Output Mode Switching for Parallel Five-bar Manipulators Using a Graph-based Path Planner

The configuration manifolds of parallel manipulators exhibit more nonlinearity than serial manipulators. Qualitatively, they can be seen to possess extra folds. By projecting such manifolds onto spaces of engineering relevance, such as an output workspace or an input actuator space, these folds cast edges that exhibit nonsmooth behavior. For example, inside the global workspace bounds of a five-bar linkage appear several local workspace bounds that only constrain certain output modes of the mechanism. The presence of such boundaries, which manifest in both input and output projections, serve as a source of confusion when these projections are studied exclusively instead of the configuration manifold itself. Particularly, the design of nonsymmetric parallel manipulators has been confounded by the presence of exotic projections in their input and output spaces. In this paper, we represent the configuration space with a radius graph, then weight each edge by solving an optimization problem using homotopy continuation to quantify transmission quality. We then employ a graph path planner to approximate geodesics between configuration points that avoid regions of low transmission quality. Our methodology automatically generates paths capable of transitioning between non-neighboring output modes, a motion which involves osculating multiple workspace boundaries (local, global, or both). We apply our technique to two nonsymmetric five-bar examples that demonstrate how transmission properties and other characteristics of the workspace can be selected by switching output modes.Comment: 7 pages, 6 figure

Homotopy Directed Optimization to Design a Six-Bar Linkage for a Lower Limb With a Natural Ankle Trajectory

This paper presents a synthesis method for the Stephenson III six-bar linkage that combines the direct solution of the synthesis equations with an optimization strategy to achieve increased performance for path generation. The path synthesis equations for a six-bar linkage can reach as many as 15 points on a curve; however, the degree of the polynomial system is 10 46 . In order to increase the number of accuracy points and decrease the complexity of the synthesis equations, a new formulation is used that combines 11 point synthesis with optimization techniques to obtain a six-bar linkage that minimizes the distance to 60 accuracy points. This homotopy directed optimization technique is demonstrated by obtaining a Stephenson III six-bar linkage that achieves a specified gait trajectory

The Kinematic Design of Six-bar Linkages Using Polynomial Homotopy Continuation

This dissertation presents the kinematic design of six-bar linkages for function, motion, and path generation by means of polynomial homotopy continuation algorithms. When no link dimensions are specified beforehand, the synthesis formulations for each design objective yield polynomial systems of degrees in the millions and billions, suggesting a large number of solutions. Complete solution sets to these systems have not yet been obtained and is the topic of this dissertation. Function generation for eleven positions is explored in most detail, in particular the Stephenson II and III function generators, for which we calculate multihomogeneous degrees of 264,241,152 and 55,050,240. A numerical reduction using homotopy estimates these systems to have 1,521,037 and 834,441 roots, respectively. For motion generation, the Watt I linkage can be specified for eight positions, producing a system of a multihomogeneous degree over 19 billion. However, for this work we focus on the smaller case of six positions, numerically reducing this system to an estimated 5,735 roots. For path generation we take a different approach. The design of path generators is formulated as RR chains constrained to have a single degree-of-freedom by attaching six-bar function generators to them. This enables us to use our results obtained on Stephenson II and III function generators to create four types of eleven position path generators: the Stephenson I linkage, two types of Stephenson II linkages, and the Stephenson III linkage

The Kinematic Design of Six-bar Linkages Using Polynomial Homotopy Continuation

This dissertation presents the kinematic design of six-bar linkages for function, motion, and path generation by means of polynomial homotopy continuation algorithms. When no link dimensions are specified beforehand, the synthesis formulations for each design objective yield polynomial systems of degrees in the millions and billions, suggesting a large number of solutions. Complete solution sets to these systems have not yet been obtained and is the topic of this dissertation. Function generation for eleven positions is explored in most detail, in particular the Stephenson II and III function generators, for which we calculate multihomogeneous degrees of 264,241,152 and 55,050,240. A numerical reduction using homotopy estimates these systems to have 1,521,037 and 834,441 roots, respectively. For motion generation, the Watt I linkage can be specified for eight positions, producing a system of a multihomogeneous degree over 19 billion. However, for this work we focus on the smaller case of six positions, numerically reducing this system to an estimated 5,735 roots. For path generation we take a different approach. The design of path generators is formulated as RR chains constrained to have a single degree-of-freedom by attaching six-bar function generators to them. This enables us to use our results obtained on Stephenson II and III function generators to create four types of eleven position path generators: the Stephenson I linkage, two types of Stephenson II linkages, and the Stephenson III linkage

The Kinematic Design of Six-bar Linkages Using Polynomial Homotopy Continuation

This dissertation presents the kinematic design of six-bar linkages for function, motion, and path generation by means of polynomial homotopy continuation algorithms. When no link dimensions are specified beforehand, the synthesis formulations for each design objective yield polynomial systems of degrees in the millions and billions, suggesting a large number of solutions. Complete solution sets to these systems have not yet been obtained and is the topic of this dissertation. Function generation for eleven positions is explored in most detail, in particular the Stephenson II and III function generators, for which we calculate multihomogeneous degrees of 264,241,152 and 55,050,240. A numerical reduction using homotopy estimates these systems to have 1,521,037 and 834,441 roots, respectively. For motion generation, the Watt I linkage can be specified for eight positions, producing a system of a multihomogeneous degree over 19 billion. However, for this work we focus on the smaller case of six positions, numerically reducing this system to an estimated 5,735 roots. For path generation we take a different approach. The design of path generators is formulated as RR chains constrained to have a single degree-of-freedom by attaching six-bar function generators to them. This enables us to use our results obtained on Stephenson II and III function generators to create four types of eleven position path generators: the Stephenson I linkage, two types of Stephenson II linkages, and the Stephenson III linkage

Numerical Synthesis of Six-Bar Linkages for Mechanical Computation

This paper presents a design procedure for six-bar linkages that use eight accuracy points to approximate a specified input-output function. In the kinematic synthesis of linkages, this is known as the synthesis of a function generator to perform mechanical computation. Our formulation uses isotropic coordinates to define the loop equations of the Watt II, Stephenson II, and Stephenson III six-bar linkages. The result is 22 polynomial equations in 22 unknowns that are solved using the polynomial homotopy software BERTINI. The bilinear structure of the system yields a polynomial degree of 705,432. Our first run of BERTINI generated 92,736 nonsingular solutions, which were used as the basis of a parameter homotopy solution. The algorithm was tested on the design of the Watt II logarithmic function generator patented by Svoboda in 1944. Our algorithm yielded his linkage and 64 others in 129 min of parallel computation on a Mac Pro with 12±2.93 GHz processors. Three additional examples are provided as well