Optimal coordinated motions for two square robots

We find the coordinated motion for two square robots that minimizes the sum of the length of the path of every robot, for every initial and final positions of both robots. We study it in an obstacle-free plane, so the unique constraint is that the robots can not collide

Geometric origin of mechanical properties of granular materials

Some remarkable generic properties, related to isostaticity and potential energy minimization, of equilibrium configurations of assemblies of rigid, frictionless grains are studied. Isostaticity -the uniqueness of the forces, once the list of contacts is known- is established in a quite general context, and the important distinction between isostatic problems under given external loads and isostatic (rigid) structures is presented. Complete rigidity is only guaranteed, on stability grounds, in the case of spherical cohesionless grains. Otherwise, the network of contacts might deform elastically in response to load increments, even though grains are rigid. This sets an uuper bound on the contact coordination number. The approximation of small displacements (ASD) allows to draw analogies with other model systems studied in statistical mechanics, such as minimum paths on a lattice. It also entails the uniqueness of the equilibrium state (the list of contacts itself is geometrically determined) for cohesionless grains, and thus the absence of plastic dissipation. Plasticity and hysteresis are due to the lack of such uniqueness and may stem, apart from intergranular friction, from small, but finite, rearrangements, in which the system jumps between two distinct potential energy minima, or from bounded tensile contact forces. The response to load increments is discussed. On the basis of past numerical studies, we argue that, if the ASD is valid, the macroscopic displacement field is the solution to an elliptic boundary value problem (akin to the Stokes problem).Comment: RevTex, 40 pages, 26 figures. Close to published paper. Misprints and minor errors correcte

Geometric interpretation of pre-vitrification in hard sphere liquids

We derive a microscopic criterion for the stability of hard sphere configurations, and we show empirically that this criterion is marginally satisfied in the glass. This observation supports a geometric interpretation for the initial rapid rise of viscosity with packing fraction, or pre-vitrification. It also implies that barely stable soft modes characterize the glass structure, whose spatial extension is estimated. We show that both the short-term dynamics and activation processes occur mostly along those soft modes, and we study some implications of these observations. This article synthesizes new and previous results [C. Brito and M. Wyart, Euro. Phys. Letters, {\bf 76}, 149-155, (2006) and C. Brito and M. Wyart, J. Stat. Mech., L08003 (2007) ] in a unified view.Comment: accepted for publication in the Journal of Chemical Physics (added discussion on RCP and ideal glass transition

Near-Optimal Multi-Robot Motion Planning with Finite Sampling

An underlying structure in several sampling-based methods for continuous multi-robot motion planning (MRMP) is the tensor roadmap (TR), which emerges from combining multiple PRM graphs constructed for the individual robots via a tensor product. We study the conditions under which the TR encodes a near-optimal solution for MRMP---satisfying these conditions implies near optimality for a variety of popular planners, including dRRT*, and the discrete methods M* and CBS when applied to the continuous domain. We develop the first finite-sample analysis of this kind, which specifies the number of samples, their deterministic distribution, and magnitude of the connection radii that should be used by each individual PRM graph, to guarantee near-optimality using the TR. This significantly improves upon a previous asymptotic analysis, wherein the number of samples tends to infinity, and supports guaranteed high-quality solutions in practice, within bounded running time. To achieve our new result, we first develop a sampling scheme, which we call the staggered grid, for finite-sample motion planning for individual robots, which requires significantly less samples than previous work. We then extend it to the much more involved MRMP setting which requires to account for interactions among multiple robots. Finally, we report on a few experiments that serve as a verification of our theoretical findings and raise interesting questions for further investigation.Comment: Submitted to the International Conference on Robotics and Automation (ICRA), 202

Near-Optimal Min-Sum Motion Planning for Two Square Robots in a Polygonal Environment

Let $\mathcal{W} \subset \mathbb{R}^2$ be a planar polygonal environment (i.e., a polygon potentially with holes) with a total of $n$ vertices, and let $A,B$ be two robots, each modeled as an axis-aligned unit square, that can translate inside $\mathcal{W}$. Given source and target placements $s_A,t_A,s_B,t_B \in \mathcal{W}$ of $A$ and $B$, respectively, the goal is to compute a \emph{collision-free motion plan} $\mathbf{\pi}^*$, i.e., a motion plan that continuously moves $A$ from $s_A$ to $t_A$ and $B$ from $s_B$ to $t_B$ so that $A$ and $B$ remain inside $\mathcal{W}$ and do not collide with each other during the motion. Furthermore, if such a plan exists, then we wish to return a plan that minimizes the sum of the lengths of the paths traversed by the robots, $\left|\mathbf{\pi}^*\right|$. Given $\mathcal{W}, s_A,t_A,s_B,t_B$ and a parameter $\varepsilon > 0$, we present an $n^2\varepsilon^{-O(1)} \log n$-time $(1+\varepsilon)$-approximation algorithm for this problem. We are not aware of any polynomial time algorithm for this problem, nor do we know whether the problem is NP-Hard. Our result is the first polynomial-time $(1+\varepsilon)$-approximation algorithm for an optimal motion planning problem involving two robots moving in a polygonal environment.Comment: The conference version of the paper is accepted to SODA 202

Kinetic and kinematic evaluation of compensatory movements of the head, pelvis and thoraco-lumbar spine associated with asymmetrical weight bearing of the pelvic limbs in dogs

The purposes of this dissertation were to 1) determine ground reaction forces of dogs with mild asymmetrical weight-bearing of the pelvic limbs while trotting and 2) use three-dimensional motion analysis to identify compensatory vertical motion of the head and pelvis, and lateral motion of the thoraco-lumbar spine in dogs with mild asymmetrical weight-bearing of the pelvic limbs while trotting, and 3) use this information to introduce a subjective grading system for the pelvic limbs in dogs. Our hypotheses were that dogs with asymmetric weight bearing demonstrate compensatory motions of the head, pelvis and thoraco-lumbar spine while trotting, and that these motions would have a positive correlation with the degree of weight bearing asymmetry. Twenty-seven dogs were included in the study. Nine were normal dogs, which had no surgical intervention, 9 dogs had a cranial cruciate ligament transection and tibial plateau leveling osteotomy 3 ± 0.5 years prior to study start, and 9 dogs had a cranial cruciate ligament transection and extracapsular lateral fabellar-tibial suture (modified retinacular imbrications technique) 7 ± 0.5 years prior to study start. A kinematic model was created so that reflective markers placed on the sagittal crest of the skull, the ischiatic tuberosity and 3 points along the thoraco-lumbar spine of each test subject could be tracked over time while trotting. Kinetic and kinematic data were used to characterize weight-bearing asymmetry between the left and right pelvic limbs, and to describe linear vertical displacement of the head and pelvis, and lateral angular displacement of the thoraco-lumbar spine. Maximum, minimum and range of motion values were analyzed for any differences between the pelvic limbs. Dogs with subtle asymmetric weight bearing of the pelvic limbs demonstrated a greater range of pelvis linear vertical displacement (PLVD) on the side with a greater peak vertical force, and greater thoraco-lumbar lateral angular displacement (TL-LAD) toward the side with a lower peak vertical force while trotting. No differences in mean head linear vertical displacement (HLVD) were detected, and there were no significant correlations between the magnitude of HLVD, PLVD and TL-LAD and the degree of asymmetrical weight bearing of the pelvic limbs