An analytical approach to determine motions/constraints of serial kinematic chains based on Clifford algebra

Determining the motions and constraints of serial kinematic chains in a concise and visual way is an inevitable step in the analysis and design of both serial and parallel mechanisms. The now most used method is the numerical approach which resorts to solving linear equations. By introducing Clifford algebra, this paper intends to propose an analytical approach to determine the unknown 6-n (n< 6) constraints (motions) from the known n motions (constraints) of serial kinematic chains in different configurations only by drawing some auxiliary points, lines, and planes. The axes and action lines of motions and constraints are characterized by the lines that would be described by Clifford algebra. These lines can be determined analytically according to the relations among points, lines, and planes, which have been expressed by using the operation rules of Clifford algebra Cl(0, 3, 1) such as inner, outer, dual, and shuffle products. Based upon the mechanics principle that the constraint does not work on the motion, the unknown 6-n constraints (motions) of serial kinematic chains from known n motions (constraints) are determined both in an analytical algebraic form and in a visual manner. Finally, four examples are given to demonstrate how to use this approach and test its validity. The merit of this approach is beneficial to the digital analysis and design of both the serial and parallel mechanisms by means of computer and programming languages.