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Home , Apollo 10, Apollo 11, Apollo 12, Apollo 13, Apollo 14, Apollo 16

1/17/21

Earth- : 27-1/2 days One side of Moon always faces

Flight to the Moon , MIT IAP 16.S585

Robert Stengel There is no “Dark Side” January 14, 2021 1 ALL SIDES are dark once a 2 1 2

The Earth and the Moon December 17, 1958 Earth = 81.4 x Moon mass Orbit eccentricity = 0.05

1st Cosmonaut Mercury 7, 1959 Class, 1959 3 4

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April 12, 1961 , 1962

John Glenn

Vostok 1

Friendship 7 Mercury- 5 6

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Project Gemini [1965-66] Lunar Missions

10 crewed II missions

June 1961 Competition among contractors for the spacecraft and launch

US takes Race Lead 7 8

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First Program Contract MIT Instrumentation Laboratory August 9, 1961

HOWEVER … Lunar technique had not been decided 9 10

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Alternative Landers

Saturn 3rd Stage

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Proposed Launch Vehicles

July 1962

Two Saturn 5s One or One Saturn 5 Ten Saturn 1s

Saturn 1 Saturn 5 Nova (Saturn 8) 13 14

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Saturn Launch Vehicles Saturn 1B Saturn 5 The Apollo Modules Earth Orbit Missions Lunar Missions

Service Command Lunar Module Module Module

North American

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First Manned Flight, , -27, 1968 October 11, 1968 • Earth-orbit mission to test LM planned • More ambitious mission was pursued st Eisele Schirra Cunningham • Repurposed to 1 manned flight to the Moon • 6-day mission, no Lunar Module

Coast Reentry

Trans- Moon’s Lunar Coast Injection “Sphere of Influence” Free-return trajectory 17 No further propulsion after Trans-Lunar Injection18

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Apollo 8 Entered , CIA KH-8 GAMBIT • Even more daring alternative Reconnaissance • 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 racing for the Moon Why the change? 19 20

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Soviet Manned Lunar Soviet Manned Lunar Spacecraft

Soyuz 7K-LOK, LK, 1-man Saturn 2-man CSM Lunar N-1 V N-1

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Apollo 10, Earth-orbit test of Lunar Module, rendezvous, and docking

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Apollo 11, Landing on the Moon Lunar Module Transfer Ellipse to , 1969 Powered Descent Initiation

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Apollo 12 Low Gate to Touchdown Nov 19, 1969 April 11-17, 1970 Pinpoint Landing [HBO dramatization, “From the Earth to the Moon”] • Public enthusiasm waned • President Nixon not a bigExplosion fan • Where’sLM as Lifeboat the ? with 3 Feb 5, 1971 , 1971 Apr 21, 1972 • Apollo 20 • ScientistsLunar Rover wanted • LunarLunar Highlandsrover and cancelled2 EVAs more flights “ Rock” science3-day packages stay • Apollo 14 flew • Congress increased weight threatened to kill 13’s mission • Saturn 5 uprated program after 14

27 • Last flights 28 devoted to science 27 28

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Apollo Guidance (AGC) vs. iPhone 5S , Dec 7-19, 1972 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

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Lunar Module Navigation, Guidance, Lunar Module Descent and Control Targeting Sequence Braking Phase (P63) Approach Phase (P64) AGC Terminal Descent Phase (P66)

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Characterize Braking Phase Lunar Module Descent Guidance Logic By Five Points (Klumpp, Automatica, 1974) • 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

2 3 4 ⎡ x(t)⎤ ⎢ ⎥ t t t r(t) = y(t) rr (t) = rt + vtt + at + jt + st ⎢ ⎥ 2 6 24 ⎣⎢ z(t)⎦⎥

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Corresponding Reference Velocity Coefficients of the Polynomials and Acceleration Vectors 2 3 2 3 4 t t t t t dr dt = vr (t) = vt + att + jt + st rr (t) = rt + vt t + a t + jt + st 2 6 2 6 24 ⎡ x⎤ ⎡ ˙ ⎤ ⎡ ⎤ 2 • r = position vector x vx t ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 2 r = y v = y˙ = v d r dt = a (t) = a + j t + s • v = velocity vector ⎢ ⎥ ⎢ ⎥ ⎢ y⎥ r t t t 2 • a = acceleration vector ⎣⎢ z⎦⎥ ⎢ z˙ ⎥ ⎢v ⎥ ⎣ ⎦ ⎣ z ⎦ • j = jerk vector (time • a (t) is the reference control vector derivative of acceleration) r ⎡a x⎤ ⎡ jx⎤ ⎡s x⎤ – Descent engine thrust / mass = a • s = snap vector (time ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ t a = a j = j s = s – Vector components controlled by derivative of jerk) ⎢ y⎥ ⎢ y⎥ ⎢ y⎥ • Attitude control required to ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ orienting yaw and pitch angles of the ⎣ az ⎦ ⎣ jz⎦ ⎣ sz ⎦ Lunar Module orient the thrust vector • Gimballed descent main engine 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)

t 2 t 2 t 3 t 4 Linear feedback guidance law a (t) = a + j t + s ⇒ r (t) = r + v t + a + j + s r t t t 2 r t t t 2 t 6 t 24 acommand (t) = ar (t) + KV [v measured (t) − vr (t)]+ KR [rmeasured (t) − rr (t)] • ... but they are not KV :velocity error gain • Therefore, feedback control is K :position error gain required to follow the reference R trajectory Nominal acceleration profile corrected for differences between actual and reference flight paths

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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 (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 Controller 41 42

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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

Thrusting away from the origin Thrusting to the origin ⎡ θ!(t) ⎤ ⎡ 0 1 ⎤⎡ θ(t) ⎤ ⎡ 0 ⎤ u(t) = u(t) ⎢ ⎥ = ⎢ ⎥⎢ ⎥ + ⎢ ⎥ +1, 0, or −1 0 0 q(t) gA / I yy ⎢ q!(t) ⎥ ⎣ ⎦⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎣ ⎦ What is the time evolution of the state while a thruster is on [u(t) = 1]?

q(t) = (gA / I yy )t + q(0)

2 θ(t) = (gA / I yy )t / 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

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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

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Switching-Curve Control with Coasting Zone Apollo Hand Controllers

Vertical Rate Switch (not 3-Axis Attitude visible) Control Assembly (ACA)

Throttle

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Apollo Angular Rate Controller Effect of ACA Sensitivity (wc/d) (ACA) on RCS Firing Times Torque and Voltage

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ACA Command Output Frequency of Manually Linear-Quadratic Attitude-Rate Scaling Controlled Rates

ω ⎡ 2 ⎤ max sgn 2 2 2 ωc = (δ)⎢δ − +(δ − ) ⎥ 51 40 ⎣ ⎦ 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

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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

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115 Lunar Missions Since 1958 Why Return to the Moon? (44 failures) • Science: lunar and astronomy Resurgent interest in lunar missions • Technological Development 60 • Educational Benefit 50 • Economic Stimulus

40 • Geopolitics: International Competition § Attempts Attempts 30 § FailuresFailures • Robots? 20 – Preliminary exploration 10 – Search for WATER and raw materials

0 • Humans? 50-59 60-69 70-79 80-89 90-99 00-09 10-present – Long-term utilization of • Proposed Missions, 2021 --- – Dealing with uncertainty Robotic: 27 (funded), 17 (TBD) 59 Crewed: 4 (funded), 4 (TBD) 60

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Where is the Water? Lunar Atmosphere and Dust Environment Explorer (2013)

§ 30-day transit § Launched from NASA § V (from Peacekeeper ICBM)

Valleys and craters near the North/South Pole 61

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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 (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

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QueQiao, Chang’e-4, 2, 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

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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

Return to the Moon

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NASA Exploration Missions, NASA (2nd Q, 2022?) Artemis 1 & 2 Uncrewed to Lunar Orbit NASA Orion Command Module, System (SLS) Interim Cryogenic Propulsion Stage (ICPS)

$16 B to date • First SLS Flight: 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 71 72

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NASA NASA Lunar Orbital Platform-Gateway Orion with 4-person crew

§ Mini 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

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Outdated Timeline to NASA Commercial 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 Lunar Lander SpaceX , #dearMoon Super Heavy Rocket (2023) • with Orion • Would launch on SLS Cancelled in 2011 Block 2 • Crew of 4 • One- on moon

Inadequate NASA Budget

Billionaire plus 6-8 artists

Crew of 1 or 2+ 77 78

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Crewed Lander Concepts

China Russia

Target Date: 2030s Target Date: 2030s

SpaceX Lockheed Martin

[email protected] http://www.stengel.mycpanel.princeton.edu 79

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