Real-time Trajectory Optimization Research Articles

Social Elastic Band with Prediction and Anticipation: Enhancing Real-Time Path Trajectory Optimization for Socially Aware Robot Navigation

AbstractAs social robots are projected to become an integral part of human life in the coming decades, their ability to adapt movement and trajectory when in proximity to people is essential for ensuring social acceptance during human-robot interaction. A key aspect of this adaptability involves predicting and anticipating human intents during robot navigation. Despite significant strides in the social navigation of autonomous robots within human environments, opportunities for advancements in related algorithms persist. This paper presents a novel real-time path trajectory optimization algorithm for socially aware robot navigation, grounded in the social elastic band concept, incorporating prediction and anticipation of human movements to adapt its forward velocity. Building upon the elastic band framework introduced in the 1990s for adapting robot trajectories in dynamic environments, our proposal of social elastic band differentiates between objects and human presence. This distinction allows for the definition of social interaction spaces and their relationship to the elastic band, facilitating the generation of socially accepted paths that rapidly adapt to environmental changes without causing a disturbance. Integrated into the SNAPE social navigation framework, the algorithm has been tested and validated through simulations and real-world experiments in various environments.

A Low-Altitude Obstacle Avoidance Method for UAVs Based on Polyhedral Flight Corridor

UAVs flying in complex low-altitude environments often require real-time sensing to avoid environmental obstacles. In previous approaches, UAVs have usually carried out motion planning based on primitive navigation maps such as point clouds and raster maps to achieve autonomous obstacle avoidance. However, due to the huge amount of data in these raw navigation maps and the highly discrete map information, the efficiency of solving the UAV’s real-time trajectory optimization is low, making it difficult to meet the demand for efficient online motion planning. A flight corridor is a series of unobstructed continuous areas and has convex properties. The flight corridor can be used as a simple parametric representation to characterize the safe flight space in the environment, and used as the cost of the collision term in the trajectory back-end optimization for trajectory solving, which can improve the efficiency of real-time trajectory solving and ensure flight safety. Therefore, this paper focuses on the construction of safe flight corridors for UAVs and autonomous obstacle avoidance algorithms for UAVs based on safe flight corridors, based on a rotary-wing UAV platform, and proposes a polyhedral flight corridor construction algorithm and realizes autonomous obstacle avoidance for UAVs based on the constructed flight corridors.

Real-Time Optimal Trajectory Planning for Autonomous Driving with Collision Avoidance Using Convex Optimization

An online trajectory planning method for collision avoidance is proposed to improve vehicle driving safety and comfort simultaneously. The collision-free trajectory for autonomous driving is formulated as a nonlinear optimization problem. A novel approximate convex optimization approach is developed for the online optimal trajectory in both longitudinal and lateral directions. First, a dual variable is used to model the non-convex collision-free constraint for driving safety and is calculated by solving a dual problem of the relative distance between vehicles. Second, the trajectory is further optimized in a model predictive control framework considering the safety. It realizes continuous-time and dynamic feasible motion with collision avoidance. The geometry of object vehicles is described by polygons instead of circles or ellipses in traditional methods. In order to avoid aggressive maneuver in the longitudinal and lateral directions for driving comfort, rates of the acceleration and the steering angle are restricted. The final formulated optimization problem is convex, which can be solved by using quadratic programming solvers and is computationally efficient for online application. Simulation results show that this approach can obtain similar driving performance compared to a state-of-the-art nonlinear optimization method. Furthermore, various driving scenarios are tested to evaluate the robustness and the ability for handling complex driving tasks.

GIS-Data-Driven Efficient and Safe Path Planning for Autonomous Ships in Maritime Transportation

Maritime transportation is vital to the global economy. With the increased operating and labor costs of maritime transportation, autonomous shipping has attracted much attention in both industry and academia. Autonomous shipping can not only reduce the marine accidents caused by human factors but also save labor costs. Path planning is one of the key technologies to enable the autonomy of ships. However, mainstream ship path planning focuses on searching for the shortest path and controlling the vehicle in order to track it. Such path planning methods may lead to a dynamically infeasible trajectory that fails to avoid obstacles or reduces fuel efficiency. This paper presents a data-driven, efficient, and safe path planning (ESP) method that considers ship dynamics to provide a real-time optimal trajectory generation. The optimization objectives include fuel consumption and trajectory smoothness. Furthermore, ESP is capable of fast replanning when encountering obstacles. ESP consists of three components: (1) A path search method that finds an optimal search path with the minimum number of sharp turns from the geographic data collected by the geographic information system (GIS); (2) a minimum-snap trajectory optimization formulation with dynamic ship constraints to provide a smooth and collision-free trajectory with minimal fuel consumption; (3) a local trajectory replanner based on B-spline to avoid unexpected obstacles in real time. We evaluate the performance of ESP by data-driven simulations. The geographical data have been collected and updated from GIS. The results show that ESP can plan a global trajectory with safety, minimal turning points, and minimal fuel consumption based on the maritime information provided by nautical charts. With the long-range perception of onboard radars, the ship can avoid unexpected obstacles in real time on the planned global course.

Flight data clustering for offline evaluation of real-time trajectory optimization framework

Real-time Trajectory Optimization (RTOP) is a machine learning-based flight path optimization framework that automatically performs in-flight re-planning continuously with the latest weather information. The framework leverages three core algorithms: (1) a supervised learning algorithm to predict winds, (2) an unsupervised learning algorithm to forecast areas of convective weather, and (3) a graph-based pathfinding algorithm to generate optimized flight trajectories. This research first proposes a methodology that uses an unsupervised learning-based clustering algorithm with the concept of Hausdorff distance to appropriately select representative flight cases to build up confidence in statistical analysis. Second, this research conducts an operational evaluation of the RTOP framework with the selected representative flight cases to highlight the statistical significance of simulation results and indicate the expected savings. The results show that optimized flight trajectories generated by the RTOP framework reduce the duration of the en-route phase up to 2% compared to the real-world flight trajectories in most cases.

Optimal Control of Dynamical Systems using Calculus of Variations

This paper explores the application of calculus of variations techniques for optimizing the control of complex dynamical systems. Determining control strategies that minimize cost metrics and satisfy constraints is critical across engineering disciplines like robotics, aerospace, and process operations. Classical optimal control methods, such as Pontryagin's Maximum Principle, transform an optimal control problem into a calculus of variations framework amenable to analytical and numerical optimization. We present a unified framework for applying these techniques, enabling dynamical systems defined by ordinary and partial differential equations to be optimized by conversion into a nonlinear programming form. Analytical approaches provide theoretical guarantees on control performance while numerical methods, like direct collocation and transcription, enable large-scale optimal control problems to be efficiently solved. The connections between dynamical systems, calculus of variations, and modern numerical optimization methods establish a holistic methodology for control engineers and applied mathematicians to design optimal controllers for physical systems. Case studies on real-time trajectory optimization, adaptive path planning, and dynamic process operations demonstrate the efficacy of the proposed optimal control framework across robotics, aerospace, power systems, and other applications.

Optimal Control of Dynamical Systems using Calculus of Variations

This paper explores the application of calculus of variations techniques for optimizing the control of complex dynamical systems. Determining control strategies that minimize cost metrics and satisfy constraints is critical across engineering disciplines like robotics, aerospace, and process operations. Classical optimal control methods, such as Pontryagin's Maximum Principle, transform an optimal control problem into a calculus of variations framework amenable to analytical and numerical optimization. We present a unified framework for applying these techniques, enabling dynamical systems defined by ordinary and partial differential equations to be optimized by conversion into a nonlinear programming form. Analytical approaches provide theoretical guarantees on control performance while numerical methods, like direct collocation and transcription, enable large-scale optimal control problems to be efficiently solved. The connections between dynamical systems, calculus of variations, and modern numerical optimization methods establish a holistic methodology for control engineers and applied mathematicians to design optimal controllers for physical systems. Case studies on real-time trajectory optimization, adaptive path planning, and dynamic process operations demonstrate the efficacy of the proposed optimal control framework across robotics, aerospace, power systems, and other applications.

DQN-Based Proactive Trajectory Planning of UAVs in Multi-Access Edge Computing

The main aim of future mobile networks is to provide secure, reliable, intelligent, and seamless connectivity. It also enables mobile network operators to ensure their customer’s a better quality of service (QoS). Nowadays, Unmanned Aerial Vehicles (UAVs) are a significant part of the mobile network due to their continuously growing use in various applications. For better coverage, cost-effective, and seamless service connectivity and provisioning, UAVs have emerged as the best choice for telco operators. UAVs can be used as flying base stations, edge servers, and relay nodes in mobile networks. On the other side, Multi-access Edge Computing (MEC) technology also emerged in the 5G network to provide a better quality of experience (QoE) to users with different QoS requirements. However, UAVs in a mobile network for coverage enhancement and better QoS face several challenges such as trajectory designing, path planning, optimization, QoS assurance, mobility management, etc. The efficient and proactive path planning and optimization in a highly dynamic environment containing buildings and obstacles are challenging. So, an automated Artificial Intelligence (AI) enabled QoS-aware solution is needed for trajectory planning and optimization. Therefore, this work introduces a well-designed AI and MEC-enabled architecture for a UAVs-assisted future network. It has an efficient Deep Reinforcement Learning (DRL) algorithm for real-time and proactive trajectory planning and optimization. It also fulfills QoS-aware service provisioning. A greedy-policy approach is used to maximize the long-term reward for serving more users with QoS. Simulation results reveal the superiority of the proposed DRL mechanism for energy-efficient and QoS-aware trajectory planning over the existing models.

FEASIBLE AND OPTIMAL TRAJECTORIES GENERATION FOR AUTONOMOUS DRIVING VEHICLES, 11-24.

Real-time feasible and optimal trajectory generation subject tothe vehicle physical constraints, the surrounding obstacles, andthe environmental conditions is one of the most important partsof autonomous driving vehicles.

Singularity Avoidance with Application to Online Trajectory Optimization for Serial Manipulators

This work proposes a novel singularity avoidance approach for real-time trajectory optimization based on known singular configurations. The focus of this work lies on analyzing kinematically singular configurations for three robots with different kinematic structures, i.e., the Comau Racer 7-1.4, the KUKA LBR iiwa 14 R820, and the Franka Emika Panda, and exploiting these configurations in form of tailored potential functions for singularity avoidance. Monte Carlo simulations of the proposed method and the commonly used manipulability maximization approach are performed for comparison. The numerical results show that the average computing time can be reduced and shorter trajectories in both time and path length are obtained with the proposed approach.

Sequential Convex Programming Methods for Real-Time Optimal Trajectory Planning in Autonomous Vehicle Racing

Optimization problems for trajectory planning in autonomous vehicle racing are characterized by their nonlinearity and nonconvexity. Instead of solving these optimization problems, usually a convex approximation is solved instead to achieve a high update rate. We present a real-time-capable model predictive control (MPC) trajectory planner based on a nonlinear single-track vehicle model and Pacejka’s magic tire formula for autonomous vehicle racing. After formulating the general nonconvex trajectory optimization problem, we form a convex approximation using sequential convex programming (SCP). The state of the art convexifies track constraints using sequential linearization (SL), which is a method of relaxing the constraints. Solutions to the relaxed optimization problem are not guaranteed to be feasible in the nonconvex optimization problem. We propose sequential convex restriction (SCR) as a method to convexify track constraints. SCR guarantees that resulting solutions are feasible in the nonconvex optimization problem. We show recursive feasibility of solutions to the restricted optimization problem. The MPC is evaluated on a scaled version of the Hockenheimring racing track in simulation. The results show that MPC using SCR yields faster lap times than MPC using SL, while still being real-time capable.

Trajectory Optimization with Complex Obstacle Avoidance Constraints via Homotopy Network Sequential Convex Programming

Space vehicles’ real-time trajectory optimization is the key to future automatic guidance. Still, the current sequential convex programming (SCP) method suffers from a low convergence rate and poor real-time performance when dealing with complex obstacle avoidance constraints (OACs). Given the above challenges, this work combines homotopy and neural network techniques with SCP to propose an innovative algorithm. Firstly, a neural network was used to fit the minimum signed distance field at obstacles’ different “growth” states to represent the OACs. Then, the network was embedded with the SCP framework, thus smoothly transforming the OACs from simple to complex. Numerical simulations showed that the proposed algorithm can efficiently deal with trajectory optimization under complex OACs such as a “maze”, and the algorithm has a high convergence rate and flexible extensibility.

Fast trajectory planning and control of a lab-scale 3D gantry crane for a moving target in an environment with obstacles

In this work, the real-time optimal trajectory planning, together with a cascaded tracking controller, is presented for a three-dimensional (3D) gantry crane in an environment with static obstacles and a dynamically moving target considering dynamic constraints and control input limits. State-of-the-art trajectory optimization-based approaches require long computation times and cannot quickly respond to changes in the target state. The focus of this paper lies on a novel trajectory planning algorithm, which consists of two steps. First, an offline trajectory planner is implemented to compute a time-optimal, collision-free, and dynamically feasible trajectory database that connects all possible initial states of the gantry crane from a predefined starting subspace to the target states in a target subspace. Second, based on linear constrained quadratic programming, the online trajectory replanner makes use of this trajectory database to generate an optimal trajectory in real time that accounts for all changes in the target state. Additionally, a trajectory tracking controller is developed to take into account the dynamic constraints of the gantry crane and to compensate for possible model inaccuracies, disturbances, and other non-modeled effects. Both simulation and experimental results are presented to demonstrate the performance of the proposed trajectory (re)planning algorithm and the control concept.

Real-time trajectory optimization for collision-free asteroid landing based on deep neural networks

For the problem of autonomous and safe landing on irregularly shaped asteroids, a real-time collision-free trajectory optimization method based on deep neural networks (DNNs) is proposed. First, in order to overcome the difficulty of solving the optimal trajectories by indirect methods caused by anti-collision path constraints and the difficulty of establishing the mapping relationship between states and initial costates by DNNs, a two-stage anti-collision trajectory optimization model is constructed, which can provide the initial costates for trajectory optimization under anti-collision constraints of each stage through the approximate analytical solution of costates. It can efficiently generate a large number of optimal trajectory samples. Second, a region division strategy is proposed for further reducing the time-consuming of DNN training. Then, the path constraints are slightly relaxed to avoid the optimal trajectories predicted by DNNs from violating the anti-collision constraints. The proposed method is applied to the scenarios of landing on asteroids 101955 Bennu and 433 Eros. The simulation results indicate that this method can estimate the initial costates efficiently and accurately, and can plan the optimal trajectories of anti-collision landing in real time.

A Real-Time Trajectory Optimization Method for Hypersonic Vehicles Based on a Deep Neural Network

Considering the high-efficient trajectory planning requirements for hypersonic vehicles, this paper proposes a real-time trajectory optimization method based on a deep neural network. First, the trajectory optimization model of the hypersonic vehicle reentry phase is developed. The pseudo-spectral method is used to perform the trajectory optimization offline, and multiple optimal trajectory data are obtained. In addition, based on the inherent relationship between the state and control variables of a trajectory, a neural network is established to predict the current control outputs. The sample library of optimal trajectory data is used to train the parameters of the deep neural network to obtain an optimal neural network model. Finally, the simulation verification of the hypersonic vehicle reentry phase is performed. The simulation results show that under the condition of the initial value deviation and environmental interference, the proposed deep learning-based method can achieve a fast generation of hypersonic vehicle optimal trajectories, while achieving the advantages of high computational efficiency and reliability. Compared to traditional trajectory optimization algorithms, the proposed method has the generalization capability that satisfies the accuracy requirements and meets the demands of online real-time trajectory optimization.

Real-time optimal trajectory planning for autonomous vehicles and lap time simulation using machine learning

Widespread development of driverless vehicles has led to the formation of autonomous racing, where technological development is accelerated by the high speeds and competitive environment of motorsport. A particular challenge for an autonomous vehicle is that of identifying a target trajectory – or, in the case of a competition vehicle, the racing line. Many existing approaches to finding the racing line are either not time-optimal solutions, or are computationally expensive – rendering them unsuitable for real-time application using on-board processing hardware. This study describes a machine learning approach to generating an accurate prediction of the racing line in real-time on desktop processing hardware. The proposed algorithm is a feed-forward neural network, trained using a dataset comprising racing lines for a large number of circuits calculated via traditional optimal control lap time simulation. The network predicts the racing line with a mean absolute error of ±0.27 m, and just ±0.11 m at corner apex – comparable to human drivers, and autonomous vehicle control subsystems. The approach generates predictions within 33 ms, making it over 9000 times faster than traditional methods of finding the optimal trajectory. Results suggest that for certain applications data-driven approaches to find near-optimal racing lines may be favourable to traditional computational methods.

Real-time trajectory optimization for powered planetary landings based on analytical shooting equations

Traditionally, numerical trajectory integration for shooting equation calculation and iterations for shooting with randomly guessed initial solutions deteriorate the real-time performance of indirect methods for on-board applications. In this study, the indirect method is improved to achieve real-time trajectory optimization of fuel-optimal powered planetary landings with the help of analytical shooting equation derivations and a practical homotopy technique. Specifically, the contributions of this paper are threefold. First, the analytical expressions for shooting equation calculation are derived to replace the traditional time-consuming trajectory integration. Consequently, the computational efficiency is significantly improved. Second, the original three-dimensional landing problem is connected with a simplified one-dimensional problem that only involves the vertical dynamics, and its analytical solution is obtained based on Pontryagin’s minimum principle. Third, starting with the analytical solution, the accurate solution of the original landing problem can be obtained through an adaptive homotopy process. Simulation results of Earth landing scenarios are given to substantiate the effectiveness of the proposed techniques and illustrate that the developed method can obtain a fuel-optimal landing trajectory in 5 ms with 100% success rate.

A Hierarchical Autonomous Driver for a Racing Car: Real-Time Planning and Tracking of the Trajectory

The aim of this study was to develop trajectory planning that would allow an autonomous racing car to be driven as close as possible to what a driver would do, defining the most appropriate inputs for the current scenario. The search for the optimal trajectory in terms of lap time reduction involves the modeling of all the non-linearities of the vehicle dynamics with the disadvantage of being a time-consuming problem and not being able to be implemented in real-time. However, to improve the vehicle performances, the trajectory needs to be optimized online with the knowledge of the actual vehicle dynamics and path conditions. Therefore, this study involved the development of an architecture that allows an autonomous racing car to have an optimal online trajectory planning and path tracking ensuring professional driver performances. The real-time trajectory optimization can also ensure a possible future implementation in the urban area where obstacles and dynamic scenarios could be faced. It was chosen to implement a local trajectory planning based on the Model Predictive Control(MPC) logic and solved as Linear Programming (LP) by Sequential Convex Programming (SCP). The idea was to achieve a computational cost, 0.1 s, using a point mass vehicle model constrained by experimental definition and approximation of the car’s GG-V, and developing an optimum model-based path tracking to define the driver model that allows A car to follow the trajectory defined by the planner ensuring a signal input every 0.001 s. To validate the algorithm, two types of tests were carried out: a Matlab-Simulink, Vi-Grade co-simulation test, comparing the proposed algorithm with the performance of an offline motion planning, and a real-time simulator test, comparing the proposed algorithm with the performance of a professional driver. The results obtained showed that the computational cost of the optimization algorithm developed is below the limit of 0.1 s, and the architecture showed a reduction of the lap time of about 1 s compared to the offline optimizer and reproducibility of the performance obtained by the driver.

Eddy current de-tumbling large geostationary debris based on feedback linearization model predictive control

In this paper, we investigate the application of Feedback Linearization Model Predictive Control (FLMPC) for the eddy current de-tumbling of space tumbling targets. Based on the Clohessy-Wiltshire (C-W) equation and the Euler rotational equations of motion, the dynamic model of the eddy current de-tumbling process is established, and the error dynamic model is further derived. After that, the safety constraint of the eddy current de-tumbling process is deduced and linearized. On this basis, real-time optimal trajectory is obtained with the optimal equilibrium solver, and FLMPC is proposed to control the system. In order to ensure the simplicity and rapidity of the controller, the feedback linearization input constraint is directly used as the global prediction constraint and the MPC constraint problem is converted into the form of Quadratic Programming (QP) which can be effectively solved. Simulation results show that the proposed algorithm can realize quick and safe de-tumbling of the target with the various constraints ensured and a very small range of steady-state errors.

Time suboptimal formation flying manoeuvres through improved magnetic charged system search

The development of fast and reliable optimization algorithms is required in order to obtain real-time optimal trajectory on-board spacecraft. In addition, the wide spread of small satellites, due to their low costs, is leading to a greater number of satellite formations in space. This paper presents an Improved version of the Magnetic Charged System Search (IMCSS) metaheuristic algorithm to compute time-suboptimal manoeuvres for satellite formation flying. The proposed algorithm exploits some strategies aimed at improving the convergence to the optimum, such as the chaotic local search and the boundary handling technique, and it is able to self-tune its internal parameters and coefficients. Moreover, the inverse dynamics technique and the differential flatness approach, through the B-splines curves, are used to approximate the trajectory. The optimization procedure is applied to the circular J2 relative model developed by Schweighart and Sedwick and to the elliptical relative motion model developed by Yamanaka and Ankersen. The results of this paper show that the convergence is better achieved by using the proposed tools, thus proving the efficiency and reliability of the algorithm in solving some space engineering problems.