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Subsections

jacobian_DLS_inv

Syntax

ReturnMatrix jacobian_DLS_inv(const Real eps, const Real lambda_max, 
                              const int ref=0);

Description

This function returns the inverse Jacobian Matrix for 6 dof manipulator based on the Damped Least-Squares scheme [10]. Using the singular value decomposition, the Jacobian matrix is


$\displaystyle J = \sum_{i=1}^6\sigma_i u_i v_i^T$     (2.52)

where $v_i$ and $u_i$ are the input and output vectors, and $\sigma_i$ are the singular values ordered so that $\sigma_i \geq \sigma_2 \geq \cdots \sigma_r \geq 0$, with $r$ being the rank of $J$. Based on the Damped Least-Squares the inverse Jacobian can be written as
$\displaystyle J^{-1} = \sum_{i=1}^6\frac{\sigma_i}{\sigma_i^2 + \lambda^2}v_iu_i^T$     (2.53)

where $\lambda$ is the damping factor. A singular region can be selected on the basis of the smallest singular value of J. Outside the region the exact solution is returned, while inside the region a configuration-varying damping factor is introduced to obtain the desired approximate solution. This region is defined as
$\displaystyle \lambda^2 = \Bigg\{
\begin{array}{cc}
0 & \textrm{if $\sigma_6 ...
...ac{\sigma_6}{\epsilon})^2 \Big)\lambda^2_{max} & \textrm{otherwise}
\end{array}$     (2.54)

Return Value

Matrix


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