NEURAL NETWORK BASED CONTROLOF UNDERACTUATED MANIPULATORS

Mareei Bergerman

Automation InstituteInfonnatics Technology Center

Campinas SP 13081-970 Brazil

Yangsheng Xu

The Robotics InstituteCarnegie Mellon UniversityPittsburgh PA 15213 USA

We propose in this work a neural network based control method to position the passive joints of a mechanicalmanipulator. Manipulators with such types of joints are known in the literature as underactuated manipulators. Astandard feedforward neural network (FNN) is trained to learn the dynamic behavior of the manipulator fromrandom state space data generated off-Iine by a variable structure controller. After learning is completed, the FNNis utilized for real time control of the underactuated manipulator's unactuated joint. Simulation results arepresented to demonstrate the validity of the proposed method .

Neste trabalho é proposta a utlização de uma rede neural para o controle das juntas passivas de um manipuladormecânico. Manipuladores equipados com este tipo de junta em sua estrutura são conhecidos na literatura comomanipuladores subatuados. Uma rede neural do tipo feedforward é treinada para aprender o comportamentodinâmico do manipulador a partir de dados aleatórios do espaço de estados do mesmo, dados estes gerados por umcontrolador com estrutura variável. Após o fim do treinamento, a rede neural é utilizada em tempo real para ocontrole da junta passiva do manipulador. Resultados numéricos comprovam a efetividade do método proposto.

1 lritroduction

In the past few years a significant research effort hasbeen directed towards modeling and development ofcontrol methods for underactuated manipulators [2],[3], [8], [10], [11] . The term underactuation refers tothe fact that not all joints are equipped with actuators.The directly actuated joints are called active joints,while those which lack actuation are called passivejoints. The class of underactuated manipulators iscomposed of manipulators with failed joint actuators,.as well as hyper-redundant snake-like robots withmore links than actuators. Interest in these systems isthen justified by a fault tolerance and a cost and energysavings point of view.

Cóntrol of all joints of an underactuatedmanipulator to anequilibrium point can be done in oneoftwo different manners. When the passive joints areequipped with brakes, one takes advantage of thedynamic coupling between these and the active joints,'and starts by applying torques at the active joints tocontrol the position of the passive ones. Once the .passive joints have converged to their set-p ôints, theycan be locked, and control of the active joints reducesto a classical robot problem . In this case

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feedback linearization techniques associated withlinear or robust controllers have been experimentallyproven to be effective for the control of underactuatedmanipulators with any number of passive joints [2],[4].

When the passive joints are completelyunactuated, one must utilize more corriplex controlmethodologies to position all joints at an equilibriumpoint. So far a few different methods have beenproposed, each one desgined for . a particularmechanical structure, and not easily generalizable fordifferent structures [1], [6], [9].

In this work we consider the first type ofunderactuated manipulators described, namely, thosewhose passive joints are equipped with brakes. Thischoice is based on the fact that most current robotmanipulators are factory-equipped with joints brakes,to be utilized either when the robot is moved from onelocation to another, or when an actuator fails and thecorresponding joint must be locked . As mentioned,control of this type of manipulators is a 2-step process,the first one (control of the passive joints via dynamiccoupling with the active ones) being the challengingone. Therefore, we will only concern ourselves in this

paper with the positioning of the passive joints to adesired set-point.

Despite ali the progress on the modeling andcontrol of underactuated manipulators, no work hasthus far addressed the application of neural networkbased control methods towards their controI. Neuralnetworks (NNs) are weIl known to be universalapproximators of nonlinear functions [12]. On theother hand, the equations describing the dynamiccoupling between the active and passive joints of anunderactuated manipulator are highly nonlinear [5].Therefore, it makes sense to consider the utilization ofNNs to map the coupling between the torques appliedat the active joints and the resulting accelerations ofthe passive ones, with the intent of controIling theIater.

We propose in thiswork the first application of aNN-based control method to control the passive jo intsof an underactuated-rnanipulator. A feedfoward NN,oroFNN, is cbosen for its simplicity and ease ofimplementation. NaturaIly, this choice does not limitthe validity ofthe method, as other tyes ofNNs couldalso be ccnsidered. The FNN weights are trained withrandom data generated off-line by a variable structurecontroller (VSC). After trainini of the FNN iscompleted, it is utilized in real time for the control ofthe passive joias. Although the VSC has been provenrobust against modeling uncerthinties and externaIdisturbances, it is a complex model based controller,

Iwhose use iu real time requires relatively fasthardware [3]. The FNN approacH facilitates the realtime implernestation of the controller, for it runsfaster than the VSC. We present simulation resultswhich show tlat the FNN effectively learns how tocontrol the pessive joint of a 2-link underactuatedmanipulator fiom any starting angle and to anydesired set-poat inside its operating range.

2 Control fi underactuated manipulators

Coriolis, centrifugaI and gravitational torques, and tis the vector of joint torques . To distinguish betweenthe motions of the active and passive joints, Equation(1) is partitioned

where the subscripts a and p respectively denotequantities related to the active and passive joints. Notethat the bottom part"of the torque vector is identicallyequal to zero, since it is-not possible to apply torquesat the passive joints.

To control the position of the passive joints (i.e.,

to control their acceleration qp) via the 'dynarniccoupling with the active torques ta' one must first

find the openloop relationship between these twovectors, and then design a c1osed-Ioop controller thatguarantees stability and set-point tracking despite thepresence of disturbances or uncertainties. The 'openloop relationship is found by factoring out the

accelerations of the active joints, qa, from the secondline.of (2), and substituting the result back into its-firstline:

-1 _ta = (Map-MaaMpaMpp)qp

- 1 (3)-«;»,»;»,A closed-loop controIler that guarantees stability

and set-poin t tracking, as well as better performancethan classical PD-type controllers, is the variablestructure controller (VSC). In this controller, theactive torques are computed as:

In (5) and (6), r and P are positive definitematrices, q; represents the desired trajectory of thepassive joints, qp = q; - qp is their position error,

and qp = q;- qp is their velocity error.The control law (4)-(6) has been successfully

utilized in the past for the real-time control of the 2-

We review in this section the main results publishedin the literatureregarding the control of underactuatedrnanipulators .ãs mentioned earlier, it is in the controlof the passise joints that we are interested, forcontrolling theactive joints when the passive ones arelocked is an exensively studied (and solved) problem.

Consider a mechanical mariipulator equippedwith n joints, 'h of them bearing actuators (the activejoints), and "r of them bearing brakes (the passiveones), The d!\Darnic model of the manipulator iswritten as:

(1)

In (1), q is themanipulator's joint vector, M(q) is its

n x n inertiamatrix, b(qA) is its n x 1 vector of

where u is an auxiliary input given by:

_ _du = rqp + qp + Psgn(s)

and s is a sliding surface defined by

(5)

(6)

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I3 Neural network based control

(7)

•••

hidden layers

Ów · · = "'0'0'I) _ 'I ) 1

Figure 2: Feedforward neural network.

In (7) , 11 is the learning rate, Oj is the network ' s error,defined as the network's output minus the torque atthe active joints that would be computed by the VSCaccording to Equations (4)-(6), and Oi is the network'soutput. This way, the network learns how to controlthe position of the passive joints fTOm the VSC,inheriting from it its nice robustness and performanceproperties.

To speed up the training procedure, we utilizedan adaptive learning rate procedure, which increases11 over smoother regions of the error surface, anddecreases it over rougher ones . Training is run untilthe network 's sum-squared-error (SSE) falls below auser defined threshold or a maximum number oftraining epochs is reached. The entire procedure isillustrated in the block diagram of Figure 3.

motorbrake

Neural networks have been shown to be universalfunction approximators [12] . Here we train afeedforward neural network (FNN) to learn thedynamic coupling between the torques applied at theactive joints of an underactuated manipulators, 'ta' and

Figure 1: Two-link underactuated rnanipulator.Joint 1 is active and joint 2 is passive.

link underactuated manipulator shown in Figure 1,whose first joint is active and second joint is passive.Its use, however, demands the availability of fastprocessors, since it requires the computation of thefull dynamic model of the manipulator at everycontrol loop . We propose then to train a neuralnetwork from random data generated by the VSC, andto utilize the NN for the real-time control of thepassive joints.

Figure 3: Training procedure of the FNN.

update NN weightsusing Equation (7)

SSE < SSEminimun ornumber of epochs > maximum?

no

compute active torque feed errors to FNN;with Equations (4)-(6) compute NN's output

the accelerations generated at the passive ones, qp.Our objective is to train the FNN so that it is able togenerate the torque vector 'ta that will bring qp to

converge to a desired q;.The network considered here is a multilayer

FNN, with sigmoidal input and hidden layers, andlinear outputs (Figure 2). The linearity of the outputneurons allows the network to generate torquesoutside the range 0-1, which is of course necessarysince the motor torques can range anywhere fromsome -'tmin to some +'tmax'

The FNN has 2np inputs, corresponding to theposition and velocity errors of the passive joints. It hasna outputs, corresponding to the vector of torques atthe active joints. Training is performed by feeding tothe FNN random position and velocity errors rangingfrom -27t to +27t (corresponding to the case when thepassive joint is far away from the setpoint), and frorn-o.I7t to +O.l7t (corresponding to the case when thepassive joint is dose to the setpoint), and updating theneurons' weights, Wij according to the delta rule forsemilinear activation fucntions :

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800 1000 1200 1400 1600 1eco 2000NW BEROF ERROR POINT5

.00 1000 1200 1400 1600 1eoo 2000NUMBER Ot=' ERROR POtNTS

aoo 1000 1200 '''00 1600 HIDO 2000NUIolBER Of TOAOUE POlPiTS

(b)

Figure 4: Generation of random datafor the FNN training procedure.

(a) position error; (b) velocity error;(cjtorque computed by the VSc.

(a)

(c)

Figure 6 presents the simulation results . One cansee that in ali four cases the passive joint convergessmoothly to its desired setpoint, controlled by thetorque computed by the FNN. Tens of othersimulat ions with different initial and final angles wererun, ali with the same success . It is interesting to note 'that the FNN was able to learn the dynamicinteraction between the active and the passive joints,and to generalize the random data presented to it

Parameter Joint J Joint 2

mass (Kg) 1.063 0.288

inertia (Kg m2) 0.031 0.020

length (m) 0.255 0.289

center of mess (m) 0.116 0.246

MatIab's Neural Netowkrs Toolbox was utilizedfor the implementation of the training code. A singleline of code takes care of the whole procedure, callingthe toolbox ' s subroutines that implement thegeneralized delta ruIe, the ca1culation of the errorgradient, and the updating of the weights. Details ofthese subroutines can be found in [7].

To demonstrate the validity of the proposed NN-based control method , we consider a 2-linkunderactuated manipulator with joint I active andjoint 2 passive. Our aim is to control the position ofthe passive joint from any starting angle to anydesired set-point. The kinematic and dynamicparameters of lhe manipulator are shown in Table 1;they corresponrl to the estimated parameters of ourexperimental setup, shown in Figure 1.Table 1: Kinematic and dvnamic parameters of the

testbed 2-link underactuated manipulator.

4 Case study

As descrêed above, to train the FNN wegenerated randnm position and velocity errors for thepassive joint, aid computed the corresponding VSCtorque accordng to (4)-(6). One thousand randompoints were geaerated for the case where the joint isfar from the sstpoint, and another thousand for thecase where it isclose to the setpoint. Figure 4 presentsthe random dita, along with the random torquecomputed by tle VSc.

The FNN for the 2-link manipulator has twoinputs (the jassive joint's position and velocityerrors) and oneoutput (the torque at the active joint).After a trial ani error procedure, we chose to build thenetwork with two hidden layers, the first orie withtwenty sigrnodal neurons, and the second one withten. The weghts and biases were initialized atrandom. We i:d to the network pairs of position/velocity errorextracted from Figure 4, computed itsoutput, and conpared it to the corresponding torque inthat figure. Aplot of the SSE as a function of thetraining epocls·is shown in Figure 5, where it can beseen that afte about 2000 epochs learning stopsimproving. Atthis point we halted the training andrecorded the lNN weights and biases.

With therecorded data we simulated the controlof the maniplator's passive joint in four differentsituations, suamarized in Table 2.

427

2110'

"",L ..:--::"::-:..,=",--::'",,.

nwHlJ(õ(I"C()lS

70

fr..'"li

20

oo t .z

5 Conclusion

. u

. ..... ..

o.,TIl,4E.ia )

JOlNl 1- 20

Figure 6: Simulation results of the FNN controllingthe position of the passive joint (joint 2).

The graphs also show the position of the active joint.

Initial angle Setpoint

90° 0°

0° 90°

-90° 0°

0° -90°

2

Experiment

during training to effectively utilize such interactionto position the passive joint precisely. Experimentalresults of the application of the proposed contraimethod on our 2-link underactuated manipulatortestbed (Figure 1) are expected to b/obtained soon . .

4

Table 2: Sumrnarv of the control experimentsrun with the FNN.

Figure 5: FNN learning: sum-squared error as afunction of the number of epochs ..

3

6 Acknowledgments

During the development of this work the first authorwas supported by a grant from the Brazilian NationalCouncil for Research and Development (CNPq).

We present in this paper the first utilization of a neuralnetwork for the control of the passive joint of anunderactuated rnanipulator. The neural network istrained with random data generated by a rabustvariable structure controller, and is able to performthe control from any initial condition to any desiredset-point. The advantage of such procedure is that isallows one to implernent.a "clone" of a kriown robustcontroller without the need for fast pracessors that cancope with the computational load required by it.Simulation results confirm the validity of the method .

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7 References

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[6] Chung ,W.; Nakaumra , Y. "Design of the chained formmanipulator." IEEE International Conference onRobotics and Automation, Albuquerque, NM, EUA,Apr. 1997.

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[8] Mukherjee, R.; Chen, D. "Control of free-flyingunderactuated space manipulators to equilibriummanifolds ." IEEE Trans. Robotics and Automation,voI. 9, no. 5, Oct. 1993, pp. 561-570.

[9] Oriolo, G.; De Luca, A.; Mattone, R. "Stabilization ofunderactuated rabots: theory and experiments for aplanar 2R manipulator." IEEE InternationalConference on Robotics and Automation,Albuquerque, NM, EUA, Apr. 1997:

[10] Oriolo, G.; Nakamura , Y. "Free-joint manipulators :motion contraI under second-order nonholonomicconstraints." IEEE International Workshop onIntelligent Robots and Systems, 1991, pp. 1248-1253.

[11] Papadopoulos, E.; Dubowsky, S. "Failure recoverycontrol for space robotic systems." American ContraiConference, vol. 2, 1991, pp. 1485-1490.

[12] Rumelhart, D.E.; Hinton, G.E.; Williams. RJ."Learning internal representations by errorprapagation." D. Rumelhart and J. McClelland,editors , Paral/el Data Processing,. voI. 1, chapter 8,MIT Press, Cambridge, MA, 1986.

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