| This thesis studies analytic solution methods for sets of nonlinear equations arising in mechanical engineering. It addresses sine-cosine and algebraic polynomials, both of which are commonly found in kinematic and geometric constraint problems. This work focuses on dialytic elimination as a solution method. This method can eliminate, in one step, all but one of the variables in a system of equations, leaving a univariate polynomial whose roots may be used to find all possible solutions of the system.; When designing a mechanical system, engineers are often faced with kinematic constraints on mechanisms; they may, for instance, wish to guide a body through a specified set of positions using a linkage, or they may wish to know the joint angles or position and orientation of the end-effector on a robot arm. Such problems lead to sets of nonlinear equations which engineers must then solve, using either iterative numeric techniques or analytic techniques to find the solutions. Since sets of nonlinear equations normally have a finite number of different solutions, knowing all solutions to such systems simplifies analysis and allows the designer to choose the best option among many.; The main contributions in this thesis are the systematic categorization of structure in sets of nonlinear equations arising in kinematics, and the identification of optimal solution methods for different categories. Sine-cosine polynomials are analyzed extensively, and alternative problem formulations based on Rodriques parameters, Euler parameters, isotropic coordinates, and direction numbers are evaluated. A new solution procedure is presented which is capable of solving the position problems of all planar linkages with revolute joints. Throughout the analysis, an effort is made to understand the behavior of different modeling choices, which can lead to more efficient solution methods. Lastly, the solutions to a variety of practical kinematics problems are presented to illustrate the application of these solution methods. |
