a unified approach for motion and force control of robot manipulators - the operational space...

7/22/2019 A Unified Approach for Motion and Force Control of Robot Manipulators - The Operational Space Formulation, Oussama Khatib

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IEEE OF ROBOTICS ANDUTOMATION, VOL. RA-3,O. 1, FEBRUARY 1987 43

A Unified Approach for Motion and Force Controlof Robot Manipulators: The Operational SpaceFormulationAbstract-A framework for the analysis and control of manipulatorsystems with respect to the dynam ic behavior of their end-effectors isdeveloped. First, issues related to the description of end-effector tasksthat involve constrained motion and active force control are discussed.The undamentals of the operational space formulation are thenpre-sented, andhe unified approach for motion and force control isdeveloped. The exten sion of this formu lation to redundant manipulator

systems is also presented,constructing the end-effector equations ofmo tion and describing their behavior with respect to joint forces. Th eseresults are used in the evelopm ent of a new and systematic approachordealing with the problems arising at kinematic singularities. t a singularconfiguration, the manipulator is treated as a mechanismhat isredundant with respect to the motion o f the end-effector in the subspaceof operational space orthogonal to the singular direction.

R I. INTRODUCTIONESEARCH in dynamics of robot mechanisms has largelyfocused on developing the equations of joint motions.These joint space dynamic models have been the basis forvarious approaches to dynamic control of .manipulators.However, task specification for motion and contact forces,dynamics, and force sensing feedback are closely linked to theend-effector. The dynamic behavior of the end-effector is oneof the most significant characteristics in evaluating theperformance of robot manipulator systems. The problem ofend-effector motion control has been investigated, and al-gorithms resolving end-effector accelerations have been devel-oped V I [111, 1221, [3Ol, 1331.The issue of end-effector dynamic modeling and control isyet more acute for tasks that involve combined motion andcontact forces of the end-effector. Precise control of appliedend-effector forces is crucial to accomplishing advanced robotassembly tasks. This is reflected by the research effort that hasbeen devoted to the study of manipulator force control.Accommodation [35], joint compliance [26], active stiffness[3 1 impedance control [9], and hybrid position/force control[28] are among the various methods that have been proposed.Force control has been generally based on kinematic andstatic considerations. While in motion, however, a manipula-tor end-effector is subject to inertial, centrifugal, and Coriolis

Manuscript receivedAugust 6, 1985; revised May 29, 1986. This work wassupported npart by the NationalScienceFoundation and inpart by theSystems Development Foundation.The a uthor s with the A rtificial Intelligence La boratory, C omputer ScienceDepartment, Cedar Hall, Stanford University, Stanford, CA 94305.IEEE Log Number 8610323.

forces. The magnitude of these dynamic forces cannot beignored when large accelerations and fast motions are consid-ered. Controlling the end-effector contact forces in somedirection can be strongly affected by the forces of couplingcreated by the end-effector motion that can take place in thesubspace orthogonal to that direction. The description of thedynamic interaction between end-effector motions and theeffects of these motions on the end-effectors behavior in thedirection of force control are basic requirements for theanalysis and design of high-performance manipulator controlsystems. Obviously, these characteristics cannot be found inthe manipulator joint space dynamic model, whichonlyprovides a description of the interaction between joint mo-tions. High-performance control of end-effector motion andcontact forces requires the description of how motions alongdifferent axes are interacting, and ow the apparent orequivalent inertia or mass of the end-effector varies withconfigurations and directions.The description, analysis, and control of manipulator sys-tems with respect to the dynamic characteristics of their end-effectors has been the basic motivation in the research anddevelopment of the operational space formulation. The end-effector equations of motion [131, [141 are a fundamental toolfor the analysis, control, and dynamic characterization [181 ofmanipulator systems. In this paper, we will discuss, from theperspective of end-effector control, the issue of task descrip-tion, where constrained motions and contact forces areinvolved. The fundamentals of the operational space formula-tion are presented, and the unified approach for the control ofend-effector motion and contact forces is developed.Treated within the framework of joint space controlsystems, redundahcy of manipulator mechanisms has beengenerally viewed as a problem of resolving the end-effectordesired motion into joint motions with respect to some criteria.Manipulator redundancy has been aimed at achieving goalssuch as the minimization of a quadratic criterion [29], [34], theavoidance of joint limits [5], [21] the avoidance of obstacles[4], [6], [20], kinematic singularities [23], or the minimizationof actuator joint forces [8]. The end-effector equations ofmotion for a redundant manipulator are established and itsbehavior with respect to generalized joint forces is described.The unified approach for motion and active force control isthen extended to these systems.Kinematic singularities is another area that has beenconsidered within the framework of joint space control and

0882-4967/87/0200-0043 0100@1987 IEEE


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44 IEEE JOURKAL OF ROBOTICS AND AUTOM ATION, VOL. RA-3, NO. 1, FEBRUARY 1

Fig, 1. Constrainedend-effector reedom of motion. a) Fivedegrees offreedom. b) Four degrees of freedom. c) Three degrees of freedom.formulated in terms of resolution of the task specifications intojoint motions. Generalized inverses and pseudo-inverses havebeen used, and recently an interesting solution based on thesingularity robust inverse has been proposed [25]. In hispaper, a new approach for dealing with the problem ofkinematic singularities within the operational space frameworkis presented. In the neighborhood of a singular configurationthe manipulator is treated as a redundant mechanismwithrespect to the motion of the end-effector in the subspace ofoperational space orthogonal to its singular direction. Controlof the end-effector for motions along the singular direction isbased on he use of the kinematic characteristic of the Jacobianmatrix.

11. GENERALIZEDASK PECIFICATIONATRICESThe end-effector motion and contact forces are among the

most important components in the planning, description, andcontrol of assembly operations of robot manipulators. Theend-effector configuration is represented by a set f mparameters, X I , xz * e , x specifying its positionndorientation in some reference frame. In free motion opera-tions, the number of end-effector degrees of freedom mo isdefined [131as the number of independent parameters requiredto specify completely, in a frame of reference Ro, its positionand orientation. A set of uch independent configurationparameters forms a system of operational coordinates.In constrained motion operations, the displacement androtations of the end-effector are subjected to a set of geometricconstraints. These constraints restrict the freedom of motion(displacements and rotations) of the end-effector. Clearly,geometric constraints will affect only the freedom of motion ofthe end-effector, since static forces andmoments at theseconstraints can still be applied. The number of degrees offreedom of the constrained end-effector is given by thedifference between mo and the number of independentequations specifying the geometric constraints, assumed to beholonomic. Examples of five-, four-, and three-degree-of-freedom constrained end-effectors are shown in Fig. 1.An interesting description of the characteristics ofend-effectors and their constraints uses a mechanical linkagerepresentation [ 5 ] , 1241. The end-effector, tool, or manipu-

lated object forms, with the fixture or constrained object, pair of two rigid bodies linked through a joint. A constrainemotion ask can be described, for instance, by a sphericaplanar, cylindrical, prismatic, or revolute joint.However, when iewed from the perspective of eneffector control, two elements of information are required foa complete description of the task. These are the vectors ototal force and moment that are to be applied to maintain thimposed constraints, and the specification of the end-effectmotion degrees of freedom and their directions.Let f d be a unite vector, in the frame of reference Ro 3 , xyo, ZO), along the direction of the force that is to be applied bthe end-effector. The positional freedom, if any, of thconstrained end-effector will therefore lie in he subspaorthogonal to f d .

A convenient coordinate frame for the description of taskinvolving constrained motion operations is a coordinate fram6if(0, f , y f , z f ) obtained from G i o by a rotation transformtion described by S f such that zf is aligned with fd. For taswhere the freedom of motion (displacement) is restricted tosingle direction orthogonal to f d , one of the axes Oxf or Owill be selected in alignment with that direction, as shown fthe task in Fig. 2.Let us define, in the coordinate @if he position specifiction matrix = f = T ;)0where ox,o;, and a, are binary numbers assigned the valuewhen a free motion is specified along the axes Ox,, Oyf, an3zf , respectively, and zero otherwise. A nonzero value ofimplies a full freedom of the end-effector position. This caof unconstrained end-effector position is integrated here focompleteness. The coordinate frame R f in this case is assumeto be identical to Bo, nd the matrix S f is the identity matriThe directions of force control are described by the forcspecification matrix zf ssociated with C f and defined by

c f =I - c f 2where I designates the 3 X 3 identity matrix.


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KHATIB: ONTROL OF ROBOTANIPULATORS 45The task specification matrices Q and a can be constant,configuration-varying, or time-varying matrices. Nonconstantgeneralized task specification matrices correspond to specifi-cations that involve changes in the direction of the appliedforce vector and/or moment vector, e.g., moving the end-effector while maintaining a normal force to a nonplanarsurface. Cl and have been expressed here with respect to theframe of reference OXo. For control systems implemented for

tasks specified with respect to the end-effector coordinateframe, these matrices will be specified with respect to thatcoordinate frame as well.111. END-EFFECTORQUATIONSF MOTION

X x0 YO Joint space dynamic models, which establish the equationsof manipulator joint motions, provide means for the analysisand control of these motions, and for the description of theig. 2. One-degree-of-freedom motion.Let us now consider the case where the end-effector taskinvolves constrained rotations and applied moments. Let 7 d bethe vector, in the frame of reference aO(O,o, yo, zo), ofmoments that are tobe applied by the end-effector, and ,(U,

x,, y,, z,) be a coordinate frame obtained from CRo O, xo,yo,zo) by a rotation S , that brings z, into alignment with themoment vector T d . In a he rotation freedom of the end-effector lies in the subspace spanned by {x,, y , ) . To a taskspecified in terms of end-effector rotations and appliedmoments in he coordinate framea e associate the rotationand momentspecification matrices C and X defined similarlyto Cf and gf.For general tasks that involve end-effector motion(bothpositionnd orientation) and contact forces (forces andmoments) described in the frame of reference Boywe definethe generalized taskspecification matrices

and

associated with specifications of motion ahd contact forces,respectively.Q and fi act on vectors described in the reference frame a0.

A position command vector, for instance, initially expressedin a0 s transformed by the rotation matrix Sf to the taskcoordinate frame The motion directions are then selectedin this frame by the application of p Finally, the resultingvector is transformed back in cRo by Sf .The construction of the generalized task specificationmatrices is motivated by the aim of formulating the selectionprocess in the same coordinate frame (reference frame CRowhere the manipulator geometric, kinematic, and dynamicmodels are formulated. This allows a more efficient imple-mentation of the control system for real-time operations.Control systems using specifications basednlyn thematrices C and X, will require costly geometric, kinematic,and dynamic transformations between the reference frame andthe task coordinate frames.

configuration dependency and interactive nature of thesemechanisms. However, the control of end-effector motion andcontact forces, or the analysis and characterization of end-effector dynamic performance requires the construction of themodel describing the dynamic behavior of this specific part ofthe manipulator system.The end-effector motion is the result of those combined ointforces that are able to act along or about the axes ofdisplacement or rotation of the end-effector. These are,indeed, the forces associated with the system of operationalcoordinates selected to describe the position and orientation ofthe end-effector. The construction of the end-effector dynamicmodel is achieved by expressing the relationships between itsoperational positions, velocities, accelerations, and the virtualoperational forces acting on it.First, let us consider the case of nonredundant manipula-tors, where a set of operational coordinates can be selected as asystem of generalized coordinates for the manipulator. Themanipulator configuration is represented by the column matrixq of n joint coordinates, and the end-effector position andorientation is described, in a frame of reference OXo, by the moX 1 column matrix x of independent configuration parame-ters, i.e., operational coordinates. With the manipulatornonredundancy assumption we have the equality n = mo.Now let us examine the conditions under which a set ofindependent end-effector configuration parameters can be usedas a generalized coordinate system for a nonredundantmanipulator. In the reference frame R0, the system of moequations expressing the components of x as functions of jointcoordinates, i.e., the geometric model, is given by

X = G ( g ) . ( 5 )Let qi and Qi be, respectively, the minimal and maximalbounds of the ith joint coordinate q i . The manipulatorconfiguration represented by the point q in joint space isconfined to the hyperparallelepiped

Obviously, for arbitrary kinematic linkages, and general jointboundaries, the set of functions G defined from Dq to the


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46 IEEE JOURNAL OF ROBOTICS ANDUTOMATION, VOL. RA-3,O. 1 , FEBRUARY 19domain 9, of the operational space given by

is not one-to-one.Different configurations of the manipulator links can, infact, be found for a given configuration of the end-effector.The restriction to a domain where G is one-to-one is thereforenecessary to construct, with the operational coordinates, asystem of generalized coordinates for the manipulator mecha-nism.In addition, for some configurations of the manipulator, theend-effector motion is restricted by the linkage constraints andits freedom of motion ocally decreases. These are the singularconfigurations, whichan be found by considering thedifferentiability characteristics of the geometric model G .Singular configurations q E are those where the Jacobianmatrix J ( q ) involved in the variational or kinematic modelassociated with G,= J(q)Aq, 8)

is singular. The end-effector behavior at singular configura-tions is treated in Section VIII.Let 5 , be the domain obtained from 9, by excluding themanipulator singular configurations and such that the vectorfunction G of 5) is one-to-one. Let a), esignate the domainS, = G(Bq ) . (9)

The independent parameters X I , x2 , * * ,xmoform a completeset of configuration parameters for a nonredundant manipula-tor, in the domaina),f operational space and thus constitute asystem of generalized coordinates for the manipulator system.The kinetic energy of the holonomic articulated mechanismis a quadratic form of the generalized operational velocities1 ( X , X) = XTA(x)X2 (10)where A ( x ) designates the mo x mosymmetric matrix of thequadratic form, i.e., the kinetic energy matrix. Using theLagrangian formalism, the end-effector equations of motionare given by

where the Lagrangian L ( x , X) isL (x, X)= T ( x , X U x) (12)

and U ( x ) epresents the potential energy due to gravity. F isthe operational force vector. Let p ( x ) be the vector of gravityforces

p ( x )= V U ( x ) . (13)The end-effector equations of motion in operational space canbe written [131, [14] in the form

A ( x ) f + p ( ~ ,) + p ( x ) = F (14)

where p ( x , x) is the vector of end-effector centrifugal anCoriolis forces given byp j ( x ,) = X ~ r I j ( x ) X , i 1, . . mo. (15

The components of the mo x mo matrices r I j ( x ) are thChristoffel symbols a i , j k given as a function of the partiaderivatives of A ( x )with respect to the generalized coordinatex by

1 ah, ahik axjka j , j k = - -+---2 xk axj axi1 . (16The equations ofmotion (14) establish the relationshipbetween positions, velocities, and accelerations of the endeffector and the generalized operational forces acting on iThe dynamic parameters in these equations are related to thparameters involved in the manipulator joint space dynammodel. The manipulator equations of motion in oint space argiven by

m ) g + b ( q , q ) + g ( q ) = r 17where b(q, q) , g ( q ) , and r represent, respectively, thCoriolis and centrifugal, gravity, and generalized forces ijoint space. A ( q ) is the n X n joint space kinetic energmatrix. The relationship between the kinetic energy matriceA ( q ) and A ( x ) corresponding, respectively, to the joint spacand operational space dynamic models can be established [13[14] by exploiting the identity between the expressions of thquadratic forms of the mechanism kinetic energy with respeto the generalized joint. and operational velocities,

A ( x ) = J - * ( q ) A ( q ) J - ( q ) . (18The relationship between the centrifugal and Coriolis forceb(q, q ) and p ( x , X ) can be established by the expansion of thexpression of p ( x , X) that results from (1 l),

p ( x , X) = h ( x ) X V T ( x , X ) . (19Using the expression of A ( x ) in (18), the components of p (x) in (19) can be written as

A ( x ) X = J - T ( q ) A 414- A(q)h(q, 4 )+ j - T ( Q ) A ( q )v m , X = J - =(q>l (q , )+j S)A 414 20

whereh(q, 4 )= j q M (21

and12l i (Q, 4 = - q=Aqi(q)q, i = l , * . - , n . (22

The subscript q i indicates the partial derivative with respect tthe ith joint coordinate. Observing from the definition of b(qq ) that

b(q1 4 > = A ( s ) 4 - 4 s , ) (23


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KHATIB: MOTION AND FORCE CONTROL OF ROBOT MANIPULATORS 47yields

A x , J- =(q)b(q,4 )-N q ) h ( q , 4 . 24)The relationship between the expressions of gravity forces canbe obtained using the identity between the functions expressingthe gravity potential energy in the two systems of generalizedcoordinates and the relationships between the partial deriva-tives with respect to these coordinates. Using the definition ofthe Jacobian matrix 8) yields

In the foregoing relations, the components involved in he end-effector equations of motion 14), i.e., A, p , p , are expressedin terms of joint coordinates. This resolves the ambiguity indefining the configuration of the manipulator corresponding toa configuration of the end-effector in the domain DX f - 7).With these expressions, the restriction to the domain 6whereG is one-to-one, then becomes unnecessary. Indeed, thedomain of definition of the end-effector dynamic model of anonredundant manipulator can be extended to the domain a),defined by a, =G( ),) 26)where 3s the domain resulting froma ,f 6) by excludingthe kinematic singular configurations.Finally, let us establish the relationship between generalizedforces, i.e., F and r. Using 18), 24), and 25) the end-effector equations of motion 14) can be rewritten as

J - T(q)rAQ ) q +M q , 4 +g q)l= F . 27)Substituting 17) yields r = J T q ) 28)which represents the fundamental relationship between opera-tional and joint forces consistent with the end-effector andmanipulator dynamic equations. This relationship is the basisfor the actual control of manipulators in operational space.

IV. END-EFFECTOROTIONONTROLThe control of a manipulator in operational space is basedon the selection of the generalized operational forces F as acommand vector. These forces are produced by submitting themanipulator to thecorresponding joint forces I obtained from

28).As with joint space control systems, the control in opera-tional space can be developed using a variety of controltechniques. In operational space control systems, however,errors, performance, dynamics, simplifications, characteriza-tions, and controlled variables are directly related to manipula-tor tasks.One of the most effective techniques for dealing with thesehighly nonlinear and strongly coupled systems is the nonlin-ear dynamic decoupling approach [36], [37], which fullyexploits the knowledge of the dynamic model structure andparameters. Within this framework of control and at the levelof the uncoupled system linear, nonlinear, robust [32], andadaptive [3] control structures can be implemented.

Nonlinear dynamic decoupling in operational space isobtained by the selection of the following control structure,F = Fm+Fccg 29)

withFm= A(x)P

where A(x), (x, x , and j (x) represent the estimates ofA x), p ( x , x), and p (x). FZ is the command vector of thedecoupled end-effector. With a perfect nonlinear dynamicdecoupling, the end-effector becomes equivalent to a. singleunitmass Z m o , moving in the mo-dimensional space. Tosimplify the notations, the symbol will be dropped in thefollowing development.At the level of the decoupled end-effector, I , variouscontrol structures can be selected. For tasks where the desiredmotionof the end-effector is specified, a linear dynamicbehavior can be obtained by selecting

P ~ = z , , X d - k P ( X - X d ) - k u ( X - X d ) . 31)where x d , i d , and are the desired position, velocity, andacceleration, respectively, of the end-effector. Zmo is the mo Xmo identity matrix. kp and k, are the position and velocity gainmatrices.An interesting approach for tasks that involve large motionto a goal position, where a particular path is not required, isbased on the selection of the decoupled end-effector commandvector F ; as

FZ = - , ( i - K i d ) 32)where

33)This allows a straight line motion of the end-effector at a givenspeed V,=. The velocity vector x is in fact controlled to bepointed toward the goal position while its magnitude islimited to V,,,. The end-effector will then travel at V,,, in astraight line, except during the acceleration and decelerationsegments. This command vector is particularly useful whenused in conjunction with the gradient of an artificial potentialfield for. collision avoidance [151.Using the relationship between generalized forces given in28), the joint forces corresponding to the operational com-mand vector F , in 29) and 30), for the end-effector dynamicdecoupling and control, can be written as

r = J = ( q ) A ( q ) F : + 4 )+ d q ) 34)where 6 q,4) is the vector of joint forces under the mappinginto joint space of the end-effector Coriolis and centrifugal


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48 IEEEOURN AL OF ROBOTICS AND AUTOMA TION, VOL. RA-3, NO. I FEBRUARY 19force vector p(x, i .o simplify the notation, A has also beenused here to designate the kinetic energy matrix whenexpressed as a function of the joint coordinate vector q. 6(q,4 ) s distinct from the vector of centrifugal and Coriolis forcesb(q, q ) in 17) hat arises whenviewing he manipulatormotion in joint space. These vectors are related by

6(q, 4 )= H q , 4 ) T ( q ) A ( q ) h ( q , ) . (35)A useful form of 6 q,q ) for real-time control and dynamicanalysis can be obtained by a separation of its dependency onposition and velocity.

The joint space centrifugal and Coriolis force vector b(q,q,of (17) can, in fact, be developed in the form

b(q7 4 )=B(q)[441+ c q)[q21 (36)where B ( q ) and C(q )are, respectively, the n X n n 1)/2and n x n matrices of the joint space Coriolis and centrifugalforces associated withb(q,4 ) . qq] and [q2]are the symbolicnotations for the n n 1)/2 X 1 and n X 1column matrices

[441=[4142 4143 ' * * 4 n - 1 @ n I T[ 4 2 ] = q 4; * * * Q i ] T . (37)With [4q] and [g2], he vectorh q,q ) can be developed in theform

h(q7 4 ) = ~ l ( 4 ) [ 9 Q 1 + ~ 2 ( ~ ) c 4 2 1 ( 3 8)where the matrices .Hl q)and H2(q)have, respectively, thedimensions n x n 1)/2 and n x n . Finally, the vectorg(q, q ) can be written as

qY 4 )=B(q)[(i@l+q)[421 (39)where B(q) and c q) re the n X n n - 1)/2 and n X nmatrices of the joint forces under the mapping into joint spaceof the end-effector Coriolis and centrifugal forces. Thesematrices are

B ( q ) = B ( q ) - J T ( g ) A ( q ) H 1 ( q )ei(41 C(q) JT(q>A(q)H2(q) (40)

With he relation 39), the dynamic decoupling of the end-effector can be obtained using the configuration dependentdynamic coefficients A ( q ) , B(q), c(q),nd g q) . The jointforce control vector (34) becomes~ = J ~ ~ ) A ~ ) F I T , + B ~ ) [ ~ ~ Ie ( m 2 i m . (41)By isolating these coefficients, end-effector dynamic decou-pling and control can be achieved in a two-level control systemarchitecture [15]. The real-time computation of these coeffi-cients can then be pacedby the rate of configuration changes,which is much lower than that of the mechanism dynamics.This leads to the following architecture for the control systema low rate dynamic parameter evaluation level: updat-a high rate servo control level: computing the commanding the end-effector dynamic parameters;vector (41) using the updated dynamic coefficients.

This approach has also been proposed [lo] for real-timdynamic control of manipulators in joint space.V. CONSTRAINEDOTION PERATIONS

The matrix s2 defined earlier specifies, with respect to thframe of reference 030, he directions of motion (displacemeand rotations) of the end-effector. Forces and moments are tobe applied in or about directions that are orthogonal to thesmotion directions. These are specified by the matrix 8.An important issue related to the specification of axes orotation and applied moments is concerned with the compatbility between these specifications and the type of representation used for the description of the end-effector orientation. Ifact, the specification of axes of rotations and applied momenin the matrices C and 2, are only compatible with descriptionof the orientation using instantaneous angular rotationsHowever, instantaneous angular rotations are not quantitiethat can be used as a set of configuration parameters for thorientation. Representations of the end-effector orientatiosuch as Euler angles, direction cosines, or Euler parametersare indeed incompatible with specifications provided by C an

Instantaneous angular rotations havebeenused for thdescription of orientation error of the end-effector. An angularotation error vector 64 that corresponds to the error betweethe actual orientation of the end-effector and its desireorientation can be formed from the orientation descriptiogiven by the selected representation [131, [22].The time derivatives of the parameters corresponding to representation of the orientation are related simply to thangular velocity vector. With linear and angular velocities iassociated the matrix Jo(q) , termed the basic Jacobiandefined independently of the particular set of parameters useto describe the end-effector configuration

E .

The Jacobian matrix J ( q ) associated with a given representation of the end-effector orientationx can then beexpressed ithe form [13]J ( q ) ExrJo 4) (43

where the matrix Ex, is simply given as a function of x .For end-effector motions specified in terms of Cartesiacoordinates and instantaneous angular rotations, the dynamidecouplingandmotion control of the end-effector can bachieved [131 byr = J,T(s>Ao(x)F: 5o(q, 4 )+ g ( 4 ) (44

where (q) and 60(q, ) are defined similarly to A ( q ) an6 q,4 ) with J ( q ) being replaced by Jo q).Using the relationship (43) similar control structures can bdesigned to achieve dynamic decoupling and motion controwith respect to descriptions using other representations for thorientation of the end-effectorThe unified operational command vector for end-effectodynamic decoupling, motion, and active force control can b


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KHATIB: MOTION AND FORCE CONTROL OF ROBOT MANIPULATORS 49

I 1

Fig. 3 . Operationalspacecontrol system architecture.written as

F = F m+Fa + Fccg (45)where Fm, Fa,nd Fccg re the operational command vectors ofmotion, active force control, and centrifugal, Coriolis, andgravity forces given by

F m = Ao(Q)Qp:F a = f i F z + A o ( q ) f i F zFccg = 60(0(4, 4 )+ g ( q ) (46)

where F: represents the vector of end-effector velocitydamping that acts in the direction of force control. The jointforce vector corresponding to F in (45) is

r = J , T ( q ) [ A o ( q ) ( Q F :+O F : ) + f i F z I + 60(q1 4 )+g(q).(47)

The control system architecture is shown in Fig. 3, where kfrepresents the force error gain and k , f denotes the velocitygain in F,*.An effective strategy for the control of the end-effector during the transition from free to constrained motionsis based ona pure dissipation of the energy at the impact. Theoperational command vector Fa during the impact transitioncontrol stage is

F a = A o ( q ) 8 F : . (48)The duration of the impact transition control is a function ofthe impact velocity and the limitations on damping gains andactuator torques (this duration is typically on the order of tensof milliseconds). Force rate feedback has also been used inF;.A more detailed description of the components involved in thiscontrol system, real-time implementation issues, and experi-mental results can be found in [19].

VI. REDUNDANTANIPULATORSThe configuration of a redundant manipulator cannot bespecified by a set of parameters that only describes the end-effector position and orientation. An independent set of end-effector configuration parameters, therefore, does not consti-tute a generalized coordinate system for a redundant

manipulator, and the dynamic behavior of the entire redundantsystem cannot be represented by a dynamic model in coordi-natesonly of the end-effector configuration. The dynamicbehavior of the end-effector itself, nevertheless, can still bedescribed, and its equations of motion n operational space canstill be established.The end-effector is affected by forces acting along or aboutthe axes of its freedom of motion. These are the operationalforces associated with the operational coordinates selected todescribe its position and orientation. Let us consider the end-effector dynamic response to the application, on the end-effector,oran operational force vector F . In this case ofredundant manipulator systems, the joint forces that can beused to produce a given operational force vector are notunique. The joint force vector

I ' = J T ( q ) F 28)represents, in fact, one of these solution.17), and the use of the relationThe application of the joint forces 28) to the manipulator

a = J ( q ) q + h ( q , 4 ) (49)allow us to establish [13] the equations of motion of the end-effector

AAq ) f + PA49 4 )+ P A 4 1 = F ( 50 )where

h , ( q ) = [ J ( q ) A - ( q ) J T ( q ) l -Pccr(Q, 4)=JT(q)b(q9 ) - ( 4 ) h ( q , 4 )

P A 4 1 = . f T ( 4 k ( q > ( 5 1)and

~ i ( 4 ) = A - ( s ) J T ( 4 ) A , ( ( I ) . ( 5 2 )J ( q ) s actually a generalized inverse of the Jacobian matrixcorresponding to the solution that minimizes the manipulator'sinstantaneous kinetic energy.Equation 50) describes the dynamic behavior of the end-effector when the manipulator is submitted to a generalized


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50 IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. RA-3,O. 1, FEBRUARY 198joint force vector of the form (28). The m X m matrix ( q ) Stability Analysiscan be interpreted as a pseudo -kinetic energy matrix corres- In the command vector (58), and with the assumption of apending to the end-effectormotion in Operational 'pace* h q 9 perfect compensation (or noncompensation) ofthe centrifu0) epresents the and forces acting On the gal and Coriolis forces, the manipulator can be considered s aend-effector and pr q) he gravity force vector.joint forces, can be determined by (50) which can be ewrittenThe effect on the end-effector of the application of arbitrary the velocity damping term ( k u i ) in F.+.. These forces are

onservative system subjected to the dissipative forces due to

Substituting 17 ) yields~ = ~ = ( q ) r . 54)

This relationship determines how the joint space dynamicforces are reflected at the level of the end-effector.Lemma: The unconstrained end-effector 50)is subjected tothe operational force F if and only if the manipulator (17) issubmitted to the generalized joint force vectorr = J T ( q ) ~ + [ z , - J T ( q ) J T ( q ) ] r o (55)

where I,, is the n X n identity matrix, J ( q ) s the matrix givenin 52) and r0 s an arbitrary joint force vector.When the applied forces r are of the form (55), it isstraightforward from 54) to verify that the only forces actingon the end-effector are the operational forces F produced bythe first term in the expression of r . Joint forces of the form[ I , , J T ( q ) J T ( q ) ] r 0orrespond in fact to a null operationalforce vector.The uniqueness of 55) is essentially linked to the use of ageneralized inverse ] ( q ) that is consistent with the dynamicequations of the manipulator and end-effector. The form of thedecomposition 55)itself is general. A joint force vector r canalways be expressed in the form of 55).

L e t P q)be a generalized inverse of J ( q ) , and let us submitthe manipulator to the joint force vector

If, for any ro,he end-effector is only subjected to F , (56)yields. I ( q ) A - ( q ) = [ J ( q ) A - ( q ) J T ( q ) 1 P T ( q )5 7 )

which implies the equivalence of P q) and J ( q ) .VII. CONTROLF REDUNDANTANIPULATORS

As in the case of nonredundant manipulators, the dynamicdecoupling and control of the end-effector can be achieved byselecting an operational command vector of the form of 29),(30):The corresponding joint forces are

where ( q , q ) is defined similarly to 6 q,4).The manipulator joint motions produced by this commandvector are those that minimize the instantaneous kinetic energyof the mechanism.

D q )= u J T ( q ) ~ r ( q ) J ( q ) .6 0 )Lyapunov stability analysis leads to the condition

qTD(q)qsO (61whichs satisfied, since D (q ) is an n x n negativesemidefinite matrix of rank m . However, the redundantmechanism can still describe movements that are solutions ofthe equation

qTD(q)q=o. (62An example of such behavior is shown in Fig. 4(a). The end-effector of a simulated three-degree-of-freedom planar manip-ulator is controlled under 58). The end-effector goal positioncoincides with its current position, while the three joints areassumed to have initially nonzero velocities 0.5 rad/s hasbeen used).Asymptotic stabilization of the system can be achieved bythe addition of dissipative joint forces [ 1 3 ] . These forces canbe selected o act in the null space of the Jacobian matrix [161This precludes any effect of the additional forces on the end-effector andmaintains its dynamic decoupling. Using 55)these additional stabilizing joint forces are of the form

r n , = [ Z , - J T ( q ) J T ( q ) I r , .6 3 )By selecting

rs= - k u , A ( q ) 4 ,6 4 )the vector r,, becomes

r n s = r s + J T ( q ) A r ( q ) F r s6 5 )with

Fr,= k u q i .6 6 )Finally, the joint force command vector can be written as

r = J ' ( q ) a r ( q ) ( F . + . + F r , ) + r , + b ; ( q , q ) + g ( q ) - 6 7 )Under this form, the evaluation of the generalized inverse ofthe Jacobian matrix is avoided. The matrix D ( q )correspond-ing to the new expression for the dissipative oint forces rdisnthe command vector (67) becomes

D ( q ) = - [ ( k , - k u , ) J T ( q ) A , ( q ) J ( q ) + k u , A ( q ) l . 6 8Now, the matrix D (q ) s negative definite and the system isasymptotically stable. Fig. 4(b) shows the effects of this


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a) b)Fig. 4. Stabilization of redundant manipulator.

51

stabilization on the previous example of a simulated three-degree-of-freedom manipulator.Constrained Motion Control

The extension to redundant manipulators of the resultsobtained in the case of nonredundancy is straightforward. Thegeneralized joint forces command vector becomesr = J~ q)[ii , , q) w; + i j~ ;F , J + i j~;]

+r,+ gr0(q,q )+g q ) 69)where &,(a) and b ; o q , q ) are defined with respect to thebasic Jacobian matrix Jo(q ) .

VII. SINGULARONFIGURATIONSA singular configuration is a configuration q at whichsome column vectors of the Jacobian matrix become linearly

dependent. The mobility of the end-effector can be defined asthe rank of this matrix [ 5 ] . In the case of nonredundantmanipulators considered here, the end-effector at a singularconfiguration loses the ability to move along or rotate aboutsome direction of the Cartesian space; its mobility ocallydecreases. Singularity and mobility can be characterized, inthis case, by the determinant of the Jacobian matrix.Singularities can be further specified by the posture.of themechanism at which they occur. Different types of singulari-ties can be observed for a given mechanical linkage. These canbe directly identified from the expression of the determinant ofthe Jacobian matrix. The expression of this determinant can, infact, be developed into a product of terms, each of whichcorresponds to a type of singularity related to the kinematicconfiguration of the mechanism, e.g., alignment of two linksor alignment of two joint axes.To each singular configuration there corresponds a singulardirection. It is in this direction that the end-effectorpresents infinite inertial mass for displacements or infiniteinertia for rotations. Its movements remain free in thesubspace orthogonal to this direction. This behavior extends,in reality, to a neighborhood of the singular configuration. Theextentof this neighborhood canbe characterized by theparticular expression s q ) in the determinant of the Jacobianmatrix that vanishes at this specific singularity. The neighbor-

hood of a given type of singularity 9, can be defined asas= ~ 1 1 s q ) l ~ ~ o l 70)

where sois positive.The basic concept in our approach to the problem ofkinematic singularities can be formulated as follows. In theneighborhood 9,of a singular configuration q , the manipula-tor is treated as a mechanism that is redundant with respect tothe motion of the end-effector in the subspace of operationalspace orthogonal to the singular direction. For end-effectormotion n hat subspace, the manipulator is controlled as aredundant mechanism. Joint forces selected.from the associ-ated null space are used for the control of the end-effectormotion along the singular direction. When moving out of thesingularity, this is achieved by controlling the rate of changeof s q ) according to the value of the desired velocity for thismotion at the configuration when Is(q)( = SO. Selecting thesign of the desired rate of change of s(q) allows the control ofthe manipulator posture among the two configurations that itcan generally take whenmovingoutof a singularity. Aposition error term on s q ) is used in the control vector fortasks that involve a motion toward goal positions located at orin the neighborhood of the singular configuration.Using polar or singular value decomposition, this approachcan be easily extended to redundant manipulator systems. Theextension to configurations where more than one singularity isinvolved can be also simply achieved. An example of asimulated two-degree-of-freedom manipulator is shown inFig. 5(a). The manipulator has been controlled to move intoandoutof the singular configuration while displaying twodifferent postures. The time-response of the motion n thesingular direction x ( t ) is shown in Fig. 5@) .

IX. SUMMARYN D DISCUSSIONA methodology for the description of end-effector con-strained motion tasks based on the construction of generalizedtask specification matrices has been proposed. For such taskswhere both motion and active force control are involved, aunified approach for end-effector dynamic control within theoperational space framework has been presented. The use ofthe generalized task specification matrix has provided a more


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52 IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. RA-3, NO. I FEBRUARY 19

Fig. 5. Control at singular configuration.efficient control structure for real-time implementations,further enhanced by a two-level control architecture.Results of the implementation of this approach have shownthe operational space formulation to be an effective means ofachieving high dynamic performance in real-time motioncontrol and active force control of robot manipulator systems.This approach has beenmplementedn an experimentalmanipulator programming system COSMOS (Control in Oper-ational Space of a Manipulator-with-ObstaclesSystem): Usinga PUMA 560 and wrist and finger sensing, basic assemblyoperations have been performed. These include contact, slide,insertion, and compliance operations [17]. With the recentimplementation of COSMOS in the multiprocessor computersystemNYMPH [2], where four National Semiconductor32016 microprocessors have been used, a low-level servo rateof 200 Hz and a high-level dynamics rate of 100Hz have beenachieved.

The impact transition control strategy was effective in theelimination of bounces at contact with a highly stiff surface.The end-effector normal velocities at impact were up to 4.0 in/s. Experiments with square wave force input have also beenconducted, and responses with rise times of less than 0.02 sandsteady force errors of less than 12 percent have beenobserved. This performance hasbeen obtained despite thelimitations in controlling the manipulator joint torques [27].Accurate identification of the PUMA 560 dynamic parameters[ l ] has contributed to a nearly perfect dynamic decoupling ofthe end-effector.For redundant manipulator systems, the end-effector equa-tions of motion have been established, and an operationalspace control system for end-effector dynamic decoupling andcontrol has been designed. The expression of joint forces ofthe null space of the Jacobian matrix consistent with the end-effector dynamic behavior has been identified and used for theasymptotic stabilization of the redundant mechanism. Theresulting control system avoids the explicit evaluation of anygeneralized inverse or pseudo-inverse of the Jacobian matrix.Joint constraints, collision avoidance [121, [151, and control ofmanipulators postures can be naturally integrated in thisframework of operational space control systems. Also, a newsystematic solution to the problem of kinematic singularitieshas been presented. This solution constitutes an effective

alternative to resolving end-effector motions into oint motiongenerally used in joint space based control systems.ACKNOWLEDGMENTS

The author would like to thank Brian Armstrong, HarlyBaker, Joel Burdick, Bradley Chen, John Craig, Ron FearingJean Ponce, and Shashank Shekhar for the discussionscomments, and help during the preparation of the manuscripREFERENCES

[l] B. Armstrong, 0 . Khatib, and J . Burdick, The explicit ynamicmodel and inertial parameters of the PUMA 560 arm , in Proc. I98ZEEE Znt. Conf. Robotics and Automation, pp. 510-518.[2]B. Chen, R. Fearing, B. Armstrong, and J. Burdick, NYMPH: Amultiprocessor for manipulation applications, in Proc. I986 ZEEZnt. Conf. Robotics and Automation, pp. 1731-1736.[3] J . J. Craig, P. Hsu, S . S . Sastry, Adaptive controlofmechanicalmanipulators, in Proc. 1986 ZEEEZnt. Conf. Robotics anAutomation, pp.190-195.[4] B. Espiau and R. Boulic, Collision avoidance for redundant obotswith roximity sensors , in Preprints 3rdZnt. Symp. RoboticResearch, 1985, pp. 94-102.[5] A. Fournier, Gt ntrat ionde mouvements en robotique-applicationdes inverses gtntralisb es t des pseudo inverses, Thts e detaMention Science, Univ. des Sciences et TechniquesduLanguedoc,Montpellier, France , Apr. 1980.[6] H. Hanafusa, T. Yoshikawa, and Y. Nakamura, Analysis and controof articulated robot arms with redundancy, in Proc. 8th ZFAC WorlCongress, vol. XIV, 1981, pp. 38-83.[7] J. R. Hewit and J . Padovan, Decoupled feedback control of a roboand manipulator arm s, in Proc. 3rd CZSM-ZFToMM Symp. Theorand Practice of Robots and Manipulators. NewYork: Elsevie

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