Position model computational complexity of walking robot with different parallel leg mechanism topology patterns

This article studies computational complexity of kinematic models for 2 patterns of 1UP-2UPS parallel mechanisms. These mechanisms are designed for hexapod walking robots applying real time control systems, which require the computing speed of kinematic models to be as swift as possible. The difference of these 2 patterns is that rotational axes of universal joints on limb 1 are reversed. In this paper, both forward and inverse kinematic models are formulated and it will be shown that both pattern I and II have analytical solutions for inverse kinematic problems. However, the forward models can be derived to univariate polynomial equations with different orders: 8th for pattern I and 4th for pattern II. It is known that the highest degree of polynomials can be solved analytically is 4, thus the computational complexity of pattern I is much higher than pattern II. In our prototypes, the forward kinematic model of pattern I costs 10 times more time than pattern II, hence pattern II is preferred for mobile robots.