Preliminary results of a new computational approach to generate aggressive trajectories in real-time 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 real-time 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: Real-time optimization, nonlinear control design, optimal control, constrained trajectory generation, guidance. 1

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 real-time motion planning of small autonomous fixed-wing UAVs. The approach divides the trajectory generation into four tasks: waypoint path planning, dynamic trajectory smoothing, trajectory tracking, and low-level 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 pop-up 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. Monte-Carlo simulations were done to analyze the performance and feasibility of the approach and determine real-time computation requirements. A planar version of the algorithm has also been implemented and tested in a low-cost micro-controller. The paper describes a custom UAV built to test the algorithms.

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 SQP-methods [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

. In recent years, general-purpose 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 general-purpose 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. large-scale 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...

. This paper gives a preliminary description of DASOPT, a software system for the optimal control of processes described by time-dependent partial differential equations (PDEs). DASOPT combines the use of efficient numerical methods for solving differential-algebraic equations (DAEs) with a package for large-scale optimization based on sequential quadratic programming (SQP). DASOPT is intended for the computation of the optimal control of time-dependent nonlinear systems of PDEs in two (and eventually three) spatial dimensions, including possible inequality constraints on the state variables. By the use of either finite-difference or finite-element 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...

. 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 boundary-value 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...

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 branch-and-bound. These are applied to a robotic multi-arm transport task, an underactuated robot arm, and a benchmark motorized traveling salesman problem. 1

Minimum time and minimum energy point-to-point 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 by-product, 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 ...

. When solving optimal control problems, indirect methods such as multiple shooting suffer from difficulties in finding an appropriate initial guess for the adjoint variables. For, this initial estimate must be provided for the iterative solution of the multipoint boundary-value problems arising from the necessary conditions of optimal control theory. Direct methods such as direct collocation do not suffer from this problem, but they generally yield results of lower accuracy and their iteration may even terminate with a non-optimal solution. Therefore, both methods are combined in such a way that the direct collocation method is at first applied to a simplified optimal control problem where all inequality constraints are neglected as long as the resulting problem is still well-defined. Because of the larger domain of convergence of the direct method, an approximation of the optimal solution of this problem can be obtained easier. The fusion between direct and indirect methods is then b...

A large class of optimal control problems for hybrid dynamic systems can be formulated as mixed-integer optimal control problems (MIOCPs). It is the intrinsic combinatorial complexity, in addition to the nonlinearity of the continuous, multi-phase 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 multi-phase optimal control problem is transcribed to a sparse, large-scale 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