Robust trajectory design using invariant manifolds. Application to (65803) Didymos L. Dell'Elce Sophia Antipolis, 22/02/2018
The AIDA mission 2
Uncertainties and nonlinearities in space missions Radar imaging of 101955 Bennu Landing of Philae 3
Are there safe orbits in the Didymos system? Nominal periodic orbit Failure of the propulsion system? Real trajectory Can we place a CubeSat? 4
Outline 1. Safe orbits in the Didymos system 2. Robust mission design using manifolds 3. Application to the Didymos system 5
Outline 1. Safe orbits in the Didymos system 2. Robust mission design using manifolds 3. Application to the Didymos system 6
1. High fidelity propagator of the Didymos system SRP 3rd body 3 axial Ellipsoid Polihedron 7
1. Binary model: full three body problem Binary Asteroid Model Orbital period ~ 1/2 day Eccentricity: max 0.03 Didymoon (103 x 79 x 66 m) Mass parameter ~ 1% Didymain (790 x 790 x 761 m) McMahon, J., and Scheeres, D. J., Dynamic limits on planar librationorbit coupling around an oblate primary, CMDA, 115(4) (2013) 8
1. Various stable orbits in the CRTBP are considered Selection rbit Selection ular Equilibrium Points of the ideal CR3BP Triangular points Planar periodic orbits L4 Planar Periodic Orbits Close toprimary the small bodies Subject to strong gravity field signals Secondary But Frequent eclipses No observation ofl5the asteroid system Terminator orbits 6 9
1. Robustness is assessed with a Monte Carlo analysis Monte Carlo: Inputs 1000 samples Uniform distributions No correlation 10
1. Triangular points are not safe rbit Selection Triangular Equilibrium Points of the ideal CR3BP 1 L4 Pfailure 0.75 Primary Secondary 0.5 L5 0.25 0 0 6 7.5 15 tfailure [day] 22.5 30 11
1. Most planar orbits are not safe Direct exterior orbits Similar results for: direct interior, direct/retrograde exterior, CSO 12
1. Retrograde interior orbits are safe! 7 13
1. Retrograde interior orbits are safe! θ(0) [deg] 360 7 Failure events 180 0 0.025 0.0625 Cr m/b [m2 / km] 0.1 14
1. Terminator orbits are safe! ar to the Sun n due to SRP ystem Side view 3D orbit 8 15
1. Terminator orbits are safe! 400 tfailure [day] θ(0) [deg] 360 180 0 0.025 0.0625 Cr /B [m2 / km] 0.1 200 0 0.025 0.0625 Cr /B [m2 / km] 0.1 16
Outline 1. Safe orbits in the Didymos system 2. Robust mission design using manifolds 3. Application to the Didymos system 17
2. Can we guarantee where the motion will evolve? Nominal periodic orbit Real trajectory Bounds 18
2. Invariant manifolds of an orbit Quasi periodic tori Unstable manifold Stable manifold Stable periodic orbit Unstable periodic orbit Numerical computation: solution of a PDE 19
2. Invariant manifolds as bounds for the motion Surface of section Initial conditions on the manifold remain on the manifold stable periodic orbit Torus section 20
2. Uncertainties can change the manifold Uncertain gravity field 21
2. Lyapunov like stability to accommodate uncertainties Take all possible realizations The motion is bounded by this union If initial conditions are in this intersection 22
2. Trade off between large intersection and narrow union min J I(p), U(p) p s.t I(p) I Injection accuracy requirements U(p) U Collision avoidance Min bounds of the motion I U Max tolerable uncertainty in initial conditions 23
Outline 1. Safe orbits in the Didymos system 2. Robust mission design using manifolds 3. Application to the Didymos system 24
3. Design of robust interior orbits in the Didymos system 20% uncertainty in mass parameter No impact 1 cm/s uncertainty 25
3. Unperturbed solution: 2d torus y [ ] x [ ] Technical need: compute manifold passing through a desired point 26
3. Safest orbit given uncertainty in the injection velocity 1.25 Objective function =U Min distance from bodies y [ ] 0 Worst case trajectories 1.25 1.25 0 x [ ] Nominal initial conditions 1.25 27
3. Maximum tolerable uncertainty in the injection velocity 1.25 Objective function = I Avoid impact y [ ] 0 Nominal initial conditions Worst case trajectories 1.25 1.25 0 x [ ] 1.25 28
3. Ballistic landing from L2 Stable manifold Landing trajectory Unstable manifold 20% uncertainty in mass parameter L2 Lyapunov orbit We consider only the first impact Envelope of asymptotic orbits 29
3. Libration point L2 (unstable) 0.6 y 0.4 0.2 0 0.2 0.8 L2 1 1.2 x 1.4 1.6 30
3. Manifolds of L2: minimum energy landing 0.6 y 0.4 0.2 Stable manifold Unstable manifold 0 0.2 0.8 L2 1 1.2 x 1.4 1.6 Unfeasible in practice (infinite time and no uncertainty ) 31
3. Lyapunov orbit (unstable) 0.6 y 0.4 0.2 0 Lyapunov orbit 0.2 0.8 1 1.2 x 1.4 1.6 32
3. Asymptotic trajectory of the Lyapunov orbit 0.6 y 0.4 0.2 Unstable manifold Stable manifold 0 0.2 0.8 1 1.2 x 1.4 1.6 33
3. Manifolds of the Lyapunov orbit 0.6 y 0.4 0.2 Unstable manifold Stable manifold 0 Lyapunov orbit 0.2 0.8 1 1.2 x 1.4 1.6 34
3. How to guarantee that a trajectory will land? If the lander is deployed from this point with the same energy of the manifold Feasible deployment directions 35
3. High energy allows to accommodate more uncertainty 3 1.5 130 1 vy vx tan 1 y y y y y y y 155 180 1.1 1.2 1.3 x 1.4 1.5 = 0.05 = 0.1 = 0.2 = 0.3 = 0.4 = 0.5 = 0.6 1.6 JLyapunov JL2 105 [deg] 80 x 10 2 0.5 0 36
3. Feasible set of the velocity relative to the manifold vy [cm / s] Min energy L2 Feasible Ma xe set ne rgy Envelope of transit orbits Max y crossing at L2 vx [cm / s] 37
3. Maximization of the uncertainty in the initial velocity Max tolerable uncertainty in velocity Guaranteed impact location 4 100 U 50 y [m] vy [cm / s] 2 0 I 2 4 4 0 50 100 2 0 vx [cm / s] 2 4 1100 1150 1200 x [m] 1250 38
3. Monte Carlo validation of a deployment from y = 0.5 0.5 y Zero velocity curves for maximum energy 0.1 0.8 x 1.6 39
Conclusions Two families of safe orbits exist: interior retrograde and terminator orbits Invariant manifolds are used to bound the motion Mission design: trade off between large initial state uncertainty and narrow bounds Way forward: accommodating binary eccentricity and uncertainties polyhedron Nonlinear Conley criterion for the landing 40
Robust trajectory design using invariant manifolds. Application to (65803) Didymos L. Dell'Elce Sophia Antipolis, 22/02/2018