Use of continuation methods for kinematic synthesis and analysis

There is an apparent need for a reliable mathematical procedure to solve kinematic synthesis and analysis problems. Quite often engineers design mechanisms using numerical methods because closed-form and graphical solutions are not always possible. Newton-Raphson is the most popular method to solve systems of nonlinear polynomial equations which arise in kinematics. This and the other currently popular methods need good initial guesses for proper convergence;The objective of this dissertation is to introduce continuation methods as an alternative for the solution of kinematic synthesis and analysis problems. Continuation methods offered several advantages over existing numerical techniques including: (1) no initial approximation for the solution is required; (2) all solutions for a system of n polynomial equations in n unknowns can be obtained; (3) systems of n equations in (n + 1) unknowns can be solved where the solution set will generate a system of curves. These features are very useful for solving kinematic problems since the equations that arise can be cast in polynomial form, and since the ability to find all the solutions provides the engineer with more options;Examples, which include four-bar, five-bar, six-bar, eight-bar and revolute-spherical dyad, demonstrate the abilities of continuation method. First, four-bar five position motion and path generation synthesis are considered. The motion generation problem can be modified to solve a path generation with prescribed timing problem. Complex mechanisms such as the geared five-bar, six-bar and eight-bar mechanisms are then synthesized by discretizing the mechanisms into combinations of dyads and triads. These components are designed independently and pieced together to form the required linkage. A seven position triad synthesis example and a revolute-spherical dyad synthesis examples illustrate the procedure followed to implement the method and to reduce the number of paths tracked. The triad was designed for motion generation with prescribed timing and the dyad for motion generation. Finally, the application of the procedure for a 3-revolute and 6-revolute robot trajectory planning is presented