Give a mathematical description of the topologies of the C-

Give a mathematical description of the topologies of the C- spaces of the following systems. Use cross products, as appropriate, of spaces such as a closed interval a, b] of a line and Rk, Sm, and T", where k, m, and n are chosen appropriately. 

(a) The chassis of a car-like mobile robot rolling on an infinite plane 

(b) The car-like mobile robot (chassis only) driving around on a spherical asteroid.

(c) The car-like mobile robot chassis only) on an infinite plane with an RRPR robot arm mounted on it. The prismatic joint has joint limits, but the revolute joints do not.

(d) A Free-Flying Spacecraft with a 6R arm is mounted on it and no joints Limits.