Page 43 - Textos de Matemática Vol. 43

PATH PLANNING FOR A DOUBLE PENDULUM USING NATURAL SPLINES ON THE TORUS
JANUSZ JAKUBIAK
Dedicated to Professor Fa´tima Silva Leite on the occasion of her 60th birthday.
Abstract. In this paper we present an algorithm to generate splines on a torus. In the task solved it is assumed that the splines connect points with given velocities in boundary positions. Results are adapted to a double pendulum robotic arm, illustrated by computer simulations and compared with regular cubic interpolation and X-splines. This paper is an extended version of the work presented at Controlo 2010.
1. Introduction
Trajectory and path planning for robotic manipulators is a well defined task and numerous research papers contributed to this problem. For references we refer to [14, 15, 7]. In robot motion planning proper smoothness of generated paths is an important requirement, enforced by the nature of mechanical systems.
One of commonly used methods to generate a smooth path for a robot is to define its path as a spline. Splines, as a method of interpolation by smooth curves, were developed over 60 years ago, and they were found useful in many applications, mainly in computer graphics [2]. However, much time had to pass until this method was generalized to non-Euclidean spaces. One of important steps on this way is a generalization of the de Casteljau algorithm to general Riemannian manifolds [11]. The fundamental idea of that work is to replace line segments which appear in the de Casteljau algorithm for Euclidean spaces with geodesics in manifolds. Motivated by this work we developed algorithms which allow generation of smooth curves on Riemannian manifolds [12, 6] and illustrated the results with splines on spheres, since in that case explicit formulas for geodesics are known.
In a number of works a task of spline construction on the torus was considered. In [4] authors developed an algorithm for B-splines on d-dimensional torus and in [8] homogeneous B-splines with symmetry property were proposed. An enhancement of sk-splines to the torus was presented in [3]. Di↵erent approaches to the problem were recently presented in [13], where authors considered spline collocations and in [1] where smoothing splines were used to approximate inaccurate data.
2000 Mathematics Subject Classification. 41A15, 53C22, 65D07.
Key words and phrases. Splines, torus, geodesics, double pendulum.
The work was supported by a statutory grant from Wroclaw University of Technology.
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