scholarly journals Deterministic sampling-based motion planning: Optimality, complexity, and performance

2017 ◽  
Vol 37 (1) ◽  
pp. 46-61 ◽  
Author(s):  
Lucas Janson ◽  
Brian Ichter ◽  
Marco Pavone
Keyword(s):  
Motion Planning ◽  
Random Sampling ◽  
Superior Performance ◽  
Asymptotically Optimal ◽  
Deterministic Sampling ◽  
Sampling Sequence ◽  
Practical Performance ◽  
Sampling Sequences ◽  
Low Dispersion ◽  
Theoretical Results

Probabilistic sampling-based algorithms, such as the probabilistic roadmap (PRM) and the rapidly exploring random tree (RRT) algorithms, represent one of the most successful approaches to robotic motion planning, due to their strong theoretical properties (in terms of probabilistic completeness or even asymptotic optimality) and remarkable practical performance. Such algorithms are probabilistic in that they compute a path by connecting independently and identically distributed (i.i.d.) random points in the configuration space. Their randomization aspect, however, makes several tasks challenging, including certification for safety-critical applications and use of offline computation to improve real-time execution. Hence, an important open question is whether similar (or better) theoretical guarantees and practical performance could be obtained by considering deterministic, as opposed to random, sampling sequences. The objective of this paper is to provide a rigorous answer to this question. Specifically, we first show that PRM, for a certain selection of tuning parameters and deterministic low-dispersion sampling sequences, is deterministically asymptotically optimal, in other words, it returns a path whose cost converges deterministically to the optimal one as the number of points goes to infinity. Second, we characterize the convergence rate, and we find that the factor of sub-optimality can be very explicitly upper-bounded in terms of the[Formula: see text] -dispersion of the sampling sequence and the connection radius of PRM. Third, we show that an asymptotically optimal version of PRM exists with computational and space complexity arbitrarily close to [Formula: see text] (the theoretical lower bound), where n is the number of points in the sequence. This is in contrast to the [Formula: see text] complexity results for existing asymptotically optimal probabilistic planners. Fourth, we discuss extending our theoretical results and insights to other batch-processing algorithms such as FMT*, to non-uniform sampling strategies, to k-nearest-neighbor implementations, and to differentially constrained problems. Importantly, our main theoretical tool is the [Formula: see text]-dispersion, an interesting consequence of which is that all our theoretical results also hold for low-[Formula: see text]-dispersion random sampling (which i.i.d. sampling does not satisfy). In other words, achieving deterministic guarantees is really a matter of i.i.d. sampling versus non-i.i.d. low-dispersion sampling (with deterministic sampling as a prominent case), as opposed to random versus deterministic. Finally, through numerical experiments, we show that planning with deterministic (or random) low-dispersion sampling generally provides superior performance in terms of path cost and success rate.

scholarly journals Why simple quadrature is just as good as Monte Carlo

2020 ◽  
Vol 26 (1) ◽  
pp. 1-16
Author(s):  
Kevin Vanslette ◽  
Abdullatif Al Alsheikh ◽  
Kamal Youcef-Toumi
Keyword(s):  
Monte Carlo ◽  
Random Sampling ◽  
Bayesian Probability ◽  
Independent Factor ◽  
Worst Case ◽  
Statistical Equivalence ◽  
Deterministic Sampling ◽  
Quadrature Methods ◽  
Quadrature Sampling ◽  
Simple Quadrature

AbstractWe motive and calculate Newton–Cotes quadrature integration variance and compare it directly with Monte Carlo (MC) integration variance. We find an equivalence between deterministic quadrature sampling and random MC sampling by noting that MC random sampling is statistically indistinguishable from a method that uses deterministic sampling on a randomly shuffled (permuted) function. We use this statistical equivalence to regularize the form of permissible Bayesian quadrature integration priors such that they are guaranteed to be objectively comparable with MC. This leads to the proof that simple quadrature methods have expected variances that are less than or equal to their corresponding theoretical MC integration variances. Separately, using Bayesian probability theory, we find that the theoretical standard deviations of the unbiased errors of simple Newton–Cotes composite quadrature integrations improve over their worst case errors by an extra dimension independent factor {\propto N^{-\frac{1}{2}}}. This dimension independent factor is validated in our simulations.

scholarly journals Energy-Dissipation Limits in Variance-Based Computing

2018 ◽  
Vol 17 (02) ◽  
pp. 1850013
Author(s):  
Sri Harsha Kondapalli ◽  
Xuan Zhang ◽  
Shantanu chakrabartty
Keyword(s):  
Energy Dissipation ◽  
Thermal Noise ◽  
Operating Conditions ◽  
Superior Performance ◽  
Energy Scavenging ◽  
Logic Device ◽  
Additional Energy ◽  
Fundamental Limits ◽  
Bit Flip ◽  
Theoretical Results

Variance-based logic (VBL) uses the fluctuations or the variance in the state of a particle or a physical quantity to represent different logic levels. In this paper, we show that compared to the traditional bi-stable logic representation, the variance-based representation can theoretically achieve a superior performance trade-off (in terms of energy dissipation and information capacity) when operating at fundamental limits imposed by thermal noise. We show that, in addition to the universal KT ln(1/[Formula: see text]) energy dissipation required for a single bit flip, a bi-stable logic device needs to dissipate at least 4.35[Formula: see text]KT/bit of energy, whereas under similar operating conditions, a VBL device reduces the additional energy dissipation requirements down to sub-KT/bit. These theoretical results are generally enough to be applicable to different instantiations and variants of VBL ranging from digital processors based on energy-scavenging or to processors based on the emerging valleytronic devices.

scholarly journals Efficient Design for Integrated Photonic Waveguides with Agile Dispersion

Sensors ◽  
2021 ◽  
Vol 21 (19) ◽  
pp. 6651
Author(s):  
Zhaonian Wang ◽  
Jiangbing Du ◽  
Weihong Shen ◽  
Jiacheng Liu ◽  
Zuyuan He
Keyword(s):  
Integrated Circuit ◽  
Frequency Comb ◽  
Design Method ◽  
Chromatic Dispersion ◽  
Nonlinear Effects ◽  
Superior Performance ◽  
Efficient Design ◽  
Linear Dispersion ◽  
Dispersion Engineering ◽  
Low Dispersion

Chromatic dispersion engineering of photonic waveguide is of great importance for Photonic Integrated Circuit in broad applications, including on-chip CD compensation, supercontinuum generation, Kerr-comb generation, micro resonator and mode-locked laser. Linear propagation behavior and nonlinear effects of the light wave can be manipulated by engineering CD, in order to manipulate the temporal shape and frequency spectrum. Therefore, agile shapes of dispersion profiles, including typically wideband flat dispersion, are highly desired among various applications. In this study, we demonstrate a novel method for agile dispersion engineering of integrated photonic waveguide. Based on a horizontal double-slot structure, we obtained agile dispersion shapes, including broadband low dispersion, constant dispersion and slope-maintained linear dispersion. The proposed inverse design method is objectively-motivated and automation-supported. Dispersion in the range of 0–1.5 ps/(nm·km) for 861-nm bandwidth has been achieved, which shows superior performance for broadband low dispersion. Numerical simulation of the Kerr frequency comb was carried out utilizing the obtained dispersion shapes and a comb spectrum for 1068-nm bandwidth with a 20-dB power variation was generated. Significant potential for integrated photonic design automation can be expected.

scholarly journals Scrubbing During Learning In Real-time Heuristic Search

2016 ◽  
Vol 57 ◽  
pp. 307-343 ◽  
Author(s):  
Nathan R. Sturtevant ◽  
Vadim Bulitko
Keyword(s):  
State Space ◽  
Real Time ◽  
Lower Bounds ◽  
Heuristic Search ◽  
Search Algorithm ◽  
The State ◽  
Worst Case ◽  
Heuristic Search Algorithm ◽  
Practical Performance ◽  
Theoretical Results

Real-time agent-centered heuristic search is a well-studied problem where an agent that can only reason locally about the world must travel to a goal location using bounded computation and memory at each step. Many algorithms have been proposed for this problem and theoretical results have also been derived for the worst-case performance with simple examples demonstrating worst-case performance in practice. Lower bounds, however, have not been widely studied. In this paper we study best-case performance more generally and derive theoretical lower bounds for reaching the goal using LRTA*, a canonical example of a real-time agent-centered heuristic search algorithm. The results show that, given some reasonable restrictions on the state space and the heuristic function, the number of steps an LRTA*-like algorithm requires to reach the goal will grow asymptotically faster than the state space, resulting in ``scrubbing'' where the agent repeatedly visits the same state. We then show that while the asymptotic analysis does not hold for more complex real-time search algorithms, experimental results suggest that it is still descriptive of practical performance.

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