pytransform3d.transformations.adjoint_from_transform¶
- pytransform3d.transformations.adjoint_from_transform(A2B, strict_check=True, check=True)[source]¶
Compute adjoint representation of a transformation matrix.
The adjoint representation of a transformation
![\left[Ad_{\boldsymbol{T}_{BA}}\right] \in \mathbb{R}^{6 \times 6}](../_images/math/650abeef71355aa7e24a8fc28ba4f8c0a1f9bcd2.png)
from frame A to frame B translates a twist from frame A to frame B
through the adjoint map![\mathcal{V}_{B}
= \left[Ad_{\boldsymbol{T}_{BA}}\right] \mathcal{V}_A](../_images/math/e037125c7686e5ec1effdc1990d32203083d7812.png)
The corresponding transformation matrix operation is
![\left[\mathcal{V}_{B}\right]
= \boldsymbol{T}_{BA} \left[\mathcal{V}_A\right]
\boldsymbol{T}_{BA}^{-1}](../_images/math/acf9c2d701183c22959d90c80e7782db2747367f.png)
We can also use the adjoint representation to transform a wrench from frame A to frame B:
![\mathcal{F}_B
= \left[ Ad_{\boldsymbol{T}_{AB}} \right]^T \mathcal{F}_A](../_images/math/b962d61f9caadbd5b44c2e4f2ca1fc27f0767824.png)
Note that not only the adjoint is transposed but also the transformation is inverted.
Adjoint representations have the following properties:
![\left[Ad_{\boldsymbol{T}_1 \boldsymbol{T}_2}\right]
= \left[Ad_{\boldsymbol{T}_1}\right]
\left[Ad_{\boldsymbol{T}_2}\right]](../_images/math/6093cacac99fd633e42403a37434a33f54429b29.png)
![\left[Ad_{\boldsymbol{T}}\right]^{-1} =
\left[Ad_{\boldsymbol{T}^{-1}}\right]](../_images/math/f8065e5964f877027ee4cbfa2018af7882cfc2a3.png)
For a transformation matrix
the adjoint is defined as
![\left[Ad_{\boldsymbol{T}}\right]
=
\left( \begin{array}{cc}
\boldsymbol R & \boldsymbol 0\\
\left[\boldsymbol{t}\right]_{\times}\boldsymbol R & \boldsymbol R\\
\end{array} \right),](../_images/math/4cbe283a29298790eaadf36fb729886bb13182fe.png)
where
![\left[\boldsymbol{t}\right]_{\times}](../_images/math/3c121068e0910a717dd95a19d5afc9827ece7345.png)
is the cross-product
matrix (see cross_product_matrix()) of the translation component.- Parameters:
- A2Barray-like, shape (4, 4)
Transform from frame A to frame B
- strict_checkbool, optional (default: True)
Raise a ValueError if the transformation matrix is not numerically close enough to a real transformation matrix. Otherwise we print a warning.
- checkbool, optional (default: True)
Check if transformation matrix is valid
- Returns:
- adj_A2Barray, shape (6, 6)
Adjoint representation of transformation matrix
