We present a new approach to the design of smooth trajectories for quadrotor unmanned aerial vehicles (UAVs), which are free of collisions with obstacles along their entire length. To avoid the non-convex constraints normally required for obstacle-avoidance, we perform a mixed-integer optimization in which polynomial trajectories are assigned to convex regions which are known to be obstacle-free. Prior approaches have used the faces of the obstacles themselves to define these convex regions. We instead use IRIS, a recently developed technique for greedy convex segmentation, to pre-compute convex regions of safe space. This results in a substantially reduced number of integer variables, which improves the speed with which the optimization can be solved to its global optimum, even for tens or hundreds of obstacle faces. In addition, prior approaches have typically enforced obstacle avoidance at a finite set of sample or knot points. We introduce a technique based on sums-of-squares (SOS) programming that allows us to ensure that the entire piecewise polynomial trajectory is free of collisions using convex constraints. We demonstrate this technique in 2D and in 3D using a dynamical model in the Drake toolbox for MATLAB.

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This paper describes a collection of optimization algorithms for achieving dynamic planning, control, and state estimation for a bipedal robot designed to operate reliably in complex environments. To make challenging locomotion tasks tractable, we describe several novel applications of convex, mixed-integer, and sparse nonlinear optimization to problems ranging from footstep placement to whole-body planning and control. We also present a state estimator formulation that, when combined with our walking controller, permits highly precise execution of extended walking plans over non-flat ter- rain. We describe our complete system integration and experiments carried out on Atlas, a full-size hydraulic humanoid robot built by Boston Dynamics, Inc.

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Here we present a closed-form solution solution to the continuous time-varying linear quadratic regulator (LQR) problem for the zero-moment point (ZMP) tracking controller. This generalizes previous analytical solutions for gait generation by allowing “soft” tracking (with a quadratic cost) of the desired ZMP, and by providing the feedback gains for the resulting time-varying optimal controller. This enables extremely fast computation, with the number of operations linear in the number of spline segments representing the desired ZMP. Results are presented using the Atlas humanoid robot where dynamic walking is achieved by recomputing the optimal controller online.

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We present a new method for planning footstep placements for a robot walking on uneven terrain with obstacles, using a mixed-integer quadratically-constrained quadratic program (MIQCQP). Our approach is unique in that it handles obstacle avoidance, kinematic reachability, and rotation of footstep placements, which typically have required non-convex constraints, in a single mixed-integer optimization that can be efficiently solved to its global optimum. Reachability is enforced through a convex inner approximation of the reachable space for the robot's feet. Rotation of the footsteps is handled by a piecewise linear approximation of sine and cosine, designed to ensure that the approximation never overestimates the robot's reachability. Obstacle avoidance is ensured by decomposing the environment into convex regions of obstacle-free configuration space and assigning each footstep to one such safe region. We demonstrate this technique in simple 2D and 3D environments and with real environments sensed by a humanoid robot. We also discuss computational performance of the algorithm, which is currently capable of planning short sequences of a few steps in under one second or longer sequences of 10-30 footsteps in tens of seconds to minutes on common laptop computer hardware. Our implementation is available within the Drake MATLAB toolbox.

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Estimates based on the strength, size, and shape of the Atlantic razor clam (Ensis directus) indicate that the animal's burrow depth should be physically limited to a few centimeters; yet razor clams can dig as deep as 70 cm. By measuring soil deformations around burrowing E. directus, we have found the animal reduces drag by contracting its valves to initially fail, and then fluidize, the surrounding substrate. The characteristic contraction time to achieve fluidization can be calculated directly from soil properties. The geometry of the fluidized zone is dictated by two commonly-measured geotechnical parameters: coefficient of lateral earth pressure and friction angle. Calculations using full ranges for both parameters indicate that the fluidized zone is a local effect, occurring between 1–5 body radii away from the animal. The energy associated with motion through fluidized substrate—characterized by a depth-independent density and viscosity—scales linearly with depth. In contrast, moving through static soil requires energy that scales with depth squared. For E. directus, this translates to a 10X reduction in the energy required to reach observed burrow depths. For engineers, localized fluidization offers a mechanically simple and purely kinematic method to dramatically reduce energy costs associated with digging. This concept is demonstrated with RoboClam, an E. directus-inspired robot. Using a genetic algorithm to find optimal digging kinematics, RoboClam has achieved localized fluidization burrowing performance comparable to that of the animal, with a linear energy-depth relationship, in both idealized granular glass beads and E. directus' native cohesive mudflat habitat.

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This work presents a novel formulation of the footstep planning problem as a mixed-integer convex optimization. The footstep planning problem involves choosing a set of footstep locations which a walking robot can follow to safely reach a goal through an environment with obstacles. Rather than attempting to avoid the obstacles, which would require non-convex constraints, we use integer variables to assign each footstep to a convex region of obstacle-free terrain, while simultaneously optimizing its pose within that safe region. Since existing methods for generating convex obstacle-free regions were ill-suited to this task, we also present IRIS (Iterative Regional Inflation by Semidefinite programming), a new method to generate such regions through a series of convex optimizations. Combining IRIS with the mixed-integer optimization gives a complete footstep planning architecture, which can produce complex footstep plans on heightmap data constructed from onboard sensors. We demonstrate the footstep planner in simulated environments and with real data sensed by the Atlas humanoid, and we discuss future applications to running robots, aerial vehicles, and robots with more than two legs.

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The DARPA Robotics Challenge Trials held in December 2013 provided a landmark demonstration of dexterous mobile robots executing a variety of tasks aided by a remote human operator using only data from the robot's sensor suite transmitted over a constrained, field-realistic communications link. We describe the design considerations, architecture, implementation and performance of the software that Team MIT developed to command and control an Atlas humanoid robot. Our design emphasized human interaction with an efficient motion planner, where operators expressed desired robot actions in terms of affordances fit using perception and manipulated in a custom user interface. We highlight several important lessons we learned while developing our system on a highly compressed schedule.

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This paper presents IRIS (Iterative Regional Inflation by Semi-definite programming), a new method for efficiently computing large polytopic and ellipsoidal regions of obstacle-free space through a series of convex optimizations. These regions can be used, for example, to efficiently optimize an objective over collision-free positions in space for a robot manipulator. The algorithm alternates between two convex optimizations: (1) a quadratic program to generate a set of hyperplanes which separate a convex region of space from the set of obstacles and (2) a semidefinite program which finds a maximum-volume ellipsoid inside the polytopic intersection of the obstacle-free half-spaces defined by those hyperplanes. Both the hyperplanes and the ellipsoid are refined over several iterations to monotonically increase the volume of the contained ellipsoid, resulting in a large convex polytope in free space. Practical applications of the algorithm are presented in 2D and 3D, and extensions to N-dimensional configuration spaces are discussed. Experiments demonstrate that the algorithm has a computation time which is linear in the number of obstacles, and our MATLAB implementation converges in seconds for environments with millions of obstacles.

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A radial magnetic bearing for magnetic bearing of a rotor has a stator which includes a magnetically conductive stator element, arranged circulating around a rotor. The stator element has recesses running in the axial direction of the stator element in which electrical lines from coils are arranged, wherein magnetic fields can be generated by the coils which hold the rotor suspended in an air gap arranged between the rotor and stator. A softer progression of the components of magnetic flow density in the radial direction is achieved by design measures on the transition from one magnetic pole to the next magnetic pole, which results in a reduction of the eddy currents induced in the rotor.

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The Atlantic razor clam (Ensis directus) burrows into underwater soil by using motions of its shell to locally fluidize the surrounding substrate. The energy associated with movement through fluidized soil – characterized by a depth-independent density and viscosity – scales linearly with depth. In contrast, moving through static soil requires energy that scales with depth squared. For E. directus, this translates to a 10X reduction in the energy required to reach observed burrow depths. For engineers, localized fluidization offers a mechanically simple and purely kinematic method to dramatically reduce burrowing energy. This concept is demonstrated with RoboClam, an E. directus-inspired robot. Using a genetic algorithm to generate digging kinematics, RoboClam has achieved localized fluidization and burrowing performance comparable to that of the animal, with a linear energy-depth relationship. In this paper, we present the critical timescales and associated kinematics necessary for achieving localized fluidization, which are calculated from soil parameters and validated via RoboClam and E. directus testing.

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Our goal is to improve the efficiency and effectiveness of natural language communication between humans and robots. Human language is frequently ambiguous, and a robot’s limited sensing makes complete understanding of a statement even more difficult. To address these challenges, we describe an approach for enabling a robot to engage in clarifying dialog with a human partner, just as a human might do in a similar situation. Given an unconstrained command from a human operator, the robot asks one or more questions and receives natural language answers from the human. We apply an information-theoretic approach to choosing questions for the robot to ask. Specifically, we choose the type and subject of questions in order to maximize the reduction in Shannon entropy of the robot’s mapping between language and entities in the world. Within the framework of the G3 graphical model, we derive a method to estimate this entropy reduction, choose the optimal question to ask, and merge the information gained from the human operator’s answer. We demonstrate that this improves the accuracy of command understanding over prior work while asking fewer questions as compared to baseline question-selection strategies.

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Muscle measurements of Ensis directus, the Atlantic razor clam, indicate that the organism only has sufficient strength to burrow a few centimeters into the soil, yet razor clams burrow to over 70cm. In this paper, we show that the animal uses the motions of its valves to locally fluidize the surrounding soil and reduce burrowing drag. Substrate deformations were measured using particle image velocimetry (PIV) in a novel visualization system that enabled us to see through the soil and watch E. directus burrow in situ. PIV data, supported by soil and fluid mechanics theory, show that contraction of the valves of E. directus locally fluidizes the surrounding soil. Particle and fluid mixtures can be modeled as a Newtonian fluid with an effective viscosity based on the local void fraction. Using these models, we demonstrate that E. directus is strong enough to reach full burrow depth in fluidized soil, but not in static soil. Furthermore, we show that the method of localized fluidization reduces the amount of energy required to reach burrow depth by an order of magnitude compared with penetrating static soil, and leads to a burrowing energy that scales linearly with depth rather than with depth squared.

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The invention relates to a radial magnetic bearing for magnetic bearing of a rotor, wherein the radial magnetic bearing has a stator, wherein the stator has a magnetically conductive stator element, arranged circulating around the rotor, wherein the stator element has recesses running in the axial direction of the stator element in which electrical lines from coils are arranged, wherein magnetic fields can be generated by the coils which hold the rotor suspended in an air gap arranged between the rotor and stator, wherein a softer progression of the components of magnetic flow density in the radial direction is achieved by design measures on the transitions from one magnetic pole to the next magnetic pole, which results in a reduction of the eddy currents induced in the rotor.

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Our goal is to build robots that can robustly interact with humans using natural language. This problem is challenging because human language is filled with ambiguity, and furthermore, due to limitations in sensing, the robot’s perception of its environment might be much more limited than that of its human partner. To enable a robot to recover from a failure to understand a natural language utterance, this paper describes an information-theoretic strategy for asking targeted clarifying questions and using information from the answer to disambiguate the language. To identify good questions, we derive an estimate of the robot’s uncertainty about themapping between specific phrases in the language and aspects of the external world. This metric enables the robot to ask a targeted question about the parts of the language for which it is most uncertain. After receiving an answer, the robot fuses information from the command, the question, and the answer in a joint probabilistic graphical model in the G3 framework. When using answers to questions, we show the robot is able to infer mappings between parts of the language and concrete object groundings in the external world with higher accuracy than by using information from the command alone. Furthermore, we demonstrate that by effectively selecting which questions to ask, the robot is able to achieve significant performance gains while asking many fewer questions than baseline metrics.

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The Atlantic razor clam (Ensis directus) reduces burrowing drag by using motions of its shell to fluidize a thin layer of substrate around its body. We have developed RoboClam, a robot that digs using the same mechanisms as Ensis, to explore how localized fluidization burrowing can be extended to engineering applications. In this work we present burrowing performance results of RoboClam in two distinctly different substrates: ideally granular 1mm soda lime glass beads and cohesive ocean mudflat soil. Using a genetic algorithm to optimize RoboClam’s kinematics, the machine was able to burrow in both substrates with a power law relationship between digging energy and depth of n = 1.17. Pushing through static soil has a theoretical energy-depth power law of n = 2, which means that Ensis-inspired burrowing motions can provide exponentially higher energy efficiency. We propose a theoretical constitutive model that describes how a fluidized region should form around a contracting body in virtually any type of saturated soil. The model predicts fluidization to be a relatively local effect, extending only two to three characteristic lengths away from the body, depending on friction angle and coefficient of lateral earth pressure, two commonly measured soil parameters.

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Razor clams (Ensis directus) are one of nature’s most adept burrowing organisms, able to dig to 70cm at nearly 1cm/s using only 0.21J/cm. We discovered that Ensis reduces burrowing drag by using motions of its shell to fluidize a thin layer of substrate around its body. We have developed RoboClam, a robot that digs using the same mechanisms as Ensis, to explore how localized fluidization burrowing can be extended to engineering applications. In this work we present burrowing performance results of RoboClam in Ensis’ habitat. Using a genetic algorithm to optimize RoboClam’s kinematics, the machine was able to burrow at speeds comparable to Ensis, with a power law relationship between digging energy and depth of n = 1.17, close to the n = 1 achieved by the animal. Pushing through static soil has a theoretical energy-depth power law of n = 2, which means that Ensis-inspired digging motions can provide exponential energetic savings over existing burrowing methods.

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Razor clams (Ensis directus) are one of nature’s most adept burrowing organisms, able to dig to 70cm at nearly 1cm/s using only 0.21J/cm. Ensis reduces burrowing drag by using motions of its shell to fluidize a thin layer of substrate around its body. Although these shell motions have an energetic cost, moving through fluidized rather than packed soil results in exponentially lower overall energy consumption. This paper describes the design and testing of RoboClam, a device that mimics Ensis digging methods to understand the limits of razor clam-inspired burrowing, how they scale for different environments and conditions, and how they can be transferred into engineering applications. Using a genetic optimization solver, we found that RoboClam’s most efficient digging motion mimicked Ensis shell kinematics and yielded a power law relationship between digging energy and depth of n = 1.17, very close to the ideal value of n = 1. Pushing through static soil has a theoretical energy-depth power law of n = 2, which means that Ensis-inspired burrowing motions can provide exponentially higher energy efficiency and nearly depth- independent drag resistance.

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