Browsing by Subject "Algebraic geometry"

Now showing 1 - 3 of 3
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    Motion induced robot-to-robot extrinsic calibration.
    (2012-05) Zhou, Xun
    Multi-robot systems, or mobile sensor networks, which have become increasingly popular due to recent advances in electronics and communications, can be used in a wide range of applications, such as space exploration, search and rescue, target tracking, and cooperative localization and mapping. In contrast to single robots, multi-robot teams are more robust against single-point failures, accomplish coverage tasks more efficiently by dispersing multiple robots into large areas, and achieve higher estimation accuracy by directly communicating and fusing their sensor measurements. Realizing these advantages of multi-robot systems, however, requires addressing certain challenges. Specifically, in order for teams of robots to cooperate, or fuse measurements from geographically dispersed sensors, they need to know their poses with respect to a common frame of reference. Initializing the robots' poses in a common frame is relatively easy when using GPS, but very challenging in the absence of external aids. Moreover, planning the motion of multiple robots to achieve optimal estimation accuracy is quite challenging. Specifically, since the estimation accuracy depends on the locations where the robots record their sensor measurements, it may take an extensive amount of time to reach a required level of accuracy, if the robots' motions are not properly designed. This thesis offers novel solutions to the aforementioned challenges. The first part of the thesis investigates the problem of relative robot pose initialization, using robot-to-robot distance and/or bearing measurements collected over multiple time steps. In particular, it focuses on solving minimal problems and proves that in 3D there exist only 14 such problems that need to be solved. Furthermore, it provides efficient algorithms for computing the robot-to-robot transformation, which exploit recent advances in algebraic geometry. The second part of the thesis investigates the problem of optimal motion strategies for localization in leader-follower formations using distance or bearing measurements. Interestingly, the robot-to-robot pose is unobservable if the robots move on a straight line and maintain their formations, hence, the uncertainty of the robots' poses increases over time. If the robots, however, deviate from the desired formation, their measurements provide additional information which makes the relative pose observable. This thesis addresses the trade-off between maintaining the formation and estimation accuracy, and provides algorithms for computing the optimal positions where the robots should move to in order to collect the most informative measurements at the next time step. By providing solutions to two important problems for multi-robot systems: motion-induced extrinsic calibration, and optimal motion strategies for relative localization, the work presented in this thesis is expected to promote the use of multi-robot teams in real-world applications.
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    On the super Mumford form in the presence of Ramond and Neveu-Schwarz punctures
    (2019-07) Diroff, Daniel
    We generalize the result of Voronov (1988) to give an expression for the super Mumford form on the moduli spaces of super Riemann surfaces with Ramond and Neveu–Schwarz punctures. In the Ramond case we take the number of punctures to be large compared to the genus. We consider for the case of Neveu-Schwarz punctures the super Mumford form over the component of the moduli space corresponding to an odd spin structure. The super Mumford form can be used to create a measure whose integral computes scattering amplitudes of superstring theory. We express it in terms of local bases of global sections of tensor powers of the Berezinian line bundle of a family of super Riemann surfaces.
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    Transceiver design and interference alignment in wireless networks: complexity and solvability
    (2013-11) Razaviyayn, Meisam
    This thesis aims to theoretically study a modern linear transceiver design strategy, namely interference alignment, in wireless networks. We consider an interference channel whereby each transmitter and receiver are equipped with multiple antennas. The basic problem is to design optimal linear transceivers (or beamformers) that can maximize the system throughput. The recent work [1] suggests that optimal beamformers should maximize the total degrees of freedom through the interference alignment equations. In this thesis, we first state the interference alignment equations and study the computational complexity of solving these equations. In particular, we prove that the problem of maximizing the total degrees of freedom for a given interference channel is NP-hard. Moreover, it is shown that even checking the achievability of a given tuple of degrees of freedom is NP-hard when each receiver is equipped with at least three antennas. Interestingly, the same problem becomes polynomial time solvable when each transmit/receive node is equipped with no more than two antennas.The second part of this thesis answers an open theoretical question about interference alignment on generic channels: What degrees of freedom tuples (d1, d2, ..., dK) are achievable through linear interference alignment for generic channels? We partially answer this question by establishing a general condition that must be satisfied by any degrees of freedom tuple (d1, d2, ..., dK) achievable through linear interference alignment. For a symmetric system with dk = d for all k, this condition implies that the total achievable DoF cannot grow linearly with K, and is in fact no more than K(M + N)/(K + 1), where M and N are the number of transmit and receive antennas, respectively. We also show that this bound is tight when the number of antennas at each transceiver is divisible by the number of data streams.