2 System description and problem   statement

2.1 Variables  in  motion equation

Figure 2 shows the prototype robotic system built at Shanghai University. According to kinematic analysis   [5],

  its motion equation is shown  as

& Shuai Guo guoshuai@shu.edu.cn

V ¼ D0W; ð1Þ

1 Shanghai Key Laboratory of Intelligent Manufacturing  and Robotics, Shanghai University, Shanghai, People’s  Republic

2VX 3

2 —R0 R0 —R0 R0    3

of China

where V ¼ 4VY 5; D0 ¼ 16

R0 R0 R0 R0    7;

44  R0

  R0

  R0         R0    5

2 Department of Aerospace Engineering, Ryerson  University, Toronto, Canada

x0 l0 þL — l0 þL — l0 þL l0 þL

Fig. 1  3D-model of the mobile   robot

2x1 3

W ¼ 6x2 7: L is half of the distance from the front axle   to

3

Fig. 3  Mecanum wheel and  roller

4 5

x4

the rear axle, l0 the transverse distance from the wheel centers which are also the contact points on the ground to the platform center line, a the mounting angle of the roller, R0 the radius of four wheels; x1, x2, x3 and x4 are angular velocities of the four wheels, respectively. All the param- eters  in   Eq.   (1)   are   defined   in   machine   design, L = 750 mm, l0  = 615 mm, R0  = 187.5 mm, a =   45°.

Figure 3 is a view of a mecanum wheel and a roller. The roller is fitted on the edge of the mecanum wheel in angle a with its central axis, which can rotate freely around its central axis. The wheel relies on the friction that the roller acts on the ground to move. Because the material of its outer rim is usually rubbers, it will deform under pressure. Figure 4 is a view of distribution of roller outer rim deformation in its transverse section. F is the pressure on the roller, T the driving torque. The radiuses of wheels reduce differently under different pressures. The greater the

Fig. 4  Distribution of roller  deformation

pressure, the greater the deformation is. The deformation zone shifts with the action of driving torque and  the shifting causes the position of wheel center to change. From the force analysis of mecanum wheel [6], we know that with the wheel rotating, a driving torque is generated by the force in roller axial direction, so the position of the contact points will shift in axial direction. With the deformation of the roller, the R0  and l0  will   change.

(a) (b)

Fig. 2   a Prototype robotic system, b motion   model

Fig. 5  Motion model of mobile   platform

center O in X-direction; R1, R2, R3 and R4 are the  radiuses of four wheels, respectively. The direction of the arrow indicates the positive  direction.