1. S ystems Analysis Laboratory Helsinki University of Technology 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

2. 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 No prerequisites about underlying mathematical methodologies Appropriate vehicle models?

3. S ystems Analysis Laboratory Helsinki University of Technology Automated approach Solve a realistic near-optimal trajectory Define the problem Compute initial iterate Compute optimal trajectory Inverse simulate optimal trajectory 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

4. 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 Boundary conditions Vehicle parameters: lift, drag, thrust,... Angular rate and acceleration, Load factor, Dynamic pressure, Stalling, Altitude,...

5. 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 Truncated planning horizon T << t * f – t 0 1.Set k = 0. Set the initial conditions. 2.Solve the optimal controls over [t k, t k + T] with direct shooting. 3.Update the state of the system using the optimal control at t k. 4.If the target has been reached, stop. 5.Set k = k + 1 and go to step 2.

6. 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 Optimize the performance measure directly subject to the constraints using a nonlinear programming solver (SNOPT) t1u1t1u1 t2u2t2u2 t3u3t3u3 t4u4t4u4 tNuNtNuN... x1x1 x3x3 xNxN T Evaluated by a numerical integration scheme

7. 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 t0u0t0u0 t1u1t1u1 t2u2t2u2 t3u3t3u3... x1x1 x2x2 x N-2 t N =t f u N t N-1 u N-1 Defect constraints

8. S ystems Analysis Laboratory Helsinki University of Technology 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 Solution by Newton’s method: Define an error function Update scheme With a good initial guess, 5. Inverse simulate optimal trajectory Matrix of scale weights Jacobian

9. S ystems Analysis Laboratory Helsinki 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 trajectories

10. 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 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)

11. S ystems Analysis Laboratory Helsinki University of Technology Ace software General data panel a/c lift coefficient profile 3-D animation

12. S ystems Analysis Laboratory Helsinki University of Technology Numerical example Minimum time climb problem,  t = 1 s Boundary conditions

13. S ystems Analysis Laboratory Helsinki University of Technology Numerical example Case  0 =0 deg Inv. simulated: Mach vs. altitude plot

14. S ystems Analysis Laboratory Helsinki University of Technology Numerical example Case  0 =0 deg, average and maximum abs. errors Velocity historiesLoad factor histories

15. S ystems Analysis Laboratory Helsinki University of Technology 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 = 1.0 h(t f )=9971,5 m, v(t f )=400 m/s

16. S ystems Analysis Laboratory Helsinki University of Technology Numerical example Case  0 =0 deg, average and maximum abs. errors Velocity historiesLoad factor histories

17. S ystems Analysis Laboratory Helsinki University of TechnologyConclusion 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