Smart Gait:A Gait Optimization Framework for Hexapod Robots

The current gait planning for legged robots is mostly based on human presets,which cannot match the flexible characteristics of natural mammals.This paper proposes a gait optimization framework for hexapod robots called Smart Gait.Smart Gait contains three modules:swing leg trajectory optimization,gait period&duty optimization,and gait sequence optimization.The full dynamics of a single leg,and the centroid dynamics of the overall robot are considered in the respective modules.The Smart Gait not only helps the robot to decrease the energy consumption when in locomotion,mostly,it enables the hexapod robot to determine its gait pattern transitions based on its current state,instead of repeating the formalistic clock-set step cycles.Our Smart Gait framework allows the hexapod robot to behave nimbly as a living animal when in 3D movements for the first time.The Smart Gait framework combines offline and online optimizations without any fussy data-driven training procedures,and it can run efficiently on board in real-time after deployment.Various experiments are carried out on the hexapod robot LittleStrong.The results show that the energy consumption is reduced by 15.9%when in locomotion.Adaptive gait patterns can be generated spontaneously both in regular and challenge environments,and when facing external interferences.

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.

Flight Range Optimization of Extended Range Guided Munitions

By the study of extended range guided munitions (ERGM) trajectory characteristics, ERGM free-flight and glide trajectory characteristics are revealed and illustrated. On the basis of free-flight trajectory mathematical model, a two-parameter optimization problem of quadrant elevation and rocket ignition time is studied. Using the atmosphere mathematical model, the best glide-starting point of the downward trajectory is determined. With an optimal control mathematical model, the ERGM optimal glide trajectory is obtained.

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.

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.

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.

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.

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.

A CALCULATION OF OPTIMAL FLIGHT TRAJECTORY USING THE PARAMETERIZED OPTIMIZATION METHOD

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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.

Practical Optimization of Low-Thrust Minimum-Time Orbital Rendezvous in Sun-Synchronous Orbits

High-specific-impulse electric propulsion technology is promising for future space robotic debris removal in sun-synchronous orbits.Such a prospect involves solving a class of challenging problems of low-thrust orbital rendezvous between an active spacecraft and a free-flying debris.This study focuses on computing optimal low-thrust minimum-time many-revolution trajectories,considering the effects of the Earth oblateness perturbations and null thrust in Earth shadow.Firstly,a set of mean-element orbital dynamic equations of a chaser(spacecraft)and a target(debris)are derived by using the orbital averaging technique,and specifically a slow-changing state of the mean longitude difference is proposed to accommodate to the rendezvous problem.Subsequently,the corresponding optimal control problem is formulated based on the mean elements and their associated costate variables in terms of Pontryagin’s maximum principle,and a practical optimization procedure is adopted to find the specific initial costate variables,wherein the necessary conditions of the optimal solutions are all satisfied.Afterwards,the optimal control profile obtained in mean elements is then mapped into the counterpart that is employed by the osculating orbital dynamics.A simple correction strategy about the initialization of the mean elements,specifically the differential mean true longitude,is suggested,which is capable of minimizing the terminal orbital rendezvous errors for propagating orbital dynamics expressed by both mean and osculating elements.Finally,numerical examples are presented,and specifically,the terminal orbital rendezvous accuracy is verified by solving hundreds of rendezvous problems,demonstrating the effectiveness of the optimization method proposed in this article.

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.

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.

An Optimization Method of Formation Flight for Minimizing Fuel Consumption

A method for formation flight trajectory optimization was established.This method aims at minimizing fuel consumption of a two-aircraft formation flight,without changing the original trajectory of the leader.Candidate flight pairs were selected from all international flights arriving at or departing from China in one day according to the requirement of the proposed method.Aircraft performance database Base of Aircraft Data(BADA)was employed in the trajectory computation.By assuming different fuel-saving percentages for the following aircraft,pre-flight plan trajectories of formation flight were optimized.The fuel consumption optimization effect under the influence of different trajectory optimization parameters was also analyzed.The results showed that the higher the fuel savings percentage,the longer the flight distance of formation flight,but the smaller the number of formation combinations that can be realized,which is limited by the aircraft performance.The following aircraft flying along the approximate actual flight trajectory can be benefited as well,and the optimal fuel-saving efficiency is related to the expected fuelsaving efficiency of formation flight.

Multi-swingby optimization of mission to Saturn using global optimization algorithms

Based on the trajectory design of a mission to Saturn, this paper discusses four different trajectories in various swingby cases. We assume a single impulse to be applied in each case when the spacecraft approaches a celestial body. Some optimal trajectories ofEJS, EMS, EVEJS and EVVEJS flying sequences are obtained using five global optimization algorithms： DE, PSO, DP, the hybrid algorithm PSODE and another hybrid algorithm, DPDE. DE is proved to be supe- rior to other non-hybrid algorithms in the trajectory optimi- zation problem. The hybrid algorithm of PSO and DE can improve the optimization performance of DE, which is vali- dated by the mission to Saturn with given swingby sequences. Finally, the optimization results of four different swingby sequences are compared with those of the ACT of ESA.