Math Concepts in Robotics and Controls

Motivation

When reading robotics and controls literature, sometimes one would encounter mathematical concepts or terminologies for which the definitions are primarily found in mathematics. Typically, authors employ them to rigorously define things, and imply the mathematical properties that come with them so that they can show and prove some new theorems. Yet, at times, these foreign and abstract concepts can cause confusion and break the flow of reading.

This post summarizes some these concepts and puts them in one place for future reference. Note that the summary here are neither rigorous nor comprehensive. They are just the bare minimum to help readers have a conceptual understanding so that they won't break their flow of reading.

In order to prevent this post from growing too long and tedious, it covers only a few items. More will be discussed in separate posts. For literature reference, see 2015@Abbott , 2016@Strang . For media reference, see Wikipedia@Set , Wikipedia@Field , Wikipedia@VecSpa , Wikipedia@Ring , MIT lecture notes on Topology.

Set

Definitions

A set is the mathematical model for a collection of different things. A set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

• Empty set - the set with no elements.
• Singleton - a set with a single element.
• Finite set - a set with finite number of elements.
• Infinite set - a set with infinite number of elements.

Sets are uniquely characterized by their elements:

• two sets that have precisely the same elements are equal and same;
• which implies that there's only one empty set.

Notations

Roster (Enumeration) notation:

• finite sets: $A=\{1,2,3\}$, $B=\{a,b,c,d\}$, $C=\{1, 2, \cdots, 10\}$;
• infinite sets: $D=\{1, 2, \cdots\}$, $E=\{\cdots, -2, -1, 0\}$, $F=\{\cdots,-2, -1, 0, 1, 2, \cdots\}$

Semantic notation:

• Let $G$ be the set of all positive integers that are less than 10.

Set builder notation:

• $H=\{x \, | \, x \in \mathbb{R}, x \leq 3 \}$

Nomenclature

• $\varnothing, \{\}$ - the empty set
• $\in, \notin$ - in, not in
• $\subseteq, \nsubseteq$ - subset, not subset
• $\subset, \not\subset$ - proper subset, not proper subset
• $\supseteq, \nsupseteq$ - superset, not superset
• $\supset, \not\supset$ - proper superset, not proper superset
• $\cap, \cup$ - intersection, union

• $\forall$ - for all
• $\wedge, \vee$ - and, or
• $:=$ - equals by definition
• $\triangleq$ - equals by definition
• $|$ - such that
• $:$ - such that
• $\exists, \nexists$ - exists, does not exist
• $\mapsto$ - maps to
• $\mathbb{Q}, \mathbb{Z}, \mathbb{R}, \mathbb{C}$ - the set of all rational numbers, integers, real numbers, complex numbers
• $\mathbb{R}^+, \mathbb{R}^-$ - the set of all positive real numbers, negative real numbers

• $\mathcal{P}(A), 2^{A} = \{ B : B \subset A \}$ - power set of $A$, a set that includes all the subsets including the empty set and the original set itself
• $A \,\triangle\, B = (A \cup B) \,\backslash\, (A \cap B)$, - disjoint union, or symmetric difference, the set of elements which are in either of the sets, but not in their intersection
• $A \,\backslash\, B = \{x : x \in A, x \notin B\}$ - difference, or relative components set, a set that has elements in A and not in B
• $A \times B = \{ (a, b) : a \in A \,\,\text{and}\,\, b \in B \}$ - Cartesian product

Functions versus Operations

Take for example binary functions and operations with some sets $A, B, C, S$:

• $f:A\times B \mapsto C$ - is a binary function,
• $g:S\times S \mapsto S$ - is a binary operation.

Field

Roughly speaking, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. Formally, a field is a set $F$ together with two binary operations, addition $+: F \times F \mapsto F$ and multiplication $\cdot: F \times F \mapsto F$ that satisfy the following axioms for arbitrary elements $a, b, c \in F$.

Under addition, $F$ satisfies:

• [1] Associativity of addition: $a+(b+c) = (a+b)+c$.
• [2] Commutativity of addition: $a+b=b+a$
• [3] Additive identity: there exists a unique element $0\in F$, such that $a+0=a$.
• [4] Additive inverses: for every $a$, there exists an element in $F$, denoted $-a$, such that $a + (-a) = 0$.

Under multiplication, $F$ satisfies:

• [5] Associativity of multiplication: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$.
• [6] Commutativity of multiplication: $a \cdot b = b \cdot a$
• [7] Multiplication identity: there exists a unique element $1\in F$, such that $a\cdot1=a$.
• [8] Multiplication inverses: for every $a \neq 0$, there exists an element in $F$, denoted by $a^{-1}$, such that $a \cdot a^{-1} = 1$.

Under mixed operations, $F$ satisfies:

• [9] Distributivity of multiplication over addition: $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$.

For example, rational numbers $\mathbb{Q}$, real numbers $\mathbb{R}$, and complex numbers $\mathbb{C}$ are fields.

Vector Space (Linear Space)

Definitions

A vector space over a field $F$ is a non-empty set $V$ together with:

• a binary operation - vector addition $g:V \times V \mapsto V$, and
• a binary function - scalar multiplication $f:F \times V \mapsto V$

that satisfy axioms listed below. Note that $\forall a\in F$, they satisfy the axioms defined in Field. Elements in $V$ are called vectors, elements in $F$ are called scalars.

Axioms

For any vectors $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$, and a unique zero vector $\mathbf{0}\in V$, the vector addition satisfies:

• [1], [2], [3], [4], same as that of a Field.

For any scalar $a, b \in F$, and a unique multiplication identity scalar $1\in F$, the scalar multiplication satisfies:

• associativity of multiplication: $a(b\mathbf{v}) = (ab)\mathbf{v}$,
• multiplication identity: $1\mathbf{v} = \mathbf{v}$.

For mixed function and operations, it satisfies:

• distributivity of scalar multiplication over vector addition: $a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}$,
• distributivity of scalar multiplication over field addition: $(a+b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}$.

Implications

With the axioms defined above, vector spaces derive further concepts, especially the linear ones, that come with many proven nice mathematical properties. Such as bases, vector coordinates, subspaces, linear combination, linear independence, linear subspace, linear span, etc.

Ring

Roughly speaking, rings are algebraic structures that generalize fields; multiplication need not be commutative and multiplicative inverses need not exist. Formally, a ring is a set $R$ equipped with two binary operations, addition and multiplication, $\forall a, b, c \in R$, satisfying the following axioms denoted in Fields.

• Under addition, $R$ satisfies [1], [2], [3], [4], same as that of a Field.
• Under multiplication, $R$ satisfies [5] and [7], different from that of a Field.
• Under mixed operations, $R$ satisfies [9], same as that of a Field.

For example, the set of all integers $\mathbb{Z}$ is a ring, but not a field, because the inverse of an integer may not be an integer. Because of the difference between a Ring and a Field, they have different mathematical properties. Note that satisfying less axioms than Fields does not mean that Rings have less properties.

Hierarchy of Mathematical Spaces

            flowchart TD

            Set[Set]
            Topological[Topological Space]
            Metric[Metric Space]
            Vector["Vector Space (Linear Space)"]
            Normed[Normed Vector Space]
            Banach[Banach Space]
            Inner[Inner Product Space]
            Hilbert[Hilbert Space]

            click Set "https://en.wikipedia.org/wiki/Set_(mathematics)"
            click Topological "https://en.wikipedia.org/wiki/Topological_space"
            click Metric "https://en.wikipedia.org/wiki/Metric_space"
            click Vector "https://en.wikipedia.org/wiki/Vector_space"
            click Normed "https://en.wikipedia.org/wiki/Normed_vector_space"
            click Banach "https://en.wikipedia.org/wiki/Banach_space"
            click Inner "https://en.wikipedia.org/wiki/Inner_product_space"
            click Hilbert "https://en.wikipedia.org/wiki/Hilbert_space"

            Set -- Openness --> Topological
            Topological -- Distance --> Metric
            Topological -- Linearity --> Vector
            Metric -- Linearity --> Normed
            Vector -- Distance --> Normed
            Normed -- Completeness --> Banach
            Normed -- Inner Product --> Inner
            Banach -- Inner Product --> Hilbert
            Inner -- Completeness --> Hilbert