Gauss Pseudospectral Method Research Articles

Energy-Efficient Rail Transit Vertical Alignment Optimization: Gaussian Pseudospectral Method

Energy efficiency is an important research topic in rail transit operations. Energy-efficient vertical alignment is a practical consideration during the line design process. Proper vertical alignment can reduce the train running energy consumption during the entire life cycle of the operation line. Most existing studies solved such problems through heuristic algorithms; exactly how much energy efficiency the vertical alignment can bring is still an unknown question. This study presents an energy-efficient line vertical alignment optimization model with the objective of minimum energy consumption and running time deviation and proposes an exact solution approach, the Gaussian pseudospectral method (GPM), to make this exploration. The optimization problem was transformed into the description form of the optimal train control problem. GPM was used to discretize the optimal control problem into a nonlinear programming problem, and the sequential quadratic programming algorithm completed the solution. The application of China’s Guangzhou Line 18 shows that the GPM has a better effect in solving energy-efficient vertical alignment than a heuristic algorithm. The results indicate that the presented approach can effectively improve the energy efficiency of the rail transit system.

The Intelligent Trajectory Optimization of Multistage Rocket with Gauss Pseudo-Spectral Method

The rapid developments of artificial intelligence in the last decade are influencing aerospace engineering to a great extent and research in this context is proliferating. In this paper, the trajectory optimization of a three-stage launch vehicle in the powering phase subject to the sun-synchronous orbit is considered. To solve the optimal control problem, the Gauss pseudo-spectral method (GPM) is used to transform the optimization model to a nonlinear programming (NLP) problem and sequential quadratic programming is applied to find the optimal solution. However, the sensitivity of the initial guess may cost the solver significant time to do the Newton iteration or even lead to the local minimum. Aiming at this issue, a good initial guess generated by combination of harmony search and GPM is introduced to help the optimizer to faster and better solve the sun-synchronous orbit (SSO) trajectory optimization. Comparative simulation tests are conducted with the proposed algorithm and popular approaches, the results indicate that with the optimized initial guess, the proposed method could achieve better performance in terms of convergence ability and convergence rate.

Fast Cooperative Trajectory Generation of Unmanned Aerial Vehicles Using a Bezier Curve-Based Shaping Method

This study uses a Bezier curve-based shaping (BCBS) approach based on the dynamic model to quickly generate 3D cooperative trajectories for unmanned aerial vehicles (UAVs). It can facilitate the solution of the flight trajectory problem that occurs after the reallocation of mission points. Trajectory coordinates are expanded based on the Bezier curves that naturally satisfy boundary conditions (BCs); trajectories with the minimum flight time and those satisfying the dynamic constraints and collision-free conditions can be obtained by adjusting the Bezier coefficients. A three-UAV flight simulation is carried out to verify the effectiveness of the proposed method. Further, the performance of the proposed method is compared with that of the Gauss pseudospectral method (GPM). The simulation results of the BCBS method are used as the initial values for GPM optimization. It is demonstrated that the BCBS method requires a lower computation time than the direct solver, which is only 0.07% of the latter, and obtains similar optimization results (3.34% difference). This is considerably important for the rapid generation of optimized trajectories with the limited computing power of onboard computers. Furthermore, this method is expected to achieve online collaborative trajectory generation for multiple UAVs in view of its high computational efficiency.

Investigation of the retrieval dynamics of the tethered satellites using ANCF-ALE variable-length element

Tether satellite system (TSS) comprises two satellites connected by a flexible tether and has wide applications, such as orbital transferring, deep space exploration, and so on. In this paper, a variable-length element, based on the absolute nodal coordinate formulation (ANCF) in the framework of arbitrary Lagrangian-Eulerian (ALE) description, is used to predict the characteristics of length variation and flexibility of the tether in the phases of retrieval and station-keeping. Then, to achieve a stable retrieval implementation and take the under-actuation property of the TSS into account, an optimal retrieval trajectory is programmed using the Gauss pseudo-spectral method based dynamic optimization. Three different retrieval schemes are implemented using the developed ANCF-ALE model. Numerical studies show that, compared with the dumbbell model, the ANCF-ALE based model can predict the tension of the tether in the phase of retrieval. Results indicate that the tether tension is sensitive to the ratio of change of the tether retrieval velocity. Improper retrieval scheme may induce the intensive oscillation in the tether tension, or a switched state of the tether between pseudo-slack and tensile. In the optimization for optima trajectory programming, weight of ratio of tether velocity in the objective function should be considered carefully to realize a robust retrieval implementation.

A hybrid hyper-heuristic whale optimization algorithm for reusable launch vehicle reentry trajectory optimization

Trajectory optimization is essentially an optimal control problem (OCP) with highly nonlinear dynamic properties and complex constraints, and a critical part of spacecraft design. In this paper, a hybrid algorithm is proposed for automatic reentry trajectory optimization of reusable launch vehicle (RLV) without providing user-specified initial guesses and a priori knowledge about the optimal trajectory. The method combines the strong robustness and global optimization properties of hyper-heuristic whale optimization algorithm (HHWOA) with the efficient and accurate features of the Gauss pseudospectral method (GPM). HHWOA works as the first-stage optimizer aims to obtain an approximate solution to provide a high-quality initial guess for the GPM, while the GPM works as the second-stage optimizer aims to accelerate the search of the optimum neighborhood to obtain an accurate optimal solution. Additionally, to enhance the progress during the evolutionary process, HHWOA is equipped with opposition-based learning, differential evolution operators, chaotic map sequences and smoothing technique strategies. The utilization of such strategies can potentially smooth the flight trajectory and improve the global convergence of the algorithm, while the three OCPs have shown their superiority in HHWOA. In order to evaluate the performance of the hybrid algorithm, complex constrained RLV maximum cross-range reentry problems with three different path constraint scenarios are investigated. Furthermore, more discussion and experiments are likewise conducted to investigate the impact of the parameters on the performance of the algorithm. The results show that the proposed hybrid algorithm can be very effective in addressing RLV reentry trajectory optimization problems.

Time-varying formation dynamics modeling and constrained trajectory optimization of multi-quadrotor UAVs

The formation of multi-quadrotor UAVs (QUAVs) in long flights will encounter many constraints and obstacles in the flight process, so it is necessary to change the formation shape to avoid these constraints. When the flight path of QUAVs is individually optimized, multiple individual dynamic problems will be faced, making the solution complicated and spending a long time. In this paper, as a whole instead of individuals considering the QUAVs, the formation is regarded as a rigid body using the Voronoi graph theory method in the flight process. The constraints of the followed QUAVs are transformed into the constraints of leader QUAV. Therefore, the formation is transformed for different constraints by changing the virtual rigid body structure or shape. A time-varying model is established to facilitate the use of optimization. The energy consumption of the leader UAV within the specified time is minimized as the trajectory optimization objective. The optimization improvement of the end heading and height constraints is proposed. A trajectory optimization method for the leader QUAV based on Gauss pseudospectral method is proposed, transforming the original optimal control problem into the nonlinear programming one. The simulation results demonstrate and verify the proposed method.

Constrained Spacecraft Attitude Optimal Control via Successive Convex Optimization

Rapid attitude path planning is the key technique in autonomous spacecraft operation missions. An efficient method is proposed for energy-optimal spacecraft attitude control in presence of constraints. Firstly, Gauss pseudospectral method is utilized to discretize and transcribe the primal continuous problem to a nonlinear programming problem. Then a set of convexification techniques are used to convexity the nonlinear programming problem to a series of second-order cone programming problems, which can be solved iteratively by the interior-point method. A solution to the nonlinear programming problem is obtained as the iteration converges. Numerical results show the method could obtain a valid energy-optimal attitude control plan more rapidly than traditional methods.

Deep learning‐based optimal tracking control of flow front position in an injection molding machine

AbstractThe product quality of injection‐molded plastic is closely related to the injection flow velocity of molten plastics. In this article, an optimal tracking control problem for the injection flow front position arising in the filling process in the injection molding machine (IMM) is considered, and an intelligent real‐time optimal control method based on deep neural networks (DNNs) is developed for the online tracking of the flow front position to improve the efficient production process of the plastics. To this end, a highly nonlinear dynamic model describing the filling process is first proposed and then an optimal control problem is formulated. A high‐fidelity numerical optimization approach known as the Gauss pseudospectral method is used to numerically obtain the optimal state‐control solutions, which can then be saved as a huge training data set for the DNNs. The DNNs architecture is then developed and offline trained using the obtained data set to determine the best state–control relationship. As a consequence, the well‐trained DNN may be utilized to produce the matching optimum feedback action on‐board. Numerical studies are also carried out to verify the performance of our proposed approach.

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.

Multi-objective optimization of dive trajectory for morphing unmanned aerial-underwater vehicle

A well-planned dive trajectory is the key for the trans-media maneuver and flight control of morphing unmanned aerial-underwater vehicles (MUAUVs). The main focus of this paper is the application of a watch-based multi-objective grey wolf optimizer (MOGWO) to dive trajectory optimization problems. A multi-objective trajectory optimization model considering the change of sweep angle and static moment is developed and parameterized by the Gauss pseudospectral method, and a multi-objective nonlinear programming problem is constructed. To overcome the defect that, in the MOGWO, the rest wolves blindly follow the best wolves, the watch strategy is introduced, which provides the wolves with the ability of independent exploration. Subsequently, the watch-based MOGWO is employed to generate the Pareto front, which is compared with that obtained through other multi-objective techniques. The simulation results demonstrated that the proposed method is more reliable and can obtain more widely distributed non-dominated solutions, indicating that the watch-based MOGWO is effective and feasible in dealing with multi-objective trajectory optimization problems with complex constraints. In addition, comparative studies on optimal trajectories demonstrated that the maneuverability and gliding ability of the MUAUV are improved through cooperation between the angle-of-attack, bank angle, and sweep angle.

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.

Multi-aircraft cooperative path planning for maneuvering target detection

<p style='text-indent:20px;'>Multi-aircraft cooperative path planning is a key problem in modern and future air combat scenario. In this paper, this problem is studied in aspect of airborne radar detection to maintain a continuous tracking of a manoeuvring air target. Firstly, the objective function is established in combination with multiple constraints considered, including Doppler blind zone constraint, radar viewing aspect constraint, baseline constraint, and so on. Then, the above optimal control problem is transformed into a nonlinear programming problem with a series of algebraic constraints by hp-adaptive Gauss pseudospectral method (HPAGPM). And it is solved by GPOPS software package based on MATLAB. Simulation results show that the optimized cooperative paths can be got to achieve continuous tracking of maneuvering air target by HPAGPM.</p>

Onboard Trajectory Generation of Hypersonic Morphing Aircraft

In this paper, a trajectory optimization strategy for the hypersonic morphing aircraft is proposed, and the AMPI method is used to generate the online trajectory with initial state errors. Firstly, the aerodynamic model and propulsion model of the hypersonic morphing aircraft were established considering the wingspan and the scramjet. Secondly, the optimization strategy was proposed via Gauss pseudospectral method considering the control variables including angle of attack (AOA) and wingspan. The optimized trajectory met the final constraints and path constraints with the objective to minimize the time of the ascent phase. Then, the AMPI method was used to generate online trajectory without solving OCP or NLP on the base of trajectory database calculated by the optimization strategy. The simulation results indicate high accuracy of AMPI method and the final errors corresponding to different initial errors were acceptable. The mean value of the CPU time of the method was about 0.1 second, which shows real-time capability.

Fast cooperative angular trajectory planning for multiple on-orbit service spacecraft based on the Bezier shape-based method

In this article, a fast cooperative obstacle avoidance angular trajectory planning method is presented for multiple on-orbit service (OOS) spacecraft, which can obtain the spatial angular trajectories of multiple spacecraft in a considerably short time. For multiple OOS spacecraft equipped with the manipulator, under the constraints of the maximum control moment, multiple obstacles, dynamics, and geometry, the Bezier shape-based (SB) method is used to rapidly generate the cooperative obstacle avoidance attitude angular trajectories of all OOS spacecraft at the same time. The superiority of the proposed method is proved by comparing the results of Bezier SB method and the finite Fourier series (FFS) SB method under the same initial conditions. To further verify the effectiveness of the proposed method, the results of the Bezier method are substituted into the Gauss pseudospectral method (GPM) as the initial values to further optimize the angular trajectories in the simulation. Simulation results show that compared with the FFS SB method, the Bezier SB method can use shorter computation time to get the results of spacecraft attitude movement faster; compared with GPM, Bezier SB method only uses about 2% of computation time to get the results of movement time difference of only 2%. The method proposed in this article can be applied to complex space OOS missions, which requires multiple OOS spacecraft to rapidly realize the space obstacle avoidance attitude maneuver to complete cooperative operation missions.

Multistage Linear Gauss Pseudospectral Method for Piecewise Continuous Nonlinear Optimal Control Problems

This article aims at proposing a multistage linear Gauss pseudospectral method (MS-LGPM) for solving the piecewise continuous nonlinear optimal control problem (OCP) with interior-point constraints and terminal constraints. First, the first-order necessary conditions for the multistage nonlinear OCP are derived, and a typical multipoint boundary value problem is obtained. Second, the original problem can be solved iteratively by integral prediction and quasi-linearization. Then Lagrange interpolation polynomials are used to approximate the state, control, and costate variables, to transfer those differential equations into a set of linear algebraic equations. Therefore, the control update coming closer to the optimal solution can be derived in an analytical manner. Additionally, those linear algebraic equations are formulated in regular banded matrix forms to make the code as concise as possible. Finally, the proposed method is applied to the midcourse guidance design for a dual-pulse missile to evaluate its performance. Simulation results show that the proposed method performs well in providing the optimal control with high accuracy and computational efficiency. Furthermore, a comparison with other typical guidance method and Monte Carlo simulations are also provided. Results demonstrate, even with some random uncertainties, the proposed method still has strong robustness and superior performance.

A Novel IGC Scheme for RHV with the Capabilities of Online Aerodynamic Coefficient Estimation and Trajectory Generation

A novel integrated guidance and control (IGC) scheme for a Re-entry Hypersonic Vehicle (RHV) is proposed with the capabilities of online aerodynamic coefficient estimation based on an Unscented Kalman Filter and online trajectory generation based on the Gaussian pseudospectral method. A linear quadratic regulator is adopted for trajectory tracking guidance and a second-layer sliding mode controller is designed for attitude control. The variation of lift and drag coefficients are modeled and estimated online, based on which a new trajectory can be generated. The commands of trajectory generation are set as moments of actuators and their extremums pose more constraints on the rate and acceleration of flow angles. Comprehensive simulations are conducted and comparable IGC performances with normal conditions are obtained under large aerodynamic coefficient errors according to online generated trajectory, which proves the effectiveness and advantages of the proposed IGC scheme.

Efficient Trajectory Planning for UAVs Using Hierarchical Optimization

Automatic generation of feasible trajectory is one of the key technologies for autonomous flying of unmanned aerial vehicles (UAVs). The existing path planning methods, such as swarm intelligence algorithm and graph-based algorithm, cannot incorporate the flying time and UAV dynamic model into evolution. To overcome such disadvantages, a hierarchical trajectory optimization scheme consisted by improved particle swarm optimization (PSO) and Gauss pseudo-spectral method (GPM) is investigated in this paper. Firstly, considering that traditional GPM is sensitive to initial values, we design an improved PSO for path planning in the first layer. By introducing adaptive parameter adjustment strategy and position mutation updating strategy, the rapidity and optimality of the improved PSO is enhanced. Then in the second layer, a fitted curve based on the path waypoints generated by improved PSO is constructed and served as the initial values for GPM. Comparing with random initial values, the designed curve can significant improve GPM efficiency. A multi-segment strategy is also put forward to further improve the efficiency. Finally, with the consideration of dynamic model and state constraints, the time minimum trajectory planning for quadrotor UAVs is solved. Plenty of simulations are carried out and the results illustrate that the proposed scheme guarantees much better efficiency.

Hierarchical Control of Trajectory Planning and Trajectory Tracking for Autonomous Parallel Parking

For the parallel parking problem in narrow space, this paper proposes a trajectory tracking control method with a novel trajectory planning layer for autonomous parallel parking based on a numerical optimization algorithm and model predictive control. In the trajectory planning layer, the vehicle kinematics model suitable for the low-velocity parking scene is established. Considering the vehicle physical constraints, boundary condition constraints, and obstacle avoidance constraints during the parking process, the parking trajectory planning task is described as an optimal control problem, further transformed into a nonlinear programming problem by Gauss pseudo-spectral method. Taking the shortest parking completion time as the optimization objective function, the parking trajectories of the large, medium and small parking spaces are obtained, respectively. A parking trajectory tracking controller based on the model predictive control algorithm is designed in the trajectory tracking control layer. The linear error model is used as the prediction model, and the quadratic programming is adopted as the rolling optimization algorithm in the tracking controller. The velocity and front-wheel swing angle are obtained as control signals for parking trajectory tracking. Through CarSim and Simulink's co-simulation, the feasibility and effectiveness of the proposed parallel parking trajectory planning and tracking control method are verified. The co-simulation results show that the maximum tracking errors of horizontal and longitudinal positions are less than 0.15m. The maximum tracking errors of heading angle are less than 2° under three different parking spaces. Real vehicle tests are carried out to verify the effectiveness of the proposed hierarchical control method. The test results show that the vehicle can park in the parking space safely, quickly and accurately when the actual parking space is detected. The proposed method can plan the parking trajectory with the constraints and the shortest time and control the vehicle to complete the parking operation accurately along the planned trajectory.

Predictive Cruise Control of Full Electric Vehicles: A Comparison of Different Solution Methods

Abstract The integration of electrification and intelligence is of great significance to alleviating range anxiety of electric vehicles. Predictive cruise control (PCC), which optimizes the longitudinal driving strategies by using the upcoming road traffic information, can further improve the vehicle economy. The paper gives the comparison of different solution methods regarding PCC problem of electric vehicles. The car-following optimization problem is formulated as a constrained nonlinear optimization problem. For ease of presentation, the car-following optimization problem is reformulated as a standard form of the optimization problem in continuous time domain. Then, the standard form of the optimization problem is transformed from a continuous form to a discrete form by using Euler method and Gauss pseudospectral method. Two common solution methods, that is dynamic programming and sequential quadratic programming, are used to solve the optimization problem in discrete form. Simulations are performed to demonstrate the comparison of different solution schemes.

Integrated Morphing Strategy and Trajectory Optimization of a Morphing Waverider and Its Online Implementation Based on the Neural Network

In previous studies, a variable-sweep-wing morphing waverider was proposed and its aerodynamic performance and dynamic model were provided. This morphing waverider can change sweep angle according to different flight scenario in order to achieve an optimal flight performance. For the traditional fixed-geometry waverider, the main objective of the trajectory planning problem is to find an optimal trajectory consisting of time-dependent variables including the angle of attack, the velocity, the altitude, the flight path angle and so on. As for this morphing waverider, an optimal morphing wing changing strategy is also required to be calculated together with an optimal trajectory. At first, to alleviate computational cost and improve trajectory-planning efficiency, aerodynamic coefficients of the morphing waverider are approximated and modeled as polynomial functions of multiple variables including the Mach number, the angle of attack and the sweep angle. Thereafter, an offline integrated optimization problem of both wing morphing strategy and trajectory based on the Gauss pseudo-spectral method is investigated, which can fulfill actual task requirements and variable constraints. Compared with four specific fixed-wing waverider configurations, this morphing waverider has a larger downrange distance and a lower angle of attack. However, the real flight trajectory of the morphing waverider may deviate from the nominal condition due to multiple uncertainties and this morphing waverider is required to have an intelligent online optimization ability. Therefore, an online optimization method based on the neural network is proposed for both wing morphing strategy and trajectory optimization. Finally, the overall performance of this method is examined by comparisons and Monte Carlo runs.