Constructing the Mathematical Model of a Mobile Manipulator and Estimating the Energy Expenditures for Moving It

Abstract

The mathematical model of a mobile robotic manipulator is investigated using the Wolfram Mathematica computer algebra system. The system constructed at the Moscow Power Engineering Institute’s Department of Theoretical Mechanics and Mechatronics for participating in international competitions was used as a prototype. The manipulator is mounted on a three-wheeled platform with one passive and two driving wheels. It is assumed that the mobile manipulator moves in a horizontal plane. It is also assumed that the wheels do not slip at the points where they come in contact with the surface. Each of the driving wheels is rotated by a DC motor. The manipulator arm has a telescopic design. At the point of its attachment to the chassis, the arm can rotate both about the horizontal and vertical axes. The arm rotary motions are effectuated by DC motors having reducers with different gear ratios. In the present study, the mobile manipulator is considered as a system of rigid bodies. Only the platform weight and the end effector mass are taken into account in the analysis. The manipulator wheels and rods are considered weightless. Transients in the drive motor are not taken into account. The no-sliding condition in the projection on the drive wheel axles (sideslip) yields the speed limit, which determines nonholonomic systems. The mobile manipulator motion equations are written in the form of Appel’s equations. Control by means of software is based on the reaction forces that arise as a result of imposing nonholonomic constraints on the movement of the platform and the manipulator’s end effector. The very formulation of the problem of using the natural reactions of mechanical constraints in control yields a close-to-optimal solution, because it corresponds to the extreme principles of classical mechanics. Numerical simulation of manipulator motion along a predetermined trajectory has been implemented. The energy consumption by the drive system is estimated based on the data obtained from the simulation. These estimates are used as a criterion in selecting the optimal gear ratios of the motor-gear unit.