Inverse Kinematics Solution to Mobile Manipulators

Abstract

This paper presents the solution at the control feedback level to the inverse kinematics problem for mobile manipulators operating in both obstacle-free task spaces and task spaces including obstacles. Using the Frechet differential of a certain criterion function, the fully specified system of algebraic and differential equations of the minimal amount has been obtained to solve the inverse kinematics problem. Based on the Lyapunov stability theory, a full differential form generating the mobile manipulator trajectory, whose attractor attained in a finite time fulfills the above system of algebraic and differential equations, has been derived. The problem of both singularity and collision avoidance is solved here based on a concept of (local) velocity perturbation which results in continuous mobile manipulator velocities near singularities and obstacles. The numerical simulation results carried out for a mobile manipulator consisting of a nonholonomic wheel and a holonomic manipulator of two revolute kinematic pairs, operating in both an obstacle-free task space and task space including obstacles, illustrate the trajectory performance of the proposed solution scheme.

Tracking the Kinematically Optimal Trajectories by Mobile Manipulators

“…It is practically desirable to generate joint trajectory q = q(t) in such a way as to minimize an objective (kinematic) function F(q). Based on the kinematic criterion F(q), which is assumed to be at least four times differentiable with respect to q, the general form for f a , proposed, e.g., in works [11,15] may be expressed as…”

“…The first approach is the extended or augmented task space formulation (including also input-output linearization techniques) of the inverse kinematics problem showed in works [2][3][4][5][6][7][8][9][10][11][12][13][14]. Extending the dimension of the task space by incorporating as many additional constraints as the degree of the redundancy is the essence of it.…”

“…The control formulations based on the augmented task space technique have some disadvantages. The controllers proposed in [2][3][4][5][6][7][8][9][10][11] require inverse of the extended Jacobian matrix that can potentially consist of algorithmic and/or kinematic singularities [15] and as a consequence, can produce the control inputs to become unbounded even though the mobile manipulator is not in a singular configuration. Rigorous definitions of singularity for mobile manipulators have been proposed in works [46,47], which introduce the traditional concept of singularity built on the definition of the extended Jacobian and the concepts of posture and configuration singularity.…”

“…Moreover, the dimensionality of the inverse kinematics problem associated with an extended Jacobian increases. Furthermore, control algorithms from [2][3][4][5][6][7][8][9][10][11] require full knowledge of the dynamic equations. The controllers offered in [12][13][14] need the knowledge of both holonomic end-effector and platform desired trajectories and are not optimal in any sense.…”

Point-to-Point Collision-Free Trajectory Planning for Mobile Manipulators

“…For this reason, in the case of trajectory planning with collision avoidance conditions it is not possible to apply for mechanical constraints the same approach as presented in Section 3.1. Therefore, in this paper collision avoidance is accomplished by perturbing the manipulator motion close to obstacles [25]. For the proposed method, scalar functions c I I i (· ), which describe collision-free conditions (6), should specify distances between a manipulator and obstacles.…”