Sandipan Bandyopadhyay

Trajectory-tracking control of parallel manipulators is more difficult than in the case of serial ones due to the presence of the loop-closure constraints and constraint forces resulting from them. One needs to eliminate these forces to get to the equation of motion, and then apply a control scheme. In this paper, such a control scheme is presented through application on a semi-regular Stewart platform manipulator (SRSPM). The manipulator has six degrees-of-freedom; however, it is modelled by a system of 18 coupled nonlinear ordinary differential equations (ODE) using the constrained Lagrangian formulation. The model is then linearised through feedback, and controlled by a linear PD servo scheme. Numerical simulations over a non-singular path show that the scheme is fairly accurate, at the cost of being computationally expensive. The scheme is general in nature, and as such, it is expected to work in the case of other parallel manipulators as well.

This paper deals with the problem of determining conclusively if a given spatial trajectory of a semi-regular Stewart platform manipulator (SRSPM) passes through singularities. The algorithm presented here is restricted to trajectories defined in terms of polynomial functions of a continuous path parameter. As such it can complement existing motion-planning strategies which use piecewise continuous cubics or other polynomials to describe the trajectories. The algorithm, however, can be extended to functions such as rational or trigonometric, which can be reduced readily to polynomial forms. The advantage of the formulation presented here is that the computation is relatively simple and the results are guaranteed theoretically. Further, if a given trajectory does pass through a singularity, the algorithm returns the set of points where it does so. The theory presented in the paper is illustrated with numerical examples involving 3-D and 6-D trajectories defined in terms of cubic polynomials in a single path parameter.

The knowledge of the singular manifold of a manipulator is essential for its design, path-planning and control. However, due to the computational complexity, it is very difficult to obtain an analytical description of the manifold in terms of the architecture as well as configuration parameters of the manipulator. In this paper, the singularity of a general hexagonal Stewart platform manipulator (GHSPM) is studied and its singular manifold is obtained. The manifolds geometric structure is analysed by projecting it to the position and orientation subspaces, respectively. An explicit geometric characterisation is obtained in the former case, where is it found that the geometric nature of the singular surface is the same as in the case of the SRSPM. The other surface, however, defies such analysis at this point due to its high degree. The theoretical results are illustrated with numerical examples and plots.

This paper compares three different methods for computing the determinant of a univariate polynomial matrix. From a systematic empirical study involving 100 trials of each numerical experiment, it appears that the method of FFT-IFT-based interpolation and evaluation performs the best in terms of both speed and accuracy of computation. The utility of this method is further demonstrated by a successful application to the classical problem of forward kinematics (FKP) of the general Stewart-Gough platform manipulator (SPM).