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Lecture 2 Kinematic Foundations Lecture 2 Kinematic Foundations Preview Definitions and basic concepts System, configuration, rigid body, displacement Lecture 2 Rotation, translation Configuration space, Kinematic Foundations degrees of freedom Metrics Group theory Groups, commutative groups Matthew T. Mason Displacements with composition as a group Noncommutativity of displacements Mechanics of Manipulation Lecture 2 Outline Kinematic Foundations Preview Preview Definitions and basic concepts System, configuration, rigid body, displacement Rotation, translation Definitions and basic concepts Configuration space, degrees of freedom System, configuration, rigid body, displacement Metrics Rotation, translation Group theory Groups, commutative Configuration space, degrees of freedom groups Displacements with composition as a group Metrics Noncommutativity of displacements Group theory Groups, commutative groups Displacements with composition as a group Noncommutativity of displacements Lecture 2 Preview Kinematic Foundations The agenda Preview Definitions and I Today: general concepts. basic concepts System, configuration, rigid 2 body, displacement I Next: rigid motions in the Euclidean plane (E ). Rotation, translation Configuration space, Then: rigid motions in Euclidean three space ( 3), degrees of freedom I E Metrics 2 and the sphere (S ). Group theory Groups, commutative groups Displacements with composition as a group Noncommutativity of Why? displacements I Why spend so much time on such fundamental concepts? Because robotics is so challenging. “Give me six hours to chop down a tree and I will spend the first four sharpening the axe.” (Abraham Lincoln) Lecture 2 Preview Kinematic Foundations The agenda Preview Definitions and I Today: general concepts. basic concepts System, configuration, rigid 2 body, displacement I Next: rigid motions in the Euclidean plane (E ). Rotation, translation Configuration space, Then: rigid motions in Euclidean three space ( 3), degrees of freedom I E Metrics 2 and the sphere (S ). Group theory Groups, commutative groups Displacements with composition as a group Noncommutativity of Why? displacements I Why spend so much time on such fundamental concepts? Because robotics is so challenging. “Give me six hours to chop down a tree and I will spend the first four sharpening the axe.” (Abraham Lincoln) Lecture 2 Ambient space, system, configuration Kinematic Foundations Definitions Preview Definition (Ambient Definitions and basic concepts space) System, configuration, rigid body, displacement Rotation, translation Let X be the ambient space, Configuration space, 2 3 2 degrees of freedom either E , E , or S . Metrics Group theory Groups, commutative A system in the Euclidean groups Definition (System) Displacements with plane composition as a group Noncommutativity of A set of points in the space displacements X. Definition (Configuration) The configuration of a system gives the location of Same system, different every point in the system. configuration Lecture 2 Rigid bodies, displacements Kinematic Foundations Definitions Preview Definitions and Definition (Displacement) basic concepts System, configuration, rigid body, displacement A displacement is a change of configuration that Rotation, translation Configuration space, preserves pairwise distance and orientation degrees of freedom (handedness) of a system. Metrics Group theory Groups, commutative groups Displacements with Definition (Rigid body) composition as a group Noncommutativity of displacements A rigid body is a system that is capable of displacements only. Rotation (rigid), dilation (not rigid), reflection (not rigid) Lecture 2 Question Kinematic Foundations Why focus on rigid bodies? Preview Definitions and basic concepts I Nothing is rigid. System, configuration, rigid body, displacement I Lots of stuff is articulated or way soft: tissue Rotation, translation Configuration space, (surgery), fluids, food, paper, books, :::. degrees of freedom Metrics Group theory Groups, commutative groups Lots of reasons Displacements with composition as a group Noncommutativity of displacements I A reasonable approximation for some objects. I Even non-rigid transformations can be factored into displacement + shape change. I Some things are invariant with respect to displacements. I Other? Lecture 2 Moving and fixed spaces Kinematic Foundations Basic convention Preview Definitions and Convention basic concepts System, configuration, rigid We will consider displacements to apply to every point in body, displacement Rotation, translation Configuration space, the ambient space. degrees of freedom Metrics Group theory Example Groups, commutative groups Displacements with For example, planar displacements are described as composition as a group Noncommutativity of motion of moving plane relative to fixed plane. displacements Fixed plane Moving plane Moving and fixed planes. Lecture 2 Rotations and translations Kinematic Foundations Definitions Preview Definitions and Definition (Rotation) basic concepts System, configuration, rigid body, displacement A rotation is a displacement with at least one fixed point. Rotation, translation Configuration space, degrees of freedom Metrics Definition (Translation) Group theory Groups, commutative groups A translation is a displacement for which all points move Displacements with composition as a group equal distances along parallel lines. Noncommutativity of displacements O Rotation Rotation about a Rotation about a about O point on the body point not on the body Lecture 2 Configuration space, degrees of freedom Kinematic Foundations Definitions Preview Definitions and basic concepts System, configuration, rigid body, displacement Definition (Configuration space) Rotation, translation Configuration space, degrees of freedom Configuration space is the space comprising all Metrics configurations of a given system. Group theory Groups, commutative groups Displacements with composition as a group Noncommutativity of Definition (Degrees of freedom (DoFs)) displacements The Degrees of Freedom (DOFs) of a system is the dimension of the configuration space. (Less precisely: the number of reals required to specify a configuration.) Lecture 2 Systems, configuration spaces, DOFs Kinematic Foundations Examples System Configuration DOFs Preview Definitions and point in plane x, y 2 basic concepts System, configuration, rigid point in space x, y, z 3 body, displacement Rotation, translation rigid body in plane x, y, θ 3 Configuration space, degrees of freedom rigid body in space x, y, z, φ, θ, 6 Metrics rigid body in 4-space ??? ??? Group theory Groups, commutative groups Displacements with composition as a group Noncommutativity of r displacements (qu ") u /J*.,^btonl 6Pu<t' a^L|*J+, 5p*<*' 4 Lecture 2 Solution: DOFs of a rigid body in . Kinematic E Foundations I Most common answer: 4 translations and 4 rotations. I Why assume four rotational freedoms? Generalizing Preview from 3? Consider: Definitions and E basic concepts System, configuration, rigid body, displacement Space Translational DOFs Rotational DOFs Rotation, translation Configuration space, 0 degrees of freedom E (point) 0 0 Metrics 1 E (line) 1 0 Group theory 2 Groups, commutative E (plane) 2 1 groups 3 Displacements with 3 3 composition as a group E Noncommutativity of E4 4 6 displacements n I The correct generalization for E is n choose 2. Identify rotational freedoms not with a single axis, which works only in E3, but with a pair of axes. I The proof is simple, after we have covered rotation matrices. 4 I Rotations in E are useful. See the lecture on quaternions. Lecture 2 Configuration spaces Kinematic Foundations Metrics I We can define a metric, or a distance function, for Preview any configuration space. What is a metric? Definitions and basic concepts System, configuration, rigid body, displacement Definition Rotation, translation Configuration space, degrees of freedom A metric d on a space X is a function d : X × X 7! R Metrics satisfying: Group theory Groups, commutative groups I d(x; y) ≥ 0 (non-negativity); Displacements with composition as a group Noncommutativity of I d(x; y) = 0 if and only if x = y; displacements I d(x; y) = d(y; x) (symmetry); I d(x; z) ≤ d(x; y) + d(y; z) (triangle inequality). I Every space has a metric! But what would make a suitable metric for configuration space? How would you devise a suitable metric for configurations of S2? E2? E3? Lecture 2 Solution: metrics for Cspaces Kinematic Foundations I Possibilities: Preview 1. Let every pair of configurations have distance 1. Definitions and 2. For 2, the minimum angle to rotate one basic concepts S System, configuration, rigid configuration to the other. body, displacement n Rotation, translation 3. For decompose displacement into rotation and Configuration space, E degrees of freedom translation, and add the distance to the angle. Metrics 4. Same idea, but scale angle by a characteristic length. Group theory Groups, commutative 5. In general: pick some finite set of points, and take groups Displacements with the maximum distance travelled by the points. composition as a group Noncommutativity of displacements I (1): useless. (2–5): if two configurations are close, in the sense that corresponding points are close in the ambient space, then distance is small. (3): dimensionally inconsistent. I A desirable property: invariance with respect to displacements. (2) achieves this for S2. Unattainable for E2 and E3. Lecture 2 Digressing for group theory Kinematic Foundations Motivation Preview Definitions and basic concepts System, configuration, rigid body, displacement Rotation, translation Configuration
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