Singularity Closeness of Stewart-Gough Platforms

This project is devoted to the evaluation of singularity closeness of Stewart-Gough (SG) platforms. To recall, a SG platform is a parallel manipulator, consisting of a moving platform, which is connected via six spherical-prismatic-spherical legs with the base, where only the prismatic joints are active. The number of applications of SG manipulators, ranging from medical surgery to astronomy, has increased enormously during the last decades due to their advantages of high speed, stiffness, accuracy, load/weight ratio, etc. One of the drawbacks of these parallel robots are their singular configurations, where the manipulator is shaky while all leg lengths are fixed. As a consequence the actuator forces can become very large, which may result in a breakdown of the mechanism. Therefore singularities have to be avoided. This reasons the high interest of the kinematic/robotic community in evaluating the singularity closeness of SG platforms, but geometric meaningful distance measures for this task are still missing. The research project closes this gap. Based on object-oriented metrics we define geometric meaningful distance measures evaluating the distance of a given SG configuration to the next singular configuration, SG design to the next architecturally singular design, where the term architecture singular is used for SG platforms, which are singular in every possible configuration. The computation of these distances is based on the homotopy continuation method using the software Bertini. Moreover the project aims to determine central configurations and designs of SG manipulators used in praxis, which are optimal with respect to these distance measures. As the proposed distances to the next singular configuration can be interpreted as radii of guaranteed singularity-free hyperspheres centered in the given configuration, they are also of interest for path-planning. In this context we study the computation of a distance field on a grid, which is a fair discretization of the 6- dimensional robot-workspace. It should be noted that the favored distance measures have a clear physical meaning for the manipulator, which is very important for their acceptance by mechanical/constructional engineers. As this research project has great potential for practical applications, we expect a big impact, which will be improved by our dissemination strategy to put all codes (including short documentations) on an online repository. The submitted project will be conducted at the Center for Geometry and Computational Design of the Vienna University of Technology, under the guidance of Georg Nawratil, who conceived and formulated all parts of the project proposal.

This project was devoted to the evaluation of singularity closeness of parallel manipulators of Stewart-Gough (SG) type. This term summarizes mechanisms, where the moving platform is connected to the base by a certain number of active prismatic (P) legs according to the robot's degree of freedom. For planar structures the legs are anchored by passive revolute (R) joints and for spatial ones by passive spherical (S) joints. One of the drawbacks of these parallel robots are their singular configurations, where the manipulator is shaky while all leg lengths are fixed. As a consequence the actuator forces can become very large, which may result in a breakdown of the mechanism. Therefore singularities have to be avoided. This also reasons the high interest of the kinematic/robotic community in the evaluation of the singularity closeness, but geometric meaningful distance measures for this task were missing. The aim of this research project was to close this gap by defining such metrics for evaluating the distance of a given configuration to the next singular configuration, design to the next architecturally singular design, where the term "architecture singular" is used for manipulators, which are singular in every possible configuration. To do so, we considered parallel manipulators of SG type as pin-jointed body-bar frameworks. By defining the combinatorial structure as well as the intrinsic metric (lengths of the bars and the shapes of the bodies), the inner geometry of the framework is fixed. But in general, this assignment does not uniquely determine the embedding of the framework into the Euclidean space, thus different incongruent realizations exist. Based on this point of view one can distinguish the following two kinds of metrics: Intrinsic metrics: The distance to the singularity is measured based on the inner metric of the manipulator. Extrinsic metrics: The distance to the singularity is measured based on the metric of the embedding space. Originally it was planned to use only extrinsic metrics to measure the distance between two configurations, but we extended the methodology also to intrinsic metrics motivated by our study of snapping realizations, which yield shakiness in the limiting case. While the worked-out theory holds in great generality for pin-jointed body-bar frameworks, we focused for the proof of the concept on the simplest parallel manipulators of SG type, which are planar 3-RPR robots. The proof of concept for computing the distance to the closest architectural singular design was done for linear pentapods (5-SPS manipulators with linear platform) as in this way we were able to compare our results with the only existing index of this kind mentioned in the literature so far. This distance was computed - as initially planned - just with respect to an extrinsic metric.