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
S ystems
Analysis Laboratory
Helsinki University of Technology
Computationally
infeasible for
sophisticated
a/c models
Appropriate
vehicle models?
No prerequisites
about underlying
mathematical
methodologies
1.
Define the problem
2.
Coarse a/c model
3.
4.
Automated
approach
Solve a realistic near-optimal trajectory
8.
Compute initial iterate
Adjust solver parameters
No
Compute optimal trajectory
7.
Delicate a/c model
5. Inverse simulate optimal trajectory
6.
Evaluate the trajectories
S ystems
Analysis Laboratory
Helsinki University of Technology
Sufficiently
similar?
Yes
9.
Realistic near-optimal trajectory
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 – t0
1.
2.
3.
4.
5.
Set k = 0. Set the initial conditions.
Solve the optimal controls over [tk, tk + T] with direct shooting.
Update the state of the system using the optimal control at tk.
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
• 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
xk 1  xk 
t k 1
 f ( x, u, t )dt
tk
Evaluated by a numerical
integration scheme
S ystems
Analysis Laboratory
Helsinki University of Technology
xN
~
max J ( xN )
...
• Optimize the performance measure directly subject to the
constraints using a nonlinear programming solver (SNOPT)
s.t. g( xk , uk )  0
x3
x1
t1
u1
t2
u2
t3
u3
t4
u4
T
...
tN
uN
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
max J ( x N )
xN-2
xN 2  xN 2   M
x2
x2  x2  1
x2
s.t. g( xk , uk )  0
h ( xk )  0
x1
t0
u0
t1
u1
S ystems
Analysis Laboratory
Helsinki University of Technology
t2
u2
t3
u3
...
tN-1
uN-1
tN=tf
uN
Defect
constraints
5. Inverse simulate optimal trajectory
• 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 x D (tk 1 )  bu(tk ) 
Matrix of scale weights
• Solution by Newton’s method:
• Define an error function εu(tk )   Wb(u(tk ))  x D (tk 1 )

• Update scheme
u ( n 1) (t k )  u ( n ) (t k )  J 1ε u ( n ) (t k )
• With a good initial guess,
ε  0 as n  .

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
Ace software
General data panel
a/c lift coefficient profile
S ystems
Analysis Laboratory
Helsinki University of Technology
3-D animation
Numerical example
• Minimum time climb problem, t = 1 s
• Boundary conditions
h0  500 m, v0  150 m/s, h f  10000 m, v f  400 m/s
 0  0, 15, 30, 45 deg,  f free
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
Mach vs. altitude plot
S ystems
Analysis Laboratory
Helsinki University of Technology
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 Wn = 1.0
• h(tf)=9971,5 m, v(tf)=400 m/s
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