## 3 Design of the Accepted Solution
The free body diagram is shown in Figure 5. From it, we can determine the equations of equilibrium as the following:  
\[\sum_{i}M_{iA}=0\to R_{bz}\cdot 0.594-G_{1}\cdot g_{1}-G_{2}\cdot g_{2}=0, \tag{2}\]  
\[\sum_{i}F_{iz}=0\to R_{bz}-R_{az}=0. \tag{3}\]  
Figure 4: The main dimensions of the travel mechanism of the platform: (**a**) when it moves in a straight line; (**b**) when it moves on a slope. Figure 3: A conceptual design of the manipulator. Solving Equations (2) and (3) we obtain:  
\[R_{bz}=\frac{G_{1}\cdot g_{1}+G_{2}\cdot g_{2}}{0.594}, \tag{4}\]  
and  
\[R_{az}=R_{bz}, \tag{5}\]  
Figure 2:0  
Figure 2:1  
The solution of Equations (8)-(10) gives  
Figure 2:2  
Figure 2:3  
Figure 2:4  
From Equation (10) we obtain:  
Figure 2:5  
The validity of the Equations (11) and (12) will be for any value of the angle \(\alpha\), if the parameter \(x\) (the distance of the gravitational force \(G_{2}\) from the vertical axis between points \(A_{r}\) and \(B_{r}\), Figure 5b) in the right direction will gain a positive value and in the left direction it will gain a negative value. Considering the dimensions of the platform of \(1.2\times 0.8\) m and the centre of gravity location, the parameter \(x\) is in the interval from \(-0.100\) to \(+0.100\) m.  
Table 1 contains all geometrical parameters and Table 2 includes all force effects, which are needed for the calculation. The maximal slope (angle \(\alpha\)) of the track including safety conditions is determined to the value of \(\alpha=47^{\circ}\). Figure 2:6 Application of the free body diagram method for the platform: (**a**) when it moves in a straight line; (**b**) when it moves on a slope. The engineering design of the platform considers the maximal carrying load capacity of 300 kg, when this load is transported up to the end station. Within the safety conditions, the platform can be loaded up to the weight of 375 kg, but, in such a case, the platform must not move. Otherwise, it could lead to the platform damage. The kerb weight of the platform is 154 kg. Substituting of values from Table 2 into Equations (1)-(10) gives results of reaction effects introduced in Table 3, where:  
\(R_{az}\)--tangential reaction of wheels of the upper trolley;  
\(R_{bz}\)--tangential reaction of wheels of the bottom trolley;  
\(R_{an}\)--normal reaction of wheels of the upper trolley;  
\(R_{bn}\)-- normal reaction of wheels of the bottom trolley;  
\(R_{bt}\)--slope resistance. Based on the performed calculation, the greatest load of the platform is in the moment, when the force \(G_{2}\) is shifted by the value of \(x\) = +0.1 m. Just this position of the load is chosen for the calculation of resistance forces. The change of the reaction forces due to the platform geometry is calculated by means of the Matlab 2017a program (MathWorks Inc., Natick, MA, USA) [26; 27; 28] and their graphs are shown in Figure 6, where a blue line represents the waveform of the reaction force \(R_{an}\), a green line depicts the reaction force \(R_{bn}\) and a red curve \(R_{sum}\) is the waveform of the sum of reaction forces. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value (m)** \\ \hline \(g_{1}\) & 0.2730 \\ \(g_{2}\) & 0.5630 \\ \(t\) & 0.4344 \\ \(x\) & \(\pm\)0.1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters for calculation. \begin{table}
\begin{tabular}{c c} \hline \hline
**Considered Weight (kg)** & **Gravitational Force (N)** \\ \hline
154 & 1510.74 \\
225 & 2207.25 \\
300 & 2943.00 \\
375 & 3678.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Determined values of external loads of the platform.