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19 lines
862 B
TeX
19 lines
862 B
TeX
\subsection{Forward Kinematics (FK)}
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$\mat{T}_{WB_n}(\vec{\theta}) = \mat{T}_{WB_0} \mat{T}_{B_0B_1}(\theta_1) \cdots \mat{T}_{B_{n - 1}B_n}(\theta_n)$.
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For 2R system: ${_W}\vec{t}_{WE} =$
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{\scriptsize
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$ \begin{bmatrix}
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L_1 \cos(\theta_1) + L_2 \cos(\theta_1 + \theta_2) \\
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L_1 \sin(\theta_1) + L_2 \sin(\theta_1 + \theta_2)
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\end{bmatrix}$\\
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}
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Workspace $W$: $\theta_1, \theta_2 \in [-\pi, \pi]$.
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\bi{Computation} Similar for $n$R sys (more angles, more lengths).
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Last dim is typically sum of angles (or equiv).
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\bi{For velocity kinematics}: See \ref{sec:manipulator-velocity-kinematics}.
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% Comp. Jacobian of this $nD$ position ($nD$ pos is vec). ${_W}\vec{t}_{WE} \mat{J}(\vec{\theta}) \cdot \vec{\dot{\theta}}$
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% \bi{Jacobian}: see \ref{sec:ms-lin}.
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\shortdefinition[Singularity] Loss of deg of Freedom $\det(\mat{J}(\vec{\theta})) = 0$
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