Reactive and Potential Field Planners David Johnson Previously • Use geometric reasoning to build path in environment – Visibility graphs – Cell decompositions • Use robot controls to generate forces to follow path • Such complete knowledge of environment is rare • May need to react to sensor data A Simple Approach for Unknown Environments • Bug algorithms – Highlights the sort of approach needed for simple robots with simple sensors • From the text – but find the errata chapter online Bug Algorithms Assumptions: The robot is modeled as a point The obstacles are bounded and are finite The robot senses perfectly its position and can measure traveled distance The robot can perfectly detect contacts and their orientations The robot can compute the direction to the goal and the distance between two points, and has small amount of memory Finish Start Bug-0 Algorithm Finish Start Bug-0 Repeat: 1. Head toward the goal 2. If the goal is attained then stop 3. If contact is made with an obstacle then follow the obstacle’s boundary (toward the left) until heading toward the goal is possible again. Is Bug-0 Guaranteed to Work? No! Finish Finish Start Start Bug-1 Algorithm Bug-1: L2 Repeat: Finish L1 Start 1. Head toward the goal 2. If the goal is attained then stop 3. If contact is made with an obstacle then circumnavigate the obstacle (by wall-following), remember the closest point Li to the goal, and return to this point by the shortest wallfollowing path Bug-2 Algorithm Bug-2: Repeat: leave point Finish hit point goal-line Start 1. Head toward the goal along the goal-line 2. If the goal is attained then stop 3. If a hit point is reached then follow the obstacle’s boundary (toward the left) until the goalline is crossed at a leave point closer to the goal than the previous hit point Path Followed by Bug-2? Finish Start Which one --- Bug-1 or Bug-2 --- does better? Bug-2 does better than Bug-1 Bug-1 does better than Bug-2 Finish Finish Start Start Bug1 vs. Bug2 Bug1 Bug2 • Exhaustive search • Optimal leave point • Performs better with complex obstacles • Opportunistic (greedy) search • Performs better with simple obstacles Kinds of sensors for Bug • Tactile sensing – Infinite number? • Goal beacon – Measure distance through • Signal strength • Time-of-flight • Phase • Wheel encoders • Orientation Potential Field Planners • Can use range information better – Also tangent bug planner in text • Can also be used in known environment – Fast – Reactive to local data • Rather than – generate forces from path – old approach – generate path from forces! Basic Idea • Model physics of robot • Attract to goal • Repulse from obstacles Basic Idea • Originally was described in terms of potentials – Potential energy is energy at a position (or configuration) q – integral of force U (q ) q F (q )dq U (q0 ) 0 – Force is derivative of potential energy dU F (q) dq • Gradient in higher dimensions U U grad (U (q)) U (q) ( , , ) q1 qn Potential Field Path Planning • Potential function guides the robot as if it were a particle moving in a gradient field. • Analogy: robot is positively charged particle, moving towards negative charge goal • Obstacles have “repulsive” positive charge • Potential functions can be viewed as a landscape • Robot moves from high-value to low-value Using a “downhill” path (i.e negative of the gradient). • This is known as gradient descent –follow a functional surface until you reach its minimum – Really, an extremum What kind of potentials/forces to use? • Want to – minimize travel time – have stability at goal – not crash Attract to goal • Force is linear with distance – Like the spring force Attractive Potential Field Repulse from Obstacles • Use inverse quadratic – 1/dist^2 – What is that force law like? Repulsive Potential Field Vector Sum of Two Fields Resulting Robot Trajectory Main Problem Some solutions to local minima • Build graph from local minima – Search graph • Random pertubation to escape – Make sure you don’t push into obstacle • Change parameters to get unstuck – Might not work • Build potential field with only one minimum – Navigation function Rotational and Random Fields • Not gradients of potential functions • Adding a rotational field around obstacles – Breaks symmetry – Avoids some local minima – Guides robot around groups of obstacles • A random field gets the robot unstuck. – Avoids some local minima. Navigation Function N(p) • A potential field leading to a given goal, with no local minima to get stuck in. • For any point p, N(p) is the minimum cost of any path to the goal. • Use a wavefront algorithm, propagating from the goal to the current location. – An active point updates costs of its 8 neighbors. – A point becomes active if its cost decreases. – Continue to the robot’s current position. Sphere Worlds • World in which Navigation Problem is solved – compact, connected subset of En – boundary formed by disjoint union of finite number of spheres – valid sphere world provided obstacle closures are contained within the workspace MATLAB Simulation 20 Kappa = 3 15 10 Target 5 0 -5 Potential Field Level Curves -10 Obstacle -15 -20 -20 -15 -10 -5 0 5 10 15 20 Plotting Navigation Function Kappa = 3 Target Obstacle Increase Kappa Kappa = 4 20 15 10 5 1 0 0.8 20 -5 0.6 -10 0.4 10 0 0.2 -15 -10 0 -20 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 -20 15 20 Sphere World: 5 Obstacles Kappa = 5.6 20 15 10 5 0 -5 -10 -15 -20 -25 No local minima -20 -15 -10 -5 0 5 10 15 20 25 More Advanced Navigation Functions • Treat C-space as fluid flow simulation • Start is fluid source • Goal is fluid sink • Run FEA Finding a Path • Start at source • Trace path along vector field • Can have non-point source and sinks • New sources are fast to compute