26th AAS/AIAA Space Flight Mechanics Meeting, Napa, CA 6th International Conference on Astrodynamics Tools & Techniques ICATT Claudio Bombardelli, Juan Luis Gonzalo, Javier Roa Space Dynamics Group Technical University of Madrid (UPM) MULTIPLE REVOLUTION LAMBERT ́S TARGETING PROBLEM: AN ANALYTICAL APPROXIMATION 6th International Conference on Astrodynamics Tools and Techniques (ICATT) March 2016 Darmstadtium, Darmstadt, Germany 1

Outline Background and motivation 1.Lambert’s Targeting Problem (LTP) 2.Applications and Challenges Optimum single-impulse time-free transfer 1.Problem formulation 2.Battin’s quartic equation D-matrix targeting 1.Review of the D matrix relation 2.D* matrix and targeting equations Performance of the method 1.Pork-chop plots comparisons 2.Speed and accuracy Conclusions

6th International Conference on Astrodynamics Tools & Techniques ICATT 1) LAMBERT’S PROBLEM Determine the finite set of (2N+1) orbits linking two position vectors in a Keplerian field with a specified transfer time (N=max number of rev) 2) LAMBERT’S TARGETING PROBLEM (LTP) Determine the minimum impulsive delta-V (and optimum N) to maneuver a S/C in order to target a specified point in space in a fixed amount of time If you can solve 1) then you have solved 2) as well We present an approximate solution for 1) (under some assumptions), not 2) STATE OF THE ART Abundant literature on solution methods for Lambert’s problem. The most popular method is probably Gooding’s (1990) Fastest methods (as far as we know) are Arora-Russell and Izzo’s (2014) Background and Motivations (1) 3

6th International Conference on Astrodynamics Tools & Techniques ICATT THE TRAJECTORY OPTIMIZATION CHALLENGE Increasing the computation speed of a LTP solver is crucial when facing complex, high-dimesional trajectory optimization problems. Background and Motivations (2) 4 o Typically intractable using brute-force approaches (grid-sampling). Bottleneck is the LTP solver CPU time. o Even with “lighter” more “intelligent” strategies a faster LTP solver always helps o Pruning capability is always welcome example: MGA with multiple deep space maneuver (MGA-DSM)

6th International Conference on Astrodynamics Tools & Techniques ICATT 5 Optimum Single-impulse time-free transfer PROBLEM FORMULATION: “A spacecraft is initially located at a point r 1 in a Keplerian gravitational field with initial velocity v 0. We seek the minimum-magnitude velocity increment  v 0 that has the spacecraft reaching, without any time constraint, a prescribed point r 2.” Solved by R.Battin using Godal’s skewed-axes velocity decomposition (radial,  and chordal, c). It reduces to solving Battin’s quartic equation: Solvable analytically (Lagrange resolvent, Ferrari, etc.) or with efficient numerical methods (Newton’s, Halley’s,…)

6th International Conference on Astrodynamics Tools & Techniques ICATT 6 Dealing with the phasing problem Think of LTP solution delta-V as the sum of Battin’s solution delta-V plus a phasing correction term: Good trajectory arcs are characterized by small phasing correction compared to the “ideal” optimum time free delta-V. When  v C <<  v B the former can be obtained, with good approximation, using a linear formulation for the dynamics (through the D matrix):

6th International Conference on Astrodynamics Tools & Techniques ICATT 7 The D matrix (1) The D matrix is an error state transition matrix for the Keplerian dynamics where time(t) is a state variable and the angle  is the indep. variable. Analytically derived in 2014 (Bombardelli CMDA) and motivated by the collision avoidance problem. Model the state as: Consider the state “error” at  =  1 : The radial position (r) and phasing (t) errors at a generic  can be approximated by:

6th International Conference on Astrodynamics Tools & Techniques ICATT 8 D matrix coefficients

6th International Conference on Astrodynamics Tools & Techniques ICATT 9 D matrix targeting The trajectory rephasing problem goes through the inversion of the D matrix, which provides the guidance matrix D*: The sought trajectory phasing correction is then: And the final analytical solution of the LTP, within the linear approximation is :

6th International Conference on Astrodynamics Tools & Techniques ICATT 10 The Ion Beam Shepherd (IBS) Concept RESULTS

6th International Conference on Astrodynamics Tools & Techniques ICATT 11 Earth Mars trajectories Arrival V_inf plots (1000x1000 points) Analytical  Numerical 

6th International Conference on Astrodynamics Tools & Techniques ICATT 12 Earth Didymos trajectories Departure C3 plots (1000x1000 points) ANALYTICAL  NUMERICAL 

6th International Conference on Astrodynamics Tools & Techniques ICATT 13 D matrix targeting The trajectory rephasing problem goes through the inversion of the D matrix, which provides the guidance matrix D*: The sought trajectory phasing correction is then: And the final analytical solution of the LTP, within the linear approximation is :

6th International Conference on Astrodynamics Tools & Techniques ICATT CPU time 14 ProblemGooding*Izzo*Arora-Russell**Analytical Earth-Mars10.0*T5.1*T4.3*TT Earth-Didymos11.1*T5.3*T4.0*TT * Fortran code from J. William Astrodynamics Toolkit git-hub repository ** Fortran code kindly provided by the authors All compiled in GNU gfortran compiler (version 5.3.1), intel i7-4770k KEY FACTS 1) CPU time for the analytical method is problem-independent 2) CPU time for numerical methods increases for multi rev solutios, Gooding’s method performance rapidly degrades with higher number of revolutions.

6th International Conference on Astrodynamics Tools & Techniques ICATT Conclusions 15 A new approximate analytical solution for the Lambert Targeting Problem was derived Preliminary testing shows that the solution retains good accuracy near low-delta-V trajectory arcs and is able to reproduce all the main features of pork-chop transfer plots. The method is extremely fast (about 1 order of magnitude faster than Gooding’s, more than 4 times faster than the best LSs available today). GPU implementation and application to more complex trajectory problems (MGA-DSM) is on the way More work needed to address the solution accuracy and obtain upper bounds. Iteration of the solution for increased accuracy is very promising.