1、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, FinlandJohn strm VTT Technical Research Center, Espoo, Finland,The problem,How to compute realistic
2、 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,Computat
3、ionally infeasible for sophisticated a/c models,No prerequisites about underlying mathematical methodologies,Appropriate vehicle models?,Automated approach,Solve a realistic near-optimal trajectory,Define the problem,Compute initial iterate,Compute optimal trajectory,Inverse simulate optimal traject
4、ory,Sufficiently 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,2. Define the problem,Mission: performance measure of the a/c Aircraft minimum time problems Missile avoidance problems
5、 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, .,3. Compute initial iterate,3-DOF models, constrained a/c rotational kinematics Receding horiz
6、on control based method a/c chooses controls at Truncated planning horizon T t*f t0,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
7、+ 1 and go to step 2.,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 Optimize the performance measure directly subject to the constraints using a nonlinea
8、r programming solver (SNOPT),t1 u1,t2 u2,t3 u3,t4 u4,tN uN,.,x1,x3,xN,.,T,Evaluated by a numerical integration scheme,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,t0 u0,t1
9、u1,t2 u2,t3 u3,.,x1,x2,xN-2,tN=tf uN,tN-1 uN-1,Defect constraints,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 Newtons method: Define an error
10、 function Update scheme With a good initial guess,5. Inverse simulate optimal trajectory,Matrix of scale weights,Jacobian,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
11、 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
12、,6. Evaluation of trajectories,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 Missiles gimbal angle Missiles tr
13、acking rate Missiles control effort Vehicle models: parameters stored in separate type files Analysis of solutions via graphs and 3-D animation,Missile vs. a/c pursuit-evasionMissiles guidance laws: Pure pursuit, Command to Line-of-Sight, Proportional Navigation (True, Pure, Ideal, Augmented),Ace so
14、ftware,General data panel,a/c lift coefficient profile,3-D animation,Numerical example,Minimum time climb problem, Dt = 1 s Boundary conditions,Numerical example,Case g0=0 deg Inv. simulated:,Mach vs. altitude plot,Numerical example,Case g0=0 deg, average and maximum abs. errors,Velocity histories,L
15、oad factor histories,Numerical example,Make the optimal trajectory easier to attain Reduce RHC step size to Dt = 0.15 s Correct the lag in the altitude by increasing Wn = 1.0 h(tf)=9971,5 m, v(tf)=400 m/s,Numerical example,Case g0=0 deg, average and maximum abs. errors,Velocity histories,Load factor
16、 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,
