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 1/17/21 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 2 1/17/21 First Apollo Program Contract MIT Instrumentation Laboratory August 9, 1961 HOWEVER … Lunar landing technique had not been decided 9 10 9 10 Alternative Landers Saturn 3rd Stage 11 12 11 12 3 1/17/21 Proposed Saturn Launch Vehicles July 1962 Two Saturn 5s One or One Saturn 5 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 4 1/17/21 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 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 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 racing for the Moon Why the change? 19 20 19 20 5 1/17/21 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 21 22 21 22 Apollo 10, May 1969 Apollo 9 March 1969 Earth-orbit test of Lunar Module, rendezvous, and docking 23 24 23 24 6 1/17/21 Apollo 11, Landing on the Moon Lunar Module Transfer Ellipse to July 20, 1969 Powered Descent Initiation 25 26 25 26 Apollo 12 Apollo 13 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 Pete Conrad • Where’sLM as Lifeboat the science? with Surveyor 3 Apollo 14 Apollo 15 Apollo 16 Feb 5, 1971 July 30, 1971 Apr 21, 1972 • Apollo 20 • ScientistsLunar Rover wanted • LunarLunar Highlandsrover and cancelled2 EVAs more flights “Genesis 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 7 1/17/21 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 8 1/17/21 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)⎦⎥ 33 34 33 34 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 35 36 9 1/17/21 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 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 39 40 39 40 10 1/17/21 A Little AGC Digital Autopilot Code Apollo Guidance Computer (AGC) Hand Controller 41 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 Thrusting away from the origin Thrusting to the origin ⎡ θ!(t) ⎤ ⎡ 0 1 ⎤⎡ θ(t) ⎤ ⎡ 0 ⎤ u(t) = u(t) ⎢ ⎥ = ⎢ ⎥⎢ ⎥ + ⎢ ⎥ +1, 0, or −1 q!(t) 0 0 q(t) gA / I 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) 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 43 44 11 1/17/21 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 12 1/17/21 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 ⎡ 2 ⎤ max sgn 2 2 2 ωc = (δ)⎢δ − +(δ − ) ⎥ 51 40 ⎣ ⎦ 52 51 52 13 1/17/21 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 55 56 55 56 14 1/17/21 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 40 • Geopolitics: International Competition § Attempts Attempts 30 § FailuresFailures • Robots? 20