## 3 Define Tip-Over Stability Constraint of WMDAR  
In this section, a new stability criterion for the overturning moment is proposed, which is obtained by improving the WMDAR. Then, the tip-over stability margin (TOSM) is defined and converted into a tip-over stability constraint. Figure 4 shows various forces/moments on the robot system. The large ellipse represents the mobile platform, \( \begin{bmatrix}\mathbf{f}_{1}&\mathbf{m}_{1}\end{bmatrix} ^{T}\); \( \begin{bmatrix}\mathbf{f}_{2}&\mathbf{m}_{2}\end{bmatrix} ^{T}\) represents the force/torque of the body acting on the left and right arm joints; \(\mathbf{f}_{A1}\),\(\mathbf{f}_{A2}\) are the gravity vectors of the loads on the left and right arm ends, respectively; \(\mathbf{f}_{g}\) is the gravitational vector of the robot human body and the moving platform; \(\mathbf{f}_{M}\) is the inertial force acting on the center of mass of body; \({}^{n}\mathbf{p}_{i}{}^{n}\mathbf{p}_{i+1}\) represents the tip-overaxis. The reaction force from joint 1 of the arm into the body with a wrench is indicated as \(-w_{1\_i}\). \[-w_{1\_i}=[f_{i}{}^{T},m_{i}^{T}]^{T}(i=1,2) \tag{18}\]  
The tip-over moment of the mobile platform around the tip-over axis can be obtained by the following formula. \[ \begin{split}\textbf{T}\textbf{O}\textbf{\emph{m}}_{ii+1}& =\big{(}f_{S}\times l_{ii+1}\big{)}\cdot a_{ii+1}+(f_{M}\times l _{ii+1})\cdot a_{ii+1}+\sum\limits_{i=1}^{k}[(m_{i}\cdot a_{ii+1})\\ +(f_{i}\times d_{ii+1})\cdot a_{ii+1}&+(f_{Ai} \times s_{ii+1})\cdot a_{ii+1}](k=1,2)\end{split}  \tag{19}\]  
In Equation (19), the first item is the weight torque of the human body and the mobile platform. The next term is the force/torque at the left and right arm joint 1 and the torque of the end load acting on the tip-over axis. \(l_{ii+1}\) represents an orthogonal vector from the point \(O_{S}\) to the tip-over axis \(a_{ii+1}\). \(d_{ii+1}\) represents an orthogonal vector from the point \(O_{L1}\) or \(O_{\textbf{R}1}\) to the tip-over axis \(a_{ii+1}\). \(s_{ii+1}\) represents an orthogonal vector from the point \(O_{LL}\) or \(O_{\textbf{R}\textbf{R}}\) to the tip-over axis \(a_{ii+1}\). In case of tipping, the robot will tip out along the tipping axis. The tipping axis is formed by two adjacent wheels, and the unit vector can be acquired by two neighboring coordinate wheel topography contact points. \[a_{ii+1}=\frac{{}^{n}p_{i+1}-{}^{n}p_{i}}{{}^{[n}p_{i+1}-{}^{n}p_{i}]} \tag{20}\]  
Figure 4: Force and moment model of the WMDAR. For the WMDAR system, the minimum moment acting on each tip-over axis of the mobile platform is:  
\[\textbf{TOM}=min\{\textbf{TOM}_{12},\textbf{TOM}_{23},\ \cdots,\textbf{TOM}_{n1}\} \tag{21}\]  
It can be seen from the above formula that when the minimum value of **TOM** of the robot human system is lower than 0. That is, the **TOM** of the robot human system along tip-over axis is outside, and the robot human system will overturn. Therefore, the tip-over stability margin (TOSM) is defined as:  
\[\psi_{ii+1}=\frac{\textbf{TOM}_{ii+1}}{\textbf{TOM}_{norm}} \tag{22}\]  
\[\psi=min\{\psi_{12},\ \psi_{23},\ \cdots,\ \psi_{n1}\} \tag{23}\]  
\(\textbf{TOM}_{norm}\) in Equation (22) is a constant value, which is the minimum **TOM** exerting on the tip-over axis when the WMDAR is in a stable condition. In other words, \(\textbf{TOM}_{norm}>0\). Thus, the minimum TOSM \(\psi\) can be used to evaluate the stability of the robot when \(\psi>0\). In other words, the minimum _TOM_ on the tip-over axis will exceed 0, and the WMDAR system will become more reliable and stable. Whereas, when \(\psi<0\), it shows that there is a moment on the flip axis, which is directed outward, and this situation represents that the system might flip. This improved tip-over stability criterion takes into account the effects of joint force/torque of the robot arm and speed/acceleration of the mobile platform. In addition, the tip-over stability margin value can be used to evaluate the stability of the robot.