Current Ballistic Missile (BM) defence systems intercept an incoming threat in its terminal phase, and to some extend during its coast phase. During the boost phase of a BM, the initial velocity of the BM is low and the hot exhaust plume is easy to spot for infrared sensors, which provides a good opportunity for interception. To do so, the Network Centric Airborne Defense Element (NCADE) is a missile in development, which is to be launched from a 5$^{\text{th}}$ generation fighter aircraft. A tracking algorithm model has already been developed at TNO, which performs the tracking and trajectory prediction of the BM, based on the location measurements performed by the IR sensors of both the aircraft and the missile. Tracking a BM and predicting its trajectory poses a challenge, as the mass, thrust and acceleration cannot be measured directly, resulting in an uncertainty in the trajectory prediction. The NCADE therefore must be able to intercept the BM in its boost phase, considering the uncertainties in the trajectory prediction. The main research goal of this thesis is to develop a guidance algorithm for the NCADE, which plans a trajectory of an air launched missile, intercepting a BM in its boost phase. To do so, the NCADE has been modelled, and the tracking algorithm has been implemented. The actual guidance is performed using trajectory optimisation.
The NCADE is designed to disable its target by means of kinetic penetration, meaning that there is no explosive warhead present. The missile is derived from the AIM120D AMRAAM, with a similar outer shape and suspension points. The NCADE consists of two stages, where the first stage is equipped with a solid booster for a fast acceleration. Control deflections are provided with aerodynamic surfaces. The second stage, also called Kill Vehicle (KV), is equipped with an IR sensor to determine the location of the target. Some sensor inaccuracy is present due to the amount of pixels used in the sensor. Control inputs on the second stage are performed using monopropellant pulses. Both control inputs and thrust of the KV use monopropellant from the same source, meaning that when the monopropellant tank is depleted, both control deflections and thrust cannot be delivered. To calculate different trajectories of the NCADE, the equations of motion are set up, where the NCADE is modelled as a 3 degrees of freedom point mass. Aerodynamic coefficients are obtained using software applying empirical methods, for which an extended database of projectiles is available. Verification of the equations is performed using a validated generic missile model, made by TNO using Simulink.
The calculation of the guidance relies on a location of the target in the future. Therefore, a trajectory prediction must be performed, for which the states of the BM must be determined. There are however only position measurements of the BM available, from which a more extended set of states of the BM must be derived. This is performed using an Extended Kalman Filter (EKF), which was developed during an earlier study. The filter initiates with a guess of the states of the BM, and continuously updates those as new readings of the BM become available. Using the Kalman states, a trajectory prediction is performed. The Kalman states require a certain tracking period to converge to the correct values, to be able to calculate usable trajectory predictions. The quality of the tracking prediction is quantified by a score, which is forwarded to the guidance algorithm to be able to take the significant uncertainties in the trajectory prediction into account. The certainty score improves when the tracking duration increases and when the trajectory prediction is nearby in the future. The certainty results are forwarded to the guidance algorithm by means of coefficients of a polynomial.
Due to the complexity of the control system of the NCADE and the uncertain target trajectory, trajectory optimisation is applied in the guidance algorithm. Trajectory optimisation aims to decrease the defined performance index, which is in this case the divert cone minus the uncertainty ellipse of the trajectory prediction, to maximise the probability for interception. The divert cone of the NCADE is a volume which the missile is able to reach on a certain time, given its states and reserve fuel. The divert cone is calculated using a separate shooting optimisation algorithm, which maximises the distance in three ENU frame directions. To maximise the probability of interception, a shooting method is applied, which uses candidate solutions in the form of functions describing the control input, to calculate the performance index. Using constraints, the missile is directed towards an interception point. Constraints are also applied to bound the magnitude of the controls, and at the trajectory itself to remain physical feasible.
To investigate the behaviour of the guidance algorithm, simulations of interceptions of the NCADE have been performed on a modelled Scud BM, using a range of launch locations and tracking settings. The optimum results are presented in control deflection functions of the NCADE, which achieve the flight with the minimised performance index. When only the tracking uncertainty is minimised, flight time is minimised, as the prediction becomes less reliable when a longer trajectory prediction is performed. When only the size of the divert cone is to be maximised, the propellant of the sustainer, used for control and propulsion, is saved to increase the divert cone. Because of this, the altitude of the flight is increased, and $t_{f}$ must become larger, because less monopropellant is applied to increase the velocity at the beginning of the sustain phase. The optimum solution is a compromise between the divert cone size and time to flight. As the duration of the tracking time increases, the target trajectory prediction becomes more reliable, so the maximisation of the divert cone becomes more prominent. However, this results in the maximum range to decrease, since there is less time for interception, and the altitude of the target has increased. When the launch location is positioned further from the target, the reduction of time to flight becomes more prominent and the divert cone decreases. In conclusion, the optimisation routine performs the compromise between the amount of reserve propellant available, and the uncertainty of the trajectory prediction.