In fact, in a particular limit, the optimal path is arithmetic.

In fact, in a particular limit, the optimal path is arithmetic.

Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing.

Time-optimal trajectories drive quadrotors to their dynamic limits, but computing such trajectories involves solving non-convex problems via iterative nonlinear optimization, making them prohibitively costly for real-time applications. In this work, we investigate learning-based models that imitate a model-based time-optimal trajectory planner to accelerate trajectory generation. Given a dataset of collision-free geometric paths, we show that modeling architectures can effectively learn the patterns underlying time-optimal trajectories. We introduce a quantitative framework to analyze local analytic properties of the learned models, and link them to the Backward Reachable Tube of the geometric tracking controller. To enhance robustness, we propose a data augmentation scheme that applies random perturbations to the input paths. Compared to classical planners, our method achieves substantial speedups, and we validate its real-time feasibility on a hardware quadrotor platform. Experiments demonstrate that the learned models generalize to previously unseen path lengths. The code for our approach can be found here: https://github.com/maokat12/lbTOPPQuad

We propose a novel, multi-layered planning approach for computing paths that satisfy both kinodynamic and spatiotemporal constraints. Our three-part framework first establishes potential sequences to meet spatial constraints, using them to calculate a geometric lead path. This path then guides an asymptotically optimal sampling-based kinodynamic planner, which minimizes an STL-robustness cost to jointly satisfy spatiotemporal and kinodynamic constraints. In our experiments, we test our method with a velocity-controlled Ackerman-car model and demonstrate significant efficiency gains compared to prior art. Additionally, our method is able to generate complex path maneuvers, such as crossovers, something that previous methods had not demonstrated.

Traditional robotic manipulation mostly focuses on collision-free tasks. In practice, however, many manipulation tasks (e.g., occluded object grasping) require the robot to intentionally collide with the environment to reach a desired task configuration. By enabling compliant robot motions, collisions between the robot and the environment are allowed and can thus be exploited, but more physical uncertainties are introduced. To address collision-rich problems such as occluded object grasping while handling the involved uncertainties, we propose a collision-inclusive planning framework that can transition the robot to a desired task configuration via roughly modeled collisions absorbed by Cartesian impedance control. By strategically exploiting the environmental constraints and exploring inside a manipulation funnel formed by task repetitions, our framework can effectively reduce physical and perception uncertainties. With real-world evaluations on both single-arm and dual-arm setups, we show that our framework is able to efficiently address various realistic occluded grasping problems where a feasible grasp does not initially exist.

Sampling from a target distribution π whose density is known up to a normalizing constant is a challenging problem in statistics and machine learning, and is currently the subject of intense interest due to applications in Bayesian statistics [19] and energy-based models in deep learning [68], among other areas. In these settings, the normalizing constant of the target distribution π is typically intractable, and Markov Chain Monte Carlo (MCMC) algorithms [58, 57] are commonly used to generate Markov chains in the ambient space, whose law eventually approximates the target distribution. Among MCMC algorithms, the Unadjusted Langevin Algorithm (ULA), which corresponds to a time discretization of a Langevin diffusion process, has attracted considerable attention due to its simplicity, theoretical grounding, and utility in practice [59, 76, 23, 66]. For example, ULA can be proven to converge quickly when the target distribution π is smooth and strongly log-concave [23]. However, many cases in practice require to sample from distributions which are not log-concave, and indeed potentially even multi-modal [54, 82]. In such settings, the convergence of ULA is governed by functional inequalities which effectively quantify the convexity, or lack thereof, of the target distribution [75]. Nonetheless, truly multi-modal target distributions generally have poor functional inequalities, thus leading to weak convergence guarantees for ULA. This phenomenon is not merely a theoretical artifact, and it is wellknown amongst practitioners that when sampling from multi-modal distributions, algorithms based on ULA can get stuck in local modes and suffer from slow convergence [20]. Tempering or annealing is a popular technique [51, 27, 69] to overcome the deficiencies of ULA and other MCMC methods in the multi-modal setting.

Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing. This means sampling successively from a path of distributions that interpolate between a tractable "proposal" distribution and the unnormalized "target" distribution. Prominent estimators in this family include annealed importance sampling and annealed noise-contrastive estimation (NCE). Such methods hinge on a number of design choices: which estimator to use, which path of distributions to use and whether to use a path at all; so far, there is no definitive theory on which choices are efficient. Here, we evaluate each design choice by the asymptotic estimation error it produces. First, we show that using NCE is more efficient than the importance sampling estimator, but in the limit of infinitesimal path steps, the difference vanishes. Second, we find that using the geometric path brings down the estimation error from an exponential to a polynomial function of the parameter distance between the target and proposal distributions. Third, we find that the arithmetic path, while rarely used, can offer optimality properties over the universally-used geometric path. In fact, in a particular limit, the optimal path is arithmetic. Based on this theory, we finally propose a two-step estimator to approximate the optimal path in an efficient way.

Kinodynamic motion planning for non-holomonic mobile robots is a challenging problem that is lacking a universal solution. One of the computationally efficient ways to solve it is to build a geometric path first and then transform this path into a kinematically feasible one. Gradient-informed Path Smoothing (GRIPS) is a recently introduced method for such transformation. GRIPS iteratively deforms the path and adds/deletes the waypoints while trying to connect each consecutive pair of them via the provided steering function that respects the kinematic constraints. The algorithm is relatively fast but, unfortunately, does not provide any guarantees that it will succeed. In practice, it often fails to produce feasible trajectories for car-like robots with large turning radius. In this work, we introduce a range of modifications that are aimed at increasing the success rate of GRIPS for car-like robots. The main enhancement is adding the additional step that heuristically samples waypoints along the bottleneck parts of the geometric paths (such as sharp turns). The results of the experimental evaluation provide a clear evidence that the success rate of the suggested algorithm is up to 40% higher compared to the original GRIPS and hits the bar of 90%, while its runtime is lower.