## 2 Materials and Methods
### Kinematics of an Omnidirectional Mobile Platform  
This section describes the general kinematic model of a platform with omni wheels, which is used to estimate its translation and rotation performances. Figure 1 shows the system definition of a mobile platform. The system is defined in a static frame \(\{s\}\). The mobile platform has its own body-attached frame \(\{b\}\), and each wheel also has its corresponding frame \(\{w_{k}\}\), being \(k\) the identifying number the wheel. The position of the platform is defined by its frame coordinates \((x_{b},y_{b})\) and orientation \(\theta\) relative to the world frame \(\{s\}\). Likewise, the position of each wheel is defined by its frame coordinates \(\left(x_{w_{k}},y_{w_{k}}\right)\) and orientation relative to the body frame \(\{b\}\). In the scientific literature, the orientation of the wheels (\(\beta_{k}\)) is usually referenced as \(x_{b}\) (Figure 1a) [47]. In this paper we propose an alternative representation that emphasizes the relative orientation of the wheels (\(\varphi_{k}\)), referenced to the perpendicular of the centerline, joining the centroids of the platform and each wheel (Figure 1b). The direction of the \(x_{b}\) axis in the platform frame defines the positive direction of translation (forwards), and the positive rotation of the platform is counterclockwise. In each wheel, \(x_{w_{k}}\) coincides with the wheel plane, and \(y_{w_{k}}\) is collinear with the wheel hub. Figure 2 shows a top view of the definition of the wheels in an omnidirectional platform. The axes of the rollers are represented in grey, with thicker lines to depict the rollers in contact with the ground. Figure 1: System definition of a mobile platform (top view): (**a**) description generally used in the scientific literature, with the orientation of the wheels referenced to the platform frame \(\{b\}\); (**b**) description proposed in this paper, with the orientation of the wheels referenced to the centerline joining the platform frame \(\{b\}\) and the wheel frame \(\{w_{k}\}\). In both cases the wheels represented are omni wheels with passive rollers with \(\gamma_{k}=0^{\circ}\). Figure 2a shows how wheels are enumerated: the first one is the closest to \(x_{b}\) in the first quadrant of the platform frame and the others are defined in the counterclockwise direction. Figure 2b shows the dimensional parameters: \(r_{k}\) is the radius of the wheel \(k\), measured from the center of the wheel to the ground; \(R_{k}\) is the centerline joining the centroids of the platform and wheel \(k\); \(x_{k}\) and \(y_{k}\) are the distances between the centroids of the platform and wheel \(k\) in the \(x_{b}\) and \(y_{b}\) directions; \(\delta_{k}\) is the angular distribution of wheel \(k\), measured between the \(x_{b}\) axis and the centerline \(R_{k}\); \(\varphi_{k}\) is the angular orientation of wheel \(k\), measured between the perpendicular of the centerline \(R_{k}\) and the \(x_{\text{\tiny{$\mathrm{{m}}$}}_{\text{\tiny{$\mathrm{{p}}$}}}}\) axis; \(\gamma_{k}\) is the angular orientation of the rollers of wheel \(k\), measured as the angle of the free rolling direction relative to the \(y_{\text{\tiny{$\mathrm{{m}}$}}_{\text{\tiny{$\mathrm{{p}}$}}}}\) axis. The target velocity of the omnidirectional motion is defined in the platform frame by the motion command \((v,\alpha,\omega)_{\{b\}}\), where \(v_{\{b\}}\) is the norm of the translational velocity of the motion, \(\alpha_{\{b\}}\) is the angular orientation of the translational velocity, and \(\omega_{\{b\}}\) is the angular velocity of the platform. The components of this command can be decomposed in the platform frame as \(\left[v_{\text{\tiny{$\mathrm{{x}}  
Equation (7) is expressed in a matrix form for wheel \(k\) and as a function of the target velocity defined in the platform frame as:  
\[\omega_{k}= \begin{bmatrix}\frac{1}{r_{k}}&\frac{\tan(\gamma_{k})}{r_{k}}\end{bmatrix}  \cdot A_{b\to w_{k}(R)}\cdot A_{b\to w_{k}(T)} \begin{bmatrix}v_{x}\\ v_{y}\\ \omega\end{bmatrix}