Global Regularization of Inverse Kinematics for Redundant 
Manipulators 
David DeMers 
Dept. of Computer Science & Engr. 
Institute for Neural Computation 
University of California, San Diego 
La Jolla, CA 92093-0114 
Kenneth Kreutz-Delgado 
Dept. of Electrical & Computer Engr. 
Institute for Neural Computation 
University of California, San Diego 
La Jolla, CA 92093-0407 
Abstract 
The inverse kinematics problem for redundant manipulators is ill-posed and 
nonlinear. There are two fundamentally different issues which result in the need 
for some form of regularization; the existence of multiple solution branches 
(global ill-posedness) and the existence of excess degrees of freedom (local ill- 
posedness). For certain classes of manipulators, learning methods applied to 
input-output data generated from the forward function can be used to globally 
regularize the problem by partitioning the domain of the forward mapping into 
a finite set of regions over which the inverse problem is well-posed. Local 
regularization can be accomplished by an appropriate parameterization of the 
redundancy consistently over each region. As a result, the ill-posed problem can 
be transformed into a finite set of well-posed problems. Each can then be solved 
separately to construct approximate direct inverse functions. 
1 INTRODUCTION 
The robot forward kinematics function maps a vector of joint variables to the end-effector 
configuration space, or workspace, here assumed to be Euclidean. We denote this mapping 
by f(.): O ' --, V " C_ '", f(O) - x, for 0 60 ' (the input space or joint space) 
and x  W m (the workspace). When m < n, we say that the manipulator has redundant 
degrees-of-freedom (doO. 
The inverse kinematics problem is the following: given a desired workspace location x, 
find joint variables 0 such that f(O) = x. Even when the forward kinematics is known, 
255 
256 DeMers and Kreutz-Delgado 
the inverse kinematics for a manipulator is not generically solvable in closed form (Craig, 
1986). This problem is ill-posed  due to two separate phenomena. First, multiple solution 
branches can exist (for both non-redundant as well as redundant manipulators). The second 
source of ill-posedness arises because of the redundant dofs. Each of the inverse solution 
branches consists of a submanifold of dimensionality equal to the number of redundant 
dofs. Thus the inverse solution requires two regularizations; global regularization to select 
a solution branch, and local regularization, to resolve the redundancy. In this paper the 
existence of at least one solution is assumed; that is, inverses will be sought only for points 
in the reachable workspace, i.e. desired x in the image of f(.). 
Given input-output data generated from the kinematics mapping (pairs consisting of joint 
variable values & corresponding end-effector location), can the inverse mapping be learned 
without making any a priori regularizing assumptions or restrictions? We show that the 
answer can be "yes". The approach taken towards the solution is based on the use of 
learning methods to partition the data into groups such that the inverse kinematics problem, 
when restricted to each group, is well-posed, after which a direct inverse function can be 
approximated on each group. 
A direct inverse function is desireable. For instance, a direct inverse is computable quickly; 
if implemented by a feedforward network were used, one function evaluation is equivalent to 
a single forward propagation. More importantly, theoretical results show that an algorithm 
for tracking a cyclic path in the workspace will produce a cyclic trajectory of joint angles 
if and only if it is equivalent to a direct inverse function (Baker, 1990). That is, inverse 
functions are necessary to ensure that when following a closed loop the ann configurations 
which result in the same end-effector location will be the same. 
Unfortunately, topological results show that a single global inverse function does not exist 
for generic robot manipulators. However, a global topological analysis of the kinematics 
function and the nature of the manifolds induced in the input space and workspace show 
that for certain robot geometries the mapping may be expressed as the union of a finite set 
of well-behaved local regions (Burdick, 1991). In this case, the redundancy takes the form 
of a submanifold which can be parameterized (locally) consistently by, for example, the 
use of topology preserving neural networks. 
2 TOPOLOGY AND ROBOT KINEMATICS 
It is known that for certain robot geometries the input space can be partitioned into disjoint 
regions which have the property that no more than one inverse solution branch lies within 
any one of the regions (Burdick, 1988). We assume in the following that the manipulator in 
question has such a geometry, and has all revolutejoints. Thus O ' - T n, the n-torus. The 
redundancy manifolds in this case have the topology of T n-m, n - m-dimensional torii. 
For O ' a compact manifold of dimensionality n, W m a compact manifold of dimensionality 
m, and f a smooth map from O ' to W , let the differential de f be the map from the tangent 
space of O ' at 0 60 ' to the tangent space of W " at f(O). The set of points in O ' which 
]Ill-posedness can arise from having either too many or too few constraints to result in a unique 
and valid solution. That is, an overconstrained system may be ill-posed and have no solutions; such 
systems are typically solved by finding a least-squares or some such minimum cost solution. An 
underconstrained system may have multiple (possibly infinite) solutions. The inverse kinematics 
problem for redundant manipulators is underconstrained. 
Global Regularization of Inverse Kinematics for Redundant Manipulators 257 
map to x 6 YV " is the pre-image of x, denoted by f- (x). The differential der has a 
natural representation given by an m by n Jacobian matrix whose elements consist of the 
first partial derivatives of f w.r.t. a basis of 0 '. Define 3 as the set of critical points of f, 
which are the set of all 0 6 0 ' such that der(o) has rank less than the dimensionality of 
)4, '". Elements of the image of 3, f(3) are called the critical values. The set 7Ea=YV"\3 
are the regular values of f. For 0  0 ', if 30*  f-(f(O)), O*  3, we call 0 a 
co-regular point of f. 
The kinematic mapping of certain classes of manipulators (with the geometry herein as- 
sumed) can be decomposed based on the co-regular surfaces which divide O" into a finite 
number of disjoint, connected regions, Ci. The image of each Ci under f is a connected 
region in the workspace, )V. We denote the kinematics mapping restricted to Cto be f; 
Locally, one inverse solution branch for a region in the workspace has the structure of 
a product space. We conjecture that (Wi,7-n-m,Ci, fICi) forms a locally trivial fiber 
bundle 2 and that the i, therefore, form regions where the inverse is unique modulo the 
redundancy. 
Given a point in the workspace, x, for which a configuration is sought, global regular- 
izafion requires choosing from among the multiple pre-image tofii. Local regularizafion 
(redundancy resolution) requires finding a location on the chosen torus. We would like to 
effectively "rood out" the redundancy manifolds by constructing an indexed one-to-one, 
invertible mapping from each pre-image manifold to a point in A 'm, and to obtain a con- 
sistent representation of the manifold by constructing an invertible mapping from itself to 
a set of n - m "location" parameters. 
3 GLOBAL REGULARIZATION 
The existence of multiple solutions for even non-redundant manipulators poses difficult 
problems. Usually (and often for plausible reasons) the manipulator's allowable configura- 
tions or task space is effectively constrained so that there exists only a single inverse solution 
(Martinetz et al., 1990; Kuperstein, 1991). This approach regularizes the problem by al- 
lowing the existence of only one-to-one data. We seek to generalize to the multi-solution 
case and to learn all of the possible solutions. 
For an all revolute manipulator there typically will be multiple pre-image torii for a particu- 
lar point in the workspace. The topology of the pre-image solution branches will generally 
be known from the type of geometry, although their number may not be obvious by inspec- 
tion, and will usually be different for different regions of the workspace. An upper bound on 
the number of inverse solutions is known (Burdick, 1988); consequently, the determination 
of the number could be made by search for the best fit among the possibilities. 
The sampling and clustering approach described in (DeMers & Kreutz-Delgado, 1992) 
can be used to partition input-output data into disjoint pre-image sets. This approach 
uses samples of the forward behavior to identify the sets in the input space which map to 
-A fiber bundle is a four-tuple consisting of a base space, a fiber (here, 7"-'), a total space and 
a projection p mapping the total space to the base space with certain properties (here, the projection 
is equivalent to fi restricted to the total space). A locally trivial fiber bundle is one for which a 
consistent parameterization of the fibers is possible. 
258 DeMers and Kreutz-Delgado 
oin 3 
Figure 1' The two pre-image manifolds for positioning the end-effector of the 3-R planar 
manipulator at a specific (x, y) location. 
within a small distance of a specific location in the workspace. The pre-image points will 
lie on the disjoint pre-image manifolds. These manifolds will typically be separable by 
clustering 3. Figures 1 shows two views of the two redundancy, or "self-motion" manifolds 
for a particular end-effector location of the 3-R planar ann. All of the input space points 
shown position the ann's end-effector near the same (x,y) location. Note that the input 
space is the 3-torus, T 3. In order to visualize the space, the torus is "sliced" along each 
dimension. Thus opposite faces of the "cube" shown are identified with each other. 
4 LOCAL REGULARIZATION 
The inverse kinematics problem for manipulators with redundant dofs is usually solved 
either by using differential methods, which attempt to exploit the redundancy by optimizing 
a task-dependent objective function, or by using learning methods which regularize at 
training time by adding constraints equal to the number of redundant dof. The former may 
be computationally inefficient in that iterative solution of derivatives or matrix inversions 
are required, and may be unsatisfactory for real-time control. The latter is unsatisfying as 
it eliminates the run-time dexterity available from the redundancy; that is, it imposes prior 
constraints on the use of any extra dofs. 
Although in practice numerical, differential methods are used for redundancy resolution, 
it has recently been shown that simple recurrent neural networks can resolve the redun- 
dancy by optimization of certain side-constraints at run-time, (Jordan & Rumelhart, 1992), 
(Kindermann & Linden, 1990). The differential methods have a number of desireable prop- 
erties. In general it is possible to iterate in order to achieve a solution of arbitrary accuracy. 
They also tend to be capable of handling very flexible constraints. Global regularizafion as 
discussed above can be used to augment such methods. For example, once a choice of a 
solution branch has been made, an initial starting location away from singularities can be 
selected, and differential methods used to achieve an accurate solution on that branch. 
Our work shows that construction of redundancy-parameterized approximations to di- 
SFor end-effector locations near the co-regular values, the pre-image manifolds tend to merge. 
This phenomenon can be identified by our methods. 
Global Regularization of Inverse Kinematics for Redundant Manipulators 259 
rect inverses are achievable. That is, the mapping from the workspace (augmented by 
a parameterization of the redundancy) to the input space can be approximated. This lo- 
cal regularization is accomplished by parameterizing the pre-image solution branch torii. 
Given (enough) samples of  points paired with their x image, a parameterization can be 
discovered for each branch. The method used exploits the fact that neighborhoods within 
each Ci map to neighborhoods, and that neighborhoods within each 142i have as pre-images 
a finite number of solution branches. 
First, all points in our sample which have their image near some initial point xo are 
found. Pullin[; back to the input space by accessing the 0 component of each of these 
(0, x) data pairs finds the points in the pre-image set of the neighborhood of xo. Now, 
because the topology of the pre-image set is known (here, the torus), a self-organizing 
map of appropriate topology can be fit to the pre-image points in order to parameterize 
this manifold. Neighboring torii have similar parameterizations; thus by repeating this 
process for a point x near xo and using the parameterization of the pre-image of xo as 
initial conditions, a parameterization of the pre-image of x qualitatively similar to that 
for xo can be constructed efficiently. By stepping between such "query points", x a set of 
parameterizations can be obtained. 
5 THE 3-R REDUNDANT PLANAR ARM 
This approach can be used to provide a global and local regularization for a three-link 
manipulator performing the task of positioning in the plane. For this manipulator, f  
T 3  I 2. The map f restricted to each connected region in the input space bounded by 
the co-regular separating surfaces defines f: T x I x S   I x S . 
The pre-image of a point in the workspace thus consists of either one or two 1-torii (the 
actual number can be no more than the number of inverse solutions for a non-redundant 
manipulator of the same type, (Burdick, 1988)). Each torus is the pre-image of one of the 
restricted mappings fi. The goal is to identify these tofii and parametefize them. Figure 2 
shows the input space and workspace of this arm, and their separating surfaces. These 
partition the workspace into disjoint annular regions, and the input space into disjoint tubular 
regions. The circles indicate workspace locations which can be reached in a kinematically 
singular configuration. The inverse image of these circles form the co-regular separating 
surfaces in the input space. For the link lengths used here (l = 5, 12 = 4, 13 = 3), there 
is a single self-motion manifold for workspace locations in regions A and C, and two 
self-motion manifolds for workspace locations in B and D. Figure 3 shows two views of 
the data points near the pre-image manifolds for an end effector location in region A, and 
its parametefizafion by a self-organizing map (using the elastic net algorithm). Such a 
parametefizafion can be made for various locations in the workspace. Inverse kinematics 
can thus be computed by first locating the nearest parameterizafion network for a given 
workspace position and then choosing a configuration on the manifold, which can be done 
at run-time. Because the kinematics map is locally smooth, interpolation between networks 
and nodes on the networks can be done for greater accuracy. For convenience, a node on 
each torus was chosen as a canonical zero-point, and the remaining nodes assigned values 
based on their normalized distance from this point. Therefore all parameter values are 
scaled to be in the interval [0,1]. 
For some end-effector locations, there are two pre-image manifolds. These first need to 
be identified by a global partitioning, then the individual manifolds parametefized. The 
260 DeMers and Kreutz-Delgado 
InputSpace 
Workspace 
Joint 2 
0 n 
Joint 3 
Figure 2: The forward la'nematics for a generic three-link planar manipulator. The sep- 
arating surfaces in the input space do not depend on the value of joint angle 1, therefore 
the input space is shown projected to the joint 2 - joint 3 space. All end-effector locations 
inside regions A and C of the workspace have a single pre-image manifold in A and C, 
respectively, of the input space. End-effector locations inside regions B and D of the 
workspace have two pre-image manifolds, one in each of B and B' (resp. D and D' ) of the 
input space. 
manifolds may belong to one of a finite set of homotopy classes; that is, because they "live" 
in an ambient space which is a torus, they may or may not wrap around one or more of the 
dimensions of the torus. Unlike in Euclidean space, where there are only two possible one- 
dimensional manifolds, there are multiple topologically distinct types (homotopy classes) 
of closed loops which can serve as self-motion manifolds. Fortunately, because physical 
robots rarely have joints with unlimited range of motion, in practice the manifolds will 
usually not have wraparound. However, we should like to be able to parameterize any 
possibility. Appropriate choice of topology for a topology-preserving net results in an 
effective parameterization. Figure 4 shows two views of a parameterization for one of the 
self-motion manifolds shown above in Figure 1, which is the pre-image for an end-effector 
location in region B of Figure 2. 
6 DISCUSSION 
The global regularization accomplished by the method described above partitions the origi- 
nal input/output data into sets for each of the distinct i regions. The redundancy parameters, 
l, obtained by local regularization can be used to augment this data, resulting in a transfor- 
marion of the (0, x) data into (Oi, (xi, t)). Let 7-: 0  t be a function that computes a 
parameter value for each 0 in the input space. Let fi(O) = (fi(0), 7-(0)). By construction, 
the regularized mapping fi ' 0  (x, ) is one-to-one and onto. Now, given examples 
from a one-to-one mapping, the inverse map f/- (x, )  0 can be direcfiy approximated 
by, e.g., a feedforward neural network. 
Global Regularization of Inverse Kinematics for Redundant Manipulators 261 
Joint 3 Angle 
Joint 2 Angle 
-1 0 1 
/ 
oint i Angle 
2.5 
Figure 3: The data points in the "self-motion" pre-image manifold of a point in the 
workspace of the 3-R planar arm, and a closed, 1-D elastic network after adaptation to 
them. This manifold is smoothly contractible to a point since it does not "wrap around" 
any of the dimensions of T 3. 
This method requires data sample sizes exponential in the number of degrees of freedom of 
the manipulator and thus will not be adequate for large dof"snake" manipulators. However, 
practical industrial robots of 7--dof may be amenable to our technique, especially if, as is 
common, it is designed with a separable wrist and is thus composable into a 4-dof redundant 
positioner plus a 3--dof non-redundant orienter. 
This work can also be used to augment the differential methods of redundancy resolution. An 
approximate solution can be found extremely rapidly, and used to initialize gradient-based 
methods, which can then iterate to achieve a highly accurate solution. Global decisions such 
as choosing between multiple manifolds and identifying criteria for choosing locations on 
the manifold can now be made at run-time. Computation of an approximate direct inverse 
can then be made in constant time. 
Acknowledgements 
This work was supported in part by NSF Presidential Young Investigator award IRI- 
9057631 and Fellowships from the California Space Institute and the McDonnell-Pew 
Center for Cognitive Neuroscience. The first author would like to thank the NIPS Founda- 
tion for providing student travel grants. 
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