Feedback control of humanoid robots: balancing and walking · 1 DLR 02/05/2012 Feedback control of...

1

DLR 02/05/2012

Feedback control of humanoid robots: balancing and walkingDr.-Ing. Christian Ott

German Aerospace Center (DLR)

Institute for Robotics and Mechatronics

Folie 2

Overview

Part 0: Short overview of (biped robots at) DLR

Part I: Modeling

Part II: Balancing

Part III: Walking Control

Folie 3

German Aerospace Center (DLR)

National research laboratoryPart of the Helmholtz associationResearch fields:

o aerospace, o space technologies, o energy and traffic

~6300 researchers13 locations29 institutes

Institute of Robotics and Mechatronics: Former director: Prof. HirzingerDirector: Prof. Albu-Schäffer

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Institute of Robotics & Mechatronics

LBR-II 1999/2000

ROKVISS 2007-2010

LBR-I,199x

Space Robotics

Manipulation

ROTEX 1993

GETEX

ESS

Torque sensors at the power outputafter the bearings

Torque sensors afterthe gears, but beforethe last bearings

Modular design,load/weight ratio ~ 1:1

LBR-M, 2003

Commercial torque controlled arm (KUKA)

Compliant ManipulationJoint torque sensing & control for manipulation

Robustness: Passivity Based Control

Performance:Joint Torque Feedback

(noncollocated)

MotorDynamics

Rigid-BodyDynamics

TorqueControl

Environ-ment

ComplianceControl

extq

um

Compliant Manipulation

Robustness: Passivity Based Control

Performance:Joint Torque Feedback

(noncollocated)

MotorDynamics

Rigid-BodyDynamics

TorqueControl

Environ-ment

ComplianceControl

extq

um

Folie 8

2009-2013

Anthropomorphic Hand-Arm System[Grebenstein, Albu-Schäffer et al, Humanoids 2010]

• Compliant actuation

• Antagonistic actuation for fingers

• Variable stiffness actuation in arm

• Robustness to shocks and impacts

Biped RobotJoint torque sensing & control for manipulation

Folie 9

Bipedal Walking Robots at DLR

DLR-Biped(2010-2012)

TORO, preliminary version (2012)

TORO (2013)TOrque controlled humanoid RObot

• Drive technology of the DLR arm Allow for position controlled walking (ZMP) and joint torque control!

• Small foot design: 19 x 9,5 cm

• Sensors:- joint torque sensors- force/torque sensors in the feet- IMU in the trunk

Folie 10

First experiments with DLR-Biped

First experiment at Automatica Fair in 06-2010: ZMP preview control [Kajita, 2003]Current approach: Walking control based on the Capture Point

[Englsberger, Ott, Roa, et al. IROS 2011]

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TORO

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Overview

Part I: Modeling• Multibody dynamics• ZMP• Simplified models for control• Capture Point• Centroidal Moment Pivot

Part II: Balancing


Models of Legged (Humanoid) Robots

Multi-Body-Models Conceptual Models

Fixed Base Models(predefined contact state)

Floating Base Models Walking Running

Dynamical Models (Mechanical)

Complexity

Specialization

Free-Floating vs. Fixed Base Models

Fixed base modelsIn each contact state the model is different:

• Single support (right, left)• Double support• Heel Off• Toe Touch Down• …

Transition between contact states

double supportover-contrained

single supportserial kin. chain

Free-floating model

Components:• Lagrangian dynamics• Constraints due to contact forces• Transition equations (impacts)

underactuated

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Free-Floating vs. Fixed Base Models

Fixed base modelsIn each contact state the model is different:

• Single support (right, left)• Double support• Heel Off• Toe Touch Down• …

Transition between contact states

double supportover-contrained

single supportserial kin. chain

Free-floating model

Components:• Lagrangian dynamics• Constraints due to contact forces• Transition equations (impacts)

underactuated

Planning & control must ensure that the considered contact state is valid! ground reaction force must fulfill constraints

Configuration Space

)3(SEHb Qq

n

111 SSST n

Configuration Space: )3(SEQ

Configuration Space

)3(SEHb Qq

n

111 SSST n

Configuration Space: )3(SEQ

Using local coordinates: n6

6bx

Folie 18

Modeling

)3(6 seFext

extT FqJqgqqqCqqM )()(),()(

,q

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Modeling

extF


extF

,q

6bx

6bF

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Modeling

extF


extF

extT

Tbb

bb

bb

qx

xqx FqJqJF

qxgqx

qxqCqx

qMqMqMqM

)()(

),(),,()()()()(

,q

6bx

6bF

Folie 21

Modeling

extF


extF

extT

Tbtb

sbbbqb

bqb FqJ

qAdWqHg

qqqC

qqMqMqMqM

)()(

),(),,()()()()(

,q

)3()3(

sexSEHx

b

sbb

body twist

Global representation!

Folie 22

Modeling

extF

rF lF

l

r

qq

q

6bx

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Modeling

extF

rF lF

lT

l

Tbl

rT

r

Tbr

bb

bb

qx

xqx FqJ

qJFqJ

qJqxg

qx

qxqCqx

qMqMqMqM

)(0

)(

0)()(

0),(),,(

)()()()(

l

r

qq

q

6bx

Folie 24

Bipedal Robot Model

rF lF

lT

l

Tbl

rT

r

Tbr

bb

bb

qx

xqx FqJ

qJFqJ

qJqxg

qx

qxqCqx

qMqMqMqM

)(0

)(

0)()(

0),(),,(

)()()()(

l

r

qq

q

6bx

Properties for control:• Underactuated• Varying unilateral constraints

(single support, double support, edge contact)

• Constraints on the state & control

Folie 25

l

Tl

Tbl

rT

r

Tbr

bb

bb

qx

xqx FqJ

qJFqJ

qJqxg

qx

qxqCqx

qMqMqMqM

)(0

)(

0)()(

0),(),,(

)()()()(

pf p p

c

Mg

lríiT

i

FqJ

Iu

MgqqCq

cqM

M

, )ˆ(00

0)ˆ,ˆ(ˆ0

ˆ)(ˆ00

p

q

),( c

bx

Bipedal Robot Model

Folie 26

l

Tl

Tbl

rT

r

Tbr

bb

bb

qx

xqx FqJ

qJFqJ

qJqxg

qx

qxqCqx

qMqMqMqM

)(0

)(

0)()(

0),(),,(

)()()()(

pf p p

c

Mg

lríiT

i

FqJ

Iu

MgqqCq

cqM

M

, )ˆ(00

0)ˆ,ˆ(ˆ0

ˆ)(ˆ00

p

q

lri

ifMgcM,

),( c

bx

Conservation of momentum:

Bipedal Robot Model

Folie 27

l

Tl

Tbl

rT

r

Tbr

bb

bb

qx

xqx FqJ

qJFqJ

qJqxg

qx

qxqCqx

qMqMqMqM

)(0

)(

0)()(

0),(),,(

)()()()(

pf p p

c

Mg

lríiT

i

FqJ

Iu

MgqqCq

cqM

M

, )ˆ(00

0)ˆ,ˆ(ˆ0

ˆ)(ˆ00

p

q

lri

ifMgcM,

),( c

bx

iMgcL Conservation of angular momentum:


Bipedal Robot Model

Folie 28

l

Tl

Tbl

rT

r

Tbr

bb

bb

qx

xqx FqJ

qJFqJ

qJqxg

qx

qxqCqx

qMqMqMqM

)(0

)(

0)()(

0),(),,(

)()()()(

pf p p

c

Mg

lríiT

i

FqJ

Iu

MgqqCq

cqM

M

, )ˆ(00

0)ˆ,ˆ(ˆ0

ˆ)(ˆ00

p

q

lri

ifMgcM,

),( c

bx

iMgcL Conservation of angular momentum:


On a flat ground:

)( MgcMpMgcLp

Bipedal Robot Model

ppfp

Folie 29

l

Tl

Tbl

rT

r

Tbr

bb

bb

qx

xqx FqJ

qJFqJ

qJqxg

qx

qxqCqx

qMqMqMqM

)(0

)(

0)()(

0),(),,(

)()()()(

pf p p

c

Mg

lríiT

i

FqJ

Iu

MgqqCq

cqM

M

, )ˆ(00

0)ˆ,ˆ(ˆ0

ˆ)(ˆ00

),( c

bx

On a flat ground:

)( MgcMpMgcLp

Bipedal Robot Model

Zero Moment Point

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Zero Moment Point[Vukobratovic and Stepanenko,1972]

)(x1x 2x

F

ZMP as a ground reference point: Distributed ground reaction force under the supporting foot can be replaced by a single force acting at the ZMP.

z

x

),( yx

y

ZMP = CoP (Center of Pressure)

p1x 2x

F0

pyx 0),( in convex hull of the support polygon.

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Some facts about the ZMP

Can ZMP leave the support polygon? NOCan ZMP location be used as a stability criterion NO

If ZMP reaches the border of the support polygon foot rotation possible.

ZMP is defined on flat contact (no uneven surface).ZMP gives no information about sliding.

)(x1x 2x

F

First usage of the ZMP

• Motion of the legs is predefined.• Upper body controls the ZMP in the center of the supporting foot

ensure proper foot contact during walking

How to obtain the ZMP?

Measurement e.g. by Force/Torque Sensor

Dynamics Computation




sss fppp )()(

z

sfs

z

zsyyzszxy

z

zsxxzszyx

ffpfpp

p

ffpfpp

p

)(

)(

sp

p




sss fppp )()(

z

zsyyzszxy

z

zsxxzszyx

ffpfpp

p

ffpfpp

p

)(

)(

pf p p

ppfp

z

sfs

sp

p



pf p p

ppfp

MgcLfMgP

c

0

0

py

px

)( MgPpMgcLp

z

xyzyy

z

yxzxx

PMgLPpMgc

p

PMgLPpMgc

p

A simplified walking model based on the ZMP

Mass concentrated model

pf p p

ppfp


pf p p

ppfp

c0

zp

cMcLcMP


pf p p

ppfp

c

z

xyzyy

z

yxzxx

PMgLPpMgc

p

PMgLPpMgc

p

0

zp

cMcLcMP

z

yzyy

z

xzxx

cgcc

cp

cgcccp

ZMP of a mass concentrated model


p

c

z

yzyy

z

xzxx

cgcc

cp

cgcccp

gcccp xz

xx

0 zz cc

xc

Cart-Table Model [Kajita]

xx cp


p

c

gcccp xz

xx

xc

Cart-Table Model [Kajita]Linear Inverted Pendulum Model [Sugihara]

xxz

x pccgc

p

c

xx pc xx cp

Capture Point(Extrapolated Center of Mass)

((Divergent Component of Motion))

Capture PointDefinition of the “Capture Point” (Pratt 2006, Hof 2008):

Point to step in order to bring the robot to stand.

constp

0

0* xxp

ptxtxttx ))cosh(1()0()sinh()0()cosh()(

ptx )(

c

p *p

cc ,

Computation of the Capture Point:

zcg

ZMP

pxx 2

Can be computed exactly for simple models, e.g. linear inverted pendulum model:

Capture Point Dynamics

xx

Coordinate transformation: ),(),( xxx

)(2 pxx p

xx

COMcapturepoint

xp

System structure: Cascaded system

exp. stableopen loopunstable

Dual use of the capture point for robotics1. step planning2. control

Capture Point in Human Measurements

Linear Inverted Pendulum Human

xx

y yData from [*]

[*] Hof, The extrapolated center of mass concept suggests a simple control of balance in walking, Human Movement Science 27, pp.112-125, 2008.

Centroidal Moment Pivot


• Observation in human data: For normal level-ground walking, the human body‘s angular momentum (and the angular excursions) about the COM remains small through the gait cycle.

• The centroidal moment pivot was introduced as a ground reference point to address the effects of angular momentum about the COM in connection with postural balance strategies.


Forces in the LIP model

p

x

z

Mg

xM

F

Effect of an additional hip torque

CMP

zgM

xM

F

pxzgx

zFpCMP

Mz

pxz

zgx

Interpretation

• The distance between CMP and ZMP corresponds to the angular momentum about the COM.

• While the ZMP cannot leave the support polygon (by definition), the CMP can leave it.

• The distance between CMP and the support polygon has been proposed as an indicator which balance strategy should dominate (via ZMP or via angular momentum).

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Overview

Part I: Modeling

Part II: Balancing1. Basics2. ZMP based balancing (concentrated mass model)3. Torque based balancing (multi body dynamics)


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Humanoid Balance

Vestibular system

Vision

Somatosensory system

IMU

Vision

force sensors

joint sensing

“Balance” is a generic term describing the ability to control the body posturein order to prevent falling.

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Humanoid Balance

Small push:Ankle strategy

force controlZMP control

angular momentum control

Medium push:Hip strategy

Large Push: Step strategy

Human

Robot

Strategies for human push recovery:

Folie 55

mass concentrated model

Strategies for gait stabilization: Effect of an additional hip torque

p

Mg

xM

F

Mz

pxzgx

1. Controlling ZMP (constraints!)

2. Angular momentum control

3. Step adaptation

Folie 56

Overview

Part I: Modeling

Part II: Balancing1. Basics2. ZMP based balancing (concentrated mass model)3. Torque based balancing (multi-body model)


Folie 57

Motivation for compliant control

completely stiff fully compliantcompliant control

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ZMP based balancing

Joint Position Control

Feedback Stabilization

dx dqrefx

refpInverse

Kinematicsdq

RobotDynamics

Forward Kinematics

ZMPComputation

p

x q

LR FF ,

position/velocity controlled robot

p

x

z

Mg

xM

F

MgFxM

zxp ext

extF

Folie 59

ZMP based balancing



dx dqrefx

refpInverse

Kinematicsdq

RobotDynamics

Forward Kinematics

ZMPComputation

p

x q

LR FF ,


)()( refPrefdXrefd ppKxxKxx

Control law for stabilization:

Stability condition [*]: 0 PX KK

Stability condition [*]: [Choi, Kim, Oh, and You, Posture/Walking Control for Humanoid Robot Based on Kinematic Resolution of CoM Jacobian With Embedded Motion, TRO, 2007].

Effective Stiffness:

zMg

KKK

P

X

1

p

x

z

Mg

xM

F

MgFxM

zxp ext

extF

ZMP Based balancing

Balancing + Vertical Motion

Balancing + Vertical Motion+ Compliant Orientation

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Overview

Part I: Modeling

Part II: Balancing1. Basics2. ZMP based balancing (concentrated mass model)3. Torque based balancing (multi-body model)


Folie 62

Balancing & Posture Control

Trunk orientation Control

)()( dDdPCOM ccKccKMgF

Mg

extF

COMF

HIPT

)3(SOR

)(),(dR

RHIP DKRVT

extT

c

IMU measurements

Compliant COM control [Hyon & Cheng, 2006]

Folie 63





Mg

extF

COMF

HIPT

)3(SOR

)(),(dR

RHIP DKRVT

extT

),( HIPCOMd TFW Desired wrench:

IMU measurements

Folie 64





)(),(dR

RHIP DKRVT

),( HIPCOMd TFW Desired wrench:

IMU measurements

dW

extFMg

Folie 65

Grasping and Balancing

Force distribution: Similar problems!

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Force Distribution in Grasping

F

FGGFGFGWO

1

111Net wrench acting on the object:

TPiOi AdG

Grasp Map

if

)3(seFC

Well studied problem in grasping: Find contact wrenches such that a desired net wrench on the object is achieved.

FCFC

)3(se

friction cone

Folie 67

Force distribution

HIPT

COMF

f

fGGWd

1

1

ii

ii Rp

RG

ˆ

3if

Relation between balancing wrench & contact forces

Constraints:• Unilateral contact: (implicit handling of ZMP constraints)• Friction cone constraints

0, zif

Cf

T

F

GG

Folie 68

Force distribution

HIPT

COMF

f

fGGWd

1

1

ii

ii Rp

RG

ˆ

3if

Relation between balancing wrench & contact forces

Constraints:• Unilateral contact: (implicit handling of ZMP constraints)• Friction cone constraints

0, zif

Formulation as a constraint optimization problem

Cf

23

22

21minarg CCTHIPCFCOMC ffGTfGFf

T

F

GG

321

Folie 69

Contact force control via joint torques

ifMgcM

3if

c

lríiT

i

FqJ

Iu

MgqqCq

cqM

M

, )ˆ(00

0)ˆ,ˆ(ˆ0

ˆ)(ˆ00

iT

i fqJ )ˆ(

Multibody robot model:COM as a base coordinate system structure with decoupled COM dynamics.

[Space Robotics], [Wieber 2005, Hyon et al. 2006]

Passivity based compliance control (well suited for balancing)

Folie 70

ForceDistribution

Torque based balancing

Force Mapping

TorqueControl

RobotDynamics

Object ForceGeneration

IMU

cf

q

for orientation control and COM computation

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Uncertain Foot Contact

[Ott, Roa, Humanoids 2011, best paper award]

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Experiments on a Perturbation Platform

Leg perturbation setup

Movable elastic platform

Experimental evaluation of the robustness with respect to disturbances (frequency & amplitude) at the foot

Out of phase disturbance

synchronous disturbance2mm, up to 8 Hz

Comparisons

1) Impact experiments2) Whole body interaction3) Singularities

Comparison 1/31) Impact experiments

Position Based Control

Torque Based Control

Comparison 1/31) Impact experiments



Observations:– Balancing after impact is comparable– Torque based controller does not control relative foot location

Comparison 2/33) Whole body interaction



Comparison 2/33) Whole body interaction



Observations:– Force sensor based controller depends on a reference frame– Torque based controller does not need information about the

point of contact

Comparison 3/34) Singular Configurations



Comparison 3/34) Singular Configurations



Observations:– Position based controller uses Inverse Kinematics, which

requires singularity handling– Torque based controller uses transposed Jacobian mapping, and

thus is not affected by singularities

Balancing: Summary

On flat floor both approaches allow for a compliant behavior

Torque based controller shows independence on precise ground contact (force mapping based on IMU information)

Admittance controller depends on a reference frame

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Overview

Part I: Modeling

Part II: Balancing

Part III: Walking Control1. Walking pattern generation2. Feedback control

Robot control based on conceptual models

Footstep Generation

Pattern Generation

cx

pZMP-COMStabilizer

dxPos. Controlled

Robot

e.g. LQR Preview Control [Kajita, 2003]

Model Predictive Control [Wieber, 2006]

realtime

F


p

c

x

Cart-Table Model [Kajita]Linear Inverted Pendulum Model [Sugihara]

pxzgx

p

c

px

xgzxp

xp


p

c

xgzxp

x

Cart-Table Model [Kajita]

xp

xxx

x

xu

py

uxx

100

000100010

xgzy

01

)(/01)(

)(2/6/

)(100

102/1

)1( 2

32

kxgcky

kuT

TT

kxTTT

kx

z

Continuous time control model

Discrete time model

LQR Preview Control

How to use future information about the reference?

LQR Preview Control

)()()()()1(

kCxkykBukAxkx

)()()()()()(0

kuRkukxQkxkeQkeJ Tx

T

ke

T

)1()()()1()()(

)()()(

kukukukxkxkx

kykyke ref

Time-discrete system:

Cost function:

• Uses differential control input integral action.

• Uses the output error compared to reference signal.

)(kyrefReference output: Assume known for N future time steps

Assume (A,B) is stabilizable & (C,A) is detectable, and [**] [**] Ensures that the system has no transmission zero at z=1.

p

n

kykx

)()(

RQe , positive definite.

npIAB

Crank

0Assumptions:

LQR Preview Control

)()()()()1(

kCxkykBukAxkx


)1(0

)()(

)(0)1(

)1(

kyI

kuB

CBkx

keA

CAIkx

keref

Modified system representation:)()()()(

kukukxkx

[**] Ensures that this system is stabilizable.

LQR Preview Control

)()()()()1(

kCxkykBukAxkx


)1(0

)()(

)(0)1(

)1(

kyI

kuB

CBkx

keA

CAIkx

keref

Modified system representation:)()()()(

kukukxkx

How to handle future reference input?

System augmentation(for next N reference input values)

)(,),1()( Nkykykx refrefd

)(

0000100000010

)1( kxkx d

A

d

d

Dynamics of the new state:

[**] Ensures that this system is stabilizable.

LQR Preview Control

)(

0)()(

)(

0000

0

)1()1(

)1(

)()1(

kuBCB

kxkx

ke

AA

ICAI

kxkx

ke

kz

dd

kz

d

Modified system representation:

)()()()()()(0

kuRkukxQkxkeQkeJ Tx

T

ke

T

Cost function:

)()()(0000000

)(0

kuRkukzQQ

kzJ T

kx

eT

Standard LQR Design for augmented system!

Preview Control

)()(

)()()(

kxkx

keKKKkKzku

d

de

N

irefd

k

irefe ikyiKkxKiyiyKku

10)()()()()()(

Control law:

Example Application: Walking Pattern Generation

Simplified Model: Cart Table Model

xxx

x

xu

p … ZMP (Zero Moment Point)loosely speaking: Point on the sole where the reduced contact force is acting.

x … Position of the CoM

)(/01)(

)(2/6/

)(100

102/1

)1( 2

32

kxgzky

kuT

TT

kxTTT

kx

py

(Kajita 2003)


Footstep planning

Walking Pattern

Generator

CoM IK


dp x dq

Preview Control:T = 5 msN = 400

Q_x = zeros(3,3);Q_e = 1.0;R = 1e-6;

x


Footstep planning

Walking Pattern

Generator

CoM IK


dp x dq

Preview Control:T = 5 msN = 400

Q_x = zeros(3,3);Q_e = 1.0;R = 1e-6;

x

Properties of Preview Control

• Efficient implementation, controller design can be computed offline

• Allows to incorporate predictive information• ZMP contstraints are not considered explicitely• Trajectory based approach

Extensions– Model predictive control (handle zmp constraints explicitely,

optimization over a finite control horizon)– Trajectory generation feedback control– Dynamic filter


Footstep planning

Walking Pattern

Generator



dp dx dqrefx

refpInverse

Kinematics

swing foot trajectory

dqRobot

Dynamics

Forward Kinematics

ZMPComputation

p

x q

LR FF ,


)()( refPrefdXrefd ppKxxKxx

Control law for stabilization:

Stability condition [*]: 0 PX KK

Stability condition [*]: [Choi, Kim, Oh, and You, Posture/Walking Control for Humanoid Robot Based on Kinematic Resolution of CoM Jacobian With Embedded Motion, TRO, 2007].

Dynamic filter

Footstep planning

Preview Control

dpdqrefx Inverse

KinematicsRobot

Dynamics

Correction of the error due to model simplificationRequires computation of the multi-body dynamics

p p

Preview Control

refx Inverse Kinematics

DLR-Biped

ZMP basierte Gangregelung

Präsentiert auf der Industriemesse Automatica, Juni 2010

Overview

Part I: Modeling

Part II: Balancing

Part III: Walking Control1. Walking pattern generation2. Feedback control

Walking Control

State of the art walking control for fully actuated robots

– Pattern Generator for desired CoM and ZMP motion– ZMP based Stabilizer

Footstep Generation

Pattern Generation

cx

pZMP-COMStabilizer

dx InverseKinematics

Position Control

dq

e.g. Preview Control [Kajita, 2003]Model Predictive Control [Wieber]

realtime

Folie 102

Walking Control used at DLR

c

p

[Englsberger, capture point control]

Walking StabilizationCore concept: Capture point controlGeneralization (3D)Stairs, etc …

Predictive Control (MPC)Reactive step adaptation

COMcapturepoint

xp

exp. stable

(Pratt 2006, Hof 2008) ),(

),(

x

xx

xx ,)(2 pxx

xx

pxx

00

open loopunstable

Folie 103

Walking Control used at DLR

c

p

[Englsberger, capture point control]

Walking StabilizationCore concept: Capture point controlGeneralization (3D)Stairs, etc …

Predictive Control (MPC)Reactive step adaptation

COMcapturepoint

xp

exp. stable

(Pratt 2006, Hof 2008) ),(

),(

x

xx

CP control

xx ,)(2 pxx

xx

pxx

00

Folie 104

COM

ZMP

Capture Point • COM velocity always points towards CP

• ZMP „pushes away“ the CP on a line

• COM follows CP

Capture Point Dynamik

Folie 105

COM

ZMP

Capture Point

Capture Point Dynamik

Folie 106

COM kinematics

Capture Point Control

xx ,

pCP control

[Englsberger, Ott, et. al., IROS 2011]

ZMPControl

RobotDynamics

CP

qTrajectoryGenerator d

ZMP projection

Folie 107

COM kinematics


xx ,

pCP control


ZMPControl

RobotDynamics

CP


ZMP projection

MPC [SYROCO 2012]

Folie 108

Position based ZMP Control

COM kinematics

xx ,

pCP control

ZMPControl

RobotDynamics

CP


ZMP projection

MPC

)(2 pxx dp

Desired ZMP implies a desired force acting on the COM:

)(2dd pxMF

Position based force control [Roy&Whitcomb,2002]:

)( FFkx dfd )(2dfd ppMkx

Position based ZMP Control

Folie 109

COM kinematics


xx ,

pCP control


ZMPControl

RobotDynamics

CP


ZMP projection

MPC [SYROCO 2012]

Applications1) Vision based walking

– stereo vision (Hirschmüller)– visual SLAM (Chilian, Steidel)– online footstep planning, collaboration with N. Perrin (IIT)

Applications2) Optimized swingfoot trajectories: collaboration with H.

Kaminaga (Univ. Tokyo)

• stride length: 70 cm• speed: 0.5 m/s• kinematically optimized swingfoot trajectory

• stride length: 70 cm• speed: 0.5 m/s• kinematically optimized torso motion

(no angular momentum conversation! slippery)

Folie 113

Summary

Modeling

Full Body ModelsSimplified Models

Balancing

Torque based BalancingZMP based balancing

Walking

Pattern generationFeedback control

Part I: Part II: Part III:

Folie 114

Thank you very much for your attention!

[email protected]

Dr. MaximoRoa

JohannesEnglsberger

AlexanderWerner

GianlucaGarofalo

Dr. Christian Ott

BerndHenze

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