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1/17/21
Earth-Moon Orbit
Orbital Period: 27-1/2 days
One side of Moon always faces Earth
Flight to the Moon
Spacecraft Attitude Control, MIT IAP 16.S585
Robert Stengel
Princeton University There is no “Dark Side”
January 14, 2021 1
ALL SIDES are dark once a month 2
1 2
The Earth and the Moon December 17, 1958
Earth mass = 81.4 x Moon mass
Orbit eccentricity = 0.05
1st Cosmonaut
Mercury 7, 1959 Class, 1959
3 4
3 4
1
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April 12, 1961 February 20, 1962
John Glenn
Vostok 1
Friendship 7
Mercury-Atlas
Yuri Gagarin
5 6
5 6
Project Gemini [1965-66]
Lunar Missions
10 crewed
Titan II missions
June 1961
Competition among contractors for
the spacecraft and launch rockets
US takes Space Race Lead 7 8
7 8
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First Apollo Program Contract
MIT Instrumentation Laboratory
August 9, 1961
HOWEVER … Lunar landing
technique had not been decided
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9 10
Alternative Landers
Saturn 3rd Stage
11 12
11 12
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Proposed Saturn Launch Vehicles
July 1962
Two Saturn 5s
One One Saturn 5
or
Nova
Ten Saturn 1s
Saturn 1 Saturn 5 Nova (Saturn 8)
13 14
13 14
Saturn Launch Vehicles
Saturn 1B Saturn 5 The Apollo Modules
Earth Orbit Missions Lunar Missions
Service Command Lunar
Module Module Module
North American Grumman
15 16
15 16
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First Manned Flight, Apollo 7 Apollo 8, December 21-27, 1968
October 11, 1968 • Earth-orbit mission to test LM planned
• More ambitious mission was pursued
Eisele Schirra Cunningham • Repurposed to 1st manned flight to the Moon
• 6-day mission, no Lunar Module
Coast
Reentry
Trans-
Lunar Moon’s
Coast
Injection “Sphere of Influence”
Free-return trajectory
17 No further propulsion after Trans-Lunar Injection18
17 18
Apollo 8 Entered Lunar Orbit August 1968, CIA KH-8 GAMBIT
• Even more daring alternative Reconnaissance Satellite
• Rocket fired on far side for Lunar-Orbit
Insertion; no free return
• Rocket had to fire again on far side to
return to Earth
N-1 Rocket: Russia was indeed
Why the change? racing for the Moon
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19 20
5
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Soviet Manned Lunar Launch Vehicle
Soviet Manned Lunar Spacecraft
Soyuz 7K-LOK, LK, 1-man
Saturn 2-man CSM Lunar Lander
N-1
V
N-1
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21 22
Apollo 10, May 1969
Apollo 9
March 1969
Earth-orbit test of Lunar Module, rendezvous, and docking
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23 24
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Apollo 11, Landing on the Moon
Lunar Module Transfer Ellipse to July 20, 1969
Powered Descent Initiation
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25 26
Apollo 12 Apollo 13
Low Gate to Touchdown Nov 19, 1969 April 11-17, 1970
[HBO dramatization, “From the Earth to the Moon”] Pinpoint Landing
• Public enthusiasm
waned
• President Nixon not a
big fan
Explosion
Pete Conrad
LM as Lifeboat
• Where’s the science?
with Surveyor 3
Apollo 14 Apollo 15 Apollo 16
Feb 5, 1971 July 30, 1971 Apr 21, 1972
• Apollo 20 • Scientists wanted • Lunar
Lunar Highlands
rover and
Lunar Rover
2 EVAs
cancelled more flights science
“Genesis Rock” 3-daypackages
stay
• Apollo 14 flew • Congress increased weight
threatened to kill • Saturn 5 uprated
13’s mission program after 14
27 • Last flights 28
devoted to science
27 28
7
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Apollo Guidance Computer (AGC)
vs. iPhone 5S
Apollo 17, Dec 7-19,
1972
Harrison Schmitt
1st Scientist on the Moon
This JPEG Image:
282,000 words
§ 16-bit computer § 64-bit computer
§ Storage: 38,332 words § A million times
§ Speed: 1 million “ticks” per sec more storage
§ Weight: 70 lb § 1,300 times faster
§ 1st integrated-circuit computer § Weight: 1/4 lb
§ Plus Inertial Measurement Unit § Including inertial
29 measurements 30
29 30
Lunar Module Navigation, Guidance, Lunar Module Descent
and Control Targeting Sequence
Braking Phase (P63)
Approach Phase (P64)
AGC Terminal Descent Phase (P66)
31 32
31 32
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Characterize Braking Phase Lunar Module Descent Guidance Logic
(Klumpp, Automatica, 1974)
By Five Points • Reference (nominal) trajectory, rr(t), from target position
back to starting point (Braking Phase example)
– Three 4th-degree polynomials in time
– 5 points needed to specify each polynomial
⎡ x(t)⎤
t2 t3 t4 ⎢ ⎥
rr (t) = rt + v t t + a t + jt + s t r(t) = ⎢ y(t)⎥
2 6 24 ⎣⎢ z(t)⎥⎦
33 34
33 34
Corresponding Reference Velocity
Coefficients of the Polynomials and Acceleration Vectors
t2 t3
t2 t3 t4 dr dt = v r (t) = v t + a t t + jt + st
rr (t) = rt + v t t + a t + jt + st 2 6
2 6 24
⎡ x⎤ ⎡x˙ ⎤ ⎡v x ⎤ t2
• r = position vector
• v = velocity vector
⎢ ⎥
r = ⎢ y⎥ ⎢ ⎥ ⎢ ⎥
v = ⎢y˙ ⎥ = ⎢v y ⎥ d 2r dt 2 = a r (t) = a t + jt t + st
• a = acceleration vector ⎢⎣ z ⎥⎦ ⎢⎣ z˙ ⎥⎦ ⎢⎣v z ⎥⎦ 2
• j = jerk vector (time
• ar(t) is the reference control vector
derivative of acceleration) ⎡ax ⎤ ⎡ jx ⎤ ⎡sx ⎤ – Descent engine thrust / mass = at
• s = snap vector (time ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
derivative of jerk) a = ⎢ay ⎥ j = ⎢ jy ⎥ s = ⎢sy ⎥ – Vector components controlled by
• Attitude control required to ⎢⎣az ⎥⎦ ⎢⎣ j z ⎥⎦ ⎢⎣sz ⎥⎦ orienting yaw and pitch angles of the
Lunar Module
orient the thrust vector
35
• Gimballed descent main engine 36
35 36
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Guidance Logic Defines Guidance Law
Desired Acceleration Vector for the Lunar
• If initial conditions, dynamic model, and thrust Module Descent
control were perfect, ar(t) would produce rr(t)
t2 t2 t3 t4 Linear feedback guidance law
a r (t) = a t + jt t + s t ⇒ rr (t) = rt + v t t + a t + jt + s t
2 2 6 24
a command (t) = a r (t) + KV [ v measured (t) − v r (t)] + K R [ rmeasured (t) − rr (t)]
• ... but they are not KV : velocity error gain
• Therefore, feedback control is
K R : position error gain
required to follow the reference
trajectory
Nominal acceleration profile corrected for differences
between actual and reference flight paths
37 38
37 38
Lunar Module Simulators Lunar Module Attitude Control
Lunar Landing Lunar Landing (R. Stengel, JSR, 8/70, W. Widnall, JSR, 1/71)
Research Vehicle Research Facility
• 16 Reaction Control System
(RCS) thrusters
– Control about 3 axes
– Redundancy of thrusters
• Gimballed descent engine
• LM Digital Autopilot
• 2,000 16-bit words of code
NASA LM Fixed-Base
Simulator • 10 samples/sec
I-Lab LM Fixed-Base
Simulator
IBM 360 Mainframe
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A Little AGC Digital Autopilot Code
Apollo
Guidance
Computer
(AGC)
Hand
41
Controller 42
41 42
Pitch-Axis Control with Constant-Thrust
Constant Thrust (Acceleration) Trajectories
q, Pitch Rate, deg/s q, Pitch Angle, deg For u = 1, For u = –1,
What if the control torque can only be turned ON or OFF? Acceleration = gA/Iyy Acceleration = –gA/Iyy
⎡ θ!(t) ⎤ ⎡ 0 1 ⎤ ⎡ θ (t) ⎤ ⎡ 0 ⎤ u (t ) = Thrusting away from the origin Thrusting to the origin
⎢ ⎥=⎢ ⎥ ⎢ q(t) ⎥ + ⎢ g / I ⎥ u(t) +1, 0, or − 1
!
⎢⎣ q(t) ⎥⎦ ⎣ 0 0 ⎦ ⎢⎣ ⎥⎦ ⎢⎣ A yy ⎥⎦
What is the time evolution of the state while a thruster is
on [u(t) = 1]?
( )
q(t) = gA / I yy t + q(0)
( )
θ (t) = gA / I yy t 2 / 2 + q(0)t + θ (0)
Neglecting initial conditions, what does With zero thrust, what does the q vs. q plot look like?
the q vs. q plot look like? 43 44
43 44
111/17/21
q vs. q Plot with Zero Thrust Switching-Curve Control
Law for On-Off Thrusters
• ORIGIN (i.e., zero rate
and attitude error) can
be reached from ANY
POINT in the state
space
• Control logic:
– Thrust in one
direction until
switching curve is
reached
– Then REVERSE
thrust
– Switch thrust OFF
How can you use this information to design when errors are zero
an on-off control law?
45 46
45 46
Switching-Curve Control with
Coasting Zone Apollo Hand Controllers
Vertical Rate
Switch (not 3-Axis Attitude
visible) Control Assembly
(ACA)
Throttle
47 48
47 48
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Apollo Angular Rate Controller Effect of ACA Sensitivity (wc/d)
(ACA) on RCS Firing Times
Torque and Voltage
49 50
49 50
ACA Command Output
Frequency of Manually Linear-Quadratic Attitude-Rate Scaling
Controlled Rates
ω max ⎡ 2 ⎤
51
ωc =
40 ⎣ (
sgn (δ ) ⎢ δ − 2 + δ − 2 ) 2⎥
⎦ 52
51 52
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Apollo Lunar Module Manual
Attitude Control Logic Lunar Module Manual Attitude
Control Law
• Coast zones conserve RCS propellant by limiting angular rate
• With no coast zone, thrusters would chatter on and off at § Rate Command / Attitude Hold
origin, wasting propellant § Angles measured by Inertial Measurement Unit (IMU)
• State limit cycles about target attitude § Rates estimated from measurements
• Switching curve shapes modified to provide robustness § Analogous to PID controller
against modeling errors (e.g., RCS thrust level, Moment of § RCS commanded ON at sampling instant
inertia) § RCS commanded OFF by timer (between sampling instants)
• Instant ON if manual command exceeds Deadband (0.6 deg/s) 53
54
53 54
Typical Phase-Plane Trajectory Simulated Rate Ramps, Time
Response and Phase Plot*
LM Ascent Module
attitude change
• With angle error, RCS turned ON until reaching OFF
switching curve
• Phase point drifts until reaching ON switching curve ________
• RCS turned OFF when rate is 0- * Instant command
response not
• Limit cycle maintained with minimum-impulse RCS firings triggered
– Amplitude = ±0.3 deg during landing
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MATLAB LM Digital Autopilot*
https://www.mathworks.com/help/simulink/slref/developing-the-apollo-
lunar-module-digital-autopilot.html Apollo 17, Dec 7-19,
1972
§ Fixed main ascent engine
§ Thrust line purposely offset
from center of mass
§ Ascent phase-plane logic biased
to produce positive RCS thrust
only
§ Pure couples not commanded
during thrusting phase
* About 25 GB, including MATLAB application 57 58
57 58
115 Lunar Missions Since 1958 Why Return to the Moon?
(44 failures)
• Science: lunar geology and astronomy
Resurgent interest in lunar missions • Technological Development
60
• Educational Benefit
50 • Economic Stimulus
• Geopolitics: International Competition
§ Attempts
40
§ Failures
Attempts
30
Failures
• Robots?
– Preliminary exploration
20
10 – Search for WATER and raw materials
0
• Humans?
– Long-term utilization of lunar resources
50-59 60-69 70-79 80-89 90-99 00-09 10-pres ent
• Proposed Missions, 2021 --- – Dealing with uncertainty
Robotic: 27 (funded), 17 (TBD)
59
Crewed: 4 (funded), 4 (TBD) 60
59 60
151/17/21
Where is the Water? Lunar Atmosphere and Dust
Environment Explorer (2013)
§ 30-day transit
§ Launched from NASA
Wallops Flight Facility
§ Minotaur V (from
Peacekeeper ICBM)
Valleys and craters near the North/South Pole
61
61 62
2019-20 Robotic Lunar Landers
Multi-Burn Lunar Transfer Chang’e 4 (1/3/19)
Pseudo-impulsive thrusting at Far Side Landing
perilune for efficient energy increase
SpaceIL Beresheet
(2/21 Launch, 4/11 Crash)
Chandrayaan-2
(7/22 Launch, 9/6 crash)
Chang’e 5 (11/20-12/16/20)
Sample Return
“Robotic Apollo” 64
63 64
161/17/21
QueQiao, Chang’e-4, Yutu 2,
Libration Points and 3-Body Problem
and Chang’e-5, 2019-20
• Earth and Moon
and Spacecraft
Earth Moon
“Gravity Wells” of the
65
Earth-Moon System
65 66
Lunar Flight Revisited
Earth Moon
“Gravity Wells” of the
Earth-Moon System
Spacecraft affected by subtle gravitational effects
Small nudge produces large effect
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Small Nudges Repositioned Spacecraft
in Lunar Orbits
Return to the Moon
70
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NASA Exploration Missions, NASA Artemis 1 (2nd Q, 2022?)
Artemis 1 & 2 Uncrewed Orion to Lunar Orbit
Space Launch NASA Orion Command Module,
System (SLS) Interim Cryogenic Propulsion Stage
(ICPS)
$16 B to date
• First SLS Flight: Europa Clipper launch scheduled for
$14 B to date Both programs are behind 11/21
schedule and over budget 71 • SLS may not be ready; Falcon Super Heavy alternative
72
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NASA Artemis 2
NASA Lunar Orbital Platform-Gateway Orion with 4-person crew
§ Mini space station in orbit about the Moon
§ Launched by two commercial rockets (TBD)
§ Easy access to lunar surface • Two ICPS burns to raise orbit
§ Reusable lunar ascent module
• One Orion rocket burn to circle the Moon
73
• Free return 74
73 74
Outdated Timeline to Artemis 3 NASA Commercial Lunar Lander
Services Program Supports Artemis
§ Robotic, launch, and
instrumentation technology
§ Nine Contracts Awarded
§ 12 Science & Technology
Investigations
• One SLS plus five commercial launch vehicles to
land two people (including a woman) on the Moon
• Seven LORs
• An optimistic, controversial approach Astrobotic Firefly
Peregrine Masten XL1 Beta
• No room for failures 75 76
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Space Tourism Lunar Fly-By Proposed Altair Lunar Lander
SpaceX Starship, #dearMoon Project
Super Heavy Rocket (2023) • Earth Orbit Rendezvous
with Orion
Constellation Program • Would launch on SLS
Cancelled in 2011 Block 2
• Crew of 4
• One-week on moon
Inadequate NASA
Budget
Billionaire Yusaku Maezawa plus 6-8 artists
Crew of 1 or 2+ 77 78
77 78
Crewed Lander Concepts
China Russia
Target Date: 2030s Target Date: 2030s
SpaceX Lockheed Blue Origin
Martin
stengel@princeton.edu http://www.stengel.mycpanel.princeton.edu
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