Summary

This dissertation presented a new controlling method of humanoid robots to give themhigh mobility and agility. Major contributions are summarized to the following threeissues.

Firstly, the reaction force manipulation was proposed in order to accelerate the wholebody and to realize agile motions of humanoid robots. A requirement for a conversion fromthe internal forces to the external reaction forces underlying on the strong nonholonomicconstraints was achieved in accordance with a simple but core lower-dimensional modelof legged. Dynamical constraints are directly dealt with by regarding the reaction forcefrom the environment as manipulated variables.

Secondly, some strategies for a decision of the reaction force were proposed based onthe similarity of dynamics of the robot to that of inverted pendulum. An assumptionof the mass-concentrated model, focusing on the linear movement of COG, substantiallysimpli�ed the problem, so that the design of controllers was encouraged in accordancewith the core dynamics of the system explained by the simple relationship between COGand ZMP.

Thirdly, it was shown that the total dynamics of humanoid robots could be approxi-mately expressed even by COG, which was originally a geometric parameter, on the as-sumption of the mass-concentrated model. And, the reconstruction method of humanoidrobot model from the mass-concentrated model was achieved using the COG Jacobian.The less requirement for the amount of computation helps realtime implementation. Inaddition, it has an enough versatility in the sense that it doesn't assume any speci�cfeatures on the kinematic structure of the robot and the type of motion, except for therequirement of a rigid multibody and fully-actuated joints.

The contributions of each chapter are summarized as follows.

In chapter 1, the previous studies on the humanoid robots and biped motion were sys-tematically classi�ed and reconsidered. And, the signi�cance of sophisticated controllerswas discussed on the achievement of the high mobility.

In chapter 2, the COG Jacobian of general legged robots, the concept, the expressionand the way of computation was introduced. Though the legged robots are structure-

87

88 7.1. SUMMARY

varying system, it doesn't require any replacement or modi�cation of the model even ascontact state varies. In any cases, the strict COG Jacobian is obtained through kinematics,so that it has an advantage in both terms of computational cost and the accuracy. It wasquantitatively revealed that COG, a kinematic parameter originally, can represent thecore dynamics of the total system and can help to design the controller of legged robots.The mass-concentrated model was associated with the precise multibody model via theCOG Jacobian in generalized form. A morph-independent calculation realized its wideapplicability.

In chapter 3, a stabilization method for a pattern-based motion control of humanoidrobots was presented. The stabilization in the case of humanoids meant the maintenanceof the consistency between the geometric pattern and the force pattern. Though they arefrequently in con ict with each other, a di�erence of the span in considering the avoidanceof short term crisis and the recovery of posture was pointed out. And, based on the fact,Dual Term Absorption of Disturbance was introduced. In other words, the robot couldbe stabilized in multiply-layered time order. The computation of the referential COG ac-celeration which realized Dual Term Absorption of Disturbance was shown. Contributionof each joint to the stabilization was coordinated by a weighting matrix.

In chapter 4, an online motion controlling method via ZMP manipulation was pre-sented based on the dynamical similarity of humanoid robots and inverted pendulums inaccordance with the mass-concentrated model. Manipulation of ZMP enabled to handledirectly severe physical constraints due to the absence of any �xation in the inertia frame.The control of dynamic motion on humanoid robots with a large number of degrees-of-freedom was achieved as well as of inverted pendulum. It doesn't require detailed motiontrajectory pattern as inputs, and a variety of motion was created only by a set of com-mand for COG and other extremities. It means that the stability margin around thereference is more expanded than a pattern-based approach, and robust behavior againstlarge disturbances is expected.

In chapter 5, an inverted pendulum model with impedance was proposed, augmentingthe inverted pendulum model. Indirect manipulation of the external reaction force, espe-cially the vertical component of the force was enabled rather easily, and then, it achievedthe transition between contact and aerial phase with impedance switching basically in ac-cordance with the physical energy conservation law. And, MIP model was also introducedin order to handle the contact state and reaction force manipulation of each groundingfoot.

In chapter 6, a miniature size humanoid robot UT-�: mighty developed towards thehigh mobility, its design, mechanics and hardware were introduced. Small but high-performance actuators and gears leaded to signi�cant and reliable results to be fed backto human-size robots. Some results of experiments were also shown. A design policy ofsoftwares in the �eld of robotics was discussed, and commonly useful dynamics calculationlibrary Z-DYNAFORM was presented. A new architecture of robot system, pow(A,4);,and the software system of mighty emulating pow(A,4); on RTLinux was shown.

CHAPTER 7. CONCLUSION 89

7.2 Known problems

The following issues were still left as known problems to be solved in short term.

Design of an inverted pendulum controller itself

In chapter 4, PID control was chosen as a controller for the decision of the referentialZMP, just for simplicity. Since the performance of proposed largely depends on thecharacteristic of this controller, it should be determined according to an insight towardsthis issue. Giving a thought to the fact that a human locomotes repeating the accelerationand deceleration of COG, a robust controller for an inverted pendulum isn't necessarilygood for humanoid robots. What is \good" for the robots should be investigated fromvarious points of view.

Foot placement

COG control through ZMP and the reaction force manipulation is a feedback-oriented. AsMiura and Shimoyama[149] and Raibert et al.[150] pointed, a feedforward-oriented footplacement also plays an important role for dynamically stable legged motion. However,that issue is not independent from the feedback control, since the foot motion a�ects onCOG movement in the case of multibody robots with unignorable mass at each leg. Thus,a compound controller of feedback/feedforward should be designed, which requires moreinvestigations.

Singularity

The robot generally su�ers from its singularity. The requirement of excessive motionrates numerically occurs around the singular posture. Although Nakamura[188] proposedSingularity-Robust inverse matrix to protect the robot from that case(see Appendix.I),it's not available when the contact condition is so severe that the reaction force has tobe strictly realized, since it causes small errors from the true motion rate to realize thedesired motion. One hint is that the advantage of proposed also lies on its exibility thatit accepts arbitrary selection of controlled variables as is shown in chapter 5. Thus, itis probably e�ective to discard some components when the controlling equations are notfull-ranked.

Dealing with friction force

In this chapter, constraints about the friction force, which is also of importance on dy-namics motion, was not taken into consideration. When the robot slips at contact points,failing to obtain enough friction, the a�ect on the movement of COG should be modeledby some means, and the controlling system should be modi�ed adaptively.

7.3 Future Works

This dissertation has focused on the very fundamental issues on the motion control,namely, manipulation of COG and contact state as the essentials of legged motion, andcoordination of the whole body motion. They are thought to be the bottom layer of ahierarchical mobility control system, corresponding to re ex and equilibratory sense ofhumans, the functions of cerebellum. In order to complete the system, the upper layerthan it should be composed as follows.

90 7.4. THE LAST REMARK

Middle layer

interprets the desired behavior into the command for the reaction force manipulator. Itimplies the necessity of a framework for the motion description.

Top layer

does decision-making to choose the behaviors matching to the task requirement. Thismeans that the high mobility control of humanoid robots naturally leads to the problemof intellects.

It seems to be a challenging issue to synthesis such an intelligent system, and shouldbe tackled from interdisciplinary standpoints. Hence, it is the long term { and, probablyan endless { future work.

7.4 The Last Remark

Robotics has been developed in terms of the functional construction of both arti�cialbody and intelligence. Those two issues originate the same interest in the analysis andsynthesis of the brain-body mechanism of human-beings. That is the reason the history ofrobot began with the humanoid robot. And they should again be combined together forthe same target as a human-like brain-body system, namely, the humanoid robot. Asidefrom the discussion that which of bipedalism and brain-evolution caused which, it is nodoubt that brain and body have been coevoluting with each other.

When the robot gains a body with the high mobility, what kind of challenge will itsbrain �nd? Or, what will the brain-body collaboration emerge? The author greatly desirethat this dissertation will contribute to �nd the answer to those questions.

Acknowledgement

This dissertation was written under the supervision of Professor Yoshihiko Nakamura. Hisprecise orientation and meaningful remarks lead me to this rewarding research to tackleto the extremely di�cult dynamical system in a down-to-earth strategy. Discussions withhim and his advice often brushed up my work.

And, other four professors gave me exact comments to improve this thesis as review-ers: Professor Masafumi Okada, Professor Isao Shimoyama, Professor Hirochika Inoue andProfessor Masayuki Inaba in the department of intelligent mechano-informatics, Gradu-ate School of University of Tokyo. I especially thank Professor Masafumi Okada forimpassioned discussions, which frequently encouraged me, and for his advice not only onresearch but on daily life as a member of the lab.

Special thanks to all ex- and current members of Nakamura & Okada Laboratory fortheir tangible and intangible supports. Professor Tetsunari Inamura, who used to be apostdoctoral fellow in the lab, encouraged me to join the lab when I was in the second yearof master course, and then gave me lots of advice. Ms. Ayami Goji had always backed meup in all works other than the study. Thanks to her, I had been able to concentrate on theresearch. Dr. Katz Yamane, assistant professor, often gave me strict and lean commentsconcerning with multibody dynamics. It was my great honor to have opportunities todiscuss with the spirited young specialist in this �eld. In addition, he is the �rst personwho attracted me to this lab. Dr. Mihoko Otake, a postdoctoral fellow, was my closestsenior researcher, so that I learned a lot from her remarkable contribution to this �eld.

The industriousness of all my conpeers and juniors often stimulated my motivation.Koji Tatani has been a nice colleague of mine. His massive e�ort to build up a newtheory to deal with a large number of degrees-of-freedom frequently inspired me. HidekiKadone has been a superior junior, who always amazed me by his deep insight into anyproblems. Tatsuro Endo, Naoyuki Hibara, Hideo Mizukawa, Motoki Odamura, HiroakiTanie and Akihiko Murai have supported me on the development of mighty. Especially,technical comments from Naoyuki Hibara always helped me, who has been lack of knowl-edge about mechatronics. And, I highly appreciate the solid comprehension of AkihikoMurai on research, steady growth of Hideo Mizukawa to have a good sense of dynamics,and the inspiration of Motoki Odamura. Not to mention, I also owe to Shigeki Ban,Yuki Kobayashi, Kazutaka Kurihara, Iwaki Toshima, Tetsuya Shinohara, Shingo Chiy-oda, Ichiro Suzuki, Vincent Pascal Penasse, Guillaume Chevant, Manabu Tange, DaisukeNakamura, Yusuke Fujita, Shigeru Kanzaki, Yusuke Yamamoto, Kennosuke Goshi, Hi-roshi Kajiyama, Takako Yasunaga and the other students. Since Shigeki Ban had been

91

92 7.4. THE LAST REMARK

concerned with motion skill of humanoids, I learned a lot from his results. Both TetsuyaShinohara and Shingo Chiyoda were good designers of mechanics, so that their ideas areapplied ubique in mighty. Manabu Tange succeeded to understand multibody dynam-ics deeply in quite short term, and discussed many signi�cant topics about it with me.Daisuke Nakamura showed me his inventions about dynamical processing, which relievedme from sticking only at embodied phenomena. An outstanding capacity for researchingof Yusuke Fujita frequently astounded me. Takako Yasunaga was not only my junior buta good friend of mine, who and I talked on a lot of things about Japanese culture together.Thanks to all of them, a fruitful life in the lab was mine.

I greatly appreciate and look up to Dr. Ken'ichiro Nagasaka, my senior researcher.I learned most of the essential things for research from him. It has been a pabulumthat he treated and discussed with me, who was lack of experiences and knowledge,as an equal researcher, even when I was an undergraduate student. In addition, hesuggested that I join Nakamura & Okada Laboratory. The very energy source of mine forresearch is a desire to be recognized by him professionally. Ms. Eri Nagasaka(Totsuka)has been a good adviser both on daily and technical subjects. Having the same intereston humanoid robots, we were able to share various information. She often encouragedme to struggle with the challenging problem. Mr. Tomonobu Kitagawa used to be mysenior in the previous lab, who was the �rst user of Z-DYNAFORM other than me andassisted me to debug it. He kindly kept his patience against this impudent junior. Mr.Mitsutaka Kabasawa gave me a signi�cant remark which wiped out my prejudice onlegged motion and inspired me to develop the new theory of the whole body manipulation.Dr. Kei Okada has been a trustworthy senior researcher for me on his personality andwealth of knowledge, and kindly helped when I confront to technical di�culties. Mr.Fuminori Yamazaki, C.E.O. of iXs research corp., and Mr. Noriaki Mitsunaga, an assistantresearcher in Osaka university also helped me a number of times to solve problems on thedevelopment of software system for mighty. Dr. Koichi Nagashima introduced up-to-datetechnology to me many times, so that I can keep up with the latest news in this �eld. Dr.James Ku�ner, Jr., a robotics research scientist in Carnegie Mellon University, was willingto respond to my consultation about my new job. Dr. Hideaki Suzuki, a spotlightednovelist Hideaki Sena, reminded me of the delight and pleasure in robotics. And, hekindly gave this immature researcher an opportunity to publish an article about robotics.I also thank to Hiroyuki Nakai, Shingo Shimoda, Minoru Sano, Tomoaki Yoshikai, RyoFukano and Masato Okabe as good friends in the last school life in University of Tokyo,and all the sta�s of the manufacturers who lent me their assistance on developing mightyover the business.

Finally, I would like to note that part of this research was supported by the \RobotBrain Project" under the Core Research for Evolutional Science and Technology (CRESTprogram) of the Japan Science and Technology Corporation (JST) through \Developmentof Machines with Brain-Like Information Processing through Dynamical Connections ofAutonomous Behavioral Primitives" (PI: Y.Nakamura, University of Tokyo).

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Appendix A

ZMP, Interpretation and Application

A.1 The Original Idea of ZMP

RCOP=ZMP

Figure A.1: De�nition of ZMP

Upper Limb(Lever)

Lower Limbs

Total ForceSagittal Plane Frontal Plane

Figure A.2: Biped Shu�e Model by Vuko-bratobi�c et al.

The original de�nition of ZMP by Vukobratovi�c et al. [112]is as follows.

As the load has the same sign all over the surface, it can be reduced to theresultant force R, the point of attack of which will be in the boundaries of thefoot. Let the point on the surface of the foot, where the resultant R passes, bedenoted as the zero-moment point, or ZMP in short.

It completely coincides with the point which has been so-called the center of pres-sure(COP), �gured by Fig.A.1, in the medical and biological �eld.

While it was named in 1972, the concept had been already introduced in 1969[110].In the paper, a kinematic model which contacted with the level ground at point feet was

117

118 A.2. SPREAD OF THE USE OF ZMP

assumed as is shown in Fig.A.2. And then, a necessary condition was pointed out to letthe system shu�e forward. That is to say, in order both to keep the supporting footstationary, and to slide the swing foot ahead, the ground reaction force must act strictlyat the supporting foot. The signi�cance of their proposal lies on the two facts. First,one can design motion trajectories which satisfy the above condition without consideringthe counteraction from the environment despite it is de�ned through the interaction withit, since the external force is dynamically equilibrated with the internal force generatedby the actuators in the system. Second, since the horizontal component of the momentaround the point is zero (as is the origin of the name of ZMP), the motion design isachieved as the result of a compensation of moment in the planning stage. Actually, fortheir �rst model in Fig.A.2, the motion trajectory of the lower extremities were givenas analytic functions, and thus, the motion of the upper limb which compensates themoment around the supporting foot was calculated by a combination of analytic solutionand iteration.

In 1970, Vukobratovi�c et al.[111] also showed a feedback control for the same system.Though the exclusion of the environment in the planning stage simpli�es the problem, itweakens the robustness of the system against disturbances caused by even small variationsof the model. The error moment should be eliminated through an online feedback. Then,they proposed a realtime motion modi�cation with a use of the sensitivity matrix ofthe moment against the variation of acceleration, in order to calculate the amount ofmodi�cation inversely from the error.

These works revealed that ZMP could be one of the criteria on the control of bipeds;it enabled to extend the conventional control of manipulators, separating the motionplanning stage and the online feedback. One should note that ZMP contains only twocomponents of the total external force, while it consists of six components, so that it neverrepresents a su�cient condition.

A.2 Spread of the Use of ZMP

A.2.1 Motion Planning based on ZMP Criterion

Vukobratovi�c et al.[112] extended their model from shu�ing system to an anthropomor-phic, and showed a control method which was basically the same with the former; theyfocused on how to design the motion trajectory in o�ine and modify it in online to achievethe desired ZMP.

The ZMP criterion in the motion planning has widely spread afterwards. However,most of those studies adopted it as a criterion to judge the dynamical feasibility of tra-jectories which appeared to realize walking motions[9], or, as a constraint to remain itwithin the supporting region [113] [119] [143] [217]. Such di�erent uses from the originalapplication for inverse problems to synthesize the motion from the desired ZMP were dueto the complexity of the system, particularly a number of degrees-of-freedom.

This problem was solved in 1990's. Yamaguchi et al.[218] proposed an o�ine motionmodi�cation method, decomposing the motion of the lower extremities into the frequencyspace via FFT, and composing the motion of the upper body which compensates the mo-

APPENDIX A. ZMP, INTERPRETATION AND APPLICATION 119

ment around the desired ZMP. Its e�cacy was shown particularly in the cases of periodicmotions. DasGupta et al.[126] proposed a conversion method of human motion capturedata to that satis�es the desired ZMP with 2-axis rotation joint at waist. Nagasaka etal.[219] developed a motion design method in o�ine using optimal gradient method. Itmodi�es the trajectory of trunk position to realize the desired ZMP trajectory. After that,they improved it[128] based on the mass-concentrated model, quantizing the equation ofmotion into tridiagonal form, and succeeded to reduce drastically the time consumption.Since it works directly in the time space, it has higher applicability for a variety of motion.Then, it was further re�ned by Kitagawa et al.[129].

On the other hand, these method didn't focus on the trajectory of ZMP, which es-sentially a�ects on the resultant motion. Sugihara et al.[220] proposed a design methodof the referential trajectory of ZMP, which concurrently works with Nagasaka's. Park etal.[221] designed it through fuzzy logic.

Hasegawa and Yoneda[222] developed an analytic planning of the trajectory of ZMPand COG from a few design parameters such as swaying magnitude, velocity and plannedfoot stamps. It is only applicable for steady walking motions.

A.2.2 Online Motion Modi�cation based on ZMP criterion

As Vukobratovi�c showed, a motion control based on referential patterns requires an onlinefeedback control to eliminate the error caused by disturbances. Some controls have beendeveloped [80] [132] [133] [82] [196] to compensate the error between the actual and desiredZMP. One should note that they don't stand for stabilizers since ZMP just represents thepoint of equilibrium in terms of d'Alembert's principle.

Hirai et al.[32] proposed ground reaction force control through modulation of anklejoint angle, and Nagasaka[128] proposed Trunk Position Compliance Control. Based onthe same principle, they function as stabilizers, indirectly manipulating ZMP.

A.2.3 ZMP Manipulation as the Input

Sano and Furusho[155] proposed an angular momentum control around the ankle jointof the supporting leg in 1989. In the paper, the limitation of input torque at the jointdue to the foot length is discussed. It implies the fact that ZMP should be consideredrather in the stage of farce manipulation, since it is strongly connected with torques orforces. Fujimoto et al.[223] proposed an indirect manipulation of the ground reactionforce, taking the limit due to ZMP and friction force into account. As are introduced inChapter4, some other studies [145] [139] [141] [147], including the authors, regard ZMPas the input variable.

In addition, Tamaki et al.[224] proposed ZMP Jacobian and the control ZMP with it.Its e�cacy should be carefully examined, since it requires precise manipulation of the jerkof joint angles, which is hardly achieved in practical controls.

120 A.3. EXTENSION OF ZMP

A.3 Extension of ZMP

Goswami[225] de�ned Foot-Rotation Indicator (FRI) point around which the momentequilibrates if the supporting foot were imaginarily �xed in the inertia frame, expectingthe possibility of being as a measure of the severity of falling-down situations by thedistance from FRI point to the supporting region, while ZMP does never leave the regionso that it cannot be such a criterion. However, a practical application of it is yet to beproposed. On the other hand, Vukobratovi�c et al. [226] pointed out the similarity betweenFRI point and the imaginary ZMP (IZMP), already applied for some conventional motionplanning methods [218] [126] [219] [129], in the sense that both of them virtually �xes thesupporting foot in the inertia frame, though the former is measured in online, while thelatter is calculated in o�ine.

Since ZMP is supposed to be on a horizontal plane, it should be augmented to beapplied for three-dimensional motions. Yoneda and Hirose[227] de�ned Tumble StabilityMargin to estimate endurance against tumbling in the legged motion planning on uneventerrains. Takeuchi[228] de�ned enhanced ZMP on a slope. Kitagawa et al.[129] alsoproposed enhanced ZMP but in di�erent style from Takeuchi's, enabling representationsof arbitrary contact states in 3-dimensional space. Kaneko et al.[229] proposed the conceptof local ZMP which enables to handle the situations that the system contacts with theenvironment at multipoints on the extremities including their hands. Saida et al.[230]insisted that the total force be necessarily considered with its six independent components,and proposed Feasible Solution of Wrench(FSW) to estimate the feasibility of motions,particularly with rotations. However, it is completely substitutable with the expressioncontaining ZMP. And thus, the advantage of FSW instead of ZMP is yet to be shown.The authors also proposed the concept of VHP (see Appendix.G).

Disappointingly, some of the above studies misunderstand that ZMP indicates a sta-bility of the system. Actually, a system is stable when it doesn't diverge from the pointof equilibrium. And, one should notice ZMP does never give any information to ensurethe stability in such a sense.

A.4 Calculation of ZMP

According to its de�nition, ZMP satis�es the following equation.

NXi=1

fmi(pGi � pZ)� (�pGi + g) + I i _!i + !i � I i!ig = nZ (A.1)

nZ � g = 0 (A.2)

where N is the number of links, mi is the mass of link i, I i is the inertia tensor of link i,pGi � [ xGi yGi zGi ]

T is the center of gravity of link i, !i is the angular velocity of linki, pZ � [ xZ yZ zZ ]T is ZMP, nZ � [ nZx nZy nZz ]

T is the moment around ZMP, and

g � [ 0 0 � g ]T is the acceleration of gravity.Let us consider the case to calculate ZMP from given pgi, �pgi and !i. Eq.(A.1) and

(A.2) consist of �ve independent equations and contains six unknowns, which means that

APPENDIX A. ZMP, INTERPRETATION AND APPLICATION 121

it has no unique solution. In conventional methods[231], Eq.(A.1)(A.2) is solved by givingthe height of the ground zZ and ignoring the inertia of each link as follows.

xZ =

Xmi fxi(�zi + g)� �xi(zi � zZ)gX

mi(�zi + g)(A.3)

yZ =

Xmi fyi(�zi + g)� �yi(zi � zZ)gX

mi(�zi + g)(A.4)

Nagasaka[128] developed a computation method of the strict ZMP in accordance withthe original equation of equilibrium of Eq.(A.1).

(p0 � pZ)� f + n0 = nZ (A.5)

where f � fxfyfz is the resultant external force and n0 � nxnynz is the resultant momentaround p0. Both f and n0 are available from the inverse dynamics. Then, we conclude:

xZ = x0 � (z0 � zZ)fx + nyfz

(A.6)

yZ = y0 � (z0 � zZ)fy � nxfz

(A.7)

Eq.(A.6)(A.7) also shows the way to compute ZMP from measured external forcesand moments at each force sensor. Suppose the set of sensor position, measured force andmoment is fpSi; fSi;nSig, the number of sensors is NS, and no other contact points. Theresultant external force and moment are calculated as follows.

f =

NSXi

fSi (A.8)

n0 =

NSXi

fnSi + (p0 � pSi)� fSig (A.9)

And from Eq.(A.6) and (A.7), we obtain ZMP.

A.5 Mass-Concentrated Model Approximation

The mass-concentrated model, which approximately ignores the e�ect of moment of inertiaaround the center of gravity, is frequently adopted in previous researches [113] [32] [128][145] [147], on account that it can reduce the complexity of the system dynamics. Underthis approximation, the resultant force is parallel to the line which connects ZMP andCOG. Hence, approximate ZMP ~pZ satis�es the following equation.

(pG � pZ)� f = 0 (A.10)

122 A.5. MASS-CONCENTRATED MODEL APPROXIMATION

And thus,

~xZ =

Xi

mixiXj

mj(�zj + g)�Xi

mi�xiXj

mj(zj � zZMP;4)Xi

mi

Xj

mj(�zj + g)(A.11)

~yZ =

Xi

miyiXj

mj(�zj + g)�Xi

mi�yiXj

mj(zj � zZMP;4)Xi

mi

Xj

mj(�zj + g)(A.12)

The validity of this model is discussed by evaluating the error between pZ and ~pZ .

~xZ � xZ =

Xi

Xj

mimj f(xj � xi)�zj � (�xj � �xi)zjgXi

Xj

mimj(�zj + g)(A.13)

~yZ � yZ =

Xi

Xj

mimj f(yj � yi)�zj � (�yj � �yi)zjgXi

Xj

mimj(�zj + g)(A.14)

Qualitatively,

(horizontal components of the moment of inertia)

(vertical component of the inertia force including gravity)(A.15)

Therefore, we conclude that ~pZ can well approximate pZ in the case that the gravity forceis dominant against the moment of inertia.

Appendix B

Micro/Macro Multipoint ContactModel in Forward Dynamics

B.1 Background

Modeling of the environment and the contact force is one of the key issues in dynamicalanalysis and forward dynamics of robots which frequently interact with the environment,contouring control of manipulators or legged motion for instance, since both the dynamicsof environment a�ects on them as largely as that of robots themselves. It is also the keyin terms of implementation since it is the part which consumes the most of computationtime, though not a few methods of forward dynamics without any contact have beenproposed [232] [233] [234] [235] [236] [237] [238] [239].

Two basic ideas are introduced to model the contact.

I.Micro Contact Model

models a micro deformation of the colliding objects. The contact force is calculatedaccording to the deformation caused by relative movements of the contact points of theobjects, so that it facilitates implementations. In addition, it allows a model of elasticmaterials such as rubbers. The micro deformation of a material, however, is basically aquite short-term phenomenon, comparing with motion of rigid multibody systems. Due tothis fact, its numerical implementation frequently requires a much smaller quantizing timestep; dynamics of ordinary multibody system is calculated on the millisecond time scale,while the micro elastic deformation of objects is usually calculated on the microsecondtime scale. A rough time step encourages a divergence of numerical integration. Moreover,it is impossible to calculate static friction force with the model in principle. Poisson'smodel [240], linear spring model, [241] [242] [243] [239] [244], nonlinear spring model [245][246] [247] [248] are applied. Yamane and Nakamura[243] proposed the use of implicitintegration method for linear spring model in order to make it less diverge.

II.Macro Contact Model

models it as a macro phenomenon based on the momentum conservation law, regard-ing the environment as a rigid object. It �rstly calculates the velocity of the collidingobject immediately after contact in accordance with rebound coe�cient and inequality

123

124 B.1. BACKGROUND

constraints of normal force and friction force. The impact applied to the system duringthe contact is calculated from a momentum variation of the colliding object. This modelenables numerically stable integration since it directly calculates a discontinuous changeof the velocity, while the micro contact model calculates the acceleration. A rigid bodymodel, however, causes a chattering at the contact points. Hyperstatics would also be atough problem. The normal force computation [249] [250], the application of quadraticprogramming to satisfy the constraints of normal force and friction force [251] [252] [140],a gradual modi�cation of supposed contact condition and constraints at each contactpoint [253] [185] have been proposed.

This chapter shows an uni�cation of those micro/macro contact model, each of whichhas some drawback and advantage in forward dynamics simulation for the following twopurposes. One is to model more complex environment which consists of various soft orhard materials. The other is to avoid both the excess penetration in the micro contactmodel and the chattering in the macro contact model simultaneously, by covering therigid model with the elastic membrane as is �gured in Fig.B.1. A similar technique toof Yamane et al. [243] is applied in the micro contact model to allow the same timestep with the macro contact model. Arbitrary contact state including edge and surfacecontact is equivalently represented as that of vertices which make up the edge or face, inorder to deal with it in the uni�ed procedure. And, a correct friction force computation isshown in the macro contact model. Yoshida et al.[250] proposed the concept of extendedinertia matrix and a contact force calculation method of two oating multibody systems,which didn't include friction force. Fujimoto et al.[140] didn't correctly deal with staticfriction force. Son et al.[254] approximately linearized the constraint of the friction coneby substituting it with a polygonal pyramid, which requires an increase of the numberof constraints and so the computational cost to make it more precise. Though Konnoet al.[255] modeled the friction force by a virtual spring-damper in which the force is inproportion to the displacement of the contact point from the point of having colliding�rst with the environment, it is less valid in terms of dynamics.

Elastic Layer

Rigid Layer

Figure B.1: Rigid object model covered by an elastic layer

APPENDIX B. MICRO/MACRO MULTIPOINT CONTACT MODEL IN FORWARDDYNAMICS 125

(A)Point contact (B)Edge contact

(C)Surface contact (D)

Figure B.2: A variety of contact states

B.2 Model of Rigid Multibody System contacting

with the Environment

B.2.1 Classi�cation of Contact State

Yamane et al.[185] classi�ed the contact states between the rigid body and the environ-ment into a point contact, an edge contact and a surface contact, �gured by (A)(B)(C)in Fig.B.2 respectively, by geometric conditions, and then, checked and corrected themby dynamical conditions. For instance, the center of the reaction force must be withinthe surface since pulling force never acts, in the case of a surface contact. In the case ofan edge contact, the moment working around the edge must be zero. And in the caseof a point contact, the moment around the point must be zero. As is shown above, itrequires an intricate procedure and geometric computation. In addition, there are somecases which cannot be classi�ed into any of those (A), (B) and (C), �gured by (D) inFig.B.2 for example. This chapter uni�es any contact states as multipoint contacts which

126B.2. MODEL OF RIGID MULTIBODY SYSTEM CONTACTING WITH THE

ENVIRONMENT

makes up the colliding edges and faces. Thanks to it, such an intricate check-and-correctphase is eluded.

B.2.2 Equation of Motion

The equation of motion of an open-chain system which interacts with the environment isrepresented by the following in generalized form [140].

H�q + b = u+Xi2C

K if i (B.1)

b = _H _q + b (B.2)

where H is the inertia matrix, q is the generalized coordinate, b is the nonlinear biasforce term involving Coriolis, centrifugal and gravity force, b is the gravity force, u is thegeneralized force, C is the set of indices for the whole contact points, f i is the three-axesforce acting at i'th contact point, Ki is a mapping matrix from f i to the equivalentgeneralized force. Although additional constraints which represent the structure of theloops is required in the case of a closed-chain [256], the extension to it is no di�cult. H,bcan be calculated by the unit vector method or by composite body method both proposedby Walker and Orin [232], which made use of the Newton-Euler's method proposed byLuh et al.[257]. Ki can also be calculated by the unit vector method or by Orin andSchrader[191]'s method in accordance with the following fact. Suppose pi is the contactpoint where the external force f i acts, the following equation is identical for any f i andin�nitesimal displacement �q from principle of virtual work.

(K if i)T �q = fTi �pi

() fTi KTi �q = fTi

@pi@q

�q (B.3)

Then, we conclude

Ki =

�@pi@q

�T

(B.4)

When u is given as an input and the external force f i is calculated by some means,the acceleration of generalized coordinate �q is obtained by Eq.(B.1). Forward dynamicssimulation is done by integrating it and updating _q and q.

B.2.3 Collision Detection and Friction

The collision check between a point pi and a face P (pi is on a link and P is a part of theenvironment, or the contrary) is achieved by the following procedure. Suppose pE is apoint on P and � i is the unit normal vector of P , the signed penetration depth di of piinto P is calculated from Eq.(B.5).

di = ��Ti (pi � pE) (B.5)


ν i

vν νi i

σ

v i

fi

v i

pivσ σi i

i

v i+∆

Figure B.3: Contact point, velocity and force

When the projection of pi to the plane including P is inside of P and di � 0, pi isin contact with P . Checking it for all the combinations of points and faces, the wholecollision points are detected. Let pi the detected collision point (i.e. i 2 C), hereafter.

The velocity of pi, vi � _pi is decomposed into the normal component to the face andthe parallel component(the slip velocity) as Fig.B.3 shows.

vi = v�i� i + v�i (B.6)

where

v�i = � _di = �Ti vi (B.7)

and v�i satis�es the following.

�Ti v�i = 0 (B.8)

Now, we de�ne the slip vector �i by Eq.(B.9).

�i �( v�i

v�i(v�i > ")

0 (v�i � ")(B.9)

where " is a tiny value, and v�i is de�ned as:

v�i �kv�ik=kvi � v�i� i k (B.10)

128 B.3. UNIFICATION OF MICRO/MACRO CONTACT MODEL

Resititution phaseCompression phase

dd + ∆d

d

−kd −k(d + ∆d) −k(d + ∆d)−kd

dd + ∆d

d

Figure B.4: Elastic force estimation in numerical integration

The external force f i is also decomposed into the normal force f�i and the frictionforce f�i as follows.

f i = f�i� i + f�i (B.11)

where

f�i � �Ti f i (B.12)

and f�i satis�es the following.

�Ti f�i = 0 (B.13)

When v�i > ", a kinetic friction force represented by the following equation works at pi.

f�i = ��Kif�i�i (B.14)

where �Ki is the coe�cient of kinetic friction. From Eq.(B.11), we get:

f i = f�i(�i � �Ki�i) (B.15)

which means that, in this case, the direction of the force is known, while the amount of itis unknown. And when v�i � ", a static force works at pi, and f�i constrains pi so longas it is less than the maximum static friction force. In this case, neither the direction northe amount is unknown.

B.3 Uni�cation of Micro/Macro Contact Model

B.3.1 Micro Contact Model based on Spring-Damper Dynamics

Here, we adopt linear spring-damper model as the micro contact model. The divergenceof numerical integration in the micro contact is mainly caused by the following reason.


Suppose the environment is modeled by a spring coe�cient ki and a damping coe�cientci, the reaction force f�i is calculated as follows.

f�i = �kidi � ci _di (B.16)

And the assumption that di varies �di ' _di�t during the time step �t implies that theforce is estimated by the minimum value in the compression phase, and by the maximumvalue in the restitution phase, as is �gured in Fig.B.4. This encourages deeper penetrationof the contact point into the environment, which causes the excess accumulation of strainenergy, and stronger rebound. Joukhadar et al.[258] proposed an adaptive modi�cationof time step �t to subdue the variation of kinetic energy within a certain amount, whichrequires some recomputation in each step, so that it is time-consuming. Here, we proposeto use the following equation instead of Eq.(B.16) in order to prevent such numericalphenomena.

f�i = �ki(di + _di�t)� ci _di (B.17)

It e�ects oppositely against the frequently used method, namely, the penetration in thecompression phase is lessen and the rebound in the restitution phase is weaken. This is inprinciple based on the same idea with implicit integration introduced in [243]. Incidentally,Eq.(B.17) is rewrote as follows.

f�i = �kidi � (ci + ki�t) _di (B.18)

One can interpret this equation as damping coe�cient is adaptively set in response to�t. It is possible to replace it with nonlinear spring model[245] or Poisson's model[240],since the restitution force in general monotonously increases as the point penetrates moredeeply. f i is obtained from Eq.(B.17), (B.9) and (B.15).

B.3.2 Macro Contact Model and Equation of Contact

Here, we group the contact points to ones based on the micro contact model fP iji 2 T gand on the macro contact model fP iji 2 Hg. Then, Eq.(B.1) becomes as follows.

H�q + b = u+Xi2H

K if i +Xi2T

K if i (B.19)

Since the external force calculated in accordance with the micro contact model is obtainedin advance from local relative movement of the point and the environment, Eq.(B.19) canbe rewrote as follows.

H�q + b = uE +Xi2H

K if i (B.20)

uE � u+Xi2T

K if i (B.21)

We assume that the variations of H, b, uE and Ki during �t are small enough to beignored against �q and f i, hereafter.


Suppose during �t the multibody system is given the impact at the contact points andthe velocity of the contact points vi and the generalized coordinate velocity _q change tovi+�vi and _q+� _q, respectively. From Eq.(B.4), one can derive the following equation.

KTi � _q = �vi (i 2 H) (B.22)

And, integrating Eq.(B.20) by �t, we get:

H� _q =

Z�t

(uE � b+Xi2H

K if i) dt

= �u+Xi2H

K i�f i (B.23)

where

�u �Z�t

(uE � b) dt ' (uE � b)�t (B.24)

�f i �Z�t

f i dt (B.25)

Eliminating � _q from Eq.(B.22) and (B.23), the following equation is derived.

KTi H

�1Xj

Kj�f j = �vi � vBi (B.26)

wherevBi �KT

i H�1�u (B.27)

Then, we consider about the conditions which �f i and �vi must satisfy. In the casethat

i) the contact point has no movement (i.e.kvi k� ")

, a static friction force acts at pi. If the point ideally remains stationary,

vi +�vi = 0 (B.28)

And, neither the amount nor the direction of �f i is known.

ii) the contact point moves at a certain velocity (i.e.kvi k> ")

a kinetic friction force acts at pi. From Eq.(B.15), �f i is represented as follows.

�f i = �f�i(� i � �Ki�i) (B.29)

And �vi is decomposed as follows.

�vi = �v�i� i +�v�i (B.30)

where�v�i = �Ti �vi; �Ti �v�i = 0 (B.31)


Let ei be the coe�cient of rebound at pi, we get:

�v�i = �(1 + ei)v�i (B.32)

And, �v�i is unknown.Suppose S and K are the set of point indices i at which the static friction force and the

kinetic friction force work, respectively. And now, we conclude as follows from Eq.(B.26)(B.28) (B.29) (B.30) (B.31) and (B.32).

when i 2 SKT

i H�1

(Xj2S

Kj�f j +Xj2K

�f�jKj(�j � �Kj�j)

)= �vi � vBi (B.33)

when i 2 K�Ti K

Ti H

�1

(Xj2S

Kj�f j +Xj2K

�f�jKj(�j � �Kj�j)

)= �(1 + ei)v�i � �Ti vBi (B.34)

This becomes simultaneous linear equations about �f i and �f�i, and is represented asfollows.

A�f = v (B.35)

Since both the dimension of �f and v is (the number of factors in S)�3+(the numberof factors in K), A is a positive semide�nite square matrix. Eq.(B.35) is the equation ofcontact of the total system.

B.3.3 Solution of Equation of Contact with Quadratic Program-ming

If A in Eq.(B.35) is a regular matrix, the impact �f is immediately obtained as follows.

�f = A�1v (B.36)

However, since the motion of a link is determined only by the motion of three points onit, it becomes a hyperstatic problem and A is a singular matrix when more than fourvertices on a link are in contact with the environment. In this case, �f is not determineduniquely. L�otstedt[251], Bara�[252], and Fujimoto et al.[140] don't take it into account,and thus they had a problem that the total external force is unnaturally distributed toall the contact points in such hyperstatic cases. In addition, Eq.(B.33) and (B.34) comesonly from local movements of each contact point, so that numerical error incidentallyleads to a geometrical inconsistency in the link movement as a rigid body. These factorsprevent Eq.(B.35) from having a strict solution. Here, we propose to calculate �f bysolving the following quadratic programming.

1

2kA�f � v k2 + 1

2� k�f k2�! minimize:

subject to �Ti �f i � 0 (i 2 H) (B.37)


where � is a tiny value, and the inequality constraint implies that pulling forces cannotwork at any contact points because of the absence of mechanical connection with theenvironment. As a result, the error between both sides of Eq.(B.35) becomes small, whilethe total external force is distributed to each point in an intuitively natural balancedmanner. We can also interpret it into the following dynamical assumption. Namely,since the micro deformation of the objects can be small enough to be ignored under theapproximation of rigid bodies, the strain energy of the objects becomes almost stationaryat any moment.

f i is computed as the average of �f i per �t as follows.

f i =�f i�t

(B.38)

B.3.4 Modi�cation of Friction Force

We derived the equation of contact in B.3.2 under the assumption that the contactpoint ideally remains stationary in the case that static friction force works. However, ifit exceeds the maximum static friction force, it should be discontinuously changed to thekinetic friction force, and then, the point will begin to slip. This transient phenomenonis mimicked by the following procedure. Firstly, we decompose f i, tentatively obtainedby the above calculation, into the normal force f�i and the friction force f�i according toEq.(B.11), where

f�i =kf�i k (B.39)

Let �Si be the coe�cient of maximum static friction at pi. If f�i and f�i satis�es thefollowing, a static friction force works, so that it has no inconsistency.

f�i � �Sif�i (B.40)

pi will slip on the contact face under the following condition.

f�i > �Sif�i (B.41)

In this case, f�i is replaced with the kinetic friction force as follows.

f�i = �Kif�i (B.42)

B.3.5 Procedure of Forward Dynamics Computation

This section shows the complete procedure of forward dynamics including a computationof contact forces.

1. Compute the inertia matrix H, the nonlinear bias force b and the gravity force b.

2. Detect collisions between the multibody system and the environment. Store the setof contact points.

3. For each pi, compute the contact velocity vi, the normal vector �i of the polygon,the slip vector �i and the force mapping matrix Ki.


4. Compute the external force modeled by the micro contact from Eq.(B.17), (B.9)and (B.15).

5. Create the equation of contact Eq.(B.35) from Eq.(B.33), (B.34) and the inputgeneralized force vector u in accordance with the macro contact model.

6. Calculate a tentative external force vector f i by solving the quadratic programming(B.37) and averaging it per �t.

7. Modify the friction force working at each f i.

8. Calculate the acceleration of the generalized coordinate �q from Eq.(B.1).

9. Integrate �q, and update _q and q.

Fig.B.5 shows the owchart. Although the total complexity of computation largelydepends on the solution of quadratic programming and thus is hardly estimated, the costfor checking of contact state and of dynamical validity is reduced against the method in[185], for example.

B.4 Simulation

This section shows some results of the simulations, using a model of the miniature anthro-pomorphic robot HOAP-1 [87], which consists of 20 actuated joints (4 for each arm and6 for each leg) and 6 underactuated joints for the root link(torso). Fig.B.6, Fig.B.7 andFig.B.8 are the snapshots of examined walking motion, falling down and jumping motion,respectively. The environment was modeled as a rigid plane(coe�cient of rebound was0, the coe�cient of maximum static friction was 3.0 and the coe�cient of kinetic frictionwas 0.8) covered by an elastic membrane(spring coe�cient was 1000[N/m], damping coef-�cient was 100[N/m�s�1] and the thickness was 1[mm]). � was set for 1.0 in the quadraticprogramming (B.37), which was solved as follows: �rstly, solve it with all constraintsremoved, using the Singularity-Robust Inverse matrix [188] (see Appendix.I), and then,attach the equational constraint �Ti �f i = 0 when �Ti �f i � 0 is not satis�ed, accordingly.Though this process possibly increases the amount of computation, compared to othermethods such as the complementary pivoting method (see Appendix J), as it takes somemodi�cation of constraints, it requires less cost in most cases. Symplectic integration[259]was adopted with its time step for 1[ms]. The above all simulations were executed withPC/AT compatible personal computer (Intel(R)Celeron 2.0GHz CPU, and 512MB RAM),and the actual average time for each step took about 1.5[ms].

When we varied � from 0.0001 to 100 by tenfold, a chattering occurred for small values,while an over penetration was led for large values because of the lack of reaction force.We also found that spring coe�cient has to be set for that can create as strong reactionforce as the macro contact model does.

134 B.4. SIMULATION

Compute

Detect Collision

START

yes

no

END

Calculate Acceleration

Compute

Modify Friction Force

Calculate tentative

Create Contact Equation

Integrate

H , b, b

Ki, vi, νi, σi i ∈ Cfor

A∆f = v

f i1

2‖A∆f − v‖2 +

1

2λ ‖∆f ‖2→ min.

subject to νTi ∆f i ≥ 0

f i = ∆f i/∆t

i ∈ C

fφi = µKifνi

fφi ≥ µSifνi

Figure B.5: Flow Chart of Forward Dynamics


Figure B.6: Walking motion

Figure B.7: Falling down

Figure B.8: Jumping

Appendix C

High Precision ZMP Manipulationwith O�set Acceleration Estimation

Chapter 2,3,4 and 5 developted discussions on the following approximate equations ofmotion in accordance with a mass-concentrated model.

�xG = !2(xG � xZ) (4.1)

�yG = !2(yG � yZ) (4.2)

And, the equivalent COG acceleration to the desired ZMP, refpZ , was calculated byputting it into Eq.(4.1) and (4.2). In fact, the actual equation of motion which takesthe moment around COG into account becomes as follows.

�xG = !2(xG � xZ) + �x (C.1)

�yG = !2(yG � yZ) + �y (C.2)

where

�x � � nGym(zG � zZ)

(C.3)

�y � nGxm(zG � zZ)

(C.4)

The presence of these o�set �x and �y causes a di�erence of the real ZMP from thedesired ZMP, even when the real COG acceleration coincides with ones in Eq.(4.1) and(4.2) against the desired ZMP. This chapter re�nes it to enable a highly precise ZMPmanipulation.

Suppose estimate o�set values of �x and �y are ~�x and ~�y, respectively, the equivalentCOG acceleration, ref �pG, to the desired ZMP is calculated as follows.

ref �xG = !2(xG �refxZ) + ~�x (C.5)ref �yG = !2(yG �refyZ) + ~�y (C.6)

137

138

Assuming that the real COG acceleration is manipulated so accurately that ref �pG ' �pG,�x and �y is calculated from Eq.(C.1)(C.2) (C.5) and (C.6) as follows.

�x = ~�x � !2(refxZ � xZ) (C.7)

�y = ~�y � !2(refyZ � yZ) (C.8)

Then, the estimate values ~�x and ~�y can be updated and re�ned from Eq.(C.7) and (C.8)by the initial values and the error between the desired and the real ZMP.

Eq.(C.5) and (C.6) implies the fact that ZMP is in the same dimension with the forcein nature, namely, the control of ZMP requires manipulation of the jerk or the torque rate,which is less preferable in practice. Note that the above idea doesn't mean the control ofZMP but a fair approximation of the COG acceleration to improve the precision of ZMPmanipulation.

Fig.C.1 shows the loci of ZMP in the same control simulation with that in section2.4.3. (A) and (D) are in the case with a simple mass-concentrated model, while (B) and(E) are the result of with an o�set estimation introduced in this chapter. One can alsoverify the e�cacy of it from (C) and (F) which show the error between the real ZMP andthe desired; the error immediately increases even after swift acceleration with the o�setestimation.

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

[m]

800 1000 1200 1400 [msec]

xrefZ

xZ

(A)

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

[m]

800 1000 1200 1400 [msec]

xrefZ

xZ

(B)

-0.004

0

0.004

0.008

0.012

[m]

800 1000 1200 1400 [msec]

xrefZ xZ

With Offset Estimation

Without Offset Estimation

(C)

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

[m]

800 1000 1200 1400 [msec]

yrefZ

yZ

(D)

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

[m]

800 1000 1200 1400 [msec]

yrefZ

yZ

(E)

-0.025

-0.02

-0.015

-0.01

-0.005

0

[m]

800 1000 1200 1400 [msec]

yrefZ yZ

With Offset Estimation

Without Offset Estimation

(F)

Figure C.1: Loci of ZMP and the error from the desired

Appendix D

Jacobian of Angular Momentum

When the robot contact with the environment at three points pF1, pF2 and pF3 at least,Eq.(2.18) is deformed from Eq.(2.45) and Eq.(2.48) as follows.

0L(0p) =n�1Xi=0

��0I i �mi(

0pG;i �0p)� (0pG;i �0pF1)�0P�1

F1;2;30JF1;2;3

+mi(0pG;i �0p)� (0JG;i �0JF1) +

0I i0J!i)

�_� (D.1)

Thus, Jacobian of angular momentum 0JAM(0p) which maps the joints angle velocity tothe angular momentum around 0p is derived as follows.

0JAM(0p) =n�1Xi=0

��0I i �mi(

0pG;i �0p)� (0pG;i �0pF1)�0P�1

F1;2;30JF1;2;3

+mi(0pG;i �0p)� (0JG;i �0JF1) +

0I i0J!i)

�(D.2)

139

Appendix E

The Whole Body Motion Planningusing the COG Jacobian

Motion planning of humanoid robots is a multidiscipline optimization problem, which re-quires simultaneous satisfactions of some conditions, geometric constraints { for avoidanceof obstacles and selfcollision or accomplishment of tasks, force constraints { torque lim-its of actuators, ZMP existence within the supporting region and vertical reaction force,energy e�ciency and even natural impression for instance. Strong nonlinearity, however,of humanoids and a variety of motion required cause a large barrier of an analytical ap-proach to solve it. Nagasaka[128] proposed the idea of Dynamics Filter in order to designthe whole body motion trajectory with desired characteristics in uni�ed way. It consistsof some \�lters" which implement partial optimization problems decomposed from theoriginal problem. In his works, a design of so-called ZMP �lter was discussed. It convertsthe input trajectory which satis�es some geometric conditions about contact with knownobjects in the environment, for instance, to the output which satis�es the condition aboutZMP, too. Kitagawa et al.[129] also developed a fast trajectory designing method basedon Nagasaka's idea.

This section introduces a pattern designing method which is an extension of the formersby the author, and enables one to design more variety of motion, using the COG Jacobian.

Suppose a geometric constraint expressed as follows is posed to the accomplishmentof the task (trajectory of hands or feet, for instance).

pC(�) =cmdpC(t) (E.1)

The method shown here calculates the command trajectory of the whole joint angles cmd�which satis�es Eq.(E.1) and lets the locus of ZMP coincide with the commanded cmdpZ(cmdpZ is supposed to be always within the supporting region).

Suppose the initial trajectory �0 which satis�es the boundary condition of posture atthe initial time and terminal time and the desired movement of feet is prepared by somemeans. It doesn't have to satisfy the force constraints in this stage, and is consideredunder an ideal situation that arbitrary external force can act to the robot. Let pG;0,pZ;0, fz;0 and nG;0 be COG, ZMP, the vertical reaction force and the moment aroundCOG, respectively, which work to the robot when it performs strictly as �0 in the ideal

141

142

environment without any disturbance. The horizontal components of Eq.(2.60) will be asfollows.

fz;0(xG;0 � xZ;0)�m(zG;0 � zZ;0)�xG;0 � nGy;0 = 0 (E.2)

fz;0(yG;0 � yZ;0)�m(zG;0 � zZ;0)�yG;0 + nGx;0 = 0 (E.3)

pZ;0 and fz;0 are available in inverse dynamics calculation. Since the physical consistencyis not taken into account, a situation that ZMP exists out of the supporting region mighthappen.

The command pattern computed as the result of the procedure also satis�es the samekind of equations with Eq.(E.2) and Eq.(E.3).

cmdfz(cmdxG �cmdxZ)�m(cmdzG �cmdzZ)

cmd�xG �cmdnGy =0 (E.4)cmdfz(

cmdyG �cmdyZ)�m(cmdzG �cmdzZ)cmd�yG +cmdnGx =0 (E.5)

Here, we assume that COG movement in the vertical direction doesn't vary in the vicinityof this procedure as follows.

zG;0 =cmdzG; fz;0 =

cmdfz (E.6)cmdfz > 0 is satis�ed in this stage. As is already noted, cmdpZ is given as an inputtrajectory. cmdxG and cmdyG are unknown at this moment. Though cmdnGx and cmdnGyare also unknown, the following two equations are obtained from Eq.(E.4)�(E.2) andEq.(E.5)�(E.3), when assuming that variation of them is less enough to be ignored thanthe linear acceleration of COG (i.e. nGx;0 ' cmdnGx,nGy;0 ' cmdnGy).

�cmd�xG;0 �cmd��cmdxG;0 = �cmd��cmdxZ;0 (E.7)

�cmd�yG;0 �cmd��cmdyG;0 = �cmd��cmdyZ;0 (E.8)

where

�cmdxG;0 � cmdxG � xG;0 (E.9)

�cmdxZ;0 � cmdxZ � xZ;0 (E.10)

�cmdyG;0 � cmdyG � yG;0 (E.11)

�cmdyZ;0 � cmdyZ � yZ;0 (E.12)

Consequently, the problem is translated to the calculation of COG trajectory cmdpG whichsatis�es Eq.(E.6)(E.7)(E.8). Quantizing Eq.(E.7) and Eq.(E.8) by �t, they becomes asfollows.

�cmdxG;0;i+1 � (2 +cmd��t2)�cmdxG;0;i +�cmdxG;0;i�1 = �cmd��cmdxZ;0;i (E.13)

�cmdyG;0;i+1 � (2 +cmd��t2)�cmdyG;0;i +�cmdyG;0;i�1 = �cmd��cmdyZ;0;i (E.14)

where �cmd�G;0;i(� = x or y) is the value of �cmd�G;0;i at the quantized time i. This is asimple tridiagonal equation, and is solved with the following boundary conditions.

�cmdxG;0;0 = 0 (E.15)

�cmdxG;0;M = 0 (E.16)

�cmdyG;0;0 = 0 (E.17)

�cmdyG;0;M = 0 (E.18)

APPENDIX E. THE WHOLE BODY MOTION PLANNING USING THE COGJACOBIAN 143

(M is the quantized terminal time). cmdpG at each moment is obtained Eq.(E.6) and thefollowing equations.

cmdxG;i = xG;0;i +�cmdxG;0;i (E.19)cmdyG;i = yG;0;i +�cmdyG;0;i (E.20)

The procedure so far is the same with that of Nagasaka's and Kitagawa's.The target command joint angles cmd� which simultaneously satis�es cmdpG and cmdpC

in Eq.(E.1) is computed by Newton-Raphson method, using the COG Jacobian and aJacobian JC de�ned by the following equation.

JC � @pC@�

(E.21)

Now, even if the robot performs strictly according to this cmd�, the locus of ZMPdoesn't necessarily coincide with cmdpZ under the in uence of ignored nG. Then, lettingthis cmd� be �0 anew, iterate the above computation until the error between cmdpZ andpZ becomes small enough.

This method enables a design of more variety of the whole body motion since it hasno restriction about how to express the geometric constraint Eq.(E.1) than the previousmethods.

One should note that the constraint about ZMP is no more than a part of requirementsto maintain the physical consistency of the robot motion. Other constraints about jointangle limitation, torque limitation and so forth should be considered additionally.

Appendix F

The Equation of Motion of InvertedPendulum

θ

-g

m

l

xc

[ , ]x zz

x

Figure F.1: Inverted pendulum

The equation of motion of an inverted pendulum shown in Fig.F.1 is represented asfollows.

�� =g

lsin � � �xc

lcos � (F.1)

where � is the angle between the vertical axis and the pendulum, l is the length ofpendulum, and xc is the position of the supporting point. The tip position is [x; z] is asfollows.

x = xc + l sin � (F.2)

z = l cos � (F.3)

145

146

Putting Eq.(F.2)(F.3) into Eq.(F.1), we get the following equation.

�� =g

l

�x� xc

l

�� xcz

l2(F.4)

Linearizing it near � ' 0, it becomes as follows.

�x = !2(x� xc) (F.5)

where ! is de�ned as follows.!2 � g

l(F.6)

Appendix G

Virtual Horizontal Plane

VHP

p1

p2

pN

p1’ p2’

...

...

real environment

xG

pN’

Figure G.1: Virtual Horizontal Plane(VHP)

Since the contact points between the robot and the environment are ranged 3-dimensionally despite ZMP is commonly de�ned as the point on the ground, a newframework is required to discuss the feasibility of ZMP. Kitagawa et al.[129] de�nedthe enhanced ZMP on a plane constructed by three points arbitrarily chosen from thecontact points, and proposed a planning method for arm-leg cooperative motion trajec-tories. However, how to choose the three points for the desired motion is nontrivial, andconsidering all of the combinations of the points necessitates a serious computational cost.

Virtual Horizontal Plane(VHP)[260] is an idea to take such 3-dimensional contact of

147

148

the robot with the environment into account. All the contact points are converted to theequivalent points on VHP in terms of the working forces to support COG. Ignoring thea�ect of the inertial force, such the equivalent point is de�ned as the intersection pointof VHP and the line which connects COG and the contact point as is shown in Fig.G.1,since the force vector accepts parallel translation along the transverse line of action. Thesupporting region on VHP is de�ned as the convex hull (see Appendix K) of those points.

In the case that some of the contact points are located above COG, we place a pointwith a �nite distance from COG on VHP in the direction towards the contact point, andsubstitute an actual unbounded region for an approximately bounded region as is shownin Fig.G.2.

VHP

contact pointabove COG

direction forthe equivalentcontact pointon VHP

pc

actual equivalentsupporting region

boundedsupporting region

bounded equivalentcontact point on VHP

Figure G.2: Contact point above COG and bounded supporting region

Appendix H

Generalized Inverse Matrix

A general form of simultaneous linear equations which consist of m equations includingn unknowns, x1 � xn, is as follows,8>>><>>>:

a11x1 + a12x2 + � � �+ a1nxn = b1a21x1 + a22x2 + � � �+ a2nxn = b2

...am1x1 + am2x2 + � � �+ amnxn = bm

(H.1)

or,Ax = b (H.2)

where

A �

26664a11 a12 : : : a1na21 a22 : : : a2n...

.... . .

...am1 am2 : : : amn

37775 (H.3)

x � [ x1 x2 : : : xn ]T (H.4)

b � [ b1 b2 : : : bn ]T (H.5)

Here, we suppose that A is a full rank matrix (i.e. rank(A) = m) for simplicity.When A is a square matrix (i.e. m = n), Eq.(H.2) is solved as follows.

x = A�1b (H.6)

We consider the cases that m 6= n, hereinafter, and note about the solution of them, usinggeneralized inverse matrix.

Case I) m < n

Eq.(H.2) has in�nite solutions, so that one cannot solve it uniquely. The general solutionforms as a summation of a particular solution xc and a homogeneous solution ~x as follows.

x = xc + ~x (H.7)

149

150

where xc and ~x satisfy the following conditions.

Axc = b; A~x = 0 (H.8)

Then, we decide xc as the weighted squared norm of x is minimized.

1

2kxk2W� 1

2xTWx (H.9)

where W � diagfwig(i = 1; � � � ; n) is a positive de�nite weighting matrix. xc is thesolution of the following quadratic programming.

E =1

2xTWx �! minimize

subject to Ax = b(H.10)

Since E is a convex quadratic function, Lagrange's multiplier method is available to solvethe problem (H.10). Suppose � is the Lagrange's multiplier, and L is the Lagrangian,de�ned as follows.

L = E + �T (Ax� b) = 1

2xTWx+ �T (Ax� b) (H.11)

Then, the optimum solution vector xc and its co-state vector �� satis�es the following.

@L

@x(xc;�

�) = 0 () Wxc +AT�� = 0 (H.12)

@L

@�(xc;�

�) = 0 () Axc � b = 0 (H.13)

Conclusively, we derive xc as:

xc = A#b (H.14)

where

A# �W�1AT (AW�1AT )�1 (H.15)

A# is called a weighted norm minimizing generalized inverse matrix. Using A#, thehomogeneous solution ~x is represented as follows.

~x = (I �AA#)� (H.16)

where � is an arbitrary vector (2 Rm) Hence, the general solution of Eq.(H.2) is:

x = A#b + (I �AA#)� (H.17)

Case II) m > n

Eq.(H.2) has no solution. Here, we de�ne the residual vector as follows.

e = Ax� b (H.18)

APPENDIX H. GENERALIZED INVERSE MATRIX 151

Then, we consider to obtain x which can minimize the weighted squared norm of e, andregard it as the solution.

1

2kek2W

=1

2eTWe (H.19)

where W � diagfwig(i = 1; � � � ;m) is a positive de�nite weighting matrix. The gen-eral solution in this case also forms as a summation of a particular solution xe and ahomogeneous solution ~x as follows.

x = xe + ~x (H.20)

xe is the solution of the following quadratic programming.

E =1

2eTWe �! minimize (H.21)

Since E is a convex quadratic function, the minimum solution of the problem (H.21)coincides with the local minimum. Thus, xe satis�es the following.

@L

@x(xe) = 0 () ATWAxe �ATWb = 0 (H.22)

Conclusively, we get xe as:xe = A�b (H.23)

whereA� � (ATWA)�1ATW (H.24)

A� is called a weighted error minimizing generalized inverse matrix. Using A�, thehomogeneous solution �x is represented as follows.

�x = (I �A�A)� (H.25)

where � is an arbitrary vector (2 Rm). Hence, the general solution of Eq.(H.2) is:

x = A�b + (I �A�A)� (H.26)

Appendix I

Singularity-Robust Inverse Matrix

Let us consider the problem to obtain the x = x� which satis�es the following nonlinearequation for a given y = y�.

y = f(x) (I.1)

It is called an inverse problem. If f�1(x), the inverse function of f(x), is analyticallyavailable, then

x� = f(y�) (I.2)

In general cases, however, how to derive such f�1(x) is not trivial. And, some itera-tive computations are applied frequently. Newton-Raphson's method, described in thefollowing paragraph, is one of the commonly used for this sake.

Linearizing Eq.(I.1) around the initial x, x0, we get:

y ' y0 + J(x0)�x (I.3)

where

J � @f

@x(I.4)

From Eq.(I.3), x� is approximately obtained as follows.

x� ' x0 +�x� (I.5)

where �x� satis�es the following equation.

J(x0)�x� = �y� � y� � y (I.6)

Putting x� into the new initial x0 and iterating the computation according to Eq.(I.5), xis expected to converge to the true solution.

Now, if J(x0) is full rank, Eq.(I.6) is solved as follows.

�x� = J#(x0)�y� (I.7)

where J#(x0) is the generalized inverse matrix of J(x0) (see Appendix.H). However, inthe case that J(x0) is not full rank, or singular, J

#(x0) doesn't exist even if Eq.(I.6) hassome feasible solutions. Further more, the most serious problem is not in the singular

153

154

case, but in the neighborhood of the singular point, where the absolute values of some ofthe singular values of J(x0) are nearly equal to zero; even a small �y

� causes an enormous�x�, which possibly leads to a disastrous instability in some applications, particularly, inthe inverse kinematics of manipulators. Nakamura et al.[188] proposed Singularity-Robustinverse matrix to overcome the above problem.

One can interpret the problems to eliminate the instability in the neighborhood ofsingular point as to minimize the norm of �x�, and to obtain �x� which satis�es Eq.(I.6)as to minimize the norm of the error �y� � J�x�, and can formulate it as the followingquadratic programming.

E =1

2k�y � J�xk2W1

+1

2k�xk2W2

�! minimize (I.8)

whereW 1 andW 2 are positive de�nite weighting matrices, and k�k2W implies the samewith that in Eq.(H.9). �x� is the optimum solution of the problem (I.8). Since E is aconvex function, �x� is calculated as follows.

@E

@�x(�x�) = 0 () JTW 1J�x

� � JTW 1�y� +W 2�x

� = 0

() �x� = (JTW 1J +W 2)�1JTW 1�y

� (I.9)

Hence, Singularity-Robust inverse matrix J? is de�ned as follows.

J? � (JTW 1J +W 2)�1JTW 1 (I.10)

Since W 1 and W 2 are positive de�nite, JTW 1J +W 2 is invariably full rank.

Appendix J

Complementary Pivoting Method

Complementary Pivoting method(Lemke's method) is known as a comparatively fast so-lution of quadratic programming which iequality constraints. This chapter shows theprinciple and algorithm of the method.

Quadratic programming problem in general is represented as the following form.

f(x) =1

2xTQx +CTx �! minimize

subject to Ax � b ; x � 0(J.1)

where Q is a positive semide�nite symmetric matrix. When the problem Pb.(J.1) has theoptimum solution x?, it is equivalent to the following problem Pb.(J.2).

Pb.(J.2):

Obtain x? and y which satisfy the following condition.

uTx? = 0; vTy = 0; u � 0; v � 0; x? � 0; y � 0 (J.2)

where u and v are de�ned as follows.

u = Qx? �AT + c (J.3)

v = Ax? � b (J.4)

Proof)

Let d 2 Rn be an arbitrary vector, and "(� 0) is a tiny value. Since x? is the localminimum in the neighborhood, the following inequality must be satis�ed.

f(x? + "d)� f(x?) =1

2"2dTQd+ "(c+Qx?)Td � 0 (J.5)

under the following conditions.

A(x? + "d) � b; x? + "d � 0 (J.6)

And thus,"(c+Qx?)Td � 0 (J.7)

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156

for an arbitrary tiny value " > 0. Let ~d � "d, and the condition (J.7) is deformed asfollows.

A~d � b�Ax?; ~d � �x? =) (c+Qx?)T ~d � 0 (J.8)

Then, let us consider the following linear programming Pb.(J.9).

(c+Qx?)T ~d �! minimize

subject to A~d � b�Ax?; ~d � �x? (J.9)

One can note that x? is the optimum solution of Pb.(J.1), and thus

b�Ax? � 0; �x? � 0 (J.10)

Therefore, Pb.(J.9) has a trivial solution ~d = 0, and the following dual problem of Pb.(J.9)also has the optimum solution y.

(b�Ax?)T � x?T (c+Qx? �ATy) �! maximizesubject to ATy � c +Qx?; y � 0

(J.11)

where

(b�Ax?)Ty � x?(c+Qx? �ATy) = 0 (J.12)

Here, let us de�ne u and v by Eq.(J.3) and (J.4), and Eq.(J.12) becomes as follows.

vTy + x?Tu = 0 (J.13)

Hence,

uTx? = 0; vTy = 0 (J.14)

because of the following conditions.

x? � 0; y � 0; u � 0; v � 0 (J.15)

Inversely, the feasible solution of Pb.(J.a) x (i.e. Ax � b; x � 0) satis�es the followingcondition.

f(x)� f(x?) =1

2(x� x?)TQ(x� x?) + (c+Qx?)T (x� x?)

� (c+Qx?)T (x� x?) ( ... Q is positive semide�nite)

= (u+ATy)T (x� x?)= uTx+ yTAx� uTx? � yTAx?� yT (b�Ax?) ( .:. uTx � 0; uTx? = 0; y � 0; Ax � b)= �yTv = �vTy = 0

Conclusively, Pb.(J.1) and Pb.(J.2) are equivalent with each other.

Q.E.D.

APPENDIX J. COMPLEMENTARY PIVOTING METHOD 157

Hereinafter, the algorithm of Lemke's Complementary Pivoting method to solvePb.(J.2) is described. Pb.(J.2) is further simpli�ed as follows.

w +Mz = q; w � 0; z � 0; wTz = 0 (J.16)

where

w ��u

v

�; z �

�x

y

�; M �

� �Q AT

�A O

�; q �

�c

�b�

(J.17)

If q � 0, Eq.(J.16) has a trivial solution w = q; z = 0, and then the problem is solved.Now, let us consider the case of q 6� 0. Firstly, let us de�ne the following set S by

introducing an arti�cial variable z0.

(w; z0; z) 2 S =

8<: (w; z0; z)�I �l M

� 24 w

z0z

35 = q; w � 0; z � 0; z0 � 0

9=;(J.18)

where I is the identity matrix, and l is de�ned as follows.

l � [ 1 1 : : : 1]T (J.19)

Pb.(J.2) comes down to the problem to �nd (w; z0; z) 2 S which satis�es wTz = 0 andz0 = 0.

Suppose qs(< 0) is the minimum component of q, and set the initial base vector as:

(w; z0;z) = (q � qsl;�qs; 0) (J.20)

Then, (w; z0; z) 2 S and wTz = 0. Since ws = 0, even if the corresponding zs is replacedwith an arbitrary positive value, it still remains (w; z0; z) 2 S and wTz = 0. Therefore,sweeping out the components around the pivot zs, one can exchange the basic feasiblevector to the contiguous feasible vector in S, which also satis�es wTz = 0. This is thebasic idea.

Here, we de�ne the following matrix called tableau.

�I �l M q

�=

266666664

11

O�l1�l2

m11

m21

: : :: : :

m1n

m2n

q1q2

. . ....

......

...

O

1. . .

1

�ls...�ln

ms1

...ms1

: : :

: : :

mnn

...mnn

qs...qn

377777775(J.21)

Suppose the factors of the base vector are x1; � � � ; xn, the pivot variable is xn+1, and theother variables are xn+2; � � � ; x2n+1, respectively. Namely,

xi = qi � lixn+1; xn+i+1 = 0 ( i = 1 � n ) (J.22)

158

Thus, sweeping out the components around lr, the new pivot, where lr is a positive valueand r minimizes qr=lr, the contiguous basic feasible vector is generated. Let us substitutexr, xn+r+1 and xn+1 for xn+r+1, xn+1 and xr, respectively. When z0 comes out of thebase vector after some successive alternation of the base variables, the base vector at themoment satis�es the following condition.

(w; z0; z) 2 S; wTz = 0; z0 = 0 (J.23)

It means that this z is the optimum solution, and Pb.(J.1) is solved. If all li are negative,one can increase xn+1 in�nitely and Pb.(J.1) has no solution.

When the tableau (J.21) is full-rank, the basic feasible vector has only two directionsto switch to the contiguous basic feasible vector as wTz = 0 remains satis�ed, so that itwill necessarily reaches the optimum solution in �nite iteration.

Appendix K

Convex Hull on a plane

(A)

pp

θ

θ

e x

e x

1

n

1k max

2k max

(B)

pp

1

n

^x-e

(C)

Figure K.1: Convex hull

This chapter shows the way to compute a convex hull which is the minimal polygonwrapping the given points on a plane as is shown in Fig.K.1(A)).

Suppose points p1 � pn on xy-plane sorted in ascending order by their x componentsare given. Note that both p1 and pn will necessarily be vertices of the convex hull.

1. Compute the upper edge as is �gured in Fig.K.1(B).

(a) Suppose i = 1 and u1 = 1.

(b) Compute �ik(� �

2� �ik � �

2) which is the angle between pk�pui and ex for

k = ui + 1 � n where ex � [ 1 0 0 ]T .(c) Find the maximum �ikmax .(d) Let i i+ 1 and ui = kmax. Terminate if ui = n.(e) Go back to (b).

2. Compute the lower edge as is �gured in Fig.K.1(C).

(a) Suppose i = 1 and l1 = 1.

(b) Compute �ik(� �

2� �ik � �

2) which is the angle between pk � pli and �ex

for k = 1 � li � 1

159

160

(c) Find the minimum �ikmin.(d) Let i i� 1 and li = kmin. Terminate if li = 1.(e) Go back to (b).

3. Connect the upper edge consisting of pui and the lower edge consisting of pli.

The complexity of this computation leans on that for sorting of the points Os and thatfor �nding of the edges Oe. Os equals to n log n, using the quick sort. And, Oe is at mostn2. Thus, Os +Oe = n2 + n log n.

List of Publications

Journal Papers

[1] COG Jacobian of Legged Robots and its Application for Motion Control, TomomichiSugihara, Yoshihiko Nakamura, Journal of the Robotics Society in Japan(in prepa-ration)

[2] Dual Term Absorption of Disturbance for the Whole Body Cooperative MotionControl of Humanoid Robot, Tomomichi Sugihara, Yoshihiko Nakamura, Journal ofthe Robotics Society in Japan(in preparation)

[3] High Mobility Control of Humanoid Robots through Inverted Pendulum ModelBased ZMP Manipulation, Tomomichi Sugihara, Yoshihiko Nakamura, Journal ofthe Robotics Society in Japan(in preparation)

[4] Computation of External Force in Forward Dynamics of Rigid Multibody basedon Micro/Macro Multipoint Contact Model, Tomomichi Sugihara, Yoshihiko Naka-mura, Journal of the Robotics Society in Japan(in preparation)

Reviewed Conference Proceedings

[1] Realtime Humanoid Motion Generation through ZMP Manipulation based on In-verted Pendulum Control, Tomomichi Sugihara, Yoshihiko Nakamura, Hirochika In-oue, IEEE International Conference on Robotics and Automation(ICRA'02), Wash-ington DC, May., 2002

[2] Whole-body Cooperative Balancing of Humanoid Robot using COG Jacobian, To-momichi Sugihara, Yoshihiko Nakamura, IEEE/RSJ International Conference onIntelligent Robots and Systems(IROS'02), Lausanne, Oct., 2002

[3] Whole-body Cooperative COG Control through ZMP Manipulation for HumanoidRobots, Tomomichi Sugihara, Yoshihiko Nakamura, 2nd International Symposiumon Adaptive Motion of Animals and Machines(AMAM2003), Kyoto, Mar., 2003

[4] Contact Phase Invariant Control for Humanoid Robot based on Variable ImpedantInverted Pendulum Model, Tomomichi Sugihara, Yoshihiko Nakamura, IEEE Inter-national Conference on Robotics and Automation(ICRA'03), Taipei, Sep., 2003

161

162

[5] Variable Impedant Inverted Pendulum Model Control for a Seamless Contact PhaseTransition on Humanoid Robot, Tomomichi Sugihara, Yoshihiko Nakamura, IEEEInternational Conference on Humanoid Robotics(Humanoids2003), Kurlsruhe, Oct.,2003

Oral Presentations(all in Japanese)

[1] Generation of Referential ZMP Trajectory Based on Minimization of Joint Torque,Tomomochi Sugihara, Ken'ichiro Nagasaka, Masayuki Inaba and Hirochika Inoue,17th Annual Conference on Robotics Society of Japan, 1999.9.

[2] Development of "Z-DYNAFORM" - library for dynamics analysis of rigid multibody,Tomomichi Sugihara, Koichi Nishiwaki, Masayuki Inaba and Hirochika Inoue, 18thAnnual Conference on Robotics Society of Japan, 2000.9.

[3] Real-Time Motion Generation of Humanoid with ZMP Actuation based on InvertedPendulum, Tomomichi Sugihara and Hirochika Inoue, 19th Annual Conference onRobotics Society of Japan, 2001.9.

[4] Whole-body Cooperative Balancing of Humanoid Robot using COG Jacobian, To-momichi Sugihara and Yoshihiko Nakamura, Annual Conference on Robotics andMechatronics, 2002, 6.

[5] Development of Z-REVICHS, Uni�ed System for Real/Virtual Robot Controlling,Tomomichi Sugihara and Yoshihiko Nakamura, Annual Conference on Robotics andMechatronics, 2002, 6.

[6] Contact Phase Invariant Controller Design for Humanoid Robot based on InvertedPendulum Model with Variable Impedance, Tomomichi Sugihara and YoshihikoNakamura, 20th Annual Conference on Robotics Society of Japan, 2002.9.

[7] Computation of Contact Force for Multibody System based on Multipoint RigidContact Model, Tomomichi Sugihara and Yoshihiko Nakamura, 8th Robotics Sym-posia, 2003.3.

[8] Responsive Legged Motion Control for Humanoid Robot via Variable ImpedantLeg Model, Tomomichi Sugihara and Yoshihiko Nakamura, Annual Conference onRobotics and Mechatronics, 2003, 5.

[9] Development of an Anthropomorphic Device for Fundamental Experiments Towardsthe Acquisition of High Mobility, Tomomichi Sugihara and Yoshihiko Nakamura,21th Annual Conference on Robotics Society of Japan, 2003.9.

Book Publications

[1] Design, Implementation and Remote Operation of the Humanoid H6, SatoshiKagami, Koichi Nishiwaki, James J. Ku�ner, Tomomichi Sugihara, Masayuki Inaba,Hirochika Inoue, In Experimental Robotics VII, Lecture Notes in Control and In-formation Sciences 271, pp.41{50, Daniela Rus and Sanjiv Singh(editors), Springer,2001.

APPENDIX K. CONVEX HULL ON A PLANE 163

[2] A Fast Dynamically Equilibrated Walking Trajectory Generation Method of Hu-manoid Robot, Satoshi Kagami, Tomonobu Kitagawa, Koichi Nishiwaki, TomomichiSugihara, Masayuki Inaba, Hirochika Inoue, Autonomous Robots , Vol.12, No.1,Kluwer Academic Publishers, pp.71{82., Jan., 2002

Patents

[1] Control of the Center of Gravity Velocity on Legged Locomotion Mechanism, To-momichi Sugihara, Japanese Patent 2003-011075

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