EE4315 / EE5325 Spring 2015 : Midterm Exam (Take home)
Instructions: This Midterm is posted on April 2 after class. There are 5 problems worth 100
points, plus 15 points of extra credit in Problem 4. You are required to work alone, meaning that
no discussion amongst yourselves or with others is allowed. You can use any materials as aids
during the exam, but you must understand (not just copy) everything you include in your
answers. There are two problems requiring written answers as well as programming. In order to
receive full credit, you must clearly explain your assumptions, reasoning, and results. You will
get more credit for having a slightly incorrect answer while showing a sound understanding of
the concepts than for posting the right answer without explaining how you arrived at the result.
You can ask for clarifications by sending email to Dr. Indika Wijayasinghe or Dr. Dan Popa at
[email protected] ; [email protected] or by stopping by at office hours. The Midterm is
due in class on Tuesday, April 7. Send your written part of the exam in scanned or typed .doc
or .pdf form as well as your .m Matlab files to [email protected] and [email protected] .
No extensions will be allowed.
Problem 1 –Robot Kinematics & Statics – 50 pts UG students, 30 pts Grad students
Consider the robotic vehicle (mobile manipulator) shown in Figure 1.
Figure 1: Schematic diagram of a mobile manipulator.
The cart of the robot has a mass M=20 Kg and two identical links with length L=0.5 meters and
mass m=5 kg. Assume the links can be approximated by cylindrical rods. The robot is also
holding a payload Mp=10 Kg. Answer the following (note that parts are worth different number
of points for undergraduate/graduate students):
A. Calculate the forward kinematic map for the manipulator (10/5 pts, all students)
B. Calculate the Jacobian of the manipulator and identify the singularities (15/10 pts, all
students).
C. Write Matlab code to calculate forces and torques necessary to balance the payload in at an
arbitrary joint configuration. Check your code at configurations [0;0;0] and [0; π/2; 0] (10/5
pts all students).
D. Simulate and animate the robot while performing movements from zero joint angles to [1;
π/4; 3π/4]. Write Matlab code to draw the 2D (x,y) manipulability ellipsoids at arbitrary
location in the workspace. (15/10 pts all students)
Problem 2 Jacobians – 20 points (Grad students only)
The Manipulator Jacobian was defined using the matrix exponentials as the mapping between the
twist of the tool coordinate frame and the joint angular velocity. That means that the tool twist
expressed in base frame evolves according to:

.
 v_   J e ( ) , where J e ( )  AdTo 1
 


AdTi1 i   Ri 1

 0
AdT1  2
...
AdTN 1  N
,

^
^
^
R
p i 1 Ri 1 i , Ti 1  e1 1 e 2  2 ...e i1  i 1   i 1

 0
Ri 1 
^
pi 1 
.
1 
A) (10 pts) Is the first joint is revolute, show that :
^ 2
q 
p1  ( 1 sin( 1 )   1 (1  cos(1 ))) q1 , where 1   1 1 . (Hint: use Rodriguez’s
 1 
^
formula, Homework 2, and the exponential of twists formula).
'
B) (10 pts) Explain why the quantity i  AdTi1i is the twist i expressed in the base frame,
after undergoing rotations and translations associated to moving the joints from 0 to joint
coordinates 1 , 2 , … N. Because of this fact, you now have a very simple way of finding
singularities for different types of manipulators, by writing:
J e ( )  1'  2' ...  N'
. Use this property to show that the following robot
configurations are singular:
 Four coplanar revolute joints
 A prismatic joint perpendicular to two parallel revolute joints.
 Three parallel coplanar revolute joints.


Problem 3 –Kinematics of Nonholonomic Vehicles – 25 points (all students)
Consider the two-wheeled vehicle shown in figure 2, also called a differential drive robot. It
consists of two wheels of radii ρ1 and ρ2, respectively (not necessarily equal), separated by an
axle of length L, and independently driven by motors that allow it to rotate the wheels by angles
φ1 and φ2. A third roller wheel (C) is passive and it is called “omnidirectional” due to the fact
that it can rotate freely in all directions without sliding.The state of the robot consists of the (x,y)
position of the axle center P, the turning angle θ, expressed with respect to a fixed coordinate
frame [O], and the two wheel angles.
A) Write down the velocity kinematic constraints of the robot, as a function of the 5-th
dimensional robot state q=[x,y, θ, φ1, φ2] (all, 10 pts).
B) Show that this robot is nonholonomic, by checking to see that integrability Pfaff conditions
are not satisfied (grad students only, 10 pts)
C) Discuss how many controls you would need to operate this robot, propose appropriate
controls, find the affine form of the kinematic equations, and comment on why it is called a
“differential drive” robot (all, 5 pts)
D) Using Labview Dani robot, or MATLAB, simulate and animate a differential drive robot
with equal wheel radii, for two different input angular velocities of the wheels, for instance
equal values, or equal values but opposite signs. (UG students only 10 pts)
Wheel 1,
radius ρ1
angle φ1
C
Robot
P(x,y)
frame
Wheel 2,
radius ρ2
angle φ2
L
θ
O
Figure 2: Diagram of a differential drive wheeled robot with an omnidirectional front wheel.
Problem 4 (Lagrangean Dynamics 15 Pts, plus 15 pts extra credit)
Consider the mechanical system in Figure 3, the so-called “cart-pendulum” system. The cart has
a moving mass M, and is connected to a linear motor via a flexible coupling with stiffness K and
damping B. An inverted pendulum of length l, negligible inertia and mass m is attached to the
cart via a rotary actuator. If the pendulum damping coefficient is b, the linear actuator force is F
and the rotary actuator torque is :
A) Form the system Langrangean (5 pts).
B) Write the dynamical equations of motion. Without performing the actual calculations.
Indicate the robot states, and the dimension of terms in the robot equation (5pts).
C) Detail the dynamical equations of motion without the use of a symbolic toolbox and put it in
the standard form (5pts).
D) Animate the cart system in MATLAB for your choice of non-zero coefficients. Plot the total
energy of the system as a function of time, and comment on the results (Extra credit 15 pts).
F
K
M
x
B
l

m
Figure 3: The cart-pendulum system with state variables x and .
Problem 5 –Parallel robots – 10 pts all students
Using Grubler’s criterion, calculate how many degrees of freedom are associated with each of
the two mechanisms shown in Figure 4. In both cases discuss implications related to their use as
telemanipulation devices.
Figure 4: Two parallel mechanisms, a planar one (left) and a 3D spatial one (right)