Design and path optimization of linear pentapods based on the geometry of their singularity varieties

Abstract

The main goal of this work is to optimize the design and path of linear pentapods via studying the geometrical properties of their related singularity varieties. Several mathematical tools are borrowed to perform the aforementioned tasks. In the first chapter, these tools are briefly reviewed in two groups of differential geometry and algebraic geometry. The chapter is peppered with several purposeful examples which are given in such a way that they would give an intuition to the reader for the coming higher dimensional counterparts. The second chapter is mainly centred around the algebraic concepts of this study, namely, the rational parametrization of the general linear pentapods’ singularity variety, topological and geometrical properties of the singularity variety and most importantly introducing a new class of linear pentapods under the name of simple pentapods which possess a simple-structured singularity variety. The third chapter views the singularity variety from a rather mathematically different discipline, namely the metric spaces. In this chapter, ideas such as pedal points on the singularity variety and object-oriented metric are investigated. The chapter ends with the main result on the generic number of pedal points on the simple pentapods’ singularity variety. The fourth chapter focuses on the concept of variational path optimization by assuming that an already singularity-free path between two non-singular poses of the simple pentapod is given. Combining the tools from chapter one with the results of chapter three paves the way for obtaining a real time path optimization. Additionally, the optimized path takes the base spherical joint and prismatic extension limits into account in such a way that during the related optimized motion the manipulator does not exceed these limits. The chapter comes to an end with a brief introduction of a graphical user interface which provides a user-friendly access to the aforementioned variational path optimization algorithm. Finally, the last chapter names and explains the related unfinished projects in the field. These include path planning, workspace analysis and the extension of the variational path optimization algorithm to the platform spherical joints limit and leg collision.