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This article presents a novel recursive smooth trajectory (RST) generation algorithm for application in robotics and in particular for unmanned aerial vehicles (UAVs). RST builds the trajectory recursively as a smooth polynomial path, thus a closed form trajectory satisfying any arbitrary dynamic limitation that translates into kinematic constraints (e.g., position, velocity, acceleration, etc.). Uncertainties and perturbations in the constraints are also discussed, modeled, and finally included in the calculation of the polynomial trajectory coefficients. Moreover, due to its recursive formulation, RST enables an immediate extension toward the solution of a trajectory optimization problem. In particular, it is shown that the minimum-snap piecewise polynomial trajectory can be interpreted as a special case of RST. The effectiveness of the proposed algorithm is demonstrated numerically via two illustrative scenarios. Its application to a UAV structure is also discussed to highlight the advantage of a smooth path over a piecewise one. Finally, the computational complexity and memory requirements are analyzed.

Numerous fast heuristic algorithms, including shooting heuristics (SH), have been developed for real-time trajectory optimization, although their optimality has not yet been quantified. This paper compares the performance between fast heuristics and exact optimization models. We investigate a core trajectory optimization problem as a building block for numerous trajectory optimization problems, i.e., guiding movements of connected automated vehicles on a one-lane highway when the arrival and departure times and velocity are given. To apply the SH algorithm to this problem, we adapt it to a fast-simplified shooting heuristic (FSSH) model to solve the trajectory smoothing problems with different arrival and departure velocities. An exact trajectory optimization (ETO) model is formulated that takes the vehicle position and velocity as the decision variables, and the fuel consumption and driving comfort as the objective function. The constraints of the model are based on the limits and safety of the vehicle dynamics between consecutive vehicles. We demonstrate the convexity of the ETO objective function, ensuring the solvability of the ETO model at the true optimum using gradient descent algorithms supplied by the MATLAB optimization toolbox. Six groups of numerical experiments using different input parameters and one experiment using real Next Generation Simulation (NGSIM) data are conducted. ETO can improve the objective values by a few to tens of percentage points. However, FSSH achieves a greater solution efficiency with an average solution time of less than 0.1 s compared to ~450 s for ETO.

For a vehicle with a combined cycle engine, an ascent guidance method based on onboard trajectory regeneration is proposed to meet the requirements of adaptability and robustness. This method combines onboard trajectory planning with reference trajectory tracking and modifies the computational model. In the onboard trajectory planning, a hierarchical receding horizon optimization is developed to update the reference trajectory depending on the current state. Further, the optimization problem subject to multiple constraints is solved by an improved segmented pseudospectral method, in which a recursive initialization strategy is introduced to improve the computational efficiency for onboard application. In reference trajectory tracking, the active disturbance rejection control theory is used to improve the robustness of the tracking process. Based on the theory, an extended state observer for the multi-input and multi-output system is designed to estimate the state and the disturbance. Then, the trajectory tracking law is derived by linearizing the model with dynamic compensation. Besides, the scale factors of the dynamic system are defined to compensate for the uncertainty of the model, and an online estimation method is used to identify the parameters. Numerical simulation validates the efficiency of the proposed method.

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.

In this study, the problem of time-optimal reconnaissance trajectory design for the aeroassisted vehicle is considered. Different from most works reported previously, we explore the feasibility of applying a high-order aeroassisted vehicle dynamic model to plan the optimal flight trajectory such that the gap between the simulated model and the real system can be narrowed. A highly-constrained optimal control model containing six-degree-of-freedom vehicle dynamics is established. To solve the formulated high-order trajectory planning model, a pipelined optimization strategy is illustrated. This approach is based on the variable order Radau pseudospectral method, indicating that the mesh grid used for discretizing the continuous system experiences several adaption iterations. Utilization of such a strategy can potentially smooth the flight trajectory and improve the algorithm convergence ability. Numerical simulations are reported to demonstrate some key features of the optimized flight trajectory. A number of comparative studies are also provided to verify the effectiveness of the applied method as well as the high-order trajectory planning model.

© 2016 IEEE. In this article we present a trajectory generation method for autonomous overtaking of unexpected obstacles in a dynamic urban environment. In these settings, blind spots can arise from perception limitations. For example when overtaking unexpected objects on the vehicle's ego lane on a two-way street. In this case, a human driver would first make sure that the opposite lane is free and that there is enough room to successfully execute the maneuver, and then it would cut into the opposite lane in order to execute the maneuver successfully. We consider the practical problem of autonomous overtaking when the coverage of the perception system is impaired due to occlusion. Safe trajectories are generated by solving, in real-time, a non-linear constrained optimization, formulated as a receding horizon planner that maximizes the ego vehicle's visibility. The planner is complemented by a high-level behavior planner, which takes into account the occupancy of other traffic participants, the information from the vehicle's perception system, and the risk associated with the overtaking maneuver, to determine when the overtake maneuver should happen. The approach is validated in simulation and in experiments in real world traffic.

In an effort to pursue more advanced missions in space, improved on-board trajectory optimization and path (re)planning capabilities are necessary. Over the past decades, numerous missions have pushed the state of the art in autonomous rendezvous and proximity operations (RPOs). Regardless of the mission, any RPO guidance algorithm must be able to react to a dynamic environment while generating a fuel-efficient trajectory. An adaptive artificial potential function (AAPF) guidance exhibiting these properties has been experimentally evaluated on a spacecraft air-bearing test bed and its performance compared to traditional APF and other real-time guidance methods.

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.

The phenomenon of stop-and-go traffic and its environmental impact has become a crucial issue that needs to be tackled, in terms of the junctions between freeway and urban road networks, which consist of freeway off-ramps, downstream intersections, and the junction section. The development of Connected and Automated Vehicles (CAVs) has provided promising solutions to tackle the difficulties that arise along intersections and freeway off-ramps separately. However, several problems still exist that need to be handled in terms of junction structure, including vehicle merging trajectory optimization, vehicle crossing trajectory optimization, and heterogeneous decision-making. In this paper, a two-stage CAV trajectory optimization strategy is presented to improve fuel economy and to reduce delays through a joint framework. The first stage considers an approach to determine travel time considering the different topological structures of each subarea to ensure maximum capacity. In the second stage, Pontryagin’s Minimum Principle (PMP) is employed to construct Hamiltonian equations to smooth vehicle trajectory under the requirements of vehicle dynamics and safety. Targeted methods are devised to avoid driving backwards and to ensure an optimal vehicle gap, which make up for the shortcomings of the PMP theory. Finally, simulation experiments are designed to verify the effectiveness of the proposed strategy. The evaluation results show that our strategy could effectively militate travel delays and fuel consumption.

Underactuated robots often require involved routines for trajectory planning due to their complex dynamics. Flapping-wing aerial vehicles have unsteady aerodynamics and periodic gaits that complicate the planning procedure. In this paper, we improve upon existing methods for flight planning by introducing a two-stage optimization routine to plan flapping flight trajectories. The first stage solves a trajectory optimization problem with a data-driven fixed-wing approximation model trained with experimental flight data. The solution to this is used as the initial guess for a second stage optimization using a flapping-wing model trained with the same flight data. We demonstrate the effectiveness of this approach with a bat robot in both simulation and experimental flight results. The speed of convergence, the dependency on the initial guess, and the quality of the solution are improved, and the robot is able to track the optimized trajectory of a dive maneuver.

In this paper, we present a motion planning framework for multi-modal vehicle dynamics. Our proposed algorithm employs transcription of the optimization objective function, vehicle dynamics, and state and control constraints into sparse factor graphs, which -- combined with mode transition constraints -- constitute a composite pose graph. By formulating the multi-modal motion planning problem in composite pose graph form, we enable utilization of efficient techniques for optimization on sparse graphs, such as those widely applied in dual estimation problems, e.g., simultaneous localization and mapping (SLAM). The resulting motion planning algorithm optimizes the multi-modal trajectory, including the location of mode transitions, and is guided by the pose graph optimization process to eliminate unnecessary transitions, enabling efficient discovery of optimized mode sequences from rough initial guesses. We demonstrate multi-modal trajectory optimization in both simulation and real-world experiments for vehicles with various dynamics models, such as an airplane with taxi and flight modes, and a vertical take-off and landing (VTOL) fixed-wing aircraft that transitions between hover and horizontal flight modes., 7 pages, 6 figures, to be included in proceedings of IEEE International Conference on Robotics and Automation 2021

To guide an automated vehicle safely through complex traffic, knowledge about the future evolution of the driving situation has to be considered. The contribution at hand proposes an approach for automated driving in structured environments. An environment representation for trajectory planning is presented that enables predictive driving by interconnecting trajectory prediction for the surrounding traffic and planning of an according ego trajectory. A maneuver-based approach with an efficient trajectory model is carried out to enable an accurate and fast estimation of future motions of other vehicles. The results are directly considered in the dynamic environment representation utilized for trajectory planning. In the trajectory optimization process, the required safety distance is hence taken into account uniquely for each future time instance. The evaluation of the trajectory prediction approach shows very good performance on a simulated as well as on a dataset recorded by a test vehicle. On account of the predictive character of the developed environment potential field, simulation experiments demonstrate the feasibility and effectiveness of the proposed method.

This paper proposes a new reentry trajectory generation method, which mainly comprises two parts: reentry profile design and trajectory optimization. Different from conventional methods that design angle-of-attack (AOA) and bank-angle profiles, a height-range profile is proposed in this paper. Using this profile, the AOA can be directly derived and the quasi-equilibrium glide condition is easily satisfied. In addition, the bank angle is analytically derived with the heading error, with only one reversal. Finally, the original problem is transformed into a parameter optimization problem and solved by an improved particle swarm optimization (PSO) method. In PSO, a comparison principle is proposed to tackle constraints and three modifications are applied to the swarm evolution to enhance the searching capacity. To conquer disturbances, an online profile updating strategy is developed based on the nominal profile, so complicated work of designing a trajectory tracker is avoided. Simulation validates the effectiveness of the proposed method and demonstrates that the improved PSO is comparable to the pseudospectral method and is better than the basic PSO.

In this article, a fast chance-constrained trajectory generation strategy incorporating convex optimization and convex approximation of chance constraints is designed so as to solve the unmanned vehicle path planning problem. A path-length-optimal unmanned vehicle trajectory optimization model is constructed with the consideration of the pitch angle constraint, the curvature radius constraint, the probabilistic control actuation constraint, and the probabilistic collision avoidance constraint. Subsequently, convexification technique is introduced to convert the nonlinear problem formulation into a convex form. To deal with the probabilistic constraints in the optimization model, convex approximation techniques are introduced such that the probabilistic constraints are replaced by deterministic ones while simultaneously preserving the convexity of the optimization model. Numerical results, obtained from a number of case studies, validate the effectiveness and reliability of the proposed approach. A number of comparative studies were also performed. The results confirm that the proposed design is able to produce more optimal flight paths and achieve enhanced computational performance than other chance-constrained optimization approaches investigated in this article.

In this paper, aiming at the discontinuous ignition reentry optimization problem of powered hypersonic vehicle, the reentry trajectory characteristics at different ignition time and ignition duration are analyzed by multiphase Gauss pseudospectral method. Given the attack angle and piecewise linear function described bank angle profiles, the problem of discontinuous ignition for powered reentry trajectory optimization is transformed into a parameter planning problem of ignition time, ignition duration, bank angle parameters and total flight time. By solving the nonlinear multi-constraint parameter optimization problem, the optimal ignition time, ignition duration and bank angle are obtained. Zero value bank angle and real-time calculation bank angle simulation examples are designed to optimize bank angle and ignition commands, respectively. The simulation results show that the discontinuous ignition reentry trajectory optimization problem of powered hypersonic vehicle can be solved by parameterized bank angle profile.

Massive machine-type communications (mMTC) is a key enabling technology for Internet of Things (IoT) services in 5G and beyond. Efficient data collection from massive machine-type communication devices (MTCDs) performing sensing tasks is an important part of the service. In this paper, we consider an unmanned aerial vehicle (UAV) being deployed to facilitate data collection from MTCDs. Taking into account the limited energy of battery-powered MTCDs, the UAV trajectory is optimized to improve the energy efficiency of data collection. By fixing the starting and ending points of the UAV trajectory, a globally optimal (GO) trajectory can be obtained, based on the assumption that the UAV’s serving radius and access capacity (number of served MTCDs) are unlimited. Interestingly, it is shown that the optimal trajectory always exists as long as the UAV flying height is greater than its service radius multiplied by a constant. However, the increase of the UAV flying height deteriorates the channel, leading to reduced efficiency of energy consumption. Alternatively, a greedy dynamic (GD) trajectory optimization scheme with limited UAV service radius and access capacity is then investigated, resulting in the optimal service location of the UAV being at a lower flying height, and the energy consumption for accomplishing the data collection task being reduced. Specifically, the UAV sorts the MTCDs within its serving radius based on the distance and selects its closest serving MTCD set. In a serving MTCD set, there is an optimal hovering location that maximizes the data collection efficiency. The UAV dynamically adjusts its service set and the optimal data collection location when MTCDs finish their data transmission and exit the service set. The process continues until all the MTCDs are served and the UAV arrives at the ending point of the trajectory. Simulation results show that both the GO and GD algorithm can improve the efficiency of overall energy consumption. In particular, the online dynamic trajectory optimization scheme is less restrictive and achieves higher efficiency.

Hybrid locomotion, which combines multiple modalities of locomotion within a single robot, enables robots to carry out complex tasks in diverse environments. This paper presents a novel method for planning multi-modal locomotion trajectories using approximate dynamic programming. We formulate this problem as a shortest-path search through a state-space graph, where the edge cost is assigned as optimal transport cost along each segment. This cost is approximated from batches of offline trajectory optimizations, which allows the complex effects of vehicle under-actuation and dynamic constraints to be approximately captured in a tractable way. Our method is illustrated on a hybrid double-integrator, an amphibious robot, and a flying-driving drone, showing the practicality of the approach., Comment: Accepted to International Conference on Intelligent Robots and Systems (IROS) 2020

Vehicles nowadays are equipped with several assistance functions for e.g. Cruise Control and Lane Keeping Assist. Classical lateral control approaches such as pure pursuit, Stanley used for lane keeping assist provide good path tracking precision. However the operational domain for these approaches is limited i.e. highway driving. An adaptation of control parameters can increase the operational domain, but its difficult to tune classical approaches for the whole range of automated driving maneuvers. Apart from these classical approaches, optimization approaches are also used for lateral trajectory stabilization. The optimization based approaches have a wider operational domain, but the feasibility and real time execution remain some open issues. In this paper, we combine a classical approach and an optimization method for lateral trajectory stabilization. We present a method to optimize the control input calculated using a classical control approach i.e. pure pursuit based on a performance criteria. The performance criteria weighs the precision and comfort requirements. A non- linear optimization based on Differential Dynamic Programming (DDP) is used to solve the optimization problem. The calculated optimal trajectory is finally evaluated using a line search method to ensure the convergence and to verify the optimization policy. The approach is demonstrated on a full scale automated truck prototype and the experimental results are discussed.

Connected and autonomous vehicles (CAV) is a key technology for this century. One of the main challenges is to define a smart interaction behavior of CAV with the other road users. The challenge is mainly raised at conflicting points where path of CAV intersects with the other users. Recent studies show interaction with humans is a big challenge. It not only requires a collision avoidance system but also more communicative behaviors of the CAV. More precisely, pedestrian needs to understand the intention of the incoming CAV whether it will cross first or not according to its speed profile. One way to overcome this issue is to design optimal trajectory control of the CAV that matches with the pedestrian expectation. However, such a design faces the non-linearity of the space sharing constraint. This paper uses sequence modeling based on Petri Net in order to overcome the problem. It also uses Hamiltonian analysis to derive the optimal control. Numerical examples of the communicative behavior of CAV at pedestrian crossing show that the proposed approach provides CAV with a kind of automatic courtesy.

This article deals with the issue of trajectory optimization of autonomous terrestrial vehicles on a specific range handled by the human driver. The main contributions of this paper are a genetic algorithm-potential field combined method for optimized trajectory planning, the definition of the multi-criteria optimization problem by including a time variable, dynamical vehicle constraints, obstacle motion for collision avoidance, and improvements on the attractive and repulsive potential field definitions. The main interests of the proposed method are its efficiency even in only arc-connected spaces with holes, trajectory optimality thanks to the genetic algorithm that minimizes multi-criteria optimization, reactivity thanks to the potential field through the consideration for nature and motion of obstacles, its orientation toward situations a human driver would consider, and finally the inclusion of constraints to avoid danger and to take into account vehicle dynamics. The global trajectory, optimized through genetic algorithm, is used as a reference in a fractional potential field, which is a reactive local path planning method. The repulsive potential field is made safer by adding fractional orders to the obstacles, and the attractive potential field is improved by creating a dynamical optimal target seen from a robust control point of view. This target replaces the unique attractive potential field point and avoids its drawbacks such as local minima. Autonomous car simulation results are given for a crossroad and an overtaking scenarios.

This paper presents an efficient, robust, comfortable, and real-time motion planning framework for on-road autonomous vehicles. This proposed framework aims to enhance the performance of motion planning in complex environments such as driving in the urban area. It uses a path velocity decomposition method to separate the motion planning problem into path planning and velocity planning. The novelty lies in the use of Historical data in the $SL$ coordinate in the framework of a tree version of Rapidly-exploring Random Graph (RRT*) technique in path planner, called HSL-RRT*, which grows the path tree efficiently by the data from previous planning cycle. The velocity planner uses a Nonlinear Model Predictive Controller (NMPC) to generate optimal velocity along the path generated from the path planner, taking account of vehicle constraints and comfort. Analytic and simulation results are presented to validate the approach, with a special focus on the robustness and efficiency of the algorithm operating in complex scenarios.

This paper introduces a methodology to optimize the trajectory of connected automated vehicles (CAVs) in roundabouts using a two-dimensional point-mass model. We formulate an optimization problem that includes vehicle dynamics and collision-avoidance constraints with explicit representation of vehicle paths. The objective function of the problem minimizes the distance of CAVs to their destinations and their acceleration magnitudes. The methodology also involves a customized solution technique that convexifies the collision-avoidance constraints and employs the alternating direction method of multipliers to decompose the convexified problem into two sub-problems. The first sub-problem only includes vehicle dynamics constraints while the second sub-problem projects the solutions of the first sub-problem onto a collision-free region. The first sub-problem is then transformed into a quadratic problem by redefining its decision variables along vehicle paths. The transformation allows solving this sub-problem with several vehicle-level problems in a distributed architecture. Moreover, we show that iterating between the two sub-problems leads to convergence to the optimal solutions of the convexified problem. The methodology is applied to a case study roundabout with different demand levels. The results show that the trajectory optimization reduced the total travel times and average delays respectively by 9.1% to 36.8% and 95.8% to 98.5% compared to a scenario with human-driven vehicles.

This paper presents an integrated design method for pedestrian avoidance by considering the interaction between trajectory planning and trajectory tracking. This method aims to reduce the need for control calibration by properly considering plant uncertainties and tire force limits at the design stage. Two phases of pedestrian avoidance-trajectory planning and trajectory tracking-are designed in an integrated manner. The available tire force is distributed to the feedforward part, which is used to generate the nominal trajectory in trajectory planning phase, and to the feedback part, which is used for trajectory tracking. The trajectory planning problem is solved not by searching through a continuous spectrum of steering/braking actions, but by examining a limited set of “motion primitives,” or motion templates that can be adopted in sequence to avoid the pedestrian. An emergency rapid random tree (RRT) methodology is proposed to quickly identify a feasible solution. Subsequently, in order to guarantee accuracy and provide safety margin in trajectory tracking with presence of model uncertainties and exogenous disturbance, a simplified LQR-based funnel algorithm is proposed. Simulation results provide insight into how pedestrian collisions can be avoided under given initial vehicle and pedestrian states.

In this chapter, the SMV trajectory optimization problem established in the previous chapter is reformulated and extended to a multi-objective continuous-time optimal control model. Because of the discontinuity or nonlinearity in the vehicle dynamics and mission objectives, it is challenging to generate a compromised trajectory that can satisfy constraints and optimize objectives. To effectively analyze the relationships between objectives and calculate the Pareto front, this chapter constructs a multi-objective optimal control solver based on the evolutionary multi-objective optimization. A dominance relationship criterion based on violation degree is also defined and used to select the new generation. Simulation results are provided to illustrate the effectiveness and feasibility of the proposed algorithm in dealing with the multi-objective SMV trajectory optimization problems. Furthermore, in order to take into account the preference requirements, different transformation techniques are also proposed and the original problem is then transcribed to a single-objective formulation. These techniques are discussed in detail in the following sections. Numerical simulations were carried out and the results indicate that the constructed strategies are effective and can provide compromised solutions for solving the multi-objective SMV trajectory design problem with the consideration of preference constraints.

For the problem of vehicle obstacle avoidance control, a steering obstacle avoidance method based on optimization idea is designed. In the research of obstacle avoidance path planning method, a design method with three-segment sinusoidal lateral acceleration is adopted, combining with various practical constraints of vehicle and safe distance, an optimization solution method of obstacle avoidance trajectory is presented, which takes obstacle avoidance time as the optimal objective. Based on the established two-degree-of-freedom vehicle dynamics model, a tracking controller with moving horizon idea is designed. Using the vehicle dynamics simulation software veDYNA, the scene of vehicle lane change and obstacle avoidance is constructed. The effectiveness and control performance of this method are illustrated by simulation.

This paper proposes a two-stage optimization framework for generating the optimal parking motion trajectory of autonomous ground vehicles. The motivation for the use of this multi-layer optimization strategy relies on its enhanced convergence ability and computational efficiency in terms of finding optimal solutions under the constrained environment. In the first optimization stage, the designed optimizer applies an improved particle swarm optimization technique to produce a near-optimal parking movement. Subsequently, the motion trajectory obtained from the first stage is used to start the second optimization stage, where gradient-based techniques are applied. The established methodology is tested to explore the optimal parking maneuver for a car-like autonomous vehicle with the consideration of irregularly parked obstacles. Simulation results were produced and comparative studies were conducted for different mission cases. The obtained results not only confirm the effectiveness but also reveal the enhanced performance of the proposed optimization framework.

We propose a numerical algorithm for multiple-vehicle motion planning that explicitly takes into account the vehicle dynamics, temporal and spatial specifications, and energy-related requirements. As a motivating example, we consider the case where a group of vehicles is tasked to reach a number of target points at the same time (simultaneous arrival problem) without colliding among themselves and with obstacles, subject to the requirement that the overall energy required for vehicle motion be minimized. With the theoretical setup adopted, the vehicle dynamics are explicitly taken into account at the planning level. This paper formulates the problem of multiple-vehicle motion planning in a rigorous mathematical setting, describes the optimization algorithm used to solve it, and discusses the key implementation details. The efficacy of the method is illustrated through numerical examples for the simultaneous arrival problem. The initial guess to start the optimization procedure is obtained from simple geometrical considerations, e.g., by joining the desired initial and final positions of the vehicles via straight lines. Even though the initial trajectories thus obtained may result in intervehicle and vehicle/obstacle collisions, we show that the optimization procedure that we employ in this paper will generate collision-free trajectories that also minimize the overall energy spent by each vehicle and meet the required temporal and spatial constraints. The method developed applies to a very general class of vehicles; however, for clarity of exposition, we adopt as an illustrative example the case of wheeled robots.

This paper focuses on the real-time trajectory planning problem for autonomous vehicles driving in realistic urban environments. To solve the complex navigation problem, we adopt a hierarchical motion planning framework. First, a rough reference path is extracted from the digital map using commands from the high-level behavioral planner. The conjugate gradient nonlinear optimization algorithm and the cubic B-spline curve are employed to smoothen and interpolate the reference path sequentially. To follow the refined reference path as well as handle both static and moving objects, the trajectory planning task is decoupled into lateral and longitudinal planning problems within the curvilinear coordinate framework. A rich set of kinematically feasible path candidates are generated to deal with the dynamic traffic both deliberatively and reactively. In the meanwhile, the velocity profile generation is performed to improve driving safety and comfort. After that, the generated trajectories are carefully evaluated by an objective function, which combines behavioral decisions by reasoning about the traffic situations. The optimal collision-free, smooth, and dynamically feasible trajectory is selected and transformed into commands executed by the low-level lateral and longitudinal controllers. Field experiments have been carried out with our test autonomous vehicle on the realistic inner-city roads. The experimental results demonstrated capabilities and effectiveness of the proposed trajectory planning framework and algorithms to safely handle a variety of typical driving scenarios, such as static and moving objects avoidance, lane keeping, and vehicle following, while respecting the traffic rules.

Summary In this paper, we investigate an optimal control problem in which the objective is to decelerate a simplified vehicle model, subject to input constraints, from a given initial velocity down to zero by minimizing a quadratic cost functional. The problem is of interest because, although it involves apparently simple drift-less dynamics, a minimizing trajectory does not exist over the admissible input trajectories. This problem is motivated by a minimum-time problem for a fairly complex car vehicle model on a race track. Numerical computations run on the car trajectory optimization problem provide evidence of convergence issues and of an apparently unmotivated ripple in the steer angle. Characterizing this ripple behavior is important to fully understand and exploit minimizing vehicle trajectories. We are able to isolate the key features of this chattering behavior in a very simple dynamics/objective setting. We show that the cost functional has an infimum, but an admissible minimizing input trajectory does not exist. We also show that the infimum can be arbitrarily approximated by bang-bang inputs with a sufficiently large number of switches. We reproduce this phenomenon in numerical computations and characterize it by means of non-existence of admissible minimizing trajectories. Copyright © 2015 John Wiley & Sons, Ltd.

The aerodynamic characteristics of air-breathing combined power flight vehicle change drastically with flight conditions during wide envelope. Using traditional trajectory optimization methods based on aerodynamic linear interpolation model, results show that the control variables and state variables change greatly with poor smoothness, which cannot meet working conditions of air-breathing dynamic model and make it difficult for control system design. In this paper, based on Hp adaptive pseudospectral method, the response surface surrogate model is put forward to gain smooth aerodynamic data, solving problems in the aerodynamic linear interpolation model. Results of ascent trajectory optimization of the Rocket-Based Combined Cycle power flight vehicle show that the angle of attack control changes smoothly and has small absolute value of the first derivative during the flight, and trajectory is relatively smooth, which provides ideal working condition for engine. Meanwhile, trajectory simulation using obtained optimal control variable is achieved, which shows small difference and further verifies the feasibility of the aerodynamic surrogate model.

In this paper we propose a local trajectory planner for front steered car-like vehicles based on a combined direct optimization of the lateral and longitudinal vehicle guidance. The planner is designed for continuously optimizing a local trajectory based on a provided reference path in a structured or semi-structured driving environment. The planner respects constraints of the driving dynamics as well as actuator lim- itations and avoids static and dynamic obstacles. It is not restricted to a limited set of maneuvers. The implementation of the planner allows an online adaptation of the resulting driving behavior to satisfy different comfort or driving style demands. After Software-in-the-Loop simulations the algorithm was tested in two different real-world driving scenarios in ika’s automated vehicle which provides interfaces for full lateral and longitudinal control.