# Energy efficiency in friction-based locomotion mechanisms for soft and hard robots: slower can be faster

## Abstract

Many recent designs of soft robots and nano-robots feature locomotion mechanisms that cleverly exploit slipping and sticking phenomena. These mechanisms have many features in common with peristaltic locomotion found in the animal world. The purpose of the present paper is to examine the energy efficiency of a locomotion mechanism that exploits friction. With the help of a model that captures most of the salient features of locomotion, we show how locomotion featuring stick-slip friction is more efficient than a counterpart that only features slipping. Our analysis also provides a framework to establish how optimal locomotion mechanisms can be selected.

### Keywords

Hybrid dynamical systems Piecewise-smooth dynamical systems Stick-slip friction Anchoring Peristaltic locomotion Worm-like motion Robotics## 1 Introduction

Recent advancements in the field of robotics include the development of soft robots [10] and micro-robots [4, 5, 20, 23]. For some soft robot designs, such as the recent pneumatic quadruped in Shepherd et al. [19], locomotion can be achieved by coordinated sticking and slipping of the limbs. A similar mechanism can be found in certain micro-robots, such as the ETH-Zürich Magmite [8, 14, 15], UT-Arlington ARRIpede [12, 13], Dartmouth scratch-drive MEMS robot [3], and magentic micro-robot from Carnegie-Mellon University [17]. At the macroscale, stick-slip locomotion is also featured in the Capsubot from Tokyo’s Denki University [9] and the Friction Board System [21]. It is also of interest to note that locomotion mechanisms featuring sticking and slipping of limbs can also be related to the limbless crawling (peristaltic locomotion [7, 11]) observed in a wide variety of species and bio-inspired robots [16, 18] where anchoring (sticking) is realized either by bristles or mucus [2].

While the majority of works in the application area of interest have addressed hardware design and fabrication, there is an ever increasing number of papers devoted to a systematic analysis of relevant theoretical models (see, e.g., [1, 6, 9, 22, 24]). Such analysis is challenging because the dynamics are governed by non-smooth hybrid dynamical systems and recourse to numerical methods is necessary. The present paper expands such efforts by examining the energetics and performance of devices that feature friction-induced locomotion.

The paper is organized as follows: In the next section, Sect. 2, a two-degree-of-freedom model for the locomotion system is described. The model is excited internally by changing the unstretched spring length \(\mathcal {L}_0(t)\). The interaction of the resulting normal and friction forces then leads to locomotion. In Sect. 3, this locomotion is classified into two types and the influence of some system parameters on the locomotion is discussed. We then turn to examining the energy efficiency of the SSL and SL mechanisms. Our numerical analysis features a range of simulations with varying system parameters. As in refs. [1, 24], we show how uneven friction force distribution can lead to locomotion of the center of mass. Our analyses conclude with a discussion of the effects of mass distribution in Sect. 5. The paper concludes with a set of design recommendations for balancing time taken by the model to travel a given distance subject to a given energy dissipation.

## 2 A simple model

While the exploitation of friction to generate locomotion is well known (see, e.g., [4]), analyzing simple models to examine the features of the implementation of this locomotion mechanism are rare. The simplest model we found that could explain the salient features and some of the challenges of SL and SSL were a two-degree-of-freedom mass-spring system shown in Fig. 4a. As can be seen from the figure, the masses are connected by a spring and are both free to move on a horizontal surface.

It is convenient to define the normalized vertical offset \(d = D/\hat{\mathcal {L}}\), where \(\hat{\mathcal {L}}\) is a suitable length scale. Possible choices of \(\hat{\mathcal {L}}\) include \(D\) and \(\mathcal L_0(t = 0) \ne 0\). As can be seen from the results shown in Fig. 4b, when \(d > 0\,(< 0)\), then the center of mass of the model moves forward (backward) and is stationary when \(d = 0\). In compiling the results shown in this figure, we choose \({\mathcal L}_0(t) =A \sin (\pi t) + {\bar{\mathcal L}}\). It is natural to ask what is the optimal \(\mathcal L_0(t)\) needed to achieve locomotion for a given average speed of the center of mass \(C\)? A related question is what is the optimal \(\mathcal L_0(t)\) to have the system perform a prescribed task with minimal power expenditure?

### 2.1 Equations of motion

Condition 1. \({\dot{x}}_i(t)=0\).

Condition 2. \(|k\left( \ell -\ell _0\right) \cos (\theta )|\le \mu _s n_i(t)\).

### 2.2 Analytical modes and natural frequencies

Mode 0. \(m_1\)—stick, \(m_2\)—stick;

Mode 1. \(m_1\)—slip, \(m_2\)—slip;

Mode 2. \(m_1\)—slip, \(m_2\)—stick;

Mode 3. \(m_1\)—stick, \(m_2\)—slip.

### 2.3 Internal excitation

## 3 Two types of locomotion: SL and SSL

## 4 Energetic considerations

In the region \(\omega /\omega _{n_1}<0.3\) where SSL is the observed locomotion mechanism, several local minima in travel time \(\tau _5\) occur with minimal changes in \(e_5\). However, there are several disadvantages for those minima in SSL region. First, these critical points are very sensitive to changes in \(\omega \) and, second, the average speed doesn’t compare to that when \(\omega \) is close to \(\omega _{n_1}\). In general, the results in Fig. 8 indicate that energy efficiency can never be achieved without lowering the average speed of the center of mass \(C\).

^{1}One feature of particular interest in Fig. 9 is that when \(a \le 0.08\), the system substantially traveled the fixed distance in the same amount of time with same amount of energy dissipated for low frequency \(\omega =0.60\omega _{n_1}\) as with a high frequency \(\omega =1.60\omega _{n_1}\). However, when \(a>0.08\), the system excited with a low frequency \(\omega =0.60\omega _{n_1}\) can travel the fixed distance in less time and with a smaller energy dissipation than the system excited with a frequency \(\omega =1.60\omega _{n_1}\). In other words, when all the other conditions are equal, the excitation with a lower frequency, namely, a longer period, appears to allow the system to take more advantage of the resultant force on the system in the \(\mathbf{E}_1\) direction than one with a high excitation frequency.

## 5 The effects of mass distribution

The motion of the system is achieved in part by varying the normal forces at the contact points with the ground. These forces are also proportional to the masses \(m_1\) and \(m_2\), respectively. Consequently, it is of interest to examine how the mass distribution \(\frac{m_1}{m_2}\) can effect the locomotion of the system. In this section, we examine how the time to travel \(\tau _5\) and the energy dissipated \(e_5\) are related to the mass parameter \(\frac{m_1}{m_2 + m_1}\) for a set of five representative excitation frequencies.

The results shown in Fig. 12 provide another way to accelerate our system when changing the excitation frequency is not possible. For \(\omega = \omega _4=16\), the trend follows what happens with \(\omega = \omega _5=23\) except the mass ratio where \(\omega = \omega _{n_1}\) changes. The third case we consider is \(\omega = \omega _3 = 9.5\). Here, as \(\omega _3\approx \omega _{n_1}\) when \(m_1=m_2\), we find that the three valleys reduced to a single wide flat valley. This is a very appealing design region for applications.

If we continue to decrease the excitation frequency to \(\omega = \omega _2 = 6\), then the first natural frequency \(\omega _{n_1}\) can never be reached regardless of the mass distribution^{2}. However, as can be seen from Fig. 12b, we still find three local minima of average velocity at \(\hat{m} = 0.27, 0.73\), and \(0.5\).

As can be seen in Fig. 12a, with the two mass distributions \(\hat{m} = 0.27\) and \(0.73\), the exciting frequencies are quite close to the approximated frequency corresponding to the mode of single mass oscillation \(\omega _{n_2} =5.85\) and \(\omega _{n_3}=5.85\), respectively. For the other minimum at \(\hat{m}= 0.5\), the excitation frequency \(\omega = \omega _2\) is the closest to \(\omega _{n_1}\). Of particular interest to us is that its corresponding energy consumption indicated by Fig. 12c is also a minimum.

The behavior when \(\omega = \omega _1=2\) follows what occurred with \(\omega = \omega _2=6\) except that it does not exhibit the two valleys for the travel time \(\tau _5\) near the frequency corresponding to a single mass oscillation. By examining numerical simulations for the case \(\omega = \omega _1\), we found that SSL was dominant during the entire motion. With one of the masses stuck, we have less energy dissipated. However, the time to reach the fixed distance 5 is longer in general compared to the other cases \(\omega _{2,3,4,5,6}\) and is not significantly improved at the minimum \(m_1 = m_2\). Finally, when \(\omega = \omega _1=2\), the system can only be set into motion in a narrow range of mass distributions near \(\hat{m} = 0.5\).

## 6 Conclusions

- 1.
SSL typically occurs only for frequencies smaller than \(\omega _{n_{1,2,3}}\).

- 2.
SSL is energy efficient, however, it is not always the fastest form of locomotion.

- 3.
During SSL, the time to travel a given distance is not very sensitive to the difference in the coefficients of static and dynamic friction.

- 4.
To achieve the same average velocity of the center of mass, especially when the excitation amplitude \(a\) is large, low frequency is better than high frequency in term of energy efficiency.

## Footnotes

- 1.
While the energy \(e_5\) dissipated for \(\omega =0.95\omega _{n_1}\) does decrease after a certain amplitude \(a\) is reached, this region in parameter space is not feasible because when the two mass are too close to each other there is a possibility that the normal force on one of them will vanish and that mass would then loose contact with the ground.

- 2.

## Notes

### Acknowledgments

Support from a Defense Advanced Research Projects (DARPA) 2012 Young Faculty Award to Carmel Majidi is gratefully acknowledged. Xuance Zhou is grateful for the support of a Anselmo Macchi Fellowship for Engineering Graduate Students and a J. K. Zee Fellowship. The authors also take this opportunity to thank an anonymous reviewer for their constructive criticisms.

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