Like any field with a big body of knowledge, robotics has its own jargon that is ever evolving and takes practice to get used to. Also, some concepts are close to each other, yet different, and they tend to be more easily confused. The goal of this page is to act like a cheat sheet, stressing the differences between concepts and giving you the pointers to dig deeper.
Lexicon
Let us first look at the most important concepts.
Configuration:
A vector q \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
\bfq q of the configuration space , that is, a
vector of generalized coordinates (a.k.a. joint coordinates) that
characterizes the positions of all points in the multi-body model of our
robot.
Orientation:
The angular coordinates of a rigid body in space. Not exactly the same
thing as a rotation, although there is an isomorphism between rotations and
orientations (similarly to the isomorphism between positions and
translations). Rotation matrices are one way to represent orientation,
others are listed below.
Pose:
Both position and orientation of a rigid body. Not exactly the same thing
as a transformation, although there is an isomorphism between poses and
their transformation group.
Position:
The linear coordinates p ∈ R 3 \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
\bfp \in \mathbb{R}^3 p ∈ R 3 of a body in space.
Not exactly the same thing as a translation, although there is an
isomorphism between the Euclidean space R 3 \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
\mathbb{R}^3 R 3 and its
translation group .
Rotation:
A transformation R ∈ S O ( 3 ) \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
R \in SO(3) R ∈ SO ( 3 ) that acts vectors e ∈ R 3 \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
\bfe \in
\mathbb{R}^3 e ∈ R 3 and preserves their lengths and the angles between them. The
rotation matrix R A B \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
\bfR_{AB} R A B represents equivalently the orientation
of frame B \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
B B in frame A \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
A A or the rotation R : B e ↦ A e \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
R : {}_B \bfe
\mapsto {}_A \bfe R : B e ↦ A e .
Translation:
A transformation t : R 3 → R 3 \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
t : \mathbb{R}^3 \to \mathbb{R}^3 t : R 3 → R 3 that maps the
position p \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
\bfp p of a rigid body to a new position t ( p ) \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
t(\bfp) t ( p ) .
Transform:
Shorthand for a Direct isometry T ∈ S E ( 3 ) \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
T \in SE(3) T ∈ SE ( 3 ) . Due to the bijection
between Poses and S E ( 3 ) \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
SE(3) SE ( 3 ) , it is common to read "the transform of a
rigid body", although more precise expressions are "the pose of a rigid
body" and "the transform from frame A to frame B".
Full package
Here comes a wider list of expressions you will find in the literature. Follow the links or scroll down for extra info on each:
Affine transformation:
A geometric transformation A : R 3 → R 3 \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
A : \mathbb{R}^3 \to \mathbb{R}^3 A : R 3 → R 3 that
preserves lines and parallelism. Direct isometries S E ( 3 ) \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
SE(3) SE ( 3 ) used to
represent the Pose of a rigid body are, in particular, affine
transformations. Affine transformations are conveniently represented by
4 × 4 \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
4 \times 4 4 × 4 matrices of Homogenous coordinates .
Attitude:
Synonym for orientation, this word is more common in aeronautics such as in
attitude and heading reference system .
Body:
Shorthand for Rigid body .
Direct isometry:
A geometric transformation T ∈ S E ( 3 ) \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
T \in SE(3) T ∈ SE ( 3 ) that preserves distances and
handedness. Direct isometries form a mathematical group S E ( 3 ) \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
SE(3) SE ( 3 )
called the special Euclidean group. They are the most commonly used
representation for the Pose of rigid bodies.
Displacement:
Synonym for Direct isometry .
Heading:
A vector that hints at the direction of motion, especially for systems such
as cars or humands that have a preferred direction of motion (forward). For
these two, a heading vector can be any vector in the intersection between
the sagittal plane and
median plane .
Homogeneous coordinates:
Convenient way to represent
an Affine transformation as a 4 × 4 \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
4 \times 4 4 × 4 matrix.
Joint:
Connection between two rigid bodies that constrains their relative degree
of freedom. The most common in robotics is the revolute joint actuated by a servomotor.
Joint coordinates:
Synonym for Configuration . Sometimes used to distinguish actuated
coordinates (e.g. a robot's joints) from unactuated degrees of freedom
(e.g. the floating base of a legged robot).
Link:
Rigid body of a robot, connected to at least one Joint .
Transformation:
In general, this expression may refer to any geometric transformation , for example an
Affine transformation . In robotics, this expression is generally used as
a shorthand for a Direct isometry T ∈ S E ( 3 ) \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
T \in SE(3) T ∈ SE ( 3 ) .
Representing orientation
Because all points on a rigid body hold, by definition, the same relative positions between each other, we can describe them all with just six numbers: the position of any one of them, and the three degrees of freedom of the body's orientation . There are several ways to represent orientations. Three common ones are:
Euler angles : only three angles, but tricky to work with because of the gimbal lock and the many different conventions (intrinsic or extrinsic rotations? upward or downward z-axis?)
Unit quaternions : four numbers, better behaved than Euler angles with only one convention to figure out (are the coordinates ordered as [ w x y z ] \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
[w\ x\ y\ z] [ w x y z ] or [ x y z w ] \def\LdG{\dot{L}_G}
\def\Ld{\dot{L}}
\def\bfA{\boldsymbol{A}}
\def\bfB{\boldsymbol{B}}
\def\bfC{\boldsymbol{C}}
\def\bfD{\boldsymbol{D}}
\def\bfE{\boldsymbol{E}}
\def\bfF{\boldsymbol{F}}
\def\bfG{\boldsymbol{G}}
\def\bfH{\boldsymbol{H}}
\def\bfI{\boldsymbol{I}}
\def\bfJ{\boldsymbol{J}}
\def\bfK{\boldsymbol{K}}
\def\bfL{\boldsymbol{L}}
\def\bfM{\boldsymbol{M}}
\def\bfN{\boldsymbol{N}}
\def\bfO{\boldsymbol{O}}
\def\bfP{\boldsymbol{P}}
\def\bfQ{\boldsymbol{Q}}
\def\bfR{\boldsymbol{R}}
\def\bfS{\boldsymbol{S}}
\def\bfT{\boldsymbol{T}}
\def\bfU{\boldsymbol{U}}
\def\bfV{\boldsymbol{V}}
\def\bfW{\boldsymbol{W}}
\def\bfX{\boldsymbol{X}}
\def\bfY{\boldsymbol{Y}}
\def\bfZ{\boldsymbol{Z}}
\def\bfalpha{\boldsymbol{\alpha}}
\def\bfa{\boldsymbol{a}}
\def\bfbeta{\boldsymbol{\beta}}
\def\bfb{\boldsymbol{b}}
\def\bfcd{\dot{\bfc}}
\def\bfchi{\boldsymbol{\chi}}
\def\bfc{\boldsymbol{c}}
\def\bfd{\boldsymbol{d}}
\def\bfe{\boldsymbol{e}}
\def\bff{\boldsymbol{f}}
\def\bfgamma{\boldsymbol{\gamma}}
\def\bfg{\boldsymbol{g}}
\def\bfh{\boldsymbol{h}}
\def\bfi{\boldsymbol{i}}
\def\bfj{\boldsymbol{j}}
\def\bfk{\boldsymbol{k}}
\def\bflambda{\boldsymbol{\lambda}}
\def\bfl{\boldsymbol{l}}
\def\bfm{\boldsymbol{m}}
\def\bfn{\boldsymbol{n}}
\def\bfomega{\boldsymbol{\omega}}
\def\bfone{\boldsymbol{1}}
\def\bfo{\boldsymbol{o}}
\def\bfpdd{\ddot{\bfp}}
\def\bfpd{\dot{\bfp}}
\def\bfphi{\boldsymbol{\phi}}
\def\bfp{\boldsymbol{p}}
\def\bfq{\boldsymbol{q}}
\def\bfr{\boldsymbol{r}}
\def\bfsigma{\boldsymbol{\sigma}}
\def\bfs{\boldsymbol{s}}
\def\bftau{\boldsymbol{\tau}}
\def\bftheta{\boldsymbol{\theta}}
\def\bft{\boldsymbol{t}}
\def\bfu{\boldsymbol{u}}
\def\bfv{\boldsymbol{v}}
\def\bfw{\boldsymbol{w}}
\def\bfxi{\boldsymbol{\xi}}
\def\bfx{\boldsymbol{x}}
\def\bfy{\boldsymbol{y}}
\def\bfzero{\boldsymbol{0}}
\def\bfz{\boldsymbol{z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\d#1{{\rm d}{#1}}
\def\defeq{\stackrel{\mathrm{def}}{=}}
\def\dim{\rm dim}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xdd{\ddot{x}}
\def\xd{\dot{x}}
\def\ydd{\ddot{y}}
\def\yd{\dot{y}}
\def\zdd{\ddot{z}}
\def\zd{\dot{z}}
[x\ y\ z\ w] [ x y z w ] ?).
Rotation matrices : nine numbers, no confusion regarding conventions, applies to position coordinates by direct matrix product.
It is best practice to avoid using Euler angles in software (colorful experimental and bug-hunting stories abound). In legged robotics, the most common convention is roll-pitch-yaw convention with upward z-axis. It corresponds to intrinsic Z-Y-X angles, or equivalently extrinsic x-y-z angles: first roll around the fixed x-axis of the parent frame, then pitch around the fixed y-axis of the parent frame, finally yaw around the fixed z-axis of the parent frame.
To go further, check out e.g. Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors by James Diebel.
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