Radau Pseudospectral Method Research Articles

Pseudospectral Real-Time Optimal Energy Control with Safety Constraints for Heavy-Haul Trains

Heavy-haul trains must be energy-efficient and safe during their operations. Owing to the multidimensional high-order nonlinear characteristic of heavy-haul trains, which include numerous cars, this paper proposes a uniform pseudospectral real-time closed-loop optimal control framework to minimize the energy consumption with control inputs and state constraints based on the Radau Pseudospectral Method (RPM). In the framework, in order to ensure safe running of the heavy-haul train, the desired in-train force and speed limit requirements are formulated as constraints of optimal control. Simultaneously, a constrained closed-loop optimal control is constructed by using the receding horizon control principle and pseudospectral observer, in which RPM is leveraged to obtain real-time optimal solutions. The effectiveness of the proposed approach is verified from simulation results.

Optimal attitude trajectory planning method for CMG actuated spacecraft

This paper presents an optimal attitude trajectory planning method for the spacecraft equipped with control moment gyros as the actuators. Both the fixed-time energy-optimal and synthesis performance optimal cases are taken into account. The corresponding nonsingular attitude maneuvering trajectories (i.e. open-loop control trajectories) with the consideration of a series of constraints are generated via Radau pseudospectral method. Compared with the traditional steering laws, the optimal steering law designed by this method can explicitly avoid the singularity from the global perspective. A linear quadratic regulator closed-loop controller is designed to guarantee the trajectory tracking performance in the presence of initial errors, inertia uncertainties and external disturbances. Simulation results verify the validity and feasibility of the proposed open-loop and closed-loop control methods.

Variable-Time-Domain Online Neighboring Optimal Trajectory Modification for Hypersonic Interceptors

The predicted impact point (PIP) of hypersonic interception changes continually; therefore the midcourse guidance law must have the ability of online trajectory optimization. In this paper, an online trajectory generation algorithm is designed based on neighboring optimal control (NOC) theory and improved indirect Radau pseudospectral method (IRPM). A trajectory optimization model is designed according to the features of operations in near space. Two-point boundary value problems (TPBVPs) are obtained based on NOC theory. The second-order linear form of transversality conditions is deduced backward to express the modifications of terminal states, costates, and flight time in terms of current state errors and terminal constraints modifications. By treating the current states and the optimal costates modifications as initial constraints and perturbations, the feedback control variables are obtained based on improved IRPM and nominal trajectory information. The simulation results show that when the changes of terminal constraints are not relatively large, this method can generate a modified trajectory effectively with high precision of terminal modifications. The design concept can provide a reference for the design of the online trajectory generation system of hypersonic vehicles.

Trajectory Optimization Based on Multi-Interval Mesh Refinement Method

In order to improve the optimization accuracy and convergence rate for trajectory optimization of the air-to-air missile, a multi-interval mesh refinement Radau pseudospectral method was introduced. This method made the mesh endpoints converge to the practical nonsmooth points and decreased the overall collocation points to improve convergence rate and computational efficiency. The trajectory was divided into four phases according to the working time of engine and handover of midcourse and terminal guidance, and then the optimization model was built. The multi-interval mesh refinement Radau pseudospectral method with different collocation points in each mesh interval was used to solve the trajectory optimization model. Moreover, this method was compared with traditional h method. Simulation results show that this method can decrease the dimensionality of nonlinear programming (NLP) problem and therefore improve the efficiency of pseudospectral methods for solving trajectory optimization problems.

A pseudospectral method for solving optimal control problem of a hybrid tracked vehicle

This study explored the feasibility of using the Radau pseudospectral method (RPM) to optimize the energy management strategy for a hybrid tracked vehicle. The engine–generator set and the battery pack of the serial hybrid tracked vehicle were modeled and validated through the bench test. A DC-DC converter was equipped between the battery pack and the DC bus in this hybrid powertrain, which increased the flexibility of energy distribution between the engine–generator set and the battery. It was simplified as a voltage regulator in the hybrid powertrain model. The power demand during the vehicle operation was calculated according to the vehicle dynamics and driving schedules. The optimal control problem was formulated to minimize the fuel consumption through regulating the power distribution properly between the engine–generator set and battery pack during a typical driving schedule. The RPM was applied to transform the optimal control problem to a finite-dimensional constrained nonlinear programming problem. A comparison of the solutions from RPM and dynamic programming showed that the former offers the higher computation efficiency and better fuel economy.

A modified pseudospectral method for solving trajectory optimization problems with singular arc

This paper presents a direct method based on Legendre–Radau pseudospectral method for efficient and accurate solution of a class of singular optimal control problems. In this scheme, based on a priori knowledge of control, the problem is transformed to a multidomain formulation, in which the switching points appear as unknown parameters. Then, by utilizing Legendre‐Radau pseudospectral method, a nonlinear programming problem is derived which can be solved by the well‐developed parameter optimization algorithms. The main advantages of the present method are its superior accuracy and ability to capture the switching times. Accuracy and performance of the proposed method are examined by means of some numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.

Energy Management Optimization for a Hybrid Tracked Vehicle Using the Radau Pseudospectral Method

This study explored the feasibility of using the Radau pseudospectral method (RPM) to optimize the energy management strategy for a hybrid tracked vehicle. The engine–generator set and the battery pack of the serial hybrid tracked vehicle were modeled and validated through the bench test. A DC-DC converter was equipped between the battery pack and the DC bus in this hybrid powertrain, which increased the flexibility of energy distribution between the engine–generator set and the battery. The optimal control problem was formulated to minimize the fuel consumption through regulating the power distribution properly between the engine–generator set and battery pack during a typical driving schedule. The RPM was applied to transform the optimal control problem to a finite-dimensional constrained nonlinear programming problem. A comparison of the solutions from RPM and dynamic programming showed that the former offers the higher computation efficiency and better fuel economy.

Fuel-optimal low-thrust formation reconfiguration via Radau pseudospectral method

This paper investigates fuel-optimal low-thrust formation reconfiguration near circular orbit. Based on the Clohessy–Wiltshire equations, first-order necessary optimality conditions are derived from the Pontryagin’s maximum principle. The fuel-optimal impulsive solution is utilized to divide the low-thrust trajectory into thrust and coast arcs. By introducing the switching times as optimization variables, the fuel-optimal low-thrust formation reconfiguration is posed as a nonlinear programming problem (NLP) via direct transcription using multiple-phase Radau pseudospectral method (RPM), which is then solved by a sparse nonlinear optimization software SNOPT. To facilitate optimality verification and, if necessary, further refinement of the optimized solution of the NLP, formulas for mass costate estimation and initial costates scaling are presented. Numerical examples are given to show the application of the proposed optimization method. To fix the problem, generic fuel-optimal low-thrust formation reconfiguration can be simplified as reconfiguration without any initial and terminal coast arcs, whose optimal solutions can be efficiently obtained from the multiple-phase RPM at the cost of a slight fuel increment. Finally, influence of the specific impulse and maximum thrust magnitude on the fuel-optimal low-thrust formation reconfiguration is analyzed. Numerical results shown the links and differences between the fuel-optimal impulsive and low-thrust solutions.

Analysis of the inertially symmetric rigid spacecraft time‐optimal three‐axis reorientation

SummaryThis paper investigates the classical time‐optimal rest‐to‐rest three‐axis reorientation of the inertially symmetric rigid spacecraft. First‐order necessary optimality conditions are derived from the Pontryagin's maximum principle. Then, control structures (i.e., switching times and control torques) for the time‐optimal solution with five, six, and seven switches are given. For any five‐switch, six‐switch, or seven‐switch time‐optimal solution, a finite number of control structures exist, and relations among the control structures and their associated time‐optimal solutions are analytically derived. By utilizing the control structure, efficient numerical optimization algorithm based on multiple‐interval Radau pseudospectral method is proposed. Numerical results show that, after rounding to integer, five‐switch and six‐switch time‐optimal solutions exist for rotation angles on the interval [1,180] deg, and s es on the interval [1,72] deg. Finally, time‐optimal solutions for typical rotation angles are given to illustrate and validate the new findings. Copyright © 2016 John Wiley & Sons, Ltd.

Indirect Radau pseudospectral method for the receding horizon control problem

To solve the receding horizon control (RHC) problem in an online manner, a novel numerical method called the indirect Radau pseudospectral method (IRPM) is proposed in this paper. Based on calculus of variations and the first-order necessary optimality condition, the RHC problem for linear time-varying (LTV) system is transformed into the two-point boundary value problem (TPBVP). The Radau pseudospectral approximation is employed to discretize the TPBVP into well-posed linear algebraic equations. The resulting linear algebraic equations are solved via a matrix partitioning approach afterwards to obtain the optimal feedback control law. For the nonlinear system, the linearization method or the quasi linearization method is employed to approximate the RHC problem with successive linear approximations. Subsequently, each linear problem is solved via the similar method which is used to solve the RHC problem for LTV system. Simulation results of three examples show that the IRPM is of high accuracy and of high computation efficiency to solve the RHC problem and the stability of closed-loop systems is guaranteed.

Optimal Point-to-Point Motion Planning of Flexible Parallel Manipulator with Multi-Interval Radau Pseudospectral Method

Optimal point-to-point motion planning of flexible parallel manipulator was investigated in this paper and the 3RRR parallel manipulator is taken as the object. First, an optimal point-to-point motion planning problem was constructed with the consideration of the rigid-flexible coupling dynamic model and actuator dynamics. Then, the multi-interval Legendre–Gauss–Radau (LGR) pseudospectral method was introduced to transform the optimal control problem into Nonlinear Programming (NLP) problem. At last, the simulation and experiment were carried out on the flexible parallel manipulator. Compared with the line motion of quantic polynomial planning, the proposed method could constrain the flexible displacement amplitude and suppress the residue vibration.

The Ascent Trajectory Optimization of Two-Stage-To-Orbit Aerospace Plane Based on Pseudospectral Method

Aerospace plane can fly in the Earth's atmosphere with hypersonic and run in the low earth orbit.According to the tactical and technical index, by adjusting power, control, pneumatic components relating to trajectory optimization, we get the optimal ascent trajectory of minimal fuel consumption at a specific separation state (height 30km, Mach 6) with the Radaupseudospectral method. The terminal state can accurately meet the requirements of the ellipse orbit of height 300km.On the basis of optimal trajectory we need find the best separation condition.The relationship between the separation height, Mach number and fuel consumption is studied.The relationship between the flight path angle and the fuel consumption of the specific separation states is also researched.

Time-optimal reorientation of the rigid spacecraft using a pseudospectral method integrated homotopic approach

This paper proposes a robust algorithm for time-optimal rigid spacecraft reorientation trajectory generation. Based on the Pontryagin's maximum principle, the first-order necessary optimality conditions are derived. These optimality conditions are numerically satisfied by adopting a pseudospectral method integrated homotopic approach to solve the associated shooting functions. First, the energy-optimal reorientation solution is obtained using the Radau pseudospectral method, which has a spectral convergence speed and can give a precise estimation of the initial costates used to start the homotopic approach. Then, a modified homotopy scheme is given to deform the associated energy-optimal solution to the desired time-optimal solution continuously. Finally, for the inertially symmetric spacecraft reorientation problem, the newly found time-optimal solutions are presented. The performance of the algorithm is illustrated by simulating a general asymmetric rigid spacecraft time-optimal reorientation problem. Copyright © 2014 John Wiley & Sons, Ltd.

A Hermite‐Lobatto Pseudospectral Method for Optimal Control

AbstractA pseudospectral (PS) method based on Hermite interpolation and collocation at the Legendre‐Gauss‐Lobatto (LGL) points is presented for direct trajectory optimization and costate estimation of optimal control problems. A major characteristic of this method is that the state is approximated by the Hermite interpolation instead of the commonly used Lagrange interpolation. The derivatives of the state and its approximation at the terminal time are set to match up by using a Hermite interpolation. Since the terminal state derivative is determined from the dynamic, the state approximation can automatically satisfy the dynamic at the terminal time. When collocating the dynamic at the LGL points, the collocation equation for the terminal point can be omitted because it is constantly satisfied. By this approach, the proposed method avoids the issue of the Legendre PS method where the discrete state variables are over‐constrained by the collocation equations, hence achieving the same level of solution accuracy as the Gauss PS method and the Radau PS method, while retaining the ability to explicitly generate the control solution at the endpoints. A mapping relationship between the Karush‐Kuhn‐Tucker multipliers of the nonlinear programming problem and the costate of the optimal control problem is developed for this method. The numerical example illustrates that the use of the Hermite interpolation as described leads to the ability to produce both highly accurate primal and dual solutions for optimal control problems.

A Comparison of Optimal Control Methods for Minimum Fuel Cruise at Constant Altitude and Course with Fixed Arrival Time

The current lack of efficiency in the use of airspace, enforced by the economic crisis an the increasing competitiveness among air transport companies are drivers for a renewed interest for the application of optimization techniques to find answers to overcame current inefficiencies.The main objective of the present paper is to assess and compare different optimal aircraft trajectories techniques applied to the minimum fuel cruise problem at constant altitude and course with fixed arrival time, International Standard Atmosphere and without wind. Four trajectory optimization methods have been used: Hermite-Simpson, 5th degree Gauss-Lobatto and Radau pseudospectral collocation methods and the singular arc solution.Hermite-Simpson and 5th degree methods have been programmed in Ampl modeling language with an IPOPT solver and Radau pseudospectral method using gpops matlab tool with SNOPT solver.5th degree Gauss-Lobatto collocation method gives the less fuel consumption solution followed by Radau pseudospectral, Hermite-Simpson and singular arc. In considering the program execution time, Hermite-Simpson collocation method is the fastest method followed by 5th degree and Radau pseudospectral. Also, taking into account the time for developing the program code the Radau pseudospectral is the most user friendly. Moreover, it has been observed that increasing the sample points in the Hermite- Simpson and 5th degree, the solution converge to the minimum fuel consumption solution. On the other hand, gpops does not show much sensitivity to the number of sample points.

Rapid design and optimization of low-thrust rendezvous/interception trajectory for asteroid deflection missions

Asteroid deflection techniques are essential in order to protect the Earth from catastrophic impacts by hazardous asteroids. Rapid design and optimization of low-thrust rendezvous/interception trajectories is considered as one of the key technologies to successfully deflect potentially hazardous asteroids. In this paper, we address a general framework for the rapid design and optimization of low-thrust rendezvous/interception trajectories for future asteroid deflection missions. The design and optimization process includes three closely associated steps. Firstly, shape-based approaches and genetic algorithm (GA) are adopted to perform preliminary design, which provides a reasonable initial guess for subsequent accurate optimization. Secondly, Radau pseudospectral method is utilized to transcribe the low-thrust trajectory optimization problem into a discrete nonlinear programming (NLP) problem. Finally, sequential quadratic programming (SQP) is used to efficiently solve the nonlinear programming problem and obtain the optimal low-thrust rendezvous/interception trajectories. The rapid design and optimization algorithms developed in this paper are validated by three simulation cases with different performance indexes and boundary constraints.

A New Method for Initial Parameters Optimization of Guided Projectiles

A new algorithm was developed for the initial parameters optimization of guided projectiles with multiple constraints. Due to the relationship between the analytical guidance logic and state variables of guided projectiles, the Radau pseudospectral method was used to discretize the differential equations expressed by control variables and state variables with multiple constraints into series algebraic equations, only expressed by state variables. The initial parameter optimization problem was transformed to a nonlinear programming problem, and the sequential quadratic programming algorithm was used to obtain the optimal combinations of initial height and range to target, and make the final velocity of guided projectiles maximum with constraints. Comparing with the appropriate initial conditions solved by Monte Carlo method and the flight characteristics solved by integrating the original differential equations in the optimal initial parameters computed by the new algorithm, the feasibility of new algorithm was validated.

A New Method for Initial Parameters Optimization of Guided Projectiles

A new algorithm was developed for the initial parameters optimization of guided projectiles with multiple constraints. Due to the relationship between the analytic guidance logic and state variables of guided projectiles, the Radau pseudospectral method was used to discretize the differential equations including control variables and state variables with multiple constraints into series algebraic equations that were expressed only by state variables. The initial parameter optimization problem was transformed to a nonlinear programming problem, and the sequential quadratic programming algorithm was used to obtain the optimal combinations of initial height and range to target for the final velocity of guided projectiles maximum with constraints. Comparing with the appropriate initial conditions solved by Monte Carlo method and the flight characteristics solved by integrating the original differential equations in the optimal initial parameters computed by the new algorithm, the feasibility of new algorithm was validated.

Re-entry trajectory optimization using an hp-adaptive Radau pseudospectral method

Aiming at increasing the convergence rate and the accuracy simultaneously, an hp-adaptive Radau pseudospectral method is presented to generate a re-entry launch vehicle’s optimal re-entry trajectory. The method determines the number of mesh intervals, the width of the each mesh interval, and the degree of the polynomial in each mesh interval iteratively until a user-specified error tolerance is satisfied. In regions of relatively high curvature, convergence is achieved by dividing a segment into more mesh intervals, while in regions of relatively low curvature, convergence is achieved by increasing the degree of the approximating polynomial within a mesh interval. Simulation results show that the optimized trajectory obtained by the method satisfies the path constraints and the boundary constraints successfully. Moreover, the hp-adaptive Radau pseudospectral method is shown to be more efficient than either a global Radau pseudospectral method or a fixed-low-order Radau pseudospectral method. The results indicate that the hp-adaptive Radau pseudospectral method can be applied for real-time trajectory generation due to its high efficiency and high precision.