ASEN 5050 SPACEFLIGHT DYNAMICS Lambert’s Problem Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 16: Lambert's Problem

Announcements Quiz after this lecture. THIS WEDNESDAY will be STK LAB 2! Wed morning lecture canceled Alan will be in Visions Wed 9-10 Alan will be in ITLL 2B10 Fri 2-3 STK Lab 2 will be due 10/17, right when the mid-term exam starts. 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. Lecture 16: Lambert's Problem

Space News Remember this? Right on! Lecture 16: Lambert's Problem

ASEN 5050 SPACEFLIGHT DYNAMICS Lambert’s Problem Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 16: Lambert's Problem

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 16: Lambert's Problem

Orbit Transfer We’ll consider orbit transfers in general, though the OD problem is always another application. Lecture 16: Lambert's Problem

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 16: Lambert's Problem

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 16: Lambert's Problem

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 16: Lambert's Problem

Lambert’s Problem Lecture 16: Lambert's Problem

Lambert’s Problem Lecture 16: Lambert's Problem

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 16: Lambert's Problem

Universal Variables Simplify Lecture 16: Lambert's Problem

Universal Variables Define Universal Variables: Lecture 16: Lambert's Problem

Universal Variables Lecture 16: Lambert's Problem

Universal Variables Use the trigonometric identity Lecture 16: Lambert's Problem

Universal Variables Now we need somewhere to go Let’s work on converting this to true anomaly, via: Lecture 16: Lambert's Problem

Universal Variables Multiply by a convenient factoring expression: Lecture 16: Lambert's Problem

Universal Variables Collect into pieces that can be replaced by true anomaly Lecture 16: Lambert's Problem

Universal Variables Substitute in true anomaly: Lecture 16: Lambert's Problem

Universal Variables Trig identity again: Lecture 16: Lambert's Problem

Universal Variables Note: Lecture 16: Lambert's Problem

Universal Variables Use some substitutions: Lecture 16: Lambert's Problem

Universal Variables Summary: Lecture 16: Lambert's Problem

Universal Variables Once you have expressions for y, A, etc., use the f and g series (remember those!?) to convert to r and v! Lecture 16: Lambert's Problem

UV Algorithm Lecture 16: Lambert's Problem

UV Algorithm Bi-section Method Lecture 16: Lambert's Problem

UV Algorithm Lecture 16: Lambert's Problem

A few details on the Universal Variables algorithm Lecture 16: Lambert's Problem

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. ψ is a function of e – it captures the orbital shape. χ requires ψ and a. So we’re ultimately testing the orbit. Applied to building a Type I transfer Lecture 16: Lambert's Problem

Single-Rev Earth-Venus Type I -4π 4π2 Lecture 16: Lambert's Problem

Single-Rev Earth-Venus Type I -4π 4π2 Lecture 16: Lambert's Problem

Note: Bisection method -4π 4π2 Lecture 16: Lambert's Problem

Note: Bisection method -4π 4π2 Lecture 16: Lambert's Problem

Note: Bisection method -4π 4π2 Lecture 16: Lambert's Problem

Note: Bisection method -4π 4π2 Lecture 16: Lambert's Problem

Note: Bisection method -4π 4π2 Lecture 16: Lambert's Problem

Note: Bisection method -4π 4π2 Lecture 16: Lambert's Problem

Note: Bisection method -4π 4π2 Lecture 16: Lambert's Problem

Note: Bisection method Time history of bisection method: Requires 42 steps to hit a tolerance of 10-5 seconds! Lecture 16: Lambert's Problem

Note: Newton Raphson method Lecture 16: Lambert's Problem

Note: Newton Raphson method Lecture 16: Lambert's Problem

Note: Newton Raphson method Lecture 16: Lambert's Problem

Note: Newton Raphson method Lecture 16: Lambert's Problem

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 16: Lambert's Problem

Note: Newton Raphson Log Method Note log scale -4π 4π2 Lecture 16: Lambert's Problem

Single-Rev Earth-Venus Type I -4π 4π2 Lecture 16: Lambert's Problem

Single-Rev Earth-Venus Type II -4π 4π2 Lecture 16: Lambert's Problem

Interesting: 10-day transfer -4π 4π2 Lecture 16: Lambert's Problem

Interesting: 950-day transfer -4π 4π2 Lecture 16: Lambert's Problem

Multi-Rev Seems like it would be better to perform a multi-rev solution over 950 days than a Type II transfer! Lecture 16: Lambert's Problem

A few details The universal variables construct ψ represents the following transfer types: Lecture 16: Lambert's Problem

Multi-Rev ψ ψ ψ ψ Lecture 16: Lambert's Problem

Earth-Venus in 850 days Type IV Lecture 16: Lambert's Problem

Earth-Venus in 850 days Heliocentric View Distance to Sun Lecture 16: Lambert's Problem

Earth-Venus in 850 days Type VI Lecture 16: Lambert's Problem

Earth-Venus in 850 days Heliocentric View Distance to Sun Lecture 16: Lambert's Problem

What about Type III and V? Type V Type III Type VI Type IV Lecture 16: Lambert's Problem

Earth-Venus in 850 days Type III Lecture 16: Lambert's Problem

Earth-Venus in 850 days Heliocentric View Distance to Sun Lecture 16: Lambert's Problem

Earth-Venus in 850 days Type V Lecture 16: Lambert's Problem

Earth-Venus in 850 days Heliocentric View Distance to Sun Lecture 16: Lambert's Problem

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 16: Lambert's Problem

Types II - VI Lecture 16: Lambert's Problem

Announcements Quiz after this lecture. THIS WEDNESDAY will be STK LAB 2! Wed morning lecture canceled Alan will be in Visions Wed 9-10 Alan will be in ITLL 2B10 Fri 2-3 STK Lab 2 will be due 10/17, right when the mid-term exam starts. 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. Lecture 16: Lambert's Problem