Andrew J. Sinclair

Optimal and Feedback Path Planning for Cooperative Attack

This paper considers cooperative path planning for aerial munitions during the attack phase of a mission against ground targets. It is assumed that sensor information from multiple munitions is available to refine an estimate of the target location. Based on models of the munition dynamics and sensor performance, munition trajectories are designed that enhance the ability to cooperatively estimate the target location. The problem is posed as an optimal control problem using a cost function based on the variances in the target-location estimate. These variances are computed by fusing the individual munition measurements in a weighted least-squares estimate. Solutions to the problem are found using a direct-shooting method. These solutions are compared with trajectories developed by an alternative suboptimal feedback-guidance law. This feedback law produces solutions with far less numerical expense and with a performance very close to the best known solutions. The reduction in target-location uncertainty associated with these trajectories could enable the attack of targets with greater precision using smaller, cheaper munitions.

Decentralized Cooperative-Control Design for Multivehicle Formations

DOI: 10.2514/1.33009 In a decentralized cooperative-control regime, individual vehicles autonomously compute their required control inputs to achieve a group objective. Controlling formations of individual vehicles is one application of decentralized cooperative control. In this paper, cooperative-control schemes are developed for a multivehicle formation problem with information flow modeled by leader–follower subsystems. Control laws are developed to drive position and velocity errors between vehicle pairs to zero. The general control law for the ith vehicle tracks its lead vehicle’s position and velocity, as well as a reference position and velocity that the whole formation follows. Rate-estimation schemes are developed for the general control law using both Luenberger-observer and passive-filtering estimation methods. It is shown that these estimation methods are complicated by the effects that the estimated rates have on formation stability. Finally, the development of a rate-free controller is presented, which does not require state informationfromothervehiclesintheformation.Thecontrolschemesaresimulatedfora five-vehicleformationand are compared for stability and formation convergence.

On the generation of exact solutions for evaluating numerical schemes and estimating discretization error

Abstract In this paper we further develop the Method of Nearby Problems (MNP) for generating exact solutions to realistic partial differential equations by extending it to two dimensions. We provide an extensive discussion of the 2D spline fitting approach which provides Ck continuity (continuity of the solution value and its first k derivatives) along spline boundaries and is readily extendable to higher dimensions. A detailed one-dimensional example is given to outline the general concepts, then the two-dimensional spline fitting approach is applied to two problems: heat conduction with a distributed source term and the viscous, incompressible flow in a lid-driven cavity with both a constant lid velocity and a regularized lid velocity (which removes the strong corner singularities). The spline fitting approach results in very small spline fitting errors for the heat conduction problem and the regularized driven cavity, whereas the fitting errors in the standard lid-driven cavity case are somewhat larger due to the singular behaviour of the pressure near the driven lid. The MNP approach is used successfully as a discretization error estimator for the driven cavity cases, outperforming Richardson extrapolation which requires two grid levels. However, MNP has difficulties with the simpler heat conduction case due to the discretization errors having the same magnitude as the spline fitting errors.

Hamel coefficients for the rotational motion of an N–dimensional rigid body

Many of the kinematic and dynamic concepts relating to rotational motion have been generalized for N–dimensional rigid bodies. In this paper a new derivation of the generalized Euler equations is presented. The development herein of the N–dimensional rotational equations of motion requires the introduction of a new symbol, which is a numerical relative tensor, to relate the elements of an N Ö N skew–symmetric matrix to a vector form. This symbol allows the Hamel coefficients associated with general N–dimensional rotations to be computed. Using these coefficients, Lagranges equations are written in terms of the angular–velocity components of an N–dimensional rigid body. The new derivation provides a convenient vector form of the equations, allows the study of systems with forcing functions, and allows for the sensitivity of the kinetic energy to the generalized coordinates.

Application of the Cayley Form to General Spacecraft Motion

The study of N-dimensional rigid-body motion is a well-developed field of mechanics. Some of the key results for describing the kinematics of these bodies come from the Cayley transform and the Cayley-transform kinematic relationship. Additionally, several forms of the equations of motion for these bodies have been developed by various derivations. By using Cayley kinematics, the motion of general mechanical systems can be intimately related to the motion of higher-dimensional rigid bodies. This is done by associating each point in the configuration space with an N-dimensional orientation. An example of this is the representation of general orbital and attitude motion of a spacecraft as pure rotation of a four-dimensional rigid body. Another example is the representation of a multibody satellite system as a four-dimensional rigid body.

Calibration of Linearized Solutions for Satellite Relative Motion

F OR a wide variety of space missions, it is desirable to describe the motion of a “deputy” satellite relative to a “chief” satellite. Cartesian coordinates and orbital-element differences are two methods to describe this motion. Cartesian coordinates offer an advantage in direct, intuitive feel for the relative motion. The equations of motion for each description are nonlinear, but can be linearized for close proximity between the chief and deputy. The linearized equations offer an advantage in simplicity. This Note is focused on two observations. First, the linearized descriptions for Cartesian coordinates and orbital-element differences share an equivalence. Second, this equivalence can be used in a simple manner to calibrate the Cartesian initial conditions such that linearization error is greatly reduced comparedwith initial conditions calculated purely fromkinematics. The first observationwas reported by Sengupta and Vadali [1]. However, this observation appears to have been underappreciated, because the second observation follows directly from the first but has not been previously published. These calibrated initial conditions greatly increase the domain of validity of the linearized approximation and provide far greater accuracy in matching the nonlinear solution over a larger range of separations. The calibration process leverages the lower nonlinearity of the orbital-element differences, which was reported by Junkins et al. [2]. These observations are related to additional work in the literature describing the degree of nonlinearity in various parameterizations of a given system [3–6]. Much of that work, however, focused on calculating an index of nonlinearity directly from properties of the nonlinear representations. This Note is focused on the linearization error in comparing nonlinear and linearized forms of each representation. In the following sections, the nonlinear and linearized descriptions of relative motion in Cartesian coordinates and orbital-element differences are reviewed. Then, the equivalence of the linearized descriptions is shown. Finally, the calibration of the Cartesian initial conditions is described and several numerical examples are shown. In the following, parameters related to the deputy’s orbit about the central body are indicated with a subscript d. For notational compactness, parameters related to the chief’s orbit about the central body are left without subscript.

Linear Feedback Control Using Quasi Velocities

A novel approach for designing feedback controllers for natural mechanical systems using quasi velocities is presented. The approach is relevant to the stabilization and regulation of finite-dimensional multibody systems. In particular the globally asymptotically stable linear feedback of Rodrigues parameters and angular velocity is taken from spacecraft attitude control and applied to a broader class of problems. This controller is shown to have good performance due to the sensitivity of the rotational kinematics to small motions near the reference configuration. Additionally, the concept is extended to systems with more than three degrees of freedom by generalizing the functional form of the three-dimensional rotational kinematics. This defines a new set of quasi velocities that allow globally asymptotically stable linear feedback for any number of degrees of freedom. Anexample shows that the use of these variables in controller design can lead to improved performance.

Simultaneous Localization and Planning for Cooperative Air Munitions

This chapter considers the cooperative control of aerial munitions during the attack phase of a mission against ground targets. It is assumed that sensor information from multiple munitions is available to refine an estimate of the target location. Based on models of the munition dynamics and sensor performance, munition trajectories are designed that enhance the ability to cooperatively estimate the target location. The problem is posed as an optimal control problem using a cost function based on the variances in the target-location estimate. These variances are computed by fusing the individual munition measurements in a weighted least squares estimate. Numerical solutions are found for several examples both with and without considering limitations on the munitions’ field of view. These examples show large reductions in target-location uncertainty when these trajectories are used compared to other naively designed trajectories. This reduction in uncertainty could enable the attack of targets with greater precision using smaller, cheaper munitions.

Nonlinearity index of the cayley form

The nonlinearity index is a measure of the nonlinearity of dynamical systems based on computing the initial-condition sensitivity of the state-transition matrix. The Cayley form is a representation for dynamical systems that relates their motion to N-dimensional rotations. The generalized coordinates of the system are used to define an N-dimensional orientation, and a set of quasi velocities is defined equal to the corresponding angular velocity. The nonlinearity index of the Cayley-form representation is computed for an elastic spherical pendulum and a planar satellite example. These results are compared to values for alternative dynamical representations. Additionally, the nonlinearity is evaluated by analyzing how well the linearized equations of each representation capture certain properties of the motion. These results show that the Cayley form can have lower nonlinearity than traditional representations, in particular those representations that suffer from kinematic singularities.

Optimization Of Spacecraft Pursuit-Evasion Game Trajectories In The Euler-Hill Reference Frame

Spacecraft pursuit-evasion game is formulated as a two-player zero-sum differential game. The Euler-Hill reference frame is used to describe the dynamics of the game. The goal is to derive feedback control law which forms a saddle point solution to the game. Both spacecraft use continuous thrust engines. The standard linear-quadratic differential game theory is applied to obtain linear control law. The state-dependent Riccati equation method is applied to extend the standard linear-quadratic differential game theory to obtain nonlinear control law. The efficacy of the nonlinear control law is found to be superior to that of the linear control law.