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.

A novel trajectory planning strategy for aircraft emergency landing using Gauss pseudospectral method

To improve the survivability during an emergency situation, an algorithm for aircraft forced landing trajectory planning is proposed. The method integrates damaged aircraft modelling and trajectory planning into an optimal control framework, in order to deal with the complex aircraft flight dynamics, a solving strategy based on Gauss pseudospetral method （GPM） is presented. A 3-DOF nonlinear mass-point model taking into account the wind is developed to approximate the aircraft flight dynamics after loss of thrust. The solution minimizes the forced landing duration, with respect to the constraints that translate the changed dynamics, flight envelope limitation and operational safety requirements. The GPM is used to convert the trajectory planning problem to a nonlinear programming problem （NLP）, which is solved by sequential quadratic programming algorithm. Simulation results show that the proposed algorithm can generate the minimum-time forced landing trajectory in event of engine-out with high efficiency and precision.

Parameters Identification of Continuous-Time Hammerstein System with Advanced Gauss Pseudo spectral Method

An advanced Gauss pseudospectral method(AGPM) was proposed to estimate the parameters of the continuous-time(CT)Hammerstein model.The nonlinear part of the Hammerstein system is approximated with pseudospectral approximation method.The linear part was written as a controllable canonical form to circumvent the high order time-derivative of the input and output(I/O) signals,which could multiply the measurement noise in the identification procession.Furthermore,an output error minimization was constructed for the CT Hammerstein model identification,which was then transcribed into a nonlinear programming(NLP) problem by AGPM.AGPM could converge to the true values of the CT Hammerstein model with few interpolated Legendre-Gauss(LG) nodes.Lastly,two illustrative examples were proposed to verify the accuracy and efficiency of the method.

Hybrid hierarchical trajectory planning for a fixed-wing UCAV performing air-to-surface multi-target attack

This paper considers the problem of generating a flight trajectory for a single fixed-wing unmanned combat aerial vehicle （UCAV） performing an air-to-surface multi-target attack （A/SMTA） mission using satellite-guided bombs. First, this problem is formulated as a variant of the traveling salesman problem （TSP）, called the dynamic-constrained TSP with neighborhoods （DCT- SPN）. Then, a hierarchical hybrid approach, which partitions the planning algorithm into a roadmap planning layer and an optimal control layer, is proposed to solve the DCTSPN. In the roadmap planning layer, a novel algorithm based on an updatable proba- bilistic roadmap （PRM） is presented, which operates by randomly sampling a finite set of vehicle states from continuous state space in order to reduce the complicated trajectory planning problem to planning on a finite directed graph. In the optimal control layer, a collision-free state-to-state trajectory planner based on the Gauss pseudospectral method is developed, which can generate both dynamically feasible and optimal flight trajectories. The entire process of solving a DCTSPN consists of two phases. First, in the offline preprocessing phase, the algorithm constructs a PRM, and then converts the original problem into a standard asymmet- ric TSP （ATSP）. Second, in the online querying phase, the costs of directed edges in PRM are updated first, and a fast heuristic searching algorithm is then used to solve the ATSP. Numerical experiments indicate that the algorithm proposed in this paper can generate both feasible and near-optimal solutions quickly for online purposes.

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.

Optimal control of stretching process of flexible solar arrays on spacecraft based on a hybrid optimization strategy

The optimal control of multibody spacecraft during the stretching process of solar arrays is investigated,and a hybrid optimization strategy based on Gauss pseudospectral method（GPM） and direct shooting method（DSM） is presented. First, the elastic deformation of flexible solar arrays was described approximately by the assumed mode method, and a dynamic model was established by the second Lagrangian equation. Then, the nonholonomic motion planning problem is transformed into a nonlinear programming problem by using GPM. By giving fewer LG points, initial values of the state variables and control variables were obtained. A serial optimization framework was adopted to obtain the approximate optimal solution from a feasible solution. Finally, the control variables were discretized at LG points, and the precise optimal control inputs were obtained by DSM. The optimal trajectory of the system can be obtained through numerical integration. Through numerical simulation, the stretching process of solar arrays is stable with no detours, and the control inputs match the various constraints of actual conditions.The results indicate that the method is effective with good robustness.

Optimal aeroassisted symmetric transfer between coplanar elliptical orbits

The problem of optimal aeroassisted symmetric transfer between elliptical orbits is concerned.The complete trajectory is assumed as consisting of two impulsive velocity changes at the beginning and the end of an interior atmospheric subarc,where the vehicle is controlled via the lift coefficient and thrust.The corresponding dynamic equations are built and bounded controls are considered.For the purpose of optimization computation,the equations are normalized.In order to minimize the total fuel consumption,the geocentric radius of initial elliptical transfer orbital perigee and controls during atmospheric flight should all be optimized.It is an optimal control problem which involves additional parameter optimization.To solve the problem,a two-level optimization method denoted by ＂genetic algorithm ＋ Gauss pseudospectral method＂ is adopted：the genetic algorithm is used for parameter optimization and the Gauss pseudospectral method is used for optimal control problems.The flow chart of simulation is given.On this basis,the issue of more realistic modeling with two finite-thrust subarcs in the nonatmospheric part of the trajectory is simultaneously addressed.The orbital transfer problem is transformed to three continuous optimal control problems,and the constraints at different times are given,which are respectively solved by using the Gauss pseudospectral method.The obtained numerical results indicate that the optimal thrust control is of bangbang type.The minimum-fuel trajectory in the atmosphere consists of aeroglide,aerocruise and aeroglide.They are compared with the results of pure impulsive model,and the conclusions that a significant fuel saving will be achieved by synergetic maneuver are drawn.

Adaptive Constrained On-board Guidance Technology forPowered Glide Vehicle

To make full use of expanded maneuverability and increased range,adaptive constrained on-board guidance technology is the key capability for a glide vehicle with a double-pulse rocket engine,especially under the requirements of desired target changing and on-line reconfigurable control and guidance.Based on the rapid footprint analysis,whether the new target is within the current footprint area is firstly judged.If not,the rocket engine ignites by the logic obtained from the analysis of optimal flight range by the method of hp-adaptive Gauss pseudospectral method(hp-GPM).Then,an on-board trajectory generation method based on powered quasi-equilibrium glide condition(QEGC)and linear quadratic regulator(LQR)method is used to guide the vehicle to the new target.The effectiveness of the guidance method consisted of powered on-board trajectory generation,LQR trajectory tracking,footprint calculation,and ignition time determination is indicated by some simulation examples.

Maneuver strategy of lifting reentry vehicle

An optimal maneuver strategy is proposed for lifting reentry vehicle to reach the maximum lateral range after reentering the atmosphere. Aiming at problems that too many co-state variables and difficulty in estimating the initial values of co-state variables,the equilibrium glide condition (EGC) is utilized to reduce the reentry motion equations and then the optimal maneuver strategy satisfied above performance index is derived. This maneuvering strategy is applied to the lifting reentry weapon platform CAV which was designed by America recently to realize both longitudinal and lateral trajectory design by controlling the attack angle and the bank angle respectively. The simulation result indicates that the maneuver strategy proposed enables CAV to reach favorable longitudinal range and lateral range.

Optimal Release Control of Companion Satellite System Using Electromagnetic Forces

Electromagnetic forces generated by the inter-action of component satellites can be used to release companion satellites. Optimal release trajectories for companion satellite system using inter-electromagnetic forces were investigated. Firstly,nonlinear relative motion dynamic equations of a two-craft electromagnetic companion satellite system were derived in spatial polar coordinates. Then principles of electromagnetic satellite formation flying were introduced. Secondly,the characteristics of the electromagnetic companion satellites release were analyzed and optimal release trajectories of companion satellites using electromagnetic forces were obtained using Gauss pseudospectral method. Three performance criteria were chosen as minimum time,minimum acceleration of the separation distance and minimum control acceleration. Finally,three release examples including expansion along separation distance, rotation in orbital plane and stable formation reconfiguration were given to demonstrate the feasibility of this method. Results indicated that the release trajectories can converge to optimal solutions effectively and the concept of release companion satellites using electromagnetic forces is practicable.

Optimal Motion Planning of the Space Manipulator for Minimum Reaction Torque to Satellite

For the problem of free⁃floating space robot(FFSR)that the motion of manipulator will cause a large disturbance to the attitude of satellite,a path planning method based on hp⁃adaptive Gauss pseudospectral method(hp⁃AGPM)is proposed in this paper.In this method,the minimum reaction torque acting on satellite is taken as the objective function,and the number of segments and the order of polynomial in each segment are determined adaptively to improve the accuracy and the efficiency of the solution.At the same time,the theoretical convergence of the designed method is innovatively proved to ensure that the solution of the discretized nonlinear programming(NLP)problem is the optimal solution to the original optimal problem.The simulation results of a planar two degree⁃of⁃freedom(2⁃DOF)space manipulator show that the proposed path planning method is more effective than the resolved acceleration control(RAC)method and the control variable parameterization(CVP)method,and is better than other pseudospectral methods both in computation speed and the number of collocation points.

Motion Planning for Vibration Reducing of Free-floating Redundant Manipulators Based on Hybrid Optimization Approach

This paper is concerned with optimal motion planning for vibration reducing of free-floating flexible redundant manipulators. Firstly, dynamic model of the system is established based on Lagrange method, and the motion planning model for vibration reducing is proposed. Secondly, a hybrid optimization approach employing Gauss pseudospectral method （GPM） and direct shooting method （DSM）, is proposed to solve the motion planning problem. In this approach, the motion planning problem is transformed into a non-linear parameter optimization problem using GPM, and genetic algorithm （GA） is employed to locate the approximate solution. Subsequently, an optimization model is formulated based on DSM, and sequential quadratic programming （SQP） algorithm is used to obtain the accurate solution, with the approximate solution as an initial reference solution. Finally, several numerical simulations are investigated, and the global vibration or residual vibration of flexible link is obviously reduced by the joint trajectory which is obtained by the hybrid optimization approach. The numerical simulation results indicate that the approach is effective and stable to the motion planning problem of vibration reducing.

Biomimetic Design and Optimal Swing of a Hexapod Robot Leg

Biological inspiration has spawned a wealth of solutions to both mechanical design and control schemes in the efforts to develop agile legged machines. This paper presents a compliant leg mechanism for a small six-legged robot, HITCR-ll, based on abstracted anatomy from insect legs. Kinematic structure, relative proportion of leg segment lengths and actuation system were analyzed in consideration of anatomical structure as well as muscle system of insect legs and desired mobility. A spring based passive compliance mechanism inspired by musculoskeletal structures of biological systems was integrated into distal segment of the leg to soften foot impact on touchdown. In addition, an efficient locomotion planner capable of generating natural movements for the legs during swing phase was proposed. The problem of leg swing was formulated as an optimal control procedure that satisfies a series of locomotion task terms while minimizing a biologically-based objective function, which was solved by a Gauss Pseudospectral Method （GPM） based numerical technique. We applied this swing generation algorithm to both a simulation platform and a robot prototype. Results show that the proposed leg structure and swing planner are able to successfully perform effective swing movements on rugged terrains.

Reentry trajectory optimization for hypersonic vehicle satisfying complex constraints

The reentry trajectory optimization for hypersonic vehicle（HV）is a current problem of great interest.Some complex constraints,such as waypoints for reconnaissance and no-fly zones for threat avoidance,are inevitably involved in a global strike mission.Of the many direct methods,Gauss pseudospectral method（GPM）has been demonstrated as an effective tool to solve the trajectory optimization problem with typical constraints.However,a series of diffculties arises for complex constraints,such as the uncertainty of passage time for waypoints and the inaccuracy of approximate trajectory near no-fly zones.The research herein proposes a multi-phase technique based on the GPM to generate an optimal reentry trajectory for HV satisfying waypoint and nofly zone constraints.Three kinds of specifc breaks are introduced to divide the full trajectory into multiple phases.The continuity conditions are presented to ensure a smooth connection between each pair of phases.Numerical examples for reentry trajectory optimization in free-space flight and with complex constraints are used to demonstrate the proposed technique.Simulation results show the feasible application of multi-phase technique in reentry trajectory optimization with waypoint and no-fly zone constraints.