The inverse kinematics of rolling contact of timelike curves lying on timelike surfaces

Authors : Mehmet AYDİNALP, Mustafa KAZAZ, Hasan Hüseyin UĞURLU

Pages : 1879-1894

Doi:10.31801/cfsuasmas.461781

View : 14 | Download : 5

Publication Date : 2019-08-01

Article Type : Research Paper

Abstract :Rolling contact between two surfaces plays an important role in robotics and engineering such as spherical robots, single wheel robots, and multi-fingered robotic hands to drive a moving surface on a fixed surface. The rolling contact pairs have one, two, or three degrees of freedom insert ignore into journalissuearticles values(DOFs); consisting of angular velocity components. Rolling contact motion can be divided into two categories: spin-rolling motion and pure-rolling motion. Spin-rolling motion has three insert ignore into journalissuearticles values(DOFs);, and pure-rolling motion has two insert ignore into journalissuearticles values(DOFs);. Further, it is well known that the contact kinematics can be divided into two categories: forward kinematics and inverse kinematics. In this paper, we investigate the inverse kinematics of spin-rolling motion without sliding of one timelike surface on another timelike surface in the direction of timelike unit tangent vectors of their timelike trajectory curves by determining the desired motion and the coordinates of the contact point on each surface. We get three nonlinear algebraic equations as inputs by using curvature theory in Lorentzian geometry. These equations can be reduced as a univariate polynomial of degree six by applying the Darboux frame method. This polynomial enables us to obtain rapid and accurate numerical root approximations and to analyze the rolling rate as an output. Moreover, we obtain another outputs: the rolling direction and the compensatory spin rate.
Keywords : Darboux frame, forward kinematics, inverse kinematics, Lorentzian 3 space, pure rolling, rolling contact, spin rolling