Mathematical Problems in Engineering

Volume 2018, Article ID 6245341, 9 pages

https://doi.org/10.1155/2018/6245341

## A Geometric Modeling and Computing Method for Direct Kinematic Analysis of 6-4 Stewart Platforms

^{1}School of Automation, Beijing University of Posts and Telecommunications, Beijing 100876, China^{2}Beijing Institute of Spacecraft System Engineering, Beijing Key Laboratory of Intelligent Space Robotic System Technology and Applications, Beijing 100094, China

Correspondence should be addressed to Ying Zhang; nc.ude.tpub@hzgniy_etaudarg

Received 1 August 2017; Accepted 28 January 2018; Published 26 February 2018

Academic Editor: Fazal M. Mahomed

Copyright © 2018 Ying Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A geometric modeling and solution procedure for direct kinematic analysis of 6-4 Stewart platforms with any link parameters is proposed based on conformal geometric algebra (CGA). Firstly, the positions of the two single spherical joints on the moving platform are formulated by the intersection, dissection, and dual of the basic entities under the frame of CGA. Secondly, a coordinate-invariant equation is derived via CGA operation in the positions of the other two pairwise spherical joints. Thirdly, the other five equations are formulated in terms of geometric constraints. Fourthly, a 32-degree univariate polynomial equation is reduced from a constructed 7 by 7 matrix which is relatively small in size by using a Gröbner-Sylvester hybrid method. Finally, a numerical example is employed to verify the solution procedure. The novelty of the paper lies in that the formulation is concise and coordinate-invariant and has intrinsic geometric intuition due to the use of CGA and the size of the resultant matrix is smaller than those existed.

#### 1. Introduction

The Stewart platform [1] is a fully parallel, six-degree-of-freedom manipulator that generally consists of a base platform, a moving platform, and six limbs connected to each other in parallel. Stewart platforms have been successfully used in a wide variety of fields and industries, ranging from astronomy to flight simulators and are becoming increasingly popular in the machine-tool industry [2]. From the 1980s, Stewart platforms have attracted wide interests from researchers and engineers due to their advantages of simplicity, high stiffness, large load capacity, quick dynamic response, and excellent accuracy.

The direct kinematic analysis of Stewart platforms has been considered a challenging problem, which leads naturally to a system of highly nonlinear algebraic equations with multiple solutions. There are two main approaches to solve these equations: numerical schemes and closed-form solutions. A closed-form solution provides more information about the geometric and kinematic behavior over a numerical solution, and the closed-form univariate polynomial equation has significant theoretical values as it is fundamental to many other kinematic problems. Hence obtaining a closed-form solution to the direct kinematic analysis is clearly preferred in most cases.

In this paper, we will revisit the direct kinematic analysis of 6-4 Stewart platforms, four of which meet the platform pairwise, while the remaining two meet both base and platform singly. Numerous researchers [3–6] have worked on this problem. Hunt (1983) [3] wrongly stated that the maximum number of assembly modes for the problem was 24 by geometrical proof. Innocenti (1995) [4] derived a suitable set of five closure equations and solved the problem by a specifically developed elimination scheme, that is, a constructed 10 by 10 matrix. The number of the solutions is 36 in view of resultant form; however, the numerical result leads to a 32-degree polynomial equation in a single variable. Liao et al. (1995) [5] formulated this problem based on the vector method from equivalent mechanisms and obtained all the 32 solutions by constructed 10 by 10 resultant matrix. The solution procedure is complex due to numerous vector computations. Zhang et al. (2012) [6] also modeled this problem based on the trilateration method and vector method from equivalent mechanisms and the solution procedure is the same as [5]. It is concluded from the above-mentioned literature that the modeling and the solution procedure either are formulated algebraically from the equivalent mechanisms or require resultant elimination. The size of the constructed resultant matrix is all 10 by 10.

Conformal geometric algebra (CGA) [7–10] is a relatively new mathematical tool for geometric representation and computation. Essentially, CGA represents various geometric entities of points, spheres, lines, planes, circles, and point pairs in a systematical hierarchy of multiple grades. More importantly, CGA provides direct algebraic operations on these geometric entities which typically lead to simple, compact, coordinate-invariant formulations and enables complicated symbolic geometric computations. The above-mentioned properties are two superior characteristics of CGA. Hence it is very efficient for geometric modeling and computation for kinematic problem of mechanisms and robotics. In recent decades, CGA has been mostly applied to solve the inverse kinematics problem of the serial mechanisms [11–14] via CGA operation of the geometric entities. In addition, Tanev [15, 16], Kim et al. [17], and Huo et al. [18] employed CGA to study the singularity analysis of PMs. Huo et al. [18] and Li et al. [19] proposed a mobility analysis approach for PMs based on geometric algebra. Zhang et al. [20, 21] and Wei et al. [22] applied CGA to solve the direct kinematics of parallel mechanisms.

In this paper, we will formulate the direct kinematic analysis problem of 6-4 Stewart platform using CGA and then construct a 7 by 7 resultant using Gröbner-Sylvester hybrid method [23, 24] which finally leads to a 32-degree univariate polynomial equation without extraneous roots. The derived coordinate-invariant equation is also applicable to other Stewart platforms or parallel mechanisms whose number of spherical joints on the moving platform is equal to 4.

The rest of the paper is organized as follows: In Section 2, the fundamentals of CGA are introduced. In Section 3, the geometric modeling for the direct kinematic analysis of 6-4 Stewart platforms is formulated based on CGA. Section 4 proposes the elimination procedure and finally reduces a 32-degree univariate polynomial equation from a constructed 7 by 7 matrix by Gröbner-Sylvester hybrid method. In Section 5, a numerical example is provided to verify our solution procedure. Finally, conclusions and future work will be given in Section 6.

#### 2. Fundamentals of Conformal Geometric Algebra

In geometric algebra, the fundamental algebraic operators are the inner product (), the outer product (), and the geometric product ().

The 5-dimensional (5D) CGA is derived from a 3D Euclidean space and a 2D Minkowski space . CGA has five orthonormal basis vectors given by with the following properties:where are the three orthonormal basis vectors in the Euclidean space and are the two orthogonal basis vectors in Minkowski space.

In addition, two null bases can now be introduced by the vectorswith the propertieswhere is the conformal origin and is the conformal infinity.

*Blades* are the basic computational elements and the basic geometric entities of the geometric algebra. The grade of a blade is simply the number of linearly independent vectors that are “wedged” together. The 5D CGA consists of blades with grades 0, 1, 2, 3, 4, and 5. A linear combination of the -blades is called a* k*-vector, and a linear combination of blades with different grades is called a multivector. The blades with the maximum grade in CGA, that is, 5-blades, are called pseudoscalars and denoted by .

According to (1)–(3), the inner () and outer () products of two 1-vectors are defined as

As extension, the inner product of an -blade with an -blade can be defined recursively bywith

We define the dual of a multivector bywhere is the inverse of and is equal to .

CGA provides the representation of primitive geometric entities for intuitive expression. The primitive geometric entities in CGA consist of spheres, points, lines, planes, circles, and point pairs. The representation of the geometric entities with respect to the inner product null space (IPNS) and the one with respect to the outer product null space are, respectively, listed in Table 1. These two representations are dual to each other and therefore can be converted by dual operator. In Table 1, the small bold character represents the point or vector in the Euclidean space, while the bold underlined character represents the basic geometric entity in the conformal space. For more information, please refer to [9, 10].