Box Approximations of Planar Linkage Configuration Spaces

Abstract

This paper presents a numerical method able to compute all possible configurations of planar linkages. The procedure is applicable to rigid linkages (i.e., those that can only adopt a finite number of configurations) and to mobile ones (i.e., those that exhibit a continuum of possible configurations).The method is based on the fact that this problem can be reduced to finding the roots of a polynomial system of linear, quadratic, and hyperbolic equations, which is here tackled with a new strategy exploiting its structure. The method is conceptually simple and easy to implement, yet it provides solutions of the desired accuracy in short computation times. Experiments are included which show its performance on the double butterfly linkage and on larger linkages formed by the concatenation of basic patterns.

Exact interval propagation for the efficient solution of position analysis problems on planar linkages

“…The second one solves the same linkage but assuming that θ 3 is a free variable, yielding a onedimensional continuum of solutions. The same benchmarks have been used previously to show the performance of elimination [15,16], continuation [17,18], and relaxation techniques [2]. We compare our results with those derived by such techniques, and employ the same linkage dimensions used in these papers.…”

“…If we set θ 3 = 75.75 • , the number of obtained solutions is six [2,15,16]. They are given in Table 1.…”

“…(4) provides real values only when the argument of the arccosine function is in the interval [−1, 1] or, equivalently, when the triangle inequalities (2) hold. In this case, the feasibility range for θ 2 consists of two points.…”

“…In recent years there has been a growing interest in the use of interval methods in Kinematics, specially for the task of finding the feasible configurations of complex linkages-the so-called position analysis problem-which, when performed analytically, may involve a large number of equations or high-degree polynomials [1][2][3]. When the solution space is continuous, interval methods can provide a discrete approximation of such space as a set of enclosing boxes, which can, at least in principle, be refined as much as desired [4].…”

“…When the box cannot be further reduced, it is split, usually into two halves (or branches), each of which is treated recursively by the same process. The pruning method may be a general interval propagation algorithm applied to the kinematic equations of the linkage [1,[4][5][6][7], or a more specific method suited to the problem [2,3,8].…”