1
CS545–Introduction to Robotics
Homework Assignment 1 (Due February 6)
In the following problems, you should use MATLAB to compute numerical results and
visualize the data, and Simulink for simulations. A handout about getting started with
MATLAB is in
http://www-clmc.usc.edu/~cs545/CS545_homework.html
This web page also contains all the files needed below. IMPORTANT: In your solutions
of the homework, also provide intermediate steps how you derived the solution to a
problem.
Attention:
There will be significant extra-credit (i.e., up to 10% of the points of this
homework) for well-typed solutions. We highly recommend going for this extra credit as
we have seen that it was usually necessary in order to achieve a good grade in this class.
1) (40 Points) You are a robot control engineer for mobile robots and your team wants to
participate in the Robot Soccer Cup 2004. In order to get started, they want you to
design a simple robot controller so that it can execute desired trajectories. Your jobs are
to model the system, design appropriate controllers, and discuss the generality of the
ideas you have applied.
The general task is shown below: you have a point-mass mobile robot for which you can
ignore the rotations. It has a mass
m
, the ground friction constant is called
D
for both
direction
y
,
z
. Your actuation allows you to apply two forces
F
y
and
F
z
in the
y
and the
z
direction, respectively.
a) (10 points) Model the system as two decoupled linear systems along the
y
and
z
directions and give the equations of motions in terms of a second order dynamics using
z
z
z
y
y
y
&
&
&
&
&
&
,
,
,
,
,
.
y
F
y
D
y
m
+
=
&
&
&
z
F
z
D
z
m
+
=
&
&
&
b) (2 points) Summarize all states in the state vector
x
, and all actions in the action vector
u
.
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
=
z
y
F
F
u
u
z
z
y
y
x
x
x
x
2
1
4
3
2
1
,
u
x
&
&
c) (5 points) Reformulate the system to achieve the canonical representation.
)
(
)
(
)
(
t
t
t
Bu
Ax
x
+
=
&
(1)
where
A
and
B
are
n
×
n
and
n
×
m
matrices, respectively, and
are
n
×
1 vectors,
while
u
is an
m
×
1 vector.
x
x,
&

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