approximation of attainable landing area of a moon … · marco sagliano [email protected]...
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APPROXIMATION OF ATTAINABLE LANDING AREA OF A MOON LANDER BY REACHABILITY ANALYSIS
DLR – German Aerospace CentreInstitute of Space SystemsGuidance, Navigation and Control Dept.Robert-Hooke-Str. 728359, Bremen, Germany
17th International Conference on Hybrid Systems: Computation and Control (HSCC 2014)15th-17th April 2014, Berlin, Germany
Method
Introduction
Discretization of Optimal Control Problem
References: [1] T. Oehlschlägel, S. Theil, H. Krüger, M. Knauer, J. Tietjen, C. Büskens, Optimal Guidance and Control of Lunar Landers with Non-throttable Main Engine, Advances in Aerospace Guidance, Navigation and Control, pp. 451-463, 2011
[2] R.Baier, M. Gerdts, I. Xausa, Approximation of Reachable Sets Using Optimal Control Algorithms, Numerical Algebra; Control and Optimization, Vol. 3, 2013
[3] M. Sagliano, S. Theil, Hybrid Jacobian Computation for Fast Optimal Trajectories Generation, AIAA Guidance, Navigation and Control(GNC) Conference, Boston, August 19-22,2013
Results
Problem Statement
Yunus Emre [email protected]
Thimo Oehlschlä[email protected]
Marco [email protected]
Stephan [email protected]
Claus [email protected]
Determination of AttainableLanding Area by Forward
Reachability
Equations of motion of the moon lander are taken from [1]. The vector of states and control inputs are defined as follows:
Initial condition and terminal conditions:
DCA Representation
Reference Frame: Downrange-Crossrange-Altitude (DCA)
0 100 200 300 400
0
100
200
Jacobian Structure
0 100 200 300 400
0
100
200
Numerical Jacobian
0 100 200 300 400
0
100
200
PseudoSpectral Jacobian
0 100 200 300 400
0
100
200
Analytical Jacobian
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1f-LGR
LG
LGRSchematic diagram of Legendre points
Acknowledgements: This research is supported by DLR (German Aerospace Center) and DAAD (German Academic
Exchange Service) Research Fellowship Programme.
Developments in space technology have paved the way for more challenging missions which require advancedguidance and control algorithms for safely and autonomously landing on celestial bodies.
Instant determination of hazards, automatic guidance during landing maneuvers and likelihood maximization ofa safe landing are of paramount importance.
Reachability analysis is used to obtain attainable landing areas for the final phase of interplanetary spacemissions given initial conditions, admissible control inputs and landing constraints.
: Downrange Rate : Downrange : Pitch
: Altitude Rate : Altitude : Yaw
: Crossrange Rate : Crossrange : Mass
: RCS Thrusters
: Pitch Rate
: Yaw Rate
Non-uniform collocation points toavoid the Runge phenomenon
0 20 40 60 80 100 120 140 160 180 200-12
-10
-8
-6
-4
-2
0
Number of Nodes
log 10
of
infin
ity n
orm
inte
rpol
atio
n er
ror
Interpolation error comparison using LGR,f-LGR,LG discretization points
LGR
f-LGRLG
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2
0
0.2
0.4
0.6
0.8
1
1.2Interpolation of a function with Lagrange Polynomials using Legendre-Gauss-Radau Polynomial roots
y
x
y = 11+ 50x2
InterpolationNodes
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Interpolation of a function with Lagrange Polynomials using Legendre-Gauss-Radau Polynomial roots
y
x
y = 11+ 50x2
InterpolationNodes
: Number of discretization points : Number of control Inputs
: Number of states : Number of constraints
Generic Mission Profile for Lunar Lander
Exponentialconvergence w.r.t. number ofcollocation points
Lagrange polynomials forinterpolation ofstates,control inputs
Jacobian of theassociated NLP isexpressed as sum of 3 different contributors. Resulting sparse is usedby commercial off-the-shelf SQP solver.
In this study, we apply an optimal-control-based algorithm for approximating nonconvex reachable sets of nonlinearsystems [2].
Discretize region of interest
Find optimal control law that steers the system from the initial condition to the target state
s.t. a.e. in
a.e. in
OCP is transcribed into NLP by SPARTAN (SHEFEX-3 Pseudospectral Algorithm for Reentry Trajectory ANalysis) [3].
-500 -400 -300 -200 -100 0 100 200 300 400 500-500
-400
-300
-200
-100
0
100
200
300
400
500
Downrange(m)
Cro
ssra
nge(
m)
Reachable Landing SiteforLunarLanderattf = 35 -Grid
Grid
Reachable
Approximate the reachable set with an error of discretization step by solving following optimal control problem (OCP) for each grid points
Discretization of State Space
Reachable Set at tf=35 (s)-Propellant Cost-
Reachable Landing Tunnel tf <35 (s)-Propellant Cost-
Reachable Set with free final time -Propellant Cost-
Reachable Set with free final time -Time Cost-
Attainable landing area and propellant&time cost map ofassociated region is computed using reachability analysis
Jacobian structure of associated NLP
Interpolation of a function with Lagrange polynomials using LGR roots
Schematic diagram of Legendre points and Interpolation Error
Condition for successful landing: