95 lines
2.9 KiB
ReStructuredText
95 lines
2.9 KiB
ReStructuredText
.. Copyright 2018 The Cartographer Authors
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.. Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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.. http://www.apache.org/licenses/LICENSE-2.0
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.. Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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==============
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Cost functions
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==============
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Relative Transform Error 2D
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===========================
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Given two poses
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:math:`\mathbf{p_i} = [\mathbf{x_i}; \theta_i] = [x_i, y_i, \theta_i]^T`
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and :math:`\mathbf{p_j} = [\mathbf{x_j}; \theta_j] = [x_j, y_j, \theta_j]^T`
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the transformation :math:`\mathbf T` from the coordinate frame :math:`j` to the
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coordinate frame :math:`i` has the following form
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.. math::
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\mathbf{T}( \mathbf{p_i},\mathbf{p_j}) =
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\left[
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\begin{array}{c}
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R(\theta_i)^T (\mathbf x_j - \mathbf x_i) \\
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\theta_j-\theta_i
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\end{array}
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\right]
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where :math:`R(\theta_i)^T` is the rotation matrix of :math:`\theta_i`.
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The weighted error :math:`f:\mathbb R^6 \mapsto \mathbb R^3` between
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:math:`\mathbf T` and the measured transformation :math:`\mathbf T_{ij}^m =
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[\mathbf x_{ij}^m; \theta_j^m]` from the coordinate frame :math:`j` to the
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coordinate frame :math:`i` can be computed as
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.. math::
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\mathbf f( \mathbf{p_i},\mathbf{p_j}) =
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\left[
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w_{\text{t}} \; w_{\text{r}}
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\right]
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\left(
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\mathbf T_{ij}^m - \mathbf T( \mathbf{p_i},\mathbf{p_j})
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\right) =
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\left[
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\begin{array}{c}
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w_{\text{t}}\left(
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\mathbf x_{ij}^m - R(\theta_i)^T (\mathbf x_j - \mathbf x_i)
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\right) \\
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w_{\text{r}}\left(
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\mathrm{clamp}(\theta_{ij}^m - (\theta_j-\theta_i))
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\right)
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\end{array}
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\right]
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where :math:`w_t` and :math:`w_r` are weights for translation and rotation
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respectively and :math:`\mathrm{clamp}: \mathbb R \mapsto [-\pi, \pi]`
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normalizes the angle difference.
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Jacobian matrix :math:`J_f` is given by:
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.. math::
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\begin{align}
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J_f( \mathbf{p_i},\mathbf{p_j}) &=
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\left[
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\frac{\partial\mathbf f}{\partial x_i} \quad
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\frac{\partial\mathbf f}{\partial y_i} \quad
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\frac{\partial\mathbf f}{\partial \theta_i} \quad
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\frac{\partial\mathbf f}{\partial x_j} \quad
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\frac{\partial\mathbf f}{\partial y_j} \quad
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\frac{\partial\mathbf f}{\partial \theta_j}
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\right] \\
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&\mathstrut \\
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&=
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\left[
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\begin{array}{cccc}
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w_{\text{t}} R^T(\theta_i)
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& -w_{\text{t}} {\frac{\mathrm d R^T(\theta_i)}{\mathrm d \theta}}(\mathbf x_j - \mathbf x_i)
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& -w_{\text{t}} R^T(\theta_i)
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& \mathbf{0} \\
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\mathbf{0}^T
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& w_{\text{r}}
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& \mathbf{0}^T
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& -w_{\text{r}}
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\end{array}
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\right]
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\end{align}
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