Ph.D Student Cohen Avraham Hyper-Dual Numbers and their Application to Rigid Body Equations of Motion Department of Mechanical Engineering Professor Moshe Shoham Abstract

The algebra of hyper-dual numbers and hyper-dual vectors of order n, developed in this Research Thesis, follows the same rules as those of dual numbers and dual vectors. By showing that the basic formulae of vectors scalar and vector multiplication, hold for dual vectors of order n and that the basic trigonometric formulas hold for dual angles of order n, we concluded, that all formulae of vector algebra and trigonometric functions that are based on the above identities also hold for dual numbers of order n. This, as a result, extends Kotelnikov’s “principle of transference” developed for dual numbers, to hyper-dual numbers of order n. Utilizing the extended principle of transference dual numbers of order 2 are then applied to multi-body kinematics. First, the hyper-dual angle that encompasses a body’s position, orientation as well as its velocity, is defined as an element of the hyper-dual transformation matrix. Then, the “automatic differentiation” feature of the dual numbers is used to obtain the second derivative of a body pose. The body’s velocity and acceleration are obtained from the elements of the hyper-dual transformation matrix by algebraic manipulations only, with no need for further derivatives of the body pose with respect to the generalized coordinates. Following the formulation for multi-body kinematics we then go on and formulate the equations of motion of a rigid body in a dual numbers of order 2 form. By introducing the hyper-dual Jacobian matrix, hyper-dual velocity, hyper-dual momentum and the hyper dual inertia operator, while taking advantage of the “automatic differentiation” feature of the dual numbers, rigid body equations of motion are derived in a compact form with no need for further differentiation with respect to the generalized coordinates. An open kinematic chain robot manipulator is presented as an exemplary application of the dual numbers of order 2 scheme to multi-body system kinematics and dynamics. Following the representation of rigid-body kinematics and dynamics in dual number form, the algebra of hyper-dual quaternion by use of quaternion and dual numbers of order 2, is first presented as an extension to Hamilton's quaternion and Clifford's dual-quaternions (bi-quaternions). First, the dual quaternion angle that encompasses a body’s position, orientation is reviewed. Then, the hyper-dual quaternion angle that encompasses a body’s position as well as its velocity, is defined as an element of the hyper-dual quaternion, while taking advantage of the “automatic differentiation” feature of hyper dual numbers, rigid body equations of motion are derived with no need for further differentiation with respect to the generalized coordinates.