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Register a new userNavigation of a Quadratic Potential with Ellipsoidal Obstacles
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Given a convex quadratic potential of which its minimum is the agents goal and a space populated with ellipsoidal obstacles, one can construct a Rimon-Koditschek artificial potential to navigate. These potentials are such that they combine the natural attractive potential of which its minimum is the destination of the agent with potentials that repel the agent from the boundary of the obstacles. This is a popular approach to navigation problems since it can be implemented with spatially local information that is acquired during operation time. However, navigation is only successful in situations where the obstacles are not too eccentric (flat). This paper proposes a modification to gradient dynamics that allows successful navigation of an environment with a quadratic cost and ellipsoidal obstacles regardless of their eccentricity. This is accomplished by altering gradient dynamics with the addition of a second order curvature correction that is intended to imitate worlds with spherical obstacles in which Rimon-Koditschek potentials are known to work. Convergence to the goal and obstacle avoidance is established from every initial position in the free space. Results are numerically verified with a discretized version of the proposed flow dynamics.
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We study the problem of existence of solutions for generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under weaker assumptions on the data. Roughly speaking we show the existence of a maximal solution for GRBSDE when the terminal condition xi is F_T-measurable, the coefficient f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z and the reflecting barriers L and U are just right continuous left limited. The result is proved without assuming any P-integrability conditions.
Navigation tasks often cannot be defined in terms of a target, either because global position information is unavailable or unreliable or because target location is not explicitly known a priori. This task is then often defined indirectly as a source seeking problem in which the autonomous agent navigates so as to minimize the convex potential induced by a source while avoiding obstacles. This work addresses this problem when only scalar measurements of the potential are available, i.e., without gradient information. To do so, it construct an artificial potential over which an exact gradient dynamics would generate a collision-free trajectory to the target in a world with convex obstacles. Then, leveraging extremum seeking control loops, it minimizes this artificial potential to navigate smoothly to the source location. We prove that the proposed solution not only finds the source, but does so while avoiding any obstacle. Numerical results with velocity-actuated particles, simulations with an omni-directional robot in ROS+Gazebo, and a robot-in-the-loop experiment are used to illustrate the performance of this approach.
Quadrupeds are strong candidates for navigating challenging environments because of their agile and dynamic designs. This paper presents a methodology that extends the range of exploration for quadrupedal robots by creating an end-to-end navigation framework that exploits walking and jumping modes. To obtain a dynamic jumping maneuver while avoiding obstacles, dynamically-feasible trajectories are optimized offline through collocation-based optimization where safety constraints are imposed. Such optimization schematic allows the robot to jump through window-shaped obstacles by considering both obstacles in the air and on the ground. The resulted jumping mode is utilized in an autonomous navigation pipeline that leverages a search-based global planner and a local planner to enable the robot to reach the goal location by walking. A state machine together with a decision making strategy allows the system to switch behaviors between walking around obstacles or jumping through them. The proposed framework is experimentally deployed and validated on a quadrupedal robot, a Mini Cheetah, to enable the robot to autonomously navigate through an environment while avoiding obstacles and jumping over a maximum height of 13 cm to pass through a window-shaped opening in order to reach its goal.
We prove the existence of maximal (and minimal) solution for one-dimensional generalized doubly reflected backward stochastic differential equation (RBSDE for short) with irregular barriers and stochastic quadratic growth, for which the solution $Y$ has to remain between two rcll barriers $L$ and $U$ on $[0; T[$, and its left limit $Y_-$ has to stay respectively above and below two predictable barriers $l$ and $u$ on $]0; T]$. This is done without assuming any $P$-integrability conditions and under weaker assumptions on the input data. In particular, we construct a maximal solution for such a RBSDE when the terminal condition $xi$ is only ${cal F}_T-$measurable and the driver $f$ is continuous with general growth with respect to the variable $y$ and stochastic quadratic growth with respect to the variable $z$. Our result is based on a (generalized) penalization method. This method allow us find an equivalent form to our original RBSDE where its solution has to remain between two new rcll reflecting barriers $overline{Y}$ and $underline{Y}$ which are, roughly speaking, the limit of the penalizing equations driven by the dominating conditions assumed on the coefficients. A standard and equivalent form to our initial RBSDE as well as a characterization of the solution $Y$ as a generalized Snell envelope of some given predictable process $l$ are also given.
We address the optimal dynamic formation problem in mobile leader-follower networks where an optimal formation is generated to maximize a given objective function while continuously preserving connectivity. We show that in a convex mission space, the connectivity constraints can be satisfied by any feasible solution to a mixed integer nonlinear optimization problem. When the optimal formation objective is to maximize coverage in a mission space cluttered with obstacles, we separate the process into intervals with no obstacles detected and intervals where one or more obstacles are detected. In the latter case, we propose a minimum-effort reconfiguration approach for the formation which still optimizes the objective function while avoiding the obstacles and ensuring connectivity. We include simulation results illustrating this dynamic formation process.
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