4958123 Decision and Control Problems in Missile Design

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Decision and control problems in missile designStéphane Le Ménec

« Autour des PBs de PB »

INRIA / Sophia-AntipolisTuesday / Wednesday, March, the 29th and the 30th



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Summary

• Decision support system in medium range air to air combat game

• Dynamic game

• Sub games, differential game barriers

• State trees

• Reprisal strategies• Markov chains

• High level management for Anti Tactical Ballistic Missiles launch

• 2 types of missiles : observers / interceptors

• Static game

• Matrix game, mixed strategies

• Bank To Turn (BTT) Optimal Guidance Laws

• In flight error estimation and compensation- Extended Kalman Filter design

• Steering law design

- Guidance Law Optimization (differential game aspect)Presentation



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Summary

• Bank To Turn (BTT) Optimal Guidance Laws• Overview on Missile Architecture Design

- Guidance, Navigation & Control (GNC)

- Missile Symmetry

- Steering Law

• Constrains and Errors

- Constrains (Acceleration, Stability, Time Response)

- Errors (Radome Aberration, Misalignments)

- Simulation Example

• Guidance and Control (roll gain) Optimization

- Proportional Navigation

- Optimal Proportional Navigation

- Differential Game Point of View

- BTT Guidance Laws

- Some Results



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Navigation, Guidance and Control

• Navigation is knowing where you are• Guidance creates the commands to take you from where you are to

where you want to be

•Control follows the commands

Guidance

+

-

ControlGuidance Control

Navigation

End Point

A t il t F ti / F Fli ht C t l (FFC)



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Autopilot Functions / Free Flight Control (FFC)

hControls the missile to follow manoeuvre demands

(within prescribed limits)

• over the missile operating envelope

• allowing for external disturbances

• allowing for variations in system characteristics(unknown but bounded)

Guidance

Law

Steering

LawAutopilot Actuators

Seeker/

Estimator Airframe

Relative

Kinematics

Instruments

Target

Motion

different strategies : roll the missile as an airplane

or angle of attack and side slip controls

S t i i il



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Symmetric missiles

hGuidance Law determines the required missile trajectory / demands

hThe steering law determines how this is achieved (interface between

guidance and autopilot). This depends on the airframe / system

configuration

Rocket Powered ≈ symmetric ⇒ Skid-to-Turn steering law

(+ possibly roll-control)

As mmetric missiles



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Asymmetric missiles

Separation Control

Separation Control Midcourse

GuidanceFree Flight

Control

Homing Guidance

Cartesian State

Estimation

Ram-Jet Powered, asymmetric, low side-slip required (engine constrains)

⇒ Bank-to-Turn steering law required

also Bank-While-Turn & STT modes

Basic Steering / Control laws



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• Skid-To-Turn (STT)

- airframe has symmetry in two axes (cruciform) and is equally capable of

being controlled in pitch and yaw.

- May be roll controlled (or not) depending on the application.

• Bank-To-Turn (BTT)

- asymmetrical airframe and has more control capability in the pitch plane.

- if control is required in another plane then the airframe must first be

rolled into the plane where it can use its pitch control. Thus, this

airframe must have roll control.

• Bank-While-Turning (BWT)

- similar to BTT but with active acceleration control in both planes

Basic Steering / Control laws

Basic Steering / Roll Control



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Basic Steering / Roll Control

φc=atan2(ayNRc, -azNRc)

BTT

φ

azRc= ac

φc=atan2(ayNRc, -azNRc)

BWT

φ

azRc

ayRc

φc=φ, 0STT

φ

azRc

ayRc

φc

ayNRc

azNRc

ac

commands in

non-rolling axes

Guidance and autopilots

Constrains on acceleration demands (STT)



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Constrains on acceleration demands (STT)

Constrains on acceleration demands (BTT)



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Constrains on acceleration demands (BTT)

Stability constrains on roll (rate) demands



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Stability constrains on roll (rate) demands

• The roll chain is designed faster than pitch and yaw

• to control acceleration in the right plane in BTT,

• to reduce side slip in BWT

• Errors on boresight angle measurements (see following slides) required

• to slow down the autopilot (more particularly to decrease the roll gain G1R)

to avoid FFC instabilities

• to decrease the guidance law gain (in particular PN gain)

• BTT more sensitive to boresight errors when• roll lag time constant small

(stability limit smaller when G1R large)

• ATd (FFC acceleration demand) small

⇒ no fast roll demands when ATd small

• Incidence lag (airframe response) large

• V M (missile velocity) small

Constrains on time response



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Constrains on time response

• The STT mode has smaller time response than BTT

BTT requires first to roll « before » applying pitch control demands.

• When Time To Go (tf – t) is small

in comparison with BTT autopilot lag time constant,

prefer STT mode

• Just before interception, the ram jet engine constrains do not apply anymore, less

restriction on side slip angle

end interception in BTT

⇒ Steering law rules

Proportional Navigation (PN) Trajectory



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target

missile

Proportional Navigation (PN) Trajectory

• Sightline angle (and rate) vary at the start of engagement

• Later the sightline moves parallel to itself so the sightline rate → 0

and a shrinking interception triangle is produced

Homing Guidance

// Lines Of Sight (LOS)

PN guidance loop



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g p

Missile/Target

Geometry Seeker Guidance

Law Autopilot

Target

Motion

Sightline

Angle

Sightline

Rate

Demanded

Accln

Achieved

Accln

Missile

Motion

MissileKinematics

• We aim for a final missile straight line motion with no sightline rate (to reach

the collision course triangle). We therefore define:

lateral acceleration (latax) demand = λ .V.ωs

where the sightline rate ωs is measured by the missile seeker

and λ is a constant (called navigation constant, gain, kinematic stiffness …)

and V a speed term

• more efficient PNs exist as

• Ideal PN (with missile longax compensation for accelerating missile) and

• Augmented PN (APN) for an accelerating target

Boresight errors / radome aberration



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g

• Radome aberration errors

•The radome is designed to protect the dish from airflow damages

• The seeker (dish) is theoretically decoupled from missile movements, but …

• Missile attitude changes imply fictive target movements (of apparent LOS)

• Radome aberration (RA) couples LOS measurements (Elevation, Circular) to

missile attitude through gimbal angles

E kgeC kgc

C ksc E kse

trueC measured C

true E measured E

..

..

++=

++=

ε ε

ε ε

antenna

true target

apparent target

radome

radome aberration angle

LOS

RA errors

Boresight errors / misalignments effects



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g g

• Inertial Measurement Unit (IMU) and Seeker misalignments

• By construction, offsets exist between IMU axis and body axis

• In the same way, the dish reference axis are not perfectly linked to body axis

⇒Errors in LOS angle and LOS rate reconstruction

Due to errors on the transfer matrix between dish, body and IMU coordinate frames

Errors occurs when no gyroscope on seeker axis (in modern missiles)

In case of 3D target maneuvers (or BTT modes)

(E)angleGimbal

2

attack)of (angle

axisDish

LOS

axis)(RadomeBody

IMU

vector)(velocityM V

measured

dish D :

1

pitch:

Reference

axisdishReference

12truemeasured computed E

Guidance Law Optimization (STT / linear model)



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• Approximation around the collision course triangle (linearization)

• Unbounded controls (penalty function through quadratic criterion)

- Linear Quadratic Theory (Bryson, Ho and all)

• PN, APN optimal under target maneuver assumptions

• OGL : first order assumption for missile and target dynamics

- Differential game point of view (Ben Asher and all)

• Trade of between miss distance and target acceleration capabilities

taking into account

• Missile lag time constant robustness,

• Imperfect target acceleration estimation

• Bounded controls, miss distance as criterion (Prof. J. Shinar and all)

- DGL/0, DGL/1 (Differential Game, 1 : first order target dynamics),

- DGL/C (delayed estimation), DGL/S (IR seeker)

Guidance Law Optimization (STT / BTT)



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• (Kinematics without small angle approximation (Prof. J. Shinar and all)

• 2D dynamics (STT context), non complete analytic solution

• Interesting for initial non collision course conditions

• BTT optimal guidance law

•Collision course assumption

• Unbounded controls approach extension (Aggarwal and all)

- Roll rate optimization

• Lyapunov approach

- Roll angle demand

Differential Game Guidance Law



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• Kinematics

• First order lag time constant for Pursuer and Evader

T M D B X A X Γ+Γ+=&

• Criterion

• 3D Guidance law expression

( )dt t yb

J

f t

T Md f ∫ Γ−Γ+=0

2222

2

1)(

2maxmin γ

term only in the one-sided optimal solution

or when adding in the game a constant maneuver

(in both case, parameter to estimate : Kalman filter)

),(',

)2

('

t t f cc

cciV c

f pilot

T M iMT icicM

−=

Γ⋅+Γ⋅+∧Ω=Γ

τ

rrrrr

Singularity checking imply a trade off

between miss distance and target maneuverability

Results on 2D simulation



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• Linear Kalman Filter to estimate LOS rate and target acceleration (for APN and OGL)

• Radome aberration without compensation (+ thermal noise, + white noise)

• APN and OGL are sensitive to bad target acceleration estimations• OGL with high guidance gain (→ ∞ at tf ) is sensitive to errors

• LQ Diff. Game : no target acceleration required, finite guidance gain

0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.0

0

5

0

5

0

5

0

5

0

5

0

miss distance [m]

radome slope R

APN

PN

LQDG

LQG

stability

small

non null miss distance

Results using a Generic 6dof simulation



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• Gen6DOF in Matlab / Simulink

• Target point mass model

• Extended Kalman Filter to estimate LOS rate and compensate radome

aberration

• 3D LQDG

• STT and BTT steering law (Steering Law : atan2)

• Results with and without errors (Radome aberrations)

• Results with different amount of Radome aberration errors

• Sensitivity larger in BTT than in STT

BTT Optimal Guidance Laws



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• Kinematics

+++= ∫ f t

t

d ca f r f r dt w Awt z t y J

0

)()()(2

1 2222 φ φ

&&

• Criterion

− Hamiltonian -> two point boundary value problem− Near optimal solution using singular perturbation method (separation

between the slow and the fast mode)

− Solution similar to OGL + control

− Game formulation as in STT ?

d φ &

BWT

φazRc

ayRcy

z

d

c

T

z z

T

y y

z r

yr

A A A

A Av

A Av

v z

v y

φ φ

ι

φ

φ &&

&

&

&

&

&

=

−=

−=

−=

=

=

sin

cos

relative positions

In inertial axis

relative velocity

In inertial axis

Conclusion



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• Long range air to air missiles will be powered by ramjets

• Ramjets tolerate small sideslip angles and small negative angle of attack

• The pitch-yaw-roll dynamics is a nonlinear (coupling effect)

• BTT / BWT roll rate has to be controlled carefully to avoid instabilities

• Most of optimal guidance laws were designed in STT context

• Steering Law optimization required

• High guidance / autopilot gains are required against highly maneuverable targets

• Low gains are required for stability reasons, errors remains

• in LOS rate estimation (thermal noise for radar seeker, radome aberration, misalignments)

• and in target acceleration estimation (delays, errors)

• First results are obtained using STT optimal guidance laws (differential game

versions)

• by design of adapted gains

•and by avoiding target acceleration estimation

4958123 Decision and Control Problems in Missile Design

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