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[AMR] updated first 4.5 sections

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janishutz
2026-08-08 12:31:01 +02:00
parent 7cdba7dca9
commit c1bdbc64bf
20 changed files with 103 additions and 55 deletions
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electives/amr/parts/01_kinematics/00_intro.tex
+8 -8
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@@ -41,17 +41,17 @@ $\begin{smallmatrix}
\varphi = \arcsin\left( R_{31} \div \sqrt{1 - R_{31}^2} \right)\\
\end{smallmatrix}$
$[\vec{n}]^\times = \begin{bmatrix}
0 & -a_3 & a_2 \\
a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0
0 & -n_3 & n_2 \\
n_3 & 0 & -n_1 \\
-n_2 & n_1 & 0
\end{bmatrix}$
For pitch axis $= \pm 90\deg$, Gimbal Lock, Jacobian singular.
For pitch axis $\theta = \pm 90\deg$, \bi{Gimbal Lock}, Jacobian \bi{singular}.
\shortdefinition[Angle-Axis]
$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); Convert to rot. mat\\
$\vec{\alpha} = \alpha \vec{n}$ ($\vec{n}$ normal); To \bi{rotation matrix}:\\
$\mat{R}(\alpha, \vec{n}) = \mat{I}_3 + \sin(\alpha)[\vec{n}]^\times + (1 - \cos(\alpha))([\vec{n}]^\times)^2$\\
To quat: $\vec{q} = [\vec{n}, \alpha]$
To \bi{quaternion}: $\vec{q} = [\vec{n}, \alpha]$
\shortdefinition[Quaternions] $q = q_w + q_x i + q_y j + q_z k$ with\\
@@ -68,14 +68,14 @@ $\vec{q} = \begin{bmatrix}
\end{bmatrix}$
}
\bi{To Rot Mat} {\scriptsize
\bi{To Rot Mat} {
$R_{AB} = \mat{I}_3 + 2a(\vec{q}_{AB}) [\vec{v}(\vec{q}_{AB})]^\times + 2([\vec{v}(\vec{q}_{AB})]^\times)^2$
}
\shortdefinition[Transf. M]
{\scriptsize
$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ R.-Handed typ; Aero: Left-H\\
$\mat{T}_{AC} = \mat{T}_{AB} \mat{T}_{BC}$ Right-Handed typically; Aero: Left-H\\
$\mat{T}_{AB} = \begin{bmatrix}
\mat{R}_{AB} & {_A}\vec{t}_B \\
\mat{0}_{1\times 3} & 1
electives/amr/parts/01_kinematics/01_forward.tex
+3 -2
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@@ -11,7 +11,8 @@ $ \begin{bmatrix}
Workspace $W$: $\theta_1, \theta_2 \in [-\pi, \pi]$.
\bi{Computation} Similar for $n$R sys (more angles, more lengths).
Last dim is typically sum of angles (or equiv).
\bi{For velocity kinematics}: Comp. Jacobian of this $nD$ position ($nD$ pos is vec). ${_W}\vec{t}_{WE} \mat{J}(\vec{\theta}) \cdot \vec{\dot{\theta}}$
\bi{Jacobian}: see \ref{sec:ms-lin}.
\bi{For velocity kinematics}: See \ref{sec:manipulator-velocity-kinematics}.
% Comp. Jacobian of this $nD$ position ($nD$ pos is vec). ${_W}\vec{t}_{WE} \mat{J}(\vec{\theta}) \cdot \vec{\dot{\theta}}$
% \bi{Jacobian}: see \ref{sec:ms-lin}.
\shortdefinition[Singularity] Loss of deg of Freedom $\det(\mat{J}(\vec{\theta})) = 0$
electives/amr/parts/01_kinematics/02_inverse.tex
+10
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@@ -13,3 +13,13 @@ $\theta_1 = \varphi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$
3. Solve for wrist joints (no infl. on pos)
\bi{For Inv. velocity}: $\vec{\dot{\theta}} = J_S^{-1}(\vec{\theta}) {_W} \vec{t}_{WE}$
% \shortdefinition[Denavit-Hartengerb (DH)] Serial chains
% \[
% \mat{T}_{L_j L_{j + 1}} = \mat{T}^{R_z}(\theta_j) \mat{T}^{t_z}(d_j) \mat{T}^{t_x}(a_j) \mat{T}^{R_x}(\alpha_j)
% \]
% with $
\shortdefinition[Underactuated] workspace $<6$D; IK solvable
\shortdefinition[Overactuated] $\8$ IK sol; Reduced singularities; Num. sol.
electives/amr/parts/01_kinematics/04_rigid-body-imu-kinematics.tex
+4 -4
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@@ -6,7 +6,7 @@
\bi{Rot. Velocity} ${_I}\vec{\omega}_{IB} = \dot{\alpha} \cdot {_I}\vec{n}$
\bi{Velocity point $P$} ${_B}\vec{v}_{IP} = {_B}\vec{v}_{IB} + {_B}\vec{\omega}_{IB} \times {_B}\vec{t}_{P}$
\bi{Vel. point $P$} ${_B}\vec{v}_{IP} = {_B}\vec{v}_{IB} + {_B}\vec{\omega}_{IB} \times {_B}\vec{t}_{P}$
\bi{Rotation Matrices}
\begin{itemize}
@@ -14,8 +14,8 @@
$\mat{\dot{R}}_{IB} = [{_I} \omega_{IB}]^\times \mat{R}_{IB}$
\item For right pertubing
$\mat{\dot{R}}_{IB} = \mat{R}_{IB} [{_I} \omega_{IB}]^\times$
\item Constant angular velocity ($\exp{[\Delta \vec{\alpha}]^\times} = \delta \mat{R}(\Delta \vec{\alpha})$)\\
$\mat{R}_{IB}(t + \Delta t) = \exp{[\Delta \vec{\alpha}]^\times} \mat{R}_{IB}(t) \quad \vec{\alpha} = {_I}\vec{\omega}_{IB} \delta t$
\item Constant angular velocity ($\exp{[\Delta \vec{\alpha}]^\times} = \delta \mat{R}(\Delta \vec{\alpha}) \quad \vec{\alpha} = {_I}\vec{\omega}_{IB} \delta t$)\\
$\mat{R}_{IB}(t + \Delta t) = \exp{[\Delta \vec{\alpha}]^\times} \mat{R}_{IB}(t)$
\end{itemize}
\bi{Quaternions}
@@ -56,4 +56,4 @@ where {\color{gray} gray parts} only IRL (in theor. models, leave out), with $\v
\bi{IMU Sensor Model}: $\vec{\tilde{z}} = \vec{b}_C + s\mat{M}\vec{z} + \vec{b} + \vec{n} + \vec{o}$
where bias $\vec{b}$ and scale $s$ often modelled time-varying $\dot{b}(t) = \sigma_C n(t)$.
$\vec{b}_C$ const. calib; $\mat{M}$ Misalignment; $\vec{n}$ noise; $\vec{o}$ other infl.
$\vec{b}_C$ const. calibration / bias; $\mat{M}$ Misalignment; $\vec{n}$ noise; $\vec{o}$ other infl.
electives/amr/parts/01_kinematics/05_rigid-body-dynamics.tex
+17 -4
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@@ -2,14 +2,12 @@
\label{sec:rigid-body-dynamics}
\shortdefinition[Newton II] For fin. body w/ mass $m$ and inertia mat. $I$, with force $\vec{F}$ and torque $\vec{T}$ on \bi{Centre of Mass} (CoM), in body frame:
\begin{align*}
{_B}\vec{F} & = \sum {_B}\vec{F}_i = m({_B} \vec{\dot{v}}_{CoM}) + m_B \vec{\omega} \times {_B}\vec{v}_{CoM} \\
{_B}\vec{T} & = \sum {_B}\vec{T}_i = \mat{I}({_B} \vec{\dot{\omega}}) + {_B} \vec{\omega} \times \mat{I}_B\vec{\omega}
{_B}\vec{F} & = m({_B} \vec{\dot{v}}_{CoM}) + m_B \vec{\omega} \times {_B}\vec{v}_{CoM} \\
{_B}\vec{T} & = \mat{I}({_B} \vec{\dot{\omega}}) + {_B} \vec{\omega} \times \mat{I}_B\vec{\omega}
\end{align*}
${_B} \vec{v}_{CoM}$ vel. of CoM, ${_B}\omega$ rot. speed; both w.r.t. inert. frame
{\scriptsize Legged robots don't need to maintain stability!}
To determine dynamics:
\bi{(1)} Define control inputs (e.g. wheel speeds),
\bi{(2)} Assumptions / Simplifications about state \& inputs
@@ -17,3 +15,18 @@ To determine dynamics:
\bi{(4)} Express as func of states / inp
\bi{(5)} Reduce to the CoM
\bi{(6)} Plug into Newton-Euler eq combined with rigid body kinematics
\subsection{Manipulator Velocity Kinematics}
\label{sec:manipulator-velocity-kinematics}
Differentiate FK $x = f(\vec{\theta})$, with $\dot{\vec{x}} = [{_W}\vec{v}, {_W} \vec{\omega}]^\top = \mat{J}(\vec{\theta})\dot{\vec{\theta}}$,
with $\mat{J} = \frac{\partial f}{\partial \vec{\theta}}$ manipulator Jacobian. Then $\dot{\vec{\theta}} = J^{-1} \vec{\dot{x}}$ or $\vec{\dot{\theta}} = J^+ \vec{\dot{x}}$,
if over-actuated, with $J^+ = J^\top(J J^\top)^{-1}$ (Moore Penrose-Inverse)
\newpage
\subsection{Legged robots}
{\scriptsize Legged robots don't need to maintain stability!}
\shortdefinition[Static Walking] 3 or more legs on ground, stable if frozen, safe, slow, inefficient.
\shortdefinition[Dynamic Walking] $<$ 3 legs on ground, falls when not moving, fast, efficient, hard.
electives/amr/parts/01_kinematics/06_wheeled-robot.tex
+11 -16
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@@ -1,11 +1,6 @@
\newpage
\subsection{Wheeled robot Kinematics}
\begin{wrapfigure}[7]{r}{0.2\columnwidth}
\includegraphics[width=0.2\columnwidth]{assets/wheel-constraints.png}
\end{wrapfigure}
\bi{Non-holonomic} systems \textbf{not integrable}, no inst. move in every direct.
\bi{Wheel constraints} $v_i = \omega_i r_i$ ($r_i$ constraints)
\bi{Wheel constraints} $v_i = \omega_i r_i$ ($r_i$ constraints, (wheel radii))
\begin{itemize}
\item \textit{Driving straight} all $\vec{v}$ equal (ICR: R.Cent.)
\item \textit{Turning} Wheel axis must intersect the \bi{Instant Centre of Rotation} (ICR) of vehicle,
@@ -40,16 +35,20 @@ $c_i = [\sin(\alpha + \beta) \div r, -\cos(\alpha + \beta) \div r, -\cos(\beta)
\end{tabular}
\end{scriptsize}
\shortdefinition[Differential Drive Kinematics]
\subsubsection{Differential Drive Kinematics}
\label{sec:diff-drive-kin}
\bi{State vec} $\vec{x} = [x_1, x_2, \theta]^\top$,
\bi{Inputs} $\vec{u} = [\omega_l, \omega_r]^\top$, radius of right (left) wheel $r_r$ ($r_l$), $w$ width of robot
\bi{Inputs} $\vec{u} = [\omega_l, \omega_r]^\top$, wheel radii $r_l, r_r$, $w$ width of robot
\bi{Gen. eq. of Motion} $\dot{x}_1 = v\cos(\theta)$, $\dot{x}_2 = v\sin(\theta)$, $\dot{\theta} = \Omega$,
with $v = 0.5\cdot(\omega_l r_l + \omega_r + r_r)$, $\Omega = \frac{\omega_r r_r - \omega_l r_l}{w}$
with $v = 0.5\cdot(\omega_l r_l + \omega_r r_r)$, $\Omega = \frac{\omega_r r_r - \omega_l r_l}{w}$
% TODO: Consider adding wheel constraints (planar) here as well (from W05 slides)
\textit{Straight}: $v = \omega_l r_l = \omega_r r_r$, $\Omega = 0$, $D = v\Delta t$.\\
\textit{Straight}: $v = \omega_l r_l = \omega_r r_r$, $\Omega = 0$, $D = v\Delta t$.
\textit{Turning}: $\Omega = (\omega_l r_l) / R_l\! =\! (\omega_r r_r) / R_r$, $R\! =\! v / \Omega$, $\Delta \theta\! =\! \Omega \Delta t$
\textbf{Discretized}: $\vec{x}_k = \vec{x}_{k - 1} + b_i$ with $i \in \{s, t\}$, respectively, with
$\vec{b}_s = {\scriptsize \begin{bmatrix}
D \cos(\theta) \\
D \sin(\theta) \\
@@ -59,11 +58,7 @@ $\vec{b}_s = {\scriptsize \begin{bmatrix}
R(\sin(\Delta \theta + \theta) - \sin(\theta)) \\
-R(\cos(\Delta \theta + \theta) - \cos(\theta)) \\
\Delta \theta
\end{bmatrix}}$
\textit{Turning}: $\Omega = (\omega_l r_l) / R_l\! =\! (\omega_r r_r) / R_r$, $R\! =\! v / \Omega$, $\Delta \theta\! =\! \Omega \Delta t$
\textbf{Discretized}: $\vec{x}_k = \vec{x}_{k - 1} + b_i$ with $i \in \{s, t\}$, respectively
\end{bmatrix}}$; $R$ dist ICR
\textbf{Swedish Wheels}: Have 3 DoF, typ. (3 pcs) equally spaced on circle.
Typ. Param. each: $\alpha, \beta, \gamma$ ($z$, $x$, $y$ axes, oriented along $x$, $z$ up)