# Energy Consumption Analysis of a Rolling Mechanism Based on a Five-Bow-Shaped-Bar Linkage
## 5 Conclusions  
The workspace of the closed five-bow-shaped-bar linkage's CoM with the rolling angle \(\phi\) ranging from 0 to \(2\pi/5\) was studied. Using different height points on the symmetric centerline of the workspace of the CoM as the via points, the CoM trajectory planning was planned by combining cubic polynomials with Bezier curves based on quadratic Bernstein polynomials. Additionally, then, the energy consumption of the rolling mechanism was investigated by the theoretical calculation through integrating the product of the generalized velocity and the generalized force versus time, where the generalized force was calculated by Lagrange's equation including the Rayleigh dissipation function. Furthermore, the CoM trajectories and rolling angle paths were investigated by the theoretical calculation and the virtual prototype simulation, respectively. Based on above studies, the significant findings are listed below.  
* The workspace of the closed five-bow-shaped-bar linkage's CoM with the rolling angle \(\phi\) ranging from 0 to \(2\pi/5\) is symmetric about \({}^{O}X_{c}=0.63R\).
* By high-order differentiable composite polynomial functions to plan the CoM trajectory, the mechanism obtains smooth joint trajectories.
* The energy consumption decreases and then increases with the height \(h\) of the via point increasing in the schemes of sinusoid acceleration and modified trapezoidal curve acceleration, where the range of \(h\) is from \(0.801R\) to \(1.183R\).
* The energy consumption of the sinusoid acceleration scheme is lowest when the height \(h\) of the via point is \(0.972R\), and the modified trapezoidal curve acceleration scheme is lowest when the height \(h\) of the via point is \(0.979R\).  
Figure 17: Theoretical and simulated trajectories of the CoM at \(h=0.120\) m.  
* The energy consumption of the sinusoid acceleration scheme is lower when the height of the via point is less than 0.967\(R\), and the modified trapezoidal curve acceleration scheme is lower when the height of the via point is higher than 0.967\(R\). This paper provides a reference for the mechanism to choose an appropriate via point, which can significantly reduce the energy consumption of the mechanism during rolling along a straight line. Future research must include the energy consumption analysis of irregular terrains and obstacles based on the workspace of the CoM. Furthermore, to enhance the adaptability of the mechanism in complex environments, a scheme of the crawling and rolling integrated mechanism will be developed and applied practically.  
All authors contributed to this work. Conceptualization, Y.Z. and L.Y.; methodology, Y.Z. and L.Y.; software and funding acquisition, X.X. and Y.W.; validation, T.Z. and N.F.; writing--original draft, Y.Z. All authors have read and agreed to the published version of the manuscript. This work has been supported by the National Natural Science Foundation of China (grant number 52275007), Hubei Provincial Department of Education Science and Technology Research Project (grant number Q20201708), and Chongqing Science and Technology Commission of China (grant number cstc2020)cyj-msxmX0242). Not applicable. Not applicable. Not applicable. Not applicable. Not applicable. Not applicable.