Sampling-based path planning: a new tool for missile guidance

P. Pharpatara, B. Hérissé, R. Pepy, Y. Bestaoui 02-06/09/2013 - Würzburg, Germany

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Missile interception −→ Target detection by the radar station. I Mission preparation (À) : trajectories calculations. −→ Missile launch. I Midcourse guidance (Á) : I

I

ensure a proper collision course to the target with an acceptable velocity ; trajectory optimization with constraints.

−→ Target detection by the embedded radar. I Terminal guidance (Â) : I

engage the target at the Predicted Intercept Point (PIP).

PIP target Á À

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 radar beam

Ref.

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Objective To generate a missile-interceptor trajectory for the midcourse guidance (off-line or/and on-line) : I Optimal control problem : I

I

Terminal constraints at the PIP : I

I

I

maximize the final velocity (hit-to-kill). lower bound of terminal speed at the PIP (to ensure the target destruction) ; related orientation between the missile and the target at the PIP (due to the clearance of the embedded sensor).

Constraints related to the system/missile (here, single-stage missile) : I I I

structural limitation ; maximum angle of attack that can be balanced using the tail fins ; variation of air density.

=⇒ Complex, non-holonomic and non-linear problem 3/18

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Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Closed-loop midcourse guidance laws I

Singular Perturbation Theory [4] ;

I

Linear Quadratic Regulators [6] ;

I

Fuzzy Logic [9] ;

I

Neural Network [13] ; Kappa guidance [8] :

I

I

I

take the final orientation into account ; maximize the final velocity (locally optimal).

-with control limitation -no control limitation

Trajectories given by Kappa guidance

Problem of analytical guidance laws (ex : kappa guidance) I

They cannot anticipate the future changes of flight conditions : I

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ex : loss of manoeuvrability at high altitude.

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Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Trajectory generation using numerical methods [2]

I

State Dependant Riccati Equation (SDRE) technique [11] ;

I

Adaptive grid methods [1] ;

I

Pseudo-spectral method [12][10].

Problems of numerical methods I

Initialisation problem : I

I

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a good first guess is recommended to ensure the acquisition of a solution.

Trade-off between accuracy and efficiency in terms of computational effort.

IFAC ACA 2013 - Würzburg, Germany

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Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Proposed solution

I

Combination of trajectory planning methods and optimal control methods, here : I

Rapidly-exploring Random Tree (RRT) [7] : I I

I

Dubins’ paths (Dubins’ metric) [5][3] : I

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it takes the future changes of flight conditions into account ; it can find a solution without any a priori guess. it is used to improve the optimality of the solution trajectory in 2D plane.

IFAC ACA 2013 - Würzburg, Germany

Ref.

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Ref.

Missile modelling in 2 Dimension ξ˙ = v,

(1)

mv˙ = fdrag (ρ, ||v||) + flift (ρ, ||v||) + fthrust − mg, θ˙ = q, I q˙ = τaero + τpert . flift fdrag

fthrust θ

α

Cg

v

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(3) (4)

I

I

fthrust lasts 20s after launch (single-stage missile) ; flift , fdrag depend on ρ and v : I

mg

(2)

I

ρ small : lost of manoeuvrability at high altitude ; v large : structural limitation.

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Mission and requirements (1) Mission I

xinit = [0 0 0 0]> with vertical launch for 1s ;

I

Xgoal = {x : kξ − ξ pip k < R, kvk > vmin , ∠(v, −vpip ) < φf } : Xgoal φf vpip φf

ξ pip vmin

R

I

vpip is in the horizontal direction ;

I

vmin = 500m/s ;

I

R = 500m ;

I

φf = π/8.

Exploration space

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I

X = {x : ξ ∈ [−20km, 20km] × [0km, 30km]} ;

I

Xfree = {x : alt(ξ) > 0, ||v(t > tboost )|| > vmin }.

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Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Mission and requirements (2) Admissible control input U(t, x) = {avc : αc 6 αmax (t, x)} with



stb struct αmax (t, x) = min αmax (t, x), αmax (t, x)



where stb (t, x) : the maximum achievable angle of attack using tail fins ; αmax struct (t, x) : the structural limit which is given by the maximum αmax lateral acceleration abmax in body frame that the missile can suffer before it breaks.

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Ref.

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Rapidly-exploring Random Trees - RRT [7]

What is RRT ? I

An incremental method designed to efficiently explore non-convex high-dimensional spaces.

Why choosing RRT ?

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I

It uses the complete model ;

I

It takes the future changes of environment into account ;

I

It can find a solution without any a priori guesses.

IFAC ACA 2013 - Würzburg, Germany

Ref.

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Rapidly-exploring Random Tree - RRT [7] RRT Algorithm

G

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Conclusions & perspectives

Ref.

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Ref.

Rapidly-exploring Random Tree - RRT [7] RRT Algorithm

uniform distribution xrand

G

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Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Ref.

Rapidly-exploring Random Tree - RRT [7] RRT Algorithm

xrand xnear (t)

Dubins’ metric

G

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Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Ref.

Rapidly-exploring Random Tree - RRT [7] RRT Algorithm

xrand xnear (t) Proportional Navigation law [14]

G

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Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Ref.

Rapidly-exploring Random Tree - RRT [7] RRT Algorithm

xrand xnew (t + ∆t) xnear (t)

Integration of complete model G with constraints, ∆t = 2s

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Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Ref.

Rapidly-exploring Random Tree - RRT [7] RRT Algorithm

xrand xnew (t + ∆t) xnear (t)

Check if xnew ∈ Xfree

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G

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Rapidly-exploring Random Tree - RRT [7] RRT Algorithm

G add xnew in G

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Conclusions & perspectives

Ref.

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

NearestNeighbor What is Dubins’ metric (based on Dubins’ curve [5][3]) ? I

Analytical method used to define the shortest path between two states while considering their orientation ; xfinal xfinal

xinit

xfinal

(i) CCC type I

xinit

xinit

(ii) CSC types

Metric function : the length of the shortest Dubins’ path.

Why choosing Dubins’ metric ?

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I

Its optimality in 2D is assured ;

I

It considers the orientations of the two states.

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Ref.

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Ref.

Results : Comparisons and Discussions (1) 1) No saturation problem 4

x 10

350

3

||av c (αmax )|| ||av kappa || ||av RRT||

phase 1 (boost) phase 2

300

2.5 250

m/s2

altitude

2 1.5

200

150

1 100

0.5 50

0 0

−2

−1.5

−1

−0.5 0 0.5 horizontal distance

1

1.5

2

0

0.1

4

x 10

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t/tf

Lateral accelerations along the solution Generated trajectories trajectories √ √ f I < π/8 and ∠(vkappa , −vpip ) < π/8 ; √ √ I ||vf || = 1807m/s > 500m/s and ||vf ; RRT kappa || = 1717m/s > 500m/s f ∠(vRRT , −vpip )

I RRT and kappa : respect the limit of control inputs. 13/18

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1

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Ref.

Results : Comparisons and Discussions (2) 2) With saturation problem 4

x 10

500

3

||av c (αmax )|| ||av kappa || ||av RRT||

450

2.5

400

350

2

m/s2

altitude

300

1.5

250

200

1 150

phase 1 (boost)

0.5

100

phase 2

50

0 0

−2

−1.5

−1

−0.5 0 0.5 horizontal distance

1

1.5

2 4

x 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t/tf

Lateral accelerations along the solution Generated trajectories trajectories √ I ∠(vf , −vpip ) < π/8 and ∠(vf kappa , −vpip ) = 0.62 > π/8X ; RRT √ I ||vf || = 1552m/s > 500m/s ; RRT I RRT anticipates the loss of manoeuvrability at the end of the trajectory but kappa doesn’t. 14/18

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1

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Conclusions

Advantages I I

Satisfaction of constraints by integrating the complete model ; It anticipates future flight conditions : I

I

for example : for the 2nd scenario, a back-turn can be found.

A solution can be found without any a priori guess.

Drawbacks

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I

Obtained solutions are not optimal ;

I

Number of iterations to find a solution can be large depending on how difficult the scenario is (ok for off-line trajectory generation).

IFAC ACA 2013 - Würzburg, Germany

Ref.

Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

Perspectives

I

Evolution of distance between RRT solution and the optimal one : I I

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maximize the final velocity ; minimize the flight time.

I

Improve the computing time by reducing X ;

I

Improve the metric and the selection of control input ;

I

2D→3D.

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Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

References I [1]

D. A. Anisi, Adaptive node distribution for on-line trajectory planning, Congress of the International Council of the Aeronautical Sciences (ICAS) (2006).

[2]

J. T. Betts, Survey of munmerical methods for trajectory optimization, Journal of Guidance, Control, and Dynamics 21 (1998), no. 2, 193–207.

[3]

J. D. Boissonnat, A. Cérézo, and J. Leblond, Shortest paths of bounded curvature in the plane, Tech. report, Institut National de Recherche en Informatique et en Automatique, 1991.

[4]

V. H. L. Cheng and N. K. Gupta, Advanced midcourse guidance for air-to-air missiles, Journal of Guidance, Control, and Dynamics 9 (1986), no. 2, 135–142.

[5]

L. E. Dubins, On curves of minimal length with a constraint on average curvature and with presribed initial and terminal position and tangents, American Journal of Mathematics 79 (1957), 497–516.

[6]

F. Imado, T. Kuroda, and S. Miwa, Optimal midcourse guidance for medium-range air-to-air missiles, Journal of Guidance, Control, and Dynamics 13 (1990), no. 4, 603–608.

[7]

S. M. LaValle, Planning algorithms, Cambridge University Press, 2006.

[8]

C. F. Lin, Modern navigation guidance and control processing, Prentice-Hall, Inc., 1991.

[9]

C. L. Lin and Y. Y. Chen, Design of fuzzy logic guidance law against high-speed target, Journal of Guidance, Control, and Dynamics 23 (2000), no. 1, 17–25.

[10] J. A. Lukacs and O. A. Yakimenko, Trajectory-shape-varying missile guidance for interception of ballistic missiles during the boost phase, AIAA (2007). [11] P. K. Menon, T. Lam, L. S. Crawford, and V. H. Cheng, Real-time computational methods for sdre nonlinear control of missiles, American Control Conference (2002), 232–237. [12] J.-W. Park, M.-J. Tahk, and H.-G. Sung, Trajectory optimization for a supersonic air-breathing missile system using pseudo-spectral method, International Journal of Aeronautical and Space Sciences 10 (2009), 112–121. 17/18

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Introduction

Problem statement

Proposed solution : RRT+Dubins’ paths

Results

Conclusions & perspectives

References II

[13] E. J. Song and M. J. Tahk, Real-time neural-network midcourse guidance, Control Engineering Practice 9 (2001), no. 10, 1145 – 1154. [14] P. Zarchan, Tactical and strategic missile guidance, second ed., Progress in Astronautics and Aeronautics, vol. 157, America Institute of Aeronautics and Astronautics, Inc. (AIAA), 1994.

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