> ## Documentation Index
> Fetch the complete documentation index at: https://aegean.ai/llms.txt
> Use this file to discover all available pages before exploring further.
# Motion Planning
> Odometry models and motion planning for mobile robots.
## Odometry Motion Model
Odometry is commonly obtained by integrating wheel encoders information; most commercial robots make such integrated pose estimation available in periodic time intervals (e.g., every tenth of a second).
Wheel displacements derived from encoder readings produce noisy pose update:
$p(x_t \mid x_{t-1}, u_t) = p(\widehat{\Delta_{\text{rot}_1}}) \, p(\widehat{\Delta_{\text{trans}}}) \, p(\widehat{\Delta_{\text{rot}_2}}),$
factorized by initial rotation, translation, final rotation.
Compute noisy components:
$$
\Delta_{\text{rot}_1} = \operatorname{atan2}(y_t - y_{t-1}, x_t - x_{t-1}) - \theta_{t-1}
$$
$$
\Delta_{\text{trans}} = \sqrt{(x_t - x_{t-1})^2 + (y_t - y_{t-1})^2}
$$
$$
\Delta_{\text{rot}_2} = \theta_t - \theta_{t-1} - \Delta_{\text{rot}_1}
$$
Sample corrupted increments:
$\widetilde{\Delta_{\text{rot}_1}} = \Delta_{\text{rot}_1} - \mathcal{N}(0, \alpha_1 \Delta_{\text{rot}_1}^2 + \alpha_2 \Delta_{\text{trans}}^2)$
and similarly for $\widetilde{\Delta_{\text{trans}}}$ and $\widetilde{\Delta_{\text{rot}_2}}$.
Updated pose:
$$
x_t = x_{t-1} + \widetilde{\Delta_{\text{trans}}} \cos(\theta_{t-1} + \widetilde{\Delta_{\text{rot}_1}})
$$
$$
y_t = y_{t-1} + \widetilde{\Delta_{\text{trans}}} \sin(\theta_{t-1} + \widetilde{\Delta_{\text{rot}_1}})
$$
$$
\theta_t = \theta_{t-1} + \widetilde{\Delta_{\text{rot}_1}} + \widetilde{\Delta_{\text{rot}_2}}
$$
## Velocity Motion Model
Given commanded $(v,\omega)$ and noise parameters $\alpha_i$:
$$
\hat{v} = v + \mathcal{N}(0, \alpha_1 v^2 + \alpha_2 \omega^2)
$$
$$
\hat{\omega} = \omega + \mathcal{N}(0, \alpha_3 v^2 + \alpha_4 \omega^2)
$$
$$
\hat{\gamma} = \mathcal{N}(0, \alpha_5 v^2 + \alpha_6 \omega^2)
$$
then sample the next pose $(x_t,y_t,\theta_t)$ via standard velocity model equations.
$$
x_t = x_{t-1} - \frac{\hat{v}}{\hat{\omega}}\sin \theta_{t-1} + \frac{\hat{v}}{\hat{\omega}}\sin(\theta_{t-1}+\hat{\omega} \Delta t)
$$
$$
y_t = y_{t-1} + \frac{\hat{v}}{\hat{\omega}}\cos \theta_{t-1} - \frac{\hat{v}}{\hat{\omega}}\cos(\theta_{t-1}+\hat{\omega} \Delta t)
$$
$$
\theta_t = \theta_{t-1} + \hat{\omega} \Delta t + \hat{\gamma} \Delta t
$$
If $|\hat{\omega}| < \epsilon$ treat as straight motion:
$$
x_t = x_{t-1} + \hat{v} \Delta t \cos \theta_{t-1}, \quad y_t = y_{t-1} + \hat{v} \Delta t \sin \theta_{t-1}, \quad \theta_t = \theta_{t-1} + \hat{\gamma} \Delta t
$$
***
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