Page 217 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 203
9. Consider the coordinate frames x y z attached to link 2 and x y z attached to link 3 in Figure
2 2 2
3 3 3
3.16. Let the length of link 2 be l .
2
1. Given that joint 2 and joint 3 axes are parallel to each other (z is parallel to z ), write the 4x4
3
2
transformation matrix that describes the coordinate frame 3 with respect to coordinate frame
2; that is, T =?
23
2. Determine T , the description of the coordinate frame 2 with respect to coordinate frame 3
32
(Hint: T = T 23 −1 ).
32
10. Consider the problem in Figure 3.20. Let l = 0.5m, l 0.25 m, and F be the weight of a payload
1
2
mass of m = 100 kg, then F = 9.81 ⋅ 100 N, in the vertical negative direction of Y . Determine the
0
joint torques necessary to hold the load in position at the following conditions. Neglect the lengths
of the links.
◦
1. = 0 , = 0 ◦
1
2
◦
2. = 30 , = 30 ◦
2
1
◦
3. = 90 , = 0 ◦
1
2
11. The objective of this problem is to illustrate the problem of backlash in motion control systems
and how to deal with it. Consider the closed loop position control system shown in Figure 10.4.
Assume that the gear motion transmission mechanism is a lead-screw type (Figure 3.3). Further, let
us model the components as follows:
the motor dynamics as inertia only (J ) with no damping, and current to torque gain is K ,

m
T
a position sensor connected to the motor that gives N count∕rev number of counts per

s1
revolution,
amplifier as a voltage to current gain (K ) with all filtering effects neglected,

a
lead-screw and the load it carries is modeled with its effective gear ratio (N = 1∕(2 p)) where

p(rev∕mm) is the pitch and mass m (rotary inertia of the lead screw is neglected). Assume there
is load force acting on the inertia (F ). In addition consider that the lead screw has a backlash
l
of x , which we will assume is a constant value.
b
control algorithm is implemented with an analog op-amp as a form of PID controller.

2
Assume the following numerical values for the system components: J = 10 −5 kg ⋅ m ,
m
K = 2.0Nm∕A, N = 2000 count∕rev, K = 2A∕V, p = 0.5rev∕mm, m = 100 kg, F = 0.0N, x =
b
T
a
s1
l
0.1 mm. For simplicity, use the following relationship for the total inertia acting on the motor (although
during the period of motion when backlash is in effect and the lead-screw is not moving the nut,
the load inertia is not coupled to the motor; but we will neglect this) as well as motor torque and
transmitted force to the moving mass,
1
J = J + ⋅ m (3.337)
t m 2
(2 p)
1
T = ⋅ F l (3.338)
l
2 p
(a) If the desired positioning accuracy of the load is 0.001 mm, draw a control system block diagram
and sensors to achieve this (Hint: due to backlash, we must have a load-coupled position sensor.
Let the resolution of that sensor be called N counts∕mm. Further, we should have a measurement
s2
accuracy that is 2 to 5 times better than desired positioning accuracy).
®
(b) Develop a dynamic model of the closed loop control system (i.e., using Simulink ). Simulate the
motion in response to a rectangular pulse, that is initial position and commanded position are at
zero until t = 1.0 s, then a step position command of 1.0 mm and back to zero position command
at t = 3.0 s, and continue simulations until t = 5.0 s. Use the motor-coupled position sensor for
velocity loop with a P-only gain, and a load-coupled position sensor in the position loop with a
PD-type control. Adjust the gains in order to achieve a good response.
(c) What happens if you use only the motor-coupled position sensor, not the load-coupled position
sensor? Show your claim with simulation results. Modify component parameters if necessary to
illustrate your point.