- Research Article
- Open Access

# Prototyping a thermal walker that can walk on a hot horizontal surface with a simple gait

- Takeru Nemoto
^{1}Email author and - Akio Yamamoto
^{1}

**Received:**19 January 2018**Accepted:**2 July 2018**Published:**16 July 2018

## Abstract

This article reports on a newly designed thermal walker that can walk on a hot horizontal surface without any motor or battery. The walker converts heat energy from the hot surface into mechanical motion by using bimetal sheets. The prototype developed in the previous study successfully demonstrated walking, but its gait was not ideal; a leg swings forward and backward a few times to achieve one step. This article designs a new walker to realize more natural walking, in which a leg simply swings forward once in each step. For this purpose, the article introduces a new design process to better estimate the swing period. Also, material used for the feet of the walker is reconsidered to realize stable continuous walking. The newly fabricated prototype has aluminum feet and can walk on a hot horizontal surface with the simple gait. The surface temperature required for walking was 110 °C, which was considerably lowered from that in the previous study, which was 170 °C.

## Keywords

- Bimetal
- Thermal deformation
- Environmental heat
- Passive dynamic walker
- Thermal
- Walker

## Introduction

### Background

A passive dynamic walker was proposed by McGeer in 1990 to realize human-like walking [1, 2]. Based on the passive dynamic walker, many walking systems have been developed. Typically, passive dynamic walkers have no actuator and walk down slopes utilizing the gravity. Some walking systems based on passive dynamic walkers, however, can efficiently walk on a horizontal surface, by adding actuators and batteries to the walkers [3–7]. A thermal walker, which is to be dealt with in this article, is categorized into such a kind of mechanism [8]. The walker, however, has no battery nor explicit actuator on it. Instead, it has bimetal sheets on the bottom surfaces of its cylindrical feet. A bimetal sheet is composed of two thin layers with different thermal expansion coefficients and is bent when heated. On the walker, the bimetal sheets attached to the feet convert the heat energy from the horizontal surface into mechanical motion, which enables the walker to walk on a horizontal surface.

In the previous work on the thermal walker, a prototype successfully walked on a hot surface heated at 170 °C [8]. By utilizing deformations of the bimetal sheets, the walker swung its body in sideways that lifted up one of its legs. The lifted leg behaved as a pendulum and thus swung forward and then landed, as the sideways swing of the body was restored. Repeating such swing motions, the thermal walker successfully walked on a hot horizontal surface. The temperature required for walking was considerably high and a lot of efforts should be made to ease the requirement of the temperature. If the temperature could be lowered in future work, the walker would be able to walk on, e.g., a hot ground surface during summer. Moreover, the mechanism would be integrated to a general bipedal robot to reduce the energy consumption; the bipedal robot would still rely on the conventional motors, but the thermal walking mechanism might help to reduce the energy consumption of the motors.

In the proposed thermal walker, the ideal, or natural walking is realized when the period of the body swing and that of the leg swing are identical; in other words, the period ratio is 1. In such a condition, the leg simply steps forward as the body leans sideways. In the previous prototype, however, the period ratio was around 3. As a result, the walker swings a leg forward, backward, and then forward again for each step. Considering the future integration of the thermal walking mechanism with a general bipedal robot, for example, such complicated gait is apparently not desired; the simple gait with the period ratio of 1 should be realized.

To shorten the period ratio to realize the ratio of 1, the leg swing period should be extended, whereas the body swing period should be shortened. However, both of them were not successfully realized in the previous work due to the constraints arising from the design process and the material. Therefore, to realize the walking with the period ratio of 1, the walker design needs to be reconsidered.

### Objective of this paper

The objective of this paper is to realize a thermal walker that can walk continuously with the period ratio of 1. The contributions of this paper are (1) proposing a new leg swing model, (2) establishing a design process for the period ratio of 1, and (3) revealing the effect of material on the continuity of self-induced oscillation.

First, by proposing a new leg swing model, the period of leg swing will be extended. The new model adopts a rigid-body pendulum that provides a wider range of swing period. With the new leg swing model, this paper proposes a new designing process that can realize the period ratio of 1. This, however, requires the body swing period to be shortened, which might cause the damped oscillation due to overheating of the bimetal. To avoid overheating, the paper investigates the effect of heat conductivity of feet materials on the oscillation. With these improvements, the paper demonstrates a thermal walker that can walk continuously with the period ratio of 1.

The remainder of this paper is structured as follows. The next section, “Thermal walker”, explains the structure and the walking principle of the thermal walker more in detail. In “Design process”, the new design process for the thermal walker is discussed to realize the period ratio of 1. “Effect of materials on self-induced oscillation” investigates how the difference of heat conductivities of materials affects the performance. In “Experiments”, a new prototype with the period ration of 1 is designed, and its gait is observed to discuss the effect of the leg length on the stride. Finally, “Conclusion” summarizes the work with some comments on the future direction of the work.

## Thermal walker

The self-induced oscillation can be used as an energy source of motion systems. In [11, 12], a linear actuator was developed using the self-induced oscillation. In the developed actuator, an eccentric cylinder integrated in the actuator continuously hit the side wall of the actuator body. Then, by the impact of the hitting, the whole actuator body placed on a heated frictional surface moved step-by-step. However, applications of the self-induced oscillation had not been extensively studied until the thermal walker was proposed in 2015.

### Period ratio

This number also indicates how many times the swing leg swings until it lands in each step [8]. Each leg is allowed to swing for \(T_{b}/2\) during each step. The largest step is obtained if the swing leg lands on the ground, \(T_{l}/2\) after it has started to swing. Therefore, the condition for the largest step is \(T_{b}/2 = T_{l}/2\), or \(r_p = 1\). When \(r_p = 2\), \(T_{b}/2\) is equal to \(T_{l}\). In this case, the stride becomes zero because \(T_{l}\) is the period of full leg swing, which means that the swing leg lands on the ground at the same place as it is lifted up. When \(r_p = 3\), \(T_{b}/2\) is equal to \((1+\frac{1}{2}) T_{l}\), which indicates that the swing leg swings one full forward swing, one full backward swing, and one more full forward swing. This period ratio also maximizes the stride but it has unnecessary leg swings, compared to the gait for \(r_p = 1\). Similarly, any odd integer numbers of \(r_p\) can realize the maximum step length, but as \(r_p\) becomes larger, the gait contains more unnecessary leg swings. Therefore, it can be said that period ratio of 1 realizes the ideal walking.

The period ratio of the prototype designed in the previous work was adjustable in a range from 1.5 to 5.4 by changing the height of weights attached to its legs. Successful walking was obtained when the period ratio was 3.2, but, at the smallest period ratio, 1.5, the body swing was not excited and the walker did not walk. The reason would be overheat of the bimetal sheets. Smaller period ratio could be obtained when the body swing was small, which, however, made the bimetal sheets contacting with the hot surface at a limited area. Then, the bimetal and the leg would be heated up too much around the limited contact area and the minute protrusion would not be created.

## Design process

In this section, a design process to realize the period ratio of 1 is discussed. First, by regarding the leg as a rigid body, the period of its swing is calculated. Then, the moments of inertia of the legs are estimated by using a simplified model. After that, we introduce the design process which can robustly realize the period ratio of 1, regardless of the height of the rotational axis.

*l*,

*r*,

*b*, and

*g*represent leg, rotational axis, body, and CoM, respectively.

### Leg swing period considering moment of inertia

### Calculation of period ratio

The period ratio \(r_p = T_b/T_l\) is evaluated by using Eqs. (2) and (3). Since these equations contain the moments of inertia, \(I_{lg}\) and \(I_{bg}\), we first need to estimate these moments. To simplify calculations, we model each leg as a thin bar, which has different constant densities above and below the CoM (see Fig. 4b1). Then \(I_{lg}\) is calculated as \(\frac{1}{3}m_{l} (L-h_{l}) h_{l}\).

### Adjusting period ratio to 1

Here we discuss a design process to achieve \(r_p = 1\). First, \(L\) needs to be decided considering a design constraint such as the desired size of the walker. For example, we adjusted \(L\) to 17 cm to design a prototype shown in the next section.

*L*, therefore \(h_{r} < L\). Second, \(h_{r}\) should be close to or larger than \(h_{l} + \kappa _{l}\), since \(T_l\) is less sensitive to the changes of \(h_r\) and \(h_l\) in this area, according to Fig. 5. Practically, this is important as \(h_l\) (and possibly \(h_r\)) can be deviated from the designed value due to the model simplification and manufacturing errors. The gray area in Fig. 6 satisfies these two conditions.

Now, we can pick up a point from the gray area to determine \(h_l\), \(h_r\), and \(T_l\). For the prototype of the next section, we picked up a set of \(h_l\) and \(h_r\) on the contour line for \(T_l=0.65~\mathrm {s}\). The chosen \(h_{l}\) and \(h_{r}\) were 4 and 12 cm, respectively.

*R*, \(\hat{m}\)) is determined to realize the period ratio of 1. The red line in Fig. 7 indicates the (

*R*, \(\hat{m}\)) relation with \(h_l\) = 4 cm and \(h_r\) = 12 cm, whilst other lines show the reference relations with different values of (\(h_{l}\), \(h_{r}\)). For the prototype, we chose \(R\) and \(\hat{m}\) as 9 cm and 0.13, respectively.

## Effect of materials on self-induced oscillation

To realize continuous walking, damped oscillation must be avoided, which occurs when the bimetal sheets are overheated. The overheat can occur with a short period of the body swing \(T_b\), which results in a small swing amplitude. As shown in Fig. 5, the leg swing period, \(T_l\), which is almost equivalent to the body swing period \(T_b\) when \(r_p\) is 1, can be set high by selecting smaller \(h_r-h_l\). In such a selection, however, \(T_l\) becomes too sensitive to a small change of \(h_r\) and \(h_l\). Therefore, such selection should be avoided. As a result, \(T_l\) and \(T_b\) should be set appropriately low with \(r_p = 1\). This imposes a risk of overheating.

The change of the oscillation amplitudes is shown in Fig. 9c, as a function of the oscillation period. The peak amplitudes, shown with circular markers, are almost the same for aluminum and nylon, and have a linear relationship with the oscillation period. Contrarily, the amplitudes at 200 s, which are shown with cross markers, are considerably different for the two materials. Whereas aluminum cylinder kept almost the same amplitude, the amplitude was damped for the nylon cylinder. Especially when the period, as well as the amplitude, were small, the nylon cylinder stopped the oscillation due to overheating. The results indicate that the self-induced oscillation can continue for long time when the material of cylinder is highly heat-conductive.

## Experiments

### Prototype

*L*is about 17 cm, and the total weight

*m*is 310 g. Each leg is composed of a thigh and a foot with an adjustable weight block. The weight of each leg \(m_{l}\) is 145 g and the height of CoM of leg \(h_{l}\) is 4 cm. The thighs are made of polyacetal (black parts in the figure). At the bottom of thighs, 3-cm-wide aluminum feet are fixed. When both feet are aligned as in Fig. 10a, the bottom of the feet comprises a part of cylindrical surface, whose radius \(R\) is 9 cm. The two thighs are connected using a rotational axis and ball bearings. Each thigh has 6 holes at different heights for the bearings, such that \(h_r\) can be changed for performance comparison. The rotational axis weighs 20 g and its length is 64 mm. When the legs are fixed on the axis, their distance \(d\) is 35 mm. In the middle of each thigh, there is a black half-ring with a white tracking marker for optical motion tracking.

### Period ratio

### Experiment on heated horizontal surface

Figure 16d–f show the corresponding stride angle. It is indicated that stride angle has a positive correlation with the body swing amplitude, although the results are scattered for \(h_r\) of 16 cm. Generally, larger weight shift resulted in smaller stride angle, which disagrees with the discussion in the previous study [8]. This might be explained by considering feet scuffing. The previous study assumed the length of the feet (in the walking direction *y*) was zero. Under such assumption, the study predicted that the stride would be proportional to the shift of CoM. This assumption might hold when the swing angles are large enough, as in the previous study. However, in this study, the swing angles were much smaller to realize \(r_p\) of 1, which would be the reason for the contradicting results. This aspect needs to be further analyzed in future studies.

## Conclusion

In this paper, a new thermal walker was designed to realize simple walking, in which each leg swings forward only once in one body swing. A design process to realize such walking has been discussed using a simplified model. The developed prototype successfully realized the simple walking motion on a hot horizontal surface heated at 110 °C or greater, in the ambient temperature of 22 °C. This temperature difference is lower than that of the previous study [8] by around 60 °C.

The temperature difference, however, is still too high for practical applications, and the future studies will aim at lowering the required temperature. One of the possible ways to reduce the temperature difference is to improve the fixation of the bimetal sheet for better contact with the feet. Since the heat in the bimetal is dissipated to the metallic feet, better contact between the bimetal and the feet will help the heat dissipation and reduce the requirement for the temperature difference. Or, amplifying the body swing by using a mechanical resonance would be also effective to decrease the requirement. If the difference between the required temperature and the ambient temperature can be smaller than, e.g., 30 °C, the walker would be able to walk on a hot ground during summer time.

## Declarations

### Authors’ contributions

TN contributed to the modeling, analysis, experiments, and drafting of the manuscript. AY took part in the conception, interpretation of data, and revising of the manuscript. Both authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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