David DeMers

Inverse Kinematics of Dextrous Manipulators

ABSTRACT Manipulators with extra, or redundant, degrees of freedom are capable of dextrous motion. However, control of such manipulators is difficult because the inverse problem; i.e., the choice of a set of control variables which positions the manipulator at a desired target location, is underdetermined. This chapter has two main points. First, it provides a taxonomy of approaches to solving the inverse problem for dextrous manipulators which outlines the relationship between the standard robotics approaches and approaches using neural networks. Second, this chapter analyzes the topological structure of the kinematics problem, showing that the inverse kinematic structure generically decomposes into a finite number of solution branches, each corresponding roughly to the notion of “posture,” such that each solution branch can be described as the product of a one-to-one inverse map and a set of free parameters describing the redundancy. Using these geometric insights, one can then generate direct inverse functions for dextrous manipulators, with which the free parameters can be used flexibly during operation to optimize any user-specified side constraint.

Canonically parameterized families of inverse kinematic functions for redundant manipulators

The workspace of a redundant manipulator can be partitioned into a finite number of invertible subsets such that the inverse image of each subset has trivial fiber bundle topology, with the manifolds of manipulator self-motion as the fibers. As a consequence we show that a canonical representation of the self-motion manifolds is possible over these well-defined subsets. The canonical representation parameterizes a family of inverse functions over the work space subsets. This parameterization can be used to approximate a continuous inverse kinematic function over each workspace subset. Setting appropriate valves of the parameters yields particular inverse functions which provide access to all solution branches and to the full range of the redundant degrees of freedom in order to satisfy any side constraints which may be imposed during operation.<<ETX>>

Issues in learning global properties of the robot kinematic mapping

The robotic kinematic mapping generally has multiple distinct solution branches for a given end-effector location, where each branch can have a nontrivial manifold structure (as in the case of a redundant manipulator). Learning techniques that exploit known topological properties of the mapping are used to determine the number and nature of these branches. Specifically, clustering of input-output data is used to map out the preimage branches. Topology preserving networks are used to learn and parameterize the topology of these branches for certain known classes of manipulators. As a practical consequence, the inverse kinematic mapping can be approximated for each branch separately.<<ETX>>

Learning Global Properties of Nonredundant Kinematic Mappings

The kinematic mapping x = f(θ) is generally many to one. For nonredundant manipulators, this means that there are a finite num ber of configurations (joint angles) that will place the end-effector at a target location in the workspace. These correspond to pos tures of the manipulator, and each configuration lies on a specific solution branch. It is shown that for certain classes of revolute joint regional manipulators (those with no joint limits and having almost everywhere a constant number of inverse solutions in the workspace), the input-output data can be analyzed by clustering methods in order to determine the number and location of the so lution branches. As a practical consequence, the inverse kinematic mapping can be directly approximated by applying neural network or other learning-based methods to each branch separately.

Operator approach to recursive Jacobian inversion and pseudoinversion

Operator forms of the inverse and Moore-Penrose pseudoinverse of the Jacobian for a nonsingular n-link serial spatial manipulator are developed. It is shown that the Jacobian pseudoinverse operation that maps an end-effector velocity to a corresponding joint-space velocity can be performed using an O(n) recursive algorithm involving one 6*6 matrix inversion. It is also shown that an alternative recursive algorithm that produces a kinematic singularity measure as a byproduct can be used to perform the Jacobian inverse operation without the need for a matrix inversion.<<ETX>>

Learning Global Topological Properties of Robot Kinematic Mappings for Neural Network-based Configuration Control

We examine the topology of the kinematics mapping and propose methods for resolving redundancy and the multiplicity of disjoint pre-image manifolds, resulting in a global regularization of the ill-posed inverse problem.. Reasonable assumptions about the kinematic mapping, based on the physical properties of a robot arm, are exploited to learn the global topology of the mapping. Heuristic algorithms are applied to the wristless Puma 560 and the 3-link planar arm. We show that the disjoint pre-image manifolds in the configuration space, each. corresponding to one of the multiple solution branches, can be identified with a high degree of precision by a combination of nearest-neighbor clustering with an adaptive classifier. We also show that one-dimensional pre-image manifolds may be parameterized in a consistent manner with topology-preserving neural networks. Thus the kinematic mapping can be globally regularized, and both the forward and inverse mappings can be completely learned. A non-linear function approximator such as a neural network can then be used to provide a solution to the inverse kinematics problem, allowing configuration control at a logical level.