UAVs cooperative task assignment and trajectory optimization with safety and time constraints

This paper proposes new methods and strategies for Multi-UAVs cooperative attacks with safety and time constraints in a complex environment.Delaunay triangle is designed to construct a map of the complex flight environment for aerial vehicles.Delaunay-Map,Safe Flight Corridor(SFC),and Relative Safe Flight Corridor(RSFC)are applied to ensure each UAV flight trajectory's safety.By using such techniques,it is possible to avoid the collision with obstacles and collision between UAVs.Bezier-curve is further developed to ensure that multi-UAVs can simultaneously reach the target at the specified time,and the trajectory is within the flight corridor.The trajectory tracking controller is also designed based on model predictive control to track the planned trajectory accurately.The simulation and experiment results are presented to verifying developed strategies of Multi-UAV cooperative attacks.

Autonomous UAV 3D trajectory optimization and transmission scheduling for sensor data collection on uneven terrains

This paper considers a time-constrained data collection problem from a network of ground sensors located on uneven terrain by an Unmanned Aerial Vehicle(UAV),a typical Unmanned Aerial System(UAS).The ground sensors harvest renewable energy and are equipped with batteries and data buffers.The ground sensor model takes into account sensor data buffer and battery limitations.An asymptotically globally optimal method of joint UAV 3D trajectory optimization and data transmission schedule is developed.The developed method maximizes the amount of data transmitted to the UAV without losses and too long delays and minimizes the propulsion energy of the UAV.The developed algorithm of optimal trajectory optimization and transmission scheduling is based on dynamic programming.Computer simulations demonstrate the effectiveness of the proposed algorithm.

Cooperative trajectory optimization of UAVs in approaching stage using feedback guidance methods

This paper mainly studies the problem of using UAVs to provide accurate remote target indication for hypersonic projectiles.Based on the optimal trajectory trends and feedback guidance methods,a new cooperative control algorithm is proposed to optimize trajectories of multi-UAVs for target tracking in approaching stage.Based on UAV kinematics and sensor performance models,optimal trajectory trends of UAVs are analyzed theoretically.Then,feedback guidance methods are proposed under the optimal observation trends of UAVs in the approaching target stage,producing trajectories with far less computational complexity and performance very close to the best-known trajectories.Next,the sufficient condition for the UAV to form the optimal observation configuration by the feedback guidance method is presented,which guarantees that the proposed method can optimize the observation trajectory of the UAV in approaching stage.Finally,the feedback guidance method is numerically simulated.Simulation results demonstrate that the estimation performance of the feedback guidance method is superior to the Lyapunov guidance vector field(LGVF)method and verify the effectiveness of the proposed method.Additionally,compared with the receding horizon optimization(RHO)method,the proposed method has the same optimization ability as the RHO method and better real-time performance.

An AutoML based trajectory optimization method for long-distance spacecraft pursuit-evasion game

Current successes in artificial intelligence domain have revitalized interest in spacecraft pursuit-evasion game,which is an interception problem with a non-cooperative maneuvering target.The paper presents an automated machine learning(AutoML)based method to generate optimal trajectories in long-distance scenarios.Compared with conventional deep neural network(DNN)methods,the proposed method dramatically reduces the reliance on manual intervention and machine learning expertise.Firstly,based on differential game theory and costate normalization technique,the trajectory optimization problem is formulated under the assumption of continuous thrust.Secondly,the AutoML technique based on sequential model-based optimization(SMBO)framework is introduced to automate DNN design in deep learning process.If recommended DNN architecture exists,the tree-structured Parzen estimator(TPE)is used,otherwise the efficient neural architecture search(NAS)with network morphism is used.Thus,a novel trajectory optimization method with high computational efficiency is achieved.Finally,numerical results demonstrate the feasibility and efficiency of the proposed method.

Re-entry trajectory optimization using a multiple- interval Radau pseudospectral method

Aiming at increasing the calculation efficiency of the pseudospectral methods, a multiple- interval Radau pseudospectral method （RPM） is presented to generate a reusable launch vehicle （RLV） ＇s optimal re-entry trajectory. After dividing the optimal control problem into many intervals, the state and control variables are approximated using many fixed- and low-degree Lagrange polyno- mials in each interval. Convergence of the numerical discretization is then achieved by increasing the number of intervals. With the application of the proposed method, the normal nonlinear program- ming （NLP） problem transcribed from the optimal control problem can avoid being dense because of the low-degree approximation polynomials in each interval. Thus, the NLP solver can easily compute a solution. Finally, simulation results show that the optimized re-entry trajectories satisfy the path constraints and the boundary constraints successfully. Compared with the single interval RPM, the multiple-interval RPM is significantly faster and has higher calculation efficiency. The results indicate that the multiple-interval RPM can be applied for real-time trajectory generation due to its high effi- ciency and high precision.

Earth-moon Trajectory Optimization Using Solar Electric Propulsion

The optimization of the Earth-moon trajectory using solar electric propulsion is presented. A feasible method is proposed to optimize the transfer trajectory starting from a low Earth circular orbit （500 km altitude） to a low lunar circular orbit （200 km altitude）. Due to the use of low-thrust solar electric propulsion, the entire transfer trajectory consists of hundreds or even thousands of orbital revolutions around the Earth and the moon. The Earth-orbit ascending （from low Earth orbit to high Earth orbit） and lunar descending （from high lunar orbit to low lunar orbit） trajectories in the presence of J2 perturbations and shadowing effect are computed by an analytic orbital averaging technique. A direct/indirect method is used to optimize the control steering for the trans-lunar trajectory segment, a segment from a high Earth orbit to a high lunar orbit, with a fixed thrust-coast-thrust engine sequence. For the trans-lunar trajectory segment, the equations of motion are expressed in the inertial coordinates about the Earth and the moon using a set of nonsingular equinoctial elements inclusive of the gravitational forces of the sun, the Earth, and the moon. By way of the analytic orbital averaging technique and the direct/indirect method, the Earth-moon transfer problem is converted to a parameter optimization problem, and the entire transfer trajectory is formulated and optimized in the form of a single nonlinear optimization problem with a small number of variables and constraints. Finally, an example of an Earth-moon transfer trajectory using solar electric propulsion is demonstrated.

Study of trajectory optimization using terminal-node adaptive-altered spline algorithm

The advantage of using a spline function to evaluate the trajectory parameters optimization is discussed. A new method that using adaptive varied terminal-node spline interpolation for solving trajectory optimization is proposed. And it is validated in optimizing the trajectory of guided bombs and extended range guided munitions （ERGM）. The solutions are approximate to the real optimization results. The advantage of this arithmetic is that it can be used to solve the trajectory optimization with complex models. Thus, it is helpful for solving the practical engineering optimization problem.

Three-Dimensional Trajectory Optimization for Secure UAV-Enabled Cognitive Communications

Unmanned aerial vehicles(UAVs)are en-visioned as a promising means of providing wireless services for various complex terrains and emergency situations.In this paper,we consider a wireless UAV-enabled cognitive communication network,where a rotary-wing UAV transmits confidential information to a ground cognitive user over the spectrum assigned to primary users(PUs),while eavesdroppers attempt to wiretap the legitimate transmission.In order to en-hance the secrecy performance of wireless communi-cations,the secrecy rate(SR)of the UAV-enabled cog-nitive communication system is maximized through optimizing UAV three-dimensional(3D)flying trajec-tory while satisfying the requirements of UAV’s initial and final locations and guaranteeing the constraint of maximum speed of UAV and the interference thresh-old of each PU.However,the formulated SR maxi-mization(SRM)problem is non-convex.For the pur-pose of dealing with this intractable problem,we em-ploy the difference of two-convex functions approxi-mation approach to convert the non-convex optimiza-tion problem into a convex one,which is then solved through applying standard convex optimization tech-niques.Moreover,an iterative 3D trajectory opti-mization algorithm for SRM scheme is proposed to achieve the near-optimal 3D trajectory.Simulation re-sults show that our proposed 3D trajectory optimiza-tion based SRM algorithm has good convergence,and the proposed SRM scheme outperforms the bench-mark approach in terms of the SR performance.

Trajectory optimization of a reentry vehicle based on artificial emotion memory optimization

The trajectory optimization of an unpowered reentry vehicle via artificial emotion memory optimization(AEMO)is discussed.Firstly,reentry dynamics are established based on multiple constraints and parameterized control variables with finite dimensions are designed.If the constraint is not satisfied,a distance measure and an adaptive penalty function are used to address this scenario.Secondly,AEMO is introduced to solve the trajectory optimization problem.Based on the theories of biology and cognition,the trial solutions based on emotional memory are established.Three search strategies are designed for realizing the random search of trial solutions and for avoiding becoming trapped in a local minimum.The states of the trial solutions are determined according to the rules of memory enhancement and forgetting.As the iterations proceed,the trial solutions with poor quality will gradually be forgotten.Therefore,the number of trial solutions is decreased,and the convergence of the algorithm is accelerated.Finally,a numerical simulation is conducted,and the results demonstrate that the path and terminal constraints are satisfied and the method can realize satisfactory performance.

Radial Basis Function Interpolation and Galerkin Projection for Direct Trajectory Optimization and Costate Estimation

This work presents a novel approach combining radial basis function(RBF)interpolation with Galerkin projection to efficiently solve general optimal control problems.The goal is to develop a highly flexible solution to optimal control problems,especially nonsmooth problems involving discontinuities,while accounting for trajectory accuracy and computational efficiency simultaneously.The proposed solution,called the RBF-Galerkin method,offers a highly flexible framework for direct transcription by using any interpolant functions from the broad class of global RBFs and any arbitrary discretization points that do not necessarily need to be on a mesh of points.The RBF-Galerkin costate mapping theorem is developed that describes an exact equivalency between the Karush-Kuhn-Tucker(KKT)conditions of the nonlinear programming problem resulted from the RBF-Galerkin method and the discretized form of the first-order necessary conditions of the optimal control problem,if a set of discrete conditions holds.The efficacy of the proposed method along with the accuracy of the RBF-Galerkin costate mapping theorem is confirmed against an analytical solution for a bang-bang optimal control problem.In addition,the proposed approach is compared against both local and global polynomial methods for a robot motion planning problem to verify its accuracy and computational efficiency.

Analysis of Controlled Trajectory Optimization for Canard Trajectory Correction Fuze

The optimization method of the canard trajectory correction fuze's controlled trajectory phase is researched by using the aerodynamics of aerocraft and the optimal control theory, the trajectory parameters of the controlled trajectory phase based on the least energy cost are determined. On the basis of determining the control starting point and the target point, the optimal trajectory and the variation rule of the normal overload with the least energy cost are provided, when there is no time restriction in the simulation process. The results provide a theoretical basis for the structure design of the canard mechanism.

A DNN based trajectory optimization method for intercepting non-cooperative maneuvering spacecraft

Current successes in artificial intelligence domain have revitalized interest in neural networks and demonstrated their potential in solving spacecraft trajectory optimization problems. This paper presents a data-free deep neural network(DNN) based trajectory optimization method for intercepting noncooperative maneuvering spacecraft, in a continuous low-thrust scenario. Firstly, the problem is formulated as a standard constrained optimization problem through differential game theory and minimax principle. Secondly, a new DNN is designed to integrate interception dynamic model into the network and involve it in the process of gradient descent, which makes the network endowed with the knowledge of physical constraints and reduces the learning burden of the network. Thus, a DNN based method is proposed, which completely eliminates the demand of training datasets and improves the generalization capacity. Finally, numerical results demonstrate the feasibility and efficiency of our proposed method.

Six-DOF trajectory optimization for reusable launch vehicles via Gauss pseudospectral method

To be close to the practical flight process and increase the precision of optimal trajectory, a six-degree-offreedom（6-DOF） trajectory is optimized for the reusable launch vehicle（RLV） using the Gauss pseudospectral method（GPM）. Different from the traditional trajectory optimization problem which generally considers the RLV as a point mass, the coupling between translational dynamics and rotational dynamics is taken into account. An optimization problem is formulated to minimize a performance index subject to 6-DOF equations of motion, including translational and rotational dynamics. A two-step optimal strategy is then introduced to reduce the large calculations caused by multiple variables and convergence confinement in 6-DOF trajectory optimization. The simulation results demonstrate that the 6-DOF trajectory optimal strategy for RLV is feasible.

Trajectory Optimization for 7-Dofs Space Manipulator

The space manipulator is always designed to have 7 degrees of freedom（Dofs）with the consideration of energy limitation,as well as the flexible moving possibility.Therefore,how to plan the trajectory is important to improve the performance of the manipulator.In this paper,the speed of the end effector is configured as a projecting parameter,when a constant acceleration is applied to adjust the velocity.To implement this trajectory planning strategy,an optimization algorithm through the pseudo inverse of Jacobin matrix is designed,which adjusts the weight functions of joints.According to the functional theory,this algorithm is analyzed and the optimal solution is found in numerous sets of planning.A MATLAB simulation platform is established and the results verity the effectiveness of the algorithm.

Trajectory Optimization Design for Morphing Wing Missile

This paper presents a new particle swarm optimization( PSO) algorithm to optimize the trajectory of morphing-wing missile so as to achieve the enlargement of the maximum range. Equations of motion for the twodimensional dynamics are derived by treating the missile as an ideal controllable mass point. An investigation of aerodynamic characteristics of morphing-wing missile with varying geometries is performed. After deducing the optimizing trajectory model for maximizing range,a type of discrete method is put forward for taking optimization control problem into nonlinear dynamic programming problem. The optimal trajectory is solved by using PSO algorithm and penalty function method. The simulation results suggest that morphing-wing missile has the larger range than the fixed-shape missile when launched at supersonic speed,while morphing-wing missile has no obvious range increment than the fixed-shape missile at subsonic speed.

Trajectory Optimization and Motion Simulation of a Flexible Polishing Industrial Robot for Watchcases

To optimize the working time of the flexible polishing industrial robot for watchcases,the polishing efficiency should be improved.Based on the quintic B-spline fitting curve trajectory planning method associated with the optimal time interval and the trajectory point angle,the trajectory route of the flexible polishing industrial robot for case parts was optimized by the Matlab software.The operation time of the flexible polishing industrial robot could reach the optimal level.The joints of the robot can be cooperated with each other to ensure that the motion track of the end-effector of the robot arm is closer to the expected motion track.Based on the Adams software,the obtained trajectory curve of multi-objective optimization was simulated,which verified the trajectory fitted after multi-objective optimization.The angular acceleration and angular plus acceleration curves were improved.Theoretical guidance was carried out for the subsequent experiment by Matlab and Adams simulation analysis.

A Unified Trajectory Optimization Approach for Long-Term and Reactive Motion Planning of Legged Locomotion

This paper proposes a unified trajectory optimization approach that simultaneously optimizes the trajectory of the center of mass and footholds for legged locomotion.Based on a generic point-mass model,the approach is formulated as a nonlinear optimization problem,incorporating constraints such as robot kinematics,dynamics,ground reaction forces,obstacles,and target location.The unified optimization approach can be applied to both long-term motion planning and the reactive online planning through the use of model predictive control,and it incorporates vector field guidance to converge to the long-term planned motion.The effectiveness of the approach is demonstrated through simulations and physical experiments,showing its ability to generate a variety of walking and jumping gaits,as well as transitions between them,and to perform reactive walking in obstructed environments.

10th China Trajectory Optimization Competition:Problem description and summary of the results

From March 20,2019 to April 30,2019,the 10th China Trajectory Optimization Competition(CTOC10)was jointly held by the Chinese Society of Theoretical and Applied Mechanics and Nanjing University of Aeronautics and Astronautics.The CTOC10 focused on trajectory optimization for Jovian exploration.The team from Harbin Institute of Technology won the first prize.In this paper,first,the history of the CTOC is presented.Subsequently,the mission of the CTOC10 is introduced,and an account of the final rankings of the competition is given.Finally,trajectory optimization methods are discussed,and suggestions for practical missions are provided.

Trajectory optimization of multiple quad-rotor UAVs in collaborative assembling task

A hierarchic optimization strategy based on the offline path planning process and online trajectory planning process is presented to solve the trajectory optimization problem of multiple quad-rotor unmanned aerial vehicles in the collaborative assembling task. Firstly, the path planning process is solved by a novel parallel intelligent optimization algorithm, the central force optimization-genetic algorithm （CFO-GA）, which combines the central force optimization （CFO） algorithm with the genetic algorithm （GA）. Because of the immaturity of the CFO, the convergence analysis of the CFO is completed by the stability theory of the linear time-variant discrete-time sys- tems. The results show that the parallel CFO-GA algorithm converges faster than the parallel CFO and the central force optimization-sequential quadratic programming （CFO-SQP） algorithm. Then, the trajectory planning problem is established based on the path planning results. In order to limit the range of the attitude angle and guarantee the fight stability, the optimized object is changed from the ordinary six-degree-of-freedom rigid-body dynamic model to the dynamic model with an inner-loop attitude controller. The results show that the trajectory planning process can be solved by the mature SQP algorithm easily. Finally, the discussion and analysis of the real-time per- formance of the hierarchic optimization strategy are presented around the group number of the wav^oints and the eoual interval time.

Reentry trajectory rapid optimization for hypersonic vehicle satisfying waypoint and no-fly zone constraints

To rapidly generate a reentry trajectory for hypersonic vehicle satisfying waypoint and no-fly zone constraints, a novel optimization method, which combines the improved particle swarm optimization （PSO） algorithm with the improved Gauss pseudospectral method （GPM）, is proposed. The improved PSO algorithm is used to generate a good initial value in a short time, and the mission of the improved GPM is to find the final solution with a high precision. In the improved PSO algorithm, by controlling the entropy of the swarm in each dimension, the typical PSO algorithm＇s weakness of being easy to fall into a local optimum can be overcome. In the improved GPM, two kinds of breaks are introduced to divide the trajectory into multiple segments, and the distribution of the Legendre-Gauss （LG） nodes can be altered, so that all the constraints can be satisfied strictly. Thereby the advan- tages of both the intelligent optimization algorithm and the direct method are combined. Simulation results demonstrate that the proposed method is insensitive to initial values, and it has more rapid convergence and higher precision than traditional ones.