ASEN 5050 SPACEFLIGHT DYNAMICS Prox Ops, Lambert Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 15: ProxOps, Lambert

Announcements Homework #5 is due next Friday 10/10 CAETE by Friday 10/17 Homework #6 will be due Friday 10/17 CAETE by Friday 10/24 Solutions will be available in class on 10/17 and online by 10/24. If you turn in HW6 early, email me and I’ll send you the solutions to check your work. Mid-term Exam will be handed out Friday, 10/17 and will be due Wed 10/22. (CAETE 10/29) Take-home. Open book, open notes. Once you start the exam you have to be finished within 24 hours. It should take 2-3 hours. Reading: Chapter 6, 7.6 Lecture 15: ProxOps, Lambert

Space News Comet Siding Spring will pass by Mars on October 19th, making things really exciting for all of the Mars vehicles. Lecture 15: ProxOps, Lambert

Space News The “Rosetta Comet” or 67P/Churyumov-Gerasimenko fired its jets last week – just published. Lecture 15: ProxOps, Lambert

ASEN 5050 SPACEFLIGHT DYNAMICS Prox Ops Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 15: ProxOps, Lambert

CW / Hill Equations Enter Clohessy/Wiltshire (1960) And Hill (1878) (notice the timeline here – when did Sputnik launch? Gemini was in need of this!) Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) Use RSW coordinate system (may be different from NASA) Target satellite has two-body motion: The interceptor is allowed to have thrusting Then So, Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) If we assume circular motion, Thus, Assume = 0 (good for impulsive DV maneuvers, not for continuous thrust targeting). CW or Hill’s Equations Lecture 15: ProxOps, Lambert

Assumptions Please Take Note: We’ve assumed a lot of things We’ve assumed that the relative distance is very small We’ve assumed circular orbits These equations do fall off as you break these assumptions! Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) The above equations can be solved (see book: Algorithm 48) leaving: So, given of interceptor, can compute of interceptor at future time. Lecture 15: ProxOps, Lambert

Applications What can we do with these equations? Estimate where the satellite will go after executing a small maneuver. Rendezvous and prox ops! Examples. First from the book. Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) Lecture 15: ProxOps, Lambert

Hubble’s Drift from Shuttle RSW Coordinate Frame Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) We can also determine DV needed for rendezvous. Given x0, y0, z0, we want to determine necessary to make x=y=z=0. Set first 3 equations to zero, and solve for . Assumptions: Satellites only a few km apart Target in circular orbit No external forces (drag, etc.) Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) Lecture 15: ProxOps, Lambert

Clohessy-Wiltshire (CW) Equations (Hill’s Equations) NOTE: This is not the Delta-V, this is the new required relative velocity! Lecture 15: ProxOps, Lambert

CW/H Examples Scenario 1: Scenario 1b: Use eqs to compute x(t), y(t), and z(t) RSW coordinates, relative to deployer. Note: in the CW/H equations, the reference state doesn’t move – it is the origin! Deployment: x0, y0, z0 = 0, Delta-V = non-zero Use eqs to compute x(t), y(t), and z(t) RSW coordinates, relative to reference. Initial state: non-zero Lecture 15: ProxOps, Lambert

Initial state in the RSW frame CW/H Equations Rendezvous For rendezvous we usually specify the coordinates relative to the target vehicle and set x, y, and z to zero Though if there’s a docking port, then that will be offset from the center of mass of the vehicle. Define RSW targets: x, y, z (often zero) Initial state in the RSW frame Lecture 15: ProxOps, Lambert Target: some constant values in the RSW frame

CW/H Equations Rendezvous For rendezvous we usually specify the coordinates relative to the target vehicle and set x, y, and z to zero Though if there’s a docking port, then that will be offset from the center of mass of the vehicle. Define RSW targets: x, y, z (often zero) This is easy if the targets are zero (Eq. 6-66) This is harder if they’re not! Velocity needed to get onto transfer Initial state in the RSW frame Lecture 15: ProxOps, Lambert Target: some constant values in the RSW frame

CW/H Equations Rendezvous For rendezvous we usually specify the coordinates relative to the target vehicle and set x, y, and z to zero Though if there’s a docking port, then that will be offset from the center of mass of the vehicle. Define RSW targets: x, y, z (often zero) This is easy if the targets are zero (Eq. 6-66) This is harder if they’re not! Velocity needed to get onto transfer Initial state in the RSW frame The Delta-V is the difference of these velocities Lecture 15: ProxOps, Lambert Target: some constant values in the RSW frame

CW / H Scenario 3: A jetpack-wielding astronaut leaves the shuttle and then returns. Lecture 15: ProxOps, Lambert Shuttle: reference frame

CW / H Scenario 3: A jetpack-wielding astronaut leaves the shuttle and then returns. Result of deployment Lecture 15: ProxOps, Lambert Shuttle: reference frame

CW / H Scenario 3: A jetpack-wielding astronaut leaves the shuttle and then returns. Rendezvous trajectory (Eq 6-66) Result of deployment Lecture 15: ProxOps, Lambert Shuttle: reference frame

CW / H Scenario 3: A jetpack-wielding astronaut leaves the shuttle and then returns. Delta-V is the difference of these velocities. Rendezvous trajectory (Eq 6-66) Result of deployment Lecture 15: ProxOps, Lambert Shuttle: reference frame

Comparing Hill to Keplerian Propagation Deviation: 100 meters 1 cm/s Position Error: ~1.4 meters/day Lecture 15: ProxOps, Lambert

Comparing Hill to High-Fidelity, e=0.15 Generally returns to 0.0 near one apsis Position Error: ~9.5 km/day Lecture 15: ProxOps, Lambert

Comparing Hill to High-Fidelity, e=0.73 Generally returns to 0.0 Position Error: ~4.2 km/day Lecture 15: ProxOps, Lambert

Analyzing Prox Ops Lecture 15: ProxOps, Lambert

Analyzing Prox Ops What happens if you just change x0? Radial displacement? Lecture 15: ProxOps, Lambert

Analyzing Prox Ops Lecture 15: ProxOps, Lambert

Analyzing Prox Ops What happens if you just change vx0? Radial velocity change? Lecture 15: ProxOps, Lambert

Analyzing Prox Ops Lecture 15: ProxOps, Lambert

Analyzing Prox Ops What happens if you just change vy0? Tangential velocity change? Lecture 15: ProxOps, Lambert

Analyzing Prox Ops Lecture 15: ProxOps, Lambert

Analyzing Prox Ops What happens if you just change vz0? Out-of-plane velocity change? Lecture 15: ProxOps, Lambert

Analyzing Prox Ops Lecture 15: ProxOps, Lambert

ASEN 5050 SPACEFLIGHT DYNAMICS Lambert’s Problem Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 15: ProxOps, Lambert

Lambert’s Problem Lambert’s Problem has been formulated for several applications: Orbit determination. Given two observations of a satellite/asteroid/comet at two different times, what is the orbit of the object? Passive object and all observations are in the same orbit. Satellite transfer. How do you construct a transfer orbit that connects one position vector to another position vector at different times? Transfers between any two orbits about the Earth, Sun, or other body. Lecture 15: ProxOps, Lambert

Lambert’s Problem Given two positions and the time-of-flight between them, determine the orbit between the two positions. Lecture 15: ProxOps, Lambert

Orbit Transfer We’ll consider orbit transfers in general, though the OD problem is always another application. Lecture 15: ProxOps, Lambert

Orbit Transfer Note: there’s no need to perform the transfer in < 1 revolution. Multi-rev solutions also exist. Lecture 15: ProxOps, Lambert

Orbit Transfer Consider a transfer from Earth orbit to Mars orbit about the Sun: Lecture 15: ProxOps, Lambert

Orbit Transfer Orbit Transfer True Anomaly Change “Short Way” Δν < 180° “Long Way” Δν > 180° Hohmann Transfer (assuming coplanar) Δν = 180° Type I 0° < Δν < 180° Type II 180° < Δν < 360° Type III 360° < Δν < 540° Type IV 540° < Δν < 720° … Lecture 15: ProxOps, Lambert

Lambert’s Problem Given: Find: Numerous solutions available. Some are robust, some are fast, a few are both Some handle parabolic and hyperbolic solutions as well as elliptical solutions All solutions require some sort of iteration or expansion to build a transfer, typically finding the semi-major axis that achieves an orbit with the desired Δt. Lecture 15: ProxOps, Lambert

Ellipse d1 d2 Lecture 15: ProxOps, Lambert

Ellipse d1 d2 d4 d3 Lecture 15: ProxOps, Lambert

Ellipse A transfer from r1 to r2 will be on an ellipse, with the central body occupying one focus. Where’s the 2nd focus? r1 r2 Lecture 15: ProxOps, Lambert

Ellipse A transfer from r1 to r2 will be on an ellipse, with the central body occupying one focus. Where’s the 2nd focus? r1 r2 Lecture 15: ProxOps, Lambert

Ellipse A transfer from r1 to r2 will be on an ellipse, with the central body occupying one focus. Where’s the 2nd focus? r1 r2 Lecture 15: ProxOps, Lambert

Ellipse A transfer from r1 to r2 will be on an ellipse, with the central body occupying one focus. Where’s the 2nd focus? r1 r2 Focus is one of these Try different a values until you hit your TOF Lecture 15: ProxOps, Lambert

Lambert’s Problem Lecture 15: ProxOps, Lambert

Lambert’s Problem Find the chord, c, using the law of cosines and Define the semiperimeter, s, as half the sum of the sides of the triangle created by the position vectors and the chord We know the sum of the distances from the foci to any point on the ellipse equals twice the semi-major axis, thus Lecture 15: ProxOps, Lambert

Lambert’s Problem Lecture 15: ProxOps, Lambert

Lambert’s Problem The minimum-energy solution: where the chord length equals the sum of the two radii (a single secondary focus) Thus, (anything less doesn’t have enough energy) Lecture 15: ProxOps, Lambert

Lambert’s Problem Lambert’s Solution Elliptic Orbits Hyperbolic Orbits Lecture 15: ProxOps, Lambert

Lambert’s Problem For minimum energy Lecture 15: ProxOps, Lambert

Universal Variables A very clear, robust, and straightforward solution. There are a few faster solutions, but this one is pretty clean. Begin with the general form of Kepler’s equation: # revolutions Transfer Duration Lecture 15: ProxOps, Lambert

Universal Variables Simplify Lecture 15: ProxOps, Lambert

Universal Variables Define Universal Variables: Lecture 15: ProxOps, Lambert

Universal Variables Lecture 15: ProxOps, Lambert

Universal Variables Use the trigonometric identity Lecture 15: ProxOps, Lambert

Universal Variables Now we need somewhere to go Let’s work on converting this to true anomaly, via: Lecture 15: ProxOps, Lambert

Universal Variables Multiply by a convenient factoring expression: Lecture 15: ProxOps, Lambert

Universal Variables Collect into pieces that can be replaced by true anomaly Lecture 15: ProxOps, Lambert

Universal Variables Substitute in true anomaly: Lecture 15: ProxOps, Lambert

Universal Variables Trig identity again: Lecture 15: ProxOps, Lambert

Universal Variables Note: Lecture 15: ProxOps, Lambert

Universal Variables Use some substitutions: Lecture 15: ProxOps, Lambert

Universal Variables Summary: Lecture 15: ProxOps, Lambert

Universal Variables It is useful to convert to f and g series (remember those!?) Lecture 15: ProxOps, Lambert

UV Algorithm Lecture 15: ProxOps, Lambert

A few details on the Universal Variables algorithm Lecture 15: ProxOps, Lambert

Universal Variables Let’s first consider our Universal Variables Lambert Solver. Given: R0, Rf, ΔT Find the value of ψ that yields a minimum-energy transfer with the proper transfer duration. Applied to building a Type I transfer Lecture 15: ProxOps, Lambert

Single-Rev Earth-Venus Type I -4π 4π2 Lecture 15: ProxOps, Lambert

Single-Rev Earth-Venus Type I -4π 4π2 Lecture 15: ProxOps, Lambert

Note: Bisection method -4π 4π2 Lecture 15: ProxOps, Lambert

Note: Bisection method -4π 4π2 Lecture 15: ProxOps, Lambert

Note: Bisection method -4π 4π2 Lecture 15: ProxOps, Lambert

Note: Bisection method -4π 4π2 Lecture 15: ProxOps, Lambert

Note: Bisection method -4π 4π2 Lecture 15: ProxOps, Lambert

Note: Bisection method -4π 4π2 Lecture 15: ProxOps, Lambert

Note: Bisection method -4π 4π2 Lecture 15: ProxOps, Lambert

Note: Bisection method Time history of bisection method: Requires 42 steps to hit a tolerance of 10-5 seconds! Lecture 15: ProxOps, Lambert

Note: Newton Raphson method Lecture 15: ProxOps, Lambert

Note: Newton Raphson method Lecture 15: ProxOps, Lambert

Note: Newton Raphson method Lecture 15: ProxOps, Lambert

Note: Newton Raphson method Lecture 15: ProxOps, Lambert

Note: Newton Raphson method Time history of Newton Raphson method: Requires 6 steps to hit a tolerance of 10-5 seconds! Note: This CAN break in certain circumstances. With current computers, this isn’t a HUGE speed-up, so robustness may be preferable. Lecture 15: ProxOps, Lambert

Note: Newton Raphson Log Method Note log scale -4π 4π2 Lecture 15: ProxOps, Lambert

Single-Rev Earth-Venus Type I -4π 4π2 Lecture 15: ProxOps, Lambert

Single-Rev Earth-Venus Type II -4π 4π2 Lecture 15: ProxOps, Lambert

Interesting: 10-day transfer -4π 4π2 Lecture 15: ProxOps, Lambert

Interesting: 950-day transfer -4π 4π2 Lecture 15: ProxOps, Lambert

UV Algorithm Lecture 15: ProxOps, Lambert

Multi-Rev Seems like it would be better to perform a multi-rev solution over 950 days than a Type II transfer! Lecture 15: ProxOps, Lambert

A few details The universal variables construct ψ represents the following transfer types: Lecture 15: ProxOps, Lambert

Multi-Rev ψ ψ ψ ψ Lecture 15: ProxOps, Lambert

Earth-Venus in 850 days Type IV Lecture 15: ProxOps, Lambert

Earth-Venus in 850 days Heliocentric View Distance to Sun Lecture 15: ProxOps, Lambert

Earth-Venus in 850 days Type VI Lecture 15: ProxOps, Lambert

Earth-Venus in 850 days Heliocentric View Distance to Sun Lecture 15: ProxOps, Lambert

What about Type III and V? Type V Type III Type VI Type IV Lecture 15: ProxOps, Lambert

Earth-Venus in 850 days Type III Lecture 15: ProxOps, Lambert

Earth-Venus in 850 days Heliocentric View Distance to Sun Lecture 15: ProxOps, Lambert

Earth-Venus in 850 days Type V Lecture 15: ProxOps, Lambert

Earth-Venus in 850 days Heliocentric View Distance to Sun Lecture 15: ProxOps, Lambert

Summary The bisection method requires modifications for multi-rev. Also requires modifications for odd- and even-type transfers. Newton Raphson is very fast, but not as robust. If you’re interested in surveying numerous revolution combinations then it may be just as well to use the bisection method to improve robustness Lecture 15: ProxOps, Lambert

Types II - VI Lecture 15: ProxOps, Lambert