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

$$v_d\int_1^t\gamma_d+\sum_1^t v_t = v_d \theta + \sum_1^t v_t$$