Polynomial Shaping of the Look Angle for Impact-Time Control

The design of missile-guidance algorithms with terminal constraints, such as the impact time, the impact angle, and both of them together, has received considerable attention recently. There HE design of missile-guidance algorithms with terminal constraints, such as the impact time, the impact angle, and both may be several reasons for resorting to such algorithms. For example, one can avoid directional defense mechanisms or reduce the collateral damage by controlling the impact angle. On the other hand, the impact-time control can provide survivability against close-in weapon systems by facilitating a simultaneous attack. It then follows that impact-time-and-angle control can offer combined advantages. In the guidance literature, more interest has been directed to the problem of impact-angle control, recent instances of which are presented in [1–5], than to the problem of impact-time control. One early example considering the impact-time problem is presented in [6], which is based on linearized engagement kinematics. The guidance command presented in the study has two parts, in which the first one is proportional navigation (PN) and the second one is a term involving the difference of the desired and estimated time-togo values. The design framework of [6] is enhanced in [7,8] by including more accurate time-to-go estimation methods accounting for nonlinear kinematics. A varying guidance gain is designed in [7], whereas [8] considers a constant gain as in [6]. In addition to such PN-based techniques, there are a number of impact-time control laws based on the nonlinear control theory. These could be categorized as Lyapunov-based [9,10] and sliding-mode-controlbased [11,12] laws. The guidance law in [9] results in an analytical solution for the impact time. However, the solution covers a limited impact-time interval because the look angle is made to decrease monotonously during the engagement. In [10], inwhich the time-togo estimation presented in [6] is used for computing the impact-time error, solutions for both two- and three-dimensional environments are presented. The work in [11] defines its switching surface as a combination of the impact-time error and the line-of-sight (LOS) rate. In contrast, the study in [12] provides a solution by using a sliding surface that is a function of the impact-time error alone. Moreover, quite a lot of effort is spent to avoid the singularity of the guidance command; a continuous nonlinear function is introduced for this purpose. The downside of the guidance laws presented in [7,10–12] is that they are likely to demand high look angles at the beginning of the flight, which results in the saturation of the guidance command. Except for [9], the performance of the impacttime control algorithmsmentioned so far, underwhich the trajectory eventually evolves into a PN trajectory regardless of the design domain, highly depends on the accuracy of the time-to-go estimate. Alternatively, in [13,14], the problem is solved without relying on the time-to-go information. In [13], a two-phased guidance scheme for indirect control of impact time is devised based on the idea of shaping the relative look-angle profile. The method in [14] uses the concept of zero-effort-miss vector to direct the total acceleration so that the target is captured at a specified time. Naturally, the impact time can also be enforced via guidance schemes that aim at simultaneous control of the impact time and angle. In [15], the guidance command is composed of two parts: the first to achieve the capture with a desired impact angle, and the second to control the impact time. A polynomial guidance law is proposed in [16], in which the three unknown coefficients of the polynomial are determined with respect to the terminal conditions. The method in [17] uses the sliding-mode-control theory for realizing the objective over LOS rate shaping. In [18], a practical guidance law is proposed to control the impact time or/and the impact angle by starting with the nonlinear optimal control framework, in which the solutions dictate the familiar PN guidance law. This study proposes an impact-time control technique based on following a look-angle profile that is a polynomial in time. Both quadratic and cubic polynomials, which satisfy the same boundary conditions on the look angle, are considered. The latter also satisfies the terminal condition ofhaving a zero look-angle rate. The guidance gain is to be calculated by numerically solving an integral equation, which can be reduced to an approximate transcendental equation in the case of the quadratic profile. The resulting guidance law, which also makes use of the LOS rate, is driven by the remaining engagement time instead of a time-to-go estimate. The design is based on the assumption of having a stationary target; however, it could be extended to nonmaneuvering targets through the use of the predicted intercept-point approach, as exemplified in [11]. The rest of the Note is organized as follows: in Sec. II, the definition of the problem is presented. Section III is reserved for the derivation of the polynomial-shaping approach. Finally, in Sec. IV, the effectiveness ofthe proposed guidance law is illustrated by means of numerical simulations, which also involve a comparison against the method in [12] and the energy optimal solution.     «

The design of missile-guidance algorithms with terminal constraints, such as the impact time, the impact angle, and both of them together, has received considerable attention recently. There HE design of missile-guidance algorithms with terminal constraints, such as the impact time, the impact angle, and both may be several reasons for resorting to such algorithms. For example, one can avoid directional defense mechanisms or reduce the collateral damage by controlling the impact angle. On the o...     »