Mechanics of Mobile Robots with Mecanum Wheels Klaus Zimmermann, Igor Zeidis, Florian Schale and Pedro Alonso Flores-Alvarez

Abstract The mechanics of a robot with Mecanum wheels is studied. A Mecanum wheel is a wheel with rollers attached to its circumference. Each roller rotates about an axis that forms an angle with the plane of the disk. In robotics, a simplified approach, in which the equations of non-holonomic kinematic constraints are solved approximately by means of a pseudo-inverse matrix, is frequently applied. Such an approximate approach leads to “holonomization” of the system and allows Lagrange’s equations of the second kind to be used. In the present paper, the equations of motion are obtained on the basis of the principles of non-holonomic mechanics. The optimal torques to be applied to the wheels and the respective voltages to be fed to the electric in order to provide a prescribed trajectory for the robot’s center of mass are defined. The theoretical results are compared with measured data obtained by experiments with a  prototype.70101

1 Introduction

For service robots and equipment for physically disabled persons, mobile systems with high maneuverability play an important role. The conventional wheel moves back into the focus of interest. Furthermore, the so-called omnidirectional wheels (e.g., “Mecanum wheels”, see Ilon 1975), generating constraints different from the conventional wheel, lead to investigations, based on the Mechanics and Control of non-holonomic systems. At the present time, vehicles with  Mecanum  wheels (Fig. 1, left) are gaining ground for various applications. These wheels have rollers that are arranged at the circumference of the wheel. The rollers are orientated at an angle to the wheel plane and they can rotate about their own axis (Fig. 1, right). As a rule, this angle is equal to 45°. Such wheels have additional kinematical possi- bilities in comparison with conventional wheels. Due to these possibilities, a vehicle with Mecanum wheels can move forward-backward, leftward-rightward and rotate in an arbitrary way. Usually a Mecanum-wheeled vehicle has four wheels. By varying the rate and the direction of rotation of each wheel, one can implement, for example, a translational motion of the vehicle in any direction, as well as arbitrary turns and rotations on the  spot.

The issues of kinematics of wheeled systems, including those with Mecanum wheels, are reviewed in Campion et al. (1996). The issues of kinematics, dynamics, and control of systems with Mecanum wheels in a non-holonomic treatment are considered, for example, in Wampfler et al. (1989), Zimmermann et al. (2009), Martynenko and Formal’skii (2007) for a number of particular cases. There are a great number of studies on robotics, in which the kinematics and dynamics of robots with four Mecanum wheels is approximately treated in terms of holonomic mechanics (see, e.g., Muir and Neumann 1990; Viboonchaicheep et al. 2003; Tsai et al. 2011). In these studies, pseudo-inverse matrices are used to resolve the kinematic constraint relations. Zimmermann et al. (2014) compared the equations derived on the basis of non-holonomic mechanics with the equations derived by using the approximate technique. In the present paper, the equations obtained by the methods of non-holonomic mechanics are used. On the basis of these equations, the optimal torques to be applied to the wheels and the respective voltages to be fed to the electric motors in order to provide a prescribed trajectory for the robot’s center of mass are calculated. The back electromotive force (EMF) effect being taken into account.

Fig. 1   A Mecanum wheel (left) and the mechanical model  (right)

2 Kinematics of a Mecanum Wheel

For a conventional wheel, the contact between the wheel and the supporting plane is characterized by the condition that the wheel is rolling without slip. This means that the velocity of the point by which the wheel contacts the plane at each current instant is equal to zero. Then the projections of the velocity of the contact point onto the direction lying in the wheel plane, as well as onto the direction perpendicular to this plane, are equal to zero. For a Mecanum wheel, only the projection of the velocity of the contact point onto the axis of rotation of the roller is equal to zero. As a model of a Mecanum wheel we will consider the rolling of a disk of radius