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Automated Solution of Realistic Near-Optimal Automated Solution of Realistic Near-Optimal Aircraft Trajectories Using Computational Aircraft Trajectories Using Computational Optimal Control and Inverse SimulationOptimal Control and Inverse Simulation

Janne Karelahti and Kai VirtanenHelsinki University of Technology, Espoo, Finland

John ÖströmVTT Technical Research Center, Espoo, Finland

S ystemsAnalysis LaboratoryHelsinki University of Technology

The problemThe problem

• How to compute realistic a/c trajectories?

• Optimal trajectories for various missions

• Minimum time problems, missile avoidance, ...

• Trajectories should be flyable by a real aircraft

• Rotational motion must be considered as well

• Solution process should be user-oriented

• Suitable for aircraft engineers and fighter pilots

Computationallyinfeasible forsophisticateda/c models

No prerequisitesabout underlyingmathematicalmethodologies

Appropriate vehicle models?

S ystemsAnalysis LaboratoryHelsinki University of Technology

AutomatedAutomatedapproachapproach

Solve a realistic near-optimal trajectory

Define the problem

Compute initial iterate

Compute optimal trajectory

Inverse simulate optimal trajectorySufficiently

similar?

Realistic near-optimal trajectory

Evaluate the trajectories

Adjust solver parameters

Coarse a/c model

Delicate a/c model

1.

2.

3.

4.

5.

6.

7.

8.

9.

No

Yes

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2. Define the problem2. Define the problem

• Mission: performance measure of the a/c

• Aircraft minimum time problems

• Missile avoidance problems

• State equations: a/c & missile

• Control and path constraints

• Boundary conditions

• Vehicle parameters: lift, drag, thrust, ...

Angular rate and acceleration,Load factor, Dynamic pressure, Stalling, Altitude, ...

S ystemsAnalysis LaboratoryHelsinki University of Technology

3. Compute initial iterate3. Compute initial iterate

• 3-DOF models, constrained a/c rotational kinematics• Receding horizon control based method• a/c chooses controls at

• Truncated planning horizon T << t*f – t0

tktk

1. Set k = 0. Set the initial conditions.

2. Solve the optimal controls over [tk, tk + T] with direct shooting.

3. Update the state of the system using the optimal control at tk.

4. If the target has been reached, stop.

5. Set k = k + 1 and go to step 2.

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Direct shootingDirect shooting• Discretize the time domain over the planning horizon T

• Approximate the state equations by a discretization scheme

• Evaluate the control and path constraints at discrete instants

• Optimize the performance measure directly subject to the

constraints using a nonlinear programming solver (SNOPT)

dttuxfxxk

k

t

t

kk ),,(1

1

t1

u1

t2

u2

t3

u3

t4

u4

tN

uN

...

x1

x3

xN...

T

Evaluated by a numericalintegration scheme

0g ),( s.t.

)(~

max

kk

N

ux

xJ

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4. Compute optimal trajectory4. Compute optimal trajectory

• 3-DOF models, constrained a/c rotational kinematics

• Direct multiple shooting method (with SQP)

• Discretization mesh follows from the RHC scheme

0h

0g

)(

),( s.t.

)(max

k

kk

N

x

ux

xJ

t0

u0

t1

u1

t2

u2

t3

u3...

x1

x2

xN-2

tN=tf

uN

tN-1

uN-1

2x 122 xx

MNN xx 22

Defectconstraints

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• 5-DOF a/c performance model

• Find controls u that produce the desired output history xD

• Desired output variables: velocity, load factor, bank angle

• Integration inverse method

• At tk+1, we have

• Solution by Newton’s method:• Define an error function

• Update scheme

• With a good initial guess,

5. Inverse simulate optimal trajectory5. Inverse simulate optimal trajectory

)())(()( 1 kDkk ttt xubWuε

)()( 1 kkD tt ubx

. as 0 nε )()()( )(1)()1(

kn

kn

kn ttt uεJuu

Matrix of scale weights

Jacobian

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• Compare optimal and inverse simulated trajectories• Visual analysis, average and maximum abs. errors

• Special attention to velocity, load factor, and bank angle

• If the trajectories are not sufficiently similar, then• Adjust parameters affecting the solutions and recompute

• In the optimization, these parameters include• Angular acceleration bounds, RHC step size, horizon length

• In the inverse simulation, these parameters include• Velocity, load factor, and bank angle scale weights

6. Evaluation of trajectories6. Evaluation of trajectories

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Example implementation: AceExample implementation: Ace• MATLAB GUI: three panels for carrying out the process• Optimization + Inverse simulation: Fortran programs• Available missions

• Minimum time climb• Minimum time flight• Capture time• Closing velocity• Miss distance• Missile’s gimbal angle• Missile’s tracking rate• Missile’s control effort

• Vehicle models: parameters stored in separate type files• Analysis of solutions via graphs and 3-D animation

Missile vs. a/c pursuit-evasion

Missile’s guidance laws:Pure pursuit,Command to Line-of-Sight,Proportional Navigation(True, Pure, Ideal, Augmented)

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Ace softwareAce software

General data panel

a/c lift coefficient profile 3-D animation

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Numerical exampleNumerical example• Minimum time climb problem, t = 1 s

• Boundary conditionsm/s 400 m, 10000 m/s, 150 m, 500 00 ff vhvh

free deg, 45 ,30 ,15 ,00 f

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Numerical exampleNumerical example

• Case 0=0 deg

• Inv. simulated:

Mach vs. altitude plot

m/s 400)(

m 2.9841)(

s 06.97

f

f

f

tv

th

t

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Numerical exampleNumerical example

• Case 0=0 deg, average and maximum abs. errors

Velocity histories

0.07n ,01.0 m/s, 30.8 m/s, 44.2 nvv

Load factor histories

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Numerical exampleNumerical example• Make the optimal trajectory easier to attain

• Reduce RHC step size to t = 0.15 s

• Correct the lag in the altitude by increasing Wn = 1.0

• h(tf)=9971,5 m, v(tf)=400 m/s

S ystemsAnalysis LaboratoryHelsinki University of Technology

Numerical exampleNumerical example

• Case 0=0 deg, average and maximum abs. errors

Velocity histories

0.045n ,003.0 m/s, 00.2 m/s, 63.0 nvv

Load factor histories

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ConclusionConclusion

• The results underpin the feasibility of the approach

• Often, acceptable solutions obtained with the default settings

• Unsatisfactory solutions can be improved to acceptable ones

• 3-DOF and 5-DOF performance models are suitable choices

• Evaluation phase provides information for adjusting parameters

• Ace can be applied as an analysis tool or for education

• Aircraft engineers are able to use Ace after a short introduction