349 research outputs found
Analytical Solutions to Spacecraft Formation-Flying Guidance Using Virtual Motion Camouflage
No full textFormation-flying has been recognized as one of the enabling technologies for a variety of space mission concepts because of the potential benefits in cost reduction, enhanced flexibility, increased observational baseline, and better survivability and reliability. One important requirement of the formation-flying system is the ability to generate trajectories in real time reliably to maximize mission return. This paper presents simple and analytical open-loop guidance solutions to formation-flying reconfiguration trajectory design problems through the bioinspired virtual motion camouflage methodology and the pseudospectral discretization approach. The algorithms investigated here can be applied to different kinds of initial and final conditions as well as different linearized models considering the eccentricity effects and/or the J2 perturbation. Simulations are used to demonstrate the capabilities of the achieved analytical solutions for two formation-flying reconfiguration scenarios. Copyright © 2010 by Yunjun Xu. Published by the American Institute of Aeronautics and Astronautics, Inc
Subspace Optimal Control and Motion Camouflage
No full textSome insects can move towards female ones or their preys while minimizing the chance of being aware from their natural background using a technique called motion camouflage. In this paper, first as a prelude, an optimal steering law for the minimum time motion camouflage under a speed constraint is proposed. The result of the optimal steering law is then compared with the ones achieved numerically using the pseudospectral collocation and nonlinear programming. The findings from this comparison thereafter leads to a much deeper discussion of the proposed virtual motion camouflage subspace optimization method, which is expected to benefit widely experienced nonlinear constrained optimal control problems. Since the optimization is progressed in the subspace, the solution optimality is justified accordingly. Through this method, the dimension of the nonlinear constrained optimal control problem can be reduced and the rapidly achieved trajectory could be used as a good and feasible initial guess for further optimization. © 2008 by Yunjun Xu
A Quadrature Based Method of Moments for Nonlinear Filtering
No full textAccording to the nonlinear filtering theory, optimal estimates of a general continuousdiscrete nonlinear filtering problem can be obtained by solving the Fokker-Planck equation, coupled with a Bayesian update rule. This procedure does not rely on linearizations of the dynamical and/or measurement models. However, the lack of fast and efficient methods for solving the Fokker-Planck equation presents challenges in real time nonlinear filtering problems. In this paper, a direct quadrature method of moments is introduced to solve the Fokker-Planck equation efficiently and accurately. This approach involves representation of the state conditional probability density function in terms of a finite collection of Dirac delta functions. The weights and locations (abscissas) in this representation are determined by moment constraints and modified using the Baye\u27s rule according to measurement updates. As compared with finite difference methods that are typically used in solving the Fokker- Planck equation, the computational cost is much lower. As emonstrated by a numerical example this approach appears to be promising in the field of nonlinear filtering. Copyright © 2008 by Yunjun Xu. Published by the American Institute of Aeronautics and Astronautics. Inc
Pre and Post Optimality Checking of the Virtual Motion Camouflage Based Nonlinear Constrained Subspace Optimal Control
No full textNonlinear constrained trajectory optimization remains an active field of research. Current popular methods end up with either solving a classical multi-dimensional two-point boundary value problem or a high dimensional nonlinear programming problem. Inspired by the motion camouflage phenomenon of insects like dragonflies, recently the authors proposed a subspace optimization method, the virtual motion camouflage method, in order to reduce the problem dimension. Thus the computational cost experienced in the widely used direct collocation and nonlinear programming methods can be reduced. The solution found through this approach is in the feasible region however the optimality of the solution is not guaranteed automatically in the original full search space. In this paper, two optimality checking ways extended from the Karush-Kuhn-Tucker necessary condition are proposed. The pre-optimality checking is proposed to judge the selection of the virtual prey motion and the reference point, while a post-optimality checking is suggested to test the solution obtained from the virtual motion camouflage method in the original search space. Two numerical examples are used to illustrate the capabilities of the new subspace optimal control algorithm. Copyright © 2009 by Yunjun Xu
Unmanned Aerial Vehicle Formation Flight via a Hierarchical Cooperative Control Approach
No full textCooperative control is crucial for networked dynamical systems, such as unmanned aerial vehicles in a formation, to respond more quickly and to be capable of working in a cluttered environment. In this paper, a divide-and-conquer hierarchical approach is proposed in three levels to handle the design challenges associated with the mission consensus, different constraints, and precise feedback tracking. In the top level, the algorithm aims to guarantee the macro cooperative behaviors, such as formation consensus of multiple agents under simplified double-integrator dynamics and obstacles. In the middle level, the virtual motion camouflage based algorithm is employed in each individual vehicle to compute the near optimal trajectory considering realistic constraints that the top-level algorithm has to neglect. To enhance the robustness of the trajectory control, a nonlinear feedback controller is used in the bottom level for each vehicle to precisely track the desired trajectory commanded by the middle level. Simulation results demonstrate the formation control capabilities of the hierarchical control strategy under obstacle-laden environments and constraints. © 2011 by Yunjun Xu, Ming Xin, Jianan Wang
SAR/InSAR Processing for Surface Deformation of the 2021 Maduo, Qinghai EQ (Xu et al., 2022, in revision)
No full textCut a release for Xu et al. (2022, in revision)
Nonlinear Bayesian Estimation via Solution of The Fokker-Planck Equation
No full textA general approach to optimal nonlinear filtering can be described by a recursive Bayesian approach. The key step in this approach is to determine the probability density function of the state vector conditioned on available measurements. However, an optimal solution to the Bayesian filtering problem can only be obtained exactly for a small class of problems such as linear and Gaussian cases. Therefore, in practice, approximate solutions, such as the extended Kalman filter, have been used.An optimal nonlinear filtering in a recursive Bayesian approach is a two-step process which consists of the prediction and the update process. In the update process, the priori conditional state probability density function (PDF) from the prediction process is updated through Bayes' rule using measurements from sensors. The prediction of conditional state PDF can be made by solving the Fokker-Planck equation (FPE) that governs the time-evolution the conditional state PDF. However, it is extremely difficult to obtain an analytical solution of the Fokker-Planck equation with the exception of a few special cases. So far this estimation method has not been employed much in practice because of the high computational cost needed in solving the FPE numerically. In this dissertation, methods to improve the efficiency of the numerical method in solving the FPE are investigated to enhance the efficiency of the nonlinear filtering.Two finite difference methods, namely i) the explicit forward method and ii) the alternating direction implicit (ADI) method, are used to solve the FPE numerically. Although the explicit forward method is much simpler to implement, the ADI method is preferred for its low computational cost. To reduce the computational cost further, as the first contribution of the dissertation, a moving domain scheme is developed to reduce the domain of integration required for solving the Fokker-Planck equation numerically. Simulation results show that the accuracy of the estimation is improved as compared with the Extended Kalman Filter, and at the same time the computational cost is significantly lower with the proposed moving grid scheme than the case without it.Recently a nonlinear filtering algorithm using a direct quadrature method of moments was proposed, where the associated Fokker-Planck equation is solved efficiently via discrete quadrature based on moment constraints. For some problems, however, this approach showed the phenomenon similar to the "degeneracy'' in a particle filter, which is the concentration of weight on particular particles. The possible cause of the phenomenon is that only the weights are updated through the modified Bayes' rule. Therefore, in this dissertation, as another contribution, a new hybrid filter is proposed where the measurement update equations in the extended or the unscented Kalman filter are used along with the direct quadrature method of moments to solve the FPE. In this way the "degeneracy'' problem can be mitigated.Then, new proposed filtering methods are applied to several challenging problems such as i) the bearing-only tracking problem, ii) the relative orbit position estimation problem, and iii) the orbit determination problem to demonstrate their advantages. Simulation results indicate that the performance of the proposed filters are better than existing nonlinear filtering methods, such as the Extended Kalman Filter especially with less measurement updates
Analytical Solutions to Formation Flying System Trajectory Guidance via the Virtual Motion Camouflage Approach
No full textFormation flying has been recognized as one of the enabling technologies for a variety of space mission concepts because of the potential benefits in cost reduction, enhanced flexibility, increased observational baseline, and better survivability and reliability. One important requirement of the formation flying system is the ability to generate trajectories in real-time reliably to maximize mission return. This paper presents simple and analytical open loop guidance solutions to formation flying reconfiguration trajectory design problems through the bio-inspired virtual motion camouflage methodology and the pseudospectral discretization approach. The algorithms investigated here can be applied to different kinds of initial and final conditions as well as different linearized models considering the eccentricity effects and/or the J2 perturbation. Simulations are used to demonstrate the capabilities of the achieved analytical solutions for two formation flying reconfiguration scenarios. Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved
Nonlinear robust stochastic control for unmanned aerial vehicles
No full textAlmost all dynamical systems experience inherent uncertainties such as environmental disturbance and sensor noise. This paper describes a new robust stochastic control methodology, which is capable of controlling the statistical nature of state variables of a nonlinear system to designed (attainable) statistical properties. First, an asymptotically stable and robust output tracking controller is designed in which discontinuous functions are not involved. Second, undetermined control parameters in the closed-loop system are optimized through nonlinear programming. In this constrained optimization, the error between the desired and actual moments of state variables is minimized subject to constraints on statistical moments. As the key point to overcome the difficulties in solving the associated Fokker-Planck equation, a direct quadrature method of moments is proposed. The advantages of the proposed method are: (1) ability to control any specified stationary moments of the states or output probability density function; (2) no need for the state process to be a Gaussian; (3) robustness with respect to parametric and functional uncertainties. © 2009 AACC
Find Modern Turning From the Rising Trend of Giving Consideration to the Han and Song Compromise School of the Academic History in the Qing Dynasty
No full textAs one of the important schools, the Han and Song compromise school was rising in the mid and late Qing Dynasty and it had lasted for hundreds of years. The reasons of the rise were very complex, but the remarks of Weng Fanggang (翁方綱) , Zhang Xuecheng (章學誠) and Xu Zongyan (許宗彥) of the Qianlong period undoubtedly enlightened the development of the school. Weng Fanggang considered that establishing a union of the Han school and the Song school was better for the both. Zhang Xuecheng thought that separating the Han school and the Song school would hurt the two. Xu Zongyan believed that the academic level of the Han school was lower and it should unite with the Song school to reach the way of sage (聖學). Although there were some differences in their opinions, they all wanted to correct the bias of the Han school and overcome shortcomings of the Song school. Their thoughts showed the changes of academic views from tradition to modernity in the Qing Dynasty. For us, studying this topic still has realistic significance
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