Highly Constrained Optimal Launch Ascent Guidance

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.

A fixed-point formulation and the associated algorithms for rapid generation of optimal ascent trajectories of launch vehicles through the atmosphere are presented. The complete ascent trajectory is tacitly divided into an endo-atmospheric and an exo-atmospheric portion. Since the exo-atmospheric trajectory can be obtained reliably and rapidly, a special form of two-point boundary value problem (TPBVP) is formulated in the endo-atmospheric portion. This special TPBVP leads to a sequence of nested fixed-point problems. Several algorithmic options to solve these problems using modified fixed-point iterations are discussed. The main advantages of the fixed-point formulation and algorithm are software simplicity and more robust convergence. Numerical results, both in open-loop trajectory generation and closed-loop simulations, are presented to demonstrate the discussions. The algorithm is validated and verified by comparing the results with the results obtained from existing method.

During the early phases of engineering design, the costs committed are high, costs incurred are low, and the design freedom is high. It is well documented that decisions made in these early design phases drive the entire design's life cycle cost. In a traditional paradigm, key design decisions are made when little is known about the design. As the design matures, design changes become more difficult in both cost and schedule to enact. The current capability-based paradigm, which has emerged because of the constrained economic environment, calls for the infusion of knowledge usually acquired during later design phases into earlier design phases, i.e. bringing knowledge acquired during preliminary and detailed design into pre-conceptual and conceptual design. An area of critical importance to launch vehicle design is the optimization of its ascent trajectory, as the optimal trajectory will be able to take full advantage of the launch vehicle's capability to deliver a maximum amount of payload into orbit. Hence, the optimal ascent trajectory plays an important role in the vehicle's affordability posture yet little of the information required to successfully optimize a trajectory is known early in the design phase. Thus, the current paradigm of optimizing ascent trajectories involves generating point solutions for every change in a vehicle's design parameters. This is often a very tedious, manual, and time-consuming task for the analysts. Moreover, the trajectory design space is highly non-linear and multi-modal due to the interaction of various constraints. When these obstacles are coupled with the Program to Optimize Simulated Trajectories (POST), an industry standard program to optimize ascent trajectories that is difficult to use, expert trajectory analysts are required to effectively optimize a vehicle's ascent trajectory. Over the course of this paper, the authors discuss a methodology developed at NASA Marshall's Advanced Concepts Office to address these issues. The methodology is two-fold: first, capture the heuristics developed by human analysts over their many years of experience; and secondly, leverage the power of modern computing to evaluate multiple trajectories simultaneously and therefore enable the exploration of the trajectory's design space early during the pre- conceptual and conceptual phases of design. This methodology is coupled with design of experiments in order to train surrogate models, which enables trajectory design space visualization and parametric optimal ascent trajectory information to be available when early design decisions are being made.

As robots become more capable in terms of hardware, and more complex tasks are considered, optimality starts playing a more important role in the design of algorithms implemented in these systems. Optimality is a guiding principle that directs the computation of feasible and efficient solutions to different robotics tasks. In control theory, this principle is implemented online as a set of efficient numerical optimization algorithms, that in addition to solving the task, purports to save a suitably defined effort or energy term. This thesis investigates trajectory generation, learning and control for dynamic tasks from the unifying point of view of optimization. As an application, we focus on Table Tennis, a chal- lenging task where robots are yet to outperform humans. We believe that the required dexterity and accuracy for this dynamical task hinges on the developments in online optimization and efficient learning algorithms. We consider trajectory generation for table tennis in the first part of the thesis. In highly dynamic tasks like table tennis that involve moving targets, planning is necessary to figure out when, where and how to intercept the target. Motion planning can be very challenging in robotic table tennis in particular, due to time constraints, dimension of the search space and joint limits. Conventional planning algorithms often rely on a fixed virtual hitting plane to construct robot striking trajectories. These algorithms, however, generate restrictive strokes and can result in unnatural strategies when compared with human playing. In this thesis, we introduce a new trajectory generation framework for robotic table tennis that does not involve a fixed hitting plane. A free-time optimal control approach is used to derive two different trajectory optimizers. The resulting two algorithms, Focused Player and Defensive Player, encode two different play- styles. We evaluate their performance in simulation and in our robot table tennis platform with a high speed cable-driven seven DOF robot arm. The algorithms return the balls with a higher probability to the opponent’s court when compared with a virtual hitting plane based method. Moreover, both can be run online and the trajectories can be corrected with new ball observations. In the second part of the thesis, we look at how such trajectories, computed on the kine- matics level, can be tracked accurately with learning control based approaches. Highly dynamic tasks like table tennis require large accelerations and precise tracking for successful perfor- mance. To track desired trajectories well, such tasks usually rely on accurate models and/or high gain feedback. While kinematic optimization allows for efficient representation and online generation of hitting trajectories, learning to track such dynamic movements with inaccurate models remains an open problem. In particular, stability issues surrounding the learning per- formance, in the iteration domain, can prevent the successful implementation of model based learning approaches. To achieve accurate tracking for these tasks in a stable and efficient way, we propose a new adaptive Iterative Learning Control algorithm that is implemented efficiently using a recursive approach. Moreover, covariance estimates of model matrices are used to ex- ercise caution during learning. We evaluate the performance of the proposed approach in our robotic table tennis platform, where we show how the performance of two Barrett WAMs can be optimized. Our implementation on the table tennis platform compares favorably with two state-of-the-art approaches. Finally, we discuss an alternative learning from demonstrations approach, where we learn sparse representations from demonstrated movements. Learning from demonstrations is an easy and intuitive way to show examples of successful behavior to a robot. However, the fact that humans optimize or take advantage of their body and not of the robot, usually called the embodiment problem in robotics, often prevents industrial robots from executing the task in a straightforward way. The shown movements often do not or cannot utilize the degrees of freedom of the robot efficiently, and typically suffer from excessive execution errors. In the last chapter, we show a new approach that can alleviate some of these difficulties by learning sparse representations of movement. Moreover, the number of learned parameters are independent of the degrees of freedom of the robot. Sparsity is a desirable feature for policy search Rein- forcement Learning algorithms that adapt the parameters of these movement primitives. By ranking the learned parameters on the Elastic Net path in terms of importance, we note that our approach could be potentially useful to combat the curse of dimensionality in robot learning applications. We show preliminary results on the real robot setup, including a successful table tennis serve using our new movement primitive representation. Throughout the thesis, we present and analyze in detail new control and learning algo- rithms. Efficient online optimization approaches are presented that can be used to solve not just table tennis problems, but they can be adapted to solve different dynamic tasks.

This paper presents a systematic method for generating optimal helicopter flight trajectories in real time for cockpit display. Helicopter flights in the event of a single engine failure are critical, and display of optimal flight paths on-board is desirable. Due to the large number of states and constraints, however, numerical solutions on a mainframe computer can take a long tune. In this paper, optimal solutions are first obtained off-line using a ground computer. For a given set of initial helicopter states and flight parameters, optimal solutions are fitted by a Fourier series. The coefficients of the Fourier approximations are then interpolated as functions of initial flight states and parameters. The approximate optimal trajectories can be quickly generated on-board using the proposed method. Introduction Engine failure is a major safety issue for all helicopters, especially during takeoff and landing operations. Due to the complexity of the situation and the dependence of safety on many parameters, optimal flight trajectories can provide useful guidance for pilot operation. In particular, on-board display of optimal trajectories is desirable. This paper presents a method for generating optimal flight trajectories in real-tune that can be used for on-board display. Depending on its capability and performance during an engine failure, a transport helicopter can be certified as Category A or Category B. The Federal Aviation Administration (FAA) grants Category B certification to a single or multi-engine helicopter with gross weight less than 20,000 Ib and capable of landing safely in case of one or all engine failures. Category A certification, on the other hand, is granted to multi-engine helicopters with independent engine systems. Category A helicopters must be able to either land safely or continue flight in case of One Engine Inoperation (OEI). This makes Category A helicopters uniquely suited for remote places where emergency landing sites are not available. I Ph.D. Candidate, Center for Control Sciences, Member AIAA * Assistant Professor, Dept. of Aerospace Eng. and Mechanics, Member AIAA + Civil Rotorcraft Group Leader, Rotorcraft and Powered Lift Branch The decision to continue the flight or to land represents an energy management problem. It has been shown''' that this decision depends on initial conditions at the time of failure as well as flight parameters. Due to the numerous initial variables involved in this decision, a method to communicate all this information to the pilot instantaneously, at engine failure, is needed. This method must keep the emergency procedure simple and manageable. In particular, on-board cockpit display of optimal flight trajectories is useful. This requires a reliable real-tune solution of optimal trajectories, on board, with limited time, limited computing power, and limited memory available. If possible, an analytical solution of optimal trajectories is desirable. Unfortunately, this can only be achieved for systems of low order and where the equations of motion and the cost function are simple. Therefore, for a complex system, an open-loop numerical procedure becomes necessary. Research efforts were made to solve optimal control problems in real-time. Schultz et al. used calculus of variation and energy state method to generate optimal trajectories on-board for the National Aerospace Plane (NASP). In his study, the equations of motion of the NASP are less complex than those of the helicopter, and an interval of time of few minutes is allowed. Bryson used neighboring control theory to generate near-optimal solutions that could be used in real-time. This method is elegant but requires a large amount of memory to store the needed perturbation matrices. Also, it is valid on a relatively small neighborhood of the nominal trajectory. Perturbation methods and table lookups are often used if only few parameters are involved. In the current problem, the equations of motion of the helicopter are complex and too many path equality and inequality constraints are present. Also, limitations on tune, computing power, and memory available make all the above methods unsuitable for this problem. This paper proposes a technique of real-time optimal trajectory generation based on approximation and

Ascent Guidance for an Aerospace Plane

Near-optimal trajectory generation, using a compound B-spline interpolation and minimum distance criterion with dynamical feasibility correction

The problem of trajectory optimization is important in all space missions. The solution of this problem enables one to specify the optimum thrust steering program which should be followed to achieve a specified mission objective, simultaneously satisfying the constraints.1 It is well known that whether or not the ascent trajectory is optimal can have a significant impact on propellant usage for a given payload, or on payload weight for the same gross vehicle weight.2 Consequently, ascent guidance commands are usually optimized in some fashion. Multi-stage vehicles add complexity to this analysis process as changes in vehicle properties in one stage propagate to the other stages through gear ratios and changes in the optimal trajectory. These effects can cause an increase in analysis time as more variables are added and convergence of the optimizer to system closure requires more analysis iterations. In this paper, an approach to simplifying this multi-stage problem through the creation of an upper stage capability boundary is presented. This work was completed as part of a larger study focused on trade space exploration for the advanced booster system that will eventually form a part of NASA s new Space Launch System.3 The approach developed leverages Design of Experiments and Surrogate Modeling4 techniques to create a predictive model of the SLS upper stage performance. The design of the SLS core stages is considered fixed for the purposes of this study, which results in trajectory parameters such as staging conditions being the only variables relevant to the upper stage. Through the creation of a surrogate model, which takes staging conditions as inputs and predicts the payload mass delivered by the SLS upper stage to a reference orbit as the response, it is possible to identify a of staging conditions which all satisfy the SLS requirement of placing 130 metric tons into low-Earth orbit (LEO).3 This identified surface represents the 130 metric ton capability boundary for the upper stage, such that if the combined first stage and boosters can achieve any one staging point on that surface, then the design is identified as feasible. With the surrogate model created, design and analysis of advanced booster concepts is streamlined, as optimization of the upper stage trajectory is no longer required in every design loop.

The analysis of an aircraft terrain-following (TF) problem and a numerical method for the optimal trajectory generation are the subjects of this paper. The TF problem is formulated as an optimal control problem that combines short flight time and path-following objectives. The aircraft dynamics are represented by point-mass equations of motion in a vertical plane. The problem is first studied analytically to reveal the bang-bang nature of the optimal thrust control in most cases. Then, an inverse dynamics approach is employed to solve the problem numerically. Numerical solutions are provided to support the analysis and demonstrate the features of the problem.

This paper presents a methodology for the concurrent first-stage preliminary design and ascent trajectory optimization, with application to a Vega-derived light launch vehicle. The reuse as first stage of an existing upper-stage (Zefiro 40) requires a redesign of the propellant grain geometry, in order to account for the mutated operating conditions. An optimization code based on the parallel running of several differential evolution algorithms is used to find the optimal law of the internal pressure during first-stage operation, together with the optimal thrust direction and other relevant flight parameters of the entire ascent trajectory. The payload injected into a target orbit is maximized, while meeting multiple design constraints, either involving the alone solid rocket motor or dependent on the actual flight trajectory. Numerical results for sun-synchronous orbit injection are presented.

Warping model predictive control for application in control of a real airborne wind energy system

Solar sailing is increasingly being considered by space agencies for future science missions. With the absence of reaction mass from the primary propulsion system arises the potential for new high-energy mission concepts in the mid to far term, such as a Solar Polar Orbiter or an Interstellar Heliopause Probe [1,2]. One of the most time consuming tasks of mission analysis is trajectory generation and optimization. Optimal trajectory generation is a complex field and many schemes exist; however, these are typically characterized as being computationally intensive systems requiring a good degree of engineering judgment [3-6].

This paper presents a novel real-time quasi-optimal trajectory generator, based on the inverse dynamics in the virtual domain (IDVD) method, to produce a reliable and efficient guidance system for autonomous underwater vehicle (AUV) docking operations. To this end, a challenging docking scenario is defined in a cluttered operating field, encompassing ocean currents and no-fly zones. Using the IDVD method, a trajectory that takes into consideration the hydrodynamic model of the vehicle is generated and the optimality of this trajectory, in regards to mission time and energy expenditure of the vehicle, is considered. Computer simulations demonstrate that the IDVD-based strategy enables the guiding of a vehicle into the dock by satisfying the final boundary conditions of the dock's position and orientation. Generated trajectories are feasible, smooth and realizable using for the vehicle's low-level auto-pilot module. In terms of computation, the solution is suitable for real-time implementation that incorporates uncertainty handling of the operating environment. For further analysis, the generated trajectory is evaluated on a high fidelity AUV simulator. The result of this latter test also demonstrates applicability of the utilized IDVD method for optimization of docking trajectories.

Multistage launch vehicles are employed to place spacecraft and satellites in their operational orbits. If the rocket aerodynamics and propulsion are modeled appropriately, optimization of their ascent trajectory consists in determining the coast duration and the thrust time history that maximize the final mass at injection. This research derives all the necessary conditions for ascent path optimization of a multistage launch vehicle. With reference to an existing rocket, the indirect heuristic method is then applied, for the numerical determination of the overall ascent trajectory. An effective approach is used with the intent of satisfying the path constraint related to the maximum dynamical pressure in the atmospheric phase. Then, the recently introduced, implicit-type variable-time-domain neighboring optimal guidance is applied to the upper stage powered arc, for the purpose of obtaining the corrective control actions in the presence of nonnominal flight conditions. The guidance approach at hand, based on the second-order analytical conditions for optimality, proves to be rather effective (in terms of propellant budget), and guarantees very accurate orbit injection in spite of perturbations.

In this paper, an L1 penalized sequential convex programming (LPSCP) method for trajectory optimization is proposed. Sequential convex methods dramatically reduce computation time for nonlinear trajectory optimizations, based on the desirable properties of convex optimization algorithms. Due to its fast convergence speed, the sequential convex method is considered as a near-future candidate for real-time optimal online guidance. However, the generic sequential convex method seldom suffers from poor robustness to crude the initial trajectory estimates and constraint perturbations, which limits its application to real-world systems. Suggested LPSCP method resolves the robustness issue of generic sequential convex methods, by addressing the convex subproblem infeasibility. While improving its robustness, proposed LPSCP method maintains its fast convergence property. Consequently, LPSCP method enhances overall versatility of sequential convex method for trajectory optimization, which includes ameliorated robustness toward initial trajectory estimates, and improved stability to state and constraint perturbations. Throughout the paper, details of LPSCP method are outlined along with an application example to missile trajectory optimization problems. Desirable properties of LPSCP method are demonstrated through several simulated examples. Optimized trajectories are compared with results from the pseudospectral method. Numerical simulation and Monte Carlo analysis show that LPSCP method accelerates optimization process dramatically (−98.2% on average) while producing the same optimal trajectory.