An information based feedback control for audio-motor binaural localization

Abstract

In static scenarios, binaural sound localization is fundamentally limited by front-back ambiguity and distance non-observability. Over the past few years, “active” schemes have been shown to overcome these shortcomings, by combining spatial binaural cues with the motor commands of the sensor. In this context, given a Gaussian prior on the relative position to a source, this paper determines an admissible motion of a binaural head which leads, on average, to the one-step-ahead most informative audio-motor localization. To this aim, a constrained optimization problem is set up, which consists in maximizing the entropy of the next predicted measurement probability density function over a cylindric admissible set. The method is appraised through geometrical arguments, and validated in simulations and on real-life robotic experiments.

Appendix

Consider the posterior state pdf $p(x_k|z_{1:k})$ of the sensor-to-source position at time k, and ${\mathcal {N}}(x_k;{\hat{x}}_{k|k},P_{k|k})$ the approximate Gaussian belief. This pdf can be mapped into the 1D Gaussian approximation ${\mathcal {N}}(z_{k+1};\hat{z}_{k+1|k},S_{k+1|k})$ of the predicted measurement pdf $p(z_{k+1}|z_{1:k})$, by using the unscented transform. The aim is then to maximize the variance $S_{k+1|k}$ so as to increase the entropy $h(z_{k+1}|z_{1:k})$. This involves the composition of several functions.

First the sigma-points $\left\{ X_{i}^{-}\right\}$ corresponding to ${p(x_{k}|z_{1:k}) = {\mathcal {N}}(x_k;{\hat{x}}_{k|k},P_{k|k})}$ are computed from the posterior mean ${\hat{x}}_{k|k}$ of the state vector at time k and the Cholesky decomposition $P_{k|k} = L_{k|k}L_{k|k}^T$ of the posterior covariance:

$$\begin{aligned} \left\{ X_{i}^{-}\right\} = {\text {Sigma}}\_{\text {points}} \left( {\hat{x}}_{k|k},L_{k|k} \right) \end{aligned}$$

(15)

The sigma-points $\left\{ X_{i}^{+}\right\}$ of the next predicted state pdf $p(x_{k+1}|z_{1:k}) = {\mathcal {N}}(x_k;{\hat{x}}_{k+1|k},P_{k+1|k})$ can be obtained by applying the translation and rotation on each sigma point in the set $\left\{ X_{i}^{-}\right\}$. Note that (2) is defined as a function of $(T_y,T_z,\phi )$, so that

$$\begin{aligned} \forall i,\ X_{i}^{+} = \Phi _{X_{i}^-}(T_y,T_z,\phi ). \end{aligned}$$

(16)

Then the set of sigma-points $\left\{ Z_{i}^+\right\}$ of the predicted measurement pdf $p(z_{k+1}|z_{1:k}) = {\mathcal {N}}(z_k;\hat{z}_{k+1|k},S_{k+1|k})$ can be obtained from $\left\{ X_{i}^+\right\}$ defined in (16) by:

$$\begin{aligned} \forall i,\ Z_{i}^+ = l\left( -\mathrm {atan2}\left( X_{i}^+(1),X_{i}^+(2) \right) \right) , \end{aligned}$$

(17)

with $X_{i}^+(1)$ and $X_{i}^+(2)$ the components of $X_{i}^+$, and $l(\cdot )$ the measurement equation used to guide the exploration. Finally the mean $\hat{z}_{k+1|k}$ and variance $S_{k+1|k}$ of $p(z_{k+1}|z_{1:k})$ are computed by

$$\begin{aligned} \hat{z}_{k+1|k}= & {} \sum _i w_m^{i}Z_{i}^+\end{aligned}$$

(18)

$$\begin{aligned} S_{k+1|k}= & {} \sum _i w_c^{i}\left( Z_{i}^+ - \hat{z}_{k+1|k}\right) ^2, \end{aligned}$$

(19)

with $\left\{ w_m^i\right\}$ and $\left\{ w_c^i\right\}$ the classic weights of the unscented transform.

The log of the variance $S_{k+1|k}$ comes as a function of the finite translation and rotation, i.e., $\log S_{k+1|k} = {F}_{k}(T_y,T_z,\phi )$. However the maximum of this function is not analytically tractable. Its gradient around ${U} = (T_y,T_z,\phi )$ is then computed as follows.

The first order Taylor expansion of the functions $\Phi _{X_{i}^-}$, $\mathrm {atan2}$, l, and $\log$, are composed around U with infinitesimal translations and rotation ${du} = (dT_y, dT_z, d\phi )^T$:

$$\begin{aligned}&\Phi _{X_{i}^-}({U} + {du}) = \Phi _{X_i^-}({U}) + {J{\Phi _{X_i^-}}}({U}) \, {du} \nonumber \\&\mathrm {atan2}(u,v){}={}\mathrm {atan2}(u_0,v_0){}+{}{{\nabla }^T\mathrm {atan2}}(u_0,v_0) \, \begin{pmatrix} u - u_0\\ v-v_0 \end{pmatrix} \nonumber \\&l(w) = l(w_0) + l'(w_0)(w-w_0) \nonumber \\&\log (r) = \log (r_0) + \frac{1}{r_0}(r-r_0) \end{aligned}$$

(20)

with ${\nabla }$ the gradient operator. $J{\Phi _{X_i^-}}({U})$ is the Jacobian of $\Phi _{X_i^-}$ at U. Then the result of the composition, noted $Z_{i}(dT_y,dT_z,d\phi )$, is used to retrieve the mean and the variance with (18) and (19). Finally, the first order Taylor expansion of $F_k(dT_y,dT_z,d\phi )$ is obtained, highlighting the gradient ${\nabla }F_k$:

$$\begin{aligned} F_k\left( {{U} + {du}}\right) = F_k\left( {U}\right) + {\nabla }^TF_k ({U}) \, {du}. \end{aligned}$$

(21)