The first part of this book is focused on the design of a compass robot as a nonlinear dynamics system. Three components including robot's structure, gear and motor are interacting during design process to achieve better performance, higher stability and lower cost. After achieving the design method, different actuators are selected for a given structure and the their performance are compared in the terms of cost, efficiency and their effect on the performance. In the second part of the book, the optimization of a nonlinear system with subject to the stability constraint is investigated. Including stability analysis in the optimization process, makes it very complicated and inefficient. Limit cycle convergence is the best solution among the existing methods. In this book, an important step is taken to upgrade the optimization. A computational algorithm is developed which obtains the convergent limit cycle in a short time. This algorithm speeds up the optimization time tremendously and allows the optimization search in a broader area. Even tough a compass robot is considered as an example of the nonlinear system, this algorithm is applicable to all nonlinear dynamics systems.