Computing mobility condition using Groebner basis

• Over-constrained mechanism
• Mobility condition of over-constrained mechanisms
• Groebner basis computation
• Variable elimination using Groebner basis

A mechanism is a set of rigid links interconnected together with ideal joints and featuring at least one degree of freedom. An overconstrained mechanism is mobile provided the links' dimensions fulfill a certain relation named the “mobility condition”. The dimensions are not independent from each other and the goal is to obtain this mobility condition. Firstly, parameters are divided into two categories: dimensional parameters and positional parameters. Dimensional parameters represent links' sizes and positional parameters represent the relative positions between the links. The closure equation models the geometric problem by capturing the relationships between the two types of parameters. The principle of the paper is to generate the mobility condition by applying Groebner basis computation to the closure equation. Three methods are presented and investigated. The first one is called the univariate polynomial method (UNIPOL); the second method is called multi-order derivation method (MOD) and the third one is called the finitely separated configurations (FISECO) method. Practical implementation of these various techniques is explained by using a standard computer algebra system. The three methods are applied on a 2D overconstrained mechanism.