Trajectory optimization for small launchers using a genetic algorithm approach

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

Investment is one of the economic growth factors of countries, especially in Indonesia. Stocks is a form of investment, which is liquid. In determining the stock investment decisions which need to be considered by investors is to choose stocks that can generate maximum returns with a minimum risk level. Therefore, we need to know how to allocate the capital which may give the optimal benefit. This study discusses the issue of stock investment based on CAPM which is estimated using covariance and Genetic Algorithm approach. It is assumed that the stocks analyzed follow the CAPM model. To do the estimation of beta parameter on CAPM equation is done by two approach, first is to be represented by covariance approach, and second with genetic algorithm optimization. As a numerical illustration, in this paper analyzed ten stocks traded on the capital market in Indonesia. The results of the analysis show that estimation of beta parameters using covariance and genetic algorithm approach, give the same decision, that is, six underpriced stocks with buying decision, and four overpriced stocks with a sales decision. Based on the analysis, it can be concluded that the results can be used as a consideration for investors buying six under-priced stocks, and selling four overpriced stocks.

Design of two-dimensional FIR digital filters for sampling structure conversion by genetic algorithm approach

Constant increase in urban road traffic is forcing traffic authorities to confront unavoidable congestion problems these days. This creates new challenges for city planners to generate traffic routes with better designs and improve the existing roads. In order to analyze and improve traffic routes, there is a need to understand movement patterns of major vehicular traffic inside the city. Origin Destination(OD) estimation is one such method for understanding the movement patterns. The objective is to find an optimal OD matrix for city traffic, which is a subtle process for classic algorithms. The solution to this problem is Genetic Algorithm (GA) approach which is a search heuristics to find optimal solution from a set of random solutions. A Genetic Algorithm approach has been developed to find an optimal OD matrix, showing the actual travel patterns of significant number of vehicles inside the city. Once the best OD matrix is obtained, it yields a big opportunity for traffic planners to analyze and improve traffic scenario. Genetic Algorithm can play a major role in solving complex problems. In this paper, one such problem of estimating OD matrix of a city is discussed and implemented. In conjunction to genetic algorithm, Mean Absolute Percentage Error(MAPE) has been used as a fitness criteria for finding the optimal OD matrix.

Clinching is a convenient and efficient cold forming process that can join two sheets without any additional part. This study establishes an intelligent system for optimizing the clinched joint. Firstly, a mathematical model which introduces the ductile damage constraint to prevent cracking during clinching process is proposed. Meanwhile, an optimization methodology and its corresponding computer program are developed by integrated finite element model (FEM) and genetic algorithm (GA) approach. Secondly, Al6061-T4 alloy sheets with a thickness of 1.4mm are used to verify this optimization system. The optimization program automatically acquires the largest axial strength which is approximately equal to 872N. Finally, sensitivity analysis is implemented, in which the influence of geometrical parameters of clinching tools on final joint strength is analyzed. The sensitivity analysis indicates the main parameters to influence joint strength, which is essential from an industrial point of view.

Multiple sourcing under supplier failure risk and quantity discount: A genetic algorithm approach

This study deals with estimation of the total and industrial sector electricity consumption based on genetic algorithm (GA) approach, and then proposes two scenarios to project future consumptions. Total electricity consumption is estimated based on gross national product (GNP), population, import and export figures of Turkey. Industrial sector electricity is calculated based on the GNP, import and export figures. Three forms of the genetic algorithm electricity demand (GAED) models for the total and two forms for the industrial electricity consumption are developed. The best-fit GAED model in terms of total minimum relative average errors between observed and estimated values is selected for future demand estimation. ‘High- and low-growth scenarios’ are proposed for predicting the future electricity consumption. Results showed that the GAED estimates the electricity demand in comparison with the other electricity demand projections. The GAED model plans electricity demand of Turkey until 2020. Copyright © 2005 John Wiley & Sons, Ltd.

A new approach based on fuzzy genetic algorithm is developed to find the time-jerk synthetic optimal trajectory of robot with a joint space scheme using cubic splines. In order to get the optimal trajectory, cubic splines are employed and derived under the constraint condition. Based on cubic splines, the mathematic model of time-jerk synthetic optimal trajectory planning is built by taking into account of both the execution time and the minimax approach of jerk with kinematics constraints expressed as upper bounds on the absolute values of velocity and acceleration. For solving the mathematic model, we designed the set of fuzzy control rules and fuzzy genetic algorithm, using real-coding and elitism approach. Finally, the proposed optimal technique is tested in simulation on a three-degrees-of-freedom glass substrate handling robot. The simulation results show the effectiveness of the algorithm to solve the contradictory problem between high production efficiency and low arm vibration.

Achieving accurate, reliable and globally consistent localization is a fundamental challenge for autonomous mobile robots. There have been numerous researches in this topic showing impressive performance in various scenarios and with various sensor combinations. In a multi-robot setting, finding the relative positions between robots is another essential issue that inhibits other high level tasks such as obstacle avoidance, path planning and collaborative tasking. These problems are especially critical for Micro Aerial Vehicles (MAVs) since any collisions with the surrounding objects or neighbor robots can be catastrophic. Furthermore, the size, weight and power constraints are much more restrictive for MAVs as opposed to ground or legged robots. In this regard, Ultra-wideband (UWB) provides many advantages compared to traditional sensors such as camera, radar or LiDAR. However, UWB has its own limitations such as being heavily reliant on line-of-sight (LoS) and being unable to provide any information about the environment. For these reasons, designing new methods that can combine UWB and other sensor modalities is a worthwhile endeavour and also the overarching goal of this thesis. Firstly, we investigate the problem of integrating UWB and a monocular camera, specifically fusing ranging and visual odometry (VO) data to estimate both the UWB anchor position and the VO's scale factor. We propose a loosely-coupled (LC) fusion scheme to address this problem, with the main advantage being its flexibility since both the VO algorithm and UWB sensor can be replaced easily. Additionally, a theoretical analysis is provided, including the derivation of Fisher Information Matrix (FIM), the FIM's determinant, an evaluation of the singular configurations and their geometric interpretations. Based on these analyses, an estimation-trajectory optimization framework is presented which consists of an estimator, which is designed based on gradient-descent or semidefinite programming (SDP) optimization, and a trajectory optimization approach that minimizes the uncertainty volume and can be easily adapted to different scenarios. Nonetheless, the VO's estimate will still drift over time as the error accumulates which cannot be addressed by the LC method. Hence, we continue to develop a tightly-coupled (TC) version that fuses both UWB and visual measurements in a joint optimization pipeline, supports any number of anchors and incorporates various other enhancements. Extensive simulation and experiments are carried out to demonstrate the effectiveness of the proposed LC and TC systems. Overall, our LC method is fast but susceptible to long-term drift while our TC method is more accurate but more computationally expensive, and the estimation results can be further improved thanks to the trajectory optimization method, even in challenging conditions. Having established the pros and cons for both LC and TC systems, the TC variant is chosen for further researches as we focus on the system performance. In our next works, we extend the fusion scheme to that of UWB and visual-inertial odometry (VIO). Existing methods discard most of the UWB data and ignore the spatial-temporal offsets between sensors. Our approach, on the other hand, addresses these issues by leveraging the propagated information readily available from the VIO pipeline and generalize the UWB processing model. As a result, our method uses the UWB data more effectively: the time-offset between measurements is compensated and all available measurements can be used. Experimental results show that our system consistently surpass previous works in localizing the anchor position and reducing the long-term odometry drift. Lastly, we explore UWB-aided methods for two important problems in multi-robot systems: 1) onboard localization, i.e. finding the position of each robot in a global frame, 2) relative localization, i.e. finding the relative position and heading between robots. Through the derivation of the FIM and its determinant, an analysis for unobservable configurations and their physical interpretation are provided. We then propose optimization-based methods to estimate the relative frame transformation accurately even with no initial guesses. These estimates are then continuously refined based on the robot's motion model and online data, while the onboard sensor data are tightly fused to correct the drift. Our method works without loop closure while requiring less exchanged data, which makes it both flexible and resource-efficient. Moreover, practical issues ignored in previous works are addressed, such as rejecting inter-robot UWB ranging outliers and checking for singular configurations. Simulations and experiments verify that our system as a whole or as its individual parts outperform previous approaches, and the theoretical results are in line with real-world observations.

Safe navigation of Micro Aerial Vehicles (MAVs) requires not only obstacle-free flight paths according to a static environment map, but also the perception of and reaction to previously unknown and dynamic objects. This implies that the onboard sensors cover the current flight direction. Due to the limited payload of MAVs, full sensor coverage of the environment has to be traded off with flight time. Thus, often only a part of the environment is covered. We present a combined allocentric complete planning and trajectory optimization approach taking these sensor visibility constraints into account. The optimized trajectories yield flight paths within the apex angle of a Velodyne Puck Lite 3D laser scanner enabling low-level collision avoidance to perceive obstacles in the flight direction. Furthermore, the optimized trajectories take the flight dynamics into account and contain the velocities and accelerations along the path. We evaluate our approach with a DJI Matrice 600 MAV and in simulation employing hardware-in-the-loop.

A method for determining optimal trajectories for tethered systems is presented based on a parameterization of the tension control input. Three separate models of the tether system are used to compare the effects of different cost functions on optimal trajectories. The parameterization of the control is based on a rigid tether model, while the optimal trajectories are determined using a model that discretizes the tether into 4 point masses connected by massless inextensible strings. The effect of various cost functions and terminal times are compared for deployment and retrieval. The effect of various cost functions are also studied for a tether-assisted sample return problem. It is shown that a cost function that minimizes the tether reel acceleration maintains the tether essentially straight when simulated in a fully flexible model of the system, whereas several other cost functions result in significant bending of the tether.

The integration of a reusable scramjet vehicle as the second stage of a multistage space launch system has the potential to reduce the cost of small-payload orbital launches. This paper determines the maximum payload to orbit trajectory of a multistage rocket–scramjet–rocket system. This trajectory is calculated by formulating the problem as an optimal control problem, and then solving it using the pseudospectral method. Using this method, it is determined that the optimal trajectory for the scramjet stage involves an initial decrease in dynamic pressure, followed by constant-dynamic-pressure flight, and finally a pullup maneuver. This optimal trajectory results in an 8.35% improvement in payload mass to orbit when compared to a constant-dynamic-pressure trajectory with minimum pullup. Furthermore, the optimal pullup maneuver decreases the maximum dynamic pressure experienced by the final rocket stage by 20.3%. The sensitivity of the trajectory is tested by varying the maximum allowable dynamic pressure and the drag produced by the vehicle. A maximum dynamic pressure variation of is shown to produce only and variations in the payload mass. A drag increase of 10% is shown to produce a similar optimal trajectory shape, indicating robustness with variation of the vehicle aerodynamics.

This paper describes a computational tool for TTRS tolerance analysis that makes use of a Genetic Algorithm approach. The Technologically and Topologically Related Surfaces (TTRS) concept represents and classifies part surfaces and mechanical assemblies. We show results for examples with position and orientation tolerances that suggest that a Genetic Algorithm approach can provide an effective means for tolerance analysis.

The authors present research results on a method to perform waypoint rendezvous in conjunction with multiple-threat avoidance with the horizontal plane trajectory optimization algorithm. A 737-300 aircraft performance model was used to study trajectory optimization with SP/HTO (singular perturbation/horizontal trajectory optimization). The cost function used by J.L. Vian and J.R. Moore (1989) was modified to include the cost of line-of-sight angle to the destination in order to generate more flyable trajectories through regions of multiple threat envelopes. Results are presented for several scenarios in which there are multiple threats and rendezvous points. The surface-to-air threat effectiveness envelope developed and used in these mission scenarios to generate optimal trajectories is discussed. It is shown that the addition of a cost of line-of-sight angle to the destination avoids a tendency of the trajectory to immediately return to the x-axis when no threats are present in the direct path to the target; instead the trajectory leads directly to the target and is a more desirable solution. The local approach to trajectory optimization is shown to be useful in short-range scenarios or in cases where there may be pop-up threats. >

This paper deals with a robust track-following system design problem with multiple constraints using genetic algorithm and LMI approach. A robust H ∞ constraint is introduced to attenuate vibration disturbance within error limits and a robust loop gain constraint is considered to minimize the variations of the tracking loop characteristics against actuator uncertainties. The robust H ∞ constraint can be expressed by a matrix inequality and the robust loop gain constraint can be treated by an objective function of a genetic algorithm. Accordingly, a robust tracking controller can be obtained by integrating a genetic algorithm and LMI approach. The proposed track-following system design method is applied to the track-following system of a DVD recording drive and is evaluated through experimental results.