EE 4315/5325 Robotics Midterm Take Home Exam, Spring 2013.

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EE

4315/
5325 Robotics

Midterm



Take Home

Exam,
Spring 2013
.


Instructions:

This Midterm is posted on March 19

after class. 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. There are two problems requiring written
answers as well as programming. In order to receive full credit, you must clearly explain
y
our 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 Prof. Dan Popa at
popa@uta.edu

or by stopping by at
office hours. The Midterm is due in class on
March 26
. Send your written part of the
exam in scanned or typed .doc or .pdf form as w
ell as your .m Matlab files to
popa@uta.edu
.
Late submissions will receive a dedu
ction of 25
% per day.



Problem 1
Manipulator

kinematics


25

points

EE 4315, 50

points EE 5325


Consider the “Stanford” manipulator shown
in Figure 1. The manipulator has six degrees
of freedom (2R+P+3R), and its end
-
effector (gripper) is mounted onto a ZYZ wrist
(similar to the wrist of PUMA 560). Your tasks are as follows:


A)

(10 Pts) Set up the kinematic equations describing the position a
nd orientation of the
center of the gripper using the DH framework. Clearly sketch your choice of
coordinate frames.




Figure1: Schematic diagram of the Stanford Manipulator with a ZYZ wrist

B)

(10

Pts) Repeat a) using the product of exponential formulas.
Show that if you have
chosen the same tool coordinate frame system in both cases, you get identical results.

(
EE 5325 students only
)

C)

(15 Pts) Calculate the 6x6 Jacobian
, and identify the singularities
.
(
EE 5325 students
only
)

D)

(15

Pts) Pick reasonable value
s for the manipulator dimensions and animate the robot
using the robotics MATLAB toolbox

or ROS/GAZEBO
. Assume that the initial joint
configuration is (0 0 0 0 0 0)
T
, and the final joint configuration is (



/3 0.5

/3

/3

/3)

T
. Animate the robot during
a pick and place sequence of a short vertical peg
between two locations of your choosing on the robot base plane.


Problem 2


Kinematics
of Nonholon
omic Vehicles


35 pts

EE 4315, 25

pts

EE 5325



Consider the two
-
wheeled vehicle shown in figure 2, also c
alled a differential drive robot.
It consists of two wheels of radii ρ
1
and ρ
2
, 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 φ
1
and φ
2
. A third rolle
r 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)

(10 pts) Write down the velocity kinematic constraints of the robot, as a function of
the 5
-
th dimensional robot state q=[x,y,

θ, φ
1
, φ
2
].

B)

(10

pts

EE 5325 students only
) Show that this robot is
nonholonomic, by checking to
see that integrability Pfaff conditions are not satisfied.

C)

(5pts) 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.

D)


(20

pts

EE 4315

students only
) Using MATLAB

or ROS/Gazebo
, 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.


Figure 2: Diagram of a differential drive wheeled robot with an omnidirectional front wheel.

P(x,y)

Wheel 1,
radius ρ
1

angle φ1

Wheel 2,
radius ρ
2

angle φ2

C

L

Robot

frame

O

θ


Problem 3

(Lagrangean

Dynamics
4
0

Pts

EE4315 students, 25 Pts EE 5325
)


Consider t
he mechanical system in Figure 3, a

“cart
-
pendulum” system. The cart has a
moving mass M, and is connected to a linear motor via a flexible coupling with damping
B. An inverted pendulum of length l, negligible inertia
, stiffness K

and mass m is
attached to the cart via a rotary actuator. I
f the pendulum damping coefficient is b, the
linear actuator force is F and the rotary actuator torque is

:

A)

Form the system Langrangean (10

pts
,
EE 4315

students only
).

B)

Write the dynamical equations of motion. Without performing the actual calculations.
I
ndicate the robot states, and the dimension o
f terms in the robot equation (10
pts
,
EE
4315

students only
).

C)

Detail the dynamical equations of motion and put it in the standard Lagrangean form
(
10
pts).

D)

Calculate the static pushing force and pendulum torque n
ecessary to balance the
pendulum at a specific location in

generalized coordinate space (10

pts
,
EE 4315

students only
).

E)

Using MATLAB or ROS/Gazebo,

a
nimate the cart system for your choice of non
-
zero coefficients. Plot the total energy of the system as a

function of time
, and
comment on the results (
15

pts
,
EE 5325 students only
).


Figure 3
: The cart
-
pendulum system
with a flexible pendulum




m

l

x

M

K

B

F