A task-driven approach to optimal synthesis of planar four-bar linkages for extended burmester problem

The classic Burmester problem is concerned with computing dimensions of planar four-bar linkages consisting of all revolute joints for five-pose problems. In the context of motion generation, each pose can be seen as a constraint that the coupler of a planar four-bar mechanism needs to interpolate or approximate through. We define extended Burmester problem as the one where all types of planar four-bars consisting of dyads of type RR, PR, RP, or PP (R: revolute, P: prismatic) and their dimensions need to be computed for n-geometric-constraints, where a geometric constraint can be an algebraically expressed constraint on the pose, or location of the fixed or moving pivots or something equivalent. In addition, we include both linear and non-linear and exact and approximate constraints. This extension also includes the problems where there is no solution to the classic Burmester problem, but designers would still like to design a four-bar that may come closest to capturing their intent. Such problems are representative of the problems that machine designers grapple with while designing linkage systems for a variety of constraints, which are not merely a set of poses. Recently, we have derived a unified form of geometric constraints of all types of dyads (RR, RP, PR, and PP) in the framework of kinematic mapping and planar quaternions, which map to generalized manifolds (G-manifolds) in the image space of planar displacements. The given poses map to points in the image space. Thus, the problem of synthesis is reduced to minimizing the algebraic error of fitting between the image points and the G-manifolds. We have also created a simple, two-step algorithm using Singular Value Decomposition (SVD) for the least-squares fitting of the manifolds, which yields a candidate space of solution. By imposing two fundamental quadratic constraints on the candidate solutions, we are able to simultaneously determine both the type and dimensions of the planar four-bar linkages. In this paper, we present 1) a unified approach for solving the extended Burmester problem by showing that all linear- and non-linear constraints can be handled in a unified way without resorting to special cases, 2) in the event of no or unsatisfactory solutions to the synthesis problem certain constraints can be relaxed, and 3) such constraints can be approximately satisfied by minimizing the algebraic fitting error using Lagrange Multiplier method. In doing so, we generalize our earlier formulation and present a new algorithm, which solves new problems including optimal approximate synthesis of Burmester problem with no exact solutions.