semi-analytical guidance algorithm for fast retargeting...
TRANSCRIPT
Semi-Analytical Guidance Algorithm for Fast
Retargeting Maneuvers Computation during
Planetary Descent and Landing
12th Symposium on Advanced Space Technologies in Robotics and Automation
Michèle LAVAGNA, Paolo LUNGHI
Politecnico di Milano,
Department of Aerospace Science and Technology (DAST)
ASTRA 2013 - ESA/ESTEC, Noordwijk, the Netherlands
Image credits: NASA/JPL-Caltech/ESA
• Several improvements in last years, but not enough
• Landing uncertainty still imposes strict requirements onto Landing Site choice
Precision Landing
• Could allow the reaching of more scientifically relevant places
• It involves the presence of an Hazard Detection and Avoidance System (HDA)
Dynamical Landing Site Selection
Autonomous, Precise and Safe Landing Capability Is a key feature for the next space systems generation.
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The Autonomous Landing Problem
• Capable to dynamically recompute and correct the landing trajectory during the descent
Development of a GUIDANCE ALGORITHM
• Computational Efficiency
• Maximum Attainable Landing Area (maximize the probability to find a safe landing site)
• Landing Accuracy
Requirements
• ESA Lunar Lander
• Demonstration of Safe Precision Landing Technology as part of preparations for participation to future human exploration of the Moon
Reference Scenario (for simulations)
Image credits: ESA
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Paper Subject
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
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Typical Landing Phases
Adaptive Guidance operates from Retargeting to Terminal Descent
Horizontal Speed
• Mass at Touchdown: 816 kg (in line with ESA Lunar Lander).
Vertical Speed
Altitude Mass
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• Planar Reference Trajectory (from a preliminary optimization)
• From Powered Descent Initiation (PDI) to Terminal Gate (TG)
• Starting point for Retargeting
Reference Trajectory
• 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
Pitch Angle Thrust Magnitude
NLS View Angle
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Reference Trajectory: Control Profile
• 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)
𝑧𝑏, 𝜓
𝑥𝑏, 𝜙 𝑦𝑏 , 𝜃
Surface-Fixed Ref. Frame
Body-Fixed Ref. Frame
Dynamic System
TARGET
𝐫 𝑡 =𝐓(𝑡)
𝑚(𝑡)+ 𝐠
𝑚 (𝑡) = −𝑇(𝑡)
𝐼𝑠𝑝𝑔0
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• Thrust vector expressed in Euler Angles
and Thrust Magnitude (Control Variables)
𝐓 𝑡 = −𝑇(𝑡)
cos𝜓(𝑡) cos 𝜃(𝑡)cos𝜓(𝑡) cos 𝜃(𝑡)
− sin𝜓(𝑡)
Adaptive Guidance: Dynamics
• Thrust dependent
acceleration:
𝐚 𝑡0 =
−𝑇0cos 𝜓 sin 𝜃
𝑚− 𝑔𝑀
−𝑇0cos 𝜓 cos 𝜃
𝑚
𝑇0sin 𝜓
𝑚
• Known Position
and Speed:
• 𝐫 𝑡0 = 𝐫0 • 𝐯 𝑡0 = 𝐯0
• Desired Position and
Speed:
• 𝐫(𝑡𝑇𝑜𝐹) = 3000
m
• 𝐯(𝑡𝑇𝑜𝐹) = −1.500
m/s
• Vertical Attitude:
• 𝐚(𝑡𝑇𝑜𝐹) = 𝐹𝑅𝐸𝐸00
Final Constraints
Initial Constraints 2 Variables: T0 , tToF
17 Boundary Constraints
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Adaptive Guidance: Boundary Constraints
• Polynomial Acceleration Profile
• Minimum degree needed to satisfy
Boundary Constraints
• Depends on 2 variables: 𝐱 =𝑇0𝑡𝑇𝑜𝐹
𝐏 𝑡 = 𝐓(𝑡)
𝑚(𝑡)=
𝑎𝑥 + 𝑔𝑀𝑎𝑦𝑎𝑧
𝐓 𝑡 = 𝐏 𝑡 𝑚(𝑡)
Thrust-to-Mass Ratio
Thrust Vector 𝑚 𝑡 = 𝑚0exp −
𝐏(𝑡)
𝐼𝑠𝑝𝑔0𝑑𝑡
𝑡
𝑡0
Solved with Pseudospectral Methods
Mass vs Time Trend
Guidance Profile
𝜃 𝑡 = tan−11
𝐓
𝑇𝑥
𝑇𝑦 𝜓 𝑡 = tan−1
1
𝐓
𝑇𝑧
𝑇𝑥2+𝑇𝑦
2
𝜙(𝑡) = 0 𝑇 𝑡 = 𝐓(𝑡)
𝑎𝑥 𝑡 = 𝑎0𝑥 + 𝐶1𝑥𝑡 + 𝐶2𝑥𝑡2
𝑎𝑦/𝑧 𝑡 = 𝑎0𝑦/𝑧 + 𝐶1𝑦/𝑧𝑡 + 𝐶2𝑦/𝑧𝑡2 + 𝐶3𝑦/𝑧𝑡
3
Acceleration Profile
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Adaptive Guidance: Descent Profile
min𝐱
𝑓 𝑥 = 𝑚0 −𝑚(𝑡𝑓) such that 𝐱𝐿 ≤ 𝐱 ≤ 𝐱𝑈𝐜𝐿 ≤ 𝐜(𝐱) ≤ 𝐜𝑈
Path Constraints are evaluated discretely with
Pseudospectral Techniques
Limited Search Domain
• Initial Thrust Magnitude
• Time-of-flight
Additional Constraint
• Final Mass
Path Constraints
• Available Thrust
• Available Control Torques
• Glide-Slope Constraint
𝐱𝐿 ≤ 𝐱 ≤ 𝐱𝑈
𝑇𝑚𝑖𝑛
0≤
𝑇0𝑡𝑇𝑜𝐹
≤𝑇𝑚𝑎𝑥
𝑡𝑚𝑎𝑥
𝐜𝐿 ≤ 𝐜 𝐱 ≤ 𝐜𝑈
𝑚𝑑𝑟𝑦
𝑇𝑚𝑖𝑛
−𝑀𝑚𝑎𝑥
−𝑀𝑚𝑎𝑥
−∞
≤
𝑚(𝑡𝑓)
𝑇(𝑡)
𝐼𝑚𝑎𝑥𝜃 (𝑡)
𝐼𝑚𝑎𝑥𝜓 (𝑡)
S𝐫(𝑡) + 𝐜𝑇𝐫(𝑡)
≤
𝑚0
𝑇𝑚𝑎𝑥
𝑀𝑚𝑎𝑥
𝑀𝑚𝑎𝑥
0
S =0 1 00 0 1
𝐜𝑇 = − tan 𝛾𝑚𝑎𝑥 0 0
Problem Formulation
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Adaptive Guidance: Constraints
Step 1 – FEASIBILITY
Step 2 – OPTIMALITY
• Unconstrained Compass Search on Modified
Cost Function:
𝜙 𝐱 = 𝑓 𝐱 + 𝜂 sgn 𝐹(𝐱 )
Unconstrained Compass Search on the Feasibility
Function:
𝐹 𝐱 = max 0, 𝑐 𝑖(𝐱 )
𝑤𝐹𝑖
𝑁𝑐
𝑖=0
Generalized Constraints vector:
𝐜 𝑥 =
𝐜𝐿 − 𝐜(𝐱 )
𝐜 𝐱 − 𝐜𝑈0 − 𝐱 𝐱 − 1
GuidALG: Compass Search Method
• Direct Optimization method: Cost function
treated as “Black Box”
• Low Computational Cost
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Adaptive Guidance: Optimization Algorithm
Mean Computation Time
Vertical Position Error
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
Performances estimated with Monte
Carlo simulations:
• Random diversion ±2000 m from
Nominal Landing Site
• 100000 samples for each value of N
• From N≥20 Error is negligible.
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Adaptive Guidance: Order of Approximation
2000 m
1500 m
1000 m
500 m
100 m
Attainable Area from
Nominal Path at
different altitude
• 4000x4000 m Test Area Centered on
Nominal Landing Site
• 100000 samples Monte Carlo Analysis
4000 m
40
00
m
200 m
20
0 m
100 m ZOOM
Feasible
Feasible, Suboptimal
Infeasible
Downrange Direction
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Adaptive Guidance: Divert Capability
• Found solution is an approximation of the optimum.
• Solution compared with alternative
methods:
• GuidALG, (proposed method)
• Polynomial formulation with
nonlinear optimization solver
(SNOPT)
• Direct Collocation Method
(more accurate, more slow)
Thrust Magnitude
Euler Angles
Mass (Cost Function)
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Adaptive Guidance: Solution Optimality
• Ideal Actuators
• ACS Thruster: constant
control torques, ±40 Nm
on each axis
• Throttleable Main Thrust 1000 2300N
• 6DoF Model • Thrust misalignment
Disturbance Torques
Lander
Dynamics
GUIDANCE
Navigation
(Error Model)
Actuators
Model
CONTROL law
Commanded
DIVERSION
Control Update
Rate:
20 Hz
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Closed-Loop Landing Simulation
Navigation System emulated by Errors added to States
Attitude
• Inertial Measurement Unit (IMU), calibrated with Star Trackers just before PDI
Position and Speed
• Visual-Based Navigation System
• Make use of Laser/radar Altimeter to rescale Images
• Zero-mean Gaussian Error
• Standard Deviation Linearly Dependent by Altitude
Property Value UoM
Scale Factor 1 Ppm
Misalignment Error 170 μrad
Bias Error 0.005 deg/h
ARW Noise Density 0.005 deg/√h
IMU Performance Properties
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
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Landing Simulation: Navigation Error Model
Trajectory Update
• Whenever a Retargeting occurs
• Every 5 seconds (minimize dispersions)
Guidance/Control Interface
• Target Quaternions
• Target Angular Speed
• Target Thrust Magnitude
Navigation
States Estimation
Hazard Detection
TLS Coordinates
Guidance Profile
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𝑇 𝑡 to Main Engine
PID
Theoretical Control Torques
ACS Thrusters Activation
Scheme
PWPF Modulation
Landing Simulation: Guidance & Control
• Landing precision appears to be independent by the
magnitude of the requested diversion.
TLS [0,+2000,+2000] m TLS [0,0,-2000] m
TLS [0,0,0] m (NLS) TLS [0,-1000,0] m
• Landing Dispersion at different Target Landing Site
• 100 Samples series of Monte Carlo Analysis
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Landing Simulation: Sensitivity to Target LS
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.
• Landing Dispersion at different Position Initial Error Standard Deviation (STD)
• 100 Samples series of Monte Carlo Analysis
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Landing Simulation: Sensitivity to Navigation Error
Navigation Camera
Hazard Detection 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: 50÷70 deg
• Visibility on TLS lost if Sightline-TLS
angle >25÷35 deg
• Oscillatory movement causes initial loss of
visibility
• Visibility recovered in 2nd half trajectory
Camera System
• Single Camera pointed towards ROLL axis
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Landing Simulation: Camera Field of View
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
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Conclusions
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
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Future Developments
Thank you for your attention
ASTRA 2013 - ESA/ESTEC, Noordwijck, the Netherlands
Pseudospectral Discretization
• Function discretized with its values at
Chebyshev-Gauss-Lobatto (CGL)
points τk :
𝜏𝑘 = −cos𝜋𝑘
𝑁, 𝑘 = 0,… , 𝑁
• Function shifted on computational domain:
𝑝 𝑡 , 𝑡 ∈ [𝑡𝑖𝑛𝑖 , 𝑡𝑒𝑛𝑑]
𝑝 𝜏 = 𝑝 𝑡(𝜏) , 𝑡 𝜏 = 𝑡𝑒𝑛𝑑 − 𝑡𝑖𝑛𝑖 𝜏 + 𝑡𝑒𝑛𝑑 + 𝑡𝑖𝑛𝑖 /2
𝑝 = 𝑝0, 𝑝1, … , 𝑝𝑘𝑇 , 𝑝𝑘 = 𝑝(𝑡 𝜏𝑘 ), 𝑘 = 0,… , 𝑁
• The function can be evaluated at each time instant with a polynomial
approximation of the form:
𝑝 𝑡 ≅ 𝑝 𝑡 = 𝑝𝑘𝜑𝑘(𝜏(𝑡))
𝑁
𝑘=0
𝜑𝑘(𝜏 𝑡 ), 𝑘 = 0,… ,𝑁 Lagrange interpolating polynomials of order N.
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Pseudospectral Derivatives
• Cebyshev Differentiation Matrix
𝑑𝑝𝑘𝑑𝜏
= 𝑝 𝜏𝑘 = 𝑝𝑗𝜑 𝑗 𝜏𝑘 =
𝑁
𝑗=0
(D𝑁)𝑘𝑗𝑝𝑗
𝑁
𝑗=0
𝑐𝑖𝑐𝑗
−1𝑖+𝑗
𝜏𝑖 − 𝜏𝑗, 𝑖 ≠ 𝑗,
−𝜏𝑗
2 1 − 𝜏𝑗2
, 1 ≤ 𝑖 = 𝑗 ≤ 𝑁 − 1,
2𝑁2 + 1
6, 𝑖 = 𝑗 = 0,
−2𝑁2 + 1
6, 𝑖 = 𝑗 = 𝑁.
D𝑁 𝑘𝑗 = 𝑐𝑗 = 1, 𝑗 = 0, 𝑁2, 1 ≤ 𝑗 ≤ 𝑁 − 1
• From Computational domain to real domain:
𝑑𝑡 =𝑡𝑒𝑛𝑑 − 𝑡𝑖𝑛𝑖
2𝑑𝜏
𝑑𝐩
𝑑𝑡=
2
𝑡𝑒𝑛𝑑 − 𝑡𝑖𝑛𝑖
𝑑𝐩
𝑑𝜏
CGL points from -1 to 1.
Differentiation Matrix Correction. 𝐷 𝑁 = −𝐷𝑁
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Pseudospectral Integration
• Clenshaw-Curtis Quadrature
• the integral is discretized to a finite sum:
𝑝 𝜏 𝑑𝜏1
−1
= 𝑤𝑐𝑘𝑝𝑘
𝑁
𝑘=0
𝑑𝑡 =𝑡𝑒𝑛𝑑 − 𝑡𝑖𝑛𝑖
2𝑑𝜏 𝑝 𝑡 𝑑𝑡
𝑡𝑒𝑛𝑑
𝑡𝑖𝑛𝑖
=𝑡𝑒𝑛𝑑 − 𝑡𝑖𝑛𝑖
2 𝑤𝑐𝑘𝑝𝑘
𝑁
𝑘=0
• 𝑤𝐶𝑘 Weights of Clenshaw Curtis Quadrature.
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