Semi-Analytical Guidance Algorithm for Fast Retargeting Maneuvers Computation during Planetary Descent and Landing

Transcription

1 ASTRA ESA/ESTEC, Noordwijk, the Netherlands Semi-Analytical Guidance Algorithm for Fast Retargeting Maneuvers Computation during Planetary Descent and Landing Michèle LAVAGNA, Paolo LUNGHI Politecnico di Milano, Department of Aerospace Science and Technology (DAST) 12th Symposium on Advanced Space Technologies in Robotics and Automation

2 The Autonomous Landing Problem Precision Landing Several improvements in last years, but not enough Landing uncertainty still imposes strict requirements onto Landing Site choice Dynamical Landing Site Selection Could allow the reaching of more scientifically relevant places It involves the presence of an Hazard Detection and Avoidance System (HDA) Image credits: NASA/JPL-Caltech/ESA Autonomous, Precise and Safe Landing Capability Is a key feature for the next space systems generation. 2

3 Development of a GUIDANCE ALGORITHM Capable to dynamically recompute and correct the landing trajectory during the descent Paper Subject Requirements Computational Efficiency Maximum Attainable Landing Area (maximize the probability to find a safe landing site) Landing Accuracy Reference Scenario (for simulations) Image credits: ESA ESA Lunar Lander Demonstration of Safe Precision Landing Technology as part of preparations for participation to future human exploration of the Moon 3

4 Typical Landing Phases Parking Orbit (LLO 100km) Deorbiting (DOI) & Coasting (100x15km) Main Brake (PDI) Constant Thrust High Gate (HG): Nominal Landing Site comes into Sensors FoV Approach Approach Gate (AG): Hazard Detection System comes into operation Once updated the Hazard Map, Thrust is reduced, and Retargeting could be commanded Throttleable Thrust after first Retargeting Low Gate (LG): Last chance of Retargeting Terminal Descent (TD) Starts at 30m altitude (Terminal Gate, TG) Vertical descent at Constant Speed -1,5m/s until Touch Down Adaptive Guidance operates from Retargeting to Terminal Descent 4

5 Planar Reference Trajectory (from a preliminary optimization) From Powered Descent Initiation (PDI) to Terminal Gate (TG) Starting point for Retargeting Reference Trajectory Horizontal Speed Vertical Speed Altitude Mass Mass at Touchdown: 816 kg (in line with ESA Lunar Lander). 5

6 Reference Trajectory: Control Profile Pitch Angle Thrust Magnitude NLS View Angle Altitude at thrust reduction (first chance of retargeting): 2000 m Hazard Detection System operation field: Altitude < 3000 m NLS View angle < 45 deg Hazard Detection available time before thrust reduction: 21.4 s 6

7 Adaptive Guidance: Dynamics Dynamic System Constant Gravity Field with flat ground (small altitude compared to planet s radius) No aerodynamic forces considered (also with low density atmosphere, speed < 100m/s) r t = T(t) m(t) + g m(t) = T(t) I sp g 0 Surface-Fixed Ref. Frame TARGET Body-Fixed Ref. Frame Thrust vector expressed in Euler Angles and Thrust Magnitude (Control Variables) T t = T(t) cosψ(t) cos θ(t) cos ψ(t) cos θ(t) sin ψ(t) y b, θ x b, φ z b, ψ 7

8 Adaptive Guidance: Boundary Constraints Final Constraints Known Position and Speed: r t 0 = r 0 v t 0 = v 0 Thrust dependent acceleration: a t 0 = T 0 cos ψ sin θ m T 0 cos ψ cos θ m T 0 sin ψ m g M Initial Constraints Desired Position and Speed: r(t ToF ) = v(t ToF ) = Vertical Attitude: a(t ToF ) = m m/s FREE Variables: T 0, t ToF 17 Boundary Constraints 8

9 Polynomial Acceleration Profile Minimum degree needed to satisfy Boundary Constraints Depends on 2 variables: x = T 0 t ToF Mass vs Time Trend P(t) m t = m 0 exp dt t 0 I sp g 0 Solved with Pseudospectral Methods Guidance Profile t Adaptive Guidance: Descent Profile Acceleration Profile a x t = a 0x + C 1x t + C 2x t 2 a y/z t = a 0y/z + C 1y/z t + C 2y/z t 2 + C 3y/z t 3 Thrust-to-Mass Ratio P t = T(t) m(t) = Thrust Vector T t a x + g M a y a z = P t m(t) θ t = tan 1 1 T T x ψ t = tan 1 1 T y T T z T x 2 +T y 2 φ(t) = 0 T t = T(t) 9

10 Adaptive Guidance: Constraints Limited Search Domain Initial Thrust Magnitude Time-of-flight Additional Constraint Final Mass Path Constraints Available Thrust Available Control Torques Glide-Slope Constraint x L x x U T min 0 T 0 t ToF T max t max c L c x c U m dry T min M max M max m(t f ) T(t) I max θ (t) I max ψ (t) Sr(t) + c T r(t) m 0 T max M max M max 0 Problem Formulation min x f x = m 0 m(t f ) such that x L x x U c L c(x) c U Path Constraints are evaluated discretely with Pseudospectral Techniques S = c T = tan γ max

11 GuidALG: Compass Search Method Direct Optimization method: Cost function treated as Black Box Low Computational Cost Step 1 FEASIBILITY Unconstrained Compass Search on the Feasibility Function: N c max 0, c i (x) F x = i=0 Step 2 OPTIMALITY w Fi Generalized Constraints vector: c L c(x) c x = c x c U 0 x x 1 Unconstrained Compass Search on Modified Cost Function: φ x = f x + η sgn F(x) Adaptive Guidance: Optimization Algorithm 11

12 The algorithm precision is affected by the choice of pseudospectral order of approximation N in mass calculation: As N increases: Computation Time Increases Precision Increases Adaptive Guidance: Order of Approximation Vertical Position Error Mean Computation Time Performances estimated with Monte Carlo simulations: Random diversion ±2000 m from Nominal Landing Site samples for each value of N From N 20 Error is negligible. 12

13 4000 m 200 m Adaptive Guidance: Divert Capability 4000 m Downrange Direction Feasible Feasible, Suboptimal Infeasible 100 m ZOOM 200 m 2000 m Attainable Area from Nominal Path at different altitude 1500 m 4000x4000 m Test Area Centered on Nominal Landing Site 1000 m samples Monte Carlo Analysis 500 m 100 m 13

14 Adaptive Guidance: Solution Optimality Solution compared with alternative methods: GuidALG, (proposed method) Polynomial formulation with nonlinear optimization solver (SNOPT) Direct Collocation Method (more accurate, more slow) Mass (Cost Function) Thrust Magnitude Euler Angles Found solution is an approximation of the optimum. 14

15 Closed-Loop Landing Simulation Actuators Model Ideal Actuators ACS Thruster: constant control torques, ±40 Nm on each axis Throttleable Main Thrust N Lander Dynamics 6DoF Model Thrust misalignment Disturbance Torques CONTROL law Navigation (Error Model) Control Update Rate: 20 Hz GUIDANCE Commanded DIVERSION 15

16 Landing Simulation: Navigation Error Model Navigation System emulated by Errors added to States Attitude Inertial Measurement Unit (IMU), calibrated with Star Trackers just before PDI Position and Speed IMU Performance Properties Property Value UoM Scale Factor 1 Ppm Misalignment Error 170 μrad Bias Error deg/h ARW Noise Density deg/ h Visual-Based Navigation System Make use of Laser/radar Altimeter to rescale Images Zero-mean Gaussian Error Standard Deviation Linearly Dependent by Altitude Altitude 2000 m 0 m Confidence Position Error ±25 m ±0.1 m 1σ Speed Error ±0.4 m/s ±0.1 m/s 1σ 16

17 Landing Simulation: Guidance & Control Hazard Detection Navigation TLS Coordinates States Estimation Theoretical Control Torques Trajectory Update Whenever a Retargeting occurs Every 5 seconds (minimize dispersions) Guidance Profile PID PWPF Modulation Guidance/Control Interface Target Quaternions Target Angular Speed Target Thrust Magnitude T t to Main Engine ACS Thrusters Activation Scheme 17

18 Landing Simulation: Sensitivity to Target LS Landing Dispersion at different Target Landing Site 100 Samples series of Monte Carlo Analysis TLS [0,0,0] m (NLS) TLS [0,-1000,0] m TLS [0,0,-2000] m TLS [0,+2000,+2000] m Landing precision appears to be independent by the magnitude of the requested diversion. 18

19 Landing Simulation: Sensitivity to Navigation Error Landing Dispersion at different Position Initial Error Standard Deviation (STD) 100 Samples series of Monte Carlo Analysis STD 45 m STD 25 m STD 10 m Exact Dispersion one order of magnitude lower with exact states estimation; Landing precision is mainly affected by Navigation errors. 19

20 Camera System Landing Simulation: Camera Field of View Hazard Detection Camera Single Camera pointed towards ROLL axis Navigation Camera Relative navigation by Landmaks tracking Tracked Landmarks pass continuously into FoV, allowing relative navigation Target LS must be into FoV for Hazard Map update Camera Aperture Angle: deg Visibility on TLS lost if Sightline-TLS angle >25 35 deg Oscillatory movement causes initial loss of visibility Visibility recovered in 2 nd half trajectory 20

21 Conclusions Low Computational Cost Obtained Computation Time <100ms in Matlab language on a PC Compatible with Control Update Rate during a real Landing Wide attainable area even with a simply optimization method Loss of Divert Capability when nominal trajectory requires High Thrust Maximum Attainable Area when Retargeting is ordered in low thrust phases Navigation Errors play a major role in determining the accuracy at Touchdown 21

22 Future Developments Improve Retargeting Strategy Dual Retargeting Increase Polynomial Order More Opt. Parameters Add Constraints Tradeoff on Opt. Algorithm Integration with Visual-Based Navigation and Hazard Detection Systems Hardware-in-the-loop Testing 22

23 Thank you for your attention

24 Pseudospectral Discretization Function discretized with its values at Chebyshev-Gauss-Lobatto (CGL) points τ k : τ k = cos πk N, k = 0,, N Function shifted on computational domain: p t, t [t ini, t end ] p τ = p t(τ), t τ = t end t ini τ + t end + t ini /2 p = p 0, p 1,, p k T, p k = p(t τ k ), k = 0,, N The function can be evaluated at each time instant with a polynomial approximation of the form: N p t p t = p k φ k (τ(t)) k=0 φ k (τ t ), k = 0,, N Lagrange interpolating polynomials of order N. B1

25 Pseudospectral Derivatives Cebyshev Differentiation Matrix dp k dτ = p τ k = p j φ j τ k = N j=0 N j=0 (D N ) kj p j c i 1 i+j, i j, c j τ i τ j τ j, 1 i = j N 1, τ j 2N 2 + 1, i = j = 0, 6 2N2 + 1, i = j = N. 6 D N kj = c j = From Computational domain to real domain: dt = t end t ini dp dτ 2 dt = 2 dp t end t ini dτ CGL points from -1 to 1. Differentiation Matrix Correction. D N = D N 1, j = 0, N 2, 1 j N 1 B2

26 Pseudospectral Integration Clenshaw-Curtis Quadrature the integral is discretized to a finite sum: 1 1 p τ dτ N = w ck p k k=0 dt = t end t ini 2 dτ t end p t dt t ini = t end t ini 2 w Ck Weights of Clenshaw Curtis Quadrature. N k=0 w ck p k B3