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Subsections

jacobian

Syntax

ReturnMatrix jacobian(const int ref=0);

Description

The manipulator Jacobian defines the relation between the velocities in joint space $\dot{\mbox{\boldmath$ q $}}$ and in the Cartesian space $\dot{\mbox{\boldmath$ \chi $}}$ expressed in frame $i$:
$\displaystyle {}^i \dot{\mbox{\boldmath$ \chi $}}$ $\textstyle =$ $\displaystyle {}^i \mbox{\boldmath$ J $}(\mbox{\boldmath$ q $}) \dot{\mbox{\boldmath$ q $}}$ (2.41)

or the relation between small variations in joint space $\delta \mbox{\boldmath$ q $}$ and small displacements in the Cartesian space $\delta \mbox{\boldmath$ \chi $}$:
$\displaystyle {}^i \delta \mbox{\boldmath$ \chi $}$ $\textstyle \approx$ $\displaystyle {}^i \mbox{\boldmath$ J $}(\mbox{\boldmath$ q $}) \delta \mbox{\boldmath$ q $}$ (2.42)

The manipulation Jacobian expressed in the base frame is given by (see [8])
$\displaystyle {}^0 \mbox{\boldmath$ J $}(\mbox{\boldmath$ q $})$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}
{}^0 \mbox{\boldmath$ J $}_1(\mbox{\bol...
...\cdots & {}^0 \mbox{\boldmath$ J $}_n(\mbox{\boldmath$ q $}) \end{array}\right]$ (2.43)

with
$\displaystyle {}^0 \mbox{\boldmath$ J $}_i(\mbox{\boldmath$ q $})$ $\textstyle =$ $\displaystyle \left[\begin{array}{c} \mbox{\boldmath$ z $}_{i-1} \times {}^{i-1...
...  \mbox{\boldmath$ z $}_{i-1} \end{array}\right]
\mbox{ for a revolute joint}$ (2.44)
$\displaystyle {}^0 \mbox{\boldmath$ J $}_i(\mbox{\boldmath$ q $})$ $\textstyle =$ $\displaystyle \left[\begin{array}{c} \mbox{\boldmath$ z $}_{i-1}   0 \end{array}\right] \mbox{ for a prismatic joint}$ (2.45)

where $\mbox{\boldmath$ z $}_{i-1}$ and ${}^{i-1} \mbox{\boldmath$ p $}_n$ are expressed in the base frame and $\times$ is the vector cross product. Expressed in the $i^{th}$ frame, the Jacobian is given by
$\displaystyle {}^i \mbox{\boldmath$ J $}(\mbox{\boldmath$ q $})$ $\textstyle =$ $\displaystyle \left[\begin{array}{cc}
({}^0 \mbox{\boldmath$ R $}_i)^T & 0   ...
... R $}_i)^T
\end{array}\right] {}^0 \mbox{\boldmath$ J $}(\mbox{\boldmath$ q $})$ (2.46)

This function returns ${}^i \mbox{\boldmath$ J $}(\mbox{\boldmath$ q $})$ ($i=0$ when not specified).

Return Value

Matrix


next up previous contents
Next: jacobian_dot Up: The Robot and mRobot Previous: inv_kin_puma   Contents
Richard Gourdeau 2004-07-06