## 3 Acoustic SLAM Based on Microphone Landmarks
### System Model  
The kinematic model was applied for the trajectory of a vehicle with two rear wheels used as the driving unit and one universal wheel as the front wheel to describe the robot motion. The vehicle was subject to rolling motion constraints (i.e., assuming zero wheel slip). The vehicle model is shown in Figure 6. In this figure, XOY is the coordinate frame of the world, X\({}^{\prime}\)O\({}^{\prime}\)Y\({}^{\prime}\) is the coordinate frame attached to the vehicle base, \(\gamma\) represents the steering angle of the front wheel, \(\theta\) is the angle between the robot's body and the \(x\)-axis of the global coordinate system.  
The kinematic model of the robot is expressed as  
\[\mathbf{X_{\mathbf{h_{k}}}}=f_{r}\big{(}\mathbf{X_{\mathbf{h_{k-1}}}},\mathbf{ u_{k}}\big{)}=\left[\begin{array}{c}x_{\tau_{k-1}}+V_{k}\Delta Tos\big{(} \theta_{\tau_{k-1}}\big{)}\\ y_{\tau_{k-1}}+V_{k}\Delta Tsin\big{(}\theta_{\tau_{k-1}}\big{)}\\ \theta_{\tau_{k-1}}+\frac{\Delta Ty_{t}\Delta t\gamma_{k}}{L}\end{array}\right] +\mathbf{w}(\mathbf{k}) \tag{11}\]  
Here the time from \(k-1\) to \(k\) was denoted \(\Delta T\), and during this period the velocity \(V_{k}\) and steering angle \(\gamma_{k}\) of the front wheel were assumed constant. \(\mathbf{u_{k}}=[V_{k},\gamma_{k}]^{T}\) was termed the \"controls\". The distance between the caster and the center of the robot was \(L\). \(\mathbf{w}(\mathbf{k})\) was the state noise of the system assumed as a Gaussian noise with a zero mean and a covariance of Q\({}_{\mathbf{k}}\).  
This paper used the distance information as the direct observation information of the EKF-SLAM, and the MTP was directly used as a feature landmark observed by the robot. \((x_{m},y_{m})\) was applied to  
Figure 5: Schematic model of the simultaneous localization and mapping (SLAM) problem.  
Figure 6: Kinematic model of the mobile robot.  
represent the coordinate of the feature landmark \"\(m\)\" in the global coordinate system, which was fixed in the global coordinate system. The coordinates of the robot's center point at the current moment was \((x_{r},y_{r})\). For the distance measurements as the observations in this paper, the observation model was established as follows.  
\[\left[\begin{array}{c}d_{1_{k}}\\ d_{2_{k}}\\ d_{3_{k}}\end{array}\right]=\left[\begin{array}{c}\sqrt{\left(x_{m}-x_{1_{k }}\right)^{2}+\left(y_{m}-y_{1_{k}}\right)^{2}}\\ \sqrt{\left(x_{m}-x_{2_{k}}\right)^{2}+\left(y_{m}-y_{2_{k}}\right)^{2}}\\ \sqrt{\left(x_{m}-x_{3_{k}}\right)^{2}+\left(y_{m}-y_{3_{k}}\right)^{2}} \end{array}\right]+\mathbf{d}(\mathbf{k}) \tag{12}\]  
where \(d_{i_{k}}(i=1,2,3)\) was the distance between a feature landmark and the three sound sources at time \(k\), \((x_{j_{k}},y_{j_{k}})(j=1,2,3)\) was the coordinate of the three sound sources, respectively, in the global coordinate system, \(\mathbf{d}(\mathbf{k})\) was the observation noise data, which was assumed to have zero mean and Gaussian noise with covariance \(\mathrm{R_{k}}\). The observation model is shown in Figure 7. In this figure, the blue circles s1, s2, and s3 represent the positions of the three sounds (sound sources), respectively, and are in the form of a regular triangle. The relative distance between them is known; the red pentagram M denotes a reference microphone, whose position coincides with the center point of the robot. The distances from the reference microphone to the three sound sources are known; the triangles m1, m2, and m3 were randomly positioned MTPs. Here, d1, d2, and d3 were the distances from the landmark \"m1\" to three sound sources, respectively, which were calculated by Equation (7), available for Equation (8) to calculate the position of the landmark.