Inverse Kinematics of 2 (3RPS) and 2 (3SPR) Serial‑Parallel Manipulators

Abstract

Serial-parallel manipulators are of great interest to academic community in recent years, especially those composed of classical parallel mechanisms. There have been many studies around 2(3RPS) and 2(3SPR) S-PMs, but unfortunately their inverse kinematics have not yet been resolved. This paper discovers that the unknown kinematic parameters of middle platform are responsible for the unresolvable of inverse kinematics, meanwhile the unknown kinematic parameters of middle platform also have huge coupling relationships. Therefore, to break through this challenges, the huge coupling relationships are decoupled layer by layer, the kinematic parameters of middle platform are solved by combining Sylvester’s elimination method, and the inverse displacements of 2(3RPS) and 2(3SPR) S-PMs are obtained subsequently. This paper not only solves the inverse kinematics of classical 2(3RPS) and 2(3SPR) S-PMs, but also reveals the essence of the inverse kinematics of general (3-DOF)+(3-DOF) 6-DOF S-PMs and proposes a corresponding solution.

1. Introduction

The serial-parallel manipulator (S-PM), which is formed by connecting parallel manipulators (PMs) in series, can promote the complementary advantages of series manipulator (SMs) and PMs, and thus has the merits of excellent motion functions and performance characteristics [1, 2]. In recent years, there have been more and more design attempts to mechanical equipment based on S-PM, such as medical surgery robot [3, 4], industrial robot [5, 6], robotic braces for dynamic trunk support [7], humanoid robot’s arm/limb [8], hip joint [9], worm-like robot [10], pointing mechanism [11], fingers of large scale reconfigurable space hand [12], mobile robot [13].

Most papers are concentrated on (3+3) type 6 DOF S-PM, i.e., both the lower and upper PM in S-PM are 3-DOF. Some of them are the lower and upper PM with some shared joints. In this regard, Hu et al. have successively studied 3PRS+2(U)PR/(S)PR [14], 3RRS+3(S)PR [15], 2SPR/RPS+2(R)PS/(S)PR [16] S-PMs. Yao et al. [10] proposed 3RPS+3(S)PR worm-like robot. Guo et al. [12] suggested 3RRS+3(S)RR fingers of a large scale reconfigurable space hand. Where, the symbols in brackets indicate shared joint. Not all structures of S-PMs are available with shared joints, so the S-PMs containing a multi-connected middle platform are more accepted, where some connections are link to lower PM and others to upper PM. In this respect, Zheng et al. [17] presented 2(3UPU) S-PM, and obtained the closed-form kinematic solutions. Gallardo-Alvarado et al. [18] analyzed the mobility, kinematics and dynamics of 2(3RPS) S-PM. Hu et al. [19] researched the complete kinematics of two tricept S-PM. Salarieh et al. [20] studied the inverse and forward dynamics of N-3RPS manipulator. Meng et al. [13] proposed a 2(3UPU) mobile robot in-pipe use.

From above papers, it is clear that the S-PMs constructed by classical PMs (such as 3UPU, 3RPS and tricept) are the mainstream of research. Among the classical PMs, the 3RPS PM proposed by Hunt [21] is the one with minimum machining and assembly requirements, and more importantly, it has better motion/force transfer capability [22]. Further, 3SPR PM which is the reverse mechanism of 3RPS PM also has the same quality motion performance [23]. Therefore, 2(3RPS) and 2(3SPR) S-PMs are studied in this paper.