Design And Experimentation Of A Jump Impact Controller - IEEE ...

Design and Experimentation of a Jump Impact Controller David K. Chiu and Sukhan Lee

A

new impadforce controller known as Jump Impact Controller (JIC) is designed and evaluated experimentally.JIC is derived by representing dynamics of collision in terms of a jump linear system where the jumps of system dynamics can be described by a state-dependent stochastic process. Thus, JIC provides robustness to the uncertainties in the environment dynamics as well as the location of the collision surface. The experimental results not only demonstrate the robustness of JIC but also help explain and interpret how robustness is associated with the JIC theory in terms of such known concepts as approach velocity and control system bandwidth.

Introduction Modern manipulators are required more and more to interact with their environment by direct contact. Thus, it is important that both the impact and steady state contact force be controlled to ensure satisfactory performance. For instance, manipulators have been used to check the conductivity of solderings on a circuit board at high speed. The impact force of the probe must be controlled so that the solderings are not damaged, and at the same time enough force must be applied to provide good contact. Previous research efforts in this area have shown that the performance of good impact control depends critically on controlling the approach velocity [l-41. While it is understood that the approach velocity is important for good impact force performance, it is unclear how to optimally control the approach velocity in the presence of collision surface location uncertainty. Optimality implies here minimal contact time as well as impact force and steady state force error. For instance, in [ 11, the optimal approach velocity (in the sense ofjust minimizing impact force and steady-state force error) for a stiff environment is expected to be low. If the collision surface location is uncertain, one may try to maintain this “optimal approach velocity” in the vicinity of the collision surface which may result in an undesirably long contact time. Another important factor that affects the performance of impact control is the knowledge of system dynamics. Explicit force controllers show good force tracking without good knowledge of the collision surface location. However, in order to dampen the impact force and maintain stability, good knowledge about the system dynamics is necessary [5,6]. Unfortunately, in most applications, we have very limited knowledge about the en-

vironment dynamics and collision surf,acelocation. It may be impractical to obtain this information exactly in every situation. Various approaches [7- 101try to circuinvent this problem but do not address the problem of environment location uncertainty. In addition to these problems, practical implementation issues, such as limited sampling rate and inability to control the motor torques due to the existing PID structure in the servo, can pose serious problems to the successful implementation of the more sophisticated control schemes. The Jump Impact Controller [11,12!] is derived from stochastic optimal control theory and jump linear system theory with the goal of creating an impadforce controller which is robust to the environment dynamics and collision surface location uncertainties. Preliminary results via simulations have shown that JIC is robust to environment dynamics and collision surface location. The goal of this article is to show that through experimental studies, one cannot only verify the robustness of JIC but also understand the reason behind the robustness in terms of some known concepts such as approach velocity and the control system bandwidth. Also, in the process of experimenting with JIC, some implementation issues can also be explored. A test bed which involves a single degree of freedom manipulator is constructed and experiments are carried out to verify the robustness of JIC. As a result of this exercise, some insight on how the JIC actually works are obtained and found to be explainable in terms of system bandwidth and careful approach velocity selection.

Jump Impact Controller Background The general derivations of JIC can be found in [11,12]. The idea behind JIC is as follows. If we a w m e that feedforward is used to linearize manipulator dynamics, the manipulator dynamics in a collision process can be described by two sets of linear systems, one for the non-contact regime and the other for the contact regime. This can be achieved by, for example, an inverse Jacobian control scheme [ 131 which compares the Cartesian po-

mr ~

71 A

ms

me

An earlier version of this article waspresented at the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, MN, 1996. The authors are with the Department of EE-Systems and Coinputer Science, University of Southern California, Los Angeles, CA 90089-0781. Sukhan Lee is also with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109.

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0272-1708/97/$10.0001997IEEE

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I

I U.3

I

A

*a

‘0

Fig. 2. Probability cuwe of P(a)

sition of the manipulator (calculated by forward kinematics) to the desired position to form an error, 6X, in Cartesian space. This error, which may be presumed small if the control system is doing its job, may be mapped into a small displacement in joint space by means of the inverse Jacobian. The resulting errors in joint space, 6 0 are then to be nulled by a PID controller in joint space. In the mean time, the gravitational and Coriolis forces of the robot are fed forward to linearize the system. Furthermore, after uncoupling the linearized manipulator dynamics in the Cartesian space, the system becomes a SISO, linear system during contact and non-contact regime. It is understood that when linearizing the system, some errors may be introduced into the system in the form of disturbance and system parameter uncertainty. As will be shown in the following, if the statistics of this uncertainty are known so that the mean of the system parameter is obtained, a robust controller to system dynamics and collision surface location uncertainties can be derived. When collision occurs, we say “there is a jump in system dynamics,” meaning that the state space description of the system dynamics changes abruptly from one to another i.e., (AI, B1) to (A2, B;?)or vice versa. Close examination of the manipulator collision dynamics [14,151indicates that the condition for the jumps depends on the states of the system. If there are uncertainties in, say, the distance between the manipulator and the collision surface, force sensor time-delay, andor the parameters of the dynamics, the time of occurrence of jumps becomes uncertain and the jump event can be represented by a stochastic process whose regime transition rates (i.e. the rate of occurrence in a probabilistic sense) is state-dependent. As a result, the manipulator collision dynamics in the presence of uncertainties can be constructed within the framework of a class of stochastic random processes called jump linear systems, whose regime transition rate is state-dependent. References [16,171 are the pioneers of the socalled JLQ problem, which refers to the problem of finding an optimal controller that minimizes the expectation of a linear quadratic loss function under the constraints of a jump linear system with a certain regime transition rate model. Good introductory references to this field can be found in [ 171 and [ 181. In most JLQ problems, the regime transition rate is independent of the states. The application of such problem is mostly in the field of failure mode controller design. References [ 19,201 generalize the JLQ problem to include state-dependent regime transition rate which model the situation when the control action affects the failure rate. References [11,121 applied the JLQ theory of [ 19,201 to manipulator impact control problem by further gener-

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alizing the result of [ 19,201 to include regime uncertainty due to collision surface location uncertainty and sensor delay and incomplete state measurement. Due to the stochastic nature of the problem, one cannot control the system in a deterministic sense, but in a mean square sense. In [20,21], it has been found that to minimize the expectation of a quadratic loss function of a jump linear system whose regime transition rate is state-dependent, one needs to maximize the expectation of the stochastic Hamiltonian of the system. As a result, the governing equations of the admissible optimal control can be obtained. In [ 113 and [ 121, by modifying the stochastic maximum principle to include incomplete state observation and regime uncertainty and assuming that the regime transition rate depends on the states of an observer, suboptimal solution especially for manipulator collision control that is known as jump impact controller (JIC) is derived. It is expected that the JIC is robust to the uncertainties in environment dynamics as well as collision surface location because we are minimizing the expectation of a LQ loss function and simulation results have demonstrated this. Derivation of JIC In the following, we shall describe the actual JIC design using some of the results from [11,12]. Shown in Fig. 1 is a linear model of the robot manipulator and its environment similar to [ 151. The generalized coordinates xr, xs, and Xe are the manipulator position, the sensor position, and the environment position, respectively, measured with respect to a frame of reference whose origin is attached to the expected location of the environment. The generalized coordinates z, A, and xe are the environment surface and environment mass positions measured with respect to a frame of reference whose origin is attached to the actual environment location. Since the actual environment location may be offset by a distance from the assumed position of the environment, we let do be the discrepancy between the frame of references of the manipulator/sensor system and the environment. It is an unknown constant describable by a certain probability distribution. The governing equations of the system dynamics described in Fig. 1 can be described by:

m& = -k,x, - beieif f=a =0

when a > 0 (contact mode) when a 2 0 (non-contact mode)

a = kl(X, + do - x,) + ko(x, + do - z )

(3)

(4)

(5)

In reality, we do not know the exact values of the system parameters and environment location do. Also, we do not have complete state information of the dynamic system such as the environment and sensor velocity. Due to force sensor time-delay, the regime is not known exactly at time of control. Therefore, let 0 be a filtered force sensor signal in the manner

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where T(s) is the filter transfer function, fm is the raw force sensor signal, and v is the threshold being set equal to the expected dynamic force of the sensor in the non-contact regime. Suppose the uncertainty which resulted from feedforward linearization can be estimated and described by some probability distribution. Let and be the approximated dynamic matrices using the mean values of the system parameters and we design an observer that describes system (1) to ( 5 ) in state space form as follows: When 8 I v, regime r = 1 and

E = $+Eu + ~ ,+hL , H ( x - ~ )

(7)

When 8 > v, regime r = 2 and

where x1 = xy,x2 = xs,x3 = xI,x4 = x,, xs = xe,XIj = x,, and x7 = z. In (7) and (8), fd is the desired contact force; Lf and Lc are estimator gain matrices; and matrix H extracts whatever information is available from the measurements such as the position of the manipulator and force measurement from the force sensor, etc. If we augment the system (1) to (5) to the observer (7), (8) and define a

i-

= d new augmented state ;

Fig. 3. Experimental setup showing aluminum beam hitting targei ( x - x) A

,the requirement of imI

pact control which is to reduce transient impact force and to track force command can be achieved by minimizing a quadratic cost functional,

J = E{l:(gQ;+

u’Ru)dt~(80,H.o,fo)}

;=1, Q;=P(a>O) when, & l O n O i v 2, Q;=P(a O n e l v 3, I$;= P ( a > O ) when & I O n 8 > u

(9)

As shown in the appendix, the augmented dynamics have four regimes according to the values of a and 8. Also, due to fd in the observer, as ; -+ 0, the contact force is expected to track fd. In order to obtain an optimal control U* which minimizes the cost functional J, we employ jump linear system theory as described in [ 191.One crucial step is to find the regime probability in terms of soine known information and thus obtain the regime transition rate. Suppose we define an estimated regime indicators same as (5) with 6= 0, & = pi,where is obtained using the mean values of the parameters. It can be shown that, as; -+ x,P(a > 0) and P(a < 0) can be approximated by the profile as shown in Fig. 2 where P(x) means the probability of event x. Intuitively speaking, this means that when the estimated states converge to the true states, the greater the value of &, the more probable a is greater than zero and vice versa. Using this regime transition model, a suboptimal infinite-time solution whose derivations are shown briefly in the appendix can be obtained as follows:

-,

A , vh = 0

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gage circuit. Thus, the beam acts as a 1 DOF manipulator with force sensor attached. The beam is driven by the DC motor to hit a big aluminum block. Also, objects of different surface properties can be mounted on the block to be hit by the beam.

1

I

Servo Controller

L

-

Rate command

Ke

Motor

1

4

-

Fig. 4. Block diagram of PID sew0 and motor

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Frequency (rad/s) Fig. 5. Frequency response of Ge(s). Remarks: (12a-d) are like Riccatti equations except for the last term, which reflects the regime transition statistics. II, can be to calculated off-line. Notice that (13), (14) requires TI and have null spaces. This can be guaranteed by incorporating integral action in the controller and estimator which make the dimensions of TIand 8 by 8. The constant CT reflects the steepness of the regime transition rate, as shown in Fig. 2, and it is a good indication of the uncertainty of the environment location. In order to correctly model the regime transition rate, it is necessary to pick stable estimator gains Lf, Lc so that i? -+x relatively fast in a given regime. If there are a lot of uncertainties, CT is small and this makes $; almost equal to 1/2 at all times, simplifying the controller of (10) to just constant gain feedback control. In fact, CJ affects the approach velocity and Q affects the control bandwidth, especially during contact, just like an LQR problem. So, when CT and Q are small, we expect a slow approach velocity and low bandwidth controller. The bias terms b; and v; are necessary to push the manipulator forward when there is force sensor time-delay or when the collision surface is farther than expected.

Experimental Setup Description of Test Bed In order to test the robustness of the JIC, we constructed a test bed which consists of a DC motor controlled by Galil DMClOOO series motion controller. Attached to the shaft of the DC motor are an encoder and a flexible aluminum beam as shown in Fig. 3. Four strain gages are attached (two gages on each side) to the two sides of the beam and connected together into a full-bridge strain

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System Identification In the previous sections, the JIC consists of an observer (7), (8) and a controller (10). The first step of implementing JIC is to get a rough approximation of the system parameters to construct the observer. First of all, similar to most other controllers, the DMClOOO motion controller is a PID servo. Thus, one can only specify position or rate and the servo, which is a PID controller operating at 1 H z ,will generate the appropriate current to drive the motor. However, in the framework described in previous section, we need to have direct control over the motor torque. Fig. 4 is a block diagram of the system when rate instead of position is commanded. In Fig. 4, output q is the angle of the motor shaft. If we let G(s) = KOs, i.e. let only the rate gain be nonzero, then the overall transfer function Ge(s) from the rate command to the position q will be

where Ki = KfKoT,KdKaKt/J, Kz = KeKoTsKdKaKJJ, and Ts is the sampling period of the servo which is 1 msec. Thus, from (16), Ge is equivalent to a mass damper system with the rate command equivalent to the force acting on a mass. We identify K1 and K2 by the frequency response method. Fig. 5 is a plot of the frequency response. From the frequency response curve, we estimate m, and b, to be 0.0028 kg and 0.154 Nmdrad. respectively. Next, we identify the force sensor dynamics as follows. First, a known mass is used to find the stiffness of the force sensor, k, = 135 “/rad. Then the force sensor is excited by an initial displacement and the frequency and time constant are recorded. Using this information, we estimate the values of m,, b, to be 0.0018 1 “/rad and 0.0002 “shad respectively. Finally, the environment is an aluminum block with known dimensions. By modeling the aluminum block as a cantilever beam, we estimated that me, be, and are 9e-3 kg, 4.62 “shad, and 2.4e5 “/rad representing the first structural mode. Values of bo, ko, kl are arbitrarily chosen to be equals to b, and k, respectively.

JIC Design and Simulations Once we obtained the system parameters, an 8th order observer is constructed. In order to increase the sampling rate, we simplified our observer as much as possible. First, we assume that there are uncertainties in our estimated system parameters and we choose a very small value of (T = le-1 1. Then the controller is simply constant gain and there is no need to evaluate $;. The norm of v; is chosen to be 1. Finally, we use only the measured position xr to update the observer. When designing our control gains, we choose small values of Q because of the high degree of system uncertainty and low sampling rate. In general, if the sampling rate is high enough, increasing Q could give better performance in terms of damping the impact transients if the system model is accurate to a high degree. In our case, due to the hardware limitations, we have to settle with small Q’s. Using these values, we obtained a controller

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with very small gains on the environment states and the integral of &, i.e. the 8th state, in both regimes. Fig. 6 shows the force measurement and control force of a simulation of this controller using previously obtained system parameters. The actual and estimated initial position of the beam are -90" and -67", respectively. The aluminum block is situated at the zero point, i.e. do = 0. The sampling period of the controller is 18 msec. The low pass filter for the force sensor has a time constant of 0.02 sec. and the threshold 2) is set equal to 0.1 N. f d and b; are 4 and 5e5 rads respectively. In the simulation, we assume that there are no uncertainties in the system dynamics. As shown in Fig. 6, the control force during contact is almost constant despite the oscillations of the beam and environment. This is due to the low bandwidth of the controller.

Experimental Results and Discussions JIC Implementation The first problem we encounter during implementation is that the sampling rate is too low if a full 8th order filter is used. In light of the observation of simulation result in the last section, we decided to throw away the environment model and reduce the observer order to four during the non-contact regime and set force command equal to a constant (i.e. fd = 4) during contact in actual implementation. Although this is the extreme case when there are many uncertainties, as will be seen in subsequent sections, we still can gain some insight about how the JIC works. Due to the reduction of observer order, a 55 Hz sampling rate can be achieved. Another problem we encounter in actual experimentation is the unexpected high oscillation of the force sensor. In the actual experiment, we found that the sensor frequency is lower than expected and shows a great deal of vibration in the non-contact regime (notice the vibratory positive and negative force sensor readings during the non-contact regime in Fig. 7). This causes many false alarms. If we only increase V,regime switching will not occur even when contact did occur. This forced us to increase the time constant of the filter T(s) to about 0.4 sec. and we also raised the value of 2) to 0.35. Fig. 7 shows the experimental result whose setting is exactly like that of the simulation in the previous section. Discussion of Nominal Case There are two observations concerning the experimental result of Fig. 7. First, the force tracks the constant command very well. Notice that we did not use the force measurement as we would in an explicit force control scheme. The force measurement merely acts as a regime indicator. During the contact regime, we simply command a constant force in an open loop fashion. Furthermore, no information about the environment dynamics was used. A question remains as to how the JIC maintained such a good force tracking performance. The answer lies in the way we modified the PID servo. In fact, no matter what environment comes in contact with the force sensor, as long as the manipulator is being stopped, the same desired force will be acting on the environment. This is because fd which is actually a rate command is given to the servo and the force is thus internally generated and does not depend on the external environment when the rate is zero. As an illustration to this point, let us go

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1 o r w ....... .j............ masuremen$........... ................... :... ........

.............. .......................................

Ud 0.6

0 Time (sec)

Fig. 6. Simulation results of controller design.

6.

0

-6

0

1.o

2.0

Time (sec)

Fig. 7. Experiment results-nominal case. back to the block diagram of Fig. 4. If the block Kds is brought into the loop, the block diagram will be reduced to that of Fig. 8. In Fig. 8, if contact occurs, even if q is, not zero, v becomes zero at steady state and f will be equal to the constant current which drives the motor to the desired torque. No external information is needed. How do we interpret this result relating to the theory behind JIC? When we design the JIC, we assume that the uncertainties are large and thus o has to be made small. From (1l), (12), this implies a low bandwidth design if Q's are also small. This is why the gains on the environment states and&are small, which means that the controller prefers to control the contact force in an openloop fashion. When there is less uncertainty, the controller will depend on the environment model somewhat. Increasing the magnitude of Q will damp the transient more, if we have a faster sampling rate and better system dynamic model. The second observation is related to the bias terms v and b in (10).The significance of these bias terms is that they compensate for the sensor time-delay and collision surface location uncertainty, and it also helps reduce the impact transient upon regime switching. In Fig. 7, due to an increase in force filter time constant, the regime did not switch until around 0.2 sec after actual contact. During this 0.2 sec. which we shall call the intermediate period, the force is maintained at around 1 N because the control force in regime 1 at the point of impact is about 1 N. This is made possible by the presence of the bias terms, b; and v;. Without them, the control force will be reduced to zero and no regime switching would occur because in regime 1,if b;, v: = 0, the controller will become a regulator driving all the states to zero, and

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once zero is reached the control will be zero as well. If the contact surface has not been reached at this state, the controller will be sitting dumb and do nothing. Having a bias term will drive the manipulator not to zero but somewhere farther than zero. This provides a chance to detect collision even if the collision surface is farther away than expected. Similarly,without b, and v; and if a low-pass filter is added to the force sensor, it will not have time to reach the threshold and signal a regime switch before the control return to zero. In light of this observation, we can appreciate the function of Q, 0,b,, v,, and @;. We believe that b, and v, are to provide the bias needed to compensate for the collision surface location uncertainty (especially when it is farther than expected) and force sensor delay. However, when the collision surface is closer than expected, the extra bias terms could increase the impact transient. That is why (T and 4; are there to regulate the approach velocities directly or through the feedback gains whose bandwidth is related to the magnitude of o and Q. Also, the JIC theory is intended to optimally select the best approach velocity and control bandwidth according to the uncertainties involved.

-I

Conclusion U Motor

1

I

Encoder

Fig. 8. Reduced block diagram of servo and motor Intermediate Period

6

Robustness Verifications To test the robustness of JIC, a series of experiments are conducted. First, we investigate the performance of the JIC when the collision surface locations are varied. Fig. 9 shows experimental results when do = +loo. The initial positions of the beam are different, but their distances from the target are about the same in both cases. The force tracking and force transient damping performances are equally good despite the collision surface location differences. Again, the presence of non-zero b; enables the manipulator to go farther than zero if no collision surface is detected. Otherwise, the position of the beam will tend to reach zero. Next, we investigate the performance of JIC when the environment dynamics are varied. Two objects of very different dynamic characteristics are mounted on the aluminum block. One is a plastic air bubble sheet usually found in mail packaging and the surface is expected to have high damping coefficients. The other object is a soft spring with stiffness equals to about 3 “/rad. and the damping is almost zero. Fig. 10 shows the experimental results. The controller shows good transient damping when the plastic sheet is used as expected, but when applied to the soft spring the transient is very oscillatory, especially during the intermediate period. However, in both cases, the force tracking is good. Notice that our environment model assumed in the controller is quite different from the actual ones. Thus, from these two sets of tests, we have demonstrated the robustness of JIC.

This article describes an experimental study of JIC, particularly, the reasons behind the robustness of JIC claimed in [l 11 and [ 121. A testbed was constructed and described. From the experimental results using the testbed, we have demonstrated the robustness and drawn some insights into how JIC works and the reason behind its robustness. We observed that the JIC theory is actually optimally choosing the control bandwidth and approach velocity according to the uncertainties in environment dynamics, collision surface location and time delay in the force sensor. It is believed that JIC has the capability to dampen impact transient if a fast enough sampling rate is available; this will be studied further in the future. A potential limitation of the JIC approach is the ability to estimate the mean of the system parameter as a result of feedforward linearization particularly for a general multi degree of freedom manipulator. This issue will also be studied further in the future.

0

Appendix Augment system (1) to (7),we have a combined system:

0

-I .a

where r = 1, when a 2 0 n 0 2 0 r = 2, when a > 0 n 0 5 0 r = 3, when a > 0 n 0 > 0 r = 4, when a 2 0 n 0 > 0 0

05

Time (sec)

1

a = [ C C]=C,G

Fig. 9. Performances when collision surface varied.

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According to Fig. 2, the probability that r jumps is expressed

6

as Prob((&( t ) + A&) = jlr(&( t ) ) = i ) =

[

h

z

qV(t)A& + o(A&)

if i # j

1+ qe( t ) A 6 + .(A&)

if i = j

v

g 0

L f (-44)

6 = [ C O]=FwG.

-6

’

I

1

0

(A4) is like the transition rate model of a Markov process r with respect to & where

2

Time (sec)

Fig. IO. Pegormances when environmen:tdynamics varies.

qu = Oand qZ1= -o,qZz= Q , if & I O n 8 I p

(Al) and (A8) are non-linear two point boundary value problem which is difficult to solve. Therefiore, we seek a suboptimal infinite-time solution as follows. The four regimes of (A4) are dealt with in similar ways. For illustration, let us consider just the case: when 6 I 0 n8 Ip. Let

qu = Oand qI1= -0,qlz = 6 ,if & > O n 8 I p qu = 0 and q43= -0,q44= B,if

6 I 0 n8 > p

qij= 0 and q33= -0,q34= B,if & > 0 n8 > p

hl

Let us assume for now that the state $is available. We transform the original problem into u’Ru)dtl&, Go,t o }

U minE{I(’($&+ 10

(A5)

under the constraints (Al) and (A4). The optimal solution to (A5) can be obtained by the stochastic minimum principle [12] and [19-211 E{H(ty

3‘

9 ‘

’ ‘)IG}

E{H(ty

$3

r7

;*>IG}

(A6)

=v

(-49)

where V is a constant vector and

Also, letQl = 0, then equation for h11of (A8) is satisfied. Thus, equation for hz of (A8) becomes

h, = -F,

I

t

h , - GzG-OC, rF$

Suppose h~takes this form,

h , = PG+ b

where

H( t ,w , r, U ) = A’[ FyG+ G,U( w , t ,r ) ] + I?&+ E’RU

(All)

(AW

As a result, if we substitute (A12) into (A1 l), we obtain the following:

Applying the stochastic minimum principle (A6), we have -

U*

1 - - 4

= - X R ‘ G ’ z h p ( r = il&) ,=I

L

2

2

where

(A8)

hi@)

E = E{FIr

= i},

Fa

=0 = E{Glr = i}

i, j = { 1,2,3,4]

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Since @a and @b are functions of &, which is a function of $, (A13) will be difficult to solve. However, if we assume that Q is small, both +a and +b will be closed to constants and $, respectively and there exist an infinite time solution such thatp = 6 = 0. Then, an infinite-time solutionfor both Pand b could be obtained

6’b - - P21G R

1 ‘G’b@,--PGR ‘G‘V@,= 0 2

(A151

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Furthermore, we want U * to depend only on (; t ) ;we can enforce the structure of P to be of the following form:

p=[:

:]

(A16).

b = [ j Also notice that (A10) is a constraint that must be satisfied. Partition and expand (AlO), with

I;[=. we have

If we let V I = Vz = Va, then (A19) and (AZO) are satisfied and -,

A, v, = 0 Substituting (A12) and (A18) using P and b from (A16) and (A17) into (A7) and (A14), the optimal control equations (10) and (12a) are obtained. Also, using (A15), we can find b. Notice that (A14) alsoimpose constraints onestimator gains LfandLc.

References [l] K. Kitagaki and M. Uchiyama, “Optimal Approach Velocity of EndEffector to the Environment,” Proc. IEEE Int. Con$ Robotics & Automation, 1992, pp. 1928-1934. [2] K. Yousef-Toumi and D.A. Gutz, “Impact and Force Control,” Proc. IEEEInt. Con$ Robotics &Automation, 1989, pp. 410-416. [3] I. Walker. “Impact Configurations and Measures for Kinematically Redundant and Multiple Armed Robot Systems,” IEEE Trans. Robotics and Automation, pp. 670-683, Oct. 1994. [4] S . Yoshikawa and K. Yamada, “Impact Estimation of a Space Robot at Capturing a Target,” Proc. IEEE Intel: Con$ on Intelligent Robots and Systems, 1994, Munich, pp. 1570-1577. [5] C. An, and J. Hollerbach. “Dynamic Stability Issues in Force Control of Manipulators,” Proc. IEEE Int. Con$ Robotics & Autonation, 1987, pp. 890-896. [6] S. Eppinger, and W. Seering,“UnderstandmgBandwidth Limitations on Robot Forcecontrol,” Proc. IEEEInt. Con$Robotics &Automation.1987.pp. 904-909. [7] R.R.Y.Zhen, and A.A.Goldenberg, “Robust Position and Force Control of Robotics Using Sliding Mode,” Proc. IEEE Oat. Con$ Robotics & Automation,1994, pp. 623-628. [8] G.T.Marth, T.J.Tam, and A.K.Bejczy, “Stable Phase Transition Control forRobot ArmMotion.” Proc. IEEElnt. Con$ Robotics &Automation, 1993, pp. 355-362. [9] J.K.Mills and D.M. Lokhorst, “Control of Robotic Manipulators During General Task Execution: A Discontinuous Control Approach,” Oat. J. Robotics Res., vol. 12, pp. 146-163, 1993.

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David K. Chiu receivedhis B.S. andM.S. degrees in mechanical engineering from the University of California, Los Angeles, in 1983 and 1985.He was with W.J. Schafer Associates Inc. from 1985 to 1987 and Hughes Aircraft Company Space and CommunicationGroup from 1987to 1993,working on thermal and optical control of chemical lasers and spacecraft attitude control. Since 1993, he has been working on his Ph.D. in electrical engineering at the University of Southem Califomia and currently is a Ph.D. candidate. His research interests are in the area of manipulatorforce control and stochasticoptimal control theory. In the summer of 1996,he was a Rand Corporation summer intem working on Global Positioning Systems. Sukhan Lee received his Ph.D. degree in electrical engineering from Purdue University, West Lafayette, IN, in 1982 and his B.S. and M.S. degrees in electrical engineering from Seoul National University, Seoul, Korea. He is currently a senior member of the technical staff at the Jet Propulsion Laboratory, Califomia Institute of Technology, and an adjunct professor of electrical engineering and computer science at the University of Southem Califomia. Lee has been active in research in the areas of robotics and automation, computer integrated manufacturing, neural networks, and intelligent systems. His research activities range from dextrous manipulation and control, advanced. madmachine systems. assembly planning, sensor fusion and planning, skill leaming, and neural networks for robotics. control, pattern recognition. and combinatorial optimization. Lee is the author and coauthor of 200 technical papers published in the scientificjournals and in the proceedings of major conferences. He has also edited a book on assembly planning. He has been awarded several patents and a dozen certificates of recognition and awards from NASA for his technical innovations. Lee has been an associate editor for the IEEE Transactions on Robotics and Autoniation, the International Journal ofAl Tools,and thelnteniational Journal oflntelligent Systems and Control. He is currently chairing the IEEE Robotics and Automation Society Technical Committee on Neural Networks and Fuzzy Systems as well as on Assembly and Task Planning.

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