1 (19)

Prévia do material em texto

Problem 2.1 OPP
Determine the dynamic equations for lateral motion of the robot in Fig. 1. Assume it has three 
wheels with a single, steerable wheel in the front where the controller has direct control of the 
rate of change of the steering angle, Usteer, with geometry as shown in Fig. 2. Assume the robot 
is going in approximately a straight line and its angular deviation from that straight line is very 
small. Also assume that the robot is traveling at a constant speed, Vo. The dynamic equations 
relating the lateral velocity of the center of the robot as a result of commands in Usteer are 
desired.
Figure 1 Robot for delivery of hospital supplies. Source: AP Images
Figure 2 Model for robot motion
Step-by-step solution
step 1 of 3
Refer FIGURE 2.46 in the textbook.
Consider the following equation for the sum of all external moments about the center of mass of 
a body.
M ^ I a .......(1)
Where.
/ is the body’s mass moment of inertia about its center of mass. 
a is the angular acceleration of the body.
Consider the following equation for the time rate of change of the steering wheel angle.
S, - U ^ .......(2)
Where.
is the control input.
The turning rate change with respect to x axis is shown in Figure 1.
Step 2 of 3
Consider the following equation for the carts turningrate of change with respect to x axis.
sin 9,
Where.
is nonzero.
L is the length of the wheel,
F^is the constant speed.
Consider is small and rearrange the above equation.
(3)
The lateral motion as a function of y is shown in Figure 2.
Step 3 of 3
Consider the following equation for the actual change in the carts lateral position.
..... (4)
Take differentiation on both sides in equation (4).
y = ......(5)
Substitute equation (3)in equation (5).
.. v^e.
Take differentiation on both sides,
y »-2—*•......(o)
Substitute equation (1 )in equation (6).
(7)
Thus, there is no dynamics come into equation (7). Therefore, there was no need to invoke 
equation (1).