Navigation and control of the motion of a rolling disk carrying a controlled translational moving rod

This work deals with the guidance and control of a system which is composed of a disk rolling on a plane, a slender translational joint attached through its center of mass to the diskʹs center, and a controlled slender rod that moves along the translational joint. The translational joint is controlled in such a manner that it is always in the intersection between the diskʹs plane and the horizontal plane. Denote by χ the displacement between the rodʹs center of mass and the diskʹs center O. Given N points Pi, i = 0,...,N−1 in the horizontal plane, N real numbers χj, j = 0,...,N−1, a finite time interval [0, tf], and a sequence of times τ0 = 0 < τ1 < ... <τN−1 = tf. Based on a dynamical model of the system, and by using the concept of path controllability, control laws are derived for the diskʹs tilting moment and pedalling moment, and for the force applied on the moving rod, such that [O,χ]will pass through [Pj, χj] at the time τj, j = 0,...,N−1, respectively.