Jacobian Inverse Kinematics Algorithm Research Articles

Approximation of Jacobian Inverse Kinematics Algorithms: Differential Geometric vs. Variational Approach

This paper addresses the approximation problem of Jacobian inverse kinematics algorithms for redundant robotic manipulators. Specifically, we focus on the approximation of the Jacobian pseudo inverse by the extended Jacobian algorithm. The algorithms are defined as certain dynamic systems driven by the task space error, and identified with vector field distributions. The distribution corresponding to the Jacobian pseudo inverse is non-integrable, while that associated with the extended Jacobian is integrable. Two methods of devising the approximating extended Jacobian algorithm are examined. The first method is referred to as differential geometric, and relies on the approximation of a non-integrable distribution (in fact: a codistribution) by an integrable one. As an alternative, the approximation problem has been formulated as the minimization of an approximation error functional, and solved using the methods of the calculus of variations. Performance of the obtained extended Jacobian inverse kinematics algorithms has been compared by means of computer simulations involving the kinematics model of the 7 dof industrial manipulator POLYCRANK. It is concluded that the differential geometric method offers a rapid, while the variational method a systematic tool for solving inverse kinematic problems.

Performance-Oriented Design of Inverse Kinematics Algorithms: Extended Jacobian Approximation of the Jacobian Pseudo-Inverse

For redundant robotic manipulators, we study the design problem of Jacobian inverse kinematics algorithms of desired performance. A specific instance of the problem is addressed, namely the optimal approximation of the Jacobian pseudo-inverse algorithm by the extended Jacobian algorithm. The approximation error functional is derived for the coordinate-free representation of the manipulator’s kinematics. A variational formulation of the problem is employed, and the approximation error is minimized by means of the Ritz method. The optimal extended Jacobian algorithm is designed for the 7 degrees of freedom (dof) POLYCRANK manipulator. It is concluded that the coordinate-free kinematics representation results in more accurate approximation than the coordinate expression of the kinematics.

Iterative learning control and the singularity robust Jacobian inverse for mobile manipulators

The method of iterative learning control, to a large extent, has been inspired by robotics research, focused on the control of stationary manipulators. In this article we deal with the inverse kinematics problem for mobile manipulators, and show that a very basic singularity robust Jacobian inverse can be derived in a natural way within the framework of iterative learning control. To achieve this objective we have exploited the endogenous configuration space approach. The introduced Jacobian inverse defines the singularity robust Jacobian inverse kinematics algorithm for mobile manipulators. A Kantorovich-type estimate of the region of guaranteed convergence of the algorithm is derived. For two example kinematics, this estimate has been computed efficiently.

Approximation of Jacobian inverse kinematics algorithms

Approximation of Jacobian inverse kinematics algorithmsThis paper addresses the synthesis problem of Jacobian inverse kinematics algorithms for stationary manipulators and mobile robots. Special attention is paid to the design of extended Jacobian algorithms that approximate the Jacobian pseudoinverse algorithm. Two approaches to the approximation problem are developed: one relies on variational calculus, the other is differential geometric. Example designs of the extended Jacobian inverse kinematics algorithm for 3DOF manipulators as well as for the unicycle mobile robot illustrate the theoretical concepts.

On Dynamic Properties of Singularity Robust Jacobian Inverse Kinematics

In the technical note we examine dynamic properties of the singularity robust Jacobian inverse kinematics algorithm for mobile manipulators. By design, this algorithm operates at regular as well as singular configurations. The main result of the technical note consists in establishing the completeness of the singularity robust algorithm, and in providing a condition for its convergence. Computer simulations illustrate the theory.

A New Inverse Kinematics Algorithm for Nonholonomic Mobile Robots

From a general form of the Jacobian right inverse for nonholonomic mobile robots we derive a specific Jacobian inverse kinematics algorithm. By design, this algorithm does not involve the mobility matrix, what makes it computationally less demanding than the most often used Jacobian pseudoinverse algorithm. The algorithm has been applied to the unicycle-type mobile robot. It has been shown that the algorithm performs efficiently unless the unicycle moves along a straight line trajectory. Performance of the new algorithm has been illustrated by computer simulations of the unicycle whose controls are synthesized in accordance with either the finite element or the finite difference method.

Optimal Extended Jacobian Inverse Kinematics Algorithms for Robotic Manipulators

Extended Jacobian inverse kinematics algorithms for redundant robotic manipulators are defined by combining the manipulator's 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.

Extended Jacobian inverse kinematics algorithm for nonholonomic mobile robots

By a generalization of the well-known extended Jacobian method for stationary manipulators, we derive the extended Jacobian inverse kinematics algorithm for nonholonomic mobile robots. Key points of the derivation consist in defining the kinematics of a mobile robot as the end-point map of a driftless control system, decomposing the space of control functions of this system into a finite and an infinite dimensional subspaces, and introducing an augmenting kinematics map subordinated to this decomposition. The original kinematics and the augmenting kinematics constitute the extended kinematics. The inverse Jacobian of the extended kinematics defines the extended Jacobian inverse kinematics algorithm. By design, the algorithm is repeatable. As an example, we derive a specific extended Jacobian inverse kinematics algorithm and illustrate its performance with the computer simulations.

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 algorithm's 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.

A HYPERBOLIC, EXTENDED JACOBIAN INVERSE KINEMATICS ALGORITHM FOR MOBILE MANIPULATORS

A contribution of this paper consists in deriving a repeatable, extended Jacobian inverse kinematics algorithm for mobile manipulators. The endogenous configuration space approach has been assumed as a guideline. An endogenous configuration of a mobile manipulator includes controls of the platform and joint positions of the onboard manipulator. After a suitable truncation of Fourier series representing platform controls a band-limited version of this algorithm is obtained. By reference to the shape of the invariant manifold in the band-limited situation the algorithm is called hyperbolic. Performance of this algorithm is illustrated with computer simulations involving a kinematic car-type mobile platform endowed with an RTR manipulator.

Extended jacobian motion planning algorithm for mobile manipulators

We define a mobile manipulator as a complex robotic system composed of a mobile platform and a stationary manipulator mounted on the platform. The platform motion is subject to nonholonomic velocity constraints. The task of the mobile manipulator amounts to placing its end effector at a prescribed position and orientation in the taskspace. The kinematics of the mobile manipulator are represented by a driftless control system with outputs. We consider a motion planning problem that consists in computing an endogenous configuration (i.e. a control function of the platform and a joint position of the manipulator) capable of driving the end effector to a desirable position and orientation, and of avoiding obstacles present in the taskspace. The motion planning problem is solved locally by the extended Jacobian inverse kinematics algorithm furnished with an extra motion component belonging to the null space of analytic Jacobian of the mobile manipulator. A performance of this motion planning algorithm is examined by computer simulations.

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.

Extended Jacobian inverse kinematics algorithms for mobile manipulators

AbstractWe consider the inverse kinematic problem for mobile manipulators consisting of a nonholonomic mobile platform and a holonomic manipulator on board the platform. The kinematics of a mobile manipulator are represented by a driftless control system with outputs together with the associated variational control system. The output reachability map of the driftless control system determines the instantaneous kinematics, while the output reachability map of the variational system plays the role of the analytic Jacobian of the mobile manipulator. Relying on a formal analogy between the kinematics of stationary and mobile manipulators we exploit the extended Jacobian construction in order to design a collection of extended Jacobian inverse kinematics algorithms for mobile manipulators. It has been proved mathematically and confirmed in computer simulations that these algorithms are capable of efficiently solving the inverse kinematic problem. Moreover, a choice of the Jacobian extension may lay down some guidelines for the platform‐manipulator motion coordination. © 2002 Wiley Periodicals, Inc.