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Chapter 5: Path Planning Hadi Moradi
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Motivation Need to choose a path for the end effector that avoids collisions and singularities Collisions are easy to define in the workspace, but need to be mapped into the configuration space for convenience
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Workspace v. configuration space Workspace: volume swept out by the end effector (in inertial frame) Configuration: location of all points on a robotic manipulator Configuration space:
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Obstacles Discrete obstacles are denoted O i (in the workspace) Denote the robot as A (q) at configuration q The configuration space obstacle, QO, is defined as: The free configuration space is the space of all collision-free configurations:
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Motion Planning for a Point Robot free space s g free path
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Problem semi-free path
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Types of Path Constraints Local constraints: lie in free space Differential constraints: have bounded curvature Global constraints: have minimal length
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Motion-Planning Framework Continuous representation Discretization Graph searching (blind, best-first, A*)
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Example: Visibility Graph (A Roadmap Method) Visibility graph Introduced in the Shakey project at SRI in the late 60s. Can produce shortest paths in 2-D configuration spaces g s
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Example: Voronoi Diagram (A Roadmap Method) Voronoi diagram Introduced by Computational Geometry researchers. Generate paths that maximizes clearance. O(n log n) time O(n) space
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Cell-Decomposition Methods Two classes of methods: Exact cell decomposition Approximate cell decomposition F is represented by a collection of non-overlapping cells whose union is contained in F Examples: quadtree, octree, 2 n -tree
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Approximate Cell Decomposition: Quad Tree
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Octree Decomposition (3D environment)
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Potential Field Methods Goal Robot Approach initially proposed for real-time collision avoidance [Khatib, 86]. Hundreds of papers published on it.
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Attractive and Repulsive fields
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Potential Fields
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Local-Minimum Issue Perform best-first search (possibility of combining with approximate cell decomposition) Alternate descents and random walks Use local-minimum-free potential (navigation function)
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Ex: 2D Cartesian manipulator The configuration space is R 2 Consider only one object in the workspace –End effector and obstacle are convex polygons What is the configuration space obstacle?
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Ex: 2D Cartesian manipulator The nice thing about this example is that the workspace and the configuration space are identical
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Ex: planar two-link manipulator What is the configuration space obstacle for a two-link manipulator
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Motivation Geometric complexity Space dimensionality
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Path planning overview Want to find a path from an initial position to a final position
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Potential fields To develop the mapping, we incrementally explore Q free Consider the manipulator (statically) as a point in the configuration space The manipulator is subject to a potential field –Attractive in the case of the goal configuration –Repulsive in the case of an obstacle
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Gradient descent In order to find minima of U, take the negative gradient:
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The attractive field We define a potential field that attracts each of the n DH coordinate frames from the initial position to the goal position
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The attractive field Simple potential field, conic well potential
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The attractive field Instead we use a continually differentiable function: parabolic well potential –Field grows quadratically with the distance from the goal configuration
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Hybrid attractive field Combine the conic well potential and parabolic well potential fields –If the i th frame is close to the workspace goal, use the parabolic well –If the i th frame is far from the workspace goal, use the conic well The distance d defines the distance from the goal that causes a transition from a conic to parabolic potential Since this is continuous everywhere, the workspace force is defined everywhere
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Hybrid attractive field Taking the gradient gives the workspace attractive force
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Ex: planar two link manipulator For the 2-link arm shown below, assume that both links have length 1
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The repulsive field Prevent collisions by creating a repulsive force in the workspace –Again, create forces that act on the origins of the n DH coordinate frames These forces should: –Repel the robot from obstacles –Do nothing of the robot is far away from obstacles
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The repulsive field Therefore, the workspace repulsive force is: To evaluate this, consider the distance function (o i (q)) as (x) where x is a three dimensional vector:
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The repulsive field So we can write this force as:
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Ex: planar two link manipulator Consider a convex obstacle close to o 2 –Obstacle is outside the distance of influence for o 1 –Again, the lengths are both 1 –Let b be the point on the obstacle closest to o 2 b = [2 0.5] T (o 2 (q s )) = 0.5 –Let 0 = 1 (no influence on o 1 ) –The initial repulsive force on o 2 is:
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Other considerations 1.what happens if either there are multiple objects, or an object is not convex?
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Other considerations 2.what if the obstacle is closest to another part of a link (i.e. not the origin of the DH frame)?
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The relation between workspace forces and joint torques
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Ex: two-link planar manipulator Consider the previous examples with an obstacle exerting a repulsive force on o 2 Find the attractive and repulsive forces on o 1 and o 2 Initial and goal configurations Obstacle location
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Ex: two-link planar manipulator To determine the joint torques, take the transpose of the Jacobians at the initial configuration
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Composing workspace forces The total joint torques acting on a manipulator is the sum of the torques from all attractive and repulsive potentials:
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Ex: two-link planar manipulator Consider again the two-link manipulator with a goal position and an obstacle near o 2 The total joint torque, due to these two potential fields is: Initial and goal configurationsObstacle location
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Gradient descent Path Planning Algorithm 1.First, determine your initial configuration 2.Second, given a desired point in the workspace, calculate the final configuration using the inverse kinematics –Use this to create an attractive potential field 3.Locate obstacles in the workspace –Create a repulsive potential field 4.Sum the joint torques in the configuration space 5.Use gradient descent to reach your target configuration
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Local minima In the absence of obstacles, the gradient descent will always converge to the global minimum (q f ) With obstacles, by proper choice of i, this will always converge to some minima
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Local minima Instead we modify the gradient descent algorithm to add a random excitation in case we are stuck in a local minima We are stuck in a local minima if successive iterations result in minimal changes in the configuration If so, perform a random walk to get out The random walk is defined by adding a uniformly distributed variable to each joint parameter
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Next class… Applications to numerically solving for the inverse kinematics Probabilistic methods
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