THREE-DIMENSIONAL TRAJECTORY OPTIMIZATION WITH DIRECT METHOD

The principle of direct method used in optimal control problem is introduced. Details of applying this method to flight trajectory generation are presented including calculation of velocity and controls histories. And capabilities of flight and propulsion systems are considered also. Combined with digital terrain map technique, the direct method is applied to the three dimensional trajectory optimization for low altitude penetration, and simplex algorithm is used to solve the parameters in optimization. For the small number of parameters, the trajectory can be optimized in real time on board.

Progress in reentry trajectory planning for hypersonic vehicle

The reentry trajectory planning for hypersonic vehicles is critical and challenging in the presence of numerous nonlinear equations of motion and path constraints, as well as guaranteed satisfaction of accuracy in meeting all the specified boundary conditions. In the last ten years, many researchers have investigated various strategies to generate a feasible or optimal constrained reentry trajectory for hypersonic vehicles. This paper briefly reviews the new research efforts to promote the capability of reentry trajectory planning. The progress of the onboard reentry trajectory planning, reentry trajectory optimization, and landing footprint is summarized. The main challenges of reentry trajectory planning for hypersonic vehicles are analyzed, focusing on the rapid reentry trajectory optimization, complex geographic constraints, and coop- erative strategies.

Joint Trajectory and Passive Beamforming Optimization in IRS-UAV Enhanced Anti-Jamming Communication Networks

This paper investigates the anti-jamming communication scenario where an intelligent reflecting surface(IRS)is mounted on the unmanned aerial vehicle(UAV)to resist the malicious jamming attacks.Different from existing works,we consider the dynamic deployment of IRS-UAV in the environment of the mobile user and unknown jammer.Therefore,a joint trajectory and passive beamforming optimization approach is proposed in the IRS-UAV enhanced networks.In detail,the optimization problem is firstly formulated into a Markov decision process(MDP).Then,a dueling double deep Q networks multi-step learning algorithm is proposed to tackle the complex and coupling decision-making problem.Finally,simulation results show that the proposed scheme can significantly improve the anti-jamming communication performance of the mobile user.

Optimal midcourse trajectory cluster generation and trajectory modification for hypersonic interceptions

The hypersonic interception in near space is a great challenge because of the target’s unpredictable trajectory, which demands the interceptors of trajectory cluster coverage of the predicted area and optimal trajectory modification capability aiming at the consistently updating predicted impact point(PIP) in the midcourse phase. A novel midcourse optimal trajectory cluster generation and trajectory modification algorithm is proposed based on the neighboring optimal control theory. Firstly, the midcourse trajectory optimization problem is introduced; the necessary conditions for the optimal control and the transversality constraints are given.Secondly, with the description of the neighboring optimal trajectory existence theory(NOTET), the neighboring optimal control(NOC)algorithm is derived by taking the second order partial derivations with the necessary conditions and transversality conditions. The revised terminal constraints are reversely integrated to the initial time and the perturbations of the co-states are further expressed with the states deviations and terminal constraints modifications.Thirdly, the simulations of two different scenarios are carried out and the results prove the effectiveness and optimality 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.

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.

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.

Robust Trajectory and Communication Design for Angle-Constrained Multi-UAV Communications in the Presence of Jammers

This paper studies a multi-unmanned aerial vehicle(UAV)enabled wireless communication system,where multiple UAVs are employed to communicate with a group of ground terminals(GTs)in the presence of potential jammers.We aim to maximize the throughput overall GTs by jointly optimizing the UAVs’trajectory,the GTs’scheduling and power allocation.Unlike most prior studies,we consider the UAVs’turning and climbing angle constraints,the UAVs’three-dimensional(3D)trajectory constraints,minimum UAV-to-UAV(U2U)distance constraint,and the GTs’transmit power requirements.However,the formulated problem is a mixed-integer non-convex problem and is intractable to work it out with conventional optimization methods.To tackle this difficulty,we propose an efficient robust iterative algorithm to decompose the original problem be three sub-problems and acquire the suboptimal solution via utilizing the block coordinate descent(BCD)method,successive convex approximation(SCA)technique,and S-procedure.Extensive simulation results show that our proposed robust iterative algorithm offers a substantial gain in the system performance compared with the benchmark algorithms.

An Energy-Efficient UAV Deployment Scheme for Emergency Communications in Air-Ground Networks with Joint Trajectory and Power Optimization

The space-air-ground integrated network(SAGIN)has gained widespread attention from academia and industry in recent years.It is widely applied in many practical fields such as global observation and mapping,intelligent transportation systems,and military missions.As an information carrier of air platforms,the deployment strategy of unmanned aerial vehicles(UAVs)is essential for communication systems’performance.In this paper,we discuss a UAV broadcast coverage strategy that can maximize energy efficiency(EE)under terrestrial users’requirements.Due to the non-convexity of this issue,conventional approaches often solve with heuristics algorithms or alternate optimization.To this end,we propose an iterative algorithm by optimizing trajectory and power allocation jointly.Firstly,we discrete the UAV trajectory into several stop points and propose a user grouping strategy based on the traveling salesman problem(TSP)to acquire the number of stop points and the optimization range.Then,we use the Dinkelbach method to dispose of the fractional form and transform the original problem into an iteratively solvable convex optimization problem by variable substitution and Taylor approximation.Numerical results validate our proposed solution and outperform the benchmark schemes in EE and mission completion time.

Evolutionary Trajectory Planning for an Industrial Robot

This paper presents a novel general method for computing optimal motions of an industrial robot manipulator （AdeptOne XL robot） in the presence of fixed and oscillating obstacles. The optimization model considers the nonlinear manipulator dynamics, actuator constraints, joint limits, and obstacle avoidance. The problem has 6 objective functions, 88 variables, and 21 constraints. Two evolutionary algorithms, namely, elitist non-dominated sorting genetic algorithm （NSGA-II） and multi-objective differential evolution （MODE）, have been used for the optimization. Two methods （normalized weighting objective functions and average fitness factor） are used to select the best solution tradeoffs. Two multi-objective performance measures, namely solution spread measure and ratio of non-dominated individuals, are used to evaluate the Pareto optimal fronts. Two multi-objective performance measures, namely, optimizer overhead and algorithm effort, are used to find the computational effort of the optimization algorithm. The trajectories are defined by B-spline functions. The results obtained from NSGA-II and MODE are compared and analyzed.

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.

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.

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.

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.

Application of Optimal-Jerk Trajectory Planning in Gait-balance Training Robot

To accommodate the gait and balance disorder of the elderly with age progression and the occurrence of various senile diseases,this paper proposes a novel gait balance training robot(G-Balance)based on a six degree-of-freedom parallel platform.Using the platform movement and IMU wearable sensors,two training modes,i.e.,active and passive,are developed to achieve vestibular stimulation.Virtual reality technology is applied to achieve visual stimulation.In the active training mode,the elderly actively exercises to control the posture change of the platform and the switching of the virtual scene.In the passive training mode,the platform movement is combined with the virtual scene to simulate bumpy environments,such as earthquakes,to enhance the human anti-interference ability.To achieve a smooth switching of the scene,continuous speed and acceleration of the platform motion are required in some scenarios,in which a trajectory planning algorithm is applied.This paper describes the application of the trajectory planning algorithm in the balance training mode and the optimization of jerk(differential of acceleration)based on cubic spline planning,which can reduce impact on the joint and enhance stability.

A CALCULATION OF OPTIMAL FLIGHT TRAJECTORY USING THE PARAMETERIZED OPTIMIZATION METHOD

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Multi-objective Optimal Trajectory and Placement of a Robot for Cotton Yarns Handling

This study deals with a robot manipulator for yarn bobbin handling in the cotton yarns lattice distortion modification system.The aim is to achieve an operation of yarn bobbin handling with minimal execution time,energy consumption and jerk in motion together.The placement of the robot,in relation to the yarn bobbin stations,is also optimized in conjunction of trajectory optimization.Three possible techniques for building the handling traj'ectory were considered:the quaternion spherical linear interpolation in Cartesian space,the quintic polynomial spline and quintic B-spline in joint space.The genetic algorithm(GA) was used to optimize the trajectories of the robot,with a penalty function to handle nonlinear constraints associated in the robot motion.Two simulations of the optimal trajectory in joint space and the placement of robot were carried out and the results obtained were presented and discussed.It is concluded that the quintic polynomial spline constructs a better trajectory in joint space and the proper placement of robot makes better performance.

Anti-Jamming Trajectory Design for UAV-Enabled Wireless Sensor Networks Using Communication Flight Corridor

This paper investigates the anti-jamming trajectory design to safeguard the effective data collection, where a unmanned aerial vehicle(UAV)is dispatched to collect data over multiple sensor nodes(SNs) in jamming environment. Under the limited power and transmission range of SNs, we aim to minimize the UAV’s flight energy consumption in a finite task period, by jointly optimizing SNs collection sequence and UAV flight trajectory. Firstly, we propose a general optimization framework which consists of path planning and trajectory optimization for the formulated non-convex problem. In the path planning phase, a dynamic programming(DP) algorithm is used to provide the initial path of the UAV, which is the shortest path to visit each SN. In the trajectory optimization phase, we introduce the concept of Communication Flight Corridor(CFC) to meet the non-convex constraints and apply a piecewise Bézier curve, based on Bernoulli polynomial, to represent the flight trajectory of the UAV, which can transform the optimization variables from infinite time variables to polynomial coefficients of finite order. Finally, we simulate the flight trajectory of UAV in hovering mode and continuous flight mode under different parameters, and the simulation results demonstrate the effectiveness of the proposed method.

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.