# Design and Analysis of a Planar Six-Bar Crank-Driven Leg Mechanism for Walking Robots
## 3 Construction, Dimensional Synthesis, and Working of the Mechanism  
As previously mentioned, the proposed mechanism features a foot path with a large height, denoted by \(h_{p}\) (Figure 2). Such an extreme point path facilitates not just the reciprocal motion of the crank link, but also equips a walking machine with the capability to step over obstacles. This mechanism is driven by input link 2 (\(O_{2}A\)). The four-bar linkage is denoted as (\(O_{2}ABEO_{4}\)) while the embedded pantograph is (\(O_{4}BCD\)). The notation of the different link lengths is provided in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Link Number** & **Denotation** & **Length** \\ \hline
1 & \(O_{2}O_{4}\) & \(d\) \\
2 & \(O_{2}A\) & \(a\) \\
3 & \(ABC\)1  & \(\overline{AB}=b\) \\ \cline{2-3}
4 & \(O_{4}B\) & \(c\) \\
5 & \(O_{4}D\) & \(f\) \\
6 & \(DCP\)1  & \(\overline{DP}=f\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Link notations and lengths. Figure 2: Construction of skew pantograph mechanism using GIM software. Link 6 (\(CDP\)) serves as an output link with a point of contact P called a foot point. The construction of this leg mechanism is carried out through the following steps:  
Construct a four-bar linkage (\(O_{2}ABEO_{4}\)) such that \(\overline{BE}=c\). The triangular coupler is \(ABE\), where \(\widehat{ABE}=\varphi\) (Note that \(\varphi\) is a design variable). Add a binary link \(O_{4}D\) such that \(\overline{O_{4}D}=f=n\times c\) (Note that \(n\) is the amplification factor while the orientation of the link \(O_{4}D\) determines the orientation of the output curve). Form a four-bar parallelogram (\(O_{4}BCD\)) by reshaping the coupler plate from \(ABE\) to \(ABC\) and adding a binary link \(CD\). Reshape link \(CD\) to a triangular link \(CDP\) such that the triangle \(\Delta CDP\) is identical to the triangle \(\Delta EBC\). With this embedded pantograph connected to the four-bar linkage at the fictitious coupler point E, the coupler curve is amplified by a factor of \(n\) and rotated by an angle \(\beta\) at point P. Here, \(n=O_{4}F/O_{4}B\) and \(\beta=\widehat{CDP}\). On a foot path, the distance between the two most extreme positions is referred to as the horizontal stride \(S_{P}\), as shown in Figure 2.  
The dimensional synthesis of a linkage involves determining the appropriate lengths of the links to achieve the desired motions. This process can take the form of quantitative synthesis if an algorithm is specified for the particular problem, or it can be qualitative synthesis if there are more variables than equations [17]. Additionally, dimensional synthesis can be accomplished using trial and error methods to fine-tune the link lengths for optimal performance. In this study, dimensional synthesis was performed using GIM software [20; 21], enabling us to achieve the foot trajectory illustrated in Figure 2. This trajectory is essential for several reasons: it ensures sufficient ground clearance during the swing phase to avoid obstacles, maintains the robot's stability, and minimizes energy consumption by optimizing the path. Additionally, its smooth curve reduces impact forces during landing, ensuring consistent and controlled foot movements, which are crucial for an efficient and reliable walking robot performance. The obtained foot trajectory is comparable to existing designs like the [PERSON] [9] and [PERSON]'s mechanisms [22], which are commonly used for such applications. The obtained leg mechanism dimensions are listed in Table 2. More discussions about this comparison and the specifics of our design will be detailed in the upcoming paragraphs of the article. The position of the walking mechanism at different crank angles is shown in Figure 3a-d. The crank is rotated at various intervals, and its initial position is \(46^{\circ}\) (measured counter-clockwise from horizontal reference), as shown in Figure 3a. This is when the leg begins its lift phase. At this position, the linkage is in a specific configuration with the rocker arms aligned accordingly.