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Paul Bunyan's brachistochrone and tautochrone

Advances in the Astronautical Sciences

Hurtado, J.E.

In this paper we concern ourselves with modified versions of the traditional brachistochrone and tautochrone problems. In the modified version of each problem the constant gravity model is replaced with an attractive inverse square law, consequently we name these the 1/r2 brachistochrone and 1/r2 tautochrone problems. With regard to the 1/r2 brachistochrone problem, we show that the shape of the minimizing curve is formally constructed from an infinite series of elliptic integrals, and we use a numerical optimal control technique to generate the trajectories. The 1/r2 tautochrone problem is solved using fractional calculus techniques and we show that the solution satisfies Lagrange's rule for tautochronous curves.

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Distributed Sensing and Cooperating Control for Swarms of Robotic vehicles

Hurtado, J.E.

DISTRIBUTED SENSING AND COOPERATING CONTROL FOR SWARMS OF ROBOTIC VEHICLES Key words: Distributed Sensing, Cooperative Control. ABSTRACT We discuss an approach to effectively control a large swarm of autonomous, robotic vehicles, as they per- form a search and tag operation. In particular, the robotic agents are to find the source of a chemical plume. The robotic agents work together through dis- tributed sensing and cooperative control. Distributed sensing is achieved through each agent sampling and sharing his information with others. Cooperative con- trol h accomplished by each agent u-sing its neighbors information to determine an update strategy. INTRODUCTION There is currently considerable interest in expanding the role of robotic vehicles in surveillance and inspec- tion; searching, following and t aggir-g and locating and identifying targets. In particular, researchers are beginning to focus on using small autonomous robotic vehicles for these tasks. This focus has been brought about largely because of the many recent advances in microelectronics and sensors, which include small, low power, CCD cameras; small microprocessors with ex- panded capabilities; autonomous navigation systems using GPS; and severrd types of small sensors. It seems likely that these technological advances will lead to in- expensive, easy to fabricate, autonomous vehicles out- fitted with an array of sensors. This, in turn, will allow researchers to consider teams, or even swarms, of these agents to perform a particular task. It is natural then to wonder how one might effectively control a team, or even a swarm, of robotic agents. In this paper, we discuss an approach to effectively control a large swarm of autonomous, robotic vehicles as they perform a search and tag operation. In par- ticular, the robotic agents are to find the source of a chemical plume. The robotic agents work together through distributed sensing and cooperative control. Distributed sensing is achieved through each agent sampling and sharing his information with others. Co- operative control is accomplished by each agent using its neighbors information to determine a control (or TECHNICAL DEVELOPMENT In this section we highlight the technical development of our distributed sensing and cooperative control ap- proach to effectively control a large swarm of au- tonomous, robotic vehicles. Recall that the agents are tasked with locating the chemical plume source within a chemical plume field. In our simulations, we assume that the agents are outfitted with a GPS sensor, which provides their cur- rent location, and a chemical "sniffer," which allows them to detect the strength of the chemical plume at their current location. Furthermore, we assume that the robots have onboard processing capability, and are able to communicate with one another via RF modems together with bit packing and error correction tech- niques, like those discussed by Lewis et al [4]. Thus, each agent is able to communicate and share informa- tion with all others (i.e., there is global communica- tion). In this mode, at a particular instant in time, the agents sample the chemical plume field and post this information and their current location for the oth- ers. The agents then assemble the information and de- termine a projected target of where they believe the chemical source is located. The position update for each agent is then based upon its current position and the position of the projected target.

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2 Results
2 Results