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Single Robot Motion Planning. Liang-Jun Zhang COMP790-058 Sep 22, 2008. Motion planning is the ability for an agent to compute its own motions in order to achieve certain goals. All autonomous robots and digital actors should eventually have this ability. Piano Mover’s Problem.
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Single Robot Motion Planning Liang-Jun Zhang COMP790-058 Sep 22, 2008
Motion planning is the ability for an agent to compute its own motions in order to achieve certain goals. All autonomous robots and digital actors should eventually have this ability
Piano Mover’s Problem • 2D or 3D rigid models
Types of Robots • Rigid robots • Articulated robots Manipulator, Video Humanoid robots
Goal of Motion Planning • Compute motion strategies, e.g.: • geometric paths • time-parameterized trajectories • sequence of sensor-based motion commands • To achieve high-level goals,e.g.: • go from A to B without colliding with obstacles • assemble product P • build map of environment E • find object O
Sense Plan Move
Basic Problem • Statement:Compute a collision-free path for a rigid or articulated object among static obstacles • Inputs: • Geometry of moving object and obstacles • Kinematics of moving object (degrees of freedom) • Initial and goal configurations (placements) • Output: Continuous collision-free path connecting the initial and goal configurations
Types of Path Constraints • Local constraints • Collision-free paths • Differential constraints • A car cannot move sideways • Have bound curvature • Global constraints • Shortest or optimal paths Path Planning Motion Planning
Is It Easy? alpha puzzle
Outline (Mon & Wed) • Path planning for a point robot • Configuration space • Approximate cell decomposition • Sampling-based motion planning
g s Path Planning for a Point Robot
g s Visibility Graphs • Introduced in the Shakey project at SRI in the late 60s • Can produce shortest paths in a point robot in 2D
Simple (Naïve) Algorithm • Install all obstacles vertices in VG, plus the start and goal positions • For every pair of nodes u, v in VG • If segment(u,v) is an obstacle edge then • insert (u,v) into VG • else • for every obstacle edge e • if segment(u,v) intersects e • then goto 2 • insert (u,v) into VG • Search VG using BFS (any other graph search scheme)
g s Complexity • A point robot in 2D using visibility graphs • Simple algorithm: O(n3) time • Rotational sweep: O(n2 log n) • Optimal algorithm: O(n2) • Space: O(n2)
Motion Planning Framework • Motion planning: a search problem in continuous space • Continuous representation Discretization • Graph search
g s Issues With Visibility Graphs • Difficult to extend from point robots to rigid or articulated robots A L-shaped robot
Outline (Mon & Wed) • Path planning for a point robot • Configuration space • Approximate cell decomposition • Sampling-based motion planning
reference direction q y reference point x Example: rigid robot in 2-D workspace workspace robot • 3-parameter specification: q = (x, y, q) with q [0, 2p). • 3-D configuration space
qn q3 q=(q1, q2,…,qn) q1 q2 Configuration Space (C-Space) • The configuration of a moving object is a specification of the position of every point on the object. • Usually a configuration is expressed as a vector of position & orientation parameters: q = (q1, q2,…,qn). • Configuration space • C-space • The set of all possible configurations. • A configuration is a point in C-space.
Dimension of C-space • The dimension of a configuration space is the minimum number of parameters needed to specify the configuration of the object completely. • It is also called the number of degrees of freedom (dofs) of a moving object.
Example: Rigid Robot in 3-D Workspace • What is the configuration space?
z z z z y f y q y y y x x x x Example: rigid robot in 3-D workspace • q = (position, orientation) = (x, y, z, ???) • Number of dofs = 6 • Euler angles • (f,q,y) 1 2 34
C = S1x S1 Φ 2p Φ θ 2p 0 θ Topology of C-Space • The topology of C is usually not that of a Cartesian space Rn. Φ θ
Example: Rigid Robot in 3-D Workspace • Number of dofs = 6 • Topology: R3 x SO(3)
Example: An Articulated Robot Number of dofs = 3 C-space is 3 dimensional
An Articulated Robot Puma 560Number of dofs = 6 An articulated object is a set of rigid bodies connected at the joints.
Obstacle C-obstacle Obstacles in C-space Workspace Configuration Space Goal Free Robot y x Start A 2D Translating Robot
Obstacles in C-space • A configuration q is collision-free, or free, if a moving object placed at q does not intersect any obstacles in the workspace. • The free spaceF is the set of free configurations. • A configuration space obstacle (C-obstacle) is the set of configurations where the moving object collides with workspace obstacles.
Disc in 2-D Workspace workspace configuration space
Polygonal Robot Translating in 2-D Workspace configuration space workspace
Articulated Robot in 2-D Workspace workspace configuration space
Fundamental Question configuration space workspace Are two given points connected by a path?
Problem: Computing C-obstacles • Input: • Polygonal moving object translating in 2-D workspace • Polygonal obstacles • Output: configuration space obstacles represented as polygons
Minkowski Sum B A
Exercise B A A B = ?
Minkowski Sum • The Minkowski sum of two sets A and B, denoted by AB, is defined asA B= { a+b | aA, bB } • Similarly, the Minkowski difference is defined as A– B= { a–b | aA, bB } q p
P-M Configuration Space Obstacle • If P is an obstacle in the workspace and M is a translating object. Then the C-space obstacle corresponding to P is P– M. C-obstacle Robot M Obstacle P Classic result by Lozano-Perez and Wesley 1979
Minkowski Sum of Convex Polygons • The Minkowski sum of two convex polygons A and B of m and n vertices respectively is a convex polygon A + B of m + nvertices. • The vertices of A + B are the “sums” of vertices of A and B.
Complexity of Minkowksi Sum • 2D convex polygons: O(n+m) • 2D non-convex polygons: O(n2m2) • Decompose into convex polygons (e.g., triangles or trapezoids), compute the Minkowski sums, and take the union • 3-D convex polyhedra: O(nm) • 3-D non-convex polyhedra: O(n3m3)
Complexity of Computing C-obstacles • 3D rigid robots with both translational and rotational DOF • 6D C-space • Arrangement of non-linear surfaces • High combinatorial complexity • Conclusion • Explicit computation of the boundary of C-obstacle is difficult and impractical for robots more than 3 DOFs
Wednesday’s Lecture • Path planning for a point robot • Configuration space • Approximate cell decomposition • Sampling-based motion planning