Multi-loop Position Analysis via Iterated Linear Programming

Abstract

Abstract-This paper presents a numerical method able to isolate all configurations that an arbitrary loop linkage can adopt, within given ranges for its degrees of freedom. The procedure is general, in the sense that it can be applied to single or multiple intermingled loops of arbitrary topology. It is also complete, meaning that all possible solutions get accurately bounded, irrespectively of whether the analyzed linkage is rigid or mobile. The problem is tackled by formulating a system of linear, parabolic, and hyperbolic equations, which is here solved by a new strategy exploiting its structure. The method is conceptually simple, geometric in nature, and easy to implement, yet it provides solutions at the desired accuracy in short computation times.

A space decomposition method for path planning of loop linkages

“…A major step towards this goal has already been taken by extending the shrink-andsplit strategy of Section III to cope with arbitrary spatial loop constraints [15]. Whereas the adaptation of the planning algorithms will not require significant changes, work is needed on the generalization of the collision detection method to the 3D case.…”

“…3. The smaller B, the tighter this polytope approximates the solution space, or, in other words, the smaller the error introduced by the circle approximations [15], [16].…”

“…The tests also show the algorithm converges quadratically to all q ∈ C, if these are isolated points, and thus exhibits a similar efficiency to that of fast single-root finding procedures like Newton-Raphson. Moreover, to deal with arbitrary linkages (either planar or spatial), only slight modifications must be introduced in its structure, maintaining these properties in all cases [15], [16].…”

“…A main obstacle in deriving a simultaneosly general and complete closed-chain path planner has been the lack of efficient methods to completely map out the self-motion set of arbitrary linkages. This problem has been extensively studied for over a decade within Robot Kinematics, where explored strategies include interval Newton [11], [12], Bernstein subdivision [13] and homotopy techniques [14], but it has not been until recently that a relatively fast and simple technique has been given to this end [15], [16]. The aim of this paper is to extend this technique to derive resolutioncomplete path planners for arbitrary closed-loop linkages.…”

Finding all valid hand configurations for a given precision grasp

“…The DenavitHartenberg parameters of this hand are given in [20]. From them, it is easy to formulate the system of equations derived in Section IV, which has been solved in all cases with the CUIK package described in [17]- [19].…”