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Figure 43: Mecanum Wheels Free Body Diagram
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- Figure 44: Mecanum Wheel Kinematics: bottom view
| Figure 43: Mecanum Wheels Free Body Diagram
The free body diagram of Figure 43 above shows that that the diagonal orientation of the mecanum wheel rollers cause a force in the direction of the red vectors when the wheels spin forward. It is important to notice that not all mecanum wheels’ rollers are oriented the same way, in fact they come in pairs that are mirror opposites of each other, and hence their resulting force vector directions are mirror opposites as well. Like any vector, each of these wheel force vectors can be broken down into standard Cartesian x and y direction vectors, shown as blue arrows in the figure. By looking at these blue arrows, it becomes apparent that mecanum wheels are made as mirror opposite pairs so that spinning them in various directions causes their x and y force components to either be added or cancelled out. For example, when the wheels spin forward, the forward moving vectors add, but the left and right vectors cancel each other out, resulting in the robot moving forward, also shown in Figure 44 below. The same idea can then be applied to create other motions such as moving left. If wheels 1 and 3 spin backward and wheels 2 and 4 spin forward, the combined forward and backward spins will cancel out, but the left pointing vectors will all add together to create an overall desired left movement. 
Figure 44: Mecanum Wheel Kinematics: bottom view In order to create more complex motions, such as moving at an angle of 30° while rotating 75°, it becomes necessary to create a free body diagram showing forces exerted on the robot (such as by the motors and by the ground). How to create a free body diagram is not within the scope of this document but is covered in a variety of standard physics or mechanical engineering textbooks. The important thing to know when creating a free body diagram is what kind of information, or rather what relationships, you need to write equations for. In the case of the ModBot, the desired relationship being investigated is, “At what speed do I need to turn my motors in order to get the desired translational and angular velocities for my robot?” Using Figure 44 above, the following inverse Jacobian equation was derived which relates the Cartesian space velocity command (i.e. the desired x and y velocity and the desired angular velocity) to the four wheel velocities, v1,v2,v3,v4 : 
is the tilted angles of rollers fixed on the wheels, for this case = π/4;  is the installed angle of the wheels, namely, the angle between the horizontal line and the line connecting the center of the wheel and the center of the robot.
is the distance from the center of the wheel to the center of mass of the robot. vx, vy are the x and y component of the velocity of the robot. ω the angular speed of the robot. It should be noted that these equations also assume the four mecanum wheels’ contact points with the ground form a perfect square. This is shown more precisely in Figure 45 below and in general is recommended in mecanum wheel designs both for performance and for simplifying the equations. 
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