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60 lines
2.6 KiB
TeX
60 lines
2.6 KiB
TeX
\subsection{Wheeled robot Kinematics}
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\begin{wrapfigure}[7]{r}{0.2\columnwidth}
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\includegraphics[width=0.2\columnwidth]{assets/wheel-constraints.png}
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\end{wrapfigure}
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\bi{Non-holonomic} systems \textbf{not integrable}, no inst. move in every direct.
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\bi{Wheel constraints} $v_i = \omega_i r_i$
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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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\end{itemize}
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\bi{Maneuverability}
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\begin{itemize}
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\item Deg. of Mobility: $\delta_m = 3 - $\#constrained directions
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\item Deg. of Steerability: $\delta_s = $\#steerable wheels
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\item Deg. of Maneuverability: $\delta_M = \delta_m + \delta_s$
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\end{itemize}
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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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\begin{scriptsize}
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\begin{tabular}{llllll}
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Bicyle & Tricycle & Ackermann & Diff. Drive & Two-Steer & Three-Steer \\
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$\delta_m = 1$ & $\delta_m = 1$ & $\delta_m = 0$ & $\delta_m = 2$ & $\delta_m = 1$ & $\delta_m = 0$ \\
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$\delta_s = 1$ & $\delta_s = 1$ & $\delta_s = 2$ & $\delta_s = 0$ & $\delta_s = 2$ & $\delta_s = 3$ \\
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$\delta_M = 2$ & $\delta_M = 2$ & $\delta_M = 2$ & $\delta_M = 2$ & $\delta_M = 3$ & $\delta_M = 3$ \\
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\end{tabular}
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\end{scriptsize}
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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} $\vec{u} = [\omega_l, \omega_r]^\top$, 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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$\vec{b}_s = {\scriptsize \begin{bmatrix}
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D \cos(\theta) \\
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D \sin(\theta) \\
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0
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\end{bmatrix}}
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\quad \vec{b}_t = {\scriptsize \begin{bmatrix}
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R(\sin(\Delta \theta + \theta) - \sin(\theta)) \\
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-R(\cos(\Delta \theta + \theta) - \cos(\theta)) \\
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\Delta \theta
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\end{bmatrix}}$
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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 $i \in \{s, t\}$, respectively
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