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 Stewart platform [
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 [
Conformal geometric algebra (CGA) [
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 [
The rest of the paper is organized as follows: In Section
In geometric algebra, the fundamental algebraic operators are the inner product (
The 5-dimensional (5D) CGA
In addition, two null bases can now be introduced by the vectors
According to (
As extension, the inner product of an
We define the dual
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
List of conformal geometric entities.
Entity | Representation | Grade | Dual representation | Grade |
---|---|---|---|---|
Point | | 1 | | 4 |
Sphere | | 1 | | 4 |
Plane | | 1 | | 4 |
Line | | 2 | | 3 |
Circle | | 2 | | 3 |
Point pair | | 3 | | 2 |
According to (
From (
In the next section, we will formulate the direct kinematics of 6-4 Stewart platforms via CGA operation and derive the univariate polynomial equation.
A 6-4 Stewart platform
The geometric model of a 6-4 Stewart platform.
As seen from Figure
According to Table
According to (
Point
For the S-joint
According to Table
Point
Please note the expression of point
According to (
However, due to the volume sign of the tetrahedron
The volume of the tetrahedron
Multiplying both sides of (
By transposing and combining the terms in (
Substituting (
Taking the square of both sides of (
Simplifying (
The derivation of (
Equation (
Equations (
In this section, the main aim is to obtain the univariate high-degree polynomial equation, from which the solutions of direct kinematics of 6-4 Stewart platforms can be obtained. In order to get the position of the four S-joints on the moving platform, first of all, we attach a reference coordinate frame
For (
By analyzing the bases in (
The vanishing of the determinant of the coefficient matrix
By expanding each element of the matrix
Therefore expanding (
Solving (
For the coordinates of S-joint
In order to validate the solution procedure, the link parameters and inputs of the numerical example are given in Table
Input data.
| | | | | | |
---|---|---|---|---|---|---|
| 0 | 5 | −2 | −3 | 6 | −3 |
| 0 | 0 | 4 | −1 | −2 | 5 |
| 0 | 1 | −1 | 1 | 2 | −1 |
The distances | | |||||
The lengths of six limbs | |
Ten real solutions.
| | | | | |
---|---|---|---|---|---|
1 | | 3.9672 | 1.6694 | 3.2853 | −1.2546 |
| 0.4233 | 1.6113 | 0.5543 | 0.9734 | |
| 4.1267 | 0.3440 | 4.8464 | −2.3100 | |
2 | | 3.9844 | 0.0656 | 4.0347 | −3.9222 |
| 0.8633 | 2.9776 | 0.9909 | 3.2402 | |
| 4.0407 | 2.9579 | 5.0312 | 3.1266 | |
3 | | 3.9990 | −0.0514 | 3.9757 | −4.0473 |
| 1.1023 | 3.0297 | 1.0191 | 2.8821 | |
| 3.9674 | 3.0296 | 4.9636 | 2.9236 | |
4 | | 4.0646 | −0.2895 | 4.5438 | −3.1386 |
| 1.7833 | 3.1288 | 2.3513 | 5.9173 | |
| 3.6396 | 3.1583 | 2.9704 | 3.4851 | |
5 | | 4.1190 | 1.6609 | 4.9908 | −0.2122 |
| 2.1540 | 1.3125 | 2.4709 | 2.5765 | |
| 3.3678 | −0.1071 | 2.9944 | −3.7077 | |
6 | | 4.6080 | 0.4144 | 4.8572 | −3.4566 |
| 3.2959 | 2.8064 | 2.3558 | 1.8759 | |
| 0.9226 | 2.7044 | 0.6897 | 2.3178 | |
7 | | 5.2094 | 1.6165 | 5.3389 | −2.2620 |
| 1.2109 | 1.7665 | 0.2194 | 1.2649 | |
| −2.0842 | 0.7055 | −2.0784 | 1.5457 | |
8 | | 5.2173 | 1.6477 | 5.8529 | 0.0286 |
| −1.1033 | 1.2742 | −0.3823 | 3.7367 | |
| −2.1238 | −0.5095 | −2.3997 | 2.1950 | |
9 | | 5.2341 | 0.9164 | 5.1481 | −2.3352 |
| −0.8230 | 0.7105 | −1.7370 | 1.8885 | |
| −2.2078 | −2.2846 | −2.6045 | −4.2944 | |
10 | | 5.2535 | 0.7010 | 5.5092 | −2.9281 |
| 0.1857 | 0.64471 | 0.9000 | 0.4219 | |
| −2.3051 | −2.5567 | −1.6533 | −0.8893 |
The paper has proposed a CGA-based formulation and solution procedure for the direct kinematic analysis of 6-4 Stewart platforms with any link parameters. Thanks to the intuitiveness of CGA, the representations of the positions of two single spherical joints have explicit geometric meaning. A coordinate-invariant polynomial equation was derived via CGA operation and it is feasible to other Stewart platforms or parallel mechanisms whose number of spherical joints on the moving platform is equal to 4. The univariate polynomial equation has been derived by constructing a 7 by 7 resultant matrix which is more compact and smaller than those published in the literature. Compared with the previous methods in the literature, the main contribution of the paper lies in that the formulation has geometric meaning due to the intuitiveness of CGA and, in addition, the size of the matrix is smaller than those existed. In future, we will extend this approach to the direct kinematics of other Stewart platforms or complicated parallel mechanisms.
The expansion of
During the process of expansion, we use the following expressions:
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors acknowledge funding provided by the National Natural Science Foundation of China (no. 51605036), Beijing Natural Science Foundation (no. L172031), and the Fundamental Research Funds for the Central Universities for this research.