True Proportional Navigation Research Articles

Analytical solution of three-dimensional realistic true proportional navigation

True proportional navigation with varying closing speed is called realistic true proportional navigation, which is implemented in practice. Our main goal is to derive the complete solutions of three-dimensional realistic true proportional navigation for nonmaneuvering and maneuvering targets. Three coupled nonlinear second-order state equations describing the relative motion are solved analytically without any linearization for performance and trajectory analysis. Properties of three-dimensional realistic true proportional navigation such as capture region, range-to-go, time-to-go, and two aspect angles within spherical coordinates are all obtained in closed form. Furthermore, the two-player game between three-dimensional realistic true proportional navigation and threedimensional ideal proportional navigation is investigated analytically in the pursuit-evasion scenario, where a realistic true proportional navigation guided missile is designed to pursue an ideal proportional navigation guided target. It is found that an ideal proportional navigation guided target is much harder to intercept than a realistic true proportional navigation guided target. I. Introduction ROPORTIONAL navigation (PN) for short-range tactical misP siles is the optimal interceptive law in the sense of producing zero miss distance with the least integral square control effort. PN has been studied since the 1940s. During the four decades that followed, proportional navigation has been explored in many different ways, such as true proportional navigation, pure proportional navigation (PPN), generalized proportional navigation, realistic true proportional navigation (RTPN), and ideal proportional navigation It has long been recognized that there exists a significant difference in the way in which PN guidance law is analyzed and in the way in which it is implemented. Most analytical studies of PN assume that the closing velocity in the PN guidance law is constant, whereas in realistic situations, the instantaneous closing speed may be continuously estimated or measured using devices such as homing seekers with Doppler radar. To remove this difference, RTPN, which adapts to varying closing speed, has recently been proposed. Performance and trajectory analysis of two-dimensional RTPN was studied by

Method for improving autopilot lag compensation in intercept guidance

References 1 Adler, F. P., Missile Guidance by Three-Dimensional Proportional Navigation, Journal of Applied Physics, Vol. 27, May 1956, pp. 500-507. Shinar, J., Rotsztein, Y., and Bezner, E., Analysis of Three-Dimensional Evasion with Linearized Kinematics, Journal of Guidance and Control, Vol. 2, No. 5, 1979, pp. 353-360. Guelman, M., and Shinar, J., Optimal Guidance Law in the Plane, Journal of Guidance, Control, and Dynamics, Vol. 7, No. 4, 1984, pp. 471476. Cochran, J. E., No, T. S., and Thaxton, D. G., Analytical to a Guidance Problem, Journal of Guidance, Control, and Dynamics, Vol. 14, No. 1,1991, pp. 117-122. Guelman, M., Closed Form Solution of True Proportional Navigation, IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-12, July 1976, pp. 472-482. Yuan, P. J., and Chern, J. S., Solutions of True Proportional Navigation for Maneuvering and Nonmaneuvering Targets, Journal of Guidance, Control, and Dynamics, Vol. 15, No. 1, 1992, pp. 268-271. 7 Yuan, P. J., and Chern, J. S., Ideal Proportional Navigation, Journal of Guidance, Control, and Dynamics, Vol. 15, No. 5, 1992, pp. 1161-1165.

Analytical solution of 3D true proportional navigation

The commanded acceleration applying in the direction normal to the line of sight (LOS) between the interceptor and its target is called true proportional navigation (TPN). The main goal of this work is to derive the complete analytical solutions of three-dimensional (3D) TPN for nonmaneuvering and maneuvering targets. Three coupled nonlinear second-order state equations describing the relative motion are solved analytically without any linearization for performance and trajectory analysis. Properties of 3D TPN such as capture region, range-to-go, time-to-go, and two aspect angles within spherical coordinates are all obtained in closed form. Furthermore, the comparison between 3D TPN and 3D realistic true proportional navigation (RTPN) is investigated analytically in the two-player game where a TPN-guided missile is commanded to pursue a RTPN-guided target, and vice versa.

Capturability of realistic generalized true proportional navigation

The capturability of a realistic generalized true proportional navigation (RGTPN) guidance law, against a nonmaneuvering target, is analyzed. The RGTPN law is obtained by relaxing the somewhat unrealistic assumption of constant closing velocity, made in all earlier studies on generalized true proportional navigation (GTPN), and incorporating the actual time-varying value in the guidance law. Closed-form solutions for the complete capture region of RGTPN is obtained in terms of both zero and acceptable non-zero miss distances. It is shown that the capture region of RGTPN in the initial relative velocity space is significantly smaller than that of GTPN, for reasonable values of navigation constant (N) and angular direction (/spl eta/) of the missile commanded latax. However, for certain values of N and /spl eta/, capturability of RGTPN is found to be better. It is also shown that if in one of the versions of GTPN, which uses constant values of both the closing velocity and the line-of-sight (LOS) angular velocity in the guidance law the corresponding realistic time-varying quantities are used, the capture region actually expands to cover the whole of the initial relative velocity space. A number of examples are given to compare the capture performance of RGTPN with other versions of the GTPN guidance laws.

Capture region for true proportional navigation guidance with nonzero miss-distance

TRUE proportional navigation (TPN) is a version of pro- portional navigation (PN) guidance law in which the missile lateral acceleration is applied perpendicular (normal) to the line of sight $(LOS)^1$ rather than to the missile velocity, as in the case of pure proportional navigation $(PPN).^2$ The TPN guidance law has been found to be analytically more tractable than the PPN guidance law. Guelman' obtained the capture region of TPN, for a nonmaneuvering target, defining capture in terms of zero miss-distance. However, guided missiles carry warheads that have a nonzero lethal radius and thus, in practical situations, it is possible to define capture in terms of an acceptable nonzero miss-distance. It is obvious that the capture region, as obtained by $Guelman,^1$ for zero miss-distance will expand further under this assumption. In a previous $paper,^3$ the capture region for TPN with nonzero miss-distance was obtained through extensive computations. In this Note we present a complete analytical solution to the problem.

On the generalization of true proportional navigation

A simple framework to define generalized true proportional navigation (GTPN) guidance laws is presented. It is shown that this framework subsumes many of the generalizations presented in the earlier literature. The capture regions of a number of GTPN guidance laws are obtained through a rigorous qualitative analysis. The method of analysis is simpler and lends itself directly to an easy geometrical interpretation. A considerable amount of misinterpretation in the previous results, arising out of certain basic misconceptions, are corrected here. It is shown that a logical application of the guidance philosophy, through a minor modification of GTPN to take into account the direction of rotation of the line-of-sight (LOS), contributes substantially to the expansion of the capture region in the relative velocity space. In particular, it is shown that the capture region also extends to the negative closing velocity region, thus making the modified GTPN almost comparable, so far as the domain of capturability of guidance laws is concerned, to the pure proportional navigation (PPN) guidance law. A number of new results on the exact bounds on the capture region are derived and illustrated through examples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

True proportional navigation with maneuvering target

We analytically obtain the capture regions for the true proportional navigation (TPN) missile guidance law against an intelligently maneuvering target. Two versions of TPN are considered. The first is the original TPN which assumes the commanded lateral acceleration to be directly proportional to the line-of-sight (LOS) rate only, the proportionality factor being an arbitrary constant or dependent only on the initial closing velocity. The other, known as RTPN (realistic TPN), assumes the commanded lateral acceleration to be directly proportional to the LOS rate and also the current closing velocity. The target is assumed to maneuver in such a way as to increase the LOS rate and thus directly oppose the proportional navigation (PN) philosophy of annulling the LOS rate. A necessary and sufficient condition for capture is derived for the original TPN, and using it, the exact capture region is obtained. A sufficient condition for capture is derived for RTPN and is used to obtain its capture region partially. Some necessary conditions for capture are also derived for RTPN and are used to obtain an upper bound on its complete capture region. Using these conditions some important results on the existence of capture regions and a comparative study of capturability of TPN laws are also presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Capture region for a realistic TPN guidance law

This work obtains the capture region for a realistic true proportional navigation (RTPN) guidance law against a nonmaneuvering target where the assumption of constant closing velocity used in the guidance law is relaxed. This is a realistic departure from earlier treatments of TPN. A capture equation is obtained and a qualitative study carried out for both zero and non-zero miss distances. The results are shown to be significantly different from those of earlier studies, showing a degradation in the performance of the guidance law in the form of reduced capture region. An example problem is computationally solved to obtain the exact capture region for various values of miss distances.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stabilization via dynamic output feedback - A numerical approach

References 1 Jacobson, R. J., McDanell, J. P., and Rinker, G. C., Use of Ballistic Arcs in Thrust Navigation, Journal of Spacecraft and Rockets, Vol. 11, No. 8, 1974, pp. 590-596. Rinker, G. C., Jacobson, R. A., and Wood, L. J., Statistical Analysis of Trim Maneuvers in Low-Thrust Interplanetary Navigation, Journal of Spacecraft and Rockets, Vol. 13, No. 8, 1976, pp. 509-512. Thornton, C. L., and Jacobson, R. A., Capability for an Ion Drive Rendezvous with Halley's Comet, Journal of the Astronautical Sciences, Vol. 26, No. 3, 1978, pp. 197-210. Noton, M., Salehi, S. V., and Elliott, C. A., Low Thrust Navigation for a Comet-Nucleus Sample-Return Mission, Second International Symposium on Spacecraft Flight Dynamics (ESA), NTIS, N8725364, Darmstadt, Germany, Oct. 1986. Jensen D. L., Kinematics of Rendezvous Maneuvers, Journal of Guidance, Control, and Dynamics, Vol. 7, No. 3, 1984, pp. 307-314. Guelman, M., Closed Form Solution of True Proportional Navigation, IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-12, 1976, pp. 472-482. Niemi, N. J., Investigation of a Terminal Guidance System for a Satellite Rendezvous, AIAA Journal, Vol. 1, No. 2, 1963, pp. 405-411. Cesari, L., Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations, Academic, New York, 1963. Carter, T., and Humi, M., Fuel-Optimal Rendezvous Near a Point in General Keplerian Orbit, Journal of Guidance, Control, and Dynamics, Vol. 10, No. 6, 1987, pp. 567-573. Wertz, R. (ed.), Spacecraft Attitude Determination and Control, D. Reidel Publishing, Dordrecht, The Netherlands, 1978.

The proportional navigation dilemma-pure or true?

Two generic classes of proportional navigation (PN) laws are compared in detail. One class consists of pursuer-velocity-referenced systems, which include pure proportional navigation (PPN) and its variants; the second category consists of line-of-sight- (LOS-) referenced systems such as true proportional navigation (TPN), generalized true proportional navigation (GTPN), and generalized guidance laws. The existing closed-form solutions are discussed in detail, and the classical linear and quasilinear analytical solutions are summarized. A critical comparison is then made with regard to the definition, implementation, and analytical aspects of the guidance laws, including the method, the nature of solution, and an appraisal of the behavior of the pursuer motion resulting from the laws. It is established that, in spite of some restricted advantages in the solvability of the equations of motion, the LOS-referenced PN schemes suffer from serious limitations in terms of implementation and trajectory behavior. Among the major drawbacks are forward velocity variation requirement, relatively large control effort requirement, restrictions on initial engagement conditions to ensure intercept, lack of robustness, and possibility of unbounded acceleration. It is concluded that PPN is a better guidance law in a practical sense than TPN and its generalizations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Generalized guidance law for homing missiles

The concept of a generalized guidance law is presented, and the closed-form solution for a homing missile pursuing a maneuvering target according to generalized guidance laws is given. It is shown that the guidance laws appearing in the literature are merely special cases of the one proposed by the authors. The derived generalized forms of capture area, missile acceleration, and homing time duration that are derived provide insight into the performance of the guidance laws being considered and lead to the discovery of new ones. The problem of finding a nonlinear optimal guidance law for a homing missile with controlled acceleration, applied so as to capture a maneuvering target with a predetermined trajectory while minimizing a weighted linear combination of time of capture and energy expenditures, is solved in closed form. The derived optimal control law is equal to the LOS (line of sight) rate multiplied by a trigonometric function of the aspect angle. Numerical simulation shows that the resulting guidance law appears to yield a significant advantage over true proportional navigation. >

The closed-form solution of true proportional navigation

The closed-form solution of the equations of motion of an ideal missile pursuing a nonmaneuvering target according to the true propotional navigation law is obtained. An analysis of the solution is performed, and the conditions necessary for the missile to reach the target are determined.

Corrections to "The Closed-Form Solution of True Proportional Navigation"

Corrections to "The Closed-Form Solution of True Proportional Navigation"