Lucas Janson

Fast marching tree

In this paper we present a novel probabilistic sampling-based motion planning algorithm called the Fast Marching Tree algorithm (FMT*). The algorithm is specifically aimed at solving complex motion planning problems in high-dimensional configuration spaces. This algorithm is proven to be asymptotically optimal and is shown to converge to an optimal solution faster than its state-of-the-art counterparts, chiefly PRM* and RRT*. The FMT* algorithm performs a ‘lazy’ dynamic programming recursion on a predetermined number of probabilistically drawn samples to grow a tree of paths, which moves steadily outward in cost-to-arrive space. As such, this algorithm combines features of both single-query algorithms (chiefly RRT) and multiple-query algorithms (chiefly PRM), and is reminiscent of the Fast Marching Method for the solution of Eikonal equations. As a departure from previous analysis approaches that are based on the notion of almost sure convergence, the FMT* algorithm is analyzed under the notion of convergence in probability: the extra mathematical flexibility of this approach allows for convergence rate bounds—the first in the field of optimal sampling-based motion planning. Specifically, for a certain selection of tuning parameters and configuration spaces, we obtain a convergence rate bound of order O(n −1/d+ρ ), where n is the number of sampled points, d is the dimension of the configuration space, and ρ is an arbitrarily small constant. We go on to demonstrate asymptotic optimality for a number of variations on FMT*, namely when the configuration space is sampled non-uniformly, when the cost is not arc length, and when connections are made based on the number of nearest neighbors instead of a fixed connection radius. Numerical experiments over a range of dimensions and obstacle configurations confirm our theoretical and heuristic arguments by showing that FMT*, for a given execution time, returns substantially better solutions than either PRM* or RRT*, especially in high-dimensional configuration spaces and in scenarios where collision-checking is expensive.

Optimal sampling-based motion planning under differential constraints: The driftless case

Motion planning under differential constraints is a classic problem in robotics. To date, the state of the art is represented by sampling-based techniques, with the Rapidly-exploring Random Tree algorithm as a leading example. Yet, the problem is still open in many aspects, including guarantees on the quality of the obtained solution. In this paper we provide a thorough theoretical framework to assess optimality guarantees of sampling-based algorithms for planning under differential constraints. We exploit this framework to design and analyze two novel sampling-based algorithms that are guaranteed to converge, as the number of samples increases, to an optimal solution (namely, the Differential Probabilistic RoadMap algorithm and the Differential Fast Marching Tree algorithm). Our focus is on driftless control-affine dynamical models, which accurately model a large class of robotic systems. In this paper we use the notion of convergence in probability (as opposed to convergence almost surely): the extra mathematical flexibility of this approach yields convergence rate bounds - a first in the field of optimal sampling-based motion planning under differential constraints. Numerical experiments corroborating our theoretical results are presented and discussed.

Effective degrees of freedom: a flawed metaphor

To most applied statisticians, a fitting procedures degrees of freedom is synonymous with its model complexity, or its capacity for overfitting to data. In particular, it is often used to parameterize the bias-variance tradeoff in model selection. We argue that, on the contrary, model complexity and degrees of freedom may correspond very poorly. We exhibit and theoretically explore various fitting procedures for which degrees of freedom is not monotonic in the model complexity parameter, and can exceed the total dimension of the ambient space even in very simple settings. We show that the degrees of freedom for any non-convex projection method can be unbounded.

Optimal sampling-based motion planning under differential constraints: The drift case with linear affine dynamics

In this paper we provide a thorough, rigorous theoretical framework to assess optimality guarantees of sampling-based algorithms for drift control systems: systems that, loosely speaking, can not stop instantaneously due to momentum. We exploit this framework to design and analyze a sampling-based algorithm (the Differential Fast Marching Tree algorithm) that is asymptotically optimal, that is, it is guaranteed to converge, as the number of samples increases, to an optimal solution. In addition, our approach allows us to provide concrete bounds on the rate of this convergence. The focus of this paper is on mixed time/control energy cost functions and on linear affine dynamical systems, which encompass a range of models of interest to applications (e.g., double-integrators) and represent a necessary step to design, via successive linearization, sampling-based and provably-correct algorithms for non-linear drift control systems. Our analysis relies on an original perturbation analysis for two-point boundary value problems, which could be of independent interest.

Sarcoma Resection With and Without Vascular Reconstruction: A Matched Case-control Study.

OBJECTIVE To examine the impact of major vascular resection on sarcoma resection outcomes. SUMMARY BACKGROUND DATA En bloc resection and reconstruction of involved vessels is being increasingly performed during sarcoma surgery; however, the perioperative and oncologic outcomes of this strategy are not well described. METHODS Patients undergoing sarcoma resection with (VASC) and without (NO-VASC) vascular reconstruction were 1:2 matched on anatomic site, histology, grade, size, synchronous metastasis, and primary (vs. repeat) resection. R2 resections were excluded. Endpoints included perioperative morbidity, mortality, local recurrence, and survival. RESULTS From 2000 to 2014, 50 sarcoma patients underwent VASC resection. These were matched with 100 NO-VASC patients having similar clinicopathologic characteristics. The rates of any complication (74% vs. 44%, P = 0.002), grade 3 or higher complication (38% vs. 18%, P = 0.024), and transfusion (66% vs. 33%, P < 0.001) were all more common in the VASC group. Thirty-day (2% vs. 0%, P = 0.30) or 90-day mortality (6% vs. 2%, P = 0.24) were not significantly higher. Local recurrence (5-year, 51% vs. 54%, P = 0.11) and overall survival after resection (5-year, 59% vs. 53%, P = 0.67) were similar between the 2 groups. Within the VASC group, overall survival was not affected by the type of vessel involved (artery vs. vein) or the presence of histology-proven vessel wall invasion. CONCLUSIONS Vascular resection and reconstruction during sarcoma resection significantly increases perioperative morbidity and requires meticulous preoperative multidisciplinary planning. However, the oncologic outcome appears equivalent to cases without major vascular involvement. The anticipated need for vascular resection and reconstruction should not be a contraindication to sarcoma resection.

EigenPrism: inference for high dimensional signal‐to‐noise ratios

Consider the following three important problems in statistical inference, namely, constructing confidence intervals for (1) the error of a high-dimensional (p > n) regression estimator, (2) the linear regression noise level, and (3) the genetic signal-to-noise ratio of a continuous-valued trait (related to the heritability). All three problems turn out to be closely related to the little-studied problem of performing inference on the [Formula: see text]-norm of the signal in high-dimensional linear regression. We derive a novel procedure for this, which is asymptotically correct when the covariates are multivariate Gaussian and produces valid confidence intervals in finite samples as well. The procedure, called EigenPrism, is computationally fast and makes no assumptions on coefficient sparsity or knowledge of the noise level. We investigate the width of the EigenPrism confidence intervals, including a comparison with a Bayesian setting in which our interval is just 5% wider than the Bayes credible interval. We are then able to unify the three aforementioned problems by showing that the EigenPrism procedure with only minor modifications is able to make important contributions to all three. We also investigate the robustness of coverage and find that the method applies in practice and in finite samples much more widely than just the case of multivariate Gaussian covariates. Finally, we apply EigenPrism to a genetic dataset to estimate the genetic signal-to-noise ratio for a number of continuous phenotypes.

Monte Carlo Motion Planning for Robot Trajectory Optimization Under Uncertainty

This article presents a novel approach, named Monte Carlo Motion Planning (MCMP), to the problem of motion planning under uncertainty, i.e., to the problem of computing a low-cost path that fulfills probabilistic collision avoidance constraints. MCMP estimates the collision probability (CP) of a given path by sampling via Monte Carlo the execution of a reference tracking controller (in this paper we consider a linear-quadratic-Gaussian controller). The key algorithmic contribution of this paper is the design of statistical variance-reduction techniques, namely control variates and importance sampling, to make such a sampling procedure amenable to real-time implementation. MCMP applies this CP estimation procedure to motion planning by iteratively (i) computing an (approximately) optimal path for the deterministic version of the problem (here, using the FMT\(^*\,\)algorithm), (ii) computing the CP of this path, and (iii) inflating or deflating the obstacles by a common factor depending on whether the CP is higher or lower than a target value. The advantages of MCMP are threefold: (i) asymptotic correctness of CP estimation, as opposed to most current approximations, which, as shown in this paper, can be off by large multiples and hinder the computation of feasible plans; (ii) speed and parallelizability, and (iii) generality, i.e., the approach is applicable to virtually any planning problem provided that a path tracking controller and a notion of distance to obstacles in the configuration space are available. Numerical results illustrate the correctness (in terms of feasibility), efficiency (in terms of path cost), and computational speed of MCMP.

Deterministic Sampling-Based Motion Planning: Optimality, Complexity, and Performance

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 independent 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. The focus is on the PRM algorithm—our results, however, generalize to other batch-processing algorithms such as \(\text {FMT}^*\). Specifically, we first show that PRM, for a certain selection of tuning parameters and deterministic low-dispersion sampling sequences, is deterministically asymptotically optimal, i.e., 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 \(\ell _2\)-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 O(n) (the theoretical lower bound), where n is the number of points in the sequence. This is in stark contrast to the \(O(n\, \log n)\) complexity results for existing asymptotically-optimal probabilistic planners. Finally, through numerical experiments, we show that planning with deterministic low-dispersion sampling generally provides superior performance in terms of path cost and success rate.

Familywise error rate control via knockoffs

We present a novel method for controlling the

An asymptotically-optimal sampling-based algorithm for Bi-directional motion planning

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