Both the forward and backward kinematics of the Gough-Stewart mechanism exhibit nonlinear behavior. It is critically important to take account of this nonlinearity in some applications such as path control in parallel kinematics machine tools. The nonlinearity of inverse kinematics is straightforward and has been first studied in this paper. However the nonlinearity of forward kinematics is more challenging to be considered as there is no analytic solution to the forward kinematic solution of the mechanism. A statistical approach including the Bates and Watts measures of nonlinearity has been employed to investigate the nonlinearity of the forward kinematics. The concept of standard sphere has been used to check the significance of the nonlinearity of the mechanism. It is demonstrated that the length of the region, defined as the linear approximation of the lifted line, has a significant impact on the nonlinearity of the mechanism.

The Gough-Stewart platform mechanism (GSPM), introduced by Gough and Whitehall [

A typical Gough-Stewart platform.

Determination of the lengths of pods and the first and second rate of change in these lengths forms basis of the inverse kinematics analysis of the mechanism. This is done by using the position and orientation, translational and angular velocities, and translational and angular accelerations of the upper platform.

The kinematic chain of the mechanism is illustrated in Figure

Kinematic chain of GSPM.

The inverse kinematics problem of the mechanism is fairly simple as discussed exhaustively by Harib and Srinivasan [

Various methods have already been applied to solve the forward displacement problem of the general Stewart mechanism. Raghavan [

Although it is known that at most 40 possible solutions exist for the forward kinematic problem of a general GSPM, only one solution corresponds to the actual pose of a physical machine [

Both the inverse and forward kinematics of GSPM exhibit nonlinear behavior. In the inverse kinematics the pods’ lengths do not change linearly with a linear path travelled by the upper platform. In the forward kinematics, when pods are actuated linearly the upper platform moves along a nonlinear path. This makes the path control and interpolation functions in Stewart-based machine tools become more complex than in conventional machine tools. Zheng et al. [

To the extent that the authors of the present paper are aware, little attention has been paid to the nonlinearity analysis of GSPM. This is especially important for the hexapod machine tools where nonlinearity of the mechanism considerably adds to the interpolation algorithms requiring an insightful analysis. The problem of path control in parallel mechanisms has recently been tackled in some studies, for example [

A concise formulation of the inverse and forward kinematics of GSPM is presented in this section for subsequent use in the analysis of nonlinearity. The vector

Local coordinate system.

As illustrated in Figure

The position and orientation of the upper platform can be represented by a 6D vector

In forward kinematics, the problem of determining the position/orientation of the upper platform from the lengths of pods leads to solving a set of nonlinear equations. These equations can be solved by using the Newton-Raphson numerical iterative method. The translational/angular velocity of the upper platform in forward kinematics can be obtained from the change rate of the pods’ lengths. It can be written from (

It is illustrated in this section that both the inverse and forward kinematics of GSPM are nonlinear. The nonlinearity of the inverse kinematics can be readily verified as follows.

The inverse kinematics of SPM is said to be nonlinear if and only if a linear relationship in the vector space of the upper platform is transformed into a nonlinear relationship in the vector space of the joints. The upper platform's center point is assumed to follow a linear path as follows:

It is obvious that (

The nonlinearity of forward kinematics is not however as straightforwardly clear. The Bates and Watts measures of nonlinearity are used to study the nonlinearity of the forward kinematics in this section.

Bates and Watts introduced the concept of relative curvature and illustrated that the relative curvature can be decomposed into intrinsic curvature and parameter-effects curvature [

The methodology proposed by Bates and Watts applies well to the nonlinearity analysis of the forward kinematics of GSPM since forward kinematics maps a linear relation in parameter space (known here as joints space,

The forward kinematic relation is mathematically represented as follows:

The maximum curvature of the solution locus in a GSPM is dependent on the position and orientation of the upper platform. It is noteworthy that both

In order to investigate the nonlinearity of the mechanism throughout its workspace, different regions in the solution locus are considered. Region is defined as the linear approximation of the lifted line. The difference between a point on the lifted line and that of the region produces the so-called kinematics error which roots in nonlinear behavior of the forward kinematics of GSPM. Regions are different in the position/orientation of the upper platform, together with their sizes and directions. A typical region can be embodied by position/orientation vector of the upper platform at the beginning, designated by

Region and lifted line. Kinematic error is due to the nonlinear behavior of the forward kinematics.

To study the effect of

Levels of

Levels | |||
---|---|---|---|

Level 1 | 0.05 | ||

Level 2 | 0.08 | ||

Level 3 | 0.1 | ||

Level 4 | 0.5 | ||

Level 5 | 1 | ||

Level 6 | 10 | ||

Level 7 | 50 | ||

Level 8 | 100 |

The maximum curvature is obtained for any permutation of

Surface plot of curvature versus

It is apparent that

Bates and Watts suggested comparing the maximum parameters-effect curvature with the curvature of

To elaborate effects of

Interaction plots of (a)

Main effect plot of

Figure

Figure

Mean values of curvature versus

Figure

Interaction plot of (a) orientation and

The above discussion leads to the following statement as a rule of thumb that for values of

Nonlinearity analysis of GSPM forward and inverse kinematics was presented and considered in this investigation. To quantitatively consider the nonlinearity of the mechanism Bates and Watts measure of nonlinearity was employed. Bates and Watts formulation was developed for the forward kinematics of the mechanism. It was implied that the most significant effect on the nonlinearity of the mechanism is that of the size of the region in the solution locus and it was shown that the nonlinearity of the mechanism is negligible if this size does not exceed 0.5 (mm). Such a conclusion is of significant importance in the interpolation of curves in hexapod machine tool. It was demonstrated that the nonlinear behavior of the mechanism is attributed to the parameter-effect curvature as Bates and Watts classified.

Bates et al. proposed a new algorithm for calculation of intrinsic and parameter-effects curvatures [

According to this algorithm, _{0} is the unit vector in the direction of which

In order to find