pubtex output 2006.11.22:0752 collection/tro... · ieee transactions on robotics, vol. 22, no. 6,...

IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006 1265

Short Papers

A Mechatronic Testbed for Revolute-JointPrototypes of a Manipulator

Farhad Aghili

Abstract—This paper describes the design and development of a mecha-tronic testbed facility which allows high-fidelity and low-cost testing of thejoint prototypes of a manipulator in a highly flexible environment. Thetestbed system consists of a set of load motors whose shafts are connectedto those of the joint prototypes through a set of torque transducers. A con-troller modifies the simple dynamics of the load motors to match the non-linear and coupled dynamics of the manipulator links. This is made pos-sible by incorporating the measurement of joint angles, velocities, and jointtorques, as well as the dynamics model of the manipulator links in a com-posite feedforward/feedback loop. The stability of the overall system andthe fidelity of the load emulator are investigated. The testbed system isconstructed to permit testing in a space-like thermal/vacuum environment.Thus, the facility is particularly useful for development and validation ofnew joints and actuators for space manipulators. The system is experimen-tally validated by comparing trajectories of the joint angles, velocities, andtorques of a set of joint prototypes obtained by installing them first on anactual robot, and then on the mechatronic testbed.

Index Terms—Emulating testbed, mechatronics, mechatronics testbed,motion control, robot control, robot simulation, space robotics.

I. INTRODUCTION

Joint servo-mechanisms consisting of actuators, sensors, and con-trollers are among the fundamentals in mechatronics and robotics. De-velopment of any new joint prototype ought to undergo extensive me-chanical, electrical, and thermal tests at different stages of the devel-opment in order to make sure that the system works as intended. Inrobotics applications, these tests can be performed using a robot proto-type built on developed joints. However, building a complete prototypeof the robotic system is an expensive and inflexible process. Moreover,due to the iterative nature of the design process, the need for multiplerobot prototypes makes it even more costly and time-consuming.

The challenge of testing space manipulators is even greater, becausethey must be tested and validated in a 1-g laboratory environment,whereas the actual robotic system will eventually work in an environ-ment with different gravity, temperature, and ambient pressure. For in-stance, one may refer to [1], which describes the development of a newtype ultrasonic motor for space robots. Another example is the ShuttleRemote Manipulator (RMS), which was designed and built for oper-ation in orbit. A ground test facility for the integrated RMS was de-veloped based on air-bearing technology to support verification of theoverall system performance [2]. In that experiment, the manipulatorwas supported on a series of air-bearing pads that could move on acushion of compressed air along a large and polished floor with a min-imum of friction. However, the main limitation of that testbed was thatit could only provide two-dimensional planar motions.

Manuscript received February 14, 2006. This paper was recommended forpublication by Associate Editor P. Rocco and Editor H. Arai upon evaluation ofthe reviewers’ comments. This paper was presented in part at the IEEE Interna-tional Conference on Robotics and Automation, Orlando, FL, May 2006. Colorversions of Figs. 5-6, and 8-10 are available online at http://ieeexplore.org.

The author is with the Canadian Space Agency, Saint-Hubert, QC J3Y 8Y9,Canada (e-mail: [email protected]).

Digital Object Identifier 10.1109/TRO.2006.882962

Simulations can be also used to verify the functionality of mecha-tronic and robotic systems. A high-fidelity simulation, however, re-quires accurate mathematical models representing many physical sub-systems whose complexities are not fully understood. In particular, ac-tuators and their gear reducers often have complex characteristics [3],which are not usually modeled. Consequently, a simulator based on asimplified model may not be useful for validation testing.

Alternatively, testing of actuators and join prototypes can be carriedout by mounting them on a dynamometer. Industrial dynamometerstend to use a flywheel and/or mechanical brake for loading the actuator[4]. However, such a simple load does not represent a real manipulator.Hence, the extent of the test result is limited, and the real performanceremains largely unknown until the actual robot becomes operational.Recently, mechatronic testbeds that use dynamometers for testing ve-hicle dynamics and control have been developed based on the conceptof hardware-in-the-loop (HIL) simulation [5]–[8]. These test methodsare limited to emulating loads with linear dynamics.

This work is aimed at building a testbed facility for testing arange of actuators, used for either space or terrestrial robots. Unlikeindustrial dynamometers that apply constant braking torques, orvehicle dynamometers that are capable of emulating linear loads,our dynamometer uses active loads that generate loading torquescorresponding to a prescribed model. This is made possible by incor-porating the measurement of joint torque, as well as joint angles andvelocities, in a composite feedforward/feedback loop. It should benoted that torque feedback used here is to modify the simple inertia ofthe rotors of the load motors so as to match the nonlinear and coupleddynamics of a manipulator’s links, whereas the application of torquesensory feedback in robot control is to improve the performance ofthe controller [9]–[14] or to compensate the friction disturbances[15]–[17]. Such a testbed system allows testing the complete jointprototypes of a manipulator without needing to construct the ma-nipulator. This can significantly reduce the cost associated with thedevelopment of a new mechatronic system. The testbed facility isespecially useful for development of space robots that are designedto operate in a space environment. In this regard, the dynamometeris specifically designed so that the joint prototypes can undergo testunder a space-like thermal/vacuum environment.

This paper is organized as follows. Section II describes thedesign of a dynamometer with active loads. Included here areSections II-B and II-C, which describe development of the loadmotors controller and the stability analysis of the entire system. Thefidelity of the emulation system in the presence of noise/disturbanceand parametric uncertainties is investigated in Section III. Section IVextends the concept for emulation of a flexible manipulator. The spec-ifications of the testbed facility built at the Canadian Space Agency(Saint-Hubert, QC, Canada) are elaborated on in Section V, followedby experimental results presented in Section VI.

NOMENCLATURE

�max Maximum eigenvalue.�min Minimum eigenvalue.s Laplace variable.

k � k Euclidean norm.q Joint angles.

1552-3098/$20.00 © 2006 IEEE

1266 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006

Fig. 1. Concept of the mechatronic testbed.

qf Flexural coordinates.M Inertia matrix of the manipulator.h Nonlinear vector.Jl Diagonal matrix of the load-motor rotor inertias.Jt Diagonal matrix of the test-motor rotor inertias.fl Friction of load motors.ft Friction of test motors.� Joint torque on the shaft.ul Torque developed by load motors.ut Torque developed by test motors.KT Diagonal matrix of load-motor torque constants.

II. METHODOLOGY

A. Dynamometer With Active Loads

The concept of the mechatronics testbed is schematically illustratedin Fig. 1; note that for simplicity, only one axis is shown in the figure.The active loads are comprised of a set of electric motors equippedwith joint-angle, velocity, and torque sensors. The control system takesthe model of the load’s dynamics and the realtime measurement of thejoint quantities, and subsequently issues commands to the load motorsso that the resulting shaft torque corresponds to the prescribed loaddynamics. In other words, the relation between the torques and the po-sitions, velocities, and acceleration on the shafts of the load motorsreplicates what would be encountered in the actual manipulator. There-fore, the set of the joint prototypes connected to the load motors shouldbe dynamically equivalent to the joints integrated in the real manipu-lator. In order to be able to analyze and study the hardware system ina hostile space environment, the actuators are mounted on a thermallyregulated cold–hot plate, and the entire dynamometer can be placed ina vacuum chamber. The thermal plate and the test actuators are ther-mally insulated to ensure that the instrumentation and the load motorsare protected from the extreme temperature.

Fig. 2. Balance of torques on the ith axis of the dynamometer.

B. Control

The equations of motion describing the link dynamics of a manipu-lator with n joints is given by [18], [19]

M(q)�q + h( _q; q) = �� (1)

where q 2 n is the vector of joint angles, �� 2 n is the vectorof joint torques, M(q) 2 n�n is the inertia matrix of the links, andh( _q; q) := C( _q; q) _q+ g(q), where C( _q; q) contains Coriolis and cen-trifugal terms, and g(q) = @Vg=@q represents potential forces.

Now, assume that the manipulator’s actuators are connected to a setof n load motors, instead of being installed in the manipulator. Fig. 2illustrates the ith axis of such a setup, where the shaft of the load motoris cut right at the junction to the joint prototype. Note that in the fol-lowing, the subscripts l and t indicate quantities associated with theload motors and the test motors, respectively. Let vectors ul 2 n and� 2 n denote the driving torques of the load motors and the subse-quent joint torques developed on the shafts, respectively. Then, writingthe balance of torques on the rotors gives

Jl�q = ul + � � fl( _q) (2)


where Jl = diagfJlig; Jl denotes the polar inertia of the ith rotor,and fl ( _qi) is the friction torque in the bearings. Note that the controlinputs ul are related to the motor current i by

ul = KT i

where KT = diagfkT g contains the torque constants of the load mo-tors.

Now, the objective is to design a feedback control law such that thejoint torque dynamics (2) is changed to match the prescribed referencemodel (1). Let �q� denote an estimation of the joint acceleration calcu-lated from the reference model (1) by taking �� = � . Then, since thevalues of �; q, and _q are available from the sensors, the acceleration canbe obtained from the direct-dynamics equations of the links

�q� := M(q)�1(�h( _q; q) + � ): (3)

The estimated acceleration is different from the actual one, becausethe equations of motions (1) and (2) are different. Hence, in general,�q 6= �q�. Defining the acceleration error by

�~q = �q � �q� (4)

and using (3), we can say

M(q)�q + h( _q; q) = � + e(q; t) (5)

where

e = M(q)�~q: (6)

Equations (1) and (5) readily imply that the perturbation e = �� can be treated as the torque error.

Now, we seek a torque-control law ul for the load motors to annihi-late the perturbation e(q; t); that is to say, � tracks ��. We will showthat this goal can be achieved by the following control law:

ul = (JlM(q)�1 � I)� + fl( _q)� JM(q)�1h(q; _q)

+Kv �q� dt� _q +Kp �q� dt� q (7)

where Kv > 0 and Kp > 0 are the feedback gains, I is the n � nidentity matrix, and �q� is obtained from (3). The gains can be ade-quately selected as Kv = 2Jl and Kp = 2Jl, where is selectedas the bandwidth of the closed-loop system. The friction can be cap-tured by one of the friction models reported in the literature [20]–[22].For steady sliding, the friction force as a function of velocity [22] isgiven by

fl ( _qi) = FC sgn( _qi) + Fs e�( )

sgn( _qi) + F� _qi

where FC ; F� ; Fs , and �i denote Coulomb friction, viscous friction,the magnitude of the Stibeck friction (frictional force at breakaway),and the characteristic velocity of the Stibeck friction for the ith joint,respectively.

Proposition 1: Consider a multiaxis dynamometer with n load mo-tors with inertias Ji and frictions fi. Also, assume that the control law(7) is applied to the motors. Then, the torque dynamics generated by

the load motors is governed by (5), and the perturbation e(q; t) expo-nentially decays to zero.

Proof: Substituting the torque-control law (7) into (2) gives theerror dynamics

�~q + 2_~q +2~q = 0: (8)

The solution of the above differential equation is �~q = (a1+a2t)e�t,

where a1 = �2_~q0 � 2~q0; a2 = 2 _~q0 + 3~q0; _~q0 and ~q0 arethe initial errors caused by the integrals in (7). It can be shown that80 2 (0;), we have

k�~qk � a0e� t (9)

where a0 = max(ka1k; ka2k=( � 0)). Moreover, since the massmatrix is bounded, 9c > 0 such that

supq

�max(M(q)) = c:

Using the above and (9) in (6), we arrive at

kek � ca0e� t (10)

which means that the magnitude of the error exponentially decays tozero, i.e.,

� ! �� as t!1:

Remark 1: It is apparent from (7) that the torque feedback is re-sponsible for changing the inertia. In other words, the force feedbackis disabled if J � M .

C. Stability Analysis

It is evident from (10) that the perturbation relaxes to zero from itsinitial value. However, we must also show that the time-varying per-turbation does not destabilize the combined system of the load and testmotors. In other words, the interactions of the two sets of the load andtest motors have to be stable.

Consider Fig. 2 again. Now, writing the balance of torques on therotors of the test motors leads to

Jt�q + ft( _q) = ut � � (11)

where Jt = diagfJt g; ft( _q), and ut are the inertia, friction, and thedriving torque of the test motors, respectively. Eliminating � from (5)and (11) gives the equations of motion of the entire load and test motorsas

(M + Jt)�q + h( _q; q) + ft( _q) = ut + e(q; t): (12)

We consider the above equation with and without perturbation e(q; t) asthe nominal system and the perturbed system, respectively. FollowingSpong’s assumptions [23] that 1) the kinetic energy of motor rotorsof a manipulator is mainly due to its own rotation, and 2) the rotor/transmission inertia is symmetric about the rotor axis of rotation, onecan show that the nominal system coincidences with that of the actual


manipulator and the test motors together [9], where M + Jt is theso-called effective inertia [19]. Now the concern is the stability of theperturbed system (12). In the following, we investigate the stability ofboth the unforced system, i.e., ut � 0, and the forced system, in whichthe test motors use feedback law ut(q; _q; t).

Proposition 2: The unforced system (12), i.e., ut = 0, is exponen-tially stable.

See Appendix I for a proof.Let vector xT = [(q � q0)

T _qT ] represent the states. Also, assumethat the servo controllers of the test motors are functions of the statesand time, i.e., ut = ut(x; t). Then, system (12) can be rewritten in thestandard form

_x = f(x; t) + g(x; t) (13)

where

g(x; t) =0

(M + J)�1M(a1 + a2t)e

� t

is viewed as the perturbation to the nominal dynamics _x = f(x; t).Note that the nominal dynamics represents the combined system ofthe test motors, their controllers, and the actual manipulator. It can beshown that the perturbation satisfies the following bound:

kg(x; t)k � � =�max(M)a0

maxf�min(M); �min(J)g:

Proposition 3: If the nominal system _x = f(x; t) is asymptoticallystable, then so is the perturbed system (13).

See Appendix II for a proof.In the case of a reference-tracking controller, the torque of the test

motors can be written as ut = ut(x; xd(t); t), where the time-varyingsignal xd(t) represents the input of the control system. Assume thatut(x; xd(t); t) be continuously differentiable, which means that theJacobian matrices [@f=@xd] and [@f=@x] are bounded. Now underthat assumption, if the assumption in Proposition 3 also holds, thenthe system (13) is locally input-to-state stable [24], [25].

Generally speaking, the above arguments show that the combinedsystem of the joint prototypes, including the test motors and their ownservo controllers, and the load motors with the proposed control laware stable, so long as the same joint prototypes are stable on the ma-nipulator whose link dynamics is being emulated by the load motors.In other words, the testbed preserves stability of the servo controllersof the test motors.

III. FIDELITY

A. Sensitivity to Noise and Disturbances

The magnitude of the the torque error e quantitatively indicates thefidelity of the system emulation. Theoretically, it was shown that theerror vanishes exponentially under the proposed control law. In prac-tice, however, the error may not be zero, due to the effects of distur-bance and sensor noise. In this section, we investigate this problem.

It is apparent from (7) that the control law comprises both feedfor-ward and feedback loops. Although one can show that incorporationof the feedforward loop alone can achieve the torque emulation, thefeedback loop is also required, as it plays an important role in rejectionof disturbances and sensor noises. Assume vector d captures lumped

modeling errors due to the torque-sensor noise and uncompensated dis-turbance. Then the equations of motion of the load motors become

Jl�q = ul + � + d� fl: (14)

Substituting the torque-control law (7) in (14), we obtain the differen-tial equation for the now nonhomogeneous error, i.e.,

�~q + 2_~q + 2~q = �J�1d

which can be written in the Laplace domain

�~q = �s2

(s+)2J�1l

d: (15)

Finally, substituting (15) in (6) and using the norm properties, we obtainthe transmissivity from the magnitude of the disturbance to that of thetorque error in the frequency domain as

kek � �!2

!2 +2kdk (16)

where � = kMJ�1k, and ! is the frequency.Equation (16) implies that the feedback can reject the disturbances

occurring at frequencies which are lower than the controller bandwidth.However, the feedback is not able to attenuate high-frequency distur-bances.

Remark 2: The effect of sensor noise and disturbance can be atten-uated if:

1) either the controller has a sufficiently high bandwidth, such that � !;

2) or � is small. In other words, using control to increase the inertiaof the system by control always tends to increase the disturbancesensitivity, and vice versa.

However, it should be pointed out that, in practice, the maximumachievable bandwidth is constrained by the sampling rate of the dis-crimination [26], and also by modeling uncertainties.

B. Parametric Uncertainty

Assume that the actual inertia of the rotors are J + �J , where �J

represents the parametric uncertainty. Then, using the control law (7)in the real system with the actual inertia leads to

�~q + 2_~q +2~q = �J�1�J �q:

The solution of the above differential equation can be written as

�~q = �J�1�J �q + J�1�J_�q (17)

where _�q = G(s) _q is the filtered version of the velocity, and G(s) =(2s2 + 2s)=((s + )2). The variables are expressed in the timedomain mixed with the Laplace domain for simplicity in notation. Now,substituting the acceleration error from (17) into (6) and then into (5),we arrive at

(M +�M )�q + (h( _q; q) + �h) = �

where �M = MJ�1�J and �h = �MJ�1�J_�q are viewed as

the uncertainties in the inertia and the nonlinear vector of the targetdynamics pertaining to the parameter uncertainty �J . Assuming abounded uncertainty, i.e., k�Jk � �, and knowing that k _�qk � 2k _qk,


we can say k�Mk � �� and k�hk � 2��k _qk. This means that theeffect of the uncertainty in the inertia of the load motors is propagatedproportionally by �, and hence, a low value of � is desirable. However,the magnitude of the uncertainty in the nonlinear vector tends toincrease proportionally with increasing the bandwidth, and that can bein conflict with the requirement for the low sensitivity to noise.

IV. EMULATION OF FLEXIBLE LOADS

So far, we have considered only emulation of rigid loads, while manyspace manipulators are flexible. In this section, we extend the conceptof the system emulation to the case where the load is structurally flex-ible, e.g., flexible-link manipulators.

The equations of motion of a flexible manipulator can be describedin the partitioned mass matrix as

Mrr Mrf

MTrf Mff

�q

�qf+

hr( _q; _qf ; q; qf)

hf( _q; _qf ; q; qf)=

��

0(18)

where q 2 n and qf 2m represent the joint angles and the flexural

generalized coordinates, respectively. The above matrix equation canbe solved for both �q and �qf . Similar to the previous case, assume that�q� represents an estimate of the joint acceleration obtained from (18)by considering �� = � . Then

�q� = �M�1

rr (� � hr �MrfM�1

ff hf) (19)

where

�Mrr = Mrr �MrfM�1

ff MTrf :

The control law (7) can be updated accordingly by replacing M and h

with Mrr and hr + MrfM�1

ff hf . Then, it can be shown that substi-tuting the estimated acceleration from (19) into the updated control lawleads to the same result as stated in Proposition 1. However, it shouldbe pointed out that estimation of acceleration from (19) requires valuesof the flexural states f _qd; qfg that, unlike the joint states f _q; qg, are notavailable by measurement. Nevertheless, the flexural states can be ob-tained by simulation. To this end, by making use of the matrix equation(18), the acceleration of the flexural coordinate can be computed from

�qf=�M�1

ff I +MTrfM

�1

rr MTrfM

�1

ff hf�M�1

ff MTrf�M�1

rr (��hr):

(20)

Thus, the flexural states can be obtained as a result of numerical in-tegration of (20). The interconnections of the simulator, accelerationestimator, and the torque-control law are schematically illustrated inFig. 1.

V. DESCRIPTION OF THE TESTBED

Fig. 3 illustrates the design of a two-axis dynamometer. The majordesign problem was the selection of appropriate actuators for the ac-tive loads. They are required to be powerful enough to overcome thetest motors’ torques, and have to operate accurately in torque-con-trol mode. In fact, the latter determines the fidelity of the load em-ulator. Direct-drive electric motors with their low friction and com-pliance and high bandwidth are suitable for this task. This is becausethey do not use any gearbox, which is the main source of mechan-ical imperfection and also friction. In fact, the motor bearing set isthe only moving part, and that creates only minimum friction. There-fore, direct-drive actuators can be modeled by a simple model withminimum uncertainty. However, since direct-drive motors suffer formlow torque capability, large-sized motors should be selected to enure

Fig. 3. Design of the two-axis dynamometer.

Fig. 4. Mechatronic testbed setup.

that their maximum torque matches those of the motors undergoingthe test. Finally, we selected brushless motors RBEH-03011-C10 andRBEH-02102-K02 (from Kollomorgen), which constitute the loads ofthe first and the second axes of the dynamometers. The load motorsare mounted on the rigid base of the dynamometer. The motors’ shaftsare connected to those of the joint prototypes undergoing testing viatorque transducers (Himmelstein 28002TS-136 and 28001TS-135) bymeans of two couplings (Huco Eng. 538.41. Long type). The couplingscome with high torsional stiffness, but relatively low bending stiff-ness. These mechanical properties allow relieving any strain caused bya misalignment of the motors’ shafts, while avoiding additional jointflexibility and the subsequent dynamics. High-resolution optical en-coders (Gurley R135S-03600Q-5L10-C18TP-04EN) are fixed to eachmotor’s housing, while the encoders shafts are connected to the motorshafts by small bellow-type couplings. Fig. 4 shows the mechatronictestbed facility developed at the Canadian Space Agency. The max-imum power, torque, and speed specifications of the dynamometer aregiven in Table I. Obviously, the maximum expected power and torqueof the motors that undergo testing should be lower than the rated valuesof the load motors. Otherwise, more powerful actuators should be usedas the loads.

The load motors are driven by power amplifiers SE03-200-2R andSE06-200-2R (from Kollmorgen Servo Star amplifier) which operate


TABLE ICHARACTERISTICS OF THE TWO-AXES DYNAMOMETER

TABLE IIRANGE OF AMBIENT TEMPERATURE/PRESSURE OF THE TESTBED FACILITY

TABLE IIIPARAMETERS OF THE LOAD MOTORS

in torque-control mode. The load motors should be able to produceeither a driving torque or a braking torque, depending on the directionof the joint torque with respect to the joint velocity. The latter situationimplies that the load motor acts as a generator, i.e., producing electricenergy. The power amplifiers are equipped with a shunt resistor circuitin order to be able to dissipate the energy when the load motors operateas generators.

The test actuators are mounted on a thermal cool/hot plate whichcan regulate the ambient temperature. A set of thermocouplers are em-bedded in the plate to measure temperature. A thick layer of teflon is putbetween the thermal plate and the base of the dynamometer to providethermal insulation. The entire setup can also be enclosed in a vacuumchamber for testing the motors and their gearboxes in a vacuum en-vironment. The ambient temperature/pressure of the undergoing testactuators can be regulated to be within the minimum and maximumranges specified in Table II. The testbed facility is also equipped withan analog data-acquisition board for measuring the motor currents andvoltages. These measurements allow the identification of electrical,mechanical, and thermal properties of the joint prototype under dif-ferent loading and ambient conditions.

The controller of the load motors was developed using Simulink.The parameters used for the control system are given in Table III. Thevalues of the inertias and the torque constants were extracted fromthe manufacturer’s manuals, and the parameters of the joint frictionwere obtained from the experimental torque-velocity characteristic ofthe load motors, shown in Fig. 5. These parameters were used for amodel-based friction compensation. Real-Time Workshop [27] gener-ated portable C code from the Simulink model, which was executedon a QNX realtime operating system. The kinematics and dynamicsmodels of the reference manipulator were developed using Symofros[28]; they were then imported to the Simulink diagram as S-functions.Matrix manipulation was carried out by using Matlab/Simulink’s DSPBlockset [29]. This architecture allows us to achieve a 1-kHz samplingrate on a 300-MHz Pentium computer. The joint prototypes have theirown servo controllers, which run on another computer.

Fig. 5. Joint frictions of the load motors.

TABLE IVCHARACTERISTICS OF THE TESTED HD MOTORS

Fig. 6. Joint prototypes and the links.

VI. EXPERIMENTAL VALIDATION

In this section, we examine the accuracy of the emulating systemby comparing the dynamic responses of two joint prototypes that wereinstalled first on the dynamometer, and then on a real manipulator. Theactuators of the joint prototypes consist of harmonic-drive (HD) motorsRH-11-3001 and RH-14-6002 (from Hi-T Drive) whose specificationsare given in Table IV. The manipulator was specially designed and builtin such a way that the HD motors constitute its joints, and that themotors can be easily dismantled/mantled. Fig. 6 shows the HD motorsand the links separately, while Fig. 7 shows their assembly in a planar-robot configuration. The HD motors had their own servo controller totrack a reference trajectory in the joint space. In this experiment, weobtained the desired trajectories of the joint servos corresponding to acircular path (with diameter of 0.2 m) of the manipulator’s end-effector.The trajectories were commanded to the joint servo-controllers whenthe joints are installed either on the actual robot or on the dynamometer.In the former case, the torques built up on the shafts are due to the


Fig. 7. Planar manipulator.

Fig. 8. Time histories of joint angles, joint velocities, and joint torques ob-tained at end-effector velocity 0.1 m/s. The solid and dashed lines correspondto experimental results obtained from the robot and from the dynamometer, re-spectively.

inertial force of the links, whereas in the latter case, the joint torquesare produced by the load motors.

A comparison of the measured quantities of the joint prototypesobtained when they are installed on the manipulator and on the dy-namometer is illustrated in Fig. 8 (where the end-effector velocity is

Fig. 9. Time histories of joint angles, joint velocities, and joint torques obtainedat end-effector velocity 0.3 m/s.

Fig. 10. Trajectories of the robot end-effector positions; the solid and dashedlines correspond to experimental results obtained from the robot and the dy-namometer, respectively.

set to 0.1 m/s). The figures demonstrate a good match between the re-sponses of the joint prototypes installed on the robot and on the dy-namometer. The perturbation in the joint torque trajectories occurringat the instants of zero velocities are attributed to the dry friction in thejoints. This experiment was repeated for a higher velocity of 0.3 m/s,and the results are illustrated in Fig. 9. The end-effector trajectoriescalculated from the joint trajectories are shown in Fig. 10.

VII. CONCLUSION

The work presented here described the development of a testbed fa-cility that allows testing of the joint prototypes of a manipulator byusing a set of controlled load motors. The controller was developedto modify the simple dynamics of the load motors to match the non-linear and coupled dynamics of the prescribed manipulator. This wasmade possible by incorporating the joint angle, velocity, and torquemeasurements, as well as the dynamics model of the manipulator, in a


composite feedforward/feedback loop. The testbeds were specificallydesigned and built so that the joint prototypes can be tested undera thermal/vacuum ambience similar to the space environment. Thismakes the facility especially useful for development of new joints forspace robots.

The stability of the entire system of the load motors connected to thetest-joint prototype, which includes motors, sensors, and servo con-trollers, was analytically investigated. It was shown that the joint pro-totypes on the testbed will retain stable if they do so on an actual robot.The fidelity of the load emulator with respect to the disturbances andsensor noise was analyzed. It was shown that the propagation of thedisturbance to the torque error is proportional to the ratio of inertias ofthe prescribed manipulator and the rotors of the load motors.

Finally, the performance of the load emulator was demonstrated ex-perimentally by comparing the trajectories of the joint positions, ve-locities, and torques of a set of joint prototypes which were installedfirst on a real robot, and then on the dynamometer. The experimentalresults showed that the two responses are close.

APPENDIX I

Assume that Vq(q0) = 0 and consider the Lyapunov function

V =1

2_qT (M + Jt) _q + Vg(q � q0): (21)

Since the potential energy is bounded, the above function satisfies thefollowing bounds:

c1k _qk2 � V � c2k _qk2 + c3: (22)

Then, using the bounds in (22), the well-known physical property that_M � 2C is a skew-symmetric matrix, and that _qT ft( _q)k � c4k _qk2,

where c4 > 0, we can show that the time derivative of (21) along (12)satisfies

_V � �c4c1V +

pV

ca0pc2e� t:

The above nonlinear inequality can be linearized by the followingchange of variable U =

pV , whose time derivative is _U = _V =2

pV ,

i.e.,

_U � � c42c1

U +ca02pc2e� t: (23)

Finally, using the comparison lemma [30, p. 222] and (22) in (23), weobtain

k _qk � c2c1k _q(0)k+ c3

c1� � e

� t+ �e� t

where � = c1ca0=((c4 � 20)pc2), and that concludes the proof.

APPENDIX II

Since x = 0 is a uniformly asymptotically stable equilibrium pointof the nominal system _x = f(x; t), there must exist a Lyapunov func-tion V (t; x) for the nominal system satisfying the following inequali-ties [31, Sec. 5.13], [32]:

�1(kxk) � V (t; x) � �2(kxk)_V � ��3(kxk); @V

@x� �4(kxk)

in the ball kxk < r, where �i(�); i = 1; 2; 3; 4, are all class K func-tions. Now consider V (t; x) for the perturbed system. Then, 8� 2(0; 1), the time derivative of V (t; x) along trajectories of (13) can bewritten as

_V � ��3(kxk) + ��4(kxk)� �(1� �)�3(kxk) 8kxk � �

where � = ��13

(�4(r)�=�). Therefore, according to the uniform ul-timate boundedness theorem of uncertain dynamic systems [33], [34,Sec. 56], the solution of the perturbed system should be bounded by

kx(t)k � �(r)

where �(r) = ��11

(�2(��1

3(�4(r)�=�))). Now since the states of the

perturbed system remain bounded, the perturbation decays gracefullyto zero, and this ensures the nominal and perturbed systems will behaveidentically. Since the nominal system is asymptotically stable, so is theperturbed system.

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Elastic Model of Deformable Fingertipfor Soft-Fingered Manipulation

Takahiro Inoue and Shinichi Hirai

Abstract—We propose a straightforward static elastic model of a hemi-spherical soft fingertip undergoing large contact deformation, as occurswhen robotic hands with the fingertips handle and manipulate objects,which is suitable for the analysis of soft-fingered manipulation because ofthe simple form of the model. We focus on formulating elastic force andpotential energy equations for the deformation of the fingers which arerepresented as an infinite number of virtual springs standing vertically.The equations are functions of two variables: the maximum displacementof the hemispherical fingertip and the orientation angle of a contactingplanar object. The elastic potential energy has a local minimum in ourmodel. The elastic model was validated by comparison with results of acompression test of the hemispherical soft fingertip.

Index Terms—Deformation model, elasticity, grasping, manipulation,robotic hand, soft fingertip.

I. INTRODUCTION

To date, various research has been done on manipulation of objectsby soft-fingered robotic hands. Most of the studies, particularly earlierstudies, focused only on contact mechanisms on various soft fingers.More recently, there has been an increase in studies on sensing mecha-nisms of human hand and designing control systems in robotic applica-tions to emulate the human capabilities which are applicable to robotichands. The conventional studies, however, have not been explicitly pro-viding any analytical exploration of the simplicity in grasping and ma-nipulating motions in terms of the soft-fingered handling. As a causeof the above-mentioned, it has been substantially difficult to derive afine elastic model of soft materials used in the fingertips.

Yokokohji et al. proposed a control scheme with visual sensorswhich can cancel the frictional twist/spin moment at the contactpoint of soft fingertips, and achieved stable grasping by spherical softfingertips [1], [2]. Maekawa et al. developed a finger-shaped tactilesensor covered with a soft and thin material, and proposed a controlalgorithm based on tactile feedback using the sensor, which needs noinformation about the geometry of the grasped object [3], [4]. Theymanaged to control the position of an object along a desired trajectory.In these papers, point-contacts were used to represent constraints ofrolling contact in their theoretical models, although the fingertips weremade from soft material such as rubber. Arimoto et al. verified thepassivity of equations of motion for a total handling system by using aLagrangian function incorporating the elastic potential energy due tothe deformation of soft fingertips [5], and compensated for the gravityeffect in three-dimensional space [6]. An elastic force model was alsoderived for the elastic potential energy of a system in which virtuallinear springs were arranged for simplicity in a radial pattern insidehemispherical soft fingertips. Doulgeri et al. discussed the problem ofstable grasping with deformable fingertips on which rolling constraintswere described as non-holonomic because of change in the effective

Manuscript received February 6, 2006; revised June 28, 2006. This paper wasrecommended for publication by Associate Editor H. R. Choi and Editor H. Araiupon evaluation of the reviewers’ comments. This work was supported by theIEEE. This paper was presented in part at the IEEE International Conference onRobotics and Automation, Barcelona, Spain, April 2005.

The authors are with the Laboratory for Integrated Machine Intelligence, De-partment of Robotics, Ritsumeikan University, Japan (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TRO.2006.886274

1552-3098/$20.00 © 2006 IEEE

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