Adaptive Control of a Robotic Manipulator with Unknown Payload

e

Cop)'riglh IFAC Motion Conlrol for }nlcUigcn. AUloms:J{,:I Perugla. hal),. OClober 27·;:9. 19<;~

ADAPTIVE CO~TROL OF A ROBOTIC ~fA1\"IPCLA TOR "'ITH L~K~O'VN PAYLOAD r. BOLZER'. G. FERRETTI and A. LOCATELLI Polilecnico d.i \1ilano. Dipanimento di Elenronica e Infonnazione Piazzale LeorurdJ da Vinci 3:, I 20133

~lilano.

lLaly

Abstraq The results ef a simulation study are presented in this paper with reference to three different adaptive cO:1:-ol schemes for a three-link rigid manipulator carrying an uncertain payload. It is shown L~:!t the satisfactory perfonnances which Are supplied by a fairly sophisticated algoriL~rn Ilke the Extended Kalman Filter can nearly be atLained through much Simpler techniques. Kt'\'\I.·ords Adaptive cm::!C'l. K:!lman Filter. Computed Torque 1.

ISTRODUCTlO~

It is widely recognized L':3t pl:::r.:iing robotic

manipulators tasks in pre~r..::e of subst2ntial uncertainties in the values Llken on by some of !.he system parameters and/or S:.1te variables may in general result in a ra!.her involved job. This fact has motivated the great interest for adaptive conLIol techniques which have indeed proved to be effective tools in overcoming such difiiCL:!ties (see. e.g .• [1)[5)). These techniques. however. often call for fairly complex algorithms which in turn email a computational effon (mainly in tenns of time) generally in contrast with the perfonnance bounds imposed to the manipulator. This paper presents the results of a simulation study of three different adaptive control schemes with the aim of showing that the satisfactory performances that can be achieved by fairly sophisticated algorithms (e.g. Extended Kalman Fillers) can nearly be attained by resorting to much simpler lechniques which negligibly increase the amount of computational effon required by a nonadaptive strategy. The here adopted framework is as follows. A rigid robot with three revolute joints involved in the transpon of an uncenain payload is studied. The object to be moved is modeled as a concentrated mass connccted to the end effector through a spherical joinL The value of the payload mass is the main source of uncenainty considered, other reasons of uncenainty originating from the knowledge of

only a simplified model of the joint friction and from noisy measurement of positions and velocities. The nonadaptive pan of the controller implements a compuled-torque control law which is modified according to an on-line estimate of the payload masS. The reason for the choice of the computed-torque control strategy is twofold: from one side su~h a control strategy is known to be particularly sensitive to uncertainties and hence suitable to enhance the (potential) benefits entailed by resorting to adaptation, while, from the other side, the simulation results show that good performances can still be recovered through adaptation by only increasing the computational effort of a relatively small amount. In this paper the payload mass is estimated according to three different techniques. The frrst of them (EK) is of the Extended Kalman Filter type. The control variables are thus generated on the basis of the knowledge of the filtered values of the payload mass and of the robot state variables as supplied by the filter. The other two estimation techniques exploit a well known propeny of robot dynamical equations, namely their linearity with respect 10 the (unknown) payload mass. In this way as many equations as the number n of the robot degrees of freedom can be solved with respect ID the payload mass which therefore can be expressed as a known function of the motion of the system in n different ways. Starting from this reformulation two different recursive estimation algorithms have been implemented, the most significant difference between them consisting

D-71

BOLZER.' P.. FERRETI1 G. . LOCAn:u..I A.

in the availability/nonavailability of the angular acceleration measurements. In the sequel they will be referred to as direct methods and labeled as D 1 and D2, respectively. The relative benefits entailed by the different adaptive control strategies are then compared wilh each other and wilh respect to a nonadaptive scheme. The performance criterion is constituted by the squared sum of lhe position deviations of the robot links from their desired values, the latter corresponding to a significant prespecified motion of the manipulator. It will be shown that, as it should be expected, EK turns out to be the best melhod under the above specified criterion even if, rather surprisingly, it does not always supply the best payload estimate. Strategies DI and D2 require a much smaller computational effort while still providing acceptable performances. Between them, D2 (essentially based on the ideas of filtered dynamics introduced in [2]) seems to be more practical as it does not call for the acceleration sensors required by D I. Contrary to most papers dealing wilh robot adaptive control (in particular, sec [1]-[5]) , the present one does not provide any theoretical result on the algorithm convergence, its main purpose consisting in an "experimental" comparison of well established approaches with na'lver ones. The paper is organized as follows . In the next section the robot model and the computed torque control law are briefly illustrated. In Section 3 the three estimation techniques considered in the paper are presented, while the simulation results are reponed in Section 4.

2. MODEL AA'D

CO~TROL

been made explicit: 1 is the vector of driving torques, 9 is the joint angles v.ector, M'(9,m) is the inertia matrix, vector ,,'(9,9,m) accounts for Coriolis, centrifugal and gravitational terms while vector 1[(9) models joint friction. Since the simulation of the highly non linear characteristic, relating joint velocities to friction torques, would imply severe numerical problems (see [6]), the following characteristic is adopted (see Fig 2), which approximates the discontinuity due to the "stiction" phenomenon (see [7]) 1fi(8) I

=M'(9,m)a + V'(9,a,m) + V8)

=atan- l (h8) + be J

J

J

(1) J

being a.J the stiction torque value relative to joint i, bJ the viscous friction coefficient and h a suitable constant (the higher can be chosen the value of h, the more accurate is the modelling of stiction) . ~

e2~1

payload \

•

,.-----+.. "--_ _ _---loP

t/ 3

~

Fig. 1 Reference manipulator

The following model based (Computed Torque) control strategy has been implemented: 't

STRA1EGY

The mechanical srructure of the considered manipulator (depicted in Fig.l) consists of three cylindrical links, connected by three revolute joints: link 1 rotates around a vertical axis with respect to the fixed base, while link 2 and link 3 lie in the same vertical plane. The payload has been assumed as a point mass, connected 10 the third link by means of a spherical joint: no torques are therefore exerted on the manipulator end effector due to payload inertia. The 3 degrees of freedom dynamic model of the manipulator, holding a payload of mass m, can be given in the form: 't

w ~re the dependence on the payload mass m has

= M'(9 ,~)'t' + V'(9 ,a m

,a

m m

A) + re m

l' =ad + K e+ K e v

p

where 9 are the measured (estimated) JOint m m . " . posluon and velocuy veclOrs, m is the estimated payload mass, 9 d is the desired joint position vector, ed - 9m' while a e is the error vector given by e . lmear characteristic has been adopted to compensate is a constant diagonal matrix for friction effects: whose entries 'Yj are chosen as shown in Sec. 4. Finally, K p = kp I and K v = k v I are the position and velocity error gains, respectively. For the sake of identification of the payload mass the dynamic model should be rewritten as follows:

=

r

't

n -72

+ JT(9)F = M(9)9 + V(9,a) + va)

ADAPTIVE CO'.I1<.OL 0" A ROBOTIC MA!\IPL:LATOR \\1TH C;';K.'OW" PA YLOAD

where v(k) accounts for the errors in the available

F = -ma + mg where F is the force exened on the end effector due to payload inenia. a is the Canesian acceleration of the end effe(:tor. g is the gravity acceleration. ~(8) is the manipulator Jacobian. M(8) and V(8 .8) are respectively the inenia matrix and the vector of Coriolis. centrifugal and gravitational terms of the unloaded manipulator. Recalling that the Cartesian acceleration of the payload can be given by:

a

x(k+ I)] [f(X(k).U(k).m)] z(k+l):= [ m(k+l) = m(k) +

.. d[J(8)]· J(8)8 + -d-[-8

=

one obtains:

..

't -

measurements y(k). The estimation of the unknown parameter m can be carried out by applying the Extended Kalman Filter te(:hniques (see, e.g., [8]) to an enlarged state space representation where the dynamics of m is also considered. Since we are interested in the case where m is constant. we will refer to the following system

.

.

M(8)8 - V(8.8 ) - \(8)

=

r .. T,EJ d r J(m l . - g]

T

W(k)] 0

+ [

:= f(z(k).u(k» + w(k)

The output transformation becomes

= m.J (8{ J(8)8 +

y(k)

or, in a compact form.

Z(,t,8,8.8 ) = mW(8,e,8)

(2)

Each one of the three equations (~) can be solved with respect to the payload mas s and used to identify the said mass from measurements and estimations of joint posilions, velocities, and acceleralions. 3. ESTlMATIO\' METHODS Three different techniques have been considered for the estimation of the unknown payload m. The first one is based on the application of the Extended Kalman Filter (EK). while the remaining two (Dl and D2). directly exploit the particular structure of the robot dynamic equations (2). Method EK By introducing the 2n-dimensional sampled state ve(:tor x(k) where

= [el(~)

l!.

el(~) ,.. en(~) en(~)JT

denotes

the

sampling

interval,

the

discretized robot model obtained from (2) by means of a suitable explicit integration te(:hnique can be written as

As customary. the noise terms wO and v(') are modeled as zero-mean white gaussian noises with covariances Q and R. respectively. The entries of R can be given values that are meaningfully linked to the precision of the sensors. while no easy criteria exists to determine the elements of Q. Therefore this matrix is intended as a design parameter to be suitably tuned. At each sampling instant. the algorithm computes the estimates x(k) and ~(k) which are exploited in the control law: ill particular, A . x(k) supplies the values for 8m (U) and m (U). The detailed equations of the Extended Kalman Filter are rather complex and are not reponed here for conciseness. The interested reader is referred to (8) where the involved computational burden can be

e

fully appreciated. Method PI The idea underlying method PI is to solve explicitly the robot dynamic equations (2) with respect to the unknown parameter m, SO yielding at each time step as many payload evaluations as the number of degrees of freedom. Such values ~. (k) are linearly I

x(k+ I) = f(x(k).u(k).m) + w(k)

=

X(k)] 0] [ m(k) + ,,'(k) := Hz(k) + v(k)

= [I

where u(k) 't(~) and the noise term wO.) has been included to take into account both model errors and the approximation intrinsic to the discretization

combined and the result J.1(k) is then filtered by a time update re(:ursion. Pre(:isely, the estimate In(k) at time ~ is given by A

m(k)

process. The measurement equation at time ~ is given by y(k)

=pm(k-I) + (l-p)J.1(k) 1\

,.

J.1(k) =

I

a.m.(k):= I I

i=1

=x(k) + v(k)

Zi('tO:.1),e(~),e(kl!.),eo:.1»

,. :=

I a .- - - - - - I

i=1

II - 73

(3)

BOLZER." P.• FERRETIl G . . LOCA TEUI A.

where the parameters p, a. .• ;= 1,2, ... ,n, are computed I on the basis of the variances of position, velocity and accc!:~ ration measurements so as to achieve an approximate minimum variance estimate. The approximation is due to the linearization of the expression for m(k) with respect to 8,8,e. The I algorithm must be initialized with an a priori estimate ~(O) and its associated variance. The details can be found in [9] . Finally it must be stressed that explicit joint acceleration information is needed either from direct measurements or estimation from velocity data. Method D2 To avoid the need for acceleration information, a variant of method D 1 has been developed by following the ideas of [2]. Precisel y, the robot equation (2) can be substituted by

Z (t ) = rnW(t )

m. /2= 1.0 rn, '3=.9 m. As for the friction terms. they are simulated by eq. (1) of Section 2 with a I =38.7 Nm, b =7 .9 Nms. a =35.1 Nm , b2=11.7 Nms. 2 l a =30.6 Nm, b 3=13.9 Nms. whereas all the 3 experimented controllers make use of simplified linear torque-velocity relationships whose slopes 1'1=17.6 Nms, 1'2=20.5 Nms, 1'3=21.5 Nms, have been chosen so as to minimize the average square error between the nonlinear characteristic of Fig. 2 and the linear one, in the range of interest of the angular velocities. E

200

z

(.J

::J

...

0

.'-'

·c <-

-200 6 0 6 angular velOCIty rao/sec)

where Z (t ) := r{S:b!(Z l"t (t).8(t).9 (t),8(l») ]

Fig . 2 Joinl friction charaaerislics ( - : ··true" I ··· : simplified )

"~'et) := r-{S~b!(\\'(8(t).9(t).8(t»)J b is a positive design parameter and! is the Laplace operator. As shown in (10). the particular structure of the robot equatio_ns make~ it possible to carry out the computation of W et ) and Z (t) using onl y position and velocity measurements. Thus, the algorithm D2 consists of eq. (3) and

~(k) :=

11

I

~.Z(k11)/H'. (k11) I I I

i=1

where the ~i's are suitable nonnegative weights, summing up to 1. In summary. the implementation of D2 calls for an initial guess ~(O) and the appropriate selection of the parameters ~.I , ;= 1.2 •...•n. and p (which detennines the speed of adaptation) on a heuristic basis.

4. SIMULATION RESULTS The simulation study has been performed with reference to a 3-link robot with the configuration presented in Section 2. The numeric values of the physical parameters can be found in [9]; however. just to give an idea of the robot size, the masses of the three links are ml1 =180 kg. m12=80 kg and m =40 kg. respectively and their lengths are 1 =.75 13

1

The position-error gains k of the error-driven pan of p the controllers are taken all equal to 200. while the velocity-error gains k have been set to k =2(k )1f1 so as to obtain critical damping. Such desi~n v.ilues correspond to a bandwith of about 14 rad/s . which appears as a reasonable choice in many applications. Accordingly, the desired motion is specified by a cubic joint reference trajectory with null initial and final velocity and duration of 1.5 s. It is assumed that the robot is equipped with position and velocity sensors in each joint. The measurements are affected b~ white noise with variances O!=8.3X10-6 and oy=8.3xlO"""' for position and velocity. respectively. For a rectangular distribution such values correspond to maximum errors of 5x1o-3 rad and 5xlo-2 rad/s. The system is sampled at 100 Hz. Finally three different situations have been considered with a payload mass of 10 kg. 5 kg, 2 kg. The first set of experiments has been concerned with the tuning of the design parameter Q for the EK method. The matrix has been taken of the fonn Q=diag(O.q,O.q.O.q.O) to emphasize in the discretized equations the role of modeling errors. Corresponding to the three different values of the payload mass (initially guessed 10 be 8 kg. 4 kg and 1.5 kg, respectively) the value of q has been sought which minimizes the perfonnance criterion Jp already

n -74

ADAP . SE CO~TROL OF A ROBOTIC MAt-.1Pl;LATOR WITH L''''K1'OW~ PA YLOAD

mentioned in the Introduction, namely the squared sum of the position deviations of the robot links from their desired values. The performed experimems (see Table 1) have shown that in the range [1CJ4,10] Jp is fairly insensitive to q and that a good choice is q= 10-3 which is the poim where the minimum is attained, no maller the value of the suspended mass was. A second set of experiments was carried out with the aim of ascertaining the relevance of the friction simplification adopted by the controller (a linear torque-velocity relationship) with respect to the tuning of Q. The extreme situation where no friction is considered by the controller was thus studied: the obtained results, collected in Table 2, show that q= 10- 3 is still a reasonable choice and that the adoption of other nearby values for q does not affect dramatically the system performance. Again spanning the whole range of possible values for the payload mass, a comparison has been performed between EK and D 1 methods. assuming the laller to be exploitable with different and beller acceleration sensors increasingly .., (c hara ctcrized by the corresponding variances thc aim being that of ascertaining whether such a

er;).

q

m=1O

m=5

direct method could overcome the other one, provided that a sufficiently good acceleration measure is available. The obtained results are collected in Table 3 and point out that. whatever the suspended mass is, the EK method s supplies beuer performances. m=5

m=2

8.3

9.4

9.6

9.6

1.I

5.4

6.9

8.5

.33

5.3

6.7

8.4

.(X133

5.3

6.7

8.4

EK

3.2

3.4

5.1

14

10-4

p

the acceleration sensors precision.

Somehow dual is the situation if the imerest is more focused on the estimation of the (unknown) payload rather than on the control performances. Indeed if an "estimation" criterion J e is defined as the squared sum of the deviations of the payload estimate from its actual value, DI has to be preferred (by far) to EK provided that the sensor is sufficiemly good (see Table 4).

m=2

3.4

10- 3

16

17

4.0

5 .8

3.2

3.4

5.1

10- 2

3.6

4.3

6.2

10- 1

3.3

4.6

6.8

I

5.1

5.8

7.3

10

6.8

6.2

7.3

I

m=1O

m=5

m=2

8.3

60000

110000

270000

1.1

420

4400

8500

.33

ISO

210

1900

.0033

0.11

0.27

1.3

EK

32000

66000

140000

e

p andm.

m=1O

",=5

",=2

10-4

12.0

16.0

21.0

10-3

8.2

11.0

14.0

10- 2

7 .4

9.5

12.0

10-1

7.0

9.5

12.0

1

6.8

9.3

12.0

10

6.8

9.3

12.0

2 a

Table 4: The value of the crit.erion J xl06 for different values of the ac:c.eleration ac:nson precision.

Table 1: The value of the allerion J xl04 for different values of q

q

•

Table 3: The value of the CTllCnon ] x104 for different values of

0

10-5

2

m=1O

0

Table 2 : The value of lh.e criterion J x104 for different values of q p and m when friction is neglected in the COI1l1OlIer.

This fact should not sound contradictory with the conclusion relying on the Jp criterion as the filtered values of the measured variables which can be exploited only by the controller connected to the Extended Kalman Filter do apparenlly make the difference. As for method D2, the first set of experiments has been performed with the aim of tuning the design parameters which are four in the considered case, precisely p, b and the three ~i 's, the latter, however, being constrained to sum up to 1. The heuristic procedure here adopted consists in flfSt restraining the auention to the case where m=5 Icg and selling p=.9 (a reasonable choice to guarantee an

n -75

BOLZER.1\ P. . FERRETn G . . LOCA TEUI A

adequate level of adaptivity) and the ~I'S all equal 10 1/3. The optimal (wilh respect to J p criterion) value . b~=.5 of b is then found. Second, Jp is minimized wllh respect to p while holding lhe ~I'S at lheir previous value and selling b=bo. Letting pO be the result of this second step (actually, pO=.97), the "optimal" triple (~~ ,~~ ,~~ ) is the one, among those listed in Table 5, which again minimizes J p for b=bo ,

p=po. Finally a nonadaptive controller (NA) has been tested in order to have a reference point in evaluating the relative advantages entailed by adaptation . For m=5 kg the perfonnance criterion attains the value J p =6.9xlO-4. Wilh reference, therefore, to the situation in which lhe suspended mass is m=5 kg and its initial estimate is ;2(0)=4 kg, we can summarize the results of the perfonned experiments as follows (see Table 6): i) Control perfonnance Jp: lhe best method is EK followed by D2; moreover lhe consequent loss of pcrfonnance can be tolerated in view of the smaller computation burden which is invol ved. The relative gain with respect to a nonadaptive control strategy justifies the use of adaptive control laws. ii ) Estimation perfonnance J e: the best method is (bv far) DJ followed by D2 and EK. ' J xl0 4

13 1

p~

P3

I

0

0

5 .0

0

1

0

5.2

0

0

I

6.4

p

.5

.5

0

5.1

.5

0

.5

5.6

0

.5

.5

5.8

113

113

1f3

5.4

Table 5: The value of the criterion J as supplied by 02 p corresponding to bO and po. J xlO 4 p

J xloS e

EK

3.4

6600

01

6.7

1.3

02

5 .0

2900

NA

6.9

_.

5. CONCLUDING REMARKS The paper has! ,ented a simulation study of three different adaptive control schemes centered on a computed torque strategy. With reference to the particular adopted framework (a three link rigid robot carrying an uncenain payload and supplied with noisy sensors) the paper does not present new lheoretical results but ralher allows one to draw the following significant conclusions: a) Adaptive control significantly Improves the control system perfonnances; b) The behavior of the system gets acceptably worse if a fairly simple adaptation algorithm is implemented in the place of a much more sophisticated one; c) A good estimate of the payload mass does not guarantee per se good control perfonnances. REFERE~CES

11 J JJ. Craig, Adaptive Co/Urol of Mechanical Manipulators, Addison- Welsey, Reading, MA, 1988. 12J R.H. Middleton and G.C Goodwin. Adaptive computed torque control for rigid link manipulations, Sysums & Co/Urol UllUS, vo1.10, pp.9-16, 1988. 131 J.J.E. Slotine and W . Li , Composite adaptive control of robot manipulators, Automatica. vol.25. pp.509-519. 1989. 14 J l\. Sadcgh and R. Horowitz. An exponcnllally slable adaptive control la ..... for robot manipulators, IEEE Trans . on Robotics aNi Automation, vol.6, pp.491.496. 1990. [5J R. Lozano LuI and C. Canudas De Wit. Passivity based adaptive control for mechanical manipulators using LS-type esumlllon.IEEE TrtJllS . on Automatic Co/Urol, vo1.35, pp.13631365. 1990. 16J S. Uran. K.. Jez.ernik and I. Troch, Coulomb friction and simulation problems. Proc. IFAClIFIPIIMACS Symp . on Robot Co/Urol. Vienna. Austria. pp.33-38, 1991. (7) G. Magnani. P. Bologna, G. Rizzi and C. Lovati. Modelling and sunuJauon of an industrial robot. Proc. IEEE lni . Conf on RobOllCs alld AutomatiOft. Philadelphia. pp.24-29. 1988. (8) B.D.O. Anderson and 1.B. Moore. Optimal FiJtui"g. Prentice· Hall, Englewood Cliffs. Nl, 1979. (9J G.A. Miani and M Pessina, Adaptive conuol of a robotic manipulator with unknown pyload. Master thesis (in l~lian) Pohtccnico di Mi1Il1o.1~ Iy, 1991. • [10]P. ~su. M . Bodson. S. Sasuy and 8 . Paden. Adaptive Idenuftcauon and conuol for manipulaton without using joint accelerauons. Proc. IEEE bSl. COfI/. Oft Robotics Gild AutomatiOft. Raleigh. NC, pp. 12J~1215. 1987. ACKNOWLEDGEMEJI,'TS This work has been supported by M.U.R.S.T. and Cenuo Teoria dei Slstenu (C.N.R.). The simulations were performed with the hdp of G. A. Miani and M. Pessma.

Table 6 : Comparison of different cxmuol strategies.

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