Trajectory Optimization Problem Research Articles

Real-Time Cooperative Vehicle Coordination at Unsignalized Road Intersections

Cooperative coordination at unsignalized road intersections, which aims to improve the driving safety and traffic throughput for connected and automated vehicles, has attracted increasing interests in recent years. However, most existing investigations either suffer from computational complexity or cannot harness the full potential of the road infrastructure. To this end, we first present a dedicated intersection coordination framework, where the involved vehicles hand over their control authorities and follow instructions from a centralized coordinator. Then a unified cooperative trajectory optimization problem will be formulated to maximize the traffic throughput while ensuring the driving safety and long-term stability of the coordination system. To address the key computational challenges in the real-world deployment, we reformulate this non-convex sequential decision problem into a model-free Markov Decision Process (MDP) and tackle it by devising a Twin Delayed Deep Deterministic Policy Gradient (TD3)-based strategy in the deep reinforcement learning (DRL) framework. Simulation and practical experiments show that the proposed strategy could achieve near-optimal performance in sub-static coordination scenarios and significantly improve the traffic throughput in the realistic continuous traffic flow. The most remarkable advantage is that our strategy could reduce the time complexity of computation to milliseconds, and is shown scalable when the road lanes increase.

Joint trajectory-resource optimization for UAV-enabled uplink communication networks with wireless backhaul

This paper studies trajectory optimization and resource allocation for unmanned aerial vehicle (UAV) enabled uplink communication networks with wireless backhaul (WB), in which a UAV provides non-orthogonal multiple access (NOMA) based uplink relaying services for multiple mobile users with stochastic movement and forwards the decoded multi-user signals to a remote BS by means of a WB link. To assure that the UAV are able to forward the decoded multi-user signals to the remote BS, we formulate a problem of maximizing the uplink average achievable sum-rate by jointly optimizing UAV trajectory, each mobile user’s transmit power, and bandwidth allocation coefficients of both wireless access and backhaul links, subject to information-causality constraint, system available bandwidth limitation and power budget of each mobile user. The challenge lies in that the time-vary successive interference cancellation (SIC) decoding order results in a strong-coupled joint NOMA decoding order design and UAV trajectory optimization problem, which makes this problem intractable. To tackle this challenge, we firstly transform the original problem into a tractable form. Then, a penalized iteration approach is developed to handle the reformulated problem. Simulation results verify the proposed scheme can achieve higher uplink average achievable sum rate than benchmarks.

Joint Optimization of Service Fairness and Energy Consumption for 3D Trajectory Planning in Multiple Solar-Powered UAV Systems

In this paper, we study the three-dimensional (3D) trajectory optimization problems of unmanned aerial vehicles (UAV) with a solar energy supply, aiming to provide communication coverage for mobile users on the ground. In general, the higher UAVs fly, the more solar energy they collect, but the smaller the range of coverage they could achieve, and vice versa. How to plan optimal trajectories for UAVs so that more users can be encompassed, while allowing UAVs to collect enough solar energy, is a challenging issue. Moreover, we also consider how geographically fair coverage for each ground user can be achieved. To solve these problems, we designed a multiple solar-powered UAV (SP-UAV) energy consumption model and a fairness model, while designed an observation space, state space, action space, and reward function. Then, we proposed a multiple SP-UAV 3D trajectory optimization algorithm based on deep reinforcement learning (DRL). Our algorithm is able to balance the energy consumption of UAVs to extend the system’s lifetime, while avoiding both collisions and flying out of communication range. Finally, we trained our model through simulation experiments and conducted comparative experiments and analysis based on real network topology data. The results show that our algorithm is superior to the existing typical algorithms in coverage, fairness, and lifetime.

Analytic Time Reentry Cooperative Guidance for Multi-Hypersonic Glide Vehicles

Aiming at the cooperative guidance problem of multi-hypersonic glide vehicles, a cooperative guidance method based on a parametric design and an analytical solution of time-to-go is proposed. First, the hypersonic reentry trajectory optimization problem was transformed into a parameter optimization problem. The parameters were optimized to determine the angle of attack profile and the time to enter the altitude velocity reentry corridor. Then, using the quasi-equilibrium glide condition, the estimation form of the remaining flight time was analytically derived to satisfy accurately the cooperative time constraint. Using the remaining time-to-go and range-to-go, combined with the heading angle deviation corridor, the bank angle command was further calculated. Finally, the swarm intelligence optimization algorithm was used to optimize the design parameters to obtain the cooperative guidance trajectory satisfying the time constraint. Simulations showed that the analytical time reentry cooperative guidance algorithm proposed in this paper can accurately meet the time constraints and cooperative flight accuracy. Monte Carlo simulation experiments verified that the proposed algorithm demonstrates a robust performance.

UAV Trajectory and Energy Efficiency Optimization in RIS-Assisted Multi-User Air-to-Ground Communications Networks

An air-to-ground downlink communication network consisting of a reconfigurable intelligent surface (RIS) and unmanned aerial vehicle (UAV) is proposed. In conjunction with a resource allocation strategy, the system’s energy efficiency is improved. Specifically, the UAV equipped with a RIS starts from an initial location, and an energy-efficient unmanned aerial vehicle deployment (EEUD) algorithm is deployed to jointly optimize the UAV trajectory, RIS phase shifts, and BS transmit power, so as to obtain a quasi-optimal deployment location and hence improve the energy efficiency. First, the RIS phase shifts are optimized by using the block coordinate descent (BCD) algorithm to deal with the nonconvex inequality constraint, and then integrated with the Dinkelbach algorithm to address the resource allocation problem of the BS transmit power. Finally, for solving the UAV trajectory optimization problem, the complex objective function is transformed into a convex function, and the optimal UAV flight trajectory is obtained. Our simulation results show that the quasi-optimal deployment location obtained by the EEUD algorithm is superior to other deployment strategies in energy efficiency. Moreover, the instantaneous energy efficiency of the UAVs along the trajectory of searching the deployment location is better than other comparison trajectories. Furthermore, the RIS-assisted multi-user air-to-ground communication network can offer up to 145% improvement in energy efficiency over the traditional amplify-and-forward (AF) relay.

An enlarged polygon method without binary variables for obstacle avoidance trajectory optimization

In this article, an Enlarged Polygon/Polyhedron (ELP) method without binary variables is proposed to represent the Convex Polygonal/Polyhedral Obstacle Avoidance (CPOA) constraints in trajectory optimization. First, the equivalent condition of a point outside the convex set is given and proved rigorously. Then, the ELP condition describing the CPOA constraints equivalently is given without introducing binary variables, and its geometric meaning is explained. Finally, the ELP method is used to transform the CPOA trajectory optimization problem into an optimal control problem without binary variables. The effectiveness and validity of ELP method are demonstrated through simulations with both simple linear dynamic model (unmanned aerial vehicle) and complex nonlinear dynamic model (hypersonic glide vehicle). Comparison indicates the computational time of ELP method is only 1%-20% of that of the traditional Mixed-Integer Programming (MIP) method.

Orthogonal Approximation of Invariant Manifolds in the Circular Restricted Three-Body Problem

Methods to parameterize and approximate the hyperbolic invariant manifolds of particular solutions in the circular restricted three-body problem (CR3BP) are presented in this paper. Analytical representations obtained from these manifold approximations are instrumental in the synthesis of optimal trajectories for cislunar transport. A multivariate Chebyshev series is used to approximate the surfaces, thereby serving as tractable parametric representations of the complex properties of motion. It is demonstrated that the continuum of ballistic capture trajectories and their associated sensitivities on the manifold can be realized in functional form using simple algebraic operations. Two applications making use of the Chebyshev manifold approximations as a terminal constraint surface are presented. The first is a low-thrust trajectory optimization problem formulated such that the optimal free final state lying on the manifold is determined as an additional set of design parameters. The second is a guidance law designed to target the manifold in the vicinity of the nominal patch point. Each of these methods takes advantage of the Chebyshev approximations to provide additional flexibility for mission design in multibody dynamic environments. These applications offer tremendous optimism about the utility of function approximation methods in arriving at a formal representation for the invariant manifolds in the three-body problem for efficient generation of optimal trajectories.

Joint Power and 3D Trajectory Optimization for UAV-Enabled Wireless Powered Communication Networks With Obstacles

Unmanned aerial vehicle (UAV)-enabled wireless powered communication networks (WPCNs) are promising technologies in 5G/6G wireless communications, while there are several challenges about UAV power allocation and scheduling to enhance the energy utilization efficiency, considering the existence of obstacles. In this work, we consider a UAV-enabled WPCN scenario that a UAV needs to cover the ground wireless devices (WDs). During the coverage process, the UAV needs to collect data from the WDs and charge them simultaneously. To this end, we formulate a joint-UAV power and three-dimensional (3D) trajectory optimization problem (JUPTTOP) to simultaneously increase the total number of the covered WDs, increase the time efficiency, and reduce the total flying distance of UAV so as to improve the energy utilization efficiency in the network. Due to the difficulties and complexities, we decompose it into two sub optimization problems, which are the UAV power allocation optimization problem (UPAOP) and UAV 3D trajectory optimization problem (UTTOP), respectively. Then, we propose an improved non-dominated sorting genetic algorithm-II with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -means initialization operator and Variable dimension mechanism (NSGA-II-KV) for solving the UPAOP. For UTTOP, we first introduce a pretreatment method, and then use an improved particle swarm optimization with Normal distribution initialization, Genetic mechanism, Differential mechanism and Pursuit operator (PSO-NGDP) to deal with this sub optimization problem. Simulation results verify the effectiveness of the proposed strategies under different scales and settings of the networks.

Trajectory Optimization for Drone Logistics Delivery via Attention-Based Pointer Network

Drone logistics delivery is a potential booster to redefine the logistics system efficiency, which has been a new special hot research field. Among that, how to optimize drone trajectory data and find optimal delivery path is a crucial problem. However, most existing studies fail to derive the feasible trajectory data due to underestimating drone energy consumption and delivery multi-variant constraints for air transportation. In this paper, we propose a novel self-driven learning procedure, named attention-based pointer network (A-Ptr-Net) model, which can solve drone delivery trajectory optimization problem. The generated A-Ptr-Net model coupled with attention mechanism is effective on adapting to the new drone trajectory data automatically, regardless of explicit distance matrix. We go into developing the convex function constraints related to drone nonlinear energy consumption, customer demand and service time windows, which is applied on A-Ptr-Net model for optimizing the drone logistics delivery. Numerical experiments indicate that the proposed method performs significantly better than classical heuristics for drone trajectory analysis and optimization.

Mixed-integer trajectory optimization with no-fly zone constraints for a hypersonic vehicle

This paper focuses on mixed-integer trajectory optimization of no-fly zones avoidance for a hypersonic vehicle. When there are multiple no-fly zones, the trajectory optimization problem has many local optimal solutions, and the performance of the solution depends on the selection of an initial guess. Essentially, the selection of the initial guess can be abstracted as a discrete path decision. To deal with this issue, this paper first integrates the discrete path decision and continuous trajectory optimization for no-fly zones avoidance and models these as a mixed-integer optimal control problem (MIOCP). As the corner stone, a creative directed graph is established to model the path decision problem, which is then formulated by integer constraints. Combined with the minimum effort optimal control problem (Problem 1), we form the MIOCP (Problem 2), thus integrating the path decision and trajectory optimization to improve global performance. Additionally, to solve the MIOCP, an effective iterative mixed-integer convex programming (IMICP) algorithm is customized by adding two ad-hoc techniques, relaxation and a proximity term, thus remedying the artificial infeasibility. Simulation results show that this approach is practically independent of the initial guess and has better performance than existing methods.

Entry trajectory optimization for hypersonic vehicles based on convex programming and neural network

Trajectory optimization is important in achieving long-range atmospheric entry hypersonic vehicles. However, the trajectory optimization problem for atmospheric entry of hypersonic vehicles is characterized by strong nonlinearity, parameter uncertainties and multiple constraints. This study proposes a novel online trajectory optimization method for hypersonic vehicles based on convex programming and a feedforward neural network. A sequential second-order cone programming (SOCP) method is obtained to describe the trajectory optimization problem after the Gauss pseudo-spectral discretization. Subsequently, multiple optimal trajectories under aerodynamic uncertainties are generated offline and classified as the training and validation datasets. Then, a multilayer feedforward neural network is trained using these datasets and to output the optimal control command online. This method yields approximately 95% shorter computation time compared with the offline SOCP method. Considering the existence of the aerodynamic uncertainties, three terminal states calculated by this method are all smaller than 4.1%. In conclusion, the proposed trajectory optimization method can provide a high-precision, robust entry trajectory for hypersonic vehicles efficiently.

Study on Energy-Saving Train Trajectory Optimization Based on Coasting Control in Metro Lines

With increasing energy consumption in urban rail transit systems, researchers have paid significant attention to energy-saving train control. In this paper, we propose an effective train trajectory optimization method to reduce the energy consumption based on coasting control, in which coasting control regimes are added to balance running time and energy consumption. For better determining the starting points of coasting control regimes, the whole train running process is divided into several subintervals. Then, aiming to achieve energy efficiency, coasting regimes are added to the subintervals with high energy-saving effects, in which more energy consumption can be reduced with the same running time addition. Based on this, a coasting control method is proposed to generate energy-saving trajectories considering train dynamics, safety, and punctuality. In addition, the proposed method can solve the multisection energy-saving train trajectory optimization problem to obtain optimal running time schemes and related trajectories. Finally, numerical examples based on one of the Beijing metro lines are implemented to verify the effectiveness of the proposed method. The results show that, for the single-section train control problem, the proposed coasting control algorithm can achieve significant energy-saving effects compared to the practical trajectory and calculate energy-saving trajectory in shorter computation times compared to the dynamic programming method. Meanwhile, for the multisection train control problem, energy consumption can be further reduced by optimizing trajectories and running times integratedly.

Distributed Trajectory Optimization and Fixed-Time Tracking Control of a Group of Connected Vehicles

This paper investigates a distributed trajectory optimization and trajectory tracking control problem for a collection of vehicles. A two-layered framework is established for distributed trajectory optimization and fixed-time tracking control. The trajectory optimization problem is solved using distributed convex optimization based on spacing errors minimization, resulting in an algorithm to provide the optimal trajectory for all following vehicles. The lower layer plays the role of tracking the designated trajectory and robustly guarantee zero steady-state spacing errors for the vehicular platoon. The controller is derived based on the tracking error system in the context of fixed-time stability and terminal sliding mode control theory. The methodology can guarantee the internal stability and string stability, as demonstrated by numerical simulations.

Train Control and Schedule Integrated Optimization With Reversible Substations

The Carbon Neutralization is a grand strategy to deal with global climate change. As a major energy consumer, urban rail transit (URT) system should contribute corresponding solutions. This paper proposes an energy-efficient train control and schedule integrated optimization method under DC traction power supply system (TPSS) with reversible substation (RSS). The dynamic coupling mechanism between the train movement process and the RSS working mode is analyzed and modeled. Aiming to minimize the total RSS energy consumption for the train traveling over multi-interstation with a given total trip time, the train trajectory and schedule optimization problem is integrated in the same level, and is formulated as a multi-phase optimal control problem (MOCP) to be solved by a pseudo-spectral method (PSM). As a supplement, a customized dynamic programming (DP) method for the problem is introduced to verify the validity of the proposed solution framework. Simulation results under the models of the RSS energy together with conventional rectifier substation energy and vehicle mechanical energy are presented and compared, indicating that the proposed methodology has great energy saving performance for the new-type DC railway system.

Direct Collocation Methods for Trajectory Optimization in Constrained Robotic Systems

Direct collocation methods are powerful tools to solve trajectory optimization problems in robotics. While their resulting trajectories tend to be dynamically accurate, they may also present large kinematic errors in the case of constrained mechanical systems, i.e., those whose state coordinates are subject to holonomic or nonholonomic constraints, like loop-closure or rolling-contact constraints. These constraints confine the robot trajectories to an implicitly-defined manifold, which complicates the computation of accurate solutions. Discretization errors inherent to the transcription of the problem easily make the trajectories drift away from this manifold, which results in physically inconsistent motions that are difficult to track with a controller. This paper reviews existing methods to deal with this problem and proposes new ones to overcome their limitations. Current approaches either disregard the kinematic constraints (which leads to drift accumulation) or modify the system dynamics to keep the trajectory close to the manifold (which adds artificial forces or energy dissipation to the system). The methods we propose, in contrast, achieve full drift elimination on the discrete trajectory, or even along the continuous one, without artificial modifications of the system dynamics. We illustrate and compare the methods using various examples of different complexity.

Thrust Integrated Trajectory Optimization for Multipulse Rocket Missiles Using Convex Programming

In this paper, a thrust integrated trajectory optimization algorithm for multipulse rocket missiles is proposed. Maximizing terminal velocity is selected as a performance index for achieving high maneuverability in the homing stage. To incorporate the optimization of thrust profile, thrust parameters are also considered as optimization variables with several thrust constraints such as impulse conservation and maximum/minimum thrust level. The original trajectory optimization problem is reformulated using new independent variables and timescale variables. This method simplifies the dynamics and makes the impulse conservation constraint disappear. Then, a control-affine system is established by defining the angle of attack and its square as new control inputs. The admissible input set is relaxed to a convex set, and mathematical proof is provided to show that the relaxation preserves the solution. The problem is discretized by applying a pseudospectral method for each subinterval. The solution is then obtained using sequential convex programming (SCP). A quadratic variable trust-region constraint is used for robust convergence of SCP. The proposed method robustly optimizes the thrust parameters and trajectory of the multipulse rocket missile within 1 s. Numerical simulation shows the feasibility and effectiveness of the proposed method.

Automatic Docking Trajectory Design-Based Time-Varying-Radius Dubins for Unmanned Surface Vessel

Facing the demand of efficient automatic docking for USV, this paper proposes a new kind of iterative varying-radius Dubins curve design. Due to the special mobility of dockyards, one novel trajectory optimization problem considering time optimum is modeled first. Then, we use the ant colony optimization (ACO) algorithm to solve such a complex problem. In addition to the global exhaustive search algorithm and ACO-based solution, a novel distributed local path planning algorithm named ADTA-TRD is proposed to obtain the solution of the above optimization problem restricted by the local perception of the USV. In essence, this paper draws lessons from the idea of the circular arc segment and straight line segment in the traditional Dubins curve, and consequently, the smoothness of the derived automatic docking trajectory can be guaranteed. Finally, the performance of proposed automatic docking path planning method is verified through simulation results and an actual sea trial.

Rotorcraft low-noise trajectories design: black-box optimization using surrogates

This paper addresses the noise-minimal trajectory optimization problem for a specific type of aircraft: rotorcraft. It relies on a realistic noise footprint computation software provided by industry that is black-box. Locally optimal trajectories are computed through a tailored solution approach based on the Mesh-Adaptive Direct Search algorithm. We propose multiple surrogates defined according to our knowledge of the problem, including a surrogate relying on the physics of the problem (approximating the rotorcraft noise model), and another based on a machine learning (neural network) method. The proposed solution approach is further enhanced by the computation of an appropriate starting guess through a path planning algorithm tailored to the problem, and by the reduction of the variable space domain. The performance of the proposed methodology both in terms of quality of the solutions (trajectories exhibiting significant noise reduction compared to those currently flown in practice) and computing time is illustrated through numerical experiments on real-world case studies.

Online Trajectory Optimization Method for Large Attitude Flip Vertical Landing of the Starship-like Vehicle

A high-precision online trajectory optimization method combining convex optimization and Radau pseudospectral method is presented for the large attitude flip vertical landing problem of a starship-like vehicle. During the landing process, the aerodynamic influence on the starship-like vehicle is significant and non-negligible. A planar landing dynamics model with pitching motion is developed considering that there is no extensive lateral motion modulation during the whole flight. Combining the constraints of its powered descent landing process, a model of the fuel optimal trajectory optimization problem in the landing point coordinate system is given. The nonconvex properties of the trajectory optimization problem model are analyzed and discussed, and the advantages of fast solution and convergence certainty of convex optimization, and high discretization precision of the pseudospectral method, are fully utilized to transform the strongly nonconvex optimization problem into a series of finite-dimensional convex subproblems, which are solved quickly by the interior point method solver. Hardware-in-the-loop simulation experiments verify the effectiveness of the online trajectory optimization method. This method has the potential to be an online guidance method for the powered descent landing problem of starship-like vehicles.

Distributed Trajectory Optimization and Platooning of Vehicles to Guarantee Smooth Traffic Flow

This paper investigates a distributed trajectory optimization and control problem for a collection of vehicles with a quadratic spacing policy. A quadratic spacing policy is introduced based on the expected platoon speed to improve flexibility of speed planning and regulation. The trajectory optimization problem is solved using distributed convex optimization based on spacing errors minimization, resulting in an algorithm to provide the optimal trajectory for all following vehicles. Then, a PID-type sliding mode controller with a double high-power reaching law is presented for speed tracking control of each follower. The methodology can guarantee the internal stability, string stability and traffic flow stability with ignorable turbulence of spacing and speed, as demonstrated by numerical simulations.