Raymond Holsapple

A modified simple shooting method for solving two-point boundary-value problems

Abstract : In the study of mechanics and optimal control, one often encounters what is called a two-point boundary-value problem (TPBVP). A couple of methods exist for solving these problems, such as the Simple Shooting Method (SSM) and its variation, the Multiple Shooting Method (MSM). In this paper a new method is proposed that was designed from the favorable aspects of both the SSM and the MSM. The Modified Simple Shooting Method (MSSM) sheds undesirable aspects of both previously mentioned methods to yield a superior, faster method for solving TPBVPs. The convergence of the MSSM is proven under mild conditions on the TPBVP. A comparison of the MSM and the MSSM is made for a problem where both methods converge. We also provide a second example where the MSM fails to converge while the MSSM converges rapidly.

New, Fast Numerical Method for Solving Two-Point Boundary-Value Problems

In this Note, we propose a class of sparse aperture interferometric satellite constellations in Earth orbit that can be used to observe astronomical bodies over the full celestial sphere. This observatory is capable of forming high-resolution images in timescales of a few hours, while completely covering the desired region of the wave number (u‐v) plane for a wide range of wavelengths. An optimization procedure is defined that supplies m pixels of resolution with a minimum number of satellites. A lower bound for the minimum number of spacecraft in the constellation is derived, and we show that for the example considered this procedure results in an observatory that is within 0‐2 satellites from this lower bound. The zonal J2 effect is used to scan the observatory across the celestial sphere. Finally, we discuss some practical implementation issues for these observatories.

Autonomous Decision Making with Uncertainty for an Urban Intelligence, Surveillance and Reconnaissance (ISR) Scenario

In this paper, we consider an urban ISR scenario where a human operator is tasked to provide feedback regarding the nature of some objects of interest. The feedback is relayed to the stochastic controller of an unmanned aerial vehicle (UAV), which must determine an appropriate mission plan. A small (unmanned) aerial vehicle (SAV) loiters at a high altitude where it may survey a large territory. An operator decides which objects in the SAV’s field of view are of interest and which are not. Then a team of micro (unmanned) aerial vehicles (MAVs) is assigned individual tours to inspect the objects of interest at a low altitude. As a MAV flies over an object of interest, the operator must decide if the object has a feature that uniquely distinguishes it as a target. The key parameters are the operator’s response and the time taken for the operator to respond. The controller uses these parameters to compute the expected information gain of a revisit. In previous studies automatic target recognition (ATR) was used for making some decisions in the SAV and the MAVs. This paper investigates the use of human feedback alone for target recognition. Different methods for calculating expected information gain are examined and compared. In addition, results from a flight test of this controller are presented.

Optimal Perimeter Patrol Alert Servicing with Poisson Arrival Rate

This paper addresses a base perimeter patrol scenario where alerts are generated from a set of stations at random intervals. A Unmanned Aerial Vehicle patrols the perimeter and responds to alerts. After arriving at an alert site, the vehicle loiters for a time to enable the operator to determine if the alert is a nuisance trip or an actual threat. The false alarms are modeled as a Poisson process. A stochastic control optimization problem is developed to determine the optimal loiter time. The optimal length of time that a vehicle can dwell at an alert site while minimizing the expected service time is a function of the size of the alert queue and the alert rate. Results from where the algorithm was ∞ight tested as part of a base defense scenario is presented.

A new method for the computation of motion from image sequences

The object of this paper is to introduce a new method for computing the linear velocity and angular velocity of an unmanned air vehicle (UAV) using only the information obtained from image sequences. In UAV applications, computational resources are limited due to payload constraints and the real-time computation requirement. Therefore, computationally intensive techniques employing feature extraction cannot be used. The alternative, in existing literature, is the computation of optical flow and the subsequent computation of motion. Both of these problems are ill-posed due to the correspondence and aperture problems. In this paper, we consider a different approach for motion estimation that is based on the spatial differentiation of an image function. We show that the solution is a well-posed problem that involves a least squares problem and nonlinear filtering. We also discuss the implementation of such a scheme on a UAV, and discuss the existence of such schemes in insects and crustaceans.

A variable step-size selection method for implicit integration schemes

Implicit integration schemes, such as Runge-Kutta methods, are widely used in mathematics and engineering to numerically solve ordinary differential equations. Every integration method requires one to choose a step-size, h, for the integration. If h is too large or too small the efficiency of an implicit scheme is relatively low. As every implicit integration scheme has a global error inherent to the scheme, we choose the total number of computations in order to achieve a prescribed global error as a measure of efficiency of the integration scheme. In this paper, we propose the idea of choosing h by minimizing an efficiency function for general Runge-Kutta integration routines. We show the efficacy of this approach on some standard problems found in the literature