# An Estimator for the Kinematic Behaviour of a Mobile Robot Subject to Large Lateral Slip
## 2 Kinematic Model
### Kinematic Model  
The kinematic model is based on the following hypothesis:  
* The model is purely kinematic.
* The model considers only planar motion on a flat surface; out-of-plane motions are neglected.
* The locomotion units are simplified considering a single equivalent wheel per side with the axis passing through the centre of mass of the module.  
Figure 3 represents the kinematic module, where \((x_{n},y_{n},\varphi_{n})\) is the pose of the n-th module (\(n=1\) for the front module, \(n=2\) for the back one), \(\delta\) is the relative yaw rotation between the front and rear module, \(i\) is the track of the module, \(a=0.132\ m\) and \(b=0.139\ m\) are the distances between the central joint and the front and rear modules respectively, and \(\varphi_{n}\) is the longitudinal speed of the n-th module in its reference frame.  
Figure 3: (**a**) Epi.q kinematic model. (**b**) The kinematic model with virtual non-slipping wheels.  
Figure 2: Schematic representation of Epi.q and its locomotion unit.  
The geometric relations between the modules impose the following constraints:  
\[x_{2}=x_{1}-b\cos\varphi_{2}-a\cos\varphi_{1} \tag{1}\]  
\[y_{2}=y_{1}-b\sin\varphi_{2}-a\sin\varphi_{1} \tag{2}\]  
\[\delta=\varphi_{1}-\varphi_{2} \tag{3}\]  
Moreover, two additional holonomic constraints can be defined:  
\[\dot{x}_{1}\sin\varphi_{1}-\dot{y}_{1}\cos\varphi_{1}=0 \tag{4}\]  
\[\dot{x}_{2}\sin\varphi_{2}-\dot{y}_{2}\cos\varphi_{2}=0 \tag{5}\]  
The velocity in the world reference frame of each module is defined as:  
\[\dot{x}_{1}=v_{1}\cos\varphi_{1} \tag{6}\]  
\[\dot{y}_{1}=v_{1}\sin\varphi_{1} \tag{7}\]  
\[\dot{x}_{2}=v_{2}\cos\varphi_{2} \tag{8}\]  
\[\dot{y}_{2}=v_{2}\sin\varphi_{2} \tag{9}\]  
Hence, the kinematic system can be defined as:  
\[\dot{q}=\left[\begin{array}{c}\dot{x}_{1}\\ \dot{y}_{1}\\ \dot{\varphi}_{1}\\ \dot{\delta}\end{array}\right]=\mathbf{A}\left[\begin{array}{c}v_{1}\\ \dot{\varphi}_{1}\end{array}\right] \tag{10}\]  
The relation of \(\dot{\delta}\) as a function of \(v_{1}\) and \(\dot{\varphi_{1}}\) can be obtained differentiating Equations (1)-(3) in time and substituting the results in Equation (5). With some manipulation, it is possible to get:  
\[\dot{\delta}=\left(\frac{a}{b}\cos\varphi_{1}+1\right)\dot{\varphi}_{1}-\frac {1}{b}\sin\delta v_{1} \tag{11}\]  
Thus, the kinematic model can be explicitly written as:  
\[\dot{q}=\left[\begin{array}{c}\dot{x}_{1}\\ \dot{y}_{1}\\ \dot{\varphi}_{1}\\ \dot{\delta}\end{array}\right]=\left[\begin{array}{ccc}\cos\varphi_{1}&0 \\ \sin\varphi_{1}&0\\ 0&1\\ -\frac{1}{b}\sin\delta&\frac{a}{b}\cos\varphi_{1}+1\end{array}\right]\left[ \begin{array}{c}v_{1}\\ \dot{\varphi}_{1}\end{array}\right] \tag{12}\]  
Considering that each module behaves as a skid steering robot, it is possible to rewrite the latter relations in an alternative form considering the two following relations:  
\[v_{1}=\frac{1}{2}\left(v_{1R}+v_{1L}\right) \tag{13}\]  
\[\dot{\varphi_{1}}=\frac{1}{i}(v_{1R}-v_{1L}) \tag{14}\]  
Therefore:  
\[\dot{q}=\left[\begin{array}{c}\dot{x}_{1}\\ \dot{y}_{1}\\ \dot{\varphi}_{1}\\ \dot{\delta}\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{2}\cos \varphi_{1}&\frac{1}{2}\cos\varphi_{1}\\ \frac{1}{2}\sin\varphi_{1}&\frac{1}{2}\sin\varphi_{1}\\ \frac{1}{1}&-\frac{1}{1}\\ \frac{a\cos\delta+b-\frac{i}{b}\sin\delta}{b!}&-\frac{a\cos\delta+b+\frac{i}{ b}\sin\delta}{b!}\end{array}\right]\left[\begin{array}{c}v_{1R}\\ \dot{\varphi}_{1L}\end{array}\right] \tag{15}\]\(\
u_{1R}\
u_{\rm IL}\) are easily related to the motor angular speeds \(\omega_{1R}\) and \(\omega_{1L}\) by knowing the wheel radius \(r_{wheel}=32\)mm and the transmission ratio of the gearing system \(\tau_{advancing}=4.1\).