An overview of higher-order kinematics of rigid body and multibody systems with nilpotent algebra

Abstract

This paper proposes a framework for a new computational method based on multidual nilpotent algebra calculus of the higher-order acceleration fields of the rigid body motion and multibody systems. A closed-form coordinate-free solution is presented, this result being generated by the morphism between the Lie group of the rigid body displacements and the Lie groups of the multidual homogenous matrix, orthogonal hyper-multidual tensors and, respectively, the hyper-multidual unit quaternions. The solution is implemented for higher-order kinematics analysis of lower-pair serial chains by a specific product of the exponential formula. A general method for studying the vector field of arbitrary higher-order accelerations is described. The “automatic differentiation” feature of the multi dual and hyper-multidual functions is used to obtain simultaneously a higher-order derivative of a rigid body pose. The methodologies are obtained without 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 multi dual homogenous matrix, or orthogonal hyper-multidual tensors, and, respectively, the unit hyper-multidual quaternions. In the case of closed kinematic chains, equations that provide higher-order kinematic constraints in the compact form are given in general form.