ferent singularity force feedback methods are defined ... Experimental results with a force feedback .... so that it always resists motion into the singularity,.
Haptic Feedback of Kinematic Conditioning for Telerobotic Applications Thavida Maneewarn and Blake Hannaford Department of Electrical Engineering University of Washington Seattle, WA 98195-2500 http://rcs.ee.washington.edu/BRL
Abstract Kinematic conditioning of robot manipulators is the problem where small motions in Cartesian space cause excessive joint velocities. This problem is signi cant in teleoperation. Haptic feedback provides the bi-directional ow of information which allows the operator to control the telerobot interactively. Haptic feedback of kinematic conditioning is proposed as a new approach to achieve better performance in telerobotic control near kinematic singularities. Three different singularity force feedback methods are de ned and studied. Experimental results with a force feedback master and simulated slave system show that teleoperation performance near singular con gurations was a ected and improved by using singularity force feedback.
1 Introduction Kinematic singularities are one of the most significant problems for robot control in Cartesian space especially in teleoperated systems. The relationship between Cartesian-space velocity and joint-space velocity, is speci ed by the Jacobian matrix (Eq. 1). q_ = J ,1 v
(1) Joint-space velocities are exceptionally high when the Jacobian matrix is singular or nearly singular even though the desired Cartesian-space velocity is small. In telerobotic application, when the slave robot is at or near a singularity, the operator may keep sending commands without any knowledge that the commands can not be attained by the slave robot. A kinematic singularity is hard to recognize visually while it constitutes a signi cant portion of the workspace for the
actual slave arms used in space telerobotics, such as Shuttle Remote Manipulator System [5] At a singular con guration where the manipulator loses one or more degrees of freedom, there are directions along (or around) which it cannot move or exert forces (or torques). Singularities occur either at extreme points of the workspace or when two or more joints move into dependent con gurations [4] There are several methods to deal with singularities such as the the damped least-square inverse [6] or the singular-robust inverse [2], on-line trajectory time scaling [3], adjoint Jacobian or null-space based approach. [5] The concept of the damped least-square inverse is to minimize the error of the joint velocities solution and the magnitude of joint velocities.The joint velocities are given as the solution of the normal equation
where
(J T J + 2 I )q_ = J T v q_ = J y v
J y = (J T J + 2 I ),1 J T = J T (JJ T + 2 I ),1
(2) (3) (4)
J y is the damped pseudoinverse Jacobian. The damping factor is a scalar representing the relationship
between joint velocity and velocity error. The larger the value of , the larger will be the position error between the commanded and actual Cartesian trajectory. However also has to be large enough to keep the system stable. The damped least-square approach for eliminating numeric problems at singularities may cause an error in Cartesian trajectory because a new trajectory generated by this approach deviates from the original path in order the keep up with command velocity. Therefore the singularity is never reached by using this method, and some portions of workspace are always
excluded. In Master-Slave application, the tracking error between the master and slave manipulator is created since the operator is not interactively informed about the deviation.The kinematic information could be useful for the operator to slow down or move in an appropriate direction near singularities so that the trajectory error can be minimized. This paper proposes to use haptic feedback to give the operator useful information about the kinematic performance of the slave robot. We are focusing on using haptic feedback to deal with kinematic singularity problem in telerobotics applications.
2 Concept of Singularity Force Feedback Our approach to apply a haptic feedback to the singularity problem is de ned as follows: 1) The operator will feel a force that pushes the end-e ector away from singularity only when the manipulator moves toward a singularity. 2) Singularity force feedback will occur only in the neighborhood of a singularity. 3) The magnitude of force will increase when the manipulater gets closer to a singular con guration. From the concept of singularity force feedback de ned above, we proposed three alternative modes of singularity force feedback as follows:
Method 1. Force feedback based on dampednull subspace of Jacobian matrix
In the neighborhood of singularity, the operator reaction towards singularity force feedback might cause the same kind of deviation as in damped least-square inverse. We can derive the singular force feedback from the trajectory deviation in damped least-square inverse. The derivations below show only a case when the Jacobian matrix is rank-de cient by one for simplicity. However cases where the rank-de ciency is greater than one can also be derived in the same fashion. We rst de ne f~sff 1 = B (~vc , ~vd )
(5) where f~sff 1 is the force feedback vector, ~vc is command velocity vector, and ~vd is the damped leastsquare velocity vector. f~sff 1 = B (~vc , JJ y~vc ) (6) f~sff 1 = B [I , JJ y ]~vc
(7) where J y is the damped least-square inverse Jacobian. It can be shown that
2 I , JJ y = 2 u uT m + 2 m m
(8)
where m is the minimum singular value of J. J is m-dimension and is rank-de cient by only one. is a damping factor. um is the mth column vector of U that corresponds to the minimum singular value which represents a direction corresponding to the damped-null 2 subspace. m2 +2 represents a magnitude of damping force which becomes larger when the minimum singular value is smaller. Damping factor () de nes the neighbourhood of singularity. also indirectly speci es the distance from singular con guration at which these methods take e ect. [I , JJ y ] can be either the same or opposite to command velocity. We constrain the force feedback so that it always resists motion into the singularity, therefore we use the dot product between f~ and ~vc to adjust the sign of force feedback. We use the sign of the instanteneous derivative of the minimum singular value dtd m to determine whether the command velocity moves the manipulator into or out of singular con gurations.
Method 2. Force feedback modeled as spring and damper force. The direction of spring force is based on damped-null subspace of Jacobian matrix In order to increase the sensation of sti ness in the singularity force feedback, the spring force is added.
f~sff 2 = f~d + f~s (9) when f~d is damper force and f~s is spring force. The
damper force is the same as method 1 (Eq. 7). Kinematic singularity is not explicitly represented as a position in Cartesian space. The distance used for the spring force computation is derived relative to the point (boundary) at which the minimum singular value drops below a threshold (bnd ). The direction of both spring and damper force comes from the dampednull subspace of Jacobian matrix. f~k =
(
~ K (bnd , m ) jff~d j if m < bnd
0
d
otherwise
(10)
Method 3. Force feedback modeled as spring and damper force. The direction of spring force is based on the direction of velocity command
The general idea is the same as in the earlier method, however the spring force in this case directly opposes the command velocity. The direction of the
spring force vector is de ned as the negative direction of the command velocity vector. f~k =
,K (bnd , m ) j~~vvcc j if m < bnd (11) 0 otherwise
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Slave simulation
~vc is command velocity vector.The singularity force feedback (f~sff 3 )is de ned by Eq. 9 and 11.
3 Teleoperation Experiment We performed an experiment to study the e ects of di erent singularity force feedback methods on teleoperation performance ( gure 2). The three singularity force feedback methods and the case without haptic feedback were used. The experiment was implemented using the High Bandwidth Force Display at the Biorobotics Laboratory, University of Washington [1], a 2 DOF Cartesian mechanism driven by brushless DC motors through steel cable transmission. It is controlled by a 266 Mhz Pentium II processer PC with an update rate of 1000 Hz. A high position resolution (0.15mm), and a high sampling rate/low latency control law enables the simulation to be very realistic. The slave manipulator is a software simulation of a 2 DOF 2R arm in which both joints can rotate up to 360 degrees. Link lengths of the simulation arm are 125 and 37.5 mm. A boundary singularity occurs when joint 2 is 0 or 180 degrees and it is the only type of singularity that exists for this mechanism. In this teleoperation experiment, the open-loop control with no force sensor and no visual feedback of the slave system is used in order to emphasize the e ect of haptic feedback. Four tasks were de ned for evaluation of teleoperation system performance as shown in gure 3. These tasks pass through the inner boundary singularity of slave mechanism. The task trajectory, start and end point are shown on the Master display unit during operation. The operator commands the manipulator to move along the task trajectory from start point to end point, in both left to right and right to left directions. In order to evaluate teleoperation performance, two performance measures were used: 1) RMS position tracking error of the slave robot. 2) variance of joint velocity. Better performance is indicated by lower values of these performance measures. Five operators, 3 males and 2 females, performed the teleoperation experiment. Each operator was given a standardized training session including information on the system and test objectives and was required to perform combinations of all tasks with di erent singularity force
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Figure 1: The diagram of teleoperation experiment system
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Figure 2: Four teloperation trajectories with workspace boundary of slave manipulator
Figure 3 demonstrates X-Y position of master and slave in task 2 operation. Without haptic feedback (method 4) ,when the large deviation occured at the slave manipulator near singularity, the operator kept moving along the desired task trajectory without keeping track of the deviation. The changes in command velocity are more distinct in method 1, 2 and 3 than method 4. When singularity force feedback was used, the magnitude of command velocity decreased as the slave manipulator approached singular con guration. Then the velocity increased when the operator tried to move away from singularity and slowed down when the operator attempted to correct back to the original path. When haptic feedback is not given, the changes of command velocity are comparatively small. Figure 5 illustrates typical joint velocities of the slave manipulator. Velocity in range of 5 to 10 rad/sec are present for methods 1 to 4. When no haptic feedback is given (method 4), the magnitude of joint velocities becomes larger when the slave manipulator is near singular con guration. The analysis of variance (ANOVA) was used to analyze the performance measures; p < 0:05 was considered statistically signi cant. The e ect of method on the RMS position tracking error is statistically signi cant(p < 0:001). Plot of RMS position tracking error ( gure 6) indicates that when no haptic feedback is given (method 4) the maximum position error occured. Among these 3 singularity force feedback methods, method 1 has the lowest position tracking error for tasks 1,2 and 3. In tasks 4, where a large portion the trajectory is unreachable, method 3 rates the lowest ( gure 6). Variance of joint velocity indicates the changes in joint velocity occurring throughout the trajectory. The e ect of method on variance of joint velocity is
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Figure 3: Master (dashed line) and Slave (solid line) x-y position. One repetition of task 2. When force feedback is absent, operator cannot correct for errors induced by damped least square inverse. statistically signi cant (p < 0:001). Method 3 has the lowest value in task 1 while method 2 has the lowest value in task 3,4 and method 1 has the lowest value in task 2. Method 4 (no haptic feedback is given) has the highest variance of joint velocity in task 2, 3 and 4.
3.2 Discussion Experimental results shows that singularity force feedback contributes to an improvement in position tracking between master and slave, especially when a portion of the trajectory goes through a singularity (task 2 and 4, gure 2, 3 and 6). With singularity force feedback, the operator command is constrained by force feedback derived from the kinematic state of the slave manipulator. Even using a numerically robust inverse, without haptic feedback, the operator cannot keep track of the deviation at the slave manipulator near singularity. Method 1 has better position tracking performance than other methods. Completion time was not signi cantly changed by any of the methods. With singularity force feedback, the maximum and
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variance of joint velocity is smaller compare to without haptic feedback case (method 4). A possible explanation is that the operator reduces command velocity when he/she feel the singularity force feedback, therefore the maximum and variance of joint velocity becomes smaller.
4 Summary and Conclusions The purpose of this study is to determine whether or not haptic feedback can provide useful information on the kinematic state of the slave manipulator in telerobotic applications. The singularity force feedback is a suggestive force that resists the operator motion in the direction of a singular con guration. The magnitude of singularity force feedback depends on the distance from singularity and the magnitude of command velocity from the operator. Three alternative methods of singularity force feedback were proposed implemented and experimentally evaluated. The results indicate that the position tracking performance near singularities was improved by using singularity force feedback. Method 1 showed better position tracking accuracy than other methods while method 3 demonstrated less variance in joint velocity and maximum joint velocity than other methods. Our study has some additional limitations which must be addressed before implementation in applications. First, we used only a two degree of freedom slave mechanism in which only a boundary singularity exists. In real applications, kinematic singularity can be much more complex. Hence, we may have to modify the singular force feedback method to be able to deal with the kinematics of a complicated system. Second, future research needs to address the e ect of singularity force feedback when visual feedback from slave manipulator is given. Finally, we need to study the e ects of singularity force feedback in contact situations where it may be superimposed with other types of force feedback information.
Acknowledgments This work was supported by Ministry of Science, Technology and Environment, Goverment of Thailand, and by Boeing Defense and Space Group. Computer equipment are donated by Intel Corp. The author would like to thank colleagues in Biorobotics Laboratory at University of Washington and all subjects who participated in the experiment.
References [1] Moreyra M.,"Design of a Planar High Bandwidth Force Display with Force Sensing," MSEE Thesis, University of Washington, Department of Electrical Engineering, August, 1996. [2] Nakamura Y.and Hanafusa H, "Inverse Kinematic Solutions with Singularity Robustness for Robot Manipulator Control,"ASME J.Dynam.Sys.Measurement and Control,vol.108,pp.163-171,1986. [3] Ola Dahl and Lars Nielsen, "Torque-Limits Path Following by On-line Trajectory Time Scaling,"IEEE Transactions on Robotics and Automation, Vol. 6, No.5 pp.554-561, October 1990. [4] Tourassis V.,Marcelo A.,"Identi cation and Analysis of Robot Manipulator Singularities," The International Journal of Robotics Research, vol.11, no.3,pp.248-259,1992. [5] Tsumaki Y.,Nenchev D.N.,Kotera S.,and Uchiyama M.,"Teleoperation Based on the Adjoint Jacobian Approach," IEEE Control System Magazine, vol.7,no.1,pp.53-61,1997. [6] WamplerII C.W.,"Manipulator Inverse Kinematic Solutions Based on Vector Formulations and Damped Least-Square Methods,"IEEE Trans.on System, Man and Cyb. vol.SMC-16,no.1,pp.93101, 1986.
