Kinematic synthesis, at its heart, involves ﬁnding the zero-dimensional solution set of a
system of polynomials. The degree of these polynomials increases rapidly as more com
plex designs are considered. The computation time required to ﬁnd the solution set has
traditionally been the bounding factor for what can been achieved in kinematic synthesis.
Homotopy continuation is typically used to ﬁnd solutions to these polynomials. Homotopy
continuation is itself an inherently parallelizable method. Graphics processing units (GPUs)
were developed were developed with a structure that makes them optimal to solve problems
in parallel. This dissertation explores the use of homotopy continuation running a GPU in
order to decrease the computation time required for kinematic synthesis.
First, we discuss the development of an algorithm for homotopy continuation that is ideal
to run on a GPU. The traditional path tracking algorithm is analyzed and then modiﬁed to
better perform on a GPU. Additionally, the endgame methods are analyzed and the more
ideal method is identiﬁed and implemented in CUDA. We outline the drawbacks of such
modiﬁcations and discuss why they are admissible in the context of kinematic synthesis.
The implementation of the new homotopy continuation algorithm on a GPU is demon
strated by solving the four-bar linkage synthesis problem. This is the ﬁrst development of
a GPU-accelerated four-bar linkage design system. Novel (non-Burmester) loop equations
are derived such that the entire mechanism is solved in one computation. These equations
are then reduced and implemented into CUDA. The entire program is outlined and then
demonstrated on a sample design problem. The results are compared with a similar CPU
implemented system and a GPU speedup of around 120 times was observed.
The results of the four-bar linkage design system were then extended to the problem of six
bar linkage synthesis. A system was developed utilizing the same GPU-based path tracker
and endgame. This has resulted in the ﬁrst known GPU-accelerated six-bar linkage design
system that presents new opportunities for multi-GPU systems capable of designing even
more complicated mechanisms.