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Effective formation reconfiguration of satellite flying is increasingly becoming important. This paper presents an alternative algorithm for satellite formation reconfiguration based on the hybridization of Legendre pseudospectral method (LPM) and co-evolutionary particle swarm optimization (CPSO), which aims to give a fuel-optimal maneuver strategy and avoid collision among satellites. The proposed method is not only able to combine the advantages of two algorithms, but also overcomes the limitations of each algorithm to some extent. More specifically, LPM is used to solve formation reconfiguration with feasible solution in the case of that the distance between any two satellites is far away. The computation time is greatly shortened because there is no collision avoidance constraint. Meanwhile, the CPSO algorithm is applied to enhance solving precision, as well as avoid collision when the satellites stay at a relatively short distance from each other. In order to fully demonstrate the efficiency of proposed method, an array of experiments with comparison to multi-phase LPM are carried out on three satellites. The numerical results verify that proposed method performs well in designing fuel-optimal strategy and ensuring the safety of satellite. It is a feasible and effective algorithm for addressing satellite formation reconfiguration problems.

This paper develops a robust controller for the attitude tracking of time-energy near-optimal angular velocity obtained by the Legendre Pseudospectral method (LPSM). The information of near-optimal reference trajectory is realized offline by employing the LPSM for the nominal spacecraft model, which is without any uncertainties. Then, the proposed robust controller is employed online to track that near-optimal path while compensating for the inertial uncertainties, external disturbances, and noises under input saturation. The robust scheme is developed by combining the finite-time robust non-linear disturbance observer (RNDO) and the adaptive non-singular fast terminal sliding mode control (NSFTSMC). The estimated output of RNDO directly attenuates lumped disturbance of moderate frequency and thus significantly helps in alleviating the chattering problem. Whereas NSFTSMC ensures the finite-time convergence of relative system states, and adaptive law solves the overestimation problem of controller gains while rejecting high-frequency disturbances. The simulation analysis is carried out to validate the proposed strategy.

This paper discusses the use of optimal control strategies for ascertaining efficient usage of infection eliminating methodologies. The mathematical model of tuberculosis is considered which is incorporated with three control terms. The preventive controls used for containing the spread of tuberculosis, include vaccinations along with treatment control for the latent, susceptible and infected population. By implementing Pseudospectral method on the Tuberculosis model, the original continuous time optimal control problem is converted into its equivalent non-linear programming problem and then solved. On application of the proposed control scheme, a minimization in the intervention cost and a significant reduction in the disease burden is expected. A comparative analysis between the widely used Legendre Pseudospectral method and Chebyshev Pseudospectral method when applied to the mathematical model of tuberculosis is included in this work. The response of the states in absence of any control effort is also included to demonstrate the effect of optimal control strategy on the disease.

Intelligent planning and accurate execution of connected automated vehicles (CAVs) enable not only improved traffic safety but also better fuel economy. This paper presents two longitudinal speed planning algorithms for fuel-saving driving on highways with varying road slopes. One is designed on the top of the model predictive control (MPC) and the other is called equivalent kinetic-energy and fuel conversion method. The MPC algorithm solves the optimal speed profile in a receding finite horizon with repeated optimization, which is numerically solved by the Legendre pseudospectral method. The latter is designed based on an instantaneous optimization, which considers vehicle kinetic energy an admissible power source, and then minimizes a weighted sum of fuel energy and kinetic energy. This strategy is capable of generating analytical rules to get the economical speed as well as the corresponding commands of the engine, transmission, and brake. The two algorithms are featured by near-global optimization and local optimization, respectively. Their performances in fuel economy and computational load are quantitatively explored and compared in order to distinguish the potential of real implementation in CAVs.

Pathogen suppressing and immune boosting drugs, are used as control measure for regulating the infections due to water borne disease. A mathematical SIWR (susceptible-infectious-recovered) model is considered in which the $W$ compartment tracks the concentration of pathogen in water. An optimal scheme for disease control is proposed in this paper by converting the original optimal control problem to a nonlinear programming problem (NLP). This transformation is done by Legendre pseudospectral method (LPSM) and the resulting NLP problem is solved. The objective of minimizing the count of infected population, along with drug dosage cost is addressed. The simulation results reported illustrates the effectiveness of the proposed optimal drug use strategy.

An optimal control scheme for attitude control of rigid body spacecraft is proposed in this paper. The dynamic optimal control problem is first converted into a nonlinear programming problem (NLP) by Legendre pseudospectral method (LPM) and then this NLP problem is solved. The time as well as energy is optimized for the rigid body spacecraft using LPM. The re-orientation problem of the rigid spacecraft is solved both as a regulation problem and then as a tracking problem. The simulation is carried out using a Matlab© toolbox called DIDO and the simulation results illustrates the desired performances of spacecraft attitude regulation and tracking.

Based on Legendre pseudospectral method for solving the optimal control problem, the tracking control of hypersonic glider is studied. In this paper, the differential form of Legendre pseudo spectral method is used to convert the optimal control problem into a nonlinear programming problem for reentry vehicle tracking. The state quantity and control amount on the LGL node are selected as the parameters to be optimized, and the optimal performance index is selected as to maximize the cross-range. Then the angle of attack in the reentry section is solved. The control rate of a universal hypersonic gliding vehicle (CAV) is calculated. The simulation results show that the proposed method has a certain precision and efficiency. In addition, the simulation process also shows that the Legendre pseudospectral method is not sensitive to the state guessing value, and the algorithm is easy to converge, which is suitable for the solution of the reentry control problem of the aircraft.

This paper presents the solution to the optimal low-thrust orbital rendezvous problem using the multiple-interval Legendre pseudospectral method. Continuous thrust equations of relative motion in the orbit of Earth are used to form the optimal control problem for minimizing the overall control input. A minimum safe distance has been used as the constraint on the optimal solution so as to avoid any collision between the target and the maneuvering spacecraft. Further, to make the problem realistic the optimal control input has been constrained to predefined feasible limits of available specific thrust. The optimal control problem is then transformed to a non-linear programming problem using the multiple-interval pseudospectral method and the reformulated problem is then solved using sequential quadratic programming. To demonstrate the effectiveness of the proposed method, numerical examples have been presented.

In this paper, the take-off trajectory optimization problem for a new vertical takeoff and landing(VTOL) unmanned aerial vehicle(UAV) is studied. The longitudinal kinematics equation and dynamics equation are established with atmospheric density, earth gravity, engine thrust and aerodynamic parameter. Taking the thrust and rudder angles as the control variables, the optimal objective function of the fuel and distance at the given time is proposed. Based on Legendre Pseudospectral method(LPM) the optimal control is transformed into the parameter optimization problem, and the optimal flight trajectory is obtained. The simulation results verify the correctness and effectiveness of the method.

A novel hybrid technique is discussed to numerically solve nonlinear stochastic optimal control (OC) problems with numerous potential applications. The hybrid technique combines a Legendre Pseudospectral Method (LPM) with a generalized Polynomial Chaos (gPC) method, which are highly accurate numerical methods for solving deterministic OC problems and stochastic differential equations (SDE), respectively. The hybrid algorithm first selects samples from the random space using collocation nodes, inserts those sample values into the differential equations of the OC problem, and then uses a pseudospectral-based deterministic solver to generate solutions for each of the resulting deterministic problems. The set of deterministic solutions (i.e., ensemble of random realizations) are then used to construct a polynomial representation of the solution to the stochastic OC problem as a function of the random inputs using a gPC method (i.e., a high-order stochastic collocation method). The hybrid technique is used to solve a nonlinear stochastic OC problem to demonstrate its utility. The algorithm is a highly accurate technique for solving nonlinear OC problems with uncertain parameters. The hybrid technique will allow the user to analyze the solution to an OC problem and understand how uncertainty on the parameters such as the states, initial conditions, and boundary conditions can affect the solution (i.e., uncertainty quantification). The algorithm may also be useful for near real-time OC since a new trajectory can be followed if the uncertainty can be estimated.

A direct optimization algorithm, which satisfies multiple path and terminal constraints, is proposed to generate the optimal trajectories for the single-stage-to-orbit space planes. Minimum fuel cost is selected as the optimality criterion in this method, and a segment strategy of the ascent trajectory is proposed according to the characteristics of the combined engines. The optimal control problem is transformed into a multi-stage optimal control problem, and a Gauss pseudo-spectral method is used to solve the parameterized control variables. Both the numerical simulations and simplified calculation based on the Gauss pseudospectral method are carried out to test the effectiveness and robustness of the proposed algorithm. Results demonstrate that the algorithm can rapidly generate the optimal ascent trajectory for single-stage-to-orbit space planes and achieve complex flight missions.

The formation reconfiguration problem of the multiple unmanned aerial vehicles (Multi-UAVs) is described as an optimal control problem in this paper. Firstly, the control model of formation flight is constructed with the three-dimensional elastic distance vector, which describes the relationship between the speed and distance of adjacent UAVs in the leader-follower type. Furthermore, the constraints of communication distance and the security anti-collision distance are taken into account, and the formation is modeled to a fuel optimal problem with terminal state restrictions. Then, the optimal control problem of UAV formation flight is transformed into a nonlinear programming problem based on Legendre Pseudospectrum Method. The advantage of LPM is to obtain a higher accuracy with fewer nodes in the process of discretization. Finally, the formation with three UAVs is simulated in Matlab and solved by Tomlab toolkit. The simulation results show that the formation can be well achieved the desired new shape with minor position error and optimal fuel. The effectiveness and feasibility of the method is verified.

The UAV formation reconfiguration problem is described as an optimal control problem in this paper. UAV formation error dynamics are established according to elastic distance vector. The terminal state, control action energy constraints and the constraint of safe distance to avoid colliding are considered during formation reconfiguration. Satisfying all these constraints, the optimal control problem of continuous UAV formation reconfiguration is converted into a nonlinear programming problem. The optimal solution is obtained by Legendre Pseudospectral Method (LPM). The advantage of LPM is to obtain a higher accuracy with fewer nodes. Finally, a three-UAV formation reconfiguration is simulated. It is shown from the results that the ideal formation can be reached within the prescribed time, formation reconfiguration task can be completed and the control standard of minimal formation error and energy consumption can be achieved. The effectiveness and feasibility of the method is verified.

A two-timescale Chebyshev pseudospectral optimal control method, which can reduce the converted nonlinear programming (NLP) significantly, is proposed. The optimal aiming maneuver problem of an advanced aircraft, which is a six-degree-of-freedom trajectory problem, is solved by this two-timescale method. It is the first time that the six-degree-of-freedom equations are used to solve optimization of the flight trajectory problems.

This paper studies the geometric pseudospectral method, which refers to the Gauss pseudospectral method that is extended to the special Euclidean group SE(3) for UAVs trajectory planning. On SE(3), a complete left invariant rigidbody dynamical UAV model is established in a body-fixed frame, including kinematical and dynamical models. For the kinematical model, the equivalent Lie algebra equation corresponding to the configuration equation is derived based on the left-trivialized tangent of the local coordinate map. Considering the Lie algebra space is isomorphic to the Euclidean space, the Gauss pseudospectral method is directly used to discretize the dynamical model. The kinematical and dynamical models are considered as the defect equality constraint used for describing the UAVs trajectory planning problem. The simulation results illustrate that the proposed method is effective for planning the UAV trajectory.

A Legendre pseudo spectral combination algorithm introduced by the theory of differential flatness is proposed, which transforms the unmanned aerial vehicles path planning problem into an optimal control problem. The vehicles kinematics state equations are built, and the optimal control problem which uses the equations as constraints and the shortest flight time as the cost is also built. By the differential flatness, the equations of motion state can be converted, that is the control variables and constraints of the optimal control problem can be simplified by the use of flatness output. Then optimal control problem after simplified can be solved through Legendre pseudo spectral method. The proposed combination algorithm has been compared with Legendre pseudo spectral method, and the simulation by Matlab is used to verify that the algorithm can reduce approximately 60% computation time with ensuring accuracy. The result has proved the high engineering value of the proposed algorithm.

The performance comparison is performed among Legendre (LPM), Gauss (GPM), and Radau (RPM) pseudospectral methods, which are utilized to optimize the low-thrust Earth-Mars rendezvous trajectories. In order to provide a fair comparison condition, a shape-based approach is implemented to provide equivalent initial guesses for all the next accurate optimization, and the SQP algorithm is used to solve the resulting nonlinear programming problem for each of the discretization. The behavior and performance of the three pseudospectral methods are compared on the mission of Earth-Mars Rendezvous in 2013.

An effective method is proposed to formulate and solve a fuel-optimal control problem of satellite formation maneuver using continuous low thrusters. To ensure long-term flying formation and to save fuel practically possible, the final satellite formation is designed as the passive and periodic relative orbits accommodating the J 2 perturbation. Based on the conditions of the passive and periodic relative trajectory and combining with the unknown final injection time, a final orbit is described by six implicit constraint equations. In order to overcome the complex and difficult computation, the nonlinear optimal control problem is converted into a nonlinear programming problem by means of the Legendre pseudospectral method. Due to the formulated problem including the free final time and the precise final constraints, the formation maneuver is very reasonable and the fuel consumption is minimized. The numerical results show that the control law works well for the optimal reconfiguration of satellite formation.

This paper presents a computational framework to solve optimal control problems (OCPs) using Legendre Pseudospectral (PS) method and its application to obtain eco-driving strategies for ground vehicles. Both control and state variables of OCPs are approximated by Lagrange interpolating polynomials at the Legendre-Gauss-Lobatto (LGL) collocation points. The OCP is converted into a nonlinear programming (NLP) problem, and numerically solved by matured optimization algorithms. To implement the PS method, we developed a computational package, called Pseudospectral Optimal control Problem Solver (POPS) in Matlab environment. Further, the POPS is applied to obtain fuel-optimized driving strategies for automated vehicles in hilly road conditions.

The objective of this paper is to investigate a reliable method to generate optimal ascent trajectory for hypersonic air-breathing vehicles. When solving optimal trajectories in endo-atmospheric flight, most of traditional indirect methods suffer from difficulties in finding an appropriate initial guess and getting a convergent solution for the two point boundary value problem (TPBVP). With improvements on on-board computer performance, direct methods such as pseudospectral method show promising potential for real-time optimal guidance. It removes the need for computing analytical gradients of aerodynamic coefficients, and remains high accurate solution similar to indirect methods. In this work, optimal ascent trajectory generation problem was formulated as a fuel-optimal control problem. Gauss pseudospectral method (GPM) was presented to generate the optimal ascent trajectories. Optimal solutions from GPM were compared with an indirect method based on finite difference method (FDM). Numerical simulations were studied with various initial conditions to investigate the optimal trajectory characteristics for hypersonic air-breathing vehicles. The results verified the validity and accuracy of GPM for ascent trajectory optimization.

To calculate feedforward control strategies by means of dynamic optimization procedures, the alternatives of direct and indirect methods are compared in this paper. The direct optimization approaches that are considered in this paper make use of the Hermite-Simpson method and of the Legendre Pseudospectral method as underlying discretization strategies. Their description is followed by a short introduction to the maximum principle of Pontryagin, i.e., the indirect method. All procedures are employed to compute feedforward control sequences for the flight control of a four-rotor UAV, which is an unstable, nonlinear multi-input multi-output system. The optimization results are compared with respect to their accuracy and applicability.

This paper presents an optimal feedback reentry guidance for the hypersonic vehicle. The proposed method implements a closed-loop feedback approach via fast computation of a series of open loop optimal control problems. At each sampling period, the optimal control problem is solved by the improved Gauss pseudospectral method (IGPM), which is an efficient method to solve the general nonlinear optimal control problem. The IGPM approximates the state and control by the polynomials at the Legendre-Gauss (LG). The costate can be obtained and the optimality of the solution is also checked easily. The proposed method can deal with the state and control constraints explicitly. The various performance can be realized by considering different cost function. Simulation results demonstrate that the hypersonic vehicle tracks the reference trajectory well in the presence of the uncertainty of aerodynamic force.

The main objective of this paper was that take Gauss pseudospectral method applied to solve maximum hypersonic unpowered glide trajectory in the presence of constraint such as dynamic pressure, overload and heat flux. In Gauss pseudospectral method, the state and control variables were approximate using a global Lagrange interpolating polynomial. First, the equations of motion and optimization model were described. Then, the optimal control problem could discrete into a nonlinear programming problem using Gauss pseudospectral method. Finally, nonlinear programming problem was solved using Sequential Quadratic Programming. Through optimization process, constrained optimal trajectory could acquire. The precision and fast convergence of this method has been demonstrated with the trajectory optimization of hypersonic unpowered glider in near space.

The optimization of lunar soft landing trajectory is an optimization control problem with non-linear free terminal time and control constrained‥ The application of the Chebyshev pseudospectral is described on the optimal control problems. This method employs Nth Lagrange polynomial approximations for the state and control variables to transform the state differential equations to algebraic equations. Finally the optimal control problem is transformed to a nonlinear programming problem. In process of solving the optimal control problem using matlab, control variables and flying time are used as optimal variables. Simulation results demonstrare the methodology for the optimal trajectory designing has strong convergence.

In this paper, we extend our previous results concerning the application of the Legendre pseudospectral method to optimal pulse design problems for open quantum systems. We now consider the more realistic case in which systems are characterized by variations in parameter values, such as relaxation rates and coupling constants. Such dispersions in system parameters motivate us to consider an ensemble of systems, each member distinct due to distinct parameter values. We demonstrate the method with systems from nuclear magnetic resonance (NMR) spectroscopy in liquid. In particular, we highlight the flexibility and robustness of the pseudospectral method approach by developing pulses that minimize the energy or total duration of the pulse sequence.

In this paper we employ pseudospectral method to solve the problem of trajectory optimization for satellites formation reconfiguration in deep space. The basic problem is all the satellites take a specified rest-to-rest maneuver synchronously without colliding with each other. First, the state and input vectors are discretized at Legendre-Gauss-Lobatto (LGL) points, and the objective function is selected as minimizing the total fuel consumption during maneuver. Then the optimization problem is converted into a parameter nonlinear programming problem with the non-convex constraints of collision avoidance as inequality equations using Legendre pseudospectral method. At the end of this paper, an example is worked out and the simulation results demonstrate that the proposed approach can produce high approximation accuracy and a small computational amount, which makes the algorithm of trajectory generation for formation reconfiguration used on-line possible.

A Gauss pseudospectral method (GPM) is presented for directly solving impulsive orbit transfer problem. The Gauss pseudospectral method is applied to the multi-phase optimal control problem; the model of multi-phase control linkage conditions and more flexible solving strategy are also presented. Two typical orbit transfer problems are studied, the simulation results are compared with the available numerical prediction, and show good accuracy, rapid convergence(time consumption is several seconds), robustness with initial guess, which means that the GPM can used to generate an optimal impulsive orbit transfer trajectory through the onboard computer.

This paper aims to design a general formulation for time optimal trajectory planning of satellites formation reconfiguration using pseudospetral method. The basic problem is all the satellites take a specified rest-to-rest maneuver synchronously without colliding with each other in a shortest time. First we discretize the state and input vectors at Legendre-Gauss-Lobatto (LGL) points and accordingly transform the model of satellite, the objective function and the non-convex constraints of collision avoidance to a parameter optimization nonlinear programming (NLP) problem using Legendre pseudospectral method. Then this NLP problem is solved by searching the state and control vectors to minimize the objective function. At the end of this paper, an example is worked out to show that the Legendre pseudospectral method is well suited for formation reconfiguration problem. The results demonstrate that our method could solve the problem in a short time with high accuracy so that it can be used as the on-line algorithm of time optimal trajectory generation for formation reconfiguration.

This paper discusses the problem of optimal motion planning for a rigid underactuated spacecraft. Firstly, we derive the dynamic and kinematic equations of the spacecraft, and for given initial and terminal states, the performance index to be optimized is selected as minimizing the total fuel consumption. Moreover, the control input saturation restriction is transformed into an inequality constraint. Then, the optimal motion planning problem can be converted into a nonlinear programming problem using Legendre pseudospectral method at a set of discretization points. Simulation results demonstrate that the obtained optimal trajectories satisfy the constraint conditions well and the proposed approach can also be used on-line in application practice with its small computational amount and low computational complexity.

We analyze a few long-standing issues related to the Chebyshev pseudospectral (PS) approximations of nonlinear constrained optimal control problems. The feasibility and consistency of a Chebyshev PS method are demonstrated. We also show that the standard primal-dual closure conditions can be mapped to primal-only closure conditions. The primal-only closure condition facilitates an easy computation of the problem covectors which can be used to verify and validate the computational solution by way of the continuous-time necessary conditions.

In this paper, we focus on the design of fuel-optimal satellite formation keeping strategy using a low thrust propulsion system. The proposed controller includes feedback control and feedforward control. For feedback control, a real-time fuel-optimal control approach is proposed. The fuel-optimal control problem is then converted into a quadratic programming problem by application of a Legendre pseudospectral method. In the optimization problem, we adopt our recently developed linear time-varying relative dynamics to accurately describe relative motion in the presence of earth oblateness effect and eccentricity effect. The control acceleration constraints are included in the optimization problem to avoid control saturation for low-thrust propulsion system. In addition, in order to meet the requirement of high-precision formation keeping for some satellite formation flying missions, a feedforward control: iterative learning control (ILC) is used for the existing close-loop feedback control system. ILC aims to improve formation keeping accuracy by eliminating the effects of periodic perturbations such as gravitational perturbation, atmospheric drag. Simulation results demonstrate the efficiency of our proposed method.

Using shifted Legendre polynomials (SLP) a recursive algorithm is presented for the analysis of linear time-invariant optimal control systems incorporating observers. An illustrative example is included to demonstrate the superiority of proposed recursive algorithm over the existing .nonrecursive approach.

Footprint generation is a necessary element of mission design in the case of aborted flight condition. Mathematically, the problem is posed as finding the maximum cross-range for any down-range position for different heading angles. Since deriving and solving the necessary optimality conditions for the full-state equations prove to be a hard task, some simplifications have to be made. Traditional footprint generators either simplify the equations of motion using energy-state approximations or use approximate (suboptimal) control laws such as Vinh's law. To remain faithful to the full-state physical model and use the optimal control law, we employ a new approach based on the direct Legendre pseudospectral method which avoids the pitfalls of the previous techniques. In this approach the optimal control problem is directly discretized and parameterized to a nonlinear programming problem. Both the reduced-state and full-state models for a vehicle example with tabular data are discretized using this method, and the results are compared.

A generic nonlinear optimal control problem with a Bolza cost functional is discretized by a Legendre pseudospectral method. According to the covector mapping theorem, the Karush-Kuhn-Tucker multipliers of the discrete problem map linearly to the spectrally discretized covectors of the Bolza problem. Using this result, it is shown that the nonlinear programming problem converges to the continuous Bolza problem at a spectral rate assuming regularity of appropriate functions.

This paper presents a parallel implementation of the discrete Legendre transform using SIMD machines, MasPar MP-1 and MP-2. Laguerre, Hermite, and binomial transforms can be implemented in a similar manner using some of the computations required to compute the discrete Legendre transform. The time required to compute the discrete Legendre transform vs. number of points for an MP-2 machine is included.