Impact Angle Constrained Guidance Using Cubic Splines

Abstract

In this thesis the cubic spline guidance law and its variants are derived. A detailed analysis is carried out to find the initial conditions for successful interception. The results are applied to three dimensional guidance design and for solving waypoint following problems. The basic cubic spline guidance law is derived for intercepting a stationary target at a desired impact angle in a surface-to-surface engagement scenario. The guidance law is obtained using an inverse method, from a cubic spline curve based trajectory. For overcoming the drawbacks of the basic cubic spline guidance law, it is modified by introducing an additional parameter. This modification has an interesting feature that the guidance command can be obtained using a single cubic spline polynomial even for impact angles greater than π/2, while resulting in substantial improvement in the guidance performance in terms of lateral acceleration demand and length of the trajectory. For imparting robustness to the cubic spline guidance law, in the presence of uncertainties and acceleration saturation, an explicit guidance expression is also derived. A comprehensive capturability study of the proposed guidance law is carried out. The capturability for the cubic spline guidance law is defined in terms of the set of all feasible initial conditions for successful interception. This set is analytically derived and its dependence on various factors, such as initial engagement geometry and interceptor capability, are also established. The basic cubic spline guidance and its variants are also derived for a three dimen- sional scenario. The novelty of the present work lies in the particular representation of the three dimensional cubic spline curve and the adoption of the analytical results available for two dimensional cubic spline guidance law. This enables selection of the boundary condition at launch for given terminal boundary condition and also in avoiding the singularities associated with the inverse method based guidance laws. For establishing the feasibility of the guidance laws in the real world, the rigid body dynamics of the interceptor is presented as a 6 degrees-of-freedom model. Further, using a simplified model, elementary autopilots are also designed. The successful interception of the target in the presence of the rigid body dynamics proves practical applicability of the cubic spline based guidance laws. Finally, the theory developed in the first part of the thesis is applied to solve the waypoint following problem. A smooth path is designed for transition of vehicle velocity from incoming to outgoing direction. The approach developed is similar to Dubins’ path, as it comprises line–cubic spline–line segments. The important feature of this method is that the cubic spline segments are fitted such that the path curvature is bounded by a pre-specified constrained value and the acceleration demand for following the smooth path obtained by this method, gradually increases to the maximum value and then decreases. This property is advantageous from a practical point of view. All the results obtained are verified with the help of numerical simulations which are included in the thesis. The proposed cubic spline guidance law is conceptually simple, does not use linearised kinematic equations, is independent of time-to-go es- timates, and is also computationally inexpensive.