`pytransform3d.uncertainty`.concat_locally_uncertain_transforms¶

pytransform3d.uncertainty.concat_locally_uncertain_transforms(mean_A2B, mean_B2C, cov_A, cov_B)[source]¶

Concatenate two independent locally uncertain transformations.

We assume that the two distributions are independent.

Each of the two transformations is locally uncertain (not in the global / world frame), that is, samples are generated through

$\boldsymbol{T} = \overline{\boldsymbol{T}} Exp(\boldsymbol{\xi}),$

where $\boldsymbol{T} \in SE(3)$ is a sampled transformation matrix, $\overline{\boldsymbol{T}} \in SE(3)$ is the mean transformation, and $\boldsymbol{\xi} \in \mathbb{R}^6$ are exponential coordinates of transformations and are distributed according to a Gaussian distribution with zero mean and covariance $\boldsymbol{\Sigma} \in \mathbb{R}^{6 \times 6}$ , that is, $\boldsymbol{\xi} \sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{\Sigma})$ .

The concatenation order is the same as in concat(), that is, the transformation B2C is left-multiplied to A2B. Note that the order of arguments is different from concat_globally_uncertain_transforms().

Hence, the full model is

$\overline{\boldsymbol{T}}_{CA} Exp(_A\boldsymbol{\xi'}) = \overline{\boldsymbol{T}}_{CB} Exp(_B\boldsymbol{\xi}) \overline{\boldsymbol{T}}_{BA} Exp(_A\boldsymbol{\xi}),$

where $_B\boldsymbol{\xi} \sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_B)$ , $_A\boldsymbol{\xi} \sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_A)$ , and $_A\boldsymbol{\xi'} \sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_{A,total})$ .

This version of Meyer et al. approximates the covariance up to 2nd-order terms.

Parameters:

mean_A2Barray, shape (4, 4): Mean of transform from A to B: $\overline{\boldsymbol{T}}_{BA}$ .
mean_B2Carray, shape (4, 4): Mean of transform from B to C: $\overline{\boldsymbol{T}}_{CB}$ .
cov_Aarray, shape (6, 6): Covariance of noise in frame A: $\boldsymbol{\Sigma}_A$ . Noise samples are right-multiplied with the mean transform A2B.
cov_Barray, shape (6, 6): Covariance of noise in frame B: $\boldsymbol{\Sigma}_B$ . Noise samples are right-multiplied with the mean transform B2C.

Returns:

mean_A2Carray, shape (4, 4): Mean of new pose.
cov_A_totalarray, shape (6, 6): Covariance of accumulated noise in frame A.

Examples using `pytransform3d.uncertainty.concat_locally_uncertain_transforms`¶

Probabilistic Product of Exponentials

pytransform3d.uncertainty.concat_locally_uncertain_transforms¶

Examples using pytransform3d.uncertainty.concat_locally_uncertain_transforms¶

`pytransform3d.uncertainty`.concat_locally_uncertain_transforms¶

Examples using `pytransform3d.uncertainty.concat_locally_uncertain_transforms`¶