Robust nonlinear feedback control forrobotic manipulators

V.D. Tourassis, DipI.Eng., M.S.E.E., and C.P. Neuman, B.S.E.E., S.M., Ph.D.

Indexing terms: Robotics, Feedback, Control systems

Abstract: Nonlinear feedback control algorithms for manipulators utilise the complete (coupled and nonlinear)dynamic model to decouple the robot joints. In this framework, we present a comparative evaluation of thecomputed-torque and direct-design control algorithms, and outline the practical problems introduced by mod-elling inaccuracies, unmodelled dynamics and parameter errors. We then formulate the a-computed-torquenonlinear feedback control algorithm which is robust in the presence of the aforementioned error sources.Numerical experiments with cylindrical robots confirm the robustness and applicability of our a-computed-torque algorithm.

1 Introduction

Robotic manipulators are highly coupled and nonlinearmultivariable mechanical systems which are designed toperform specific tasks [1, 2]. The robot control problemcentres around the computation of the control forces andtorques which are required to execute these tasks. Thecomplexity of robot dynamics poses challenging problemsfor the control engineer and creates the need for sophisti-cated controllers. The robot control problem is compli-cated by the practical fact that the desired motion of theend effector and feedback information (from such intelli-gent sensors as vision systems) are frequently transmittedin a Cartesian co-ordinate system, rather than in thenatural joint co-ordinates of the robot. The real-time co-ordinate transformations (from the Cartesian co-ordinatesto the joint co-ordinates) impose an additional computa-tional burden on the control computer [1, 3].

The continuously increasing demands for enhanced pro-ductivity and improved precision have imposed specialrequirements on the control of industrial robots andcaused a shift of emphasis towards the dynamic behaviourof robot manipulators. This shift has led to the develop-ment of nonlinear feedback controllers for robots [4]. Theunderlying principle is to:

(i) design a nonlinear feedback algorithm that trans-forms the highly coupled and nonlinear robot dynamicsinto equivalent, decoupled linear systems (one for eachdegree of freedom);

(ii) synthesise linear controllers in the framework of clas-sical control engineering [5]. The applied control signal isthe sum of a nominal feedforward control signal to cancelthe nonlinear dynamics and an augmented feedbackcontrol signal to specify the closed-loop response.

The first nonlinear feedback approach to robot control,which utilised the complete dynamic robot model [6], wasinvestigated by Paul [1] and Raibert and Horn [7] andwas called the computed-torque controller by Markiewicz[8] and Bejczy [9]. In this approach, the input torques arecomputed as functions of the desired trajectory (and itsvelocity and acceleration) in joint co-ordinates. Theapproach requires on-line evaluation of the robotdynamics which led to implementation problems. Therecent development of customised algorithms and dedi-cated hardware [10, 11, 12], which compute (in real-time)the robot dynamics, has revitalised interest in computed-

Paper 3864D (C8, C9), first received 21st November 1984 and in revised form 26thFebruary 1985The authors are with the Department of Electrical and Computer Engineering,Robotics Institute, Carnegie-Mellon University, Pittsburgh, PA 15213, USA

torque control. The nonlinear torque controller of Sahbaand Mayne [13] involves essentially the same structure asthe one proposed by Raibert and Horn [7], although, inthe former case, on-line calculations rather than tablelookup are used to obtain the relevant terms. Resolved-acceleration control [6] extends the computed-torqueconcept to robot control in end-effector co-ordinates.

A related approach is the nonlinear direct-designmethod [14, 15] which is based upon nonlinear feedbackdecoupling [16] and arbitrary pole placement. Time-optimal torque control [17] is a variation of the direct-design principle. The direct-design method has also beenextended to robot control in end-effector co-ordinates[18].

The computed-torque and direct-design methods arenonlinear feedback control strategies, and, as such, theyhave a number of similarities. For instance, both achievedynamic decoupling by nonlinear feedback and theirimplementation requires exact knowledge of the robotdynamics. Nevertheless, they differ dramatically in theirphilosophy (with regard to closed-loop pole/zeroplacement) and performance.

The objectives of this paper are to:(i) Present a comparative evaluation of the computed-

torque and direct design methods in the coherent frame-work of nonlinear feedback control.

(ii) Outline the practical problems associated with non-linear feedback in the presence of modelling inaccuracies,unmodelled dynamics and parameter errors.

(iii) Apply the computed-torque concept to formulaterobust feedback control algorithms (that are insensitive tothe aforementioned error sources) for manipulator control.

The paper is organised as follows: in Section 2, we reviewthe robot control problem and, in Section 3, we introduce(as a case study) the prototype cylindrical robot. We evalu-ate (in Section 4) the computed-torque and direct-designapproaches under the assumption of perfect modelling. InSection 5, we identify the practical problems associatedwith the implementation of nonlinear feedback controllers.We develop (in Section 6) robust control algorithms forrobotic manipulators and, in Section 7, we evaluate theperformance of these algorithms. Finally (in Section 8), wesummarise our results and identify areas for futureresearch.

2 The robot control problem

2.1 IntroductionIn this Section, we review robot dynamics and outline ourassumptions about commercially available robotic actu-

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ators and sensors. We then formulate the robot controlproblem and establish the framework for our comparativeevaluation of nonlinear feedback controllers.

2.2 Robot dynamicsIn the robotics literature, the physical laws of classicalmechanics are applied to formulate the dynamic equationsof motion of robotic manipulators. Upon neglectingmechanical dissipation, the standard matrix-vector formu-lation of the closed-form dynamic robot model is [1]

D(q)q + H(q, q) = F(t) (1)

Standard robotic terminology is utilised throughout thispaper [1, 2]. In eqn. 1

q(q, q) is the iV-vector of co-ordinates (velocities,accelerations);

D(q) is the symmetric, positive-definite inertial(N x N) matrix;

H(Q> O) is the iV-vector of coupling forces/torques whichis the sum of the centrifugal, Coriolis and gravita-tional force/torque iV-vectors; and

F(t) is the JV-vector of applied joint forces/torques.

Robot dynamics are characterised by the system of Nsecond-order nonlinear differential equations in eqn. 1whose parameters are explicit functions of the instanta-neous configuration (co-ordinates and velocities) of themanipulator.

2.3 Actuators and sensorsAdvances in actuator and sensor technology motivatepractical assumptions which simplify manipulator model-ling for controller design. In this Section, we outline ourassumptions. Actuator dynamics relate the applied jointforces/torques F(t) to the actuator inputs. For control-engineering applications, each actuator is modelled by alow-order system [19, 20] and the actuator dynamics areincorporated directly into eqn. 1. The constant actuatorinertias (scaled by the squares of the gear ratios) are super-imposed on the diagonal elements of the inertial matrixD(q), and the dissipation forces/torques are added toH{q, q).

Although the dynamic robot model in eqn. 1 can readilyaccommodate actuator dynamics, recent advances in robotactuator technology have shown that the actuatordynamics are negligible in comparison with the manipula-tor dynamics [21]. In particular, there are commercially-available disc motors and harmonic drive precision gearsfor robotic applications. Disc motors [22] employ a flat-disc ironless armature. The lack of rotating iron in thearmature results in small mechanical time constants andthe ability to handle high pulse torques for fast acceler-ations. Furthermore, disc motors have extremely lowinductance (typically less than 85 fiH) and low losses forhigh efficiency. Disc motors can be coupled with harmonicdrive gears [23] for precise motion control. Natural gearpreload and almost pure radial tooth engagement allowstandard harmonic drive gears to operate with negligiblebacklash*.

The availability of such components leads to our actu-ator assumptions:

We henceforth neglect gear backlash and friction. (A.I)

We neglect actuator dynamics for control engineering

* Private discussion with Professor H. Van Brussel, Katholieke Universiteit Leuven,Heverlee, Belgium, August 17, 1984.

applications and focus on the dynamic robot model ineqn. 1. (A.2)

We refer to (A.I) as the quasidirect-drive [24] assumption.Robot actuators have force/torque limits which deter-

mine the realisability of task trajectories. For a set of actu-ator limits and a proposed trajectory, there is only a rangeof joint speeds that is achievable. The time-scaling pro-perty of robot dynamics [25] can indicate whether aplanned trajectory is dynamically realisable and, if not,how to modify the trajectory to satisfy the actuator con-straints. Throughout the paper, we assume that these actu-ator limits have been incorporated in the trajectoryplanning process.

Optical position encoders (which have achievedresolutions as fine as 0.5 /mi at 300 kHz), tachometers andaccelerometers for robotic applications provide superiornoise immunity [26, 27]. Through this demonstrated per-formance, sensor errors are negligible in comparison withmodelling and parameter errors. The availability of suchcomponents leads to our sensor assumption:

We assume that the controller processes the actual co-ordinates, velocities and accelerations q, q and q of themanipulator. (A.3)

The proliferation of electronic and mechanical components(such as Burr-Brown's 16-bit DAC 710/711 digital-to-analog converter [28]) designed specifically for roboticapplications reinforces our practical modelling assump-tions A.1-A.3 for manipulator controller design.

2.4 Control issuesIndustrial robots are positioning mechanisms. The robotcontrol problem is to design a stable and robust algorithmto control the robot to follow a specified trajectory. Thedesired trajectory is typically a sequence of points, inworld (Cartesian) co-ordinates, which specify the locationand orientation of the end effector. Recently developedalgorithms (which can be incorporated in the planningprocess) compute the profile of joint positions, velocitiesand accelerations along the desired trajectory [29]. Thecontrol problem then is to design a control algorithm, injoint co-ordinates, to generate the specified profiles.

Robotic manipulators consist of two subsystems: thepositioning arm and the end-effector wrist. In an N degree-of-freedom (DOF) robot, the first M (M < N) DOF consti-tute the arm and the remaining (N-M) DOF constitute thewrist. In a typical N — 6 DOF industrial manipulator,M = 3. Although the dynamic robot model in eqn. 1 doesnot distinguish symbolically between the arm and thewrist, the two subsystems differ substantially in their taskassignments, dynamic characteristics, controller require-ments and supporting devices. The arm is responsible foroositioning the wrist at a specified point in the workplace.The requirements for such a task are rapid execution andprecise final position. Deviations from the prescribed tra-jectory, which do not affect the precision of the final posi-tion, are tolerable in an obstacle-free environment. Incontrast to the arm, the wrist is responsible for fine anddelicate motion in the workplace. To execute a task prop-erly, the wrist is equipped with intelligent sensors (such asvision, compliance and force feedback) which can achievethe precision requirements of the control problem. Therobot control problem deals primarily with the positioningof the arm, while artificial intelligence methodology can beapplied to control the end effector [30]. This interpretationof the robot control problem leads to our decompositionassumption

IEE PROCEEDINGS, Vol. 132, Pt. D, No. 4, JULY 1985 135

In this paper, we focus on the control of the positioningarm and, in particular, on the design of controllers whichactuate the arm to follow the prescribed trajectory. (A.4)

Since the mass of the end-effector wrist is negligible incomparison with the mass of the positioning system, thewrist can be viewed as an additional mass which isattached to the end of the last link of the arm. This physi-cal insight motivates neglecting the dynamic coupling fromthe end effector to the positioning system. Although thedynamics of the end effector have a negligible effect on thepositioning system, the mass of the wrist is included in theformulation of the dynamic robot model of the arm [30].

3 The cylindrical robot

3.1 IntroductionWithin this control engineering framework, emphasis isplaced upon designing a robot which reduces the inertial,centrifugal, Coriolis and gravitational coupling and non-linearities, thus yielding a model that is accurate andsimple [30, 31, 32].

In a 1984 survey of 149 industrial robots [33], the fourgeometrical configurations of robots are distributed asfollows:

(i) Cartesian prototype robots —31%(ii) Cylindrical prototype robots—19%(iii) Spherical prototype robots — 9%(iv) Articulated robots —41%

Industrial robots which are particularly suitable for robotcontrol design are the prototype robots [31]. A prototyperobot is a robot whose physical structure results in adiagonal (co-ordinate-dependent) inertial matrix D(q),which thereby decouples the accelerations in eqn. 1. Proto-type robots operate in world co-ordinates and eliminatethe need for transformations between the world and jointco-ordinate systems. Prototype robots thus become thefocal point for the development of nonlinear feedback con-trollers. In particular, the cylindrical robot, which has beenused to develop and evaluate robot controllers [2, 14, 17],is the focus of our simulation experiments outlined in thisSection. Cylindrical robots, which have an abundance ofindustrial applications (in parts handling, machineloading/unloading, and palletising), move objects along ahorizontal path which is parallel to the ground. Thisprototype is ideally suited when the tasks to be performedor machines to be serviced are located radially from therobot. Even though we focus on the cylindrical robot, our

approach is directly applicable to the four geometricalconfigurations of robots.

3.2 Dynamic robot modelThe cylindrical robot, depicted schematically in the liter-ature [2, 14, 15, 17, 31] consists of 3 DOF: a rotation 6, avertical translation z, and a radial translation r. These jointco-ordinates, which are the cylindrical world co-ordinates,are measurable with commercially available instrumenta-tion. The dynamic model of the cylindrical robot [2, 31],and the symbols and typical values [2] of the parametersare compiled in Table 1.

The inertial matrix D(q) of the cylindrical robot isdiagonal and depends explicitly upon the radial displace-ment. The cylindrical robot is thus a prototype robot [31].The radial dependence of the inertial matrix leads to aCoriolis torque in the angular 6 equation of motion and acentrifugal force in the radial r equation of motion. Thefirst element of the coupling H(q, q) vector is the Coriolistorque acting on the angular link, and the third element isthe centrifugal force acting on the radial link. The rota-tional (0-co-ordinate) and radial (r-co-ordinate) motionsare thus coupled and nonlinear. The second (constant)element Mg of H(q, q) is the gravitational force acting onthe vertical link and indicates that the vertical motion (z-co-ordinate) is characterised by an uncoupled doubleintegrator. Even though the cylindrical robot exhibits arelatively simple dynamic model, it preserves all of theinherent coupling and nonlinear characteristics of robotdynamics. While the vertical motion can potentially becontrolled by a conventional servomechanism, the coupledrotational and radial DOF introduce challenging robotcontrol engineering problems. In our simulation experi-ments, we thus maintain the vertical z-co-ordinate con-stant (by constraining the vertical motion).

3.3 Simulation experimentsTo compare the performances of the control algorithmsevaluated in this paper, we have designed an industrially-oriented continuous-path trajectory. The robot is requiredto follow the desired Cartesian path shown in Fig. 1. Thedesired joint co-ordinates are computed from therectangular-to-polar transformation (x,}/)—> (r, 6). Only theangular and radial joints are driven in the simulation.

The robot is initially at rest at the origin, with the radiallink fully retracted. The robot must reach the final position(x = — 0.406 m, y = 0.909 m) in 2 s, with zero velocity andthe radial link fully extended (r = 1 m). For this trajectory

Table 1: Dynamic model of the cylindrical robot

Dynamic robot modelCo-ordinate vectorInertial matrix

Coupling vector

Exogenous vectorJ = 10 kgm2

Rr) = {mR + mL)r2 - Rr

dj(r)

drm* = 2 kgmL = 5kg

M = 10 kg

H(q,q)=F(t)q = [8zr]T

D(q) = diagonal [J +j(r) M

(1)

+ mL)]

= 2(mR + mL)r-mR R =

H(q, q) = —— rd Mg - - -^p- d2

]_ dr 2 dr J

= constant inertia of the vertical column= co-ordinate-dependent inertia of the radial link which

is a quadratic function of the radial displacement

co-ordinate-dependent centrifugal and Coriolis

coupling coefficient between the 0 and r DOF= mass of the radial link= mass of the payload which is concentrated

at the tip of the radial link; i.e. at r = R= vertically translated mass (i.e. the sum of the masses

of the vertical column, radial link mR and payload mL).= length of the radial link= external joint forces/torques that actuate the

6, z and r DOF, respectively

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design, the desired joint accelerations exhibit a bang-bangprofile, which is typical in industrial applications. We

1-Or

0.2 0.3-0.5 -0.4

Fig. 1 Desired trajectory in Cartesian co-ordinatesInitial position (f = 0 s): r = 0 m, 0 = 0 radFinal position (t = 2 s): r = 1 m, 6 — 2 radInitial and final velocities are zero

integrate the acceleration profiles to generate the desiredjoint velocities and co-ordinates. The desired radial andangular co-ordinates evolve smoothly (as parabolic curves)over time [1, 2].

In our simulation algorithm, we select controller gains(in Sections 4 and 5) to specify a critically-dampedresponse for the cylindrical robot. We evaluate the per-formance of all of the control algorithms discussed in thispaper by their deviation from the desired trajectory shownin Fig. 1. At each time instant t, we compute the trackingerror | s(t) | according to the law of cosines

{r2 + rj-2rrd cos (6> - 0d)}1/2

4 Nonlinear feedback control

4.1 IntroductionIn this Section, we compare the computed-torque anddirect-design methods in the framework of nonlinear feed-back control for robots. In control engineering, design of anonlinear feedback algorithm is formidable (and some-times impossible), and is the subject of extensive research[16]. Direct-design methods [14, 15] are based upon anonlinear concept which has evolved from this research.Fortunately, a decoupling feedback control algorithm forrobots can be obtained by inspection of eqn. 1. We outlinethis control algorithm, which is the basis of the computed-torque and direct-design methods, under the assumption ofperfect knowledge of the robot dynamics. In Section 5 westudy the practical problems that are introduced by imper-fect dynamic robot models.

4.2 The decoupling principleTo decouple the dynamic robot model in eqn. 1, we applyassumptions A.1-A.3 to generate the vector of externaljoint forces/torques

F(t) = D(q)u(t) • , q) (2)

In eqn. 2, the caret" signifies the estimated inertial matrixD(q) and coupling vector H(q, q), and u(t) is the com-manded acceleration. We incorporate these estimates (ofthe dynamic robot model) in the controller because ofinevitable modelling inaccuracies and/or parameter errors.Modelling inaccuracies are introduced by unmodelleddynamics or simplified models that are designed to reduce

the computational requirements of the controller [19].Parameter errors arise from practical limitations in thespecification of numerical values for the kinematic anddynamic robot parameters, or from payload variations.

The nonlinear feedback control algorithm in eqn. 2leads to the closed-loop system

q = u(t) - [£(?)]"'{lD(q) - D(q)Jq

+ [H(q, q) - H(q, *)]} (3)

If the robot dynamics are known exactly; i.e. if D(q) <- D(q)and H(q, q) *- H(q, q), eqn. 3 becomes

q = u(t) (4)

The commanded acceleration u(t) is designed to be thesuperposition of a nominal feedforward control signal anda linear state-variable feedback control signal

u(t) = r(t) - Kpq - Kvq (5)

where r(t) = [r,(t)] is the reference signal vector, and Kp =[fcp/] and Kv = [fcui] are diagonal feedback matrices of con-stant position and velocity gains. The commanded acceler-ation in eqn. 5 thus decouples the axes of the closed-loopsystem in eqn. 4:

q + Kvq + K.q = r(t) (6)

In summary, the nonlinear feedback control algorithm ineqns. 2 and 5 results in the system of uncoupled, linear,second-order differential equations in eqn. 6. The controlengineer selects the feedback gains kpi and kvi to guaranteethe stability of eqn. 6 and actuate the joint co-ordinatesq(t) to track the reference signal r(t).

4.3 Direct design against computed torqueThe direct-design and computed-torque algorithms differin their specification of the reference signal r{t) in eqn. 6. Indirect design, the reference signal is proportional to thedesired joint co-ordinate vector [14, 15]:

•it) = (7)

where A = [A,] is a diagonal feedforward gain matrix. Thischoice of the reference signal results in the system ofuncoupled transfer functions

Qi(s)

kvisfor i = 1, 2, . . . , N (8)

The gains ?H, kvi and kpi are chosen to insure that the char-acteristic polynomials in eqn. 8 are stable and that thespecified performance requirements are met.

In contrast to direct design, computer torque defines thereference signal as a linear combination of the desired jointposition, velocity and acceleration vectors [2, 8, 6]:

r(t) = qd + Kvqd + Kpqd ' (9)

The commanded acceleration in eqn. 5 becomes

u(t) = qd + Kv[_qd -q] + Kp[qd - q~] (10)

The reference signal in eqn. 9 introduces zeros to cancelthe poles of the closed-loop system in eqn. 6 and leads tothe unity transfer functions

" - ! tori-1,2 N (11)

The closed-loop transfer functions eqn. 11 indicate that thecomputed-torque algorithm is ideally-suited for trajectorytracking applications.

Upon substituting eqn. 9 into eqn. 6, the computed-

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torque design results in the system of uncoupled, homoge-neous second-order error equations

e + Kve + Kpe = 0 (12)

where e(t) = qd(t) — q(t) is the closed-loop tracking errorbetween the desired trajectory qd{t) and actual trajectoryq(t). In contrast to e(t), the scalar tracking error | e(t) \, ourperformance measure, is the Cartesian distance betweenthe desired and actual end-effector positions. The errorequations in eqn. 12 characterise the temporal evolution ofthe closed-loop tracking error e{t). The engineer specifiesthe diagonal feedback gain matrices Kp and Kv to controlthe transient response of e(t).

We simulated both the direct-design and computed-torque algorithms (for the cylindrical robot in Table 1 anddesired trajectory in Fig. 1) with the set of feedback gainsKv = kvl and Kp = kp I. We implemented a critically-damped* design (kl = 4kp) for the linear systems in eqns. 8and 11. To place the closed-loop poles at s = — 20 s"1, forexample, we specified the position gain kp = 400 s~2 andthe velocity gain kv = 40 s"1. The decoupled axes of theclosed-loop system are thus designed to respond with theequivalent time constant [34]:

= 0.107s ({ = !)

the robot control problem. The coupled error equationsbecome

KDe=WCT(t) (14)

Our simulation results indicate, that the direct-design algo-rithm leads to a maximum tracking error | s(t) | of 14% ofthe length of the radial link, even under the assumption ofperfect knowledge of the robot dynamics. Computedtorque, with the same feedback gains, results in perfect tra-jectory tracking [19]. On the basis of these initial simula-tion experiments, we evaluate (in Section 5) theperformance of only the computed-torque algorithm in thepresence of modelling inaccuracies, unmodelled dynamicsand parameter errors.

5 Practical problems

5.1 IntroductionIn an ideal situation, in which the manipulator modelimplemented in the controller is exact, computed torquewould suffice and the robot would track the desired trajec-tory. In industrial applications, however, the dynamicrobot model is only relatively accurate. Specification andcomputation of the parameters in the dynamic robotmodel in eqn. 1 is a formidable task, and estimatednumerical values are sometimes far from the actual ones.We investigate, in this Section, the impact of modellinginaccuracies, unmodelled dynamics and parameter errorson the performance of the computed-torque nonlinearfeedback control algorithm.

5.2 Closed-loop system dynamicsIf the robot dynamics in the controller are inexact, theclosed-loop robot system is described by eqn. 3 with thecommanded acceleration u(i) defined in eqn. 5 and the ref-erence signal r(t) defined in eqn. 9. The nonlinear vector ineqn. 3

WCT(t)=lD(q)yl{lD(q)-D(q)Jq

+ [H(q, q) - H(q, q)-]} (13)

couples the robot joints and retains the nonlinear nature of

* Under the assumption of perfect modelling of the robot dynamics, the closed-looprobot response in eqn. 6 is critically damped. When the dynamic robot model isimperfect, the closed-loop robot response will not necessarily be critically-damped.

Modelling inaccuracies, unmodelled dynamics and par-ameter errors in the nonlinear feedback control algorithmare reflected in the driving vector WCT(t) in eqn. 14. Thisnonlinear vector drives the error equation and the per-formance of the computed-torque algorithm degrades asWCT(t) increases. We highlight our simulation results in thefollowing Section.

5.3 Numerical expert men tsIn this Section, we evaluate the performance of thecomputed-torque algorithm in the presence of payloadvariations (to exemplify parameter errors). We implement-ed the computed-torque algorithm for the cylindrical robot(as described in Section 3.3) with three different estimates(mL = 0, 10 and 20 kg) of the mL = 5 kg payload. Thesimulation results, which are displayed in Fig. 2, revealthat significant tracking errors (of up to 20 mm) can arise,even if the only error is the incorrect payload estimate.

In Fig. 3 we display the temporal evolution of the com-ponents WCT e and WCT r of the driving vector WCT(t) for

25r

Fig. 2 Computed-torque algorithm (payload error)

kv = 40 s"1, kp = 400 s"2 ; actual payload 5 kgassumed payload 0 kg, assumed payload 10 kg,

assumed payload 20 kg

-10L

Fig. 3 Time profile of the driving vector

kv = 40s" 1 , kp = 400s" 2 ; actual payload 5 kg, assumed payload 0 kg

the worst-performance case (mL = 0 kg) in Fig. 2. Thedriving vector (in Fig. 3) produces the significant trackingerror (in Fig. 2), as the error eqn. 14 suggests.

Interestingly, the tracking error is smaller when weoverestimate the payload (mL = 10 and 20 kg) than whenwe underestimate the payload (mL = 0 kg). Hewit andBurdess [35] made a similar observation by modelling the

138 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 4, JULY 1985

robot as a single lumped mass. We attribute this phenome-non to the presence of the inverse of the modelled inertialmatrix in eqn. 13. For the cylindrical robot, the inertialmatrix is diagonal, and WCT e and WCT r can be writtenexplicitly as the product of a dynamic factor and a para-metric (inertia-ratio) factor

and

L - mL)/{(mR + mL)r2 -mRRr + J}] (15)

- mL)/(mR + mj] (16)

When we underestimate the payload, the denominators ofthe parametric factors in eqns. 15 and 16 decrease andthereby increase the driving vector. For moderate par-ameter errors (mL — mL), overestimating the payloadincreases the denominators of the parametric factors ineqns. 15 and 16, thereby reducing the driving vector. Whenwe overestimate the payload, and the parameter errorbecomes significant, the numerators of the parametricfactors in eqns. 15 and 16 dominate. Accordingly, the mod-erate case of mL = 10 kg (in Fig. 2) performs better thanthe extreme case of mL = 20 kg.

5.4 Nonlinear feedback is not robustThe driving vector WCT{t) in the computed-torque erroreqn. 14 is proportional to the difference between the actualand the assumed robot dynamics. Errors in modelling D(q)and H(q, q) degrade performance and can lead to instabil-ity [13, 36]. Nonlinear feedback control is thus not robustin the presence of modelling inaccuracies, unmodelleddynamics and parameter errors.

Nevertheless, by selecting high feedback gains kp and kv,we can force the error of eqn. 14 to be relatively insensitiveto modelling inaccuracies, unmodelled dynamics and par-ameter errors, and thereby approach the homogeneouserror equation in eqn. 12. The choice of high gains [37]depends explicitly upon the (unknown) nonlinear(configuration-dependent) driving vector in eqn. 13. Thefeedback gains must be selected for a worst-case configu-ration and the resulting high-gain computed-torque algo-rithm is a conservative design.

High-gain designs lead to singularly perturbed systems[38]. High feedback gains increase the bandwidth of theclosed-loop system and make the design more sensitive tonoise [37]. High gains can also cause instability incomputer-control systems, and this problem can beameliorated only by decreasing the sampling period [37].In the following Section, we introduce a robust nonlinearfeedback control algorithm that is insensitive to modellinginaccuracies, unmodelled dynamics and parameter errors,and preserves the bandwidth of the closed-loop system.

6 a-computed torque

6.1 IntroductionIn this Section we apply the nonlinear feedback controlconcept to formulate our a-computed-torque algorithmwhich is insensitive to modelling inaccuracies, unmodelleddynamics and parameter errors. We then highlight oursimulation experiments for the cylindrical robot to illus-trate the performance of our design.

6.2 MotivationIn the presence of modelling inaccuracies, unmodelleddynamics and parameter errors, we seek to add a compen-sating control signal Au(t) to the commanded acceleration

in eqn. 5 according to

uJLt) = u{t) + Au{t) (17)

so that the closed-loop joint dynamics and error equationwill be specified by eqns. 6 and 12, respectively. Ideally, wecan design A*i(t) as

Au(t) = uJLt) - q(t) (18)

Substitution of Au(t) from eqn. 18 into eqn. 17 leads to eqn.6 and consequently eqn. 11. Unfortunately, we cannotimplement eqn. 18 because ua(t) is unknown. Instead, weapproach controller design from a classical control engin-eering point of view.

The fundamental principle of the a-computed-torquealgorithm is to reduce the driving vector WCT(t) of thelinear computed-torque error eqn. 14. Forcing eqn. 14 toapproach the homogeneous linear error eqn. 12 diminishesthe tracking error e(t) and thereby enhances the per-formance of the closed-loop system. Our simulationexperiments (highlighted in Section 6.4) demonstrate thatthe a-computed-torque algorithm accomplishes this task.

A block diagram implementation of the a-computed-torque algorithm is depicted in Fig. 4. (When a = 1, the

Fig. 4 a-computed-torque block diagram

block diagram realises the conventional computed-torquealgorithm [8]). The desired trajectory {qd, qd, qd} is gener-ated by a trajectory planner, and we assume that position,velocity and acceleration sensors are available.

The compensating action of the a-computed-torquealgorithm is achieved by incorporating a negative feedbackloop around the conventional computed-torque controller.This third feedback loop processes the commanded accel-eration error \_u{t) — q] with a feedforward gain of (a — 1),for large values of a. The commanded acceleration signalua(t) of the a-computed-torque controller is

uJLt) = u(t) + (a - l)[ii(t) ~ *] = «[«W - *] + 1 (19)

From eqns. 17 and 19, the compensating control signal ofthe a-computed-torque algorithm is

Au(t) = (a - l)[«(t) - q\ =a - 1

Mt) - q] (20)

For large values of the feedforward gain (a — 1), the com-pensating signal Au(t) in eqn. 20 approaches the ideal

IEE PROCEEDINGS, Vol. 132, Pt. D, No. 4, JULY 1985 139

signal in eqn. 18, and the commanded acceleration signalua(t) in eqn. 19 approaches its conventional computed-torque counterpart u(t) in eqn. 17. When a = 1, the com-pensating signal Au(t) is zero and ua(t) = u(t). Henceforth,for expository convenience, we refer to the feedforwardgain (a — 1) as a.

6.3 Control algorithmFor the commanded acceleration signal in eqn. 19, theclosed-loop linear joint dynamics in eqn. 4 become

q = u(t) - i Wcl{t)

and the error eqn. 14 becomes

where

= ^ Wcr{t) = WaC7{t)

= \ [D(q)Yl{[D(q) - D(q)Jq

(21)

(22)

+ [H(q, q) - H(q, q (23)

The driving input WaCT(t) in eqn. 23 is the commandedacceleration error in eqn. 21. In the block diagram of Fig.4 we note the real signal WaC1{t) which can be processed tomonitor the performance of the a-computed-torque con-trolled system.

The driving inputs W'crM a nd WaCT(t) m t n e error eqns.14 and 22, respectively, are related symbolically by thescale factor a. This symbolic relationship holds only underthe assumption that the computed-torque and a-computed-torque algorithms actuate the robot to followthe same trajectory. In the following Section, we relate thedriving inputs WCT(t) and WaCT(t) in the more realistic casewhen the computed-torque and a-computed-torque algo-rithms actuate the robot along different trajectories.

As the feedforward gain a is increased, the drivingvector WCT(t) is reduced by a factor of a, and the linearjoint dynamics in eqn. 21 and the error eqn. 22 approachtheir ideal counterparts in eqns. 4 and 12, respectively. Ifthe robot dynamics are modelled perfectly, i.e. if D(q) <-D(q) and H(q, q)i«- H(q, q) in eqn. 13, both WCT(t) andWaC1{t) are identically zero for all values of a. In this idealcase, a-computed torque and conventional computedtorque are identical nonlinear feedback control algorithms,and the actuating signal is computed according to eqn. 10.

Note: Dwyer et al. [17] identified a cyclic problem,similar to the one in eqn. 18, in the design of optimal con-trollers for robotic arms with modelling errors. To circum-vent the problem, they introduced (in the hardwareimplementation of their time-optimal controller) the com-pensating signal

Au(t) = [1 - s]u(t) ~ 4(t)

where e is a small positive quantity. The commandedacceleration in eqn. 17 becomes

m = - wo - *]£

and the error eqn. 14 takes the form

e + Kve + Kpe = e[WCT(t) + q(t)-]which resembles eqn. 22 if s <- I/a. The drawback of thisapproach is that, when the robot dynamics are modelledexactly, i.e. when fVCT(t) is zero, the error equation isdriven by the nonzero acceleration eq(t). For all nonzero

values of £, the performance of the optimal controller isinferior to that of the computed-torque controller.

6.4 Simulation experimentsTo investigate the response characteristics of our design,we simulated the a-computed-torque algorithm (with dif-ferent values of a) for the parameter error case of mL =0 kg. We display in Fig. 5 the tracking error for a = 10,

2.0r

1.5

.1.0

0.5

0.5 1.0t.s

1.0 2.0

Fig. 5 a-computed-torque (payload error)

Actual payload 5 kg, assumed payload 0 kga-computed torque kv = 40s" 1 , kp = 400s"2

high-gain computed torque kv = 400 s" ' , kp = 40000-s"2

100 and 1000. For comparison, we include the high-gaincomputed-torque design (with closed-loop poles at s =— 200 s"1) whose performance is comparable to that ofour a-computed-torque algorithm with a = 100. In theseexperiments, the a-computed-torque and high-gaincomputed-torque designs produced identical actuatingsignal profiles. We observe (in Fig. 5) that the a-computed-torque algorithm decreases significantly the tracking error,even for relatively low values of a. For a = 10, the trackingerror is less than 1.7 mm; i.e. the maximum tracking errorin the corresponding conventional computed-torqueexperiment in Fig. 2 is reduced by a factor of 13. In oursimulation experiments, the a-computed-torque controllersmooths the control signal of the conventional computed-torque controller.

In Figs. 6 and 7, respectively, we portray the temporal

W,r«CT.9'rad/s2

0.04r

0.02

0

^-0.02

^-0.04

-0.06

-0.08

-0.10

Fig. 6 Time profile of the a-driving vector

a-computed torque, kv = 40 s"1, kp = 400 s"2, a = 100; actual payload 5 kg,assumed payload 0 kg

140 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 4, JULY 1985

evolution of the driving vectors WCT(t) and WaCr(0 ineqns. 13 and 23. Upon comparing these plots with Fig. 3,

-6 L

Fig. 7 Time profile of the high-gain driving vector

High-gain computer torque, kv = 40 s"1, kp = 400 s'2, a = 100; actual payload5 kg, assumed payload 0 kg

we see that the high-gain computed-torque algorithm doesnot alter the driving vector of the conventional computed-torque algorithm, whereas the a-computed-torque algo-rithm (with a = 100) reduces this vector by a factor ofa = 100.

In a companion experiment, we implemented thereduced [19] a-computed-torque algorithm by neglectingthe coupling forces/torques in the controller, i.e. we setH{q, q) — 0 in eqn. 2, in addition to the payload error. Wethereby investigate the practical case of unmodelleddynamics and parameter errors.

Fig. 8 reveals that the tracking error remains small and

0.20r

£ 0.15E

o

0.05

0.5 1.0t ,s

1.5 2.0

Fig. 8 a-computed torque (unmodelled dynamics)kv = 40 s~', kp = 400 s~2, a = 100; actual payload 5 kg, assumed payload 0 kg. Thecoupling centrifugal force and the Coriolis torque are neglected in the controller

bounded (less than 0.17 mm) with the same feedback gains(kp = 400 s"2 and kv = 40 s"1) and the same value of thefeedforward gain (a = 100) used in the parameter errorexperiments (in Fig. 5). The corresponding driving inputs^aCT,eif) an<3 WaCT r(t) in Fig. 9 are scaled replicas (by afactor of a = 100) of their computed-torque counterpartsin Fig. 3. This scaling effect becomes even more pro-nounced when the feedforward gain is increased toa = 1000.

For the simulation experiments highlighted in thisSection, the a-computed-torque algorithm (as a isincreased) reduces the tracking error e(t) and drivingvector WCT(t) by a factor of a and replicates the actuatingsignals of the high-gain computed-torque algorithm ofsimilar performance. We interpret this robust property ofthe a-computed-torque algorithm in the following Section.

0.02

0

J-0.02

-0.04

-0.06

-0.08

W o fCT.6 . r a d / s 2

W,rafCT. r ,m/s2

Fig. 9 Time profile of the oc-driving vector

kv = 40 s" ' , kp = 400 s~2, at = 100; actual payload 5 kg, assumed payload 0 kg. Thecoupling centrifugal force and the Coriolis torque are neglected in the controller

7 Interpretations

7.1 IntroductionIn this Section, we apply our controller design and simula-tion experiments (in Section 6) to interpret (in Section 7.2)the robust performance of our a-computed-torque algo-rithm. We examine (in Section 7.3) the qualitative behav-iour of the error eqns. 14 and 23, and outline (in Section7.4) empirical guidelines for selecting the feedforwardgain a.

7.2 Disturbance rejection characteristicsTo highlight the disturbance rejection characteristics ofour controller, we compare the conventional computed-torque and a-computed torque feedback designs. In Fig. 10we depict the block-diagram implementation of the con-ventional computed-torque and a-computed-torque algo-rithms for each joint. The block diagram illustrates thenonlinear feedback control system in eqn. 3. The referencesignal r(t) = [/*,(£)] is defined in eqn. 9 and the disturbanceWCT(t) = [WiitJ] is defined in eqn. 13. From Fig. 10, wecompile (in Table 2) the input/output transfer functions of

Table 2: Input/output transfer functions of conventionalcomputed-torque (a = 1) and a-computed-torque controlledjoints (a > 1 )

AlgorithmDisturbance

Transfer function rejection ratio

Q,(

* , (

Q,(s)

1Computedtorque (o = 1) s2 + kus + kn

a-computedtorque (a > 1)

1

+ kvs + kp as2 + kvs +

Wj(t)

r,(t) q;(t)

Fig. 10 Block diagram of conventional computer torque (a = /) and ct-computed-torque controlled joints (a > /)

1EE PROCEEDINGS, Vol. 132, Pt. D, No. 4, JULY 1985 141

conventional computed-torque and a-computed-torquecontrolled joints.

We observe (in Table 2) that the a-computed-torquealgorithm reduces (by the feedforward gain a) the dis-turbance rejection ratio of the conventional computed-torque algorithm, while preserving the bandwidth of theclosed-loop system. With this bandwidth, the system isrobust to unmodelled high frequency dynamics. This is thenotable improvement over a high-gain computed-torquedesign that incorporates only position and velocity feed-back. The successful operation of our a-computed-torquealgorithm exemplifies the property that [39]

'. . . the extent of disturbance reduction is given by theamount of gain [a] which precedes the disturbance in theloop . . .'

7.3 Scaling of the driving vectorUpon comparing Figs. 2 and 4, we observe that the track-ing error of the a-computed-torque system (in the presenceof parameter errors) is approximately I/a times the track-ing error of the computed-torque algorithm (with the sameset of feedback gains). We apply the principle of super-position to compare the transient responses of the linearerror equations in eqns. 14 and 22. Scaling the drivinginput in the computed-torque error eqn. 14 by the feed-forward gain a scales the tracking error of the a-computed-torque algorithm by a factor of a.

In the case of modelling errors, a comparison of Figs. 2and 4 reveals that there is a time-varying scaling of thetracking errors. We explain this observation by relating thedriving inputs Wcl(t) and WaCT(t) when the computed-torque and a-computed-torque algorithms actuate therobot along different trajectories.

We assume that (with a proper choice of the feed-forward gain a) the a-computed-torque algorithm leads tothe trajectory {q, q, q) that follows the desired trajectory{tfd> 9d, 4d) with negligible tracking error. Upon replacingthe actual trajectory in eqn. 23 by the desired trajectory,we approximate the driving input WaC1{t) in eqn. 23 as

x {[D(qd) - D(qd)Jqd + \H{qd, qd) - H(qd, q (24)

For the same set of feedback gains kp and kv, thedriving input Wcl{t) in the computed-torque error of eqn.14 is specified by eqn. 13 in which the trajectory is

, <l, 4} = {id, 4d, 4d) (25)

The trajectory {dq, dq, dq] in eqn. 25 is the deviation of therobot motion (actuated by the computed-torquealgorithm) from the desired trajectory. We expand thecomputed-torque driving input WCT(t) in eqn. 13 about thedesired trajectory according to

(26)

Upon substituting eqn. 26 into eqn. 13 and evaluating thea-computed-torque driving input in eqn. 23 along thedesired trajectory, we obtain

= D(qd) + SD(q)

D(q) =

= H(qd,qd) + 3H(q,q)

x {[_D{qd) - D(qd)-\ dq + [5D(q) - SD(q)Jqd

+ [dH(q, q) - 3H(q, *)]} + d[D(q)y'

x [D(qd) - D(qd)Jqd + \_H(qd, qd) - H(qd, q^]}

+ higher-order terms (27)

The error sources (of modelling inaccuracies, unmodelleddynamics and parameter errors) are interwoven in eqn. 27.

When the dynamic robot model implemented in thecontroller approximates the manipulator (i.e. for practicalparameter errors and/or modelling inaccuracies), thesecond and third terms in eqn. 27 become negligible incomparison with the first term and eqn. 27 leads to

(28)

Eqn. 28 illustrates that the driving inputs WCT(t) andfVaCT(t) are scaled by the feedforward gain a. This relation-ship illuminates the robust performance of the a-computed-torque algorithm over that of the conventionalcomputed-torque algorithm (with the same set of feedbackgains).

In the case of unmodelled dynamics or significant mod-elling inaccuracies and parameter errors, eqn. 27approaches eqn. 28 as a is increased. As our simulationexperiments in Section 6 illustrate, increasing a reduces theeffect of the aforementioned error sources on the trackingaccuracy of the closed-loop system. Choice of the feed-forward gain a thus becomes the practical design issuewhen there are significant modelling errors.

7.4 Feedforward gain designImplementation of the a-computed-torque algorithmrequires a judicious selection of the feedback gains kv andkp and the feedforward gain a. The designer specifies thefeedback gains kv and kp to control the transient responseof the tracking error e(t) in eqn. 12.

Choice of the feedforward gain a depends upon theunknown driving vector fVaCT(t) in eqn. 23. This, in turn,depends upon the manipulator configuration and desiredtrajectory, modelling inaccuracies, unmodelled dynamicsand parameter errors. When the robot model is accurateand we implement a simpler model in the controller (as inthe case of reduced computer torque [19]), we cancompute the maximum value of WaCi{t) for the desired tra-jectory and specify a accordingly.

When we lack sufficient knowledge of the manipulatordynamics, application of classical analysis and a conserva-tive norm evaluation of eqn. 23 leads to high-gain feedbackand weak performance bounds [37]. In this case, wespecify a by synthesising hardware experiments and com-puter simulations (for specific manipulators andtrajectories) with our interpretations and design guidelinesfor the a-computed-torque algorithm.

Dynamic robot models are nonlinear in the kinematicparameters [1] (link angles, lengths, distances and twists),and are linear in the dynamic parameters [40] (the linkmasses, and the elements of the inertia tensors and center-of-mass vectors). Recently developed identification algo-rithms [41] can estimate both the kinematic and dynamicparameters of a manipulator. The dynamic parameters ofthe payload thus become the unknown parameters in thedriving vector in eqn. 23. The driving vector WaCT{t) islinear in the dynamic payload parameter errors; i.e. \_D{q)— D(qJ] and [H(q, q) — H(q, qj], as we have illustrated in

eqns. 15 and 16 for the cylindrical robot. Compensation

142 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 4, JULY 1985

for unmodelled dynamics remains an engineering designand implementation issue. In this framework we specify anumerical value of the feedforward gain a for the entirerange of anticipated payloads.

8 ConclusionsIn this paper, we have presented a comparative evaluationof nonlinear feedback control strategies for robotic manip-ulators. These algorithms are based upon methods of clas-sical control engineering and lead to high-gain feedbackdesigns and conservative performance bounds. We appliedthe computed-torque concept to formulate the a-computed-torque algorithm which is insensitive to model-ling inaccuracies, unmodelled dynamics and parametererrors. Our experiments with cylindrical robots illustratethe efficacy and applicability of the a-computed-torquealgorithm. From the practical point of view, we design thea-computed-torque algorithm for the closed-form model ineqns. 2, 5 and 19, and implement the controller with thecomputationally efficient Newton-Euler algorithm [10].

Our research efforts are focusing on applications of thea-computed-torque algorithm to articulated manipulators.Current robot engineering activities include:

(i) Development of systematic approaches for choosingthe feedforward gain a which exploit the physical andstructural characteristics of dynamic robot models [30, 31,32].

(ii) Digital redesign [42] of our a-computed-torquealgorithm, utilising our recently introduced discretedynamic robot model [43]. (This discrete a-computed-torque algorithm eliminates the need for accelerometers inour implementation).

(iii) Comparative performance evaluation of a-computed-torque control algorithms for articulated robots.

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