S ystems Analysis Laboratory Helsinki University of Technology Near-Optimal Missile Avoidance Trajectories via Receding Horizon Control Janne Karelahti, Kai Virtanen, and Tuomas Raivio Systems Analysis Laboratory Helsinki University of Technology, Finland

S ystems Analysis Laboratory Helsinki University of Technology Goal: avoid a closing missile Criterion: max capture time min closing velocity max miss distance max guidance effort max gimbal angle max tracking rate Criterion: max capture time min closing velocity max miss distance max guidance effort max gimbal angle max tracking rate Controls wanted in a feedback form: Receding Horizon Control + cost-to-go approximation Controls wanted in a feedback form: Receding Horizon Control + cost-to-go approximation Select the most suitable criterion Select the most suitable criterion

S ystems Analysis Laboratory Helsinki University of Technology Problem overview Some physical constraints of the missile system: Lags in the missile guidance system dynamics Seeker head’s gimbal angle limit Seeker head’s tracking rate limit Limited energy supply of the missile control system Target aircraft Gimbal angle

S ystems Analysis Laboratory Helsinki University of Technology Problem overview Assumptions The vehicles receive perfect state information about each other The vehicles are modeled as 3-DOF point-masses Target aircraft’s angular velocities and accelerations are limited The missile utilizes proportional navigation The missile has a single lag guidance system

S ystems Analysis Laboratory Helsinki University of Technology Optimal Control Problem The target aircraft minimizes/maximizes subject to State equations Control/state constraints Terminal constraint

S ystems Analysis Laboratory Helsinki University of Technology Receding horizon control scheme The target makes decisions at Let’s define and 1.Set k = 0. Set the initial state x 0 and initial controls u 0. 2.Solve the optimal controls over [ t k, t k +T ] by min./maximizing 3.Set and solve by implementing for 4.If the missile has reached its target set, stop. 5.Set k = k + 1 and go to step 2.

S ystems Analysis Laboratory Helsinki University of Technology Criteria & ctg. approximation 1)Capture time: 2)Closing velocity: 3)Miss distance: 4)Control effort: 5)Gimbal angle: 6)Tracking rate:

S ystems Analysis Laboratory Helsinki University of Technology Optimal controls over t k +T The direct shooting method: Time discretization Explicit integration of the state at by using The resulting NLP problem is solved by SNOPT SQP-solver

S ystems Analysis Laboratory Helsinki University of Technology Numerical results Capture time maximization:

S ystems Analysis Laboratory Helsinki University of Technology Numerical results Miss distance maximization:

S ystems Analysis Laboratory Helsinki University of Technology Numerical results Capture time maximizationMiss distance maximization

S ystems Analysis Laboratory Helsinki University of TechnologyConclusions The scheme provides near-optimal feedback controls in virtually real-time Launch state maps provide a way for selecting the most effective performance measure for the current state Additional research topics: Evaluation of the realism of the optimal trajectories by inverse simulation Uncertainty about the missile establishes a need for automatic identification of the missile parameters (e.g. guidance law)