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.

Figure 4 shows (a) a schematic of the thermal walker and (b) a simplified model for calculating the periods. Subscript of *l*, *r*, *b*, and *g* represent leg, rotational axis, body, and CoM, respectively.

### Leg swing period considering moment of inertia

In the previous study [8], the leg was considered as a point-mass pendulum when calculating its period. It simplifies the calculation and makes a good enough estimation in some situations. For more precise estimation in broader ranges of parameters, we model the leg as a rigid-body pendulum considering the moment of inertia. By using the parameters in Fig. 4a, the leg swing period is described as

$$\begin{aligned} T_{l}&= 2 \pi \sqrt{\frac{I_{lc}}{m_{l} g (h_{r}-h_{l})}}, \end{aligned}$$

(2)

where \(I_{lc}\) is the moment of inertia of each leg about the rotation axis and is equal to \(I_{lg} + m_{l} (h_{r}-h_{l})^2\), \(m_{l}\) is the weight of each leg, \(g\) is the acceleration of gravity, \(h_{r}\) is the height of the rotational axis from the ground, \(h_{l}\) is the height of the leg’s CoM from the ground, and \(I_{lg}\) is the moment of inertia of the leg about its CoM.

Figure 5 shows the change of the leg swing period \(T_{l}\) against the distance between the axis and the leg’s CoM (\(h_{r}-h_l\)). The shortest swing period is obtained at \(h_r - h_{l} = \kappa _{l}\), where \(\kappa _l\) is the radius of gyration of the leg and equals \(\sqrt{I_{lg}/m_l}\). As \(h_{r}-h_l\) gets closer to zero, the period increases to infinity. On the other hand, for \(h_{r} -h_l\) larger than \(\kappa _{l}\), the period gradually converges to the period calculated by the point-mass pendulum approximation [8] which is \(2 \pi \sqrt{(h_{r}-h_{l})/g}\). This comparison tells that the point-mass pendulum approximation in the previous work is precise enough only when \(h_{r}-h_l\) is large enough. The plot also shows that the period \(T_l\) is less sensitive to the change of \(h_{r}\) and/or \(h_l\) when \(h_{r}-h_l\) is close to or larger than \(\kappa _{l}\), compared to when \(h_{r}-h_l\) is close to zero. This characteristics will be exploited in the design process described later.

### Calculation of period ratio

For calculating the period of the body swing, we approximate the whole body as an eccentric cylinder. The period is then calculated using the parameters in Fig. 4a as:

$$\begin{aligned} T_{b} = 2 \pi \sqrt{\frac{m h ^2 + I_{bg}}{m g (R- h)}}, \end{aligned}$$

(3)

where \(m\) is the total weight of the walker, \(h\) is the height of the CoM of the entire walker, measured from the ground, \(I_{bg}\) is the moment of inertia of the walker about the walker’s CoM, and \(R\) is the curvature radius of the bottom surface of the feet.

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 \(\frac{1}{3}m_{l} (L-h_{l}) h_{l}\).

By the same assumption, the moment of inertia of the body, \(I_{bg}\), is also estimated (see Fig. 4b2). First, the moment of inertia of the two legs is calculated as \(2 I_{lg} + 2m_{l}\left\{ (h-h_{l})^2 + (d/2)^2\right\}\), where \(d\) is the distance between the two legs. However, due to the complexity of calculation, \(d\) is regarded as zero in the following calculation. Then, adding the moment of inertia of the rotational axis which is calculated as \(1/12 m_{r}d^2 + m_{r}(h-h_{r})^2\), gives the total moment \(I_{bg}\). With these assumptions, the period ratio is calculated as

$$\begin{aligned} r_p =&\frac{1}{2}\sqrt{\frac{num}{den}}, \end{aligned}$$

(4)

$$\begin{aligned} num =\ &(h_{r}-h_{l}) \left[ 16(2+\hat{m}) h_{l}^2 \right. \\&\left. +\, 8 \{(2-5\hat{m})L +6\hat{m}h_{r}\} h_{l} + 24\hat{m}L^2 \right. \\&\left. +\,12\hat{m}^2 h_{r}^2 + (\hat{m}^2 + 8\hat{m} +12) d^2\right] \\ den =\ &(2+\hat{m})R -(2+\hat{m})\{2h_{l} -\hat{m}h_{r}\} \\&\{2h_{l}^2 + (L-6h_{r}) h_{l} + 3 h_{r}^2\} \end{aligned}$$

where \(\hat{m}\) is the normalized mass and equals \(m_r/m_l\).

### 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.

Figure 6 shows the relationship among \(h_{l}\), \(h_{r}\), and \(T_l\), calculated using Eq. (2). Solid contour lines show constant values of \(T_l\). In this plot, \(h_r\) must satisfy the following two conditions. First, \(h_r\) which is the height of the axis, must be smaller than the total length of the walker, *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.

Finally, by using Eq. (4), a pair of (*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.