Optimal Trajectory Design for SIMONE Mission Study

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.

Helicopters' Vertical Take-Off and Landing (VTOL) capabilities are essential for maritime operations, especially for small-deck naval vessels. Unmanned Aerial Vehicles (UAVs) offer a cheaper, expendable, and efficient alternative for certain tasks, such as reducing pilot risk and lowering fuel consumption. While the procedures to approach and land on (moving) ships are standardized and bound to established operational limits in the case of crewed helicopters, UAVs lack such guidelines. This study investigates optimal rotary-wing UAV approach trajectories to a moving ship, for varying wind conditions and relative initial positions, and for different objectives. The goal is to provide preliminary guidelines for maritime UAV recovery operations, and a preliminary estimation of performance-based operational limits. The optimal trajectories are obtained using a global path-performance optimization framework based on Optimal Control Theory. The trajectories are compared to each other and to reference cases using the Longest Common SubSequence (LCSS) similarity measure, revealing how the unmanned helicopter adjusts its path to exploit the wind direction and profile for more efficient ground speeds. The violation of performance and/or geometric constraints is used to preliminarily indicate the presence of operational boundaries. The control effort and energy consumption are used to identify optimal starting positions for the helicopter approach phase for a given wind profile and intensity.

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

This paper gives a brief list of commonly used direct and indirect efficient methods for the numerical solution of optimal control problems. To improve the low accuracy of the direct methods and to increase the convergence areas of the indirect methods we suggest a hybrid approach. For this a special direct collocation method is presented. In a hybrid approach this direct method can be used in combination with multiple shooting. Numerical examples illustrate the direct method and the hybrid approach.

A technique of searching for optimal trajectories with gravity assisted maneuvers (GAMs) for interplanetary transfers of spacecraft (SC) with an electric propulsion system (EPS) is proposed. In this case, the indirect optimization method is used. A distinctive feature of this technique is the combination of optimality conditions at the point of GAMs within a single boundary value problem for two cases, when the height of the flyby hyperbola with the GAM is less than or equal to the maximum one. This approach makes it possible to considerably reduce the volume of necessary calculations in optimizing SC interplanetary trajectories that include GAMs. It considers end-to-end trajectory optimization with an analysis of the full set of optimality conditions at the point of the GAM. The efficiency of the proposed approach is demonstrated by the example of optimization of interplanetary trajectories from Earth to Mercury with a GAM in the vicinity of Venus and from Earth to Jupiter with a GAM near Earth.

Traditionally, power amplifier behavioral modeling and digital predistortion have been based on polynomials or Volterra series. Recently, however, there has been a trend to substitute polynomial representations by interpolated lookup tables (LUTs) because of the known problems associated with the series' identification. These problems, commonly recognized as ill-conditioning, are generally related to the poor orthogonality verified in the polynomial basis functions for many common input signals. However, in this paper we will show that polynomials can be reformulated to achieve the identification quality of interpolated LUTs, while maintaining their beneficial properties. At the same time we will provide a unified view between polynomial and spline interpolation basis functions. By doing this we will present polynomials that share the benefits of spline interpolated LUTs and argue that both descriptions can be used interchangeably and with similar advantages.

Low-thrust spacecraft trajectory optimization for the many-revolution orbital transfer problem is especially challenging due to the high problem dimension and perturbing accelerations that prevent or complicate accurate analytical solutions. This analysis seeks to simplify the calculation of the nonaveraged spiral trajectory through the use of the well-known Q-Law guidance algorithm. Here, Q-Law is used to seed phases of the direct trajectory optimization problem. We explore the trade space between the fast calculation speed of Q-Law and the time or mass optimality of the full optimization problem using the common example of a geostationary transfer orbit to geostationary transfer. We find that a computationally efficient, near-optimal solution can be achieved by using Q-Law for an initial fraction of a spiral trajectory and direct optimization for the remainder of the trajectory. This reduces the direct optimization problem to a lower dimension and enables the spacecraft to reach a specific final state. Q-Law is particularly effective for portions of the trajectory where there are many eclipses, which can challenge direct optimization methods.

Multimode electric propulsion system is widely used in deep space exploration. Its thrust is output in stages with the change of input power, which also means that continuous power changes lead to discontinuous thrust changes. In the design and optimization of low-thrust trajectory, the discontinuity of thrust is often neglected and regarded as a continuous parameter. The optimal design of low-thrust transfer trajectory constrained by multimode electric propulsion is studied, and the influence of thrust grading characteristics on the trajectory is analyzed in this paper. A specific optimization strategy is proposed, including three main steps: model discretization based on direct method, fast generation of initial value based on particle swarm optimization (PSO), and optimization based on sequential quadratic programming (SQP). The dynamic model and optimization model of low-thrust transfer trajectory is established. Then, the optimization model is discretized based on the direct method and the original optimal control problem is transformed into a parameter optimization problem in the form of nonlinear programming (NLP). A hybrid optimization algorithm based on PSO and SQP is proposed to solve the new NLP problem. PSO is used to search the initial value efficiently and guarantee the global performance of the initial value and SQP is used for local optimization based on initial value, which can give full play to the excellent global optimization performance of PSO and the outstanding local optimization performance of SQP. The effectiveness of the proposed optimization strategy is verified by a simulation example of the main belt asteroid Metis (1974 QU2) rendezvous mission with once Mars gravity assist. The results show that the optimal control curve obtained is in accordance with the optimal control form of bang-bang control derived from the theory, and the thruster has just two situations at any time: shutdown or startup with maximum thrust. In the working power range, increasing the number of thrust stages can improve the rate of energy utilization, thereby improving the efficiency of thruster, shortening the startup time and reducing fuel consumption. Meanwhile, ineffective grading should be avoided in the non-working power range.

Unsteady mixed convection about a rotating circular cylinder with small fluctuations in the free-stream velocity

E LECTRIC propulsion has been selected in several interplanetary missions due to its higher efficiency compared with chemical propulsion. Perturbations such as maneuver execution errors, navigation uncertainties, operational constraints, uncertain parameters (e.g. initial state), and disturbance forces, produce deviations from the reference trajectory that the guidance and control functions must cancel at the expense of additional propellant mass. NASA Deep Space 1 mission proved an autonomous low-thrust guidance method [1] based on linearization of the terminal state in terms of a certain parameterization of the thrust profile. The terminal controller [2] was solved via Lagrange multipliers. This method has been refined [3] including a line search during the improvement step in order to assure convergence in case of poorly defined reference trajectory (e.g. sparse nodes, low numerical resolution). An interesting finite-thrust guidance and control method has been applied to small-body proximity operations [4]. It is based on pseudowaypoints generation and linear matrix inequality based feedback control. Waypoint-based, autonomous guidance with lowthrust propulsion was implemented for the approach phase of rendezvous missions to small asteroids [5]. The thrust profile was parameterized using a simple law that reduces the guidance problem to solve a system of nonlinear equations. Another autonomous guidance algorithm [6] used the Pontryaginminimumprinciple [2] to derive the optimal corrections to the nominal thrust profile minimizing the propellant consumption. Receding horizon control has been analyzed [7] using direct transcription to discretize the optimal control problem and an interior-point algorithm to solve numerically the resulting nonlinear programming problem. The optimal guidance and control methods presented in this paper focus on closed-form algorithms as in Deep Space 1 autonomous guidance [1], with special emphasis on the propagation and computation of the required partial derivatives matrices. The guidance function is a terminal controller that aims to reach the next waypoint at a specific time. A receding horizon control maintains the spacecraft in the proximity of the reference trajectory during the thrusting arc. The main differences with the aforementioned methods are that the optimal guidance and control laws are derived on a closed-form solution and accommodate thrust constraints in magnitude and direction. These qualities make them suitable for operational use even in autonomous guidance, navigation and control systems [8].

In this paper, the application of direct parameter optimization method to reentry vehicle three-dimension skipping trajectory optimization is introduced. Firstly, the model of trajectory optimization control problem to reentry vehicle skipping trajectory is established, while motion equation of the state variables is a three degree-of-freedom model. Performance is selected to minimize the sum of heat, overload and dynamic pressure. The control variables are angle-of-attack and bank angle. During the flight, reentry vehicle is subjected to the heat rate and overload constraints. Terminal state variables constraints are path angle, altitude and latitude constraints. Then, the optimal control problem is discretized into nonlinear programming problem (NLP) using direct parameter optimization method. The control variables are selected as optimal parameters at all nodes. Parameter optimization problem is solved using POWELL algorithm. The simulation result shows that the optimal solutions of skipping trajectory optimization problem are in real-time using direct parameter optimization method. Therefore, direct parameter optimization method is a viable approach to the reentry vehicle skipping trajectory optimization problem.

This paper describes a recently developed method for the determination of optimal, e g minimum-fuel or minimum-time, trajectories for spacecraft. The method will be introduced after a brief discussion of methods previously used by researchers for trajectory optimization. However, the objective here is not to discuss all such methods or provide a comprehensive description of them but rather to provide sufficient information regarding extant methods that what is novel and beneficial about the solution method we have developed may be recognized. Several example optimal trajectories for spacecraft using lowthrust propulsion will be presented.KeywordsOptimal Control ProblemOptimal TrajectoryPrime VectorAdjoint VariableSystem Differential EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Thecompletesetofthestateand adjoint differentialequationsforthenonsingularequinoctialorbitelementsexpressed in polar coordinatesand written in terms of the truelongitudeispresented. Previous formulations adopted the equinoctial frame as the orbital frame for component resolution of the various perturbation accelerations. The consideration of the position dependency of the J2 acceleration in the context of precision integrated orbital transfer trajectories led us to adopt the more convenient polar frame coupled with the use of the true longitude as the accessory variable needed in the description of the variational equations, inasmuch as it is more dife cult to generate the contribution of the J2 perturbation to the adjoint equations if the variational equations are left in terms of the eccentric longitude. I. Introduction T HEconsideration of thenonsingularequinoctialorbitelements has been of great benee t in trajectory propagation and optimization.References1 ‐7 madeuseofthecorresponding variational equations resolved in the equinoctial orbital coordinate system. In Refs. 8‐12, modern numerical methods based on the techniques of collocation and nonlinear programming are applied to the solutionoflow-thrustEarth-boundandinterplanetarytrajectories.These direct methods have introduced robustness characteristics to the optimization process and were e rst used with great success by launch vehicle trajectory optimization and performance analysts. In particular, Betts 9,10 uses a variety of equinoctial element sets within the framework of the direct method. Following the more traditional approach based on the calculus of variations and the indirect shooting method,thispaper developsthe differentialequationsforthe adjoint or multiplier variables to be numerically integrated simultaneously withthestatedifferentialequationstosolvethetwo-point-boundaryvalue orbit transfer problem. As noted by Betts, the elements of the matrix of partial derivatives of the Hamiltonian with respect to the equinoctial elements derived here in analytic form can be used to developacertainanalyticJacobianmatrixinconnectionwithnonlinear programming (NLP) solvers used with direct methods, thereby saving considerable computation time with respect to the adoption ofa numerically determined Jacobian. The variational equations are usually expressed in terms of the eccentric longitude, which can either be left as an accessory variable or be adopted as the fastor sixth orbital element. In Ref. 3, the averaged J2 perturbation effect was accounted for in the design of the optimal transfer trajectory. However, when the exact or rather position-dependent J2 effect must be considered in the context of precision integrated transfer solutions, the J2 perturbation acceleration components become dependent on the eccentric longitude. Because these components are given in the rotating Euler‐Hill or polar frame, it is much simpler to use this frame as the orbital frame instead of the equinoctial framebecause otherwise the transformed expressions for these components become more complicated. These expressions are still quite complicated if expressed in terms of the eccentric longitude even if the polar frame is used. This is important because we must take the partial derivatives of these expressions with respect to the elements to generate the adjoint differential equations. This task is made easier by expressing the variational equations for the J2 perturbation in terms of the true

To expand mission capabilities needed for explorati on of the Solar System, optimal lowthrust trajectories must be found. However, low-thr ust, multiple gravity-assist trajectories pose significant optimization challenges because of their expansive, multimodal design space. Here, a novel technique is developed for global, lo w-thrust, interplanetary trajectory optimization through the hybridization of a genetic algorithm and a gradient-based direct method (GALLOP). The hybrid algorithm combines the effective global search capabilities of a genetic algorithm with the robust convergence and constraint handling of the local, calculus-based direct method. The automated approach alleviates the difficulty and biases of initial guess generation and provides near globally optimal solutions. The technique is applied to several complex low-thrust, gravity-assi st trajectory scenarios, generating previously unpublished optimums. Nomenclature ck = Chebyshev series coefficient bi = bit sting length of i th design variable f(x) = objective function gj(x) = inequality constraint function hk(x) = equality constraint function mf = final spacecraft mass, kg pm = mutation percentage Pa

The Unified State Model is a method for expressing orbits using a set of seven elements. The elements consist of a quaternion and three parameters based on the velocity hodograph. The equations of this model and the background theory necessary to understand them have been shown here. Numerical simulations comparing the Unified State Model with the traditional Cartesian coordinates have been carried out for perturbed orbits, orbits with low-thrust propulsion, and a solar sailing trajectory. The Unified State Model outperforms Cartesian coordinates for all cases in terms of accuracy and computational speed, except for highly eccentric perturbed orbits. The performance of the Unified State Model is exceptionally better for the case of orbits with continuous low-thrust propulsion with CPU simulation time being an order of magnitude lower than for the simulation using Cartesian coordinates. This makes the Unified State Model especially suited for use in trajectory simulators and optimizers.