Initialisation
At T=0s, the distribution of predicted target trajectories will be calculated from an initial target state with components [x, vx, y, vy]0. Each predicted target tra-
jectory will be calculated for a maximum weapon flight duration of 40s. For the optimisation algorithm to be effective, each random weapon trajectory used to determine the intercept probability must have a scan area which overlaps part of the predicted target trajectory distribution. If the target trajectory distribution is compared with the reachable set of the weapon, it is apparent that it will only occupy a very small part of the reachable set. An example of this comparison is shown in Figure 5.9 with an initial target state of [10000,-26,350,0]’.
The target distribution in this case is actually bounded by an area defined from a trajectory with an off-boresight angle of 9◦ which is applied to the weapon at T=0s, and a trajectory with an off-boresight angle of−5◦ which is applied at the same time step. Only trajectories where the weapon seeker area will be enclosed by these boundaries over the distribution of predicted target trajectories will be capable of achieving an intercept probability. The area bounded by two trajecto- ries calculated from respective off-boresight commands applied at launch can be defined as the feasible trajectory solution area.
Figure 5.9: Feasible Trajectory Solution Area
The first stage of the weapon initialisation process in the fire control system is to calculate the feasible solution area for each target trajectory distribution which is performed as follows:
1. The reachable set of the weapon is overlaid on the distribution of predicted target trajectories. The number of individual target locations which form each predicted target trajectory, that are enclosed by the boundaries of the reachable set is determined.
2. The reachable set is bounded by trajectories with respective initial off- boresight commands of±40◦ which are programmed into the weapon while on the launcher.
3. The number of target locations is then re-evaluated based on a reachable set which is bounded by weapon trajectories with off-boresight commands of 39◦ and −40◦. If the number of target locations is the same, the process is repeated reducing the positive off-boresight command trajectory in steps of 1◦, i.e. producing a reachable set defined by 38◦, −40◦, 37◦, −40◦. The process continues until the number of target locations is below the initial value defined from a full reachable set.
4. The process is then performed considering a reachable defined by the bound- aries of off-boresight trajectories 40◦, −39◦, the same procedure is followed
but with with the negative off-boresight command increased in steps of 1◦, i.e 40◦,−38◦ etc.
The final off-boresight limits which define the feasible solution area are then±1◦ of the calculated boundaries in the respective direction. Once the feasible solution area has been calculated, 3420 weapon trajectories are randomly generated based on 4 off-boresight commands which are applied at T=0s, T=10s, T=20s, T=30s. In order for the random trajectories to lie within the feasible solution area, each off-boresight command is then generated between the limits of the feasible solu- tion area. An example of the calculation of the random weapon trajectories is shown in Figure 5.10. −2000 0 2000 4000 6000 8000 10000 12000 −1000 −500 0 500 1000 1500 X/m Y/m
Missile Trajectory XY Data P.T T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 1 2 3 4 5 6 7 8 9 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Trajectory Number Probability of Intercept
(b) − Probability of Intercept for 10 Random Weapon Trajectories
Figure 5.10: Feasible Trajectory Solution Area
trajectories are then calculated by generating 4 off-boresight commands which will each have a random value between−5◦ and 9◦. The off-boresight commands will be applied at T=0s, T=10s, T=20s and T=30s. 10 example trajectories generated under these conditions are shown in Figure 5.10 (a). In Figure 5.10 (b) the probability of intercept obtained for each of the 10 generated trajectories is displayed. By generating trajectories using these off-boresight constraints, each weapon trajectory has an associated target intercept probability >0.
Once the random weapon trajectories have been generated, the last stage of the weapon initialisation phase is to run the simulated annealing optimisation algorithm to determine the initial optimal weapon trajectory.
The simulated annealing algorithm will optimise the trajectory based on the tar- get trajectory distribution up to a prediction time of 35s as this is the maximum time by which it can confidently be assumed that the seeker will actually be on. The assumption is made that provided a target detection is made up to 35s, a successful interception will occur within the remaining 5s of the 40s flight time of the weapon.
The results of a simulated annealing optimisation run for this particular dis- tribution of target trajectories are presented in Figure 5.11
0 500 1000 1500 2000 2500 3000 3500 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Update Run Probability of intercept
Current Probability of Intercept Best Probability of Intercept found
Figure 5.11: Typical Simulated Annealing Results at Weapon Initialisation
The probability of intercept for the initial optimal weapon trajectory will be fairly small at T=0s as indicated by Figure 5.11. This is due to a significant predic- tion into the future of the target behaviour, resulting in a large distribution of possible target trajectories with associated lower probability peaks. As more in- formation is obtained about the target through manoeuvre detections, possible target trajectories will be eliminated resulting in a smaller probability distribu- tion with larger peaks. The simulation begins with the first off-boresight angle programmed into the weapon at T=0s.