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37
Direct Trajectory Optimization Using a Variable LowOrder Adaptive Pseudospectral Method
 AIAA Journal of Spacecraft and Rockets
"... A variableorder adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval until a specified error tolerance is satisfied. In regions of relatively high cur ..."
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A variableorder adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval until a specified error tolerance is satisfied. In regions of relatively high curvature, convergence is achieved by refining the mesh, while in regions of relatively low curvature, convergence is achieved by increasing the degree of the polynomial. An efficient iterative method is then described for accurately solving a general nonlinear optimal control problem. Using four examples, the adaptive pseudospectral method described in this paper is shown to be more efficient than either a global pseudospectral method or a fixedorder method. Nomenclature C = path constraint function D = N N 1 Radau pseudospectral differentiation matrix E = maximum absolute solution error Fd = magnitude of drag force, N Fg = magnitude of gravity force, N Fl = magnitude of lift force, N
Costate Estimation Using MultipleInterval Pseudospectral Methods
 Journal of Spacecraft and Rockets
, 2011
"... A method is presented for costate estimation in nonlinear optimal control problems using multipleinterval collocation at Legendre–Gauss or Legendre–Gauss–Radau points. Transformations from the Lagrange multipliers of the nonlinear programming problem to the costate of the continuoustime optimal co ..."
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Cited by 3 (3 self)
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A method is presented for costate estimation in nonlinear optimal control problems using multipleinterval collocation at Legendre–Gauss or Legendre–Gauss–Radau points. Transformations from the Lagrange multipliers of the nonlinear programming problem to the costate of the continuoustime optimal control problem are given. When the optimal costate is continuous, the transformed adjoint systems of the nonlinear programming problems are discrete representations of the continuoustime firstorder optimality conditions. If, however, the optimal costate is discontinuous, then the transformed adjoint systems are not discrete representations of the continuoustime firstorder optimality conditions. In the case where the costate is discontinuous, the accuracy of the costate approximation depends on the locations of the mesh points. In particular, the accuracy of the costate approximation is found to be significantly higher when mesh points are located at discontinuities in the costate. Two numerical examples are studied and demonstrate the effectiveness of using the multipleinterval collocation approach for estimating costate in continuoustime nonlinear optimal control problems. Nomenclature C = path constraint function D = Gauss or Radau pseudospectral state differentiation matrix
A phMesh Refinement Method for Optimal Control
"... A mesh refinement method is described for solving a continuoustime optimal control problem using collocation at LegendreGaussRadau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative er ..."
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Cited by 2 (2 self)
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A mesh refinement method is described for solving a continuoustime optimal control problem using collocation at LegendreGaussRadau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of the state and a LegendreGaussRadau quadrature integration of the dynamics within a mesh interval. The derived relative error estimate is then used to decide if the degree of the approximating polynomial within a mesh should be increased or if the mesh interval should be divided into subintervals. The degree of the approximating polynomial within a mesh interval is increased if the polynomial degree estimated by the method remains below a maximum allowable degree. Otherwise, the mesh interval is divided into subintervals. The process of refining the mesh is repeated until a specified relative error tolerance is met. Three examples highlight various features of the method and show that the approach is more computationally efficient and produces significantly smaller mesh sizes for a given accuracy tolerance when compared with fixedorder methods. 1
Optimal Control of a Surface Vehicle to Improve Underwater Vehicle Network Connectivity
"... The use of an autonomous surface vehicle as an auxiliary agent to improve connectivity in a network of autonomous underwater vehicles is investigated. An algorithm is developed that consists of an optimal waypoint generator and a minimumtime guidance law that is used to steer the vehicle to the way ..."
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Cited by 1 (0 self)
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The use of an autonomous surface vehicle as an auxiliary agent to improve connectivity in a network of autonomous underwater vehicles is investigated. An algorithm is developed that consists of an optimal waypoint generator and a minimumtime guidance law that is used to steer the vehicle to the waypoint. This algorithm is used together with a communication architecture to improve underwater vehicle connectivity in a region of interest. The approach is simulated using various underwater vehicle configurations, both with and without the autonomous surface vehicle, where it is found that network connectivity is improved significantly via the inclusion of the autonomous surface vehicle. An inwater hardware test is then performed and is shown to be consistent with the simulation results. The results of this study show that adding an autonomous surface vehicle to an underwater vehicle network can improve the connectivity of the network. I.
ThreeDimensional Aircraft Path Planning based on Nonconvex Quadratic Optimization
"... Abstract — In this paper, we examine the threedimensional aircraft path planning problems under fieldofview constraints using a nonconvex quadratic optimization method. The aircraft is assumed to be flying at constant speed with small angle of attack. We focus on determining the attitude of the ..."
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Abstract — In this paper, we examine the threedimensional aircraft path planning problems under fieldofview constraints using a nonconvex quadratic optimization method. The aircraft is assumed to be flying at constant speed with small angle of attack. We focus on determining the attitude of the aircraft when planning the optimal paths. Under this venue, the aircraft kinematics are expressed as quadratic functions in terms of unit quaternions and the path planning problem is reformulated as a general quadratically constrained quadratic programming (QCQP) problem. A semidefinite programming method is then applied to relax the nonconvex QCQP problem to obtain the bounds on the optimal value. Subsequently, an iterative rank minimization approach is proposed to find the optimal solution. Simulation results for planned paths using the proposed method are presented and compared with those obtained from the other method.
Costate Approximation in Optimal Control Using Integral Gaussian Quadrature Orthogonal Collocation Methods
"... Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at LegendreGauss and LegendreGaussRadau points. It is shown that the derivative of the costate of the continuoustime optimal control problem is equal to the negative ..."
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Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at LegendreGauss and LegendreGaussRadau points. It is shown that the derivative of the costate of the continuoustime optimal control problem is equal to the negative of the costate of the integral form of the continuoustime optimal control problem. Using this continuoustime relationship between the differential and integral costate, it is shown that the discrete approximations of the differential costate using LegendreGauss and LegendreGaussRadau collocation are related to the corresponding discrete approximations of the integral costate via integration matrices. The approach developed in this paper provides a way to approximate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LegendreGauss and LegendreGaussRadau collocation methods. The methods are demonstrated on two examples where it is shown that both the differential and integral costate converge exponentially as a function of the number of LegendreGauss or LegendreGaussRadau points. 1
MINIMUMTIME LOWEARTH ORBIT TO HIGHEARTH ORBIT LOWTHRUST TRAJECTORY OPTIMIZATION
"... The problem of many revolution lowthrust Earthorbit trajectory optimization is considered. The objective is to transfer a spacecraft from a parking orbit to a desired terminal orbit in minimum time. The minimumtime orbital transfer problem is posed as a nonlinear optimal control problem, and the ..."
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The problem of many revolution lowthrust Earthorbit trajectory optimization is considered. The objective is to transfer a spacecraft from a parking orbit to a desired terminal orbit in minimum time. The minimumtime orbital transfer problem is posed as a nonlinear optimal control problem, and the optimal control problem is solved using a direct transcription variableorder Gaussian quadrature collocation method. It is found that the thrusttomass ratio holds a power relationship to the transfer time, final mass, and final true longitude. Using these power relationships obtained through regression on the collected results, it is possible to estimate the performance of a given thrusttomass ratio without solving for the optimal trajectory. In addition, the key structure of the optimal orbital transfers are identified. The results presented provide insight into the structure of the optimal performance for a range of small thrusttomass ratios and highlight the interesting features of the optimal solutions. Finally, a discussion of the performance of the variableorder Gaussian quadrature collocation method is provided.
AA Source Transformation via Operator Overloading Method for the Automatic Differentiation of Mathematical Functions in MATLAB
"... A source transformation via operator overloading method is presented for computing derivatives of mathematical functions defined by MATLAB computer programs. The transformed derivative code that results from the method of this paper computes a sparse representation of the derivative of the function ..."
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A source transformation via operator overloading method is presented for computing derivatives of mathematical functions defined by MATLAB computer programs. The transformed derivative code that results from the method of this paper computes a sparse representation of the derivative of the function defined in the original code. As in all source transformation automatic differentiation techniques, an important feature of the method is that any flow control in the original function code is preserved in the derivative code. Furthermore, the resulting derivative code relies solely upon the native MATLAB library. The method is useful in applications where it is required to repeatedly evaluate the derivative of the original function. The approach is demonstrated on several examples and is found to be highly efficient when compared with well known MATLAB automatic differentiation programs.
MinimumFuel LowEarthOrbit Aeroassisted Orbital Transfer of Small Spacecraft
"... The problem of small spacecraft minimumfuel heatrateconstrained aeroassisted orbital transfer between two lowEarth orbits with inclination change is considered. Assuming impulsive thrust, the trajectory design is described in detail and the aeroassisted orbital transfer is posed as a nonlinear op ..."
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The problem of small spacecraft minimumfuel heatrateconstrained aeroassisted orbital transfer between two lowEarth orbits with inclination change is considered. Assuming impulsive thrust, the trajectory design is described in detail and the aeroassisted orbital transfer is posed as a nonlinear optimal control problem. The optimal control problem is solved using an hpadaptive pseudospectral method, and the key features of the optimal trajectories are identified. It was found that the minimum impulse solutions are obtained when the vehicle enters the atmosphere exactly twice. Furthermore, even for highly heatrateconstrained cases, the final mass fraction of the vehicle was fairly large. Finally, the structural loads on the vehiclewere quite reasonable, even in the caseswhere the heating rate was unconstrained. Nomenclature A = vehicle reference area, m2 a = semimajor axis, m CD = coefficient of drag CL = coefficient of drag CL = maximum allowable coefficient of lift D = drag specific force magnitude, m=s2