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

Janne Karelahti and Kai Virtanen

Helsinki University of Technology, Espoo, Finland

John Öström

VTT Technical Research Center, Espoo, Finland

S ystems Analysis Laboratory Helsinki University of Technology

The 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 Computationally infeasible for sophisticated a/c models Appropriate vehicle models?

No prerequisites about underlying mathematical methodologies

S ystems Analysis Laboratory Helsinki University of Technology

1.

Solve a realistic near-optimal trajectory

Automated approach

3.

2.

Define the problem Coarse a/c model Compute initial iterate 8.

Adjust solver parameters 5.

4.

Compute optimal trajectory Delicate a/c model Inverse simulate optimal trajectory 6.

Evaluate the trajectories No 7.

Sufficiently similar?

9.

Yes Realistic near-optimal trajectory

S ystems Analysis Laboratory Helsinki University of Technology

2. 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

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

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

S ystems Analysis Laboratory Helsinki University of Technology

3. Compute initial iterate

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

t k



k



t

Truncated planning horizon

T << t * f – t

0 1.

2.

3.

4.

5.

Set

k

= 0. Set the initial conditions.

Solve the optimal controls over [

t k , t k + T

] with direct shooting .

Update the state of the system using the optimal control at

t k

.

If the target has been reached, stop.

Set

k = k

+ 1 and go to step 2.

S ystems Analysis Laboratory Helsinki University of Technology

Direct shooting

x k

 1  • 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)

x k



t k t k

  1

f

(

x

,

u

,

t

)

dt x N

max ~

J

(

x N

) s.t.

g

(

x k

,

u k

) 

0

Evaluated by a numerical integration scheme

x

3

x

1

t

1

u

1

t

2

u

2

t

3

u

3

t

4

u

4

...

t N u N T

S ystems Analysis Laboratory Helsinki University of Technology

4. Compute optimal trajectory

• • • 3-DOF models, constrained a/c rotational kinematics Direct multiple shooting method (with SQP) Discretization mesh follows from the RHC scheme

x N-2 x

2

x

2

x

1

t

0

u

0

S ystems Analysis Laboratory Helsinki University of Technology

t

1

u

1

t

2

u

2

x

2 

x

2   1

t

3

u

3

...

x N

 2 

x N

 2  

M

max

J

(

x N

) s.t.

g

(

x k

,

u k

)

h

(

x k

) 

0



0

t N-1 u N-1 t N =t f u N

Defect constraints

5. Inverse simulate optimal trajectory

• 5-DOF a/c performance model • Find controls

u

that produce the desired output history

x

D

• Desired output variables: velocity, load factor, bank angle • Integration inverse method • At

t k+1

, we have

x

D

(

t k

 1 ) 

b



u

(

t k

)  Matrix of scale weights • Solution by Newton’s method: • Define an error function

ε



u

(

t

• Update scheme • With a good initial guess,

u

(

ε

n

 1 )

k

 (

t k

) )   

u W



b

0 as

n

(

n

) (

t



k

(

u

(

t k

)  

J

))  1

ε

 

x u

 .

(

D n

) (

t

(

t k k

)  1  )  Jacobian

S ystems Analysis Laboratory Helsinki University of Technology

6. Evaluation of trajectories

• 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

S ystems Analysis Laboratory Helsinki University of Technology

Example 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

Missile vs. a/c pursuit-evasion Missile’s guidance laws:

Pure pursuit, Command to Line-of-Sight, Proportional Navigation (True, Pure, Ideal, Augmented) • Vehicle models: parameters stored in separate type files • Analysis of solutions via graphs and 3-D animation

S ystems Analysis Laboratory Helsinki University of Technology

General data panel a/c lift coefficient profile

S ystems Analysis Laboratory Helsinki University of Technology

3-D animation

Ace software

Numerical example

• Minimum time climb problem, 

t

= 1 s • Boundary conditions

h

 0 0   500 m,

v

 150 0 0 , 15 , 30 , 45 deg, 

f

m/s, free

h f

 10000 m,

v f

 400 m/s

S ystems Analysis Laboratory Helsinki University of Technology

Numerical example

• Case  0 =0 deg • Inv. simulated:

t f

 97 .

06 s

h

(

t f

)  9841 .

2 m

v

(

t f

)  400 m/s

S ystems Analysis Laboratory Helsinki University of Technology

Mach vs. altitude plot

Numerical example

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

v

 2 .

44 m/s, 

v

 8 .

30 m/s, 

n

 0 .

01 ,  n  0.07

Velocity histories

S ystems Analysis Laboratory Helsinki University of Technology

Load factor histories

Numerical 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

W n h

(

t f

)=9971,5 m,

v

(

t f

)=400 m/s = 1.0

S ystems Analysis Laboratory Helsinki University of Technology

Numerical example

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

v

 0 .

63 m/s, 

v

 2 .

00 m/s, 

n

 0 .

003 ,  n  0.045

Velocity histories

S ystems Analysis Laboratory Helsinki University of Technology

Load factor histories

Conclusion

• 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

S ystems Analysis Laboratory Helsinki University of Technology