HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
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AER506H1F Spacecraft Dynamics and Control I
Problem Set #3
Due: Wednesday, October 22, 2014
1. Show how the potential function for the nonspherical earth would be used in conjunction with Cowells method. Write down the six first-order differential equatio

Bias-Momentum Stabilization
15
15.1
1
Bias-Momentum Stabilization
Introduction
This represents an attempt to combine the benefits of passive spin stabilization with those
of active attitude control. The bulk of the spacecraft makes one turn per orbit abou

1
Orbital Maneuvers
5
5.1
Orbital Maneuvers
Introduction
For earth satellites, the launch vehicle may insert a spacecraft into a desired orbit. This is
equivalent to establishing a particular v at a particular r.
~
~
Prevalent use of the Space Shuttle and

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AER506H1F
Spacecraft Dynamics and Control I
A course presented at the
University of Toronto
Notes prepared by
C. J. Damaren
1
1
Kinematics of Spacecraft Motion
Spacecraft are free bodies; that is they are free to translate and rotate. Mathematically,
they

AER506H1F Spacecraft Dynamics and Control I
Problem Set #6
Due: Wednesday, November 26, 2014
1. In the notes, we derived the gravity-gradient torque and subsequent motion equations for the case of a principal axis frame that is only slightly displaced fro

AER506H1F Spacecraft Dynamics and Control I
Problem Set #5
Due: Wednesday, November 12, 2014
1. A rectangular lamina of dimensions a and b is rotating about a diagonal with a
constant angular rate .
(a) Show that with respect to F 2 ,
I11 =
1
1
ma2 , I22

AER506H1F Spacecraft Dynamics and Control I
Tutorial #3
1. Using q1 = r and q2 = as generalized coordinates, determine the differential
equations of motion for the planar two-body problem using Lagranges equations:
d
dt
L
qi
!
L
= 0, i = 1, 2
qi
2. The i

AER506H1F Spacecraft Dynamics and Control I
Problem Set #2
Due: Wednesday, October 8, 2014
1. A satellite is in a Keplerian (two-body) orbit with a period of 270 min and
eccentricity e = 0.5. It has passed perigee and is now at a point in which the orbit

1
The Restricted Three-Body Problem
7
7.1
The Restricted Three-Body Problem
Formulation
Consider three masses that interact gravitationally:
h
m2
r2
~
r m3
r = r3
FI
~ ~
~
1h m1
6
r1
~
O
Assumptions
1. The gravitational effect of m3 on m2 and m1 is

AER506H1F Spacecraft Dynamics and Control I
Problem Set #1
Due: Wednesday, September 24, 2014
1. Show that C21 r
1 C12 = (C21 r1 ) . Hint: Consider expressing the components of
r s in two different ways where r = F T1 r1 = F T2 r2 and s = F T2 s2 .
~ ~
~

Orbital Perturbations
4
1
Orbital Perturbations
4.1
Classification of Perturbations
Consider the two-body problem which addresses the motion of a point mass m1 in the
gravitational field of another point mass m2 (the primary). The two-body problem defines

1
Gravity-Gradient Stabilization
12
12.1
Gravity-Gradient Stabilization
Reference Frames
At this point, three important reference frames are introduced. The first, F I , is an inertial
frame which is centred at the earth. This frame does not rotate with t

1
Interplanetary Spacecraft
6
6.1
Interplanetary Spacecraft
Introduction
Consider an interplanetary trajectory from earth to mars. Near earth, the spacecraft is
primarily under the influence of the earths gravitational field. For most of the journey, it i

1
Spin Stabilization
9
Spin Stabilization
At this point, we have a general statement for the rotational behaviour of a rigid body,
Eulers equations. In the absence of external torques,
I1 1 + (I3 I2 )2 3 = 0
I2 2 + (I1 I3 )1 3 = 0
I3 3 + (I2 I1 )1 2 = 0
(

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1
Orbital Dynamics
3
3.1
Orbital Dynamics
The Two-Body Problem
In this section we consider a system of two particles with masses m1 and m2 which experience
only the mutual gravitation acting between them. Although only point masses are covered
by this tre

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Spin Stabilization Revisited
13
1
Spin Stabilization Revisited
13.1
The Thomson Equilibrium
Recap
In the absence of energy dissipation, major and minor axis spins are stable. With energy
dissipation accounted for, minor axis spins are unstable.
Question
W

AER506H1F Spacecraft Dynamics and Control I
Tutorial #1
1.(a) Which of the following is a valid rotation matrix? Why?
1 0
1
0 0
1 0
0
0
(i) 2/2 0
2/2 , (ii) 0 1 1 , (iii) 0 2/2 2/2
0 1 1
0
0
1
0
2/2
2/2
(b) Which of the following will uniquely determin

AER506H1F Spacecraft Dynamics and Control I
Tutorial #9
1. Consider the open-loop roll-yaw equations for the bias-momentum situation. Under the approximation hs Ii o , i = 1, 2, 3, show that the natural frequencies are
o and p = hs / I1 I3 .

1
Disturbance Torques
11
Disturbance Torques
In this section, we relax the torque-free assumption and consider the rotational dynamics of
an earth-orbiting spacecraft.
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AER506H1F Spacecraft Dynamics and Control I
Problem Set #2
Due: Thursday, October 8, 2015
1. A satellite is in a Keplerian (two-body) orbit with a period of 270 min and
eccentricity e = 0.5. It has passed perigee and is now at a point in which the orbit
p

AER506H1F Spacecraft Dynamics and Control I
Problem Set #1
Due: Monday, September 28, 2015
1. Show that C21 r C12 = (C21 r1 ) . Hint: Consider expressing the components of
1
r s in two dierent ways where r = F T r1 = F T r2 and s = F T s2 .
1
2
2
2. Show