On the Displacement for Covering a d−d-dimensional Cube with Randomly Placed Sensors

Consider $n$ sensors placed randomly and independently with the uniform distribution in a $d-$dimensional unit cube ($d\ge 2$). The sensors have identical sensing range equal to $r$, for some $r >0$. We are interested in moving the sensors from their initial positions to new positions so as to ensure that the $d-$dimensional unit cube is completely covered, i.e., every point in the $d-$dimensional cube is within the range of a sensor. If the $i$-th sensor is displaced a distance $d_i$, what is a displacement of minimum cost? As cost measure for the displacement of the team of sensors we consider the $a$-total movement defined as the sum $M_a:= \sum_{i=1}^n d_i^a$, for some constant $a>0$. We assume that $r$ and $n$ are chosen so as to allow full coverage of the $d-$dimensional unit cube and $a > 0$. The main contribution of the paper is to show the existence of a tradeoff between the $d-$dimensional cube, sensing radius and $a$-total movement. The main results can be summarized as follows for the case of the $d-$dimensional cube. If the $d-$dimensional cube sensing radius is $\frac{1}{2n^{1/d}}$ and $n=m^d$, for some $m\in N$, then we present an algorithm that uses $O\left(n^{1-\frac{a}{2d}}\right)$ total expected movement (see Algorithm 2 and Theorem 5). If the $d-$dimensional cube sensing radius is greater than $\frac{3^{3/d}}{(3^{1/d}-1)(3^{1/d}-1)}\frac{1}{2n^{1/d}}$ and $n$ is a natural number then the total expected movement is $O\left(n^{1-\frac{a}{2d}}\left(\frac{\ln n}{n}\right)^{\frac{a}{2d}}\right)$ (see Algorithm 3 and Theorem 7). In addition, we simulate Algorithm 2 and discuss the results of our simulations

Analysis of the Threshold for Energy Consumption in Displacement of Random Sensors

Consider $n$ mobile sensors placed randomly in $m-$dimensional unit cube for fixed $m\in\{1,2\}.$ The sensors have identical sensing range, say $r.$ We are interested in moving the sensors from their initial random positions to new locations so that every point in the unit cube is within the range of at least one sensor, while at the same time each pair of sensors is placed at interference distance greater or equal to $s.$ Suppose the displacement of the $i-$th sensor is a distance $d_i$. As a \textit{energy consumption} for the displacement of a set of $n$ sensors we consider the $a-$total displacement defined as the sum $\sum_{i=1}^n d_i^a,$ for some constant $a> 0.$ The main contribution of this paper can be summarized as follows. For the case of unit interval we \textit{explain a threshold} around the sensing radius equal to $\frac{1}{2n}$ and the interference distance equal to $\frac{1}{n}$ for the expected minimum $a-$total displacement. For the sensors placed in the unit square we \textit{explain a threshold} around the square sensing radius equal to $\frac{1}{2 \sqrt{n}}$ and the interference distance equal to $\frac{1}{\sqrt{n}}$ for the expected minimum $a-$total displacement

An emperical study of diffusion: The role of particle size, particle shape, gravity, and edge effects

The theories of Brownian motion and diffusion are often treated as separate idea, though Brownian motion is believed to cause diffusion. Both theories were initially studied separately in this project via two experiments; the theories were then studied in conjunction. The Brownian motion experiment examined milk fat globules and Cabosil particles in colloidal suspensions under a microscope, following their trajectory for 10 seconds. A flow model was developed to determine the diffusion coefficients. Drag coefficients revealed the geometry is correlated with the diffusion coefficients. The Diffusion experiment used a custom-made, thermally isolated chamber to study diffusion in the horizontal and vertical planes. Diffusion in the vertical direction took approximately 100 times longer than the horizontal. the two theories were then linked by using the diffusion data to determine the microscopic water molecule radius. The horizontal data agreed within uncertainty of the accepted value for the water molecule radius, while the vertical direction data did not. Gravity, particle geometry, and edge effects are suspected of affecting the rate of diffusion as suggested by its demonstrated influence on the Brownian motion of colloidal particles

Realization And Evaluation Of A 3-Degrees-Of-Freedom Mouse Model

Kinesiology research has shown that translation and rotation are inseparable actions in the real world. Motivated by this fact, this thesis explores a model for the computer mouse, the new addition being rotational input about vertical axis of a mouse. We realize our model through Mushaca, a 3-degrees-of-freedom mouse (3DOF mouse) that can sense rotation, in addition to sensing XY planar translation. The thesis presents two realizations of Mushaca - namely a MEMS version that uses accelerometer and gyroscope, and an optical sensor version that uses two optical sensors. Through a controlled user study we try to find out if that rotation is an useful input modality in pointing devices. The user study shows that in general rotation is a useful input modality, but it excels a standard mouse only in certain scenarios. Through the user study we also study the effect of the rotating coordinate system of the mouse and also how users adapt to this changing frame of reference through kinesthetic learning

Natural freehand grasping of virtual objects for augmented reality

Grasping is a primary form of interaction with the surrounding world, and is an intuitive interaction technique by nature due to the highly complex structure of the human hand. Translating this versatile interaction technique to Augmented Reality (AR) can provide interaction designers with more opportunities to implement more intuitive and realistic AR applications. The work presented in this thesis uses quantifiable measures to evaluate the accuracy and usability of natural grasping of virtual objects in AR environments, and presents methods for improving this natural form of interaction. Following a review of physical grasping parameters and current methods of mediating grasping interactions in AR, a comprehensive analysis of natural freehand grasping of virtual objects in AR is presented to assess the accuracy, usability and transferability of this natural form of grasping to AR environments. The analysis is presented in four independent user studies (120 participants, 30 participants for each study and 5760 grasping tasks in total), where natural freehand grasping performance is assessed for a range of virtual object sizes, positions and types in terms of accuracy of grasping, task completion time and overall system usability. Findings from the first user study in this work highlighted two key problems for natural grasping in AR; namely inaccurate depth estimation and inaccurate size estimation of virtual objects. Following the quantification of these errors, three different methods for mitigating user errors and assisting users during natural grasping were presented and analysed; namely dual view visual feedback, drop shadows and additional visual feedback when adding user based tolerances during interaction tasks. Dual view visual feedback was found to significantly improve user depth estimation, however this method also significantly increased task completion time. Drop shadows provided an alternative, and a more usable solution, to dual view visual feedback through significantly improving depth estimation, task completion time and the overall usability of natural grasping. User based tolerances negated the fundamental problem of inaccurate size estimation of virtual objects, through enabling users to perform natural grasping without the need of being highly accurate in their grasping performance, thus providing evidence that natural grasping can be usable in task based AR environments. Finally recommendations for allowing and further improving natural grasping interaction in AR environments are provided, along with guidelines for translating this form of natural grasping to other AR environments and user interfaces

Three variants Particle Swarm Optimization technique for optimal cameras network two dimensions placement

This paper addresses the problem of optimal placement in two-dimensions of the cameras network for the motion capture (MoCap) system. In fact, the MoCap system is a three- dimensional representation environment used mainly to reconstruct a real motion by using a number of fixed cameras (in position and pose). The main objective is to find the optimal placement of all cameras in a minimal time under a major constraint in order to capture each reflector that must be seen by at least three cameras in the same frame in a sequence of a random motion. The two-dimensional representation is only used to solve the problem of reflector recovery. The choice of two-dimensional representation is to reduce the resolution of a three- dimensional recovery problem to a simple two-dimensional recovery, especially if all the cameras have the same height. With this strategy, the placement of cameras network is not treated as an image processing problem. The use of three variants optimization techniques by Particle Swarm Optimization (Standard Particle Swarm Optimization, Weight Particle Swarm Optimization and Canonical Particle Swarm Optimization), allowed us to solve the problem of cameras network placement in a minimal amount of time. The overall recovery objective has been achieved despite the complexity imposed in the third scenario by the Canonical Particle Swarm Optimization variant