P.N. Dwivedi

Suboptimal Midcourse Guidance of Interceptors for High-Speed Targets with Alignment Angle Constraint

Using the recently developed computationally efficient model predictive static programming and a closely related model predictive spread control concept, two nonlinear suboptimal midcourse guidance laws are presented in this paper for interceptors engaging against incoming high-speed ballistic missiles. The guidance laws are primarily based on nonlinear optimal control theory, and hence imbed effective trajectory optimization concepts into the guidance laws. Apart from being energy efficient by minimizing the control usage throughout the trajectory (minimum control usage leads to minimum turning, and hence leads to minimum induced drag), both of these laws enforce desired alignment constraints in both elevation and azimuth in a hard-constraint sense. This good alignment during midcourse is expected to enhance the effectiveness of the terminal guidance substantially. Both point mass as well as six-degree-of-freedom simulation results (with a realistic inner-loop autopilot based on dynamic inversion) are presented in this paper, which clearly shows the effectiveness of the proposed guidance laws. It has also been observed that, even with different perturbations of missile parameters, the performance of guidance is satisfactory. A comparison study, with the vector explicit guidance scheme proposed earlier in the literature, also shows that the newly proposed model-predictive-static-programming-based and model-predictive-spread-control-based guidance schemes lead to lesser lateral acceleration demand and lesser velocity loss during engagement.

Generalized State Estimation and Model Predictive Guidance for Spiraling and Ballistic Targets

An extended Kalman filter based generalized state estimation approach is presented in this paper for accurately estimating the states of incoming high-speed targets such as ballistic missiles. A key advantage of this nine-state problem formulation is that it is very much generic and can capture spiraling as well as pure ballistic motion of targets without any change of the target model and the tuning parameters. A new nonlinear model predictive zero-effort-miss based guidance algorithm is also presented in this paper, in which both the zero-effort-miss as well as the time-to-go are predicted more accurately by first propagating the nonlinear target model (with estimated states) and zero-effort interceptor model simultaneously. This information is then used for computing the necessary lateral acceleration. Extensive six-degrees-of-freedom simulation experiments, which include noisy seeker measurements, a nonlinear dynamic inversion based autopilot for the interceptor along with appropriate actuator and sensor models and magnitude and rate saturation limits for the fin deflections, show that near-zero miss distance (i.e., hit-to-kill level performance) can be obtained when these two new techniques are applied together. Comparison studies with an augmented proportional navigation based guidance shows that the proposed model predictive guidance leads to a substantial amount of conservation in the control energy as well.

Estimation of Ballistic Coefficient of Reentry Vehicle with Divided Difference Filtering using Noisy RF Seeker Data

Estimation of relative kinematic states of the reentry vehicle from noisy seeker measurements is a nonlinear filtering problem. In case of nonlinear filtering problem, choice of any estimator is scenario dependent. Recently few nonlinear estimation techniques such as Unscented Kalman filter (UKF), Divided Difference Filter (DDF) and other techniques promise to be performing some what better than Extended Kalman Filters (EKF), although the claim depends on particular nonlinear problem. In this paper, application of Divided Difference Filter is examined in estimating relative kinematic parameters of reentry vehicle. The application of the proposed estimator on noisy measurement data available from seeker is demonstrated and comparison results are shown along with EKF and UKF. These estimators becomes more accurate than estimators based on Taylor approximation like EKF. Basic DDF implementation takes very high computational time because of many state propagation equation are solved online. For present problem, special numerical solution is presented, which makes DDF computation almost as fast as that of EKF for a chosen number of states. This is a new solution to present problem and can be utilized in new genre of filtering with present problem.

Computationally Efficient Suboptimal Mid Course Guidance Using Model Predictive Static Programming (MPSP)

For a homing interceptor, suitable initial condition must be achieved by mid course guidance scheme for its maximum effectiveness. To achieve desired end goal of any mid course guidance scheme, two point boundary value problem must be solved online with all realistic constrain. A Newly developed computationally efficient technique named as MPSP (Model Predictive Static Programming) is utilized in this paper for obtaining suboptimal solution of optimal mid course guidance. Time to go uncertainty is avoided in this formulation by making use of desired position where midcourse guidance terminate and terminal guidance takes over. A suitable approach angle towards desired point also can be specified in this guidance law formulation. This feature makes this law particularly attractive because warhead effectiveness issue can be indirectly solved in mid course phase.

Generalized estimation and predictive guidance for evasive targets

A generalized and unified target state estimation formulation is presented for accurately estimating the states of an airborne target, which can either be nonmaneuvering or undergoing various maneuvers. A nonlinear predictive zero effort miss-based guidance algorithm is also presented here. Extensive high-fidelity six degrees-of-freedom simulation experiments show that near-zero miss distance (i.e., hit-to-kill level performance) can be obtained when these two new techniques are applied together. Moreover, there is a substantial saving in control energy.

Quaternion error free estimation of reentry ballistic targets

A common approach followed in state estimation of reentry ballistic targets is to formulate the problem where the process model is written in the Earth fixed inertial frame, where as the measurement model is written in the body frame of the interceptor. In such a formulation, to relate the states with the measurement vector, a transformation matrix is required to transform the position and velocity vectors from inertial frame to body frame and vice-versa. This matrix is constructed using the attitude quaternion information of the interceptor as obtained from its inertial navigation system. In a practical situation, however, when the time of flight is high, the navigation error build up is usually large, which introduces large errors in the transformation matrix. Due to this, the performance of the estimator is found to be not very good, even though the seeker gives good measurement data. To avoid this problem, a new estimation formulation is proposed in this paper, where the process model is written in the body frame of the interceptor. This formulation needs body rate information of the interceptor instead of the attitude quaternion, which does not degrade much with the time of flight. Hence, this new formulation is quite effective in estimating the target states accurately, thereby improving the mission performance substantially, even in presence of large navigation errors in the attitude quaternion. Extensive simulation results show the effectiveness of the proposed approach.

Terminal-Lead-Angle-Constrained Generalized Explicit Guidance

Besides enhancing the warhead effectiveness, the terminal lead angle constraint ensures that the target remains within the field of view of the radio proximity fuse, which is very critical for the success of a mission. This paper presents substantial enhancement of the generalized explicit guidance to include the terminal lead angle constraint in three dimension (3-D).. With the availability of the desired total lead angle in 3-D, it computes the necessary terminal flight path and heading angles so that it not only satisfies the lead angle constraint, but also leads to the minimum among all minimum control effort solutions for various possible terminal flight path and heading angle combinations. With no lead angle constraint, it also gracefully degenerates to the standard proportional navigation guidance.

Modified Generalized Explicit Guidance for Midcourse with Near-Zero Lateral Acceleration in Terminal Phase

Abstract By appropriately modifying the generalized explicit guidance and developing it further, an optimal midcourse guidance law is proposed in this paper to ensure terminal impact angle constrained engagement of missiles with near-zero lateral acceleration in the terminal phase. This guidance law generates a desired intermediate point in space as well as the desired angles which needs to be achieved at the end of the midcourse. By reaching this intermediate point with the desired intermediate angles, it ensures that the missile will intercept the target with very less (near zero) acceleration demand in terminal phase with desired impact angles. When both the terminal impact angles are specified, the guidance law solves the constraint equations and find out the intermediate point along with the angles, whereas if only one terminal angle is specified (or both are left free), it generates the intermediate point with missing angle (both angles) which can lead to minimum control effort and then solves that problem.

Effect of Choice of Basis Functions in Neural Network for Capturing Unknown Function for Dynamic Inversion Control

The basic requirement for an autopilot is fast response and minimum steady state error for better guidance performance. The highly nonlinear nature of the missile dynamics due to the severe kinematic and inertial coupling of the missile airframe as well as the aerodynamics has been a challenge for an autopilot that is required to have satisfactory performance for all flight conditions in probable engagements. Dynamic inversion is very popular nonlinear controller for this kind of scenario. But the drawback of this controller is that it is sensitive to parameter perturbation. To overcome this problem, neural network has been used to capture the parameter uncertainty on line. The choice of basis function plays the major role in capturing the unknown dynamics. Here in this paper, many basis function has been studied for approximation of unknown dynamics. Cosine basis function has yield the best response compared to any other basis function for capturing the unknown dynamics. Neural network with Cosine basis function has improved the autopilot performance as well as robustness compared to Dynamic inversion without Neural network.

Improved Capturability of Terminal Angles with Modified Suboptimal Mid-Course Guidance

Abstract Successful destruction of reentry targets, which can enter into atmosphere with different reentry angles, needs a near-perfect operation of terminal guidance. However, since such targets usually approach with very high velocities, if suitable terminal angle is not ensured by mid course guidance, the terminal guidance may not operate successfully. Hence, capturability of the terminal angle is an important performance measure of the mid-course guidance. In this paper, two algorithmic improvements are proposed in the framework of model predictive static programming (MPSP) based suboptimal mid-course guidance. In one of these methods, the desired terminal angle is successively varied and in the other method a learning rate coefficient is introduced while updating the lateral acceleration command history. Both of these techniques result in substantial improvement of capturability of the alignment angles.