IK: Numerical methods
Several possible approaches to solve \(\mathrm{J}\,\Delta \theta = \Delta p\)
- - Pseudo Inverse
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- \(\Delta \theta = J^{+}\;\Delta p\;\), with \(\;\;\mathrm{J} \mathrm{J}^{+} = \mathrm{I}\)
- \(\mathrm{J} ^ {+} = \mathrm{J} ^ {T}\left(\mathrm{J}\,\mathrm{J}^{T}\right)^{-1} \)
- - Can also be computed using SVD: \(\mathrm{J} ^ {+} = \mathrm{V} \Sigma^{+}\,\mathrm{U}^T\)
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- \(\Sigma_{ii} = \sigma_{i}\), \(\Sigma^+_{ii}=1/\sigma_{i}\) if \(\sigma_i\neq 0\), \(0\) otherwise.
- - Adding damping to compensate for singularities
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- \(\Delta \theta = \mathrm{J} ^ {T}\left(\mathrm{J}\,\mathrm{J}^{T}+\lambda^2\,\mathrm{I}\right)^{-1}\;\Delta p\)