9 AUTONOMOUS VEHICLE MISSION DESIGN, WITH A SIMPLE BATTERY MODEL13
9
Autonomous Vehicle Mission Design, with a Simple
Battery Model
An autonomous land robot carries its own energy in the form of a E(t = 0) = 700W h
(Watt-hour) battery. The robot can stay
2.004 MODELING DYNAMICS AND CONTROL II
Spring 2003
Problem Set No. 4
Problem 1
A ladder AB of length 1m slides down a wall. At the instant shown below, the velocity of
point B touching the oor is vB = 2 m/s. Determine the velocity of point A touching the
2.004 MODELING DYNAMICS AND CONTROL II
Spring 2003
Problem Set No.1
Problem 1
An experimental apparatus to determine the friction characteristics of various materials
is shown in Figure 1. A mass m is released with a known velocity and slides on the test
4 BRETSCHNEIDER SPECTRUM DEFINITION
4
5
Bretschneider Spectrum De nition
The formula for the Bretschneider (one-sided) ocean wave spectrum is
S(!) =
4
5 !m 2
H e
16 ! 5 1/3
5! 4 /4! 4
where ! is frequency in radians per second, !m is the modal (most likel
13 NUMERICAL SOLUTION OF ODES
13
28
Numerical Solution of ODEs
In simulating dynamical systems, we frequently solve ordinary dierential equations. These
are of the form
dx
= f (t, x),
dt
where the function f has as arguments both time and the state x. The
2.004 MODELING DYNAMICS AND CONTROL II
Spring 2003
Problem Set No. 2
Problem 1
A fender is mounted to a vehicle via two shock absorbers as depicted in the sketch.
Shock absorber
Fender
Vehicle
Barrier
Figure 1
Each shock absorber can be approximately desc
2.004 MODELING DYNAMICS AND CONTROL II
Spring 2003
Problem Set No. 7
Problem 1. Pendulum mounted on elastic support. A collar of mass m slides
without friction on a horizontal rigid rod and is restrained by a pair of identical springs
with spring constant
2.004/MODELING DYNAMICS AND CONTROL II Spring 2003
Problem Set No. 3
Problem 1
The two masses on the right of the sketch below are slightly separated and at rest initially;
the left mass is incident with speed vo. Assuming head-on perfectly elastic collis
1
1
2
3
Doing Justice embodies three principles:
Offenders will be held fully accountable for their actions
The rights of persons who have contact with the system will be protected.
Like offenses will be treated alike
1
Controlling crime- designed to cont
10 SIMULATION OF A SYSTEM DRIVEN BY A RANDOM DISTURBANCE
10
17
Simulation of a System Driven by a Random Disturbance
1. Simulate the second-order system:
x + ax + bx = d(t)
with a = 0.4 and b = 2.25.
Figure 1 below shows responses to the same disturbance
3 FOURIER SERIES
3
4
Fourier Series
Compute the Fourier series coecients A0 , An , and Bn for the following signals on the
interval t = [0, 2]:
1. f (t) = 4 sin(t + /3) + cos(3t)
First, write this in a fully expanded form: y(t) = 4 sin(t) cos(/3) + 4 cos(
2 CONVOLUTION
2
3
Convolution
The step function s(t) is dened as zero when the argument is negative, and one when the
argument is zero or positive:
0 if t < 0
1 if t 0
s(t) =
For the LTI systems whose impulse responses are given below, use convolution to
Problem Set 1 Solutions
1.
Vm(t)
Ffriction
m
dv m
= f friction = Bv m
dt
vm = v0 e
B
t
m
m
B
1
m = 10kg, v m = v0 at t = 5 sec
2
B
5
1
v 0 = v0 e 10
2
B = 1.386
Time constant =
2.
d
d d
d2
d2
f friction = B = B
= K B , where K B = B
2
2 2
4
4
= K i I s
2.004 MODELING DYNAMICS AND CONTROL II
Spring 2003
ProblemSet No. 5
Problem 1
A particle of mass m slides without friction on a smooth inclined plane M which, itself,
is freeto slidewithout frictionon asmo oth horizontal surface.
(a)Select acompleteand in
1 LINEAR TIME INVARIANCE
1
1
Linear Time Invariance
1. For each system below, determine if it is linear or non-linear, and determine if it is
time-invariant or not time-invariant (adapted from Siebert 1986).
(a) y(t) = u(t + 1)
The system is linear time-i
15 BOUNCING ROBOT
15
37
Bouncing Robot
You are asked to explore the simple dynamics of a bouncing robot device, using simulation.
There is a single mass with a very light helical spring attached on the bottom of it; the mass
is 40kg, and the spring consta
14 PENDULUM DYNAMICS AND LINEARIZATION
14
35
Pendulum Dynamics and Linearization
Consider a single-link arm, with length l and all the mass m concentrated at the end. A
motor at the xed pivot point supplies a controllable torque . As drawn, a positive tor
11 SEA SPECTRUM AND MARINE VEHICLE PITCH RESPONSE
11
21
Sea Spectrum and Marine Vehicle Pitch Response
1. Make a plot of the spectrum for about one hundred frequencies from zero to 4 rad/s,
with modal frequency !m = 1 rad/s, and signicant wave height H1/3
2.004 MODELING DYNAMICS AND CONTROL II
Spring 2003
Solutions to Problem Set No. 4
Problem 1 This problem can be solved by using the equation to get the velocity at any
point in the 2D rigid body. The equation in this problem is
vA = vA + w r
Because the l
6 CONVOLUTION OF SINE AND UNIT STEP
6
9
Convolution of Sine and Unit Step
The sine function q(t) has a zero value before zero time, and then is a unit sine wave
afterwards:
q(t) =
0 if t < 0
sin(t) if t 0
For the LTI systems whose impulse responses h(t) a
2.004 MODELING DYNAMICS AND CONTROL II
Spring 2003
Problem Set No. 6
Problem 1
Rod sliding down wall. The two ends of the rigid rod of mass M and length L are in
contact with the oor and a vertical wall. Assuming that friction is negligible, derive the
eq
12 RANGING MEASUREMENTS IN THREE-SPACE
12
24
Ranging Measurements in Three-Space
The global positioning system (GPS) and some acoustic instruments provide long-baseline
navigation - wherein a number of very long range measurements can be used to triangula