Abstract
This paper addresses computational aspects of the Jacobian motion planning algorithms for non-holonomic robotic systems. The motion planning problem is formulated in terms of a control problem in the control affine system representing the system’s kinematics. Jacobian motion planning algorithms are derived by means of the continuation (homotopy) method applied to the inverse kinematics problem in the space of control functions. The solution of the motion planning problem is obtained as the limit solution of a functional differential equation involving the control function. Two methods of representing the control functions are studied: parametric and non-parametric. The parametric method parametrizes the control functions by truncated orthogonal series. The non-parametric method can manage without the parametrization. The functional differential equation can be solved using either the Euler method of integration or higher order methods. The paper focuses on the non-parametric Jacobian pseudo inverse motion planning algorithms incorporating a higher order integration method. Performance of this algorithm is illustrated by the numeric solution of a motion planning problem for the rolling ball kinematics.
Highlights
The motion planning problem of robotic systems can be regarded as a special instance of the inverse kinematics problem, and solved by means of the continuation or homotopy method [1]
For the kinematics (35) with output (36), the following motion planning problem will be addressed: Find a control u(t) that drives the ball from the initial q0 = (0, 0, 0, π/4, 0)T to the desired point yd = (1, 1, 0)T in the task space over the time interval [0, 2]
This paper has been devoted to computational aspects of the Jacobian pseudo inverse motion planning algorithm of nonholonomic robotic systems
Summary
The motion planning problem of robotic systems can be regarded as a special instance of the inverse kinematics problem, and solved by means of the continuation or homotopy method [1]. The motion planning algorithm returns coefficients of the control function with respect to an orthogonal base in the control space. Such an algorithm will be called parametric. In the context of endogenous configuration space approach a comparative study of motion planning algorithms employing diverse orthogonal bases, such as trigonometric functions, Legendre, Gegenbauer and Tchebyshev polynomials as well as Haar functions, can be found in [12]. A further improvement of the numerical properties of motion planning algorithms could be obtained after replacing the Euler (first order) method by
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