Lecture 2 Kinematic Foundations

Home , Euclidean group, Translation (geometry)

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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

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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

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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

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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

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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

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

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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, ψ 6 Metrics rigid body in 4-space ??? ??? Group theory Groups, commutative groups Displacements with composition as a group Noncommutativity of r displacements

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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

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Definitions and basic concepts System, configuration, rigid body, displacement Rotation, translation Configuration space, degrees of freedom I Why? If you can show that your mathematical Metrics Group theory construct is a group, then you can use algebra—you Groups, commutative groups can write and solve equations. Displacements with composition as a group Noncommutativity of I Displacements are a group, and we need to use displacements algebra on them. Lecture 2 Group Kinematic Foundations Definition

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A group is a set of elements X and a binary operator ◦ Definitions and basic concepts satisfying the following properties: System, configuration, rigid body, displacement I Closure: Rotation, translation Configuration space, degrees of freedom for all x and y in X, x ◦ y is in X. Metrics

I Associativity: Group theory Groups, commutative groups for all x, y, and z in X, (x ◦ y) ◦ z is equal to Displacements with composition as a group x ◦ (y ◦ z). Noncommutativity of displacements I Identity: there is some element, called 1, such that for all x in X x ◦ 1 = 1 ◦ x = x.

I Inverses: for all x in X, there is some element called x−1 such that x ◦ x−1 = x−1 ◦ x = 1. Lecture 2 Commutativity of groups Kinematic Foundations

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Definitions and basic concepts System, configuration, rigid body, displacement Rotation, translation Configuration space, degrees of freedom I Note that commutativity is not required for a group. Metrics Some are, some are not. Group theory Groups, commutative groups I A commutative group is also known as an Abelian Displacements with composition as a group Noncommutativity of group. displacements Lecture 2 Examples of groups Kinematic Foundations Example

Preview I Let the elements be the integers Z; Definitions and basic concepts System, configuration, rigid I let the operator be ordinary addition +. body, displacement Rotation, translation I Is it a group? Verify the properties. Configuration space, degrees of freedom Metrics I Is it commutative? Group theory Groups, commutative groups Displacements with Example composition as a group Noncommutativity of displacements I Let the elements be the reals R;

I Let the operator be multiplication ×.

I Is it a group?

I Other examples: positive rationals with multiplication (commutative); nonsingular k by k real matrices with matrix multiplication (noncommutative). Lecture 2 Displacements with composition are a group Kinematic Foundations

Preview I Every displacement D is an operator on the ambient Definitions and space X, mapping every point x to some new point basic concepts 0 System, configuration, rigid D(x) = x . body, displacement Rotation, translation Configuration space, I The product of two displacements is the composition degrees of freedom of the corresponding operators, i.e. Metrics Group theory (D2 ◦ D1)(·) = D2(D1(·)). Groups, commutative groups Displacements with I The inverse of a displacement is just the operator composition as a group Noncommutativity of that maps every point back to its original position. displacements

I The identity is the null displacement, which maps every point to itself. In other words: The displacements, with functional composition, form a group. Lecture 2 SE(2), SE(3), and SO(3) Kinematic Foundations Special Euclidean and Special Orthogonal groups

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Definitions and basic concepts System, configuration, rigid body, displacement These groups of displacements have names: Rotation, translation Configuration space, I SE(2): The Special Euclidean group on the plane. degrees of freedom Metrics 3 I SE(3): The Special Euclidean group on E . Group theory Groups, commutative groups I SO(3): The Special Orthogonal group. Displacements with composition as a group Noncommutativity of Whence the names? displacements I Special: they preserve orientation / handedness.

I Orthogonal: referring to the connection with orthogonal matrices, which will be covered later. Lecture 2 Noncommutativity of rotations. Kinematic Foundations

Does SO(3) commute? NO! No, no, no. Preview Definitions and z y basic concepts System, configuration, rigid body, displacement Rotation, translation Configuration space, degrees of freedom y y Metrics x z z x Group theory x Groups, commutative groups Displacements with composition as a group Noncommutativity of displacements

z y

z z y x x y x Lecture 2 Do displacements commute? Kinematic Foundations

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Definitions and basic concepts System, configuration, rigid body, displacement Rotation, translation Configuration space, degrees of freedom Does SE(3) commute? Metrics Group theory Does SE(2) commute? Groups, commutative groups Displacements with Does SO(2) commute? composition as a group Noncommutativity of displacements