Abstract — Recent advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of stability for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses these results to efficiently combine locally valid linear quadratic regulator (LQR) controllers into a nonlinear feedback policy which probabilistically covers the reachable area of a (bounded) state space with a region of stability, certifying that all initial conditions that are capable of reaching the goal will stabilize to the goal. We investigate the properties of this systematic nonlinear feedback control design algorithm on simple underactuated systems and discuss the potential for control of more complicated control problems like bipedal walking. I.

In this paper, we analyze second-order Runge-Kutta approximations to a nonlinear optimal control problem with control constraints. If the optimal control has a derivative of bounded variation and a coercivity condition holds, we show that for a special class of Runge-Kutta schemes, the error in the discrete approximating control is O(h ) where h is the mesh spacing.

Introduction The first direct multiple shooting method for optimal control problems was developed by Bock and Plitt [2,7] in 1983. Their code MUSCOD was successfully applied, e.g., in robot trajectory optimization [5,6,10]. Other direct SQP methods based on collocation have been developed in the sequel [1,3,11]. However, the solution of QP subproblems becomes a bottleneck in large scale applications, in specific when fine discretizations are needed. Exploitation of sparseness is then crucial. But general-purpose sparse solvers have several drawbacks: Analyzing the sparseness pattern is expensive, the fill-in is unknown a priori, and reasons for failure of a factorization may be hard to detect. Therefore we have developed a structured approach for control problems, based on the following general strategy. The nonlinear problem is set up in a way that reduces nonlinear coupling. Usually this means introducing additional variables and cond

An approach toward the automated solution of aircraft trajectory optimization problems is introduced and implemented in an interactive program called visual interactive aircraft trajectory optimization (VIATO). This MS Windowscompatible software produces minimum time trajectories to a fixed or moving target. It is easy to use by nonexperts as no previous knowledge of the methods of optimal control theory or mathematical modeling are needed. VIATO consists of a graphical user interface, an optimization server, and a model server. In VIATO, different aircraft types are represented by a set of parameters. The equations of motion and state as well as control constraints are fixed in advance. Since the objective function is also specified, the user avoids the modeling and explicit formulation of optimal control problems. Reliable convergence to an approximate optimal solution is achieved by converting the original optimal control problem into a finite dimensional optimization problem. The parameterized problem is solved using nonlinear programming.

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...

. This paper presents a numerical investigation of an optimal reentry maneuver under several control and control-state constraints. The essential aim of the optimization is the minimization of the maximal skin temperature of an orbiter. It is demonstrated that interaction of different solution techniques is indispensable in order to successfully treat such a highly constrained problem. The reduction of the skin temperature is significant. Moreover, the maximum heat flux and the integrated heat flux are also reduced considerably by the optimization. Key Words Minimax optimal control problems multiple control-state constraints multiple shooting direct collocation 1 1. Introduction Since the early sixties, space missions and especially manned space missions have been in the focus of common interest. In particular the launch phases and the reentry maneuvers are still receiving broad coverage in all kinds of media, since these maneuvers represent the most critical moments for man and m...

Optimal control problems which describe realistic technical applications exhibit various features of complexity. First, the consideration of inequality constraints leads to optimal solutions with highly complex switching structures including bang-bang, singular, and control- and state-constrained subarcs. In addition, also isolated boundary points may occur. Techniques are surveyed for the computation of optimal trajectories with multiple subarcs. If the precise computation of the switching structure holds the spotlight, the indirect multiple shooting method is top quality. Second, the differential equations describing the dynamics may be so complicated that they have to be generated by a computer program. In this case, direct methods such as direct collocation are generally superior. Third, the task is often given in applications to solve many optimal control problems, either for parameter homotopies in the course of the solution process itself or for sensitivity investigations of the...

We develop a numerically efficient algorithm for computing controls for nonlinear systems that minimize a quadratic performance measure. We formulate the optimal control problem in discrete-time, but many continuous-time problems can also be solved after discretization. Our approach is similar to sequential quadratic programming for finite-dimensional optimization problems in that we solve the nonlinear optimal control problem using sequence of linear quadratic subproblems. Each subproblem is solved efficiently using the Riccati difference equation. We show that each iteration produces a descent direction for the performance measure and that the sequence of controls converges to a solution that satisfies the well-known necessary conditions for the optimal control. We also show that the algorithm is a Gauss-Newton method, which means it inherits excellent convergence properties. We demonstrate the convergence properties of the algorithm with two numerical examples. 1

Abstract — The ability of birds to perch robustly and effectively is a powerful demonstration of the capabilities of nature’s control systems. Their apparent robustness to gust disturbances is particularly remarkable because when the airspeed approaches zero just before acquiring a perch, the influence of aerodynamic forces, and therefore potentially the control authority, is severely compromised. In this paper we present a simplified closed-form model for a fixed-wing aircraft which closely agrees with experimental indoor perching data. We then carefully examine the LTV controllability along an optimized perching trajectory for three different actuation scenarios- a glider (no powerplant), a fixed propeller, and a propeller with thrust vectoring. The results reveal that while all three vehicles are LTV controllable along the trajectory, the additional actuators allow the perch to be more easily acquired with less control surface deflections. However, in all three cases, disturbances experienced just before reaching the perch cannot be effectively rejected. I.