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47
A New Computational Approach to RealTime Trajectory Generation for Constrained Mechanical Systems
, 2000
"... Preliminary results of a new computational approach to generate aggressive trajectories in realtime for constrained mechanical systems are presented. The algorithm is based on a combination of nonlinear control theory, spline theory, and sequential quadratic programming. It is demonstrated that rea ..."
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Cited by 74 (19 self)
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Preliminary results of a new computational approach to generate aggressive trajectories in realtime for constrained mechanical systems are presented. The algorithm is based on a combination of nonlinear control theory, spline theory, and sequential quadratic programming. It is demonstrated that realtime trajectory generation for constrained mechanical systems is possible by mapping the problem to one of finding trajectory curves in a lower dimensional space. Performance of the algorithm is compared with existing optimal trajectory generation techniques. Numerical results are reported using the NTG software package. Keywords: Realtime optimization, nonlinear control design, optimal control, constrained trajectory generation, guidance. 1
Autonomous Vehicle Technologies for Small Fixed Wing UAVs
 AIAA JOURNAL OF AEROSPACE COMPUTING, INFORMATION, AND COMMUNICATION
, 2003
"... Autonomous unmanned air vehicle flight control systems require robust path generation to account for terrain obstructions, weather, and moving threats such as radar, jammers, and unfriendly aircraft. In this paper, we outline a feasible, hierarchal approach for realtime motion planning of small aut ..."
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Cited by 43 (15 self)
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Autonomous unmanned air vehicle flight control systems require robust path generation to account for terrain obstructions, weather, and moving threats such as radar, jammers, and unfriendly aircraft. In this paper, we outline a feasible, hierarchal approach for realtime motion planning of small autonomous fixedwing UAVs. The approach divides the trajectory generation into four tasks: waypoint path planning, dynamic trajectory smoothing, trajectory tracking, and lowlevel autopilot compensation. The waypoint path planner determines the vehicle 's route without regard for the dynamic constraints of the vehicle. This results in a significant reduction in the path search space, enabling the generation of complicated paths that account for popup and dynamically moving threats. Kinematic constraints are satisfied using a trajectory smoother which has the same kinematic structure as the physical vehicle. The third step of the approach uses a novel tracking algorithm to generate a feasible state trajectory that can be followed by a standard autopilot. MonteCarlo simulations were done to analyze the performance and feasibility of the approach and determine realtime computation requirements. A planar version of the algorithm has also been implemented and tested in a lowcost microcontroller. The paper describes a custom UAV built to test the algorithms.
Numerical solution of optimal control problems by direct collocation’, Ed
 In Optimal Control  Calculus of Variation, Optimal Control Theory and Numerical Methods, 111, 129143, International Series of Numerical Mathematics, Birkhäuser
, 1993
"... By an appropriate discretization of control and state variables, a constrained optimal control problem is transformed into a finite dimensional nonlinear program which can be solved by standard SQPmethods [10]. Convergence properties of the discretization are derived. From a solution of this method ..."
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Cited by 41 (2 self)
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By an appropriate discretization of control and state variables, a constrained optimal control problem is transformed into a finite dimensional nonlinear program which can be solved by standard SQPmethods [10]. Convergence properties of the discretization are derived. From a solution of this method known as direct collocation, these properties are used to obtain reliable estimates of adjoint variables. In the presence of active state constraints, these estimates can be significantly improved by including the switching structure of the state constraint into the optimization procedure. Two numerical examples are presented. 1 Statement of problems Systems governed by ordinary differential equations arise in many applications as, e. g., in astronautics, aeronautics, robotics, and economics. The task of optimizing these systems leads to the optimal control problems investigated in this paper. The aim is to find a control vector u(t) and the final time tf that minimize the functional
SQP Methods And Their Application To Numerical Optimal Control
, 1997
"... . In recent years, generalpurpose sequential quadratic programming (SQP) methods have been developed that can reliably solve constrained optimization problems with many hundreds of variables and constraints. These methods require remarkably few evaluations of the problem functions and can be shown ..."
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Cited by 19 (0 self)
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. In recent years, generalpurpose sequential quadratic programming (SQP) methods have been developed that can reliably solve constrained optimization problems with many hundreds of variables and constraints. These methods require remarkably few evaluations of the problem functions and can be shown to converge to a solution under very mild conditions on the problem. Some practical and theoretical aspects of applying generalpurpose SQP methods to optimal control problems are discussed, including the influence of the problem discretization and the zero/nonzero structure of the problem derivatives. We conclude with some recent approaches that tailor the SQP method to the control problem. Key words. largescale optimization, sequential quadratic programming (SQP) methods, optimal control problems, multiple shooting methods, single shooting methods, collocation methods AMS subject classifications. 49J20, 49J15, 49M37, 49D37, 65F05, 65K05, 90C30 1. Introduction. Recently there has been c...
Nonlinear optimal control via occupation measures and LMI relaxations
 SIAM Journal on Control and Optimization
, 2008
"... Abstract. We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control constraints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state an ..."
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Cited by 16 (13 self)
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Abstract. We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control constraints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI (linear matrix inequality)relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments. 1.
Numerical Optimal Control Of Parabolic PDEs Using DASOPT
, 1997
"... . This paper gives a preliminary description of DASOPT, a software system for the optimal control of processes described by timedependent partial differential equations (PDEs). DASOPT combines the use of efficient numerical methods for solving differentialalgebraic equations (DAEs) with a package ..."
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Cited by 11 (6 self)
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. This paper gives a preliminary description of DASOPT, a software system for the optimal control of processes described by timedependent partial differential equations (PDEs). DASOPT combines the use of efficient numerical methods for solving differentialalgebraic equations (DAEs) with a package for largescale optimization based on sequential quadratic programming (SQP). DASOPT is intended for the computation of the optimal control of timedependent nonlinear systems of PDEs in two (and eventually three) spatial dimensions, including possible inequality constraints on the state variables. By the use of either finitedifference or finiteelement approximations to the spatial derivatives, the PDEs are converted into a large system of ODEs or DAEs. Special techniques are needed in order to solve this very large optimal control problem. The use of DASOPT is illustrated by its application to a nonlinear parabolic PDE boundary control problem in two spatial dimensions. Computational resu...
OffLine and OnLine Computation of Optimal Trajectories in the Aerospace Field
, 1991
"... . The present paper is an introductory and survey paper of the treatment of realistically modelled optimal control problems from applications in the aerospace field. Especially those problems are considered which include different types of constraints. In the tutorial part of the paper, recipes are ..."
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Cited by 9 (3 self)
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. The present paper is an introductory and survey paper of the treatment of realistically modelled optimal control problems from applications in the aerospace field. Especially those problems are considered which include different types of constraints. In the tutorial part of the paper, recipes are given for the treatment of optimal control problems for which, among other constraints, control and/or state variable inequality constraints are to be taken into account. Optimal control problems having singular subarcs and/or discontinuities are also investigated. The discussion of the necessary conditions aims to the subsequent application of the multiple shooting method, which is known to be a very precise and efficient method for the solution of those multipoint boundaryvalue problems that arise from these necessary conditions. Homotopy techniques as well as the fusion of direct collocation and multiple shooting techniques are described. Both approaches facilitate the construction of an...
Decomposition of mixedinteger optimal control problems using branch and bound and sparse direct collocation
 IN PROCEEDINGS OF ADPM 2000 – AUTOMATION OF MIXED PROCESSES: HYBRID DYNAMIC SYSTEMS
, 2000
"... A large class of optimal control problems for hybrid dynamic systems can be formulated as mixedinteger optimal control problems (MIOCPs). It is the intrinsic combinatorial complexity, in addition to the nonlinearity of the continuous, multiphase optimal control problems that is largely responsible ..."
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Cited by 8 (3 self)
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A large class of optimal control problems for hybrid dynamic systems can be formulated as mixedinteger optimal control problems (MIOCPs). It is the intrinsic combinatorial complexity, in addition to the nonlinearity of the continuous, multiphase optimal control problems that is largely responsible for the challenges in the theoretical and numerical solution of MIOCPs. We present a new decomposition approach to numerically solving fairly general MICOPs with binary control variables. A Branch and Bound (B&B) technique is applied to efficiently search the entire discrete solution space performing a truncated binary tree search for the discrete variables maintaining upper and lower bounds on the performance index. The partially relaxed binary variables at an inner node define an optimal control problem with dynamic equations defined in multiple phases. Its global solution provides a lower bound on the performance index for all nodes of the subtree. If the lower bound for a given subtree is greater than the current global upper bound then that entire subtree need no longer be searched. The many optimal control problems with nonlinear, continuous state dynamics defined in multiple phases subject to nonlinear constraints are solved most efficiently by a sparse direct collocation transcription. Hereby, the multiphase optimal control problem is transcribed to a sparse, largescale nonlinear programming problem being solved efficiently by a tailored SQP method. Despite the high efficiency of the sparse direct collocation method, the efficiency of the decomposition technique for MIOCPs strongly depends on
Optimal Control of the Industrial Robot Manutec r3
, 1994
"... Minimum time and minimum energy pointtopoint trajectories for an industrial robot of the type Manutec r3 are computed subject to state constraints on the angular velocities. The numerical solutions of these optimal control problems are obtained in an efficient way by a combination of a direct coll ..."
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Cited by 8 (2 self)
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Minimum time and minimum energy pointtopoint trajectories for an industrial robot of the type Manutec r3 are computed subject to state constraints on the angular velocities. The numerical solutions of these optimal control problems are obtained in an efficient way by a combination of a direct collocation and an indirect multiple shooting method. This combination links the benefits of both approaches: A large domain of convergence and a highly accurate solution. The numerical results show that the constraints on the angular velocities become active during large parts of the time optimal motion. But the resulting stress on the links can be significantly reduced by a minimum energy trajectory that is only about ten percent slower than the minimum time trajectory. As a byproduct, the reliability of the direct collocation method in estimating adjoint variables and the efficiency of the combination of direct collocation and multiple shooting is demonstrated. The highly accurate solutions ...
Nonlinear hybrid dynamical systems: modeling, optimal control, and applications
 in Modelling, Analysis and Design of Hybrid Systems, ser. Lecture Notes in Control and Information
, 2002
"... Abstract. Nonlinear hybrid dynamical systems are the main focus of this paper. A modeling framework is proposed, feedback control strategies and numerical solution methods for optimal control problems in this setting are introduced, and their implementation with various illustrative applications are ..."
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Cited by 8 (7 self)
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Abstract. Nonlinear hybrid dynamical systems are the main focus of this paper. A modeling framework is proposed, feedback control strategies and numerical solution methods for optimal control problems in this setting are introduced, and their implementation with various illustrative applications are presented. Hybrid dynamical systems are characterized by discrete event and continuous dynamics which have an interconnected structure and can thus represent an extremely wide range of systems of practical interest. Consequently, many modeling and control methods have surfaced for these problems. This work is particularly focused on systems for which the degree of discrete/continuous interconnection is comparatively strong and the continuous portion of the dynamics may be highly nonlinear and of high dimension. The hybrid optimal control problem is defined and two solution techniques for obtaining suboptimal solutions are presented (both based on numerical direct collocation for continuous dynamic optimization): one fixes interior point constraints on a grid, another uses branchandbound. These are applied to a robotic multiarm transport task, an underactuated robot arm, and a benchmark motorized traveling salesman problem. 1