Robot Motion Planning and Computer Vision
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Robot Motion Planning and Computer Vision

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Questions and Answers

What is the primary focus of this chapter?

  • Impact of uncertainty in motion planning
  • Robot arms and their assembly tasks
  • Operative intelligence in computers
  • Overview of robot motion planning (correct)

    • What concept is a unifying concept throughout the book?

    Configuration space

    Motion planning involves ________________ of particular planning methods.

    analyzing the complexity

    The basic motion planning problem includes dynamic properties of the robot.

    <p>False</p> Signup and view all the answers

    What is the workspace of a robot typically denoted as?

    <p>W</p> Signup and view all the answers

    What does the obstacle region, denoted by 'B', represent?

    <p>The union of all obstacles</p> Signup and view all the answers

    Motion planning is simply about collision checking.

    <p>False</p> Signup and view all the answers

    In motion planning, the robot decides automatically what motions to execute in order to achieve a task specified by initial and goal spatial ____________ of physical objects.

    <p>arrangements</p> Signup and view all the answers

    How is the orientation '0' described in terms of parameters?

    <p>N x N matrix whose columns are the components, in Fw, of the unit vectors along the axes of FA</p> Signup and view all the answers

    What is an atlas of C in this context?

    <p>A set of charts covering C</p> Signup and view all the answers

    The Whitney Embedding Theorem states that any r-dimensional manifold can be embedded in R2r.

    <p>True</p> Signup and view all the answers

    In the parameterization of SO(2), an orientation is represented by a single angle ___.

    <p>theta</p> Signup and view all the answers

    What is one of the ultimate goals in robotics?

    <p>Creating autonomous robots</p> Signup and view all the answers

    Define motion planning in the context of robotics.

    <p>Motion planning is the process through which a robot decides what motions to perform in order to achieve goal arrangements of physical objects.</p> Signup and view all the answers

    Motion planning primarily involves collision checking and avoidance.

    <p>False</p> Signup and view all the answers

    The concept of __________ space is used throughout the book to organize various facets of motion planning.

    <p>configuration</p> Signup and view all the answers

    What is the mathematical representation used to denote the inner product in the principal axis basis?

    <p>Denoted by '&lt; , &gt;' in Tq(C)</p> Signup and view all the answers

    What does the six-dimensional vector F represent?

    <p>Force F in the principal axis basis of Tq(C)</p> Signup and view all the answers

    An orthonormal basis is used when N = 3 in the principal axis basis.

    <p>True</p> Signup and view all the answers

    If It ∩ h = I 0, then N1 and N2 are connected by a ____ in G.

    <p>link</p> Signup and view all the answers

    What limitations of the freeway method are mentioned in the content?

    <p>All of the above</p> Signup and view all the answers

    What method is suggested as an alternative to the freeway method?

    <p>Silhouette method</p> Signup and view all the answers

    The Silhouette method can be used for solving problems involving multiple robots.

    <p>True</p> Signup and view all the answers

    What is the definition of a convex polygonal decomposition K of Cfree?

    <p>A finite collection of convex polygons (cells) whose interiors do not intersect and the union of all cells is equal to Cfree.</p> Signup and view all the answers

    How are two cells, K and K', in K defined as adjacent?

    <p>If K ∩ K' is a line segment of non-zero length.</p> Signup and view all the answers

    The output of the algorithm for planning a free path connecting initial configuration qinit and goal configuration qgoal is a sequence K1,..., Kp of cells such that qinit ∈ K1, qgoal ∈ Kp and for every j ∈ [1, p - 1], Kj and ___ are adjacent.

    <p>Kj+1</p> Signup and view all the answers

    What is the worst-case number of links in the reduced visibility graph when the C-obstacle region consists of c disjoint convex polygons with n vertices?

    <p>c * n</p> Signup and view all the answers

    What is the key method suggested for transforming a semi-free path into a free path in a two-dimensional configuration space with polygonal C-obstacles and a manifold boundary?

    <p>Propose a simple method</p> Signup and view all the answers

    What is a Voronoi cell in the context of point sets and how is it defined?

    <p>Voronoi cell is a set bounded by arcs of the Voronoi diagram and straight segments joining nodes</p> Signup and view all the answers

    What does the roadmap of a one-dimensional set represent?

    <p>The set itself</p> Signup and view all the answers

    The Minkowski sum of two convex generalized polygons is not a convex generalized polygon.

    <p>False</p> Signup and view all the answers

    What is the applicability condition of the type A contact between E;t and bj?

    <p>iif(q)· (b j - 1 - bj ) ~ 0 and iif(q)· (bj+1 - bj ) ~ 0</p> Signup and view all the answers

    Which type of contact is defined by the expression: fi~(q) = z7t(q).(bj - ai(q))?

    <p>Type A contact</p> Signup and view all the answers

    What is the purpose of the C-obstacle CB being completely within the closed half-space determined by J/j(q)?

    <p>To maintain the type A contact between Ef and bj</p> Signup and view all the answers

    A type B contact between ai and Ef is dependent on both q and e.

    <p>True</p> Signup and view all the answers

    The surface along which the robot's configuration moves during the displacement is called a C-surface of type ___.

    <p>B</p> Signup and view all the answers

    Study Notes

    Robot Motion Planning Book Overview

    • The book focuses on the central theme of motion planning, which is necessary for autonomous robots to achieve goal arrangements of physical objects.
    • Motion planning involves deciding what motions to perform to achieve a goal, which is a challenging problem in robotics.

    Configuration Space Concept

    • The concept of configuration space is used throughout the book to organize the various facets of motion planning in a coherent framework.
    • Configuration space treats the robot as a point in an appropriate space, representing the geometry of the task.
    • Physical concepts, such as force and friction, can be represented in this space as additional geometrical constructs.

    Book Structure

    • The book consists of 11 chapters, with the first half focusing on the basic motion planning problem and the second half exploring extensions of this problem.
    • Chapters 2 and 3 develop the notion of configuration space for a rigid object that can translate and rotate freely among obstacles in a two- or three-dimensional workspace.
    • Chapters 4-7 describe four computational approaches for solving the basic motion planning problem: the roadmap approach, exact and approximate cell decomposition approaches, and the potential field approach.
    • Chapters 8-11 present various extensions of the basic motion planning problem, including moving obstacles, multiple robots, nonholonomic kinematic constraints, uncertainty, and motion planning with movable objects.

    Key Concepts and Techniques

    • The book covers key concepts and techniques in motion planning, including:
      • Collision-free path planning
      • Obstacles in configuration space
      • Roadmap methods
      • Exact and approximate cell decomposition
      • Potential field methods
      • Kinematic constraints
      • Dealing with uncertainty
      • Motion planning with movable objects

    Assumed Knowledge

    • The book assumes good "undergraduate-level" knowledge in mathematics (geometry, topology, algebra) and algorithms.
    • A glossary of mathematical definitions is provided in Appendix A for the reader's convenience.
    • Background in robotics and mechanics is also assumed.### Notations and Definitions
    • An element of a configuration (Le.a configuration) is denoted by q
    • The region of the workspace occupied by the robot A at configuration q is denoted by A(q)
    • An obstacle B in the workspace maps to a region called C-obstacle and denoted by CB
    • The free space in configuration space is denoted by Cfree, the contact space by Ccontact, and Cvalid = Cfree U Ccontact is the valid space
    • A path is denoted by r, and it is usually a function of a parameter denoted by s that takes its values in the interval [0,1]
    • A vector is denoted by a symbol with an arrow on top of it, e.g. v, and its modulus is written ||v||
    • The inner product of two vectors v1 and v2 is denoted by or v1.v2, and the outer product of v1 and v2 is denoted by v1 A v2
    • The angle between v1 and v2 is written angle(v1, v2)
    • The distance between two points x and y of a Euclidean space is denoted by ||x - y||
    • An interval in R bounded by a and b is denoted by [a, b] if it is closed at both ends, and by (a, b) if it is open at both ends
    • The symbols Ef) and e are the Minkowski operators for affine set addition and subtraction, respectively
    • Let E be a topological space and F be a subset of it. cl(F), int(F), and B(F) denote the closure, the interior, and the boundary of F, respectively
    • Most of the time, calligraphic letters, e.g. S, denote sets
    • Symbols in typewriter type style are used to denote boolean expressions or predicates, e.g. CB, Achieve

    Aspects of Motion Planning

    • A robot is a versatile mechanical device, e.g. a manipulator arm, a multi-joint multi-fingered hand, a wheeled or legged vehicle, a free-flying platform, or a combination of these
    • A robot operates in a workspace within the real world, populated by physical objects and subject to the laws of nature
    • The robot performs tasks by executing motions in the workspace
    • The capability of planning its own motions, i.e. deciding automatically what motions to execute in order to achieve a task, is a major undertaking in robotics

    Basic Problem

    • The basic motion planning problem is defined as:
      • Let A be a single rigid object - the robot - moving in a Euclidean space W, called workspace, represented as R^N, with N = 2 or 3
      • Let B1, ..., Bq be fixed rigid objects distributed in W
      • The problem is: Given an initial position and orientation and a goal position and orientation of A in W, generate a path r specifying a continuous sequence of positions and orientations of A avoiding contact with the Bi's, starting at the initial position and orientation, and terminating at the goal position and orientation
      • Report failure if no such path exists

    Configuration Space Formulation

    • The configuration space of A is the space C of all the configurations of A
    • A configuration q of A is a specification of the position T and orientation θ of FA with respect to Fw
    • The subset of W occupied by A at configuration q is denoted by A(q)
    • The point a on A at configuration q is denoted by a(q) in W
    • C is a differentiable manifold of dimension m, where m = N(N + 1)

    Manifold Structure

    • C is a manifold of dimension m, where m = N(N + 1)
    • C can be covered by a finite number of charts
    • The charts in an atlas are C^∞-related, which allows us to extend the differential properties established in a chart of the atlas to all the other charts
    • The Whitney Embedding Theorem states that every r-dimensional manifold can be embedded in R^2r### Introduction to Configuration Space
    • Configuration space (C) is a space where a robot's position and orientation are represented as a single point
    • The concept of C-obstacles is introduced, which are regions in C where the robot collides with obstacles in the workspace

    Case of a Polygonal Workspace

    • The case of a polygonal workspace is examined, where a polygon A moves freely in the plane among obstacles modeled as polygonal regions
    • Two representations of C-obstacles are constructed: a predicate CB and an explicit description of the boundary of CB

    Polygonal Regions and Polygons

    • A convex polygonal region in R2 is defined as the intersection of a finite number of closed half-planes
    • A polygonal region is a subset of R2 obtained by taking the union of a finite number of convex polygonal regions
    • A polygon is a polygonal region that is homeomorphic to the closed unit disc
    • A polygon has a boundary that is a closed-loop sequence of line segments forming a Jordan curve

    Type A and Type B Contacts

    • Two types of contact between the robot A and an obstacle B are defined:
      • Type A contact: an edge of A contains a vertex of B
      • Type B contact: a vertex of A is contained in an edge of B
    • The type of contact is determined by the constraint that the interiors of A and B do not overlap
    • Type A contact requires a subrange of orientations of A for feasibility

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    This quiz covers topics related to robot motion planning, computer vision, and manipulation. It includes questions on robotic grasping, fine manipulation, shadows, and silhouettes in computer vision.

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