Krzysztof Tchoń

Endogenous configuration space approach to mobile manipulators: A derivation and performance assessment of Jacobian inverse kinematics algorithms

By a mobile manipulator we mean a robotic system composed of a non-holonomic mobile platform and a holonomic manipulator fixed to the platform. A taskspace of the mobile manipulator includes positions and orientations of its end effector relative to an inertial coordinate frame. The kinematics of a mobile manipulator are represented by a driftless control system with outputs. Admissible control functions of the platform along with joint positions of the manipulator constitute the endogenous configuration space. Endogenous configurations have a meaning of controls. A map from the endogenous configuration space into the taskspace is referred to as the instantaneous kinematics of the mobile manipulator. Within this framework, the inverse kinematic problem for a mobile manipulator amounts to defining an endogenous configuration that drives the end effector to a desirable position and orientation in the taskspace. Exploiting the analogy between stationary and mobile manipulators we present in the paper a collection of regular and singular Jacobian inverse kinematics algorithms. Their performance is evaluated on the basis of intense computer simulations.

Singular inverse kinematic problem for robotic manipulators: a normal form approach

In this article, a new method of solving the singular inverse kinematic problem for nonredundant robotic manipulators is proposed, based on normal forms of the manipulator kinematics. A complete solution to the inverse problem is presented under assumption that the kinematics are equivalent to the quadratic normal form. Comparisons with other existing methods, including the singularity-robust inverse, are made that reveal advantages of the normal form approach. Experimental results concerned with singular tracking in the AdeptOne manipulator are reported.

Instantaneous kinematics and dexterity of mobile manipulators

We consider the kinematics of a mobile manipulator that consists of a nonholonomic (or holonomic) mobile platform and a holonomic manipulator mounted atop of the platform. At any fixed time instant and with fixed initial posture of the platform these kinematics are viewed as a map from a Hilbert space of controls and manipulator joint positions into a task-space of the mobile manipulator, and called the instantaneous kinematics. A derivative of this map is referred to as the instantaneous analytic Jacobian of the mobile manipulator. We derive an instantaneous dexterity matrix and a dexterity measure of the mobile manipulator. The dexterity matrix is used in order to introduce an instantaneous dexterity ellipsoid of the mobile manipulator. Concepts are illustrated with examples of mobile manipulators equipped with both holonomic and nonholonomic mobile platforms.

Differential topology of the inverse kinematic problem for redundant robot manipulators

In this paper a program of singularity theory is pro claimed to be of systematic use in robotics. Complete lists of normal forms are proposed and are regarded as candi date models of kinematics of robot manipulators. Argu ments for the applicability of candidate normal forms to manipulator kinematics are provided. Singularities and bifurcation diagrams of the normal forms are examined and consequences derived for the inverse kinematic prob lem in redundant kinematics with singularities.

Repeatability of inverse kinematics algorithms for mobile manipulators

We introduce and examine the property of repeatability of inverse kinematics algorithms for mobile manipulators. Similarly to stationary manipulators, repeatability of mobile manipulators is defined by requiring that a closed path in the task space should be transformed by the inverse kinematics algorithm into a closed path in the configuration space. In a simply connected, singularity-free region of the task space, a necessary and sufficient condition for repeatability is derived as the integrability condition of a distribution associated with the inverse kinematics algorithm.

Singularities of nonredundant robot kinematics

This article is concerned with an investigation of kinematic singularities of nonredundant robot manipulators, conducted toward describing the kinematics by simple mathematical mod els called normal forms. The adopted approach comes from the singularity theory of maps. The analysis is concentrated on kinematic singularities of codimension 1. The kinematics at singular configurations are characterized by a number called the differential degree. It is proved that, if this degree is finite, then the kinematics can be transformed to normal forms called pre-Morin or Morin forms. If the differential degree becomes infinite, the kinematics can be given either the prehyperbolic or the hyperbolic normal form. Special attention has been paid to the kinematics of the Unimation PUMA manipulator. A sin gularity analysis of the PUMA kinematics has shown that in a neighborhood of the shoulder and elbow singular configura tions, the kinematics assume a simple quadratic (Morin) form, whereas at the wrist singularity, the kinematics are represented by the hyperbolic normal form.

On inverting singular kinematics and geodesic trajectory generation for robot manipulators

In this paper, a new method is proposed of solving the inverse kinematic problem for robot manipulators whose kinematics are allowed to possess singularities. The method is based upon the so-called generalized Newton algorithm, introduced by S. Smale, and can be adopted to both nonredundant and redundant kinematics. Moreover, given a pair of points in the external space of a manipulator, the method is capable of generating a minimum-length trajectory joining the points (a geodesic), in particular a straight-line trajectory. Results of representative computer experiments, including those with the PUMA 560 kinematics, are reported in order to illustrate the performance of the method.

Optimal Extended Jacobian Inverse Kinematics Algorithms for Robotic Manipulators

Extended Jacobian inverse kinematics algorithms for redundant robotic manipulators are defined by combining the manipulators kinematics with an augmenting kinematics map in such a way that the combination becomes a local diffeomorphism of the augmented taskspace. A specific choice of the augmentation relies on the optimal approximation by the extended Jacobian of the Jacobian pseudoinverse (the Moore-Penrose inverse of the Jacobian). In this paper, we propose a novel formulation of the approximation problem, rooted conceptually in the Riemannian geometry. The resulting optimality conditions assume the form of a Poisson equation involving the Laplace-Beltrami operator. Two computational examples illustrate the theory.

A repeatable inverse kinematics algorithm with linear invariant subspaces for mobile manipulators

On the basis of a geometric characterization of repeatability we present a repeatable extended Jacobian inverse kinematics algorithm for mobile manipulators. The algorithms dynamics have linear invariant subspaces in the configuration space. A standard Ritz approximation of platform controls results in a band-limited version of this algorithm. Computer simulations involving an RTR manipulator mounted on a kinematic car-type mobile platform are used in order to illustrate repeatability and performance of the algorithm.

Repeatable, extended Jacobian inverse kinematics algorithm for mobile manipulators

Abstract An inverse kinematics algorithm for a robotic system is called repeatable, if it transforms closed paths in the task space into closed paths in the configuration space. In this paper we assume the endogenous configuration space approach and define a repeatable inverse kinematics algorithm for mobile manipulators, patterned on the extended Jacobian scheme. An examination of a dynamic system associated with this algorithm gives new insight into the mechanism of repeatability. As an illustration of the general design strategy an extended Jacobian algorithm is applied to a kinematic car-type platform carrying the POLYCRANK, 7 DOF on-board manipulator.