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Questions and Answers
What is the primary focus of this chapter?
What is the primary focus of this chapter?
What concept is a unifying concept throughout the book?
What concept is a unifying concept throughout the book?
Configuration space
Motion planning involves ________________ of particular planning methods.
Motion planning involves ________________ of particular planning methods.
analyzing the complexity
The basic motion planning problem includes dynamic properties of the robot.
The basic motion planning problem includes dynamic properties of the robot.
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What is the workspace of a robot typically denoted as?
What is the workspace of a robot typically denoted as?
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What does the obstacle region, denoted by 'B', represent?
What does the obstacle region, denoted by 'B', represent?
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Motion planning is simply about collision checking.
Motion planning is simply about collision checking.
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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.
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.
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How is the orientation '0' described in terms of parameters?
How is the orientation '0' described in terms of parameters?
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What is an atlas of C in this context?
What is an atlas of C in this context?
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The Whitney Embedding Theorem states that any r-dimensional manifold can be embedded in R2r.
The Whitney Embedding Theorem states that any r-dimensional manifold can be embedded in R2r.
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In the parameterization of SO(2), an orientation is represented by a single angle ___.
In the parameterization of SO(2), an orientation is represented by a single angle ___.
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What is one of the ultimate goals in robotics?
What is one of the ultimate goals in robotics?
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Define motion planning in the context of robotics.
Define motion planning in the context of robotics.
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Motion planning primarily involves collision checking and avoidance.
Motion planning primarily involves collision checking and avoidance.
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The concept of __________ space is used throughout the book to organize various facets of motion planning.
The concept of __________ space is used throughout the book to organize various facets of motion planning.
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What is the mathematical representation used to denote the inner product in the principal axis basis?
What is the mathematical representation used to denote the inner product in the principal axis basis?
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What does the six-dimensional vector F represent?
What does the six-dimensional vector F represent?
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An orthonormal basis is used when N = 3 in the principal axis basis.
An orthonormal basis is used when N = 3 in the principal axis basis.
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If It ∩ h = I 0, then N1 and N2 are connected by a ____ in G.
If It ∩ h = I 0, then N1 and N2 are connected by a ____ in G.
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What limitations of the freeway method are mentioned in the content?
What limitations of the freeway method are mentioned in the content?
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What method is suggested as an alternative to the freeway method?
What method is suggested as an alternative to the freeway method?
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The Silhouette method can be used for solving problems involving multiple robots.
The Silhouette method can be used for solving problems involving multiple robots.
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What is the definition of a convex polygonal decomposition K of Cfree?
What is the definition of a convex polygonal decomposition K of Cfree?
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How are two cells, K and K', in K defined as adjacent?
How are two cells, K and K', in K defined as adjacent?
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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.
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.
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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?
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?
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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?
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?
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What is a Voronoi cell in the context of point sets and how is it defined?
What is a Voronoi cell in the context of point sets and how is it defined?
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What does the roadmap of a one-dimensional set represent?
What does the roadmap of a one-dimensional set represent?
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The Minkowski sum of two convex generalized polygons is not a convex generalized polygon.
The Minkowski sum of two convex generalized polygons is not a convex generalized polygon.
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What is the applicability condition of the type A contact between E;t and bj?
What is the applicability condition of the type A contact between E;t and bj?
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Which type of contact is defined by the expression: fi~(q) = z7t(q).(bj - ai(q))?
Which type of contact is defined by the expression: fi~(q) = z7t(q).(bj - ai(q))?
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What is the purpose of the C-obstacle CB being completely within the closed half-space determined by J/j(q)?
What is the purpose of the C-obstacle CB being completely within the closed half-space determined by J/j(q)?
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A type B contact between ai and Ef is dependent on both q and e.
A type B contact between ai and Ef is dependent on both q and e.
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The surface along which the robot's configuration moves during the displacement is called a C-surface of type ___.
The surface along which the robot's configuration moves during the displacement is called a C-surface of type ___.
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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
Studying That Suits You
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Description
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.
