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

Year 2019, Volume: 68 Issue: 2, 1879 - 1894, 01.08.2019

Mehmet Aydinalp Mustafa Kazaz , Hasan Hüseyin Uğurlu

https://doi.org/10.31801/cfsuasmas.461781

Cited By: 2

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 (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 (DOFs), and pure-rolling
motion has two (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

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