Optimal Guidance Problem Research Articles

Cooperative guidance with multiple constraints using convex optimization

In the existing closed-loop cooperative guidance approaches for salvo attack of multiple missiles, the multiple constraints and time-variant velocity basically cannot be effectively considered. Therefore, two closed-loop cooperative guidance methods are developed in this paper, by employing the efficient convex optimization technique and receding horizon control (RHC) strategy. During each guidance cycle of RHC, the system coordination target is updated and then broadcasted to each missile as a constraint. Subsequently, the convex optimization technique is utilized to solve the multi-constraint optimal proportional guidance problem of each missile online to achieve the consensus on time-to-go among missiles. Simulation results show that for three cases with different conditions of velocity, the cooperative simultaneous attack under multiple constraints can be effectively carried out using each of the two proposed cooperative guidance laws, which verify their effectiveness and feasibility.

A New Command Shaping Guidance Law using Lagrange Multiplier

This article presents a new command shaping guidance law by change of Lagrange multiplier (LM), called CSGL-LM. The Schwarz inequality approach is used to solve the optimal guidance problems considering both terminal constraints on interception and impact angle control. LM is introduced to combine two terminal constraints into a single equation. The main idea of this paper is to use LM as a design parameter for shaping the guidance command as well as controlling the terminal constraints. The guidance command of CSGL-LM is given a unified functional form of the time-to-go, the state variables, and LM. Therefore, through an appropriate choice of LM, we can achieve various shapes of the guidance commands for the interception case, as well as the impact angle control case. As illustrative examples, this paper also shows that a class of previous guidance laws is just one of particular solutions of CSGL-LM. Numerical simulations are performed to validate the properties of CSGL-LM, compared with the conventional guidance law.

Optimal terminal guidance for exoatmospheric interception

In this study, two optimal terminal guidance (OTG) laws, one of which takes into account the final velocity vector constraint, are developed for exoatmospheric interception using optimal control theory. In exoatmospheric interception, because the proposed guidance laws give full consideration to the effect of gravity, they consume much less fuel than the traditional guidance laws while requiring a light computational load. In the development of the guidance laws, a unified optimal guidance problem is put forward, where the final velocity vector constraint can be considered or neglected by properly adjusting a parameter in the cost function. To make this problem analytically solvable, a linear model is used to approximate the gravity difference, the difference of the gravitational accelerations of the target and interceptor. Additionally, an example is provided to show that some achievements of this study can be used to significantly improve the fuel efficiency of the pulsed guidance employed by the interceptor whose divert thrust level is fixed.

LINEAR QUADRATIC OPTIMAL GUIDANCE WITH ARBITRARY WEIGHTING FUNCTIONS

In this article, the linear quadratic (LQ) optimal guidance laws with arbitrary weighting functions are introduced. The optimal guidance problems in conjunction with the control effort weighed by arbitrary functions are formulated, and then the general solutions of these problems are determined. Based on these investigations, we can know a lot of previous optimal guidance laws belong to the proposed results. Additionally, the proposed results are compared with other results from the generalization standpoint. The potential importance on the proposed results is that a lot of useful new guidance laws providing their outstanding performance compared with existing works can be designed by choosing weighting functions properly. Accordingly, a new optimal guidance law is derived based on the proposed results as an illustrative example.

Optimal guidance for hypersonic reentry using inversion and receding horizon control

This study investigates the optimal guidance problem for reentry vehicles with focus on novel methods for nominal trajectory generation and tracking. To reduce the dimensionality of reentry trajectory optimisation problem, the inversion-based approach is utilised to partly eliminate the differential equations by reformulating the states and controls with some desired outputs. These outputs are parameterised with a mapped Chebyshev pseudospectral method, which is improved by conformal map and barycentric rational interpolation techniques in order to enhance the numerical accuracy for high-order derivatives of desired outputs. On the basis of the obtained nominal trajectory, a robust closed-loop guidance scheme is proposed using receding horizon control, whereas the associated two-point boundary value problem in each guidance cycle is readily solved by differential transformation method. Numerical simulations show that the proposed guidance scheme is feasible and effective for atmospheric reentry.

Optimal On–Off Cooperative Maneuvers for Long-Term Satellite Cluster Flight

When a group of satellites is equipped with a particularly simple propulsion system (e.g., cold-gas thrusters), constraints on the thrust level and total propellant mass renders cluster keeping extremely challenging. This is even more pronounced in disaggregated space architectures, in which a satellite is formed by clustering a number of heterogonous free-flying modules. The research described in this paper develops guidance laws aimed at keeping the relative distances between the cluster modules bounded for long mission lifetimes, typically more than a year, while using constant-magnitude low thrust, with a characteristic on–off profile. A cooperative guidance law capable of cluster establishment and maintenance under realistic environmental perturbations is developed. The guidance law is optimized for fuel consumption, subject to relative distance constraints. Some of the solutions found to the optimal guidance problem require only a single maneuver arc to keep the cluster within relatively close distances for an entire year.

Partial Integrated Guidance and Control of Interceptors for High-Speed Ballistic Targets

A new partial integrated guidance and control design approach is proposed in this paper, which combines the benefits of both integrated guidance and control as well as the conventional guidance and control design philosophies. The proposed technique essentially operates in a two-loop structure. In the outer loop, an optimal guidance problem is formulated considering the nonlinear six degrees-of-freedom equation of motion of the interceptor. From this loop, the required pitch and yaw rates are generated by solving a nonlinear suboptimal guidance formulation in a computationally efficient manner while simultaneously assuring roll stabilization. Next, the inner loop tracks these outer loop body rate commands. This manipulation of the six degrees-of-freedom dynamics in both loops preserves the inherent time scale separation property between the translational and rotational dynamics, while retaining the philosophy of integrated guidance and control design as well. Because of this, the tuning process is quite straightforward and nontedious as well. Extensive six degrees-of-freedom simulations studies have been carried out, considering three-dimensional engagement geometry, to demonstrate the effectiveness of the proposed new design approach engaging high-speed ballistic targets. A variety of comparison studies have also been carried out to demonstrate the effectiveness of the proposed approach.

Optimal guidance law in quantum mechanics

Following de Broglie’s idea of a pilot wave, this paper treats quantum mechanics as a problem of stochastic optimal guidance law design. The guidance scenario considered in the quantum world is that an electron is the flight vehicle to be guided and its accompanying pilot wave is the guidance law to be designed so as to guide the electron to a random target driven by the Wiener process, while minimizing a cost-to-go function. After solving the stochastic optimal guidance problem by differential dynamic programming, we point out that the optimal pilot wave guiding the particle’s motion is just the wavefunction Ψ(t,x), a solution to the Schrödinger equation; meanwhile, the closed-loop guidance system forms a complex state–space dynamics for Ψ(t,x), from which quantum operators emerge naturally. Quantum trajectories under the action of the optimal guidance law are solved and their statistical distribution is shown to coincide with the prediction of the probability density function Ψ∗Ψ.

Optimal impact angle control guidance law considering the seeker’s field-of-view limits

A new optimal guidance problem with impact angle constraint and seeker’s field-of-view limits is investigated for a missile with a strapdown seeker. Impact angle control to satisfy the terminal flight path angle constraint causes missile trajectories to be very curved. Since a strapdown seeker has a narrower field-of-view than a gimbaled seeker, a missile with a strapdown seeker using the impact angle control guidance law can miss its target within the field-of-view during the homing phase, leading to the failure of the mission. Therefore, considering the seeker’s field-of-view limits is a key issue for missiles with strapdown seekers when implementing the impact angle control guidance law. To handle the seeker’s field-of-view limits in the homing guidance problem, a look angle, which is defined as the angle between the velocity vector and the line-of-sight, is considered as an inequality constraint. Based on the optimal control theory with a state variable inequality constraint, the optimal impact angle control guidance law with the seeker’s field-of-view limits is designed. The proposed guidance law is made up of three types of optimal acceleration commands: the first command is to reach the maximum look angle of the seeker, the second is to keep the seeker look angle constant on the constraint boundary, and the third is to intercept the target with the desired impact angle. Nonlinear simulations are performed to validate the proposed approach. In addition, comparisons with other guidance laws considering the limitation of the seeker look angle are carried out via nonlinear simulations, and the results show that the proposed guidance law is more efficient in terms of the control energy.

Optimal Guidance Solution for Impact Angle Control Based on Linearization about Collision Triangle

AbstractIn this paper, an optimal guidance problem is investigated where specific missile-target geometry at the time of the intercept of, or a terminal impact angle with respect to, a maneuvering target is prescribed. The kinematics is linearised to obtain a closed-form optimal solution and the proposed approach addresses the question of how to accomplish the linearization in a right manner. We use, as the basis of linearization, the perfect (or zero effort) collision triangle for the impact angle control problem which varies depending on the value of the prescribed impact angle, and solve a linear optimal guidance problem. The performance of the resultant guidance law is illustrated by performing some numerical simulations and the selection of a guidance variable that effectively capture the distance between the missile and the zero effort collision triangle is discussed.

Asymptotically Optimal Nonlinear Filtering: Theory and Examples with Application to Target State Estimation

The State-Dependent Riccati Equation (SDRE) filter, which is derived by constructing the dual of the well-known SDRE nonlinear regulator control design technique, has been studied in various papers, with mainly practical investigations of the filter. Until recently, theoretical aspects of the filter had not been fully investigated, leaving many unanswered questions, such as stability and convergence of the filter. The authors conducted an investigation of the conditions under which the state estimate given by this algorithm converges asymptotically to the first order minimum variance estimate given by the extended Kalman filter (EKF). Conditions for determining a region of stability for the SDRE filter were also investigated. The analysis was based on stable manifold theory and Hamilton-Jacobi-Bellman (HJB) equations. In this paper, the motivation for introducing HJB equations is justified with mathematical rigor, which is given by reference to the maximum likelihood approach to deriving the EKF. The application of the SDRE filter is then demonstrated on challenging examples to illustrate the theoretical aspects and design flexibility (additional degrees of freedom) of the algorithm when loss of observability is encountered. In particular, a realistic and detailed evaluation of the filter is carried out for the problem of target state estimation in an advanced tactical missile guidance application for analysis in the optimal guidance problem for air-air engagements using only passive sensor (angle-only) information. Simulation results are presented which show dramatic tracking improvement using the SDRE target tracker.

New Trends in Astrodynamics and Applications: Optimal Trajectories for Space Guidance

This paper represents recent results on the development of optimal analytic solutions to the variation problem of trajectory optimization and their application in the construction of on-board guidance laws. The importance of employing the analytically integrated trajectories in a mission design is discussed. It is assumed that the spacecraft is equipped with a power-limited propulsion and moving in a central Newtonian field. Satisfaction of the necessary and sufficient conditions for optimality of trajectories is analyzed. All possible thrust arcs and corresponding classes of the analytical solutions are classified based on the propulsion system parameters and performance index of the problem. The solutions are presented in a form convenient for applications in escape, capture, and interorbital transfer problems. Optimal guidance and neighboring optimal guidance problems are considered. It is shown that the analytic solutions can be used as reference trajectories in constructing the guidance algorithms for the maneuver problems mentioned above. An illustrative example of a spiral trajectory that terminates on a given elliptical parking orbit is discussed.

Optimal Guidance and Tracking Control of Autonomous Underwater Vehicle under Flow Disturbances

Numerical solution algorithm for the optimal guidance law of AUV (Autonomous Underwater Vehicle) is presented when it transfers between two pre-designated points in the region of flow disturbances. In the formulation of optimal guidance problem, position vector of the vehicle relative to the inertial frame is qualified as the state vector, while the vehicle's heading as the input. Total amount of time to reach the terminal point is the performance index to be minimized. By solving the optimal guidance law, optimized sequence of vehicle's heading is obtained in time domain, which is to be fed to the heading control system as the reference. Proposed algorithm is applicable to the time-varying flow disturbances as well as time-invariant ones, if only the distribution of them is known a priori. Strategy of optimal guidance with tracking control is applied to the navigation of r2D4, a newly developed AUV by IIS, the university of Tokyo. Effectiveness of optimal guidance law shows its primary dependency on the local distribution of flow disturbances.

A combined genetic algorithms-shooting method approach to solving optimal control problems

In this paper, an alternative method for solving optimal control problems is presented. By applying calculus of variations, the optimal control problem can be reduced to solving a two-point boundary value problem. Here, the solution is generated with a combination of two methods genetic algorithms (GA) and the shooting method. An estimate of the optimal solution is first obtained using GA. This solution is in turn used as the initial guess for the shooting method. This combined method is applied to an optimal missile guidance problem. The performances of the combined method and GA are evaluated by simulation and compared. The results clearly show that the proposed combined method is able to locate the optimal solution more efficiently than GA. The results also show that the combined method never fails to correctly determine the optimal solution. Therefore, it proves to be more robust than the shooting method whose convergence is not always guaranteed.

Implementation of Optimal Guidance Laws Using Predicted Missile Velocity Profiles

A class of weighted control-effort minimizing guidance laws are derived for missiles of varying velocity. As a practical weighting function, we consider a function of air density and missile velocity parameterized by positive real numbers. The resulting optimal guidance problem can be interpreted as the drag minimization problem for subsonic or supersonic missiles, depending on what parameters are used. This approach is extended easily to solve the drag minimization of a typical antiaircraft missile system with an arbitrary velocity proe le and arbitrary drag characteristics, as demonstrated by a simulation study. We also present analytical results on how the guidance gain of the optimal law varies according to the values of the parameters. Because the optimal guidance laws make use of the future missile velocity proe le, one critical issue is how to implement the laws. To avoid the dife culty that an inaccurately predicted missile velocity proe le causes the guidance command to blow up in the last part of the engagement, we suggest two simple on-line velocity-proe le updating schemes, which considerably alleviate the problem.

Analysis of generalized guidance laws for homing missiles

It is shown that the optimal guidance problem can be handled as a Bolza-type problem which can be solved by straightforward calculation. The solution proves to be different from that given by Yang, et al. (1989) and is verified to be a minimum performance index solution. Also presented is an alternate unified approach to the solution of the nonlinear system of differential equation of the generalized proportional navigation law discussed earlier by Yang, et al. This unified equation has been solved for all the different cases of proportional navigation for which closed-form solutions have been published by earlier authors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Trajectory optimization using parallel shooting method on parallel computer

The efficiency of a parallel shooting method on a parallel computer for solving a variety of optimal control guidance problems is studied. Several examples are considered to demonstrate that a speedup of nearly 7 to 1 is achieved with the use of 16 processors. It is suggested that further improvements in performance can be achieved by parallelizing in the state domain. 10 refs.

The Closed-Form Solution and its Approximation of the Optimal Guidance Law

In this paper, the optimal homing guidance problem is investigated for the general missile/target models described in the state-space. The closed-form solution of the optimal guidance law is derived, and its asymptotic properties are studied as the time-to-go goes to infinity or zero. Futhermore, several polynomial approximations of the optimal guidance law are suggested for real-time applications

The Homicidal Chauffeur

P differential games had their beginning in the pioneering work of Isaacs. These differential games are a generalization of the optimal guidance problem, in the sense that guidance strategies are to be found for each of two players having conflicting objectives. The players' strategies, respectively, are found to minimize and to maximize a specific criterion, which is often taken to be the time to termination, or capture. The second-order differential equations describing the dynamics of the homicidal chauffeur game include two parameters, which are the speed ratio and the ratio of capture-radius to pursuer's minimum-turn radius. It is the purpose of this paper to show that this apparently simple, physically motivated differential game can lead to novel and remarkably complex solutions. It is shown that the classical necessary conditions often fall far short of providing the complete solution to the problem. On the contrary, a variety of exceptional lines are found to occur in the state space of the game. These lines are not predicted by the theory and they are of fundamental importance, since they separate regions of differing optimal strategies for either or both of the players. Isaacs discovered the barrier, universal line (singular arc), equivocal line, and a pursuer's dispersal line in his study of the problem. Other types of lines are found to include new dispersal lines, switch lines, switch envelopes, focal lines, and safe-contact motion. The switch envelope and focal line phenomena are believed to be new to the general theory of differential games.