# Energy Consumption Analysis of a Rolling Mechanism Based on a Five-Bow-Shaped-Bar Linkage
## 2 Trajectory Planning of the Mechanism's CoM
### Kinematics Model  
According to the mechanical structure, the kinematics analytical model is established in Figure 2, in which coordinates and DH parameters [41] are also labeled. As shown in Figure 2, the contact point between the bow-shaped module and the ground is marked as \(P\), while the joints used to connect adjacent modules are successively defined as \(A\), \(B\), \(C\), \(D\), and \(E\). \(XOY\) represents the global coordinate system with the initial grounded point \(O\) that is consistent with joint \(A\), while the \(X\)-axis is parallel to the ground and the \(Y\)-axis is perpendicular to the ground. The point \(o\) is located at the center of the curvature circle that is composed of the outer contour of the grounded module. Link coordinate systems \(x_{i}y_{i}\) (\(i=1,\ 2,\dots,\ 5\)) are attached at the joints \(i\) of each module, and the directions of \(x_{i}\) are consistent with the lines between adjacent joints. Then, the joint angle \(\theta_{i}\) is defined to describe the rotation of link \(i\) with respect to link \(i-1\), and the rolling angle \(\phi\) represents the angle between the lines \(\overline{oA}\) and \(\overline{oP}\). Despite a range of [0, 2\(\pi\)] for the rolling course while different modules contact the ground in succession, we can only discuss the case  
Figure 1: Mechanical structure of the rolling mechanism. of \(\phi\in[0,\,2\pi/5]\) for any one is grounded due to the symmetrical configuration of the mechanical structure. According to the geometrical relationship in Figure 2, the angle between the grounded module and the ground, denoted as \(\alpha\), can be expressed as  
\[\alpha=\frac{\pi}{5}-\phi \tag{1}\]  
The coordinates of joint \(A\) with respect to \(XOY\) can be denoted as  
\[\left\{ \begin{array}{l}{{}^{\mathcal{O}}}x_{a}=-R\cdot\sin(\phi)+R\phi\\ {{}^{\mathcal{O}}}y_{a}=-R\cdot\cos(\phi)+R\end{array} \right. \tag{2}\]  
where \(R\) is the radius of the curvature circle that is composed of the outer contour of the grounded module, and \({{}^{\mathcal{O}}}x_{a}\) and \({{}^{\mathcal{O}}}y_{a}\) represent the coordinates of joint \(A\) in frame \(XOY\). Additionally, then, based on [9], the homogeneous transformation matrixes of adjacent coordinate systems could be written as  
\[\left\{ \begin{array}{l}{{}^{\mathcal{O}}}T=\text{trans}\big{(}{{}^{-\mathcal{ O}}}x_{a},{{}^{-\mathcal{O}}}y_{a},0\big{)}\cdot\text{rot}(z_{1},\alpha)\\ \frac{1}{2}T=\text{trans}(l,\,0,\,0)\cdot\text{rot}(z_{2},\theta_{2})\\ \frac{2}{2}T=\text{trans}(l,0,0)\cdot\text{rot}(z_{3},\theta_{3})\\ \frac{3}{4}T=\text{trans}(l,0,0)\cdot\text{rot}(z_{4},\theta_{4})\\ \frac{4}{5}T=\text{trans}(l,0,0)\cdot\text{rot}(z_{5},\theta_{5})\end{array} \right. \tag{3}\]  
where \({{}^{\mathcal{O}}}T\) is the homogeneous transformation matrix of \(x_{1}y_{1}\) with respect to \(XOY_{i}^{\;\;j}T\) is a homogeneous transformation matrix representing the coordinate transformation from \(x_{i}y_{i}\) to \(x_{j}y_{j}\); \(l\) represents the chord length of the bow-shaped module, and \(l=2R\sin\big{(}\frac{\pi}{5}\big{)}\). The coordinate of the CoM for module \(i\) with respect to \(XOY\) can be expressed as  
\[\left({{}^{\mathcal{O}}}x_{ci},{{}^{\mathcal{O}}}y_{ci},0\right)^{T}={{}^{ \mathcal{O}}}T\cdot\prod_{j=2}^{i}j^{-1}T\cdot\left(\frac{l}{2},0,0\right)^{T} \tag{4}\]  
Figure 2: Kinematics analytical model. where \({}^{O}x_{ci}\) and \({}^{O}y_{ci}\) denote the first and second components of the coordinate. Subsequently, the coordinate of the CoM for the whole rolling mechanism can be defined by [9]. \[\left\{ \begin{array}{l}{{}^{O}X_{c}=\frac{1}{5m_{t}}\sum\limits_{i=1}^{5}{{}^{O }x_{ci}m_{i}}}\\ {{}^{O}Y_{c}=\frac{1}{5m_{t}}\sum\limits_{i=1}^{5}{{}^{O}y_{ci}m_{i}}}\\ \end{array} \right. \tag{5}\]  
where \(m_{i}\) denotes the mass of module \(i\), and \(m_{t}=\sum_{i=1}^{5}m_{i}\). By substituting Equations (1)-(4) into Equation (5), the coordinates of the mechanism's CoM can be described in another form as  
\[\left\{