Semi-Analytical Guidance Algorithm for Fast Retargeting ...robotics.estec.esa.int/ASTRA/Astra2013/Presentations/Lunghi_28245… · TLS [0,0,0] m (NLS) TLS [0,-1000,0] m • Landing

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.

2 ASTRA 2013 - ESA/ESTEC, Noordwijck, the Netherlands

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


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


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


• 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


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


• 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


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


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


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


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.


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


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)


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


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


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


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


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


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


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


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


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.

B1 ASTRA 2013 - ESA/ESTEC, Noordwijck, the Netherlands

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


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.


Documents

Semi-Analytical Guidance Algorithm for Fast Retargeting ...robotics.estec.esa.int/ASTRA/Astra2013/Presentations/Lunghi_28245… · TLS [0,0,0] m (NLS) TLS [0,-1000,0] m • Landing