Advances Representation of Higher-Order Kinematics of Motion. Hypercomplex Lie Groups and Lie Algebras

This paper proposes a new computational method based on vector and quaternionic calculus and the properties of dual and multidual algebra for analysis of the higher-order acceleration field of rigid body and spatial kinematics chains. The hypercomplex commutative nilpotent algebras represent simultaneous higher-order kinematic invariants of vector field of higher-order acceleration. The solution is implemented for higher-order kinematics analysis of lower-pair serial chains. Then, a general method for studying the vector field of arbitrary higher-order accelerations is described. The “automatic differentiation” feature of the multidual and hyper-multidual functions is used to obtain the higher-order derivative of a rigid body pose. This is obtained without requiring further differentiation of the body pose concerning time. It is proved that all information regarding the properties of the distribution of higher-order accelerations is contained in the specified unit hyper-multidual quaternion.