CATCH ME IF YOU CAN!

Learning mathematics outside the classroom is not enrichment, it is at the core of empowering an individuals understanding of the subject. The three activities described in this article can all be used outside the classroom in a maths lesson. Teaching mathematical concepts in this way engages and reinforces learning. It puts the ideas learnt into a setting and allows time for those ideas to be developed without any of the maths hang-ups which can occur in the classroom. By taking maths beyond the classroom, we can more clearly illustrate the connections between the real world and what they are studying in school. In so doing students and teachers alike are enthused by the wealth of resources they have all around them in their own environments

Space pirates: a pursuit curve problem involving retarded time

We revisit the classical pursuit curve problem solved by Pierre Bouguer in the 18th century, taking into account that information propagates at a finite speed. To a certain extent, this could be seen as a relativistic correction to that problem, though one does not need Einstein's theory of relativity in order to derive or understand its solution. The discussion of this generalized problem of pursuit constitutes an excellent opportunity to introduce the concept of retarded time without the complications inherent to the study of electromagnetic radiation (where it is usually seen for the first time), which endows the problem with a clear pedagogical motivation. We find the differential equation which describes the problem, solve it numerically, compare the solution to Bouguer's for different values of the parameters, and deduce a necessary and sufficient condition for the pursuer to catch the pursued, complementing previous work by Hoenselaers.Comment: 15 pages, 7 figures, submitted to the American Journal of Physic

Optimal strategies for driving a mobile agent in a guidance by repulsion model

We present a guidance by repulsion model based on a driver-evader interaction where the driver, assumed to be faster than the evader, follows the evader but cannot be arbitrarily close to it, and the evader tries to move away from the driver beyond a short distance. The key ingredient allowing the driver to guide the evader is that the driver is able to display a circumvention maneuver around the evader, in such a way that the trajectory of the evader is modified in the direction of the repulsion that the driver exerts on the evader. The evader can thus be driven towards any given target or along a sufficiently smooth path by controlling a single discrete parameter acting on driver's behavior. The control parameter serves both to activate/deactivate the circumvention mode and to select the clockwise/counterclockwise direction of the circumvention maneuver. Assuming that the circumvention mode is more expensive than the pursuit mode, and that the activation of the circumvention mode has a high cost, we formulate an optimal control problem for the optimal strategy to drive the evader to a given target. By means of numerical shooting methods, we find the optimal open-loop control which reduces the number of activations of the circumvention mode to one and which minimizes the time spent in the active~mode. Our numerical simulations show that the system is highly sensitive to small variations of the control function, and that the cost function has a nonlinear regime which contributes to the complexity of the behavior of the system, so that a general open-loop control would not be of practical interest. We then propose a feedback control law that corrects from deviations while preventing from an excesive use of the circumvention mode, finding numerically that the feedback law significantly reduces the cost obtained with the open-loop control

Chasing Puppies: Mobile Beacon Routing on Closed Curves

We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others' work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time.Comment: Full version of a SOCG 2021 paper, 28 pages, 27 figure

Pursuit problems: basic models and their analysis

Tato práce se zabývá sestavením modelů a analýzou základních úloh o pronásledování. Konkrétně je zde obsažena Perraultova úloha, Bouguerova úloha, Hathawayova úloha, úloha o hlemýždi na popruhu a Baileyho úloha. Všechny úlohy jsou opatřeny analytickým nebo numerickým řešením, včetně kvalitativní analýzy.This thesis is focused on creation of mathematical models and analysis of basic pursuit problems. In particular, the Perrault's problem, the Bouguer's problem, the Hathaway’s problem, the snail–racehorse problem and the Bailey's problem are involved. Each of the problems is solved analytically or numerically, including qualitative analysis.

Fast interceptor of a dynamic object

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 99-100).This thesis presents a path planning and control strategy that enables an unmanned non-holonomic vehicle to intercept a fast moving object. The path planning is performed under model uncertainty, with respect to the vehicle's maneuverability, as well as uncertainty in the estimation of the object's future trajectory and position. This problem involves the tracking of the dynamic object in a cluttered environment and the accurate estimation of its future position in the presence of noisy measurements. The ground vehicle (interceptor) is required to intercept the dynamic object at a predicted (catch) location in a finite amount of time. This time restriction presents quite a challenge given the inherent limitation in the vehicle's steering and maneuverability. The solution strategy is divided into three sub-problems: 1) prediction, 2) path planning and 3) control. The prediction of the parameters that describe the dynamic's object in space is accomplished via Kalman Filtering which, in conjunction with an impact predictor, provide the waypoints needed to construct a reference path that will place the interceptor on a collision course with the dynamic object (target.) A pure pursuit algorithm was used to steer the interceptor along a reference trajectory, which was designed to make the vehicle engage the dynamic object on a near tail-on aspect. In the endgame, the pure pursuit algorithm was modified to ensure arrival to the catch point while a position controller was added to ensure timely arrival to the predicted catch location. The problem statement was then augmented to include obstacle avoidance.(cont.) The dynamic object was required to navigate around fixed obstacles in order to catch the dynamic object. Results will show that the proposed strategy performed very well in the absence of obstacles and was well suited to handle the maneuverability constraints of the non-holonomic vehicle. Results also will show that, with minor modifications to the path planner, the interceptor successfully managed to avoid obstacles and catch the dynamic object although at a slightly lower success rate. The proposed solution was first demonstrated in simulation and then tested using MIT's RAVEN testbed.by Sergio A. Cafarelli.S.M

Implementation and Assessment of State Estimation Algorithms in Simulation and Real-World Applications

Title from PDF of title page viewed July 12, 2021Thesis advisor: Travis FieldsVitaIncludes bibliographical references (pages 86-91)Thesis (M.S.)--School of Computing and Engineering. University of Missouri--Kansas City, 2021State estimation algorithms are important mathematical tools for engineers and are capable of improving system modeling capabilities and scenario outcomes. Typically, systems utilize a variety of sensors to measure certain system states such as velocity or position; however, these sensors suffer from noise and biases which contaminate the states. Through implementing a state estimation algorithm, noise from low-cost sensors may be mitigated to provide better system state estimates. Thus, a need exists to apply these algorithms while using low-cost sensors and to assess the performance in different scenarios. The algorithms that were selected for analysis in this research were the Kalman Filter and Extended Kalman Filter, which were implemented in two separate experiments. The first experiment encompassed a pursuer-evader scenario where the initial starting position of a pursuer and tracking measurement uncertainty of an evader were varied. Position of the evader was determined through two methods: raw tracking sensor measurements and estimates from a Kalman Filter. In both cases, the tracking sensor uncertainty was parameterized as a single term to represent combined uncertainty from all possible noise sources. This experiment showed that an increase in sensor measurement uncertainty led to an increase in the mean miss distance for the pursuer for both the raw tracking sensor method and Kalman Filter method. However, most engagement resulted in the Kalman Filter method providing an improved position estimate of the evader, reducing the average miss distance by upwards of 50%. In the second experiment, an Extended Kalman Filter was applied to an aircraft that experienced a multitude of free-flight hardware failures such as control surface and aerial delivery failures. The Extended Kalman Filter was designed to estimate the aircraft’s stability and control derivatives along with the aircraft’s dynamic states. Fixed-position aileron failures, ranging from -30° to 30°, were assessed and showed a loss of effectiveness in the control derivative state estimates. An aerial delivery failure from a bay located near the center of gravity resulted in small changes in some of the control derivatives and noise characteristics of the aircraft; however, a payload release failure from a bay located on the wing resulted in every state estimate changing. Post-failure state estimate changes indicate the potential of implementing fault isolation control schemes to mitigate the failure after initial detection. This research explored the usage of applying two state estimation algorithms to expand the modeling capabilities of two systems. Not only did the algorithms provide improved estimates for system states, the states that did not have direct sensor measurements were accurately estimated. Furthermore, the algorithms were tailored to each scenario and successfully utilized low-cost sensors to improve the scenario results.Introduction -- Literature review -- State estimation in a tracking-based pursuer-evader scenario -- Assessment of fixed-winged UAV system identification models during actuator and payload drop failures -- Conclusions -- Appendix A. Additional result