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[AMR] Improve kinematics summary
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@@ -4,7 +4,7 @@
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\end{wrapfigure}
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\bi{Option}: Solve Forward Kinematics for angles.\\
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\bi{Better}: Law of cosine with polar coordinates. Compute angle using cosine rule,\\
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$\theta_1 = \phi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$
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$\theta_1 = \varphi \pm \alpha$, $\theta_2 = \pm(\pi - \beta)$
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(Positive for {\color{ForestGreen} Elbow Down}, Negative for {\color{red} Elbow Up})
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@@ -1,14 +1,10 @@
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\subsection{Temporal Models}
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For \bi{Cont-time n.-lin. system of ODE} $\dot{\vec{x}} = \vec{f}_C(\vec{x}(t), \vec{u}(t))$, with measurements $\vec{z}(t) = \vec{h}(\vec{x}(t)) + \vec{v}(t)$.\\
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Need linearised (around $\vec{f}_C(\vec{\overline{x}}, \vec{\overline{y}}) = 0$, at \bi{equilibrium}):\\
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Model \bi{robot dyn} as \bi{Cont-time n.-lin. system of ODE} $\dot{\vec{x}} = \vec{f}_C(\vec{x}(t), \vec{u}(t), \vec{w}(t))$, with meas. $\vec{z}(t) = \vec{h}(\vec{x}(t)) + \vec{v}(t)$.\\
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Linearize around $\vec{f}_C(\vec{\overline{x}}, \vec{\overline{y}}) = 0$, at \bi{equilibrium}:\\
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$\delta \vec{\dot{x}}(t) = \vec{f}_C(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F}_C \delta \vec{x}(t) + \mat{G}_C \delta \vec{u}(t) + \mat{L}_C \vec{w}(t)$\\
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$\delta \vec{z}(t) = \mat{H} \delta \vec{x}(t) + \vec{v}(t)$.
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Herein, $\mat{H}$ is measurements, $\mat{F}_C$ system, $\mat{G}$ input gain, $\vec{w}$ process noise, $\vec{v}$ measurement noise, both zero-mean \bi{Gaussian White Noise Process}.
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For \bi{n-lin. cont-time system}:
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$\vec{\dot{x}}(t) = \vec{f}_C(\vec{x}(t), \vec{u}(t), \vec{w}(t))$\\
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$\vec{z}(t) = \vec{h}(\vec{x}(t)) = \vec{v})(t)$,
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linearization is the same
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$\mat{H}$ is meas., $\mat{F}_C$ system, $\mat{G}$ input gain, $\vec{w}$ process noise, $\vec{v}$ measurement noise,
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both zero-mean \bi{Gaussian White Noise Process}.
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To \bi{discretize}, integrate from $t_{k - 1}$ to $t_k$:\\
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$\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$
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@@ -17,7 +13,7 @@ $\vec{z}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$,
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$\delta \vec{x}_k = \vec{f}(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F} \delta \vec{x}_{k - 1} + \mat{G}_k \delta \vec{u}_k + \mat{L}_k \vec{w}_k$;
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$\delta \vec{z}_k = \mat{H}_k \delta \vec{x}_k$
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\bi{Trapezoidal num. int}
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\shortdefinition[Trapezoidal num. int]
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$\Delta \vec{x}_1 = \Delta t \vec{f}_C (\vec{x}_{k - 1}, \vec{u}_{k - 1}, t_{k - 1})$\\
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$\Delta \vec{x}_2 = \Delta t \vec{f}_C (\vec{x}_{k - 1} + \Delta \vec{x}_1, \vec{u}_{k}, t_{k})$, then:\\
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$\vec{x}_k = \vec{x}_{k - 1} + 0.5 \cdot (\Delta \vec{x}_1 + \Delta \vec{x}_2)$
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@@ -3,9 +3,9 @@
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\begin{wrapfigure}[5]{r}{0.3\columnwidth}
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\includegraphics[width=0.3\columnwidth]{assets/rigid-body-6d.png}
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\end{wrapfigure}
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\bi{Velocity} ${_I}\vec{v}_{IB} = \diff{t} ({_I}\vec{t}_B)$
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\bi{Velocity} ${_I}\vec{v}_{IB} = {_I}\dot{\vec{t}}_B$
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\bi{Rot. Velocity} ${_I}\vec{\omega}_{IB} = \diff{t} (\alpha)\; {_I}\vec{t}$
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\bi{Rot. Velocity} ${_I}\vec{\omega}_{IB} = \dot{\alpha} \cdot {_I}\vec{n}$
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\bi{Velocity point $P$} ${_B}\vec{v}_{IP} = {_B}\vec{v}_{IB} + {_B}\vec{\omega}_{IB} \times {_B}\vec{t}_{P}$
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@@ -15,35 +15,45 @@
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$\mat{\dot{R}}_{IB} = [{_I} \omega_{IB}]^\times \mat{R}_{IB}$
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\item For right pertubing
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$\mat{\dot{R}}_{IB} = \mat{R}_{IB} [{_I} \omega_{IB}]^\times$
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\item Constant angular velocity ($\exp{[\Delta \alpha]^\times} = \delta \mat{R}(\Delta \alpha)$)\\
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$\mat{R}_{IB}(t + \Delta t) = \exp{[\Delta \alpha]^\times} \mat{R}_{IB}(t)$
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\item Constant angular velocity ($\exp{[\Delta \vec{\alpha}]^\times} = \delta \mat{R}(\Delta \vec{\alpha})$)\\
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$\mat{R}_{IB}(t + \Delta t) = \exp{[\Delta \vec{\alpha}]^\times} \mat{R}_{IB}(t) \quad \vec{\alpha} = {_I}\vec{\omega}_{IB} \delta t$
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\end{itemize}
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\bi{Quaternions}
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\begin{itemize}
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\item For left pertubing
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$\displaystyle \vec{\dot{q}}_{IB} = \frac{1}{2} \begin{bmatrix}
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{_I}\vec{\omega}_{IB} \\
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0
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\end{bmatrix}
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$\vec{\dot{q}}_{IB} =
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\frac{1}{2}
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{\scriptsize
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\begin{bmatrix}
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{_I}\vec{\omega}_{IB} \\
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0
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\end{bmatrix}
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}
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\otimes \vec{q}_{IB}$
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\item For right pertubing
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$\displaystyle \vec{\dot{q}}_{IB} = \frac{1}{2} \vec{q}_{IB} \otimes \begin{bmatrix}
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{_B}\vec{\omega}_{IB} \\
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0
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\end{bmatrix}$
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$\vec{\dot{q}}_{IB} =
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\frac{1}{2} \vec{q}_{IB} \otimes
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{\scriptsize
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\begin{bmatrix}
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{_B}\vec{\omega}_{IB} \\
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0
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\end{bmatrix}
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}$
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\end{itemize}
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\bi{IMU} (Outputs {\color{blue} ${_S}\vec{\tilde{a}}$} (accel.), {\color{red} ${_S}\vec{\tilde{\omega}}$} (rot. accel.))\\
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${_W}\vec{\dot{t}}_S = {_W} \vec{v}$,
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$\displaystyle \vec{\dot{q}}_{WS} = \frac{1}{2} \vec{q}_{WS} \otimes
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\begin{bmatrix}
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{\color{red}{_S}\vec{\tilde{\omega}}} {\color{gray} + \vec{w}_g - \vec{b}_g} \\
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0
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\end{bmatrix}$
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${_W}\vec{\dot{t}}_S = {_W} \vec{v}$;
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$\quad \displaystyle \vec{\dot{q}}_{WS} = \frac{1}{2} \vec{q}_{WS} \otimes
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{\scriptsize
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\begin{bmatrix}
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{\color{red}{_S}\vec{\tilde{\omega}}} {\color{gray} + \vec{w}_g - \vec{b}_g} \\
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0
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\end{bmatrix}
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}$
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${_W}\vec{\dot{v}} = \mat{R}_{WS}\; ({\color{blue}{_S}\vec{\tilde{a}}} {\color{gray} + \vec{w}_a - \vec{b}_a}) + {_W}\vec{g}$
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where {\color{gray} gray parts} only IRL (in theor. models, leave out), with $\vec{\dot{b}}_g = \vec{w}_{b_g}$ and $\vec{\dot{b}}_a = \vec{w}_{b_a}$
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${_W}\vec{\dot{v}} = \mat{R}_{WS}\; ({\color{blue}{_S}\vec{\tilde{a}}} {\color{gray} + \vec{w}_a - \vec{b}_a}) + {_W}\vec{g}$
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where {\color{gray} gray parts} only IRL (in theor. models, leave out), with $\vec{\dot{b}}_g = \vec{w}_{b_g}$ and $\vec{\dot{b}}_a = \vec{w}_{b_a}$
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\bi{IMU Sensor Model}: $\vec{\tilde{z}} = \vec{b}_C + s\mat{M}\vec{z} + \vec{b} + \vec{n} + \vec{o}$
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where bias $\vec{b}$ and scale $s$ often modelled time-varying $\dot{b}(t) = \sigma_C n(t)$.
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@@ -6,4 +6,4 @@
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{_B}\vec{T} &= \sum {_B}\vec{T}_i = \mat{I}({_B} \vec{\dot{\omega}}) + {_B} \vec{\omega} \times \mat{I}_B\vec{\omega}
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\end{align*}
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${_B} \vec{v}_{CoM}$ vel. of CoM, ${_B}\omega$ rot. speed; both w.r.t. world frame
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${_B} \vec{v}_{CoM}$ vel. of CoM, ${_B}\omega$ rot. speed; both w.r.t. inert. frame
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@@ -9,7 +9,7 @@
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\begin{itemize}
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\item \textit{Driving straight} all $\vec{v}$ equal
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\item \textit{Turning} Wheel axis must intersect the \bi{Instant Centre of Rotation} (ICR),
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speeds: $v_i \div R_i = \Omega$ ($R_i$ dist. wheel-ICR, $\Omega$, vehicle body rotation rate)
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speeds: $v_i \div R_i = \Omega$ ($R_i$ = dist. wheel-ICR; $\Omega$: vehicle body rotation rate)
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\end{itemize}
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\bi{Maneuverability}
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@@ -19,7 +19,8 @@
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\item Deg. of Maneuverability: $\delta_M = \delta_m + \delta_s$
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\end{itemize}
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\bi{Wheel Configurations}
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\newpage
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\shade{ForestGreen}{Wheel Configurations}
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\includegraphics[width=1\columnwidth]{assets/wheel-config.png}
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@@ -32,27 +33,27 @@
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\end{tabular}
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\end{scriptsize}
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\bi{Differential Drive Kinematics}
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\shortdefinition[Differential Drive Kinematics]
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\bi{State vec} $\vec{x} = [x_1, x_2, \theta]^\top$,
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\bi{Inputs} $\vec{u} = [\omega_l, \omega_r]^\top$, $r_r$ radius of right wheel, $w$ width of robot
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\bi{Inputs} $\vec{u} = [\omega_l, \omega_r]^\top$, radius of right (left) wheel $r_r$ ($r_l$), $w$ width of robot
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\bi{Gen. eq. of Motion} $\dot{x}_1 = v\cos(\theta)$, $\dot{x}_2 = v\sin(\theta)$, $\dot{\theta} = \Omega$,
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with $v = 0.5\cdot(\omega_l r_l + \omega_r + r_r)$, $\Omega = \frac{\omega_r r_r - \omega_l r_l}{w}$
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% TODO: Consider adding wheel constraints (planar) here as well (from W05 slides)
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\textit{Straight}: $v = \omega_l r_l = \omega_r r_r$, $\Omega = 0$, $D = v\Delta t$.\\
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$\vec{b}_s = \begin{bmatrix}
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$\vec{b}_s = {\scriptsize \begin{bmatrix}
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D \cos(\theta) \\
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D \sin(\theta) \\
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0
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\end{bmatrix}$
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$\vec{b}_t = \begin{bmatrix}
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R(\sin(\Delta \theta + \theta) - \sin(\theta))\\
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-R(\cos(\Delta \theta + \theta) - \cos(\theta))\\
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\Delta \theta
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\end{bmatrix}$
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\end{bmatrix}}
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\quad \vec{b}_t = {\scriptsize \begin{bmatrix}
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R(\sin(\Delta \theta + \theta) - \sin(\theta)) \\
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-R(\cos(\Delta \theta + \theta) - \cos(\theta)) \\
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\Delta \theta
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\end{bmatrix}}$
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\textit{Turning}: $\Omega = (\omega_l r_l) / R_l\! =\! (\omega_r r_r) / R_r$, $R\! =\! v / \Omega$, $\Delta \theta\! =\! \Omega \Delta t$
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\textbf{Discretized}: $\vec{x}_k = \vec{x}_{k - 1} b_i$ with $i \in \{s, t\}$. ($\int \ldots \dx \Delta t$)
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\textbf{Discretized}: $\vec{x}_k = \vec{x}_{k - 1} b_i$ with $i \in \{s, t\}$, respectively
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