# Rotation and Head Direction

### Path Integration

$$
d\_t = \gamma\_d \delta\_t+ v\_t
$$

where `\delta_d` is the true velocity, `\gamma_d` is the velocity gain, and `\v_t` is the noise

$$
P(\theta\_t|d\_{1:t-1})=\mathcal{N}(\theta\_t|m\_t,s\_t^2)
$$

thus d\_t is integral from 1 to t-1

$$
\sum\_1^t d\_t = \sum\_1^t(\gamma\_d \delta\_t+ v\_t)
$$

because gamma is constant

$$
\sum\_1^t(\gamma\_d \delta\_t+ v\_t) = v\_d\int\_1^t\gamma\_d+\sum\_1^t v\_t
$$

because the initial heading angle is 0, thus the sum velocity is the current heading angle

$$
\int\_1^t\gamma\_d = path =\theta
$$

and we get

$$
v\_d\int\_1^t\gamma\_d+\sum\_1^t v\_t = v\_d \theta + \sum\_1^t v\_t
$$


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