Explicit Algorithm for Robot Manipulator Dual Dynamics

Kinematicians have used dual numbers to obtain rigid body kinematics in a compact three-dimensional form by substituting dual for real numbers in the equation of rotational motion. No such simple relation, known as ‘principle of transference’, existed however, for dynamics. The commonly used inertia binor by which dual momentum is calculated, raises the dual dynamic equations to six dimensions. In fact, the inertia binor does not act on the dual vector as a whole, but rather on its real and dual parts as two distinct real vectors. The recently introduced dual mass operator can serve as the missing link between the dual kinematic and the dual dynamic equations. It gives the mass a dual property which has a complementary sense of Clifford’s dual unit, namely, it reduces a motor to a rotor proportional to the vector part of the motor. With this definition of mass, the same equation of momentum and its time derivative, which holds for a linear motion, holds for both linear and angular motion of a rigid body if dual force, dual velocity, and dual inertia replace their real counterparts. Application of the dual inertia operator and motor transformation rule permits derivation of an explicit dynamic algorithm of a serial manipulator which has several advantages over the more conventional Newton-Euler and Lagrange formulations. Firstly, all the expressions of this algorithm are explicit parts of the dual transformation matrices and the constant link-attached inertia parameters. Secondly, this algorithm is an explicit, not a recursive one and does not require derivative of any one of its terms. It rather gives all coefficients of the dynamic equations in a simple and compact form of determinants and vector scalar product.