Hp-adaptive Pseudospectral Method Research Articles

Application of the hp-adaptive pseudospectral method in spacecraft orbit pursuit-evasion game

In this paper, a Cartesian model of the orbit pursuit-evasion game under J2 perturbations is constructed, and a step-by-step solution method applicable to this problem, namely the Hp-Adaptive Pseudospectral Hybrid Optimal Algorithm, is proposed by analyzing the principle of the pseudospectral method and the costate mapping principle. We first use the hp-adaptive Radau pseudospectral method to solve the one-sided interception problem to obtain a tentative guess of the costate and then get the saddle-point solution of the game problem by solving the two-point boundary value problem through the Newton method. The numerical results show that the hybrid method proposed in this work can efficiently find the optimal trajectory of interception and the optimal control laws of the game problem and have high accuracy. In addition, the effects of J2 perturbations and the initial orbital altitude of the evader on the game results and the algorithm itself are discussed in this paper.

Distributed safe trajectory optimization for large-scale spacecraft formation reconfiguration

This paper presents a distributed trajectory optimization algorithm for the reconfiguration of large-scale spacecraft formation, which uses a low thrust propulsion system to safely guide the spacecraft formation to a desired formation. Formation reconfiguration is formulated as a trajectory optimization problem with complex constraints, and the relative motion is accurately described by a nonlinear relative dynamics considering J2 perturbation. The collision avoidance constraint is convexified into a tangent plane to ensure the accuracy of the solution. The resulting nonlinear trajectory optimization problem is solved by applying the hp-adaptive pseudospectral method to convert it into a nonlinear programming problem. In order to overcome the disadvantage of huge computation of centralized algorithm considering collision avoidance, “predicted trajectory” is introduced to transmit information between spacecraft, and the parallel computing is implemented. Finally, a numerical simulation is given to verify the computational efficiency and collision avoidance effectiveness of the proposed distributed algorithm.

Boost-Phase Trajectory Planning with the Nonregular Reachable Area Constraints

During emergency return or mission change, the boost-glide vehicle needs to meet the nonregular reachable area constraints (NRACs). Therefore, this paper introduced NRACs into boost-phase trajectory planning to extend the maneuvering range and enhance the mission adaptability of boost-glide vehicle. Firstly, the library of the reachable area boundaries was constructed with the fast computation method under the deviations of boost terminal states, and the polynomials were used to fit the boundary parameters. Secondly, the nonlinear mapping relationship between the reachable area boundary parameters and the deviations of boost terminal states was obtained by the deep neural networks (DNNs). Then, the new boundary parameters were obtained by the constraint transformation rules under NRACs and the separation window constraints and transformed into the constraints of the boost-phase terminal states by DNNs. Finally, the hp-adaptive pseudospectral method (hpPM) was adopted to complete the trajectory planning considering the path and terminal constraints. The simulation results showed that the proposed trajectory planning method considering NRACs had high trajectory planning accuracy and good deviation adaptability and exhibited excellent performance in the adjustment of reachable area. This study provides theoretical support for integrate mission decisions and trajectory planning of boost-glide vehicles.

Trajectory optimization of morphing aircraft based on multi-fidelity model

Morphing aircraft can flexibly change its aerodynamic shape to adapt to the varying flight conditions during a flight. Compared with the traditional fixed shape aircraft, it has a very obvious advantage. This paper proposed a solution flow based on the multi-fidelity model for the morphing aircraft with morphing wings, and the optimal trajectory and morphing rules are studied. The angle of attack, Mach number, sweep angle and axial position of the morphing wing are defined as variables for generating training data for building the multi-fidelity Kriging model, which is used to predict the aerodynamic performance of the aircraft. Based on the hp-adaptive pseudospectral method, the model is used as aerodynamic input to establish the optimization process of morphing rules, and the trajectory optimization is carried out for the contrast fixed wing aircraft and morphing aircraft with the goal of minimum fuel consumption, respectively. The control parameters such as morphing parameters, angle of attack and engine control parameters are optimized simultaneously while meeting the flight mission requirements. The results show that the morphing aircraft has higher climbing and descending efficiency, and the optimal trajectory has obvious advantages. Moreover, the research flow proposed in this paper is universal, which can effectively reduce the CFD calculation cost and improve the efficiency of trajectory optimization of the variable shape vehicle.

A modified HP-adaptive pseudospectral method for multi-UAV formation reconfiguration

The muti-UAV formation reconfiguration is a challenging problem due to the state constraints. This is because state constraints usually lead to a large-dimensional nonlinear program (NLP), which highly increases the computational burden. Inspired by the idea of an exact penalty function scheme, a novel computational method is developed by modifying hp-adaptive pseudospectral method (PM). More specifically, hp-adaptive PM is introduced to transform the UAVs reconfiguration control problem into an optimization problem, while the exact penalty function is utilized to remove the inter-UAV distance state constraints. The original problem is then converted into a standard NLP problem which can be handled by the gradient-based method with the off-the-shelf software packages. Comparing with the hp-adaptive PM, the proposed method dramatically decreases the computational overhead. Numerical simulations are provided for illustrating the effectiveness of the proposed scheme.

UAV trajectory optimization using chance-constrained second-order cone programming

It is challenging to generate optimal trajectories for nonlinear dynamic systems under external disturbances. In this brief, we present a novel approach for planning safe trajectories of the chance-constrained trajectory optimization problems with nonconvex constraints. First, the chance constraints are handled by deterministic ones which show its availability. We derive an iterative convex optimization method to solve the optimal control problem. Then the chance-constrained optimal control (CCOCP) problem is reformed to be a nonlinear programming problem (NLP) through the hp-adaptive pseudospectral method. An iterative successive linearization algorithm is detailed to convex the NLP to be a convex optimization one which described as a second-order cone programming problem. We demonstrate the proposed approach on a 3-DoF of unmanned aerial vehicle system under chance-constrained. The simulation results show reliable solutions for the UAV chance-constrained trajectory optimization problem.

Optimum Path Tracking Control for the Inverse Problem of Vehicle Handling Dynamics Based on the hp-Adaptive Gaussian Pseudospectral Method

We propose a vehicle path-tracking method based on the hp-adaptive Gaussian pseudospectral method (GPM), which tackles the problem of the slow convergence speed of the optimal control of vehicle path tracking. First, we establish a kinematic vehicle model by considering the path constraints and boundary constraints during the process of tracking the described path of a vehicle. Subsequently, finding the minimum error of the lateral distance between the prescribed path and the expected trajectory is set as the performance objective function. Finally, the vehicle path tracking problem is transformed into an optimal control problem. The optimization algorithm is combined with the sequential quadratic programming algorithm to optimize problems related to the control and state variables and the boundary and path constraints. The simulated results show that the hp-adaptive GPM can improve the convergence rate of the optimal control problem for vehicle path tracking. The proposed hp-adaptive pseudospectral method has a higher solving efficiency compared with traditional approaches for the vehicle path-tracking problem. Concurrently, the verification results of a real vehicle test indicate the feasibility of the proposed algorithm for solving the vehicle path tracking problem. This research provides valuable insight into the design work of lane changes and is an important step towards the design of feedback control laws for path tracking.

Optimal lateral control for autonomous vehicle based on multi-stage handling inverse dynamics

In this paper, an optimal lateral control method for autonomous vehicle based on multi-stage handling inverse dynamics is proposed. By applying this method, the vehicle can be controlled to complete the corresponding driving task without tracking a certain ideal state variable trajectory. Firstly, a separate-stage handling inverse dynamics problem is described; and by setting the connections between each stage, a multi-stage handling inverse dynamics model is developed; then we convert the inverse dynamics problem into an optimal control problem (OCP). Secondly, an improved hp-adaptive Radau pseudospectral method ( hp-RPM) is applied to discrete the continuous OCP; and then the sequential quadratic programming is used to solve the problem. In addition, the results of contrast verification show that, when solving non-smooth problem, the improved hp-RPM has higher computational efficiency than hp-RPM. Finally, simulations under three typical cases and bench tests illustrate the effectiveness of the proposed method.

Deriving a Control-Oriented Model for an Axisymmetric Vehicle With the Power-Law Revolution Nose

The purpose of this paper is to construct a new general vehicle model as an open fundamental material for the guidance and control research. In this study, parameterized configuration, aerodynamics calculation, control-oriented modeling, stability analysis, and nominal trajectory design are performed for the general vehicle model. First, the aerodynamic configuration is parameterized as an axisymmetric body with a power-law revolution nose. Then, an engineering method considering inviscid flow, base drag and skin friction is used for the aerodynamics calculation, and a control-oriented fitting model of longitudinal aerodynamics is established based on the analysis of the correlation between aerodynamic force and the parameters of Mach number, attack angle, elevator deflection and height. Next, the aerothermodynamic environment prediction of power-law revolution axisymmetric hypersonic vehicle (PRAHV) is discussed, and the nose heating rate formula of PRAHV is established. The stability analysis and nominal trajectory design of PRAHV is performed based on the fitting model and the heating rate formula. The stability analysis shows that both the static stability and dynamic stability of the vehicle are unstable. The nominal trajectory of unpowered longitudinal maneuvering is achieved by the hp-adaptive pseudospectral method, which demonstrated that the availability of the control-oriented model established in this paper. In conclusion, this work provides a fundamental object for further study of vehicle guidance, control, and evaluation.

Modeling and Energy-Optimal Control for High-Speed Trains

This article presents a microscopic simulation model and corresponding energy-optimal control method for high-speed trains. The vehicle-line-coupled longitudinal dynamics model and nonlinear power loss model of the traction transmission system (TTS) for different traction force and speed pairs are discussed. Based on these detailed models, an energy-optimal control problem is formulated by systematically considering speed limitations, punctuality, and control force constraints in neutral sections of contact wire. Then, the formulation is transformed into a multiple phases’ optimal control model based on some key positions, and the hp-adaptive pseudospectral method is applied to obtain the optimal results. The proposed dynamics model and the energy consumption model are validated by the comparison of model predictions and measured data. The simulation results show that the energy efficiency of the optimal results with a nonlinear power loss model is improved compared to the traditional method with constant efficiency of the TTS. The energy consumption can be reduced by 12.67%, comparing optimal speed profiles against measured results on commercial lines of China. Finally, the influences of gradients are analyzed, and the coast-traction pairs can replace cruising strategy on downhill sections.

Optimal Design of Cooperative Penetration Trajectories for Multiaircraft

At present, two kinds of shortages exist in the research on cooperative combat. One is that radar detection threat (which cannot be ignored) is rarely considered. The other is that limited efforts have been made on the cooperative penetration trajectories under the conditions of long distance, vast airspace, and wide speed range. In order to offset the shortages of the research on cooperative combat, the penetration trajectory optimization method considering the influence of aircraft radar cross-section (RCS) and the cooperative penetration strategy is proposed in this study. Firstly, the RCS data are calculated by the physical optics (PO) method. The radar detection threat model is established considering the influence of the aircraft RCS. Then, a trajectory optimization framework with the dynamic model, constraint conditions, and optimal objectives is formed. Using the hp-adaptive Radau pseudospectral method, the optimal control problem for a single aircraft flight is solved. Finally, a cooperative penetration strategy is proposed to solve the cooperative penetration problem of multiaircraft. The impact time and angle constraints are given, and the virtual target point is introduced for terminal guidance. Two cases are simulated and verified. Simulation results demonstrate that the proposed method is effective. The single aircraft can effectively penetrate, and the multiaircraft can fulfill the requirement of cooperative impact time and angle under the condition of meeting the minimum threat of radar detection.

Path Planning of Autonomous Parking Based on hp-adaptive Pseudospectral Method

Path Planning of Autonomous Parking Based on hp-adaptive Pseudospectral Method

Mission analysis and optimal control for cislunar mission with spinning tether system in hyperbolic orbits

This paper presents a concept of using spinning tether system for interstellar mission, and designs the optimal trajectory and the controller required for the interstellar mission. The interstellar mission carried out by a two-body spinning tether system relies on the momentum-exchange techniques, which equipped with ion thrusters can greatly economize the propellant. In order to ensure that the system can achieve a state of being able to complete the mission under the given optimal conditions, an optimal trajectory is designed by using hp-adaptive pseudospectral method. Then, an adaptive robust sliding mode controller is given based on radial basis function neural network to make the system with uncertainty reach the optimal trajectory. Finally, numerical simulations are presented to evaluate the effectiveness of the optimal trajectory and the controller.

Study on the optimal lane changing steering input under different road adhesion coefficients

To study the influence of road adhesion coefficient on lane changing steering input, vehicle handling inverse dynamics based on hp-adaptive Radau pseudospectral method is proposed in this article. First, a steering inverse dynamics model is established and then the inverse dynamics problem is converted into an optimal control problem. Second, by applying the hp-adaptive Radau pseudospectral method, the optimal control problem is converted into a nonlinear programming problem, which is then solved through sequential quadratic programming. Besides, two contrast verifications are carried out to validate the correctness and advantages of the proposed method. The simulation results show that the obtained control input satisfies the vehicle dynamic constraints. And comparing with Gauss Pseudospectral Method (GPM), hp-adaptive Radau pseudospectral method has higher computational accuracy and computational efficiency when solving non-smooth problem. The study of optimal steering input can provide guidance for driver’s lane changing behavior, so that the accident rate caused by human factors can be decreased. The proposed method can provide a reference value into the active safety of manned and unmanned vehicles.

Mars entry trajectory planning using robust optimization and uncertainty quantification

Due to the existence of obvious uncertainties within Mars atmospheric entry, design and analysis of the optimal trajectory under uncertainties play a significant role in reducing flight risk and maintaining guidance performance and delivery accuracy. In this paper, using robust optimization strategy and uncertainty quantification technique, we developed an innovative approach for planning the Mars entry trajectory under uncertainties. The main contribution of this work is twofold: one is considering both dynamics parameter uncertainty and initial state uncertainty, and the other is simultaneously focusing on the reliability of constraint satisfaction and insensitivity of performance index towards uncertainties. First, Mars entry trajectory planning problem under uncertainties is formulated as a robust optimization problem. Second, the uncertainty propagation with regard to Mars entry dynamics is derived, and the quantification of objective function and constraints under uncertainties is conducted. Third, the robust trajectory optimization problem is transformed into an equivalent optimal control problem in the expanded higher-dimensional state space by polynomial chaos expansion, and the solution is obtained by the hp-adaptive pseudospectral method. Finally, the effectiveness of the proposed method is verified through two Mars entry optimal trajectory cases. And the obtained optimal trajectories are reliable and evidently robust to uncertainties compared to traditional deterministic optimization and reliability-based optimization.

Attitude Maneuver of Dual Rigid Bodies Spacecraft Using hp-Adaptive Pseudo-spectral Method

The spatial attitude optimal control problem of a spacecraft system with dual rigid bodies interconnected by a frictionless universal joint is considered. Based on the conservation of angular momentum, the dynamical model of the system is deduced using Quaternion. For there exists non-square matrix in the dynamical equation caused by the representation of Quaternion, the first-order derivative of the scalar section in Quaternion is employed to serve as a pseudo-control input for the system. Then the motion planning problem of the system can be discretized into a nonlinear programming problem based on hp-adaptive Radau pseudo-spectral method. Comparing with the previous works, the hp-adaptive method and the pseudo-control input is combined together to deal with the potential non-square matrix. Also as a contrast to the use of pseudo-control input employed in hp-adaptive method, another method (the Radau pseudo-spectral method) and the theory of pseudo-inverse matrix is adopted. The effectiveness of hp-adaptive method on the system expressed by pseudo-control input and the rationality of the Radau pseudo-spectral method on the system expressed by pseudo-inverse matrix is demonstrated and verified perfectly through numerical simulation.

Time-Optimal Spacecraft Reorientation for the Observation of Multiple Asteroids

Abstract To observe multiple asteroids in a short time, the time-optimal reorientation is investigated here for the rest-to-rest reorientation of a generic rigid spacecraft. First, the problem is formulated and solved using the hp-adaptive pseudospectral method. It is found that there exist not merely bang-bang but also singular solutions for the problem. Then, the bang-bang and singular solutions are discussed by various cases. The results reveal that the optimal solution is essentially a balance between the larger torque, the shorter angular path and the less moment of inertia. For bang-bang solutions, the total number of switches varies with the reorientation angle and the moment of inertia. The number of switches is usually 5 or 6 and does not exceed 8. For singular solutions, either one and two controls singular are possible. The singular optimal solution is probably optimal for the slender rod-like spacecraft. Finally, an analytical estimation method is proposed for determining the range of the optimal time. The results of the simulations indicate that the proposed method is real-time and highly accurate. All formulas of this paper are derived in canonical units and therefore apply to any rigid spacecraft.

Trajectory Planning of the Unmanned Aerial Vehicles with Adaptive Convex Optimization Method

Abstract In this paper, a constrained unmanned aerial vehicles trajectory optimization problem is formulated. We present an iterative convex programming algorithm to solve the optimal control problem, which consists of both nonlinear dynamics and path constraints. A hp-adaptive pseudospectral method is introduced to transform the original problem into nonlinear programming. Furthermore, in order to improve the computational ability, nonlinear dynamics and path constraints are convexified by convexification techniques. The trajectory planning problem is formulated as a convex programming problem. We evaluate the proposed strategy by some numerical results. The results indicate that the algorithm can get better performance compared with other typical optimal control solvers.

Optimal Maneuver Trajectory for Hypersonic Missiles in Dive Phase Using Inverted Flight

To determine the optimal trajectory for the dive phase of a hypersonic missile, a maneuver strategy using inverted flight is presented. The combat scene that the hypersonic missile attacks the fixed target on the ground is considered in this paper. Particularly, a maneuvering form named inverted flight is first applied to the hypersonic missiles. Afterward, an optimal trajectory is designed by minimizing the attack time with a preset terminal flight path angle, where the constraints of the angle of attack, dynamic pressure, heating transfer rate, and normal overload are taken into account. In order to solve the designed trajectory optimization problem, an improved hp-adaptive pseudospectral method with mesh size reduction is presented. The simulation results show that the proposed algorithm can significantly reduce the mesh scale with satisfactory accuracy and the trajectory obtained by the proposed algorithm is in accordance with the actual flight law. Furthermore, contrast simulation demonstrates that the inverted flight has better trajectory performance than normal flight.

Integrated guidance for Mars entry and powered descent using reinforcement learning and pseudospectral method

Traditional studies on Mars entry, descent, and landing usually take a divide-and-conquer approach where each phase is investigated separately. This paper proposed a new approach to achieve optimal integrated guidance for mid-lift high-mass Mars entry and powered descent. First, optimal Mars atmospheric entry and optimal powered descent are respectively modeled. Second, optimal handover problem is formulated to integrate the optimal Mars entry and powered descent problems in a collaborative optimization manner. Third, hp-adaptive pseudospectral method is employed to capture optimal entry guidance and propellant-optimal powered descent guidance. Reinforcement learning methodology is adopted to obtain the optimal handover, according to a large amount of trial results obtained from solving optimal Mars entry and powered descent problems. Finally, numerical simulations verified the optimality, robustness, and accuracy of the proposed technique through comparing with successive convexification approach under the nominal conditions and uncertainties, and the comparison results demonstrated the significant benefit of the integrated optimal guidance.