Human-Centered Robotics and Interactive Haptic Simulation

O. Khatib, O. Brock, K.C. Chang, D. Ruspini, L. Sentis, S. VijiRobotics Laboratory

Department of Computer ScienceStanford University, Stanford, California 94305

email:�khatib, oli, kcchang, ruspini, lsentis, viji � @robotics.stanford.edu

Abstract

A new field of robotics is emerging. Robots are todaymoving towards applications beyond the structured en-vironment of a manufacturing plant. They are mak-ing their way into the everyday world that people in-habit. The paper focuses on models, strategies, andalgorithms associated with the autonomous behaviorsneeded for robots to work, assist, and cooperate withhumans. In addition to the new capabilities they bringto the physical robot, these models and algorithms andmore generally the body of developments in robotics ishaving a significant impact on the virtual world. Hap-tic interaction with an accurate dynamic simulationprovides unique insights into the real-world behaviorsof physical systems. The potential applications of thisemerging technology include virtual prototyping, ani-mation, surgery, robotics, cooperative design, and ed-ucation among many others. Haptics is one area wherethe computational requirement associated with the res-olution in real-time of the dynamics and contact forcesof the virtual environment is particularly challenging.The paper describes various methodologies and algo-rithms that address the computational challenges asso-ciated with interactive simulations involving multiplecontacts and impacts between human-like structures.

1. Introduction

The successful introduction of robotics into human en-vironments will rely on the development of compe-tent and practical systems that are dependable, safe,and easy to use. To work, cooperate, assist, andinteract with humans, the new generation of robotmust have mechanical structures that accommodatethe interaction with the human and adequately fit inhis unstructured and sizable environment. Human-compatible robotic structures must integrate mobility(legged or wheeled) and manipulation (preferably bi-manual), while providing the needed access to percep-

tion and monitoring (head camera) (Hirai et al. 1998;Takanishi et al. 1998; Khatib et al. 1999; Asfouret al. 1999; Nishiwaki et al. 2000). These require-ments imply robots with branching structures - tree-like topology involving much larger numbers of de-grees of freedom than those usually found in conven-tional industrial robots. The substantial increase in thedimensions of the corresponding configuration spacesof these robots renders the set of fundamental problemsassociated with their modeling, programming, plan-ning, and control much more challenging.

The first of these challenges is the whole-robot mod-eling, motion coordination, and dynamic control. Forrobots with human-like structures, tasks are not limitedto the specification of the position and orientation of asingle effector. For these robots, task descriptions mayinvolve combinations of coordinates associated withone or both arms, the head-camera, and/or the torsoamong others. The remaining freedom of motion is as-signed to various criteria related to the robot postureand its internal and environmental constraints.

There is a large body of work devoted to the study ofmotion coordination in the context of kinematic redun-dancy. In recent years, algorithms developed for redun-dant manipulators have been extended to mobile ma-nipulation robots (Cameron et al. 1993; Umetani andYoshida 1989; Ullman and Cannon 1989; Papadopou-los and Dubowsky 1991). Typical approaches to mo-tion coordination of redundant systems rely on the useof pseudo or generalized inverses to solve an under-constrained or degenerate system of linear equations,while optimizing some given criterion. These algo-rithms are essentially driven by kinematic considera-tions and the dynamic interaction between the end ef-fector and the robot’s self motions are ignored.

Our effort in this area has resulted in a task-orientedframework for whole-robot dynamic coordination andcontrol (Khatib et al. 1996). The dynamic coordi-nation strategy we developed is based on two modelsconcerned with the task dynamics (Khatib 1987a) and

Figure 1: Manipulation and Posture Behaviors: a sequence of three snapshots from the dynamic simulation of a 24-degree-of-freedom humanoid system, whose task is generated from simple manipulation and posture behaviors.

the robot posture behavior. The task dynamic behav-ior model is obtained by a projection of the robot dy-namics into the space associated with the task, whilethe posture behavior is characterized by the comple-ment of this projection. To control these two behav-iors, a consistent control structure is required. The pa-per discusses these models and presents a unique con-trol structure that guarantees dynamic consistency anddecoupled posture control (Khatib 1995), while pro-viding optimal responsiveness for the task. The paperalso presents recently developed recursive algorithmswhich efficiently address the computational challengesassociated with branching mechanisms. Dynamic sim-ulation of virtual environments is another importantarea of applications of these algorithms. The paperalso discusses our ongoing effort for the developmentof a general framework for interactive haptic simula-tion that addresses the problem of contact resolution.

A robotic system must be capable of sufficient levelof competence to avoid obstacles during motion. Evenwhen a path is provided by a human or other intelligentplanner, sensor uncertainties and unexpected obstaclescan make the motion impossible to complete. Our re-search on the artificial potential field method (Khatib1986) has addressed this problem at the control level toprovide efficient real-time collision avoidance. Due totheir local nature, however, reactive methods (Khatib1986; Krogh 1984; Arkin 1987; Latombe 1991) arelimited in their ability to deal with complex environ-ments. Using navigation functions (Koditschek 1987)the problems arising from the locality of the potentialfield approach can be overcome. These approaches,however, do not extend well to robots with many de-grees of freedom, such as mobile manipulators (Car-riker, Khosla, and Krogh 1989; Seraji 1993; Ya-

mamoto and Yun 1995). Our investigation of a frame-work to integrate real-time collision avoidance capa-bilities with a global collision-free path has resulted inthe elastic band approach (Quinlan and Khatib 1993),which combines the benefits of global planning and re-active systems in the execution of motion tasks. Theconcept of elastic bands was also extended to nonholo-nomic robots (Khatib et al. 1997). The paper discussesour ongoing work in this area and presents extensionsto the elastic strip approach (Brock and Khatib 1997),which enable real-time obstacle avoidance in a task-consistent manner. Task behavior can be suspendedand resumed in response to changes in the environmentto ensure collision avoidance under all circumstances.

2. Whole-Robot Control: Task and Pos-ture

Human-like structures share many of the characteris-tics of macro/mini structures (Khatib 1995): coarseand slow dynamic responses of the mobility system(the macro mechanism), and the relatively fast re-sponses and higher accuracy of the arms (the mini de-vice). Inspired by these properties of macro/mini struc-tures, we have developed a framework for the coor-dination and control of robots with human-like struc-tures. This framework provides a unique control struc-ture for decoupled manipulation and posture control,while achieving optimal responsiveness for the task.This control structure is based on two models con-cerned with the task dynamic behavior and the robotposture behavior. The task behavior model is obtainedby a projection of the robot dynamics into the spaceassociated with the effector task, and the posture be-

havior model is characterized by the complement ofthis projection. We first present the basic models asso-ciated with the task. In a subsequent section we presentthe whole-robot coordination strategy and posture con-trol behavior.

2.1. Task Dynamic Behavior

The joint space dynamics of a manipulator are de-scribed by ������������ �����������������������

(1)

where�

is the � joint coordinates,�������

is the ����� ki-netic energy matrix,

� ����������is the vector of centrifugal

and Coriolis joint forces,�������

is the vector of gravity,and�

is the vector of generalized joint forces.

The operational space formulation (Khatib 1987b)provides an effective framework for dynamic mod-eling and control of branching mechanisms (Rus-sakow, Khatib, and Rock 1995), with multiple oper-ational points. The generalized torque/force relation-ship (Khatib 1987b; Khatib 1995) provides the decom-position of the total torque,

�(equation 1) into two

dynamically decoupled command torque vectors: thetorque corresponding to the task behavior commandvector and the torque that only affects posture behaviorin the null space:� �!�#"�$&%(')*�,+.-/%0"214365

(2)

For a robot with a branching structure of 7 effectors oroperational points, the task is represented by the 8�79�):vector, ; , and the 8<7���� Jacobian matrix is = ����� . ThisJacobian matrix is formed by vertically concatenatingthe 7>8?�@� Jacobian associated with the 7 effectors.

The task dynamic behavior is described by the opera-tional space equations of motion (Khatib 1995)AB� ; � �; �C#� ; ���; ���D � ; ���FE (3)

where ; , is the vector of the 8<7 operational coordi-nates describing the position and orientation of the 7effectors,

AB� ; � is the 8�7G� 8�7 kinetic energy ma-trix associated with the operational space.

C#� ; ���; � ,D � ; � , andE

are respectively the centrifugal and Cori-olis force vector, gravity force vector, and generalizedforce vector acting in operational space.

The joint torque corresponding to the task commandvector

E, acting in the operational space is

�,H�IKJ2LM� =�N �����(E (4)

The task dynamic decoupling and control is achievedusing the control structureEOH�I/J2LM�QPAB� ; �RE ST,U H�V UKWX PC#� ; ���; �� PD � ; � (5)

where,E SH�IKJ2L

represents the inputs to the decoupledsystem, and P Y represents estimates of the model param-eters.

2.2. Posture Behavior

An important consideration in the development of pos-ture behaviors is the interactions between the postureand the task. It is critical for the task to maintain itsresponsiveness and to be dynamically decoupled fromthe posture behavior. The posture can then be treatedseparately from the task, allowing intuitive task andposture specifications and effective whole-robot con-trol. The overall control structure for task and postureis

���F�#H�I/J�L���#Z U J[H�\&]�^(6)

where

�#Z U J[H�\.]2^ �!_ N �����(��`&^aJ2V ]2^a`<bcZ U J2H�\&]2^ (7)

with _*�������edgfMh = ����� = �����0i (8)

where = ����� is the dynamically consistent generalizedinverse (Khatib 1995), which minimizes the robot ki-netic energy

= �������F�bkj ����� =�N �����RAX����� (9)

and AB�����#�ml = �����R�b�j ����� = N �����0n

bkj(10)

This relationship provides a decomposition of jointforces into two control vectors: joint forces corre-sponding to forces acting at the task, = N E , and jointforces that only affect the robot posture,

_ N � Z U J2H�\&]2^ .For a given task this control structure produces jointmotions that minimize the robot’s instantaneous ki-netic energy. As a result, a task will be carried outby the combined action of the set of joints that reflectthe smallest effective inertial properties.

To control the robot for a desired posture, the vector� `.^aJ2V ]2^a`�b�Z U J[H�\.]2^will be selected as the gradient of a

potential function constructed to meet the desired pos-ture specifications. The interference of this gradientwith the task dynamics is avoided by projecting it intothe dynamically consistent null space of = N ����� , i.e._ N �����(��`&^aJ2V ]2^a`<bcZ U J2H�\&]2^ .Dynamic consistency is the essential property for thetask behavior to maintain its responsiveness and tobe dynamically decoupled from the posture behaviorsince it guarantees not to produce any coupling accel-eration in the operational space given any oqp 1qrsr . In

Figure 2: Dynamic Consistency and Posture Behaviors: a sequence of snapshots from the dynamic simulation of a 24-degree-of-freedom humanoid system. On the left, the task is to maintain a constant position for the two hands, while achieving hand-eye coordination. The posture motion has no effect on the task. On the right, the task also involves hand-eye coordinationand motion of the common hand position. This position is interactively driven by the user. The posture is to maintain therobot total center-of-mass along the � -axis.

figure 2 (left), the robot (a 24-degree-of-freedom hu-manoid system) was commanded to keep the positionof both hands constant (task behavior) while movingits left and right in the null space (posture behavior).Notice that dynamic consistency enables task behaviorand posture behavior to be specified independently ofeach other, providing an intuitive control of complexsystems.

For instance, the robot posture can be controlled tomaintain the robot total center-of-mass aligned alongthe ��� axis of the reference frame. This posture can besimply implemented with a posture energy function

� Z U J[H�\&]�^6bc^ W ^a]���� ������� :� ���� � U�� ��� � U�� �(11)

where is a constant gain,� � U��

, and� � U��

are the�

and�

coordinates of the center of mass. The gradientof this function� `.^aJ2V ]2^a`�b�Z U J[H�\&]2^ � = N� U�� �(h�� � Z U J[H�\.]2^6bc^ W ^a]���� �

(12)

provides the required attraction to the � axis of therobot center of mass. This is illustrated in the simu-lation shown in Figure 2 (right), whose task involvedthree operational point associated with the arms andhead.

Collision avoidance can be also integrated in the pos-ture control as discussed in section 4.. With this pos-ture behavior, the explicit specification of the associ-ated motions is avoided, since desired behaviors aresimply encoded into specialized potential functions forvarious types of operations.

More complex posture behaviors can be obtained bycombining various posture energies. We are currentlyexploring the generation of human-like natural mo-tion from motion capture of human and the extractionof motion characteristics using human biomechanicalmodels.

2.3. Cooperative Manipulation

The development of effective cooperation strategiesfor multiple robots platforms is an important issuefor both the operations in human environments andthe interaction with humans. Human guided mo-tions may involve tightly constrained cooperation per-formed through compliant motion actions or less re-stricted tasks executed through simpler free-space mo-tion commands. Several cooperative robots, for in-stance, may support a load while being guided bythe human to an attachment, or visually following theguide to a destination.

Our approach is based on the integration of two ba-sic concepts: The augmented object (Khtaib 1988) andthe virtual linkage (Williams and Khatib 1993). Thevirtual linkage characterizes internal forces, while theaugmented object describes the system’s closed-chaindynamics. For systems of a mobile nature, a decentral-ized control structure is needed to address the difficultyof achieving high-rate communication between plat-forms. In the decentralized control structure, the objectlevel specifications of the task are transformed into in-dividual tasks for each of the cooperative robots. Lo-

Figure 3: Cooperative Manipulation with the Stanford Robotic Platforms

cal feedback control loops are then developed at eachgrasp point. The task transformation and the design ofthe local controllers are accomplished in consistencywith the augmented object and virtual linkage models(Khtaib 1988; Williams and Khatib 1993). This ap-proach has been successfully implemented on the Stan-ford robotic platforms (see Figure 3) for cooperativemanipulation and human-guided motions.

2.4. Efficient Operational Space Algorithms

Early work on efficient operational space dynamic al-gorithms has focused on open-chain robotic mecha-nisms. An efficient O(n) recursive algorithm was de-veloped using the spatial operator algebra (Rodriguez,Kreutz, and Jain 1989; Kreutz-Delgado, Jain, and Ro-driguez 1991) and the articulated-body inertias (Feath-erstone 1987). A different approach that avoided theextra computation of articulated inertias also resultedin an O(n) recursive algorithm for the operational spacedynamics (Lilly 1992; Lilly and Orin 1993) Build-ing on these early developments, our effort was aimedat algorithms for robotic mechanisms with branchingstructures that also address the issue of redundancy anddynamics in the null space.

The most computationally expensive element in theoperational space whole-body control structure (equa-tion 6) is the posture control, which involves the ex-plicit inversion operation of the � � � joint space in-

ertia matrix�

of (equation 9), which requires� � ��� � .

We have developed a computationally more efficientoperational space control structure that eliminates theexplicit computation of the joint space inertia matrixand its inverse. This elimination was achieved by com-bining the dynamically consistent null space controland the operational space control in a computationallymore efficient dynamic control structure.

Using this control structure, we have developed a re-cursive algorithm for computing the operational spacedynamics of an � -joint branching redundant articulatedrobotic mechanism with 7 operational points (Changand Khatib 2000). The computational complexity ofthis algorithm is

� � �k7 7 � � , while existing sym-bolic methods require

� � � � 7 � � . Since 7 can beconsidered as a small constant in practice, this algo-rithm attains a linear time

� � � � as the number of linksincreases. This work was extended for the dynamics ofclosed-chain branching mechanisms with an efficient� � �k7 7 � � algorithm (Chang, Holmberg, and Khatib2000).

3. Interactive Haptic Simulation

Beyond their immediate application to physical robots,these efficient dynamic algorithms are making a sig-nificant impact on the simulation and interaction withthe virtual world. The computational requirements as-sociated with the haptic interaction with complex dy-

Figure 4: A frame of an animation showing the dynamic interaction of multiple articulated/rigid bodies (left). A similarsequence in which direct haptic interaction is permitted between the user and the objects in the environment (right).

namic environments are quite challenging. In addi-tion to the need for real-time free-motion simulationof multi-body systems, contact and impact resolutionand constrained motion simulation are also needed.

Building on the operational space formulation, wedeveloped a general framework (Ruspini and Khatib1999) for the resolution of multi-contact between artic-ulated multi-body systems. A contact point is treatedas an operational point and a contact space is defined.Similarly to the operational space inertia matrix, a con-tact space inertia matrix

Ais introduced to provide the

effective masses seen at all the contact points and tocharacterize the dynamic relationships between them.Computing the contact space inertia matrices

Afor a

number of 7 contact point on a branching mechanismis achieved with an efficient

� � �k7 7 � � recursivealgorithm.

The contact space representation allows the interactionbetween groups of dynamic systems to be describedeasily without having to examine the complex equa-tions of motion of each individual system. As such, acollision model can be developed with the same ease asif one was considering interaction only between simplebodies. Impact and contact forces between interactingbodies can then be efficiently solved to prevent pene-tration between all the objects in the environment.

This framework was integrated with our haptic ren-dering system (Ruspini, Kolarov, and Khatib 1997) toprovide a general environment for interactive hapticdynamic simulation. Figure 4 illustrates some of thevirtual environments that have been modeled with thissystem. Figure 4(left) is one frame from an anima-

tion consisting of two puma560 manipulators (6 d.o.f.each) on which a rain of large blocks is allowed to fall.A total of 366 d.o.f. are modeled.

In Figure 4(right) a similar environment where twopuma560 manipulators and two rigid bodies (16 d.o.f)is modeled. Direct haptic interaction is permitted viaa 3 d.o.f PHANToM haptic manipulator. The user isallowed to push and attach oneself to any of the ob-jects in the environment and feel the force and impactcreated by their interaction.

4. Task-Consistent Elastic Plans

The control methods presented in Section 2. allow theconsistent control of task and posture for robots withcomplex mechanical structures, such as human-likerobots. To perform or assist in the execution of com-plex actions, however, these control structures have tobe linked with motion generated by a planner. Further-more, since unstructured environments can be highlydynamic, such an integration has to accommodate un-foreseen obstacle motion in real time, while conform-ing to constraints imposed by the task. We have devel-oped algorithms that perform task-consistent, real-timepath modification to address this issue.

4.1. Real-Time Path Modification

Motion planners generally perform a global search inconfiguration space to determine a collision-free mo-tion accomplishing a given task. Due to the high di-mensionality of the configuration space of the class

Figure 5: Top images show from left to right: real-time obstacle avoidance without task consistency, with task consistency,and transitioning between task-consistent and non task-consistent behavior. Lines indicate trajectories of the base, elbow,and end effector. The graphs show end effector error and base deviation from the task for the respective experiment.

of robots we are concerned with in this paper, plan-ning operations are too computationally complex tobe performed in real time. As a consequence, motionin dynamic environments cannot adequately be gener-ated by those planners. The elastic band framework(Quinlan and Khatib 1993) was developed to allowreal-time modification of a previously planned path,effectively avoiding a costly planning operation in re-action to changes in the environment. More recently,this framework was complemented by the elastic stripframework (Brock and Khatib 1997).

The elastic strip framework augments the representa-tion of a path computed by a planner with a descrip-tion of free space around that path. Collision avoidancecan be guaranteed, if the work space volume swept bythe robot along its path is contained within the freespace. Real-time path modification is implemented bysubjecting the entire path to an artificial potential field(Khatib 1986), keeping the path at a safe distance fromobstacles. The modification of the path in accordancewith those potentials is performed while ensuring thatthe volume swept by the robot along the path is alwayscontained within the representation of local free space.This results in “elastic” paths, which deform in reac-tion to approaching obstacles, while maintaining theglobal properties of the path.

In addition to the repulsive, external potential, we alsoapply internal forces to consecutive configurations ofthe robot along the path. This shortens and smoothes

the path. The overall behavior of of a path representedin the elastic strip framework can be compared to astring of elastic material: as obstacles approach, thepath is locally modified or “stretched” by repulsiveforces; once the obstacle moves further away, internalforces shorten and smoothen the path. Such behav-ior is shown in the top left image of Figure 5. Theend effector of the robot is commanded to move on astraight line. The original path generated by the plan-ner is a simple straight-line motion, as no obstacles arepresent. The image shows the path after two mobileobstacles move into the path.

The elastic strip framework scales to robots with manydegrees of freedom and with many operational points,as it avoids a costly search for collision-free motion inconfiguration space. Instead, it employs simple workspace-based potential fields in conjunction with afore-mentioned control structures to modify a previouslyplanned motion in real time.

4.2. Task-Consistent Path Modification

In dynamic environments it is desirable to integrate re-active obstacle avoidance with task behavior. To ac-complish this we extend the overall control structurefor task and posture behavior (equation 2) by addingtorques

� - � %0"�$�� r 5 representing desired obstacle avoid-ance behavior:

� ����"�$ %(')���+.-/%a"21q365#�� - � %a"�$�� r 5

Figure 6: Task-consistent, real-time obstacle avoidance: despite the base changing its trajectory to avoid the moving obstacle,the end effector performs a straight-line trajectory

Both,� +.-/%0"214365

and� - � %0"�$�� r 5 have been mapped into

the nullspace of the task, as shown in equation 7. Us-ing this control structure, the task-consistent obstacleavoidance behavior shown in the second image of Fig-ure 5 is achieved. Note how without task-consistencythe end effector deviates significantly from the re-quired straight-line trajectory. Using task-consistentobstacle avoidance, the end effector only deviates min-imally from the task, as can be seen in the graphsshown in Figure 5. A task-consistent motion execu-tion on the Stanford Assistant Manipulator can be seenin Figure 6.

This approach of integrating task and obstacle avoid-ance behavior can fail, however, when the torques re-sulting from mapping

� - � %0"�$ � r 5 into the nullspace yieldinsufficient motion to ensure obstacle avoidance. Insuch a situation it would be desirable to suspend taskexecution and to realize obstacle avoidance with all de-grees of freedom of the robot.

Let� N � = �����/�B� d f�h = N �������= N ����� i be the dynam-

ically consistent nullspace mapping of the Jacobian= ����� associated with the task. The coefficient

� ��� _ N � = �����R�,�O- � %0"�$ � r 5 �� � - � %0"�$�� r 5 �

corresponds to the ratio of the magnitude of the torquevector

� - � %0"�$�� r 5 mapped into that nullspace to its un-mapped magnitude. This coefficient is an indication ofhow well the behavior represented by

� - � %a"�$�� r 5 can beperformed inside the nullspace of the task. We experi-mentally determine a value � % at which it is desirable tosuspend task execution in favor of the behavior previ-ously mapped into the nullspace. Once the coefficient� assumes a value ����� % , a transition is initiated. Dur-ing this transition task behavior is gradually suspended

and previous nullspace behavior is performed using alldegrees of freedom of the manipulator. The motion ofthe manipulator is now generated using the equation� � = N �����#E

d fMh = N ������= N ����� i � - � %0"�$ � r 5 � - � %a"�$�� r 5

where�� l Y Y : n is a time-based transition variable,

transitioning between 1 and 0 during task suspensionand between 0 and 1 during resumption of the task,and �m� : h��� is defined as the complement of

.

The experimental results, performed on the StanfordAssistant Manipulator, for such transistioning behav-ior can be seen in Figure 5. The image on the top rightshows how despite task-consistent obstacle avoidancethe task has to be suspended to ensure obstacle avoid-ance. Below, the graph shows how the base deviatessignificantly from the straight line in response to theobstacle. The end effector, however, maintains the taskuntil it has to be suspended. The graph also shows thatthe task is resumed in a smooth manner, after the basehas passed the obstacle

5. Conclusion

Advances toward the challenge of robotics in humanenvironments depend on the development of the ba-sic capabilities needed for both autonomous operationsand human/robot interaction. In this article, we havepresented methodologies for whole-robot coordinationand control, cooperation between multiple robots, in-teractive haptic simulation with contact, and the real-time modification of collision-free path to accommo-date changes in the environment.

For the whole-robot coordination and control, we pre-sented a framework which provides the user with twobasic task-oriented control primitives: task control andposture control. The major characteristic of this con-trol structure is the dynamic consistency it provides inimplementing these two primitives: the robot posturebehavior has no impact on the end-effector dynamicbehavior. While ensuring dynamic decoupling and im-proved performance, this control structure provides theuser with a higher level of abstraction in dealing withtask specifications and control.

Addressing the computational challenges of human-like robotic structures, we presented efficient

� � �k7 7 � � recursive algorithms for the operational space dy-namics of mechanisms involving branching structuresand closed chains. Building on the operational spaceformulation, we also developed a framework for theresolution of multi-contact between articulated multi-body systems. The computational efficiency of the dy-namic algorithms developed for physical robots pro-vided the interactivity needed for haptic simulation ofcomplex virtual environments.

The elastic strip framework allows the seamless inte-gration of reactive, real-time obstacle avoidance andthe task-oriented control structure. It provides for real-time motion generation that combines obstacle avoid-ance and task execution. When kinematic or externalconstraints imposed by obstacles make it impossible tomaintain the task, task-consistent obstacle avoidanceis suspended and all degrees As the constraints are re-laxed, the task is resumed in a smooth manner. Us-ing the elastic strip framework, motion for complexkinematic structures can be generated very efficiently,as the required computations are mostly performed inwork space and as a result are independent of the num-ber of degrees of freedom of the mechanism.

Acknowledgments

The financial support of Honda Motor Company andNSF (grants IRI-9320017) is gratefully acknowledged.Many thanks to Alan Bowling, Arancha Casal, Fran-cois Conti, Robert Holmberg, Jaeheung Park, CostasStratelos, James Warren, and Kazuhito Yokoi for theirvaluable contributions to the work reported here.

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