Direct Shooting Method Research Articles

Robust Trajectory Planning of Gliding-Guided Projectiles with Weak Maneuverability

Due to constraints in launch platforms and cost, the maneuverability of gliding-guided projectiles is limited, necessitating a rational design of their trajectory schemes. To reduce the sensitivity of trajectory schemes to uncertainties while ensuring compatibility between flight schemes and guidance control systems and fully exploiting the control capability of the projectile, a closed-loop robust trajectory planning method is proposed. Models of major uncertain factors and state deviation at the control start point are established. Based on the NIPCE method, the stochastic dynamic model is transformed into a high-dimensional deterministic model with PCE coefficients as state variables, and the uncertainty propagation law is obtained. A PID algorithm is employed to design a tracking guidance law based on position error feedback, and open-loop and closed-loop robust trajectory planning models are established accordingly. The optimal control problem is solved by transforming it into a nonlinear programming problem using the direct shooting method. Our simulation results indicate that the NIPCE method can significantly improve the computational efficiency of uncertainty propagation while ensuring accuracy; compared with parallel MCS, the computation time is reduced by 96.8%. Open-loop robust planning can effectively mitigate the sensitivity of gliding trajectories to uncertainties (the standard deviations of terminal altitude and lateral deviations are reduced by 23.6% and 35.3%, respectively, compared to deterministic planning) but cannot completely eliminate terminal dispersion. Closed-loop robust planning effectively improves control effort consumption on the basis of open-loop planning.

Transition Flight Trajectory Optimization for a Flapping-Wing Micro Air Vehicle with Unsteady Vortex-Lattice Method

Flapping-wing micro air vehicles (FWMAVs) have the capability of performing various flight modes like birds and insects. Therefore, it is necessary to understand the various flight modes of FWMAVs in order to fully utilize the capability of the vehicle. The unique flight modes of FWMAVs can be studied through the trajectory optimization. This paper proposes a trajectory optimization framework of an FWMAV. A high-fidelity simulation model is included in the framework to sufficiently consider the complicated dynamics of the FWMAV. The unsteady aerodynamics are modeled with the unsteady panel method (UPM) and the unsteady vortex-lattice method (UVLM). The effect of wing inertia is also considered in the simulation model. In this study, transition flight trajectories are searched with the proposed framework. An optimal control problem is formulated for the transition flight from hovering to forward flight and transcribed to the parameter optimization problem with the direct shooting method. The cost function is defined as energy consumption. The same converged solution can be obtained with different initial guesses. The optimization results show that the FWMAV utilizes the pitch-up maneuver to increase altitude, although the forward speed is reduced. This pitch-up maneuver is performed more actively when the target velocity of transition is low, or the wind condition is favorable to acceleration.

Single-stage to orbit ascent trajectory optimisation with reliable evolutionary initial guess

In this paper, the ascent trajectory optimization of a lifting body Single-Stage To Orbit (SSTO) reusable launch vehicle is investigated. The work is carried out using a Direct Multiple Shooting method to solve the Optimal Control problem. The crucial initialisation of the optimisation process is performed by using a combination of two evolutionary algorithms, namely a Multi-Objective Parzen-based Estimation of Distribution (MOPED) algorithm and a Multi-Population Adaptive Inflationary Differential Evolution Algorithm (MP-AIDEA). Multi-Objective Parzen-based Estimation of Distribution (MOPED) belongs to the class of Estimation of Distribution Algorithms (EDAs) and it is used in the first phase of the initial guess research to explore the search space, then Multi-Population Adaptive Inflationary Differential Evolution Algorithm (MP-AIDEA) is used to refine the obtained results, and better fulfill the imposed constraints. The initial guesses obtained with this evolutionary framework were tested on different multiple shooting configurations. The importance of the continuity properties of the employed mathematical models was also quantitatively addressed.

Hybrid Trajectory Optimization Method and Tracking Guidance for Variable-Sweep Missiles

In this paper, an offline hybrid trajectory optimization approach is proposed for variable-sweep missiles to explore the superiority in the diving phase. Aiming at the maximal terminal velocity with the impact angle constraint, the trajectory optimization model is formulated under multiple constraints, and the aerodynamic analysis in different sweep angles is discussed. Unlike only the attack angle used for the optimization process traditionally, the two-variable optimization scheme on both the attack angle and sweep angle is investigated for variable-sweep missiles. Then, the trajectory optimization problem is transformed into the nonlinear programming problem via a hybrid optimization strategy combining the Gauss pseudospectral method and direct shooting method to obtain the high precision and fast convergence solution. Finally, to verify the feasibility of the optimal trajectory under uncertainties, the tracking guidance law is designed on basis of the gain scheduled linear quadratic regulator control. Numerical simulation results reveal not only of the proposed hybrid optimization strategy but also of the superiority of variable-sweep missiles compared with traditional missiles.

Microcanonical rates from ring-polymer molecular dynamics: Direct-shooting, stationary-phase, and maximum-entropy approaches.

We address the calculation of microcanonical reaction rates for processes involving significant nuclear quantum effects using ring-polymer molecular dynamics (RPMD), both with and without electronically non-adiabatic transitions. After illustrating the shortcomings of the naive free-particle direct-shooting method, in which the temperature of the internal ring-polymer modes is set to the translational energy scale, we investigate alternative strategies based on the expression for the microcanonical rate in terms of the inverse Laplace transform of the thermal reaction rate. It is shown that simple application of the stationary-phase approximation (SPA) dramatically improves the performance of the microcanonical rates using RPMD, particularly in the low-energy region where tunneling dominates. Using the SPA as a Bayesian prior, numerically exact RPMD microcanonical rates are then obtained using maximum entropy inversion of the thermal reaction rates for both electronically adiabatic and non-adiabatic model systems. Finally, the direct-shooting method is revisited using the SPA-determined temperature for the internal ring-polymer modes, leading to a simple, direct-simulation method with improved accuracy in the tunneling regime. This work suggests a general strategy for the extraction of microcanonical dynamical quantities from RPMD (or other approximate thermal) simulations.

Influence of the Earth Rotation on Trajectory of a Returnable Hypersonic Cruise Vehicle

It is of high difficulty for a hypersonic vehicle to implement a hypersonic cruise flight and return to a scheduled airport without power. Accuracy of mission simulation and analysis is more crucial at this time. Under this circumstance, it is meaningful to discuss whether the earth rotation should be considered in the dynamic model and trajectory design. In this research, the dynamic equations in which the earth rotation is taken into consideration are derived for a hypersonic cruise vehicle. The direct shooting method is utilized for parameterization and discretization of the trajectory, so that the trajectory can be analyzed and simulated through numerical integration. Then, two groups of trajectory simulation results are presented. In each group, the only difference between the two trajectories is that whether the earth rotation is involved, and the two trajectories are compared with each other. For a returnable hypersonic cruise vehicle which returns to the ground without power, whether to involve the earth rotation in the dynamic model is discussed on the basis of the numerical results, and relevant suggestions are given for reference.

A Fast Nonlinear Model Predictive Control Method Based on Discrete Mechanics

Abstract Nonlinear Model Predictive Control (NMPC) is an advanced control technique that often relies on a computationally demanding optimization problem and numerical integration algorithms. This paper proposes and investigates a novel method with less computational effort to improve the efficiency of NMPC using a formulation based on discrete mechanics (DM). In contrast to classical NMPC formulations, the proposed method merges the two stages, for solving both the initial value problem (IVP) for prediction as well as the nonlinear programming problem (NLP) for optimization, into a single stage for solving an optimal boundary value problem (BVP) using NLP techniques. By exploiting the structural features of DM a symbolic solution set of the equations of motion are derived offline on each discretization node along the whole optimization horizon. Within an NLP, the optimal solution is efficiently obtained online under the consideration of the boundary constraints. As a benchmark, the widely used NMPC formulation based on direct multiple shooting (MS) method is served to assess the convergence and the excellent real-time performance of this method. The closed-loop performance is demonstrated by the swing-up of an unstable numerical experiment.

Stackelberg均衡下での追跡回避型ゲーム問題に関する数値解法

This study presents the algorithm of a numerical analysis for pursuit-evasion type's differential games under the Stackelberg equilibrium. The Semi-DCNLP method is introduced as the optimizer for the solution of the flight path optimization under the Stackelberg equilibrium. For optimal flight path of the evader for an optimally pursuit spacecraft under the Stackelberg equilibrium, the Semi-DCNLP finds the proper trajectories which are corresponded with the trajectories obtained using the direct shooting method.

Optimization and Analysis on Trajectory with Multiple Constraints for Hypersonic Air-vehicle

The trajectory optimization technology is one of the key technologies for hypersonic air-vehicle. There are multiple constraints in the process of hypersonic flight, such as uncertainty of flight environment, thermal current, dynamic pressure and overload. The trajectory optimization of hypersonic air-vehicle is facing with a great challenge. This article studies the direct shooting method, the Gauss pseudo spectral method and sequential gradient-restoration algorithm, among which the direct shooting method simply makes the control variables discrete in the time domain, and obtains the status value by explicit numerical integration; Gauss pseu- do spectral method makes the status variable and control variable discrete in a series of Gauss points, and con- structs multinomial to approximate to the status and control variable by taking the discrete points as the nodes; sequential gradient-restoration algorithm uses iteration to meet the constraints and minimize the increment of initial value of control and status variable in order to constantly approximate to the optimal solution on condition that the constraints meet first order approximation. Finally this article conducts a numerical simulation by taking the diving segment of hypersonic air-vehicle as an example for comparative analysis on those three algorithms respectively from, such as, the initial value selection, constraint handling, convergence speed and calculation accuracy. The simulation result indicates Gauss pseudo spectral method is a method with fairly good comprehen- sive performance.

Technical Note: Space Station Zero Propellant Maneuver Path Planning Considering SGCMG Saturation and Singularity

The zero-propellant maneuver is an advanced concept of the spacecraft attitude control. The problem of zero-propellant maneuver using single gimbal control moment gyroscopes is researched in this paper. The saturation and singularity problems of single gimbal control moment gyroscopes are analyzed, and the performance indexes to measure saturation and singularity are proposed. The path planning model considering the gimbal steering law is established using modified rodrigues parameter. Then, a hybrid approach combining the pseudospectral method and direct shooting method is proposed to solve the nonlinear path planning problem with complicated constraints. Finally, the numerical simulation is executed to validate the effectiveness of the proposed method. It is shown that an optimized attitude maneuver path satisfying the saturation and singularity constraints can be obtained and the zero-propellant maneuver using single gimbal control moment gyroscopes can be successfully achieved.

Study of the Optimal Design for the Gliding Trajectory

The optimal design for the gliding trajectory is studied. The particle trajectory model in the longitudinal plane is established. Aiming at solving the problems in numerical solution ,the direct shooting method based on the interpolating function of Akima is put forward .In direct collocation method, the parameters of the trajectory with approximate maximum lift-drag ratio is adopted as the starting value of the optimal variables. The trajectory with approximate maximum lift-drag ratio could be obtained by the search method. The transfer approach of turning the problem of trajectory optimization to the problem of parameter optimization by the direct shooting method is illustrated. To one kind of guided vehicles in simulation, the simulation results prove the practicability of the direct shooting method.

Trajectory Optimization for Hypersonic Vehicle Satisfying Maneuvering Penetration

A trajectory optimization method for hypersonic vehicle in glide phase satisfying maneuvering penetration is proposed. Divide the dangerous zones that the hypersonic vehicle may encounter during glide phase into avoidable no-fly zones and avoidless no-fly zones. Take the avoidable no-fly zones as path constraints to join the trajectory optimization. To penetrate the avoidless no-fly zones, trajectory is programmed by some maneuvering policy. Direct shooting method is used to discretize the control variable to piecewise constant functions. So the optimal control problem is transferred to a nonlinear programming (NLP) problem, and solved by the serial quadratic program (SQP) method.

Motion Planning for Vibration Reducing of Free-floating Redundant Manipulators Based on Hybrid Optimization Approach

This paper is concerned with optimal motion planning for vibration reducing of free-floating flexible redundant manipulators. Firstly, dynamic model of the system is established based on Lagrange method, and the motion planning model for vibration reducing is proposed. Secondly, a hybrid optimization approach employing Gauss pseudospectral method (GPM) and direct shooting method (DSM), is proposed to solve the motion planning problem. In this approach, the motion planning problem is transformed into a non-linear parameter optimization problem using GPM, and genetic algorithm (GA) is employed to locate the approximate solution. Subsequently, an optimization model is formulated based on DSM, and sequential quadratic programming (SQP) algorithm is used to obtain the accurate solution, with the approximate solution as an initial reference solution. Finally, several numerical simulations are investigated, and the global vibration or residual vibration of flexible link is obviously reduced by the joint trajectory which is obtained by the hybrid optimization approach. The numerical simulation results indicate that the approach is effective and stable to the motion planning problem of vibration reducing.

Translational zero-disturbance curve and its application to zero-disturbance motion planning of space manipulator system

A new motion planning method is proposed for enlarging the solvable space of zero-disturbance motion planning for the space free-floating manipulator system. First, a class of translational zero-disturbance curves is put forward for the first time. The equation of translational zero-disturbance curve is deduced using the nonholonomic constraint of the manipulator system, and its characteristics are also discussed. Second, the zero-disturbance curve of the whole operating process is divided into two segments. The first one is a translational zero-disturbance curve which passes through the target point. Another one is a common zero-disturbance curve which passes through the original point and intersects with the translational zero-disturbance curve. Finally, the common zero-disturbance curve is obtained by a hybrid programming strategy based on Gauss pseudo-spectral method(GPM) and direct shooting method(DSM). The numerical simulation results indicate that the proposed method is effective, and that the solvable space of this method almost covers the whole work space of the manipulator system.

Trajectory optimization for hypersonic boost‐glide missile considering aeroheating

PurposeThe purpose of this paper is to optimize the downrange for hypersonic boost‐glide (HBG) missile under near‐real condition, and to validate the suitability of proposed wall cooling materials.Design/methodology/approachThe trajectory optimization problem is characterized by a boost phase followed by a glide phase. A multi‐phase trajectory optimization tool is adopted to optimize the downrange. The associated optimal control problem has been solved by selecting a direct shooting method. The dynamics has been transcribed to a set of nonlinear constraints and the arising nonlinear programming problem has been solved through a sequential quadratic programming solver. An aerothermodynamics analysis method is introduced to calculate the aerodynamic heating at nose, leading edge, and ventral centerline regions.FindingsHBG missile is suitable for long‐range attack, and the optimal trajectory solved is a novel boost‐glide‐skip trajectory, i.e. boost firstly, glide secondly, and skip at last. The proposed wall materials are valid.Originality/valueThis paper provides further study on the methods of trajectory design and aerothermodynamics analysis for HBG missile.

Optimal and Feedback Path Planning for Cooperative Attack

This paper considers cooperative path planning for aerial munitions during the attack phase of a mission against ground targets. It is assumed that sensor information from multiple munitions is available to refine an estimate of the target location. Based on models of the munition dynamics and sensor performance, munition trajectories are designed that enhance the ability to cooperatively estimate the target location. The problem is posed as an optimal control problem using a cost function based on the variances in the target-location estimate. These variances are computed by fusing the individual munition measurements in a weighted least-squares estimate. Solutions to the problem are found using a direct-shooting method. These solutions are compared with trajectories developed by an alternative suboptimal feedback-guidance law. This feedback law produces solutions with far less numerical expense and with a performance very close to the best known solutions. The reduction in target-location uncertainty associated with these trajectories could enable the attack of targets with greater precision using smaller, cheaper munitions.

Near-Optimal Missile Avoidance Trajectories via Receding Horizon Control

The paper introduces a receding horizon control scheme for obtaining near-optimal controls in a feedback form for an aircraft trying to avoid a closing air-to-air missile. The vehicles are modeled as point-masses. Rotation kinematics of the aircraft are taken into account by limiting the pitch and roll rates as well as the angular accelerations of the angle of attack and the bank angle. The missile utilizes proportional navigation and it has a boost-sustain propulsion system. In the proposed scheme, the optimal controls of the aircraft over a short planning horizon are solved on-line by the direct shooting method at each decision instant. Thereafter, the state of the system is updated by using only the first controls in the sequence, and the process is repeated. The performance measure defining the objective of the aircraft can be chosen freely. In this paper, six performance measures consisting of the capture time, closing velocity, miss distance, gimbal angle, tracking rate, and control effort of the missile are considered. The quality of the receding horizon solutions computed by the scheme is validated by comparing them to the off-line computed optimal open-loop solutions.

Optimal control and real‐time optimization of mechanical multi‐body systems

AbstractNumerical methods for the solution and real‐time optimization of parametric optimal control problems for mechanical multi‐body systems are considered. The equations of motion are given as a differential‐algebraic equation (DAE) system of index 3 augmented by additional index 1 constraints. It turns out that the sensitivity analysis of parametric nonlinear programming problems is of central importance for the numerical solution of the optimal control problem by a direct shooting method and its real‐time optimization. On the one hand, a sensitivity analysis of the discretized optimal control problem provides a real‐time approximation of the optimal solution of a perturbed optimal control problem. On the other hand, a sensitivity analysis of the projection method used for consistent initialization yields consistent initial values of sensitivity matrices needed within the iterative solution process of the direct shooting method. Two examples conclude the article.

A moving horizon technique for the simulation of automobile test‐drives

AbstractThe test‐drive of an automobile along a given test‐course can be modeled by formulation of a suitable optimal control problem. For the numerical solution the optimal control problem is discretized by a direct shooting method and transformed into a finite dimensional nonlinear optimization problem. With increasing length of the test‐course, the dimension of the nonlinear optimization problem increases as well and its numerical solution becomes very difficult due to stability reasons. Therefore a moving horizon technique with reduced range of vision for the test‐driver is introduced. Instead of treating the complete test‐course, a comparatively short local sector is considered on which a corresponding local optimal control problem can be solved comfortably. The local solutions are then combined by suitable transient conditions. A numerical example with a realistic car model is given.

Trajectory optimization involving sloshing media

AbstractThis paper is concerned with the optimization of the transport motion of an open topped fluid filled container within a warehouse environment. In particular, optimal trajectories of the motion of the driver–container system in two‐dimensional space will be investigated via numerical solutions of the model equations using sequential quadratic programming. The fluid and the mechanical facility that moves the container are subject to several constraints. The objective of the optimization is the time to transport the container from an initial position to its final destination within the warehouse. Optimization criteria are investigated to control the movement of the fluid within the container. The systems of ordinary and partial differential equations, representing the dynamics of the models are solved numerically using a direct shooting method. The resulting non‐linear programming problem is solved using sequential quadratic programming (SQP). Copyright © 2002 John Wiley & Sons, Ltd.