By enabling coordinated task execution and movement, robotic swarms can achieve efficient exploration of unknown environments. In this paper, we propose a relative localization sensor system using Ultra-wideband (UWB) radio technology for ranging. This system is lightweight and relatively indifferent to the types of surrounding environments. Infrastructure dependency such as the requirement of beacons at known locations is eliminated by making an array of sensors on a swarm agent. In this paper, a novel algorithm is implemented on hardware with limited resources and compared to a more traditional trilateration approach. Both utilize Particle Swarm Optimization (PSO) to be more robust against noise and to achieve similar accuracy. The experimental results show that the proposed algorithm runs up to ten times faster than the existing trilateration approach. The sensor array which forms the localization system weighs only 56g, and achieves around 0.5m RMSE with a 10Hz update rate. Experiments show that the accuracy can be further improved if the rotational bias observed in the UWB devices are compensated for.
Integer relation algorithms can convert numerical results for Feynman integrals to exact evaluations, when one has reason to suspect the existence of reductions to linear combinations of a basis, with rational or algebraic coefficients. Once a tentative reduction is obtained, confidence in its validity is greatly increased by computing more decimal digits of the terms and verifying the stability of the result. Here we give examples of how the PSLQ and LLL algorithms have yielded remarkable reductions of Feynman integrals to multiple polylogarithms and to the periods and quasi-periods of modular forms. Moreover, these algorithms have revealed quadratic relations between Feynman integrals. A recent application concerning black holes involves quadratic relations between combinations of Feynman integrals with algebraic coefficients.
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