Grid-Based Topology Optimization of Rigid Body Mechanisms Using Different Problem Formulations

In the design process of rigid body mechanisms two main development steps can be distinguished, namely type (topology) and dimensional synthesis. Whereas much work has already been done on solving the problem of dimensional synthesis, optimization based approaches to topology design of rigid body mechanisms are rare.

Unlike solving the discrete combinatorial problem of optimal mechanism topology by means of genetic algorithms [

1

], we investigate in this work a ground structure approach similar to [

2

] but based solely on rigid bars. A relaxed formulation of the kinematic constraint equations allows an almost straightforward kinematic analysis despite the over-determined system of equations due to redundant bars in the ground structure. Similar to cross sectional areas in topology optimization of truss structures, the bars in the ground structure are parameterized by continuous design variables that can have intermediate values between 0 and 1. This continuous description allows a solution with efficient gradient-based optimization methods. However, the problem is of discrete (binary) nature and intermediate values are physically meaningless so that appropriate problem formulations must be found in order obtain a 0-1 design.

This work investigates different problem formulations and solution techniques and their ability to solve the intrinsically discrete problem of mechanism topology optimization. All presented formulations are using the continuously parameterized truss-like structure with rigid bars, but they are based on different (separation) constraints in order to achieve a 0-1 design. This includes a simple quadratic penalization as well as the power-law (SIMP) method for the solution of the path generation and output maximization problem. The functionality, the advantages and drawbacks of the grid structure approach with respect to the problem of rigid body mechanism design are discussed and illustrated by example problems.