Abstract: The displacement analysis is the basis of kinematics and dynamics research for spatial mechanisms. Though there exit many methods to solve the displacement analysis, the solution of multi-dimensional high-order algebraic equations is still a difficult problem in the research of mechanism and robotics due to the strong nonlinearity of input and output displacement parameters of spatial mechanisms. The basic process of traditional displacement analysis theory of spatial mechanisms is analyzed, three common geometric algebraic systems and the basic theoretical framework of conformal geometric algebra (CGA) are expounded. CGA is introduced into the displacement analysis of spatial mechanism and new algorithms are proposed for displacement analysis of a spatial 6R spatial mechanism and a spatial RRSRR mechanism. The revolute joints are determined by the description of rigid motion in CGA. The cosine values of all joint rotating angles and the input-output polynomial equation with a single unknown can be derived by the inner product between and all closed-form solutions are obtained. The algorithm is the pure geometric computation in a clear and coordinates-free way and avoids the use of rational angles or matrices, and complex computations for nonlinear and multivariable equations. Finally, two numerical examples are given to demonstrate the efficiency of the algorithm and the results show that this algorithm can simplify the complexity of computation with strong geometrical intuition based on Maple16.

Key words: conformal geometric algebra, spatial linkage mechanism, displacement analysis, closed-form solutions