Abstract This paper presents an implementation of a homotopy path tracking algorithm for polynomial numerical continuation on a graphical processing unit (GPU). The goal of this algorithm is to track homotopy curves from known roots to the unknown roots of a target polynomial system. The path tracker solves a set of ordinary differential equations to predict the next step and uses a Newton root finder to correct the prediction so the path stays on the homotopy solution curves. In order to benefit from the computational performance of a GPU, we organize the procedure so it is executed as a single instruction set, which means the path tracker has a fixed step size and the corrector has a fixed number iterations. This trade-off between accuracy and GPU computation speed is useful in numerical kinematic synthesis where a large number of solutions must be generated to find a few effective designs. In this paper, we show that our implementation of GPU-based numerical continuation yields 85 effective designs in 63 s, while an existing numerical continuation algorithm yields 455 effective designs in 2 h running on eight threads of a workstation.
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A Study on Finding Finite Roots for Kinematic Synthesis
This study presents new results on a method to solve large kinematic synthesis systems termed Finite Root Generation. The method reduces the number of startpoints used in homotopy continuation to find all the roots of a kinematic synthesis system. For a single execution, many start systems are generated with corresponding startpoints using a random process such that startpoints only track to finite roots. Current methods are burdened by computations of roots to infinity. New results include a characterization of scaling for different problem sizes, a technique for scaling down problems using cognate symmetries, and an application for the design of a spined pinch gripper mechanism. We show that the expected number of iterations to perform increases approximately linearly with the quantity of finite roots for a given synthesis problem. An implementation that effectively scales the four-bar path synthesis problem by six using its cognate structure found 100% of roots in an average of 16,546 iterations over ten executions. This marks a roughly six-fold improvement over the basic implementation of the algorithm.
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- Award ID(s):
- 1636302
- NSF-PAR ID:
- 10062514
- Date Published:
- Journal Name:
- ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference Volume 5B: 41st Mechanisms and Robotics Conference
- Page Range / eLocation ID:
- V05BT08A083
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1115/DETC2017-68341
