Aaron Becker

June 10th, 2013

Aaron Becker

Biography

Aaron Becker’s passion is robotics and control. Aaron completed his PhD in the Electrical & Computer Engineering department at the University of Illinois at Urbana-Champaign August 2013.
 His dissertation, Ensemble Control of Robotic Systems, derived an approximate steering algorithm for two classical robotic systems—the nonholonomic unicycle and the plate-ball manipulator—in the presence of model perturbation that scales all inputs by an unknown but bounded constant.

Aaron wanted a robot ever since the day in first grade he wrote a badly spelled letter to Santa Claus requesting one.  In the RMS lab that dream was lived out daily.   His focus is on techniques for handling uncertainty in robotics. Applications include manipulating robotic swarms, directing self-assembled microbots, robotic coverage, and state-estimation for novel prosthesis.

Projects

Ensemble Control of Robotic Systems (main page)

A differential-drive robot that thinks it has smaller wheels than it actually does Six steel balls of decreasing diameters with the letters "ORIENT" written one letter per ball 6 differential-drive robots that all receive exactly the same control inputs

Force Field Control (main page)

Robotic Coverage under Uncertainty (main page)

Self-Assembled Swimmers (main page)

Medical orthotics and state estimation of a human walker and orthotic system

Master’s thesis:  particle filtering, Gaussian-processes, dead-reckoning

Publications

Journal Publications

  • A. Becker and T. Bretl, "Approximate Steering of a Unicycle Under Bounded Model Perturbation Using Ensemble Control," IEEE_J_RO, vol. 28, iss. 3, pp. 580-591, 2012.
    [bibtex] [pdf] [video] [code]

    @article{Becker2012, Abstract = {This paper considers the problem of steering a nonholonomic unicycle despite model perturbation that scales both the forward speed and the turning rate by an unknown but bounded constant. We model the unicycle as an ensemble control system, show that this system is ensemble controllable, and derive an approximate steering algorithm that brings the unicycle to within an arbitrarily small neighborhood of any given Cartesian position. We apply our work to a differential-drive robot with unknown but bounded wheel radius, and validate our approach with hardware experiments.},
      Author = {Aaron Becker and Timothy Bretl},
      Date-Added = {2012-02-25 14:24:49 -0600},
      Date-Modified = {2012-06-18 23:34:00 -0500},
      Journal = IEEE_J_RO, Month = June, Number = {3},
      Pages = {580--591},
      Pdf = {Becker2012.pdf},
      Title = {Approximate Steering of a Unicycle Under Bounded Model Perturbation Using Ensemble Control},
      Video = {http://www.youtube.com/watch?v=8yYD_KMwfaM},
      Code = {http://www.mathworks.com/matlabcentral/fileexchange/38502-approximate-steering-of-a-unicycle-under-bounded-model-perturbation-using-ensemble-control},
      Volume = {28},
      Year = {2012}
    }
  • D. Y. Li, A. Becker, A. Shorter, T. Bretl, and E. T. Hsiao-Wecksler, "Estimating System State During Human Walking With a Powered Ankle-Foot Orthosis," IEEE/ASME Transactions on Mechatronics, vol. 16, iss. 5, pp. 835-844, 2011.
    [bibtex] [pdf] [video]

    @article{David-Yifan-Li2011, Abstract = {This paper presents a state estimator that reliably detects gait events during human walking with a portable powered ankle-foot orthosis (AFO), based only on measurements of the ankle angle and of contact forces at the toe and heel. Effective control of the AFO critically depends on detecting these gait events. A common approach detects gait events simply by checking if each measurement exceeds a given threshold. Our approach uses cross correlation between a window of past measurements and a learned model to estimate the configuration of the human walker, and detects gait events based on this estimate. We tested our approach in experiments with five healthy subjects and with one subject that had neuromuscular impairment. Using motion capture data for reference, we compared our approach to one based on thresholding and to another common one based on k-nearest neighbors. The results showed that our approach reduced the RMS error by up to 40% for the impaired subject and up to 49% for the healthy subjects. Moreover, our approach was robust to perturbations due to changes in walking speed and to control actuation.},
      Annote = {Li, D.; Becker, A.; Shorter, K.A.; Bretl, T.; Hsiao-Wecksler, E.A.; , "Estimating System State During Human Walking With a Powered Ankle-Foot Orthosis," Mechatronics, IEEE/ASME Transactions on , vol.16, no.5, pp.835-844, Oct. 2011 doi: 10.1109/TMECH.2011.2161769 URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5985534&isnumber=6006642 },
      Author = {David Yifan Li and Aaron Becker and Alex Shorter and Timothy Bretl and Elizabeth T. Hsiao-Wecksler},
      Date-Added = {2012-02-25 14:24:49 -0600},
      Date-Modified = {2012-02-25 14:24:49 -0600},
      Journal = {IEEE/ASME Transactions on Mechatronics},
      Month = {Oct},
      Number = {5},
      Pages = {835--844},
      Pdf = {Li2011.pdf},
      Title = {Estimating System State During Human Walking With a Powered Ankle-Foot Orthosis},
      Video = {http://www.youtube.com/watch?v=oWtR8n2EwOQ},
      Volume = {16},
      Year = {2011}
    }

Dissertation and Thesis

  • A. Becker, "Ensemble Control of Robotic Systems," PhD Thesis , 2012.
    [bibtex] [pdf] [video]

    @phdthesis{Becker2012a, Abstract = {In this dissertation, we apply the framework of ensemble control theory to derive an approximate steering algorithm for two classical robotic systems---the nonholonomic unicycle and the plate-ball manipulator---in the presence of model perturbation that scales all inputs by an unknown but bounded constant. The basic idea is to maintain the set of all possible configurations and to select inputs that reduce the size of this set and drive it toward some goal configuration. The key insight is that the evolution of this set can be described by a family of control systems that depend continuously on the unknown constant. Ensemble control theory provides conditions under which it is possible to steer this entire family to a neighborhood of the goal configuration with a single open-loop input trajectory. For both the nonholonomic unicycle and the plate-ball manipulator, we show how to construct this trajectory using piecewise-constant inputs. We also validate our approach with hardware experiments, where the nonholonomic unicycle is a differential-drive robot with unknown wheel size, and the plate-ball manipulator is a planar motion stage that uses magnetic actuation to orient a sphere of unknown radius. We conclude by showing how the same framework can be applied to feedback control of multi-robot systems under the constraint that every robot receives exactly the same control input. We focus on the nonholonomic unicycle, instantiated in experiment by a collection of differential-drive robots. Assuming that each robot has a unique wheel size, we derive a globally asymptotically stabilizing feedback control policy. We show that this policy is robust to standard models of noise and scales to an arbitrary number of robots. These results suggest that our approach may have possible future application to control of micro- and nano-scale robotic systems, which are often subject to similar constraints. },
      Author = {Aaron Becker},
      Date-Added = {2012-06-11 21:48:36 -0500},
      Date-Modified = {2012-06-11 21:48:36 -0500},
      Month = Aug, School = {University of Illinois at Urbana-Champaign},
      Title = {Ensemble Control of Robotic Systems},
      Pdf = {Becker2012a.pdf},
      Video = {http://www.youtube.com/watch?v=_v4KUBfzbv0},
      Year = {2012}
    }
  • A. Becker, "Mobile Robot Motion-Planning Using Wireless Signals for Localization," Master's Dissertation , 2008.
    [bibtex] [pdf] [video]

    @mastersthesis{Becker2008, Abstract = {Many applications for autonomous agents require the agent to have accurate position data. Signal-strength based localization attempts to determine an agent's location in {R^n} from a scalar sensor reading z. This is a nontrivial problem because the mapping from {R^n} to z is noninvertible. Indoor Localization attempts to solve this problem in an indoor environment. This adds challenges. Among these are the effects of signal interference; signal dropout due to walls, doors, and humans; and the expense of data collection. This project implements a localization system on an autonomous system, the Seg-Monster. The SegMonster is a human-sized robot that rides a commercial Segway personal transporter. Local data from wheel-mounted encoders are combined with signal-strength based localization that interprets wireless signal strength using Gaussian processes to calculate a global position estimate.},
      Author = {Aaron Becker},
      Date-Added = {2012-02-25 14:24:34 -0600},
      Date-Modified = {2012-02-25 14:24:34 -0600},
      Pdf = {Becker2008.pdf},
      School = {University of Illinois at Urbana-Champaign},
      Title = {Mobile Robot Motion-Planning Using Wireless Signals for Localization},
      Video = {http://www.youtube.com/watch?v=TTtliHC7siY},
      Year = {2008}
    }

Peer-Reviewed Conference Publications

  • A. Becker and T. Bretl, "Approximate Steering of a Plate-Ball System Under Bounded Model Perturbation Using Ensemble Control," in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2012. Accepted.
    [bibtex] [pdf] [video]

    @inproceedings{Becker2012j, Abstract = {In this paper we revisit the classical plate-ball system and prove this system remains controllable under model perturbation that scales the ball radius by an unknown but bounded constant. We present an algorithm for approximate steering and validate the algorithm with hardware experiments. To perform these experiments we introduce a new version of the plate-ball system based on magnetic actuation. This system is easy to implement and, with our steering algorithm, enables simultaneous manipulation of multiple balls with different radii.},
      Author = {Aaron Becker and Timothy Bretl},
      Booktitle = {IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)},
      Date-Added = {2012-06-20 10:40:55 -0500},
      Date-Modified = {2012-07-09 10:44:40 -0500},
      Month = oct, Title = {Approximate Steering of a Plate-Ball System Under Bounded Model Perturbation Using Ensemble Control},
      Year = {2012. Accepted},
      Pdf={Becker2012j.pdf},
      Video={http://www.youtube.com/watch?v=nPGz0Nd3QzE},
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  • C. Orduño, A. Becker, and T. Bretl, "Motion Primitives for Path Following with a Self-Assembled Robotic Swimmer," in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2012. Accepted.
    [bibtex] [pdf] [video]

    @inproceedings{Carlos-Orduno2012, Abstract = {This paper presents a control strategy based on model learning for a self-assembled robotic ``swimmer''. The swimmer forms when a liquid suspension of ferro-magnetic micro-particles and a non-magnetic bead are exposed to an alternating magnetic field that is oriented perpendicular to the liquid surface. It can be steered by modulating the frequency of the alternating field. We model the swimmer as a unicycle and learn a mapping from frequency to forward speed and turning rate using locally-weighted projection regression. We apply iterative linear quadratic regulation with a receding horizon to track motion primitives that could be used for path following. Hardware experiments validate our approach.},
      Author = {Carlos Ordu{\~n}o and Aaron Becker and Timothy Bretl},
      Booktitle = {IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)},
      Date-Added = {2012-06-20 10:44:57 -0500},
      Date-Modified = {2012-07-09 10:45:05 -0500},
      Month = oct, Title = {Motion Primitives for Path Following with a Self-Assembled Robotic Swimmer},
      Year = {2012. Accepted},
      Pdf = {Carlos-Orduno2012.pdf},
      Video = {http://www.youtube.com/watch?v=dMqPhriLXnA}
    }
  • A. Becker, C. Onyuksel, and T. Bretl, "Feedback Control of Many Differential-Drive Robots with Uniform Control Inputs," in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2012. Accepted.
    [bibtex] [pdf] [video] [code]

    @inproceedings{Becker2012k, Abstract = {In this paper, we derive a globally asymptotically stabilizing feedback control policy for a collection of differential-drive robots under the constraint that every robot receives exactly the same control inputs. We begin by assuming that each robot has a slightly different wheel size, which scales its forward speed and turning rate by a constant that can be found by offline or online calibration. The resulting feedback policy is easy to implement, is robust to standard models of noise, and scales to an arbitrary number (even a continuous ensemble) of robots. We validate this policy with hardware experiments, which additionally reveal that our feedback policy still works when the wheel sizes are unknown and even when the wheel sizes are all approximately identical. These results have possible future application to control of micro- and nano-scale robotic systems, which are often subject to similar constraints.},
      Author = {Aaron Becker and Cem Onyuksel and Timothy Bretl},
      Booktitle = {IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)},
      Date-Added = {2012-06-20 10:43:59 -0500},
      Date-Modified = {2012-07-09 10:44:15 -0500},
      Month = oct, Title = {Feedback Control of Many Differential-Drive Robots with Uniform Control Inputs},
      Year = {2012. Accepted},
      Pdf = {Becker2012k.pdf},
      Video={http://www.youtube.com/watch?v=50gb5WMqJbY},
      Code={http://www.mathworks.com/matlabcentral/fileexchange/38369-feedback-control-of-many-differential-drive-robots-with-uniform-control-inputs},
      Bdsk-File-1 = {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}
    }
  • C. Das, A. Becker, and T. Bretl, "Probably Approximately Correct Coverage for Robots with Uncertainty," in Intelligent Robots and Systems, 2011. IROS 2011. IEEE/RSJ International Conference on, 2011, pp. 12-13.
    [bibtex] [pdf] [video]

    @conference{Das2011, Abstract = {The classical problem of robot coverage is to plan a path that brings a point on the robot within a fixed distance of every point in the free space. In the presence of significant uncertainty in sensing and actuation, it may no longer be possible to guarantee that the robot covers all of the free space all the time, and so it becomes unclear what problem we are trying to solve. We will restore clarity by adopting a ``probably approximately correct'' measure of performance that captures the probability 1-delta" of covering a fraction 1-epsilon of the free space. The problem of coverage for a robot with uncertainty is then to plan a feedback policy that achieves a given value of epsilon and delta. Just as solutions to the classical problem are judged by the resulting path length, solutions to our problem are judged by the required execution time. We will show the practical utility of our performance measure by applying it to several examples in simulation.},
      Author = {Colin Das and Aaron Becker and Timothy Bretl},
      Booktitle = {Intelligent Robots and Systems, 2011. IROS 2011. IEEE/RSJ International Conference on},
      Date-Added = {2012-02-25 14:24:49 -0600},
      Date-Modified = {2012-02-25 14:24:49 -0600},
      Pages = {12-13},
      Pdf = {Das2011.pdf},
      Title = {Probably Approximately Correct Coverage for Robots with Uncertainty},
      Video = {http://www.youtube.com/watch?v=5e3zz8ezxXo},
      Volume = {1},
      Year = {2011}
    }
  • I. R. D. Pablo, A. Becker, and T. Bretl, "An optimal solution to the linear search problem for a robot with dynamics," in Intelligent Robots and Systems (IROS), 2010 IEEE/RSJ International Conference on, 2010, pp. 652-657.
    [bibtex] [pdf] [doi] [video]

    @inproceedings{Pablo2010, Abstract = {In this paper we derive the control policy that minimizes the total expected time for a point mass with bounded acceleration, starting from the origin at rest, to find and return to an unknown target that is distributed uniformly on the unit interval. We apply our result to proof-of-concept hardware experiments with a planar robot arm searching for a metal object using an inductive proximity sensor. In particular, we show that our approach easily extends to optimal search along arbitrary curves, such as raster-scan patterns that might be useful in other applications like robot search-and-rescue.},
      Author = {Irene Ruano De Pablo and Aaron Becker and Timothy Bretl},
      Booktitle = {Intelligent Robots and Systems (IROS), 2010 IEEE/RSJ International Conference on},
      Date-Added = {2012-02-25 14:24:49 -0600},
      Date-Modified = {2012-02-25 14:24:49 -0600},
      Doi = {10.1109/IROS.2010.5653185},
      Keywords = {arbitrary curves;inductive proximity sensor;linear search problem;planar robot arm;point mass;proof-of-concept hardware experiments;raster-scan patterns;robot search-and-rescue;emergency services;robot dynamics;search problems;sensors;service robots;},
      Month = {oct},
      Pages = {652--657},
      Pdf = {De-Pablo2010.pdf},
      Title = {An optimal solution to the linear search problem for a robot with dynamics},
      Video = {http://www.youtube.com/watch?v=HVxfTqSy1RA},
      Year = {2010},
      Bdsk-Url-1 = {http://dx.doi.org/10.1109/IROS.2010.5653185}
    }
  • A. Becker and T. Bretl, "Motion planning under bounded uncertainty using ensemble control," in Robotics: Science and Systems (RSS), 2010.
    [bibtex] [pdf] [video]

    @inproceedings{Becker2010, Abstract = {This paper considers the problem of motion planning for a nonholonomic unicycle despite uncertainty that scales both the forward speed and the turning rate by an unknown but bounded constant. We model the unicycle as an ensemble control system, show that the position of this ensemble is controllable, and derive motion planning algorithms to steer this position between a given start and goal. We apply our work to a differential-drive robot with unknown but bounded wheel radius, and validate our approach with hardware experiments.},
      Author = {Aaron Becker and Timothy Bretl},
      Booktitle = {Robotics: Science and Systems (RSS)},
      Date-Added = {2010-08-30 11:42:26 -0500},
      Date-Modified = {2010-08-30 13:59:31 -0500},
      Month = {February},
      Pdf = {Becker2010.pdf},
      Title = {Motion planning under bounded uncertainty using ensemble control},
      Url = {http://www.roboticsproceedings.org/rss06/p38.pdf},
      Video = {http://www.youtube.com/watch?v=8yYD_KMwfaM},
      Year = {2010},
      Bdsk-Url-1 = {http://www.roboticsproceedings.org/rss06/p38.pdf}
    }
  • A. Becker, R. Sandheinrich, and T. Bretl, "Automated manipulation of spherical objects in three dimensions using a gimbaled air jet," in Intelligent Robots and Systems, 2009. IROS 2009. IEEE/RSJ International Conference on, 2009, pp. 781-786.
    [bibtex] [pdf] [doi] [video]

    @inproceedings{Becker2009, Abstract = {This paper presents a mechanism and a control strategy that enables automated non-contact manipulation of spherical objects in three dimensions using air flow, and demonstrates several tasks that can be performed with such a system. The mechanism is a 2-DOF gimbaled air jet with a variable flow rate. The control strategy is feedback linearization based on a classical fluid dynamics model with state estimates from stereo vision data. The tasks include palletizing, sorting, and ballistics. All results are verified with hardware experiments.},
      Author = {Becker, A. and Sandheinrich, R. and Bretl, T.},
      Booktitle = {Intelligent Robots and Systems, 2009. IROS 2009. IEEE/RSJ International Conference on},
      Doi = {10.1109/IROS.2009.5354427},
      Keywords = {2-DOF gimbaled air jet;air flow;automated noncontact manipulation;ballistics;classical fluid dynamics model;feedback linearization;palletizing;sorting;spherical objects;state estimation;stereo vision data;variable flow rate;feedback;fluid dynamics;manipulator dynamics;state estimation;},
      Month = {oct.},
      Pages = {781 -786},
      Pdf = {Becker2009.pdf},
      Title = {Automated manipulation of spherical objects in three dimensions using a gimbaled air jet},
      Url = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5354427},
      Video = {http://www.youtube.com/watch?v=HkhMCCOHFmM},
      Year = {2009},
      Bdsk-Url-1 = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5354427},
      Bdsk-Url-2 = {http://dx.doi.org/10.1109/IROS.2009.5354427}
    }
  • T. T. Dang, T. D. Vu, and A. Becker, "A lossless coding scheme for images using cross-point regions for modeling," in Electro/Information Technology, 2007 IEEE International Conference on, 2007, pp. 96-101.
    [bibtex] [pdf] [doi]

    @inproceedings{Dang2007, Abstract = {This paper presents CRIC (Cross-Point Regions for Lossless Image Compression), a scheme for losslessly encoding and decoding images, especially medical images, by optimizing on the probability of cross points that neighbor points of grey levels 2n. The base of this statement is the effect of Gray coding on cross points. At first, the effect of Gray codes is determined on an adjacent data set because images characteristically contain data that does not change much in a specific area; then this effect is generalized for real data without losing generality for their statistical properties. This is especially true for medical images that have many regions with the same grey levels. The Gray code transformation makes the bit states of cross points change from the original data bits, so first the probabilities of data bits on specific bit planes in cross point regions and then the entropies of the messages are changed. These probabilities are estimated and compared with the probabilities of the original data bits. This change of probability has important effects on the encoding and decoding processes in lossless medical image compression.},
      Author = {Dang, T.T. and Vu, T.D. and Becker, A.},
      Booktitle = {Electro/Information Technology, 2007 IEEE International Conference on},
      Doi = {10.1109/EIT.2007.4374440},
      Keywords = {Gray coding;cross-point regions;grey levels;lossless coding scheme;lossless image compression;medical image compression;data compression;image coding;},
      Month = {may.},
      Pages = {96 -101},
      Pdf = {Becker2007.pdf},
      Title = {A lossless coding scheme for images using cross-point regions for modeling},
      Url = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4374440},
      Year = {2007},
      Bdsk-Url-1 = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4374440},
      Bdsk-Url-2 = {http://dx.doi.org/10.1109/EIT.2007.4374440}
    }

Peer-Reviewed Software

  • A. Becker, ``Rolling a Sphere around a Circle without Slipping'', Wolfram Demonstrations Project, 2012.
    [bibtex] [video]

    @misc{Becker2012i,
      author = {Aaron Becker},
      Date-Added = {2012-06-18 23:52:16 -0500},
      Date-Modified = {2012-06-18 23:52:16 -0500},
      Month = jan, Title = {{``Rolling a Sphere around a Circle without Slipping'', Wolfram Demonstrations Project}},
      Url = {http://demonstrations.wolfram.com/RollingASphereAroundACircleWithoutSlipping/},
      Video = {http://demonstrations.wolfram.com/RollingASphereAroundACircleWithoutSlipping/},
      Year = {2012},
      Bdsk-Url-1 = {http://demonstrations.wolfram.com/RollingASphereAroundACircleWithoutSlipping/}
    }
  • A. Becker, ``Re-Orient a Sphere with Two Straight Rolls'', Wolfram Demonstrations Project, 2012.
    [bibtex] [video]

    @misc{Becker2012c, Annote = {Aaron Becker "Re-Orient a Sphere with Two Straight Rolls" http://demonstrations.wolfram.com/ReOrientASphereWithTwoStraightRolls/ Wolfram Demonstrations Project Published: June 11, 2012},
      Author = {Aaron Becker},
      Date-Added = {2012-06-18 23:52:16 -0500},
      Date-Modified = {2012-06-18 23:52:16 -0500},
      Month = jun, Title = {{``Re-Orient a Sphere with Two Straight Rolls'', Wolfram Demonstrations Project}},
      Url = {http://demonstrations.wolfram.com/ReOrientASphereWithTwoStraightRolls/},
      Year = {2012},
      Video = {http://demonstrations.wolfram.com/ReOrientASphereWithTwoStraightRolls/},
      Bdsk-Url-1 = {http://demonstrations.wolfram.com/ReOrientASphereWithTwoStraightRolls/}
    }
  • A. Becker, ``Morph Sphere to Disc'', Wolfram Demonstrations Project, 2012.
    [bibtex] [video]

    @misc{Becker2012h,
      author = {Aaron Becker},
      Date-Added = {2012-06-18 23:52:16 -0500},
      Date-Modified = {2012-06-18 23:52:16 -0500},
      Month = Feb, Title = {{``Morph Sphere to Disc'', Wolfram Demonstrations Project}},
      Url = {http://demonstrations.wolfram.com/MorphSphereToDisc/},
      Video = {http://demonstrations.wolfram.com/MorphSphereToDisc/},
      Year = {2012},
      Bdsk-Url-1 = {http://demonstrations.wolfram.com/MorphSphereToDisc/}
    }
  • A. Becker, ``Tangent Plane to a Sphere'', Wolfram Demonstrations Project, 2012.
    [bibtex] [video]

    @misc{Becker2012g,
      author = {Aaron Becker},
      Date-Added = {2012-06-18 23:52:16 -0500},
      Date-Modified = {2012-06-18 23:52:16 -0500},
      Month = feb, Title = {{``Tangent Plane to a Sphere'', Wolfram Demonstrations Project}},
      Url = {http://demonstrations.wolfram.com/TangentPlaneToASphere/},
      Video = {http://demonstrations.wolfram.com/TangentPlaneToASphere/},
      Year = {2012},
      Bdsk-Url-1 = {http://demonstrations.wolfram.com/TangentPlaneToASphere/}
    }
  • A. Becker, ``Curvature of the Projection of a Trefoil Knot'', Wolfram Demonstrations Project, 2012.
    [bibtex] [video]

    @misc{Becker2012f,
      author = {Aaron Becker},
      Date-Added = {2012-06-18 23:52:16 -0500},
      Date-Modified = {2012-06-18 23:52:16 -0500},
      Month = feb, Title = {{``Curvature of the Projection of a Trefoil Knot'', Wolfram Demonstrations Project}},
      Url = {http://demonstrations.wolfram.com/CurvatureOfTheProjectionOfATrefoilKnot/},
      Video = {http://demonstrations.wolfram.com/CurvatureOfTheProjectionOfATrefoilKnot/},
      Year = {2012},
      Bdsk-Url-1 = {http://demonstrations.wolfram.com/CurvatureOfTheProjectionOfATrefoilKnot/}
    }
  • A. Becker, ``Dark Fraction of the Moon'', Wolfram Demonstrations Project, 2012.
    [bibtex] [video]

    @misc{Becker2012e,
      author = {Aaron Becker},
      Date-Added = {2012-06-18 23:52:16 -0500},
      Date-Modified = {2012-06-18 23:52:16 -0500},
      Month = mar, Title = {{``Dark Fraction of the Moon'', Wolfram Demonstrations Project}},
      Url = {http://demonstrations.wolfram.com/DarkFractionOfTheMoon/},
      Video = {http://demonstrations.wolfram.com/DarkFractionOfTheMoon/},
      Year = {2012},
      Bdsk-Url-1 = {http://demonstrations.wolfram.com/DarkFractionOfTheMoon/}
    }
  • A. Becker, ``Sampling a Uniformly Random Rotation'', Wolfram Demonstrations Project, 2012.
    [bibtex] [video]

    @misc{Becker2012d,
      author = {Aaron Becker},
      Date-Added = {2012-06-18 23:52:16 -0500},
      Date-Modified = {2012-06-18 23:52:16 -0500},
      Month = may, Title = {{``Sampling a Uniformly Random Rotation'', Wolfram Demonstrations Project}},
      Url = {http://demonstrations.wolfram.com/SamplingAUniformlyRandomRotation/},
      Video = {http://demonstrations.wolfram.com/SamplingAUniformlyRandomRotation/},
      Year = {2012},
      Bdsk-Url-1 = {http://demonstrations.wolfram.com/SamplingAUniformlyRandomRotation/}
    }
  • A. Becker, ``Coverage of a Unit Square by Random Discs'', Wolfram Demonstrations Project, 2011.
    [bibtex] [video]

    @misc{Becker2011,
      author = {Aaron Becker},
      Booktitle = {Wolfram Demonstrations Project},
      Date-Added = {2012-06-18 23:52:16 -0500},
      Date-Modified = {2012-06-18 23:52:16 -0500},
      Month = may, Title = {{``Coverage of a Unit Square by Random Discs'', Wolfram Demonstrations Project}},
      Url = {http://demonstrations.wolfram.com/CoverageOfAUnitSquareByRandomDiscs/},
      Video = {http://demonstrations.wolfram.com/CoverageOfAUnitSquareByRandomDiscs/},
      Year = {2011},
      Bdsk-Url-1 = {http://demonstrations.wolfram.com/CoverageOfAUnitSquareByRandomDiscs/}
    }
  • A. Becker, ``Fraction of a Circle Covered by Arcs of a Given Length'', Wolfram Demonstrations Project, 2011.
    [bibtex] [video]

    @misc{Becker2011a,
      author = {Aaron Becker},
      Date-Added = {2012-06-18 23:52:16 -0500},
      Date-Modified = {2012-06-18 23:52:16 -0500},
      Month = may, Title = {{``Fraction of a Circle Covered by Arcs of a Given Length'', Wolfram Demonstrations Project}},
      Url = { http://demonstrations.wolfram.com/FractionOfACircleCoveredByArcsOfAGivenLength/},
      Video = { http://demonstrations.wolfram.com/FractionOfACircleCoveredByArcsOfAGivenLength/},
      Year = {2011},
      Bdsk-Url-1 = {http://demonstrations.wolfram.com/FractionOfACircleCoveredByArcsOfAGivenLength/}
    }

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