janne karelahti and kai virtanen helsinki university of technology, espoo, finland john Öström
DESCRIPTION
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. The problem. - PowerPoint PPT PresentationTRANSCRIPT
S ystemsAnalysis LaboratoryHelsinki University of Technology
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
S ystemsAnalysis LaboratoryHelsinki University of Technology
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
S ystemsAnalysis LaboratoryHelsinki University of Technology
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
S ystemsAnalysis LaboratoryHelsinki University of Technology
• 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
S ystemsAnalysis LaboratoryHelsinki University of Technology
• 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
S ystemsAnalysis LaboratoryHelsinki University of Technology
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
S ystemsAnalysis LaboratoryHelsinki University of Technology
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
S ystemsAnalysis LaboratoryHelsinki University of Technology
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