# Energy Consumption Analysis of a Rolling Mechanism Based on a Five-Bow-Shaped-Bar Linkage
## 3 Dynamics and Energy Consumption Analysis  
### Dynamics Model  
[PERSON] et al. [9] analyzed the dynamics of the closed five-bow-shaped-bar linkage using [PERSON]'s equation ignoring joint friction. However, during the rolling process, the revolute joints are affected by viscous damping, generating joint friction torque, which hinders the mechanism from rolling. Therefore, the influence of joint friction must be taken into account in the dynamics. [PERSON] [43] presented a method for modeling damping forces using [PERSON]'s dissipation function. Taking the viscous friction into consideration, the [PERSON]'s dissipation function [43] is defined as  
\[\psi=\frac{1}{2}\sum_{i=1}^{5}f_{i}\dot{\theta_{i}}^{2} \tag{17}\]  
where \(f_{i}\) is the coefficient of viscous friction damping in the joint \(i\).  
The mechanism dynamics is modeled by the [PERSON]'s equation including the dissipative function [43] as  
\[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)-\frac{\partial L} {\partial q}+\frac{\partial\psi}{\partial\dot{q}}=\tau \tag{18}\]  
Figure 8: The upper and lower limits of the CoM trajectories.  
where \(q=[\theta_{1},\theta_{2},\phi]^{T}\) and \(\dot{q}=\left[\dot{\theta_{1}},\dot{\theta_{2}},\dot{\phi}\right]^{T}\) represent the generalized coordinates and generalized velocity, respectively; \(\tau=\left[\tau_{1},\tau_{2},\tau_{\phi}\right]^{T}\) represents the generalized force; \(L\) represents the Lagrange capability.  
To define the ground as the zero plane of potential energy, the mechanism's potential energy \(U\) can be expressed as  
\[U=m_{t}g^{O}Y_{c} \tag{19}\]  
where g is the gravitational acceleration.  
As a typical multi-rigid-body system, the mechanism's kinetic energy consists of the kinetic energy of each component module. Thus, the mechanism's kinetic energy \(T\) can be denoted as  
\[T=\frac{1}{2}\sum\limits_{i=1}^{5}\left(m_{i}{}^{O}\dot{x}_{ci}{}^{2}+m_{i}{}^ {O}\dot{y}_{ci}{}^{2}+J_{i}\dot{\varphi}_{i}{}^{2}\right) \tag{20}\]  
where \(\dot{\varphi}_{i}\) is the angular velocity for the CoM of the bow-shaped module \(i\); \(J_{i}\) is the rotational inertia of the bow-shaped module \(i\) about its CoM; \({}^{O}x_{ci}\) and \({}^{O}y_{ci}\) are the first and second elements of the CoM coordinates of each module with respect to the global coordinate system.  
Additionally, then, the Lagrange capability could be computed out by \(L=T-U\). Based on Equation (1), the angular velocity for the CoM of the \(I\) module can be given as  
\[\dot{\varphi}_{i}=\left\{\begin{array}{ll}\dot{\alpha},&i=1\\ \dot{\alpha}+\sum\limits_{j=2}^{i}\dot{\theta}_{j},&2\leq i\leq 5\end{array}\right. \tag{21}\]  
Substituting Equation (17) and Equations (19)-(21) into Equation (18), the Lagrange's equation could be written as  
\[M(q)\ddot{q}+H\big{(}q,\dot{q}\big{)}+G(q)+D(q)\dot{q}=\tau \tag{22}\]  
where \(M(q)_{[3\times 3]}\) is the inertia matrix; \(H\big{(}q,\dot{q}\big{)}_{[3\times 1]}\) is the moment matrix generated by the influence of Coriolis force and centripetal force terms; \(G(q)_{[3\times 1]}\) is the moment matrix generated by the influence of gravity; \(D(q)_{[3\times 3]}\) is the damping coefficient matrix; \(\tau\) is the generalized force, and \(\tau=[\tau_{1},\tau_{2},0]^{T}\) due to the passive degree of freedom of the rolling angle \(\phi\).