is greater than the dimension m of x, but we do not impose any globally applied assumptions regarding the dimension of the variable 9. We wish to study models ...
Kinematically Redundant Robots WW Flexible Components J. Baillieul
The robot manipulators studied here simultaneously exhibit features of joint elasticity and kinematic redundancy. More specifically, manipulators in which there are more controlled (joint space) degrees of freedom than task space degrees of freedom and in which at least one of these controlled joints is connected to an adjacent link through an elastic coupling are discussed. It will be shown that there is a wide class of examples for which it is impossible to find joint space motions which track a given end effector trajectory and d o not at the same time excite elastic dynamics. No assumptions on the smoothness of the prescribed end effector trajectory or the degree of redundancy alter this conclusion. For this class of robot manipulator geometries, the most we can in general hope to do is to plan motions which move the end effector between prescribed endpoints without any net storage of elastic energy in the joints. For certain congenial manipulator geometries (including cases in which all joints are revolute). it is shown that under certain assumptions it is possible to plan motions which store no elastic energy. These motions
Presented at the 1992 IEEE International Conjerence on Robotics and Automation, Nice, France, Mriy 1992. The author is vt’ith the Department of Aerospace and Mechanical Engineering. Boston, MA 02215. This work was supported by the U.S. Air Force under Ci-ant AFOSR90-0226.
February 1993
involve constraints relating accelerations of the end effector along the work space path to joint space velocity and configuration variables. In a typical example, it is shown how trajectories interpolating prescribed endpoints in the work space may be planned using fifth order splines. A brief discussion of the problem of exactly tracking end effector trajectories with minimum average elastic energy storage is also provided. The problem of planning motions which interpolate between given points in the work space is similar to recently studied problems in the control theory of nonholonomic systems. Interesting open questions involve the existence and multiplicity of interpolating work space paths, the geometry of the reachable work space, and the characterization of algorithmically singular behavior.
Elastic Links and Joints There is widespread interest within the research community in the design and control of robot arms with elastic links and joints. In principle. such devices have a number of advantages. By admitting the possibility of elastic deformations, it is possible to design lighter robots which are more efficient to operate and which have a more favorable ratio of maximum payload to weight. Moreover, such mechanisms
0272- 1708/93/$03.000 I993IEEE
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exhibit passive compliance which may be exploited in a variety of manipulation tasks. Robots with elastic components can, of course, “store energy.” and while this is usually perceived as something to be avoided through the active damping of vibrations, one can show that such energy storage enriches the set of possible modes of behavior of the mechanism and may be exploited to achieve control objectives which would not be possible with rigid-link mechanisms. The present paper will be concerned, however. only with the problems of utilizing robot kinematics to avoid storing elastic energy. We shall consider open kinematic chain mechanisms consisting of a sequence of rigid or elastic links joined together by single degree of freedom joints which may, depending on the model in question, undergo elastic deformation. The principal object of our analysis will be robot manipulators, and hence we assume that there is a “base” of any kinematic chain under investigation as well as a “tip” to which a tool or end effector is attached. The forward kinematics of such mechanisms associates to each joint space configuration a description of the position and orientation of the end effector relative to the chosen base frame. The simplest models of robots with elasticity involve joints which are coupled to adjacent links through simple linear springs (i.e., springs obeying a linear Hooke’s law), and such models will be the primary focus of this paper. (cf. [I].) This case is depicted for a revolute joint in Fig. 1. In general, we shall assume that the relative position between two links in the mechanisms we study are given in terms of two variables: 0, to which controls may be applied and 9,measuring the elastic deformations. A kinematics function then describes the position and orientation of the end effector (with respect to the base) in terms of the controlled joint variables 0 and the deformation variables 9:
Here we assume thatx is a suitably chosen vector quantity describing the position and orientation of the end effector of the manipulator. and 0 and Q are vectors of controlled joint variables and elastic deformations respectively as described above. We assume the dimension. say n , of 0 is greater than the dimension m of x, but we do not impose any globally applied assumptions regarding the dimension of the variable 9.We wish to study models in which some components of the mechanism may be more flexible than others, and to permit treatment of the widest possible variety of cases. we allow for the possibility that 9 will have fewer
Fig. I . A revolute joint with elastic defonnution. Or is the controlled variable; describes elastic deformation.
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Fig. 2. A redi~nduntmunipulatorwith one-dimensional work space. The end effector is coupled to thefinal actuutedjoint through a simple linear spring of stiffness k. dimensions than 0 (in the case that some joints in the manipulator are relatively stiff), or that I$could have more dimensions than 0, say in the case that we needed to model more than a single mode ofelastic behavior in a certain set ofjoints. We shall not deal with link elasticity here. Link elasticity is typically studied using distributed parameter models, some recent work on which is reported in [2] and [3]. We wish to draw a distinction between the subject matter at hand and the growing technical literature on the dynamics and control of robots with elastic components (e.&. [I], [4]-[6]). Little appears to have been written regarding the use of kinematic redundancy to mitigate the undesirable effects of elasticity, although very recent work reported by Nguyen et al. ([7]) suggests that interest in the area may be increasing. The idea studied in [7] is to use redundant degrees of freedom to alter the dynamic response in a robot manipulator with elastic components. A simple example serves to illustrate the nature of the problem and to suggest the intrinsic limits to be expected in its solution.
E-rumple I The simplest possible context in which to explore whether kinematic redundancy can be exploited to mitigate the effects of elasticity is provided by the one dimensional “robot” model depicted in Fig. 2. Here the hand of the robot holds an object which together with the hand has mass n i l . Actuation is applied to the mass designated mi, and this is coupled to the hand through 3 simple linear spring of stiffness k. There may be any number of actuated prismatic joints t o the left of the mass m i . To simplify the problem, it is assumed that there are no elastic deformations in the joints between mass in1 and the base. We wish to investigate the possibility of making the end effector follow a prescribed path in which the payload moves between the positions x = 0 and x = I in one unit of time. We assume the beginning and ending velocities will be zero. It is straightforward to write down the relevant dynamics:
First Obserimtion It is impossible to execute the desired motion ~ i t h o u thaving energy stored in the spring at some point along the trajectory. This may be verified using the second equation. Not storing energy in the spring implies ~ ( t- ). x l ( t ) = 0. But then = 0, and combining this with the left hand boundary conditions x z ( 0 ) = i 2 ( 0 ) = 0 requires s i ( t ) = 0 rendering it impossible to satisfy the right hand boundary condition.
/ € EControl Systems
This very simple observation shows that no matter how many degrees of freedom are present, it may be impossible to avoid exciting the elastic dynamics. Given this observation, we might ask whether it is possible to achieve the more modest objective of having no energy stored in elastic components at the end of the trajectory. With some qualification, the answer is affirmative, and it is again noted that for the simple example at hand, the answer does not depend on whether there are redundant degrees of freedom.
Second Observcition It i s possible to control the system (2) so that motions may be prescribed by polynomial interpolating functions meeting the endpoint conditions x?(O) = 0, x 2 ( 1 ) = 1 while satisfying the additional constraint that there is no resulting energy stored i n the spring at the end of the motion. Motions of this type may be achieved if .xz(.) is prescribed by a seventh degree polynomial. To see this, note that.ri( .) is completely determined by the second equation in (2). once .u?(.) is prescribed. The necessary endpoint conditions x2(0) = 0. X?( 1j = I
k, (0j = .i2(0) = .Ii ( I ) = i. 1( )=0
together with the conditions for non-storage of energy: .xl(O) = 0. . x i ( 1 ) = I are eight in number. In order to have enough parameters to satisfy these conditions, apolynomial specifying .x.(.)
must clearly have degree
2 7, and a straightforward calculation shows that in fact all boundary
conditions may be met using polynomials of degree ';even. Using degree seven polynomial interpolation of x ? ( ,) with the given boundary conditions results in the curves depicted in Fig. 3.
Fig. 3. Typical trc,i(O)) = (0.0) and ( x ~ ( l ) , y ~ ( l ) ) = ( l , Writing.ul(r)=rIo+rrii+...+n,,t” O). andyl(t)= bo+bl t+...+bfit", i t follows from the four (inherited) boundary conditions x i ( 0 ) =.\-i(O) = ji(0) =?.1(0)= Othat C I O = N I =bo=hi = O . The boundary conditions ( q ( I ) , y i ( l ) ) = (l,O), (.i-~(l),?.l(l))= (0,O) provide four additional constraints. Let O(j = O(0). Since ( I 2) must be satisfied at the initial time t = 0, it follows that
-0.5
-1
t
I
1.5
W
p = 1.0
F, = 0.404267
E.
= 0.684224
((.I Differentiating ( 1 2) we find that
The requirement that 0(O) = 0 iniplies
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Fig. 6. Fifth-order ptitl1.c in ~.ilzic~h the erzd-effector moves from (.x/(0),~i(O)) = (0,O) to (x/(l),j(I ) ) = (1.0) without e.wcitiiig elastic “\riai?iics. Note the /nrLyrinotioris required of the hub.
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Reninrk 2 Actually more information can be immediately deduced about such polynomial trajectories. It follows from (13) and (15) that there exist real numbers h.p such that
Under the constraints we have enumerated. we can detennme the nonLero polynomial coetficients a? = A cos Bo hi = h sin Bo q = y cos 80 173 = g sin 80 LIJ
=5
rig = -3
(3h + 2y) cos 80 ha = -(3h + 2p) sin eo + (?A + 17711) cos 80 bs = (2h + p ) sin 80).
-
The final constraint determines a polynotnial relationship between 1 and m:
References 1 I ] A. De Luca. "Dynamic control of robots with joint elasticity," in Proc. IY88 /EE€ / f 7 f . Conf Rohorics and Auromulion, Philadelphia, PA, Apr. 24-29. pp. 152-158. 1988. [?I J.C. Sinio. J.E. Marsden, and P.S. Krishnaprasad, "The Hamiltonian structure of nonlinear e h i c i t y : The material and convective representationofrods, plates. and shells."Arc.k. RurioncdMech. Anul., vol. 104. pp. 125-183, 1988. 131 J . Baillieul and M. Levi. "Constrained relative motions in rotational iiiechanics."Alr,/~.Rtrrionri/ Mrch. A n d . . vol. 115. pp. 101-135, 1991. 141 S. Cetinkunt and W.J. Book, "Performance limitations ofjoint variable-feedback controllers due to manipulator structural flexibility." /EE€ Truns. Robotics ri/7t/Airronztrrion. vol. 6, no. 2. pp. 2 19-23 I . 1990.
[SI A. De Luca and B. Siciliano. "Closed-form dynamic model of planar Man, Cjkvrn., Feb. 1991. multi-link lightweight robots." IEEE Truns. SXSZ.. 161 K. Khomsani. "Adaptive control of flexible joint robots," in Proc. IY91 I€€€ Au/rJnifirihi, Sacramento. CA, Apr. 9- 1 1. 1991, pp. 2 117-21 34.
/ ~ 7 r . COJ:~.' Rohoricx tind
L.A. Nguyen. I.D. Walker. and R.J.P. DeFigueiredo, "Dynamic control of llexible, hinetnatically redundant robot manipulators." /€E€ Trrtrzs.Robotics and [ 71
With fifth order spline interpolation. there is thus ;I $ingle degree of freedom (given BO) in specifying work space trajectories which satisfy the boundary conditions and along which (12) holds. Once s i ( / ) and yl(f) have been specified. we solve (12) for 8(f). Altematively. we can solve (14) for e(/)in terms of the parameters A,p and initial value ($0) = Bo. x ( f )and y ( t ) are then determined froin ( I O ) . Fig. 6 exhibits motions taking the end-effector from (.I 1(0),y1(0)) = (0.0) to (xi( i),yi( 1 )) = ( I ,0) for different values of hp satisfying the equation ( I 7). Note that as h,y --f 0 along the constraint locus. the end-effector follows a path which increasingly closely approximates the straight line segment. For 8 #O. x. however. it is not possible for the end-effector to exactly follow the straight line segment from (0.0)to ( I ,O), and this is retlected in the polynomial paths we have constructed requiring very large velocities 8 along certain segments of the motion when h.y - 0.
,\iironwri(in,
1993.
181 J. Baillieul. "The behavior of single-input wper-articulated mechanisms," in Pro(, lYY/A/n
