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```									                                                               EE5106/MCH5209/ME5402

NATIONAL UNIVERSITY OF SINGAPORE

FACULTY OF ENGINEERING

EXAMINATION FOR
(Semester II: 2002/2003)

April / May 2003 - Time Allowed: 2 ½ Hours

INSTRUCTIONS TO CANDIDATES:

1. This examination paper contains four (4) questions and comprises seven (7) pages.

2. Answer all four (4) questions.

3. All questions carry equal marks.

4. This is an open-book examination.

Q.1   (i) Figure 1.1 is the schematic diagram of a robot with one translational joint (along
the yu axis) followed by three rotational joints. The first rotational joint following
the translational joint has an axis of rotation parallel to zu. The 2nd and 3rd
rotational joint axes are parallel to each other and to the xy plane of Frame U. The
joint limits of travel are indicated in the figure. Note the direction of the positive
motion for each joint axis.

a) How many frames are needed to completely describe the kinematics of this
robot?
(2 marks)

b) Define all needed frames according to the Denavit-Hartenberg convention
such that the position and orientation of adjacent frames are defined by the
four kinematic parameters (, r, d, ):

Ti  Rot ( z , i )Trans (0, 0, ri )Trans (d i , 0, 0) Rot ( x,  i )
i 1

(7 marks)

c) Provide the values for all the kinematic parameters and indicate which
parameter is the joint coordinate (variable) for each link. That is, fill in the
following table.
1
2
etc
(6 marks)

d) Indicate the range of possible motions for each joint coordinate (qi) of the
robot, where qi is the kinematic parameter that is variable and defined

(4 marks)

XE
1m

3m
90 to 180

ZE

60 to 300
ZU

1 to 10 m          2m
0 to 360 (no joint limit)

XU
YU
Figure 1.1

Q.1   (ii) Frame B is attached to a moving body. Frames A and C are fixed to the ground
with ATC known, as shown in Figure 1.2. The body B is initially at a known
position and orientation with respect to Frame A, and defined by the 4  4
homogenous transformation matrix ATB. The body undergoes the following
motion in the indicated sequence:

1st:       Rotation about X axis of Frame A by 30
2nd:       Rotation about Y axis of Frame C by 40
3rd:       Translation along Frame B (itself) by (1,2,3) m.

Determine the new position and orientation of Frame B is Frame A, ATB, after the
three motions above. You need not evaluate nor simplify the expression for ATB.

(6 marks)

B

C
A

Figure 1.2

Q.2   (i) Figure 2.1 is the schematic diagram of a robot with three rotational joints whose
axes of rotation are all parallel to the Z axis of Frame 0. The joint configuration of
the robot is indicated by (q1, q2, q3) whose zero positions and positive directions
of motions are shown in Figure 2.1. The position of the end-effector expressed in
Frame 0 is (x,y), and the orientation of the end effector is desribed by the angle 
between the end-effector (last link) and the positive x axis of Frame 0.

a) Does the robot have singularities with respect to the task (x, y, )? If so,
determine the joint coordinates that correspond to each singular configuration.
(5 marks)

b) Does the robot have singularities with respect to the task (x, y)? This means
that the robot is programmed to position its end-effector at (x, y) and its
orientation is not important. If there are singularities, determine the joint
coordinates that correspond to each singular configuration.
(5 marks)
c) Does the robot have singularities with respect to the task (x)? This means that
the robot is programmed to position its end-effector to satisfy the desired x-
coordinate position only. The y-coordinate position and orientation of the
end-effector are not important. If there are singularities, determine the joint
coordinates that correspond to each singular configuration.
(5 marks)

q3
(x,y)

q2               
Y0

q1

Z0                          X0
Figure 2.1

Q.2   (ii) Figure 2.2 shows a schematic diagram of a robot with n rotational joints hanging
on a ceiling and carrying a load of mass m kg. At a certain robot configuration,
the following are known:

        Centre of gravity of the load is at coordinates (a, b, c) m with respect to
Frame E, where Frame E is the frame attached to the end-effector.

        The position and orientation of the end-effector in Frame 0 is given by 3
 1 vector 0pE and 3  3 rotation matrix 0RE.

        The n  1 robot joint configuration q.

        The 6  n manipulator Jacobian relating the velocity of Frame E with
respect to Frame 0, 0JE.

Determine the n joint actuation torques needed to maintain the load at the robot
configuration q. Assume that the mass of the robot is negligible compared to the
(10 marks)

Y0
Centre of
X0                                               gravity of

End Effector
Frame E

Figure 2.2

Q.3   Figure 3.1 shows a two link (RP) robot in the vertical plane under the influence
of gravity. Assume that I 1 is the moment of inertia of link 1 about its rotating
axis, m2 is the equivalent lumped mass of link 2, m3 is the mass of the
payload, l1 is the distance between joint 1 and the starting point of link 2,  1 is
the driving torque at the joint 1, and f 2 is the applied force for link 2.

(i)     Sketch a set of generalized coordinates, and derive the positions and
velocities of m2 and m3 with respect to a base coordinate system.
(5 marks)

(ii)    Find the kinetic energy K and the potential energy V (q) of the robot,
and derive equations of motion using Langrangian formulation.
(10 marks)

l2

m3
m2
l1

Figure 3.1

(iii)   Assuming the drive chain of the second link is flexible as shown in
Figure 3.2, where m0 is the equivalent mass of the rotor of the motor,
k is the sping constant, and the rests of the system remain the same as
in Figure 3.1. Derived the corresponding equations of the motion using
any method you are comfortable with.

l2

m3
m2

l1
k
m0

Figure 3.2

(10 marks)

Q.4   Figure 4 is a schematic drawing of a single link robot, where  m and  are the motor
and load torques, I m and I the inertias of the motor and load, bm and b the viscous
damping coefficients at the motor and load bearings,  m and  the joint angles of the
motor and load respectively, and  is the ratio of an ideal belt transmission
mechanism with no slippage.

(i)     If the inertial load varies between 5 and 9 kgm2 , the rotor inertia is
I m  0.01kgm2 ,         the   viscous       damping       coefficients    are
bm  0.01kgm2 s 1 and b  0.02kgm2 s , the transmission ratio   10 , and
the system posses unmodelled resonances at 8.0, 12.0 and 20.0
radians/second, design a PD controller with feedback from  such that the
system is never underdamped, will not excite the resonances, but is as stiff as
possible.
(7 marks)
independent joint control such as the one in part (i). Explain the situation
where there is a need for integral control.
(5 marks)
(iii)   Explain why most industrial robot control systems work well though
independent PID control has been used.
(5 marks)

(iv)    Assume that all the parameters ( I m , I , bm , b and  ) are unknown. Design
an adaptive controller that can guarantee the stability of the system, and
briefly explain the main ideas.
(8 marks)

I


m
Im                             b
m                         
bm

END OF PAPER

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