2.1. Related Works

Robotic locomotion has been studied using walking robots and their characteristics, such as control methods and body structures ^10,11,12,13. The central pattern generator (CPG) was proposed as a model for the generation of rhythmic locomotion in living organisms based on physiological findings from animal experiments and has been studied by applying a rhythmic generator in the spine as a controller ^{14,15,16,17,18,19,20,21,22}. The CPG was proposed to be located in the vertebrae of animals ²³, starting from the work of Brown et al. in the decerebrate cat ²⁴, which is a predecessor of the CPG. The gait pattern was thought to be generated according to the vibration mode to move the body according to a predetermined rhythm.

Passive walking has been proposed ^25,26,27 using a robot that incorporates interactions between the body and the environment, and it has been particularly applied to walking robots. This passive walking robot uses positional energy to walk ²⁸. Passive walking robots use positional energy as a driving source and achieve walking via a mechanism that generates a stable gait. Passive walking robots rely heavily on body dynamics, and the environment in which their gaits are observed is limited. Therefore, although they are vulnerable to environmental changes, they unprecedentedly perform efficient locomotion ^{29,30,31,32,33}.

Recently, robots that can change gait patterns have been proposed ^{19,20,21,22,34}. Many of these robots incorporate the CPG, which are mechanisms that generate gait patterns with no explicit predesign via reflexive rhythm generation. Spontaneous gait transitions were achieved by sensory feedback to the CPG using external factors, such as environmental load. Thus, not only the body but also mechanical interactions with the environment contribute to the generation of rhythm for gait generation and transitions in animals.

Furthermore, gait movements that emerge owing to complex interactions between animals can be viewed as nonlinear phenomena. Using a compass-type passive walking robot, a bifurcation phenomenon was discovered, where the walking cycle diverges from one to two or four cycles as the slope angle increases ^35,36. It was shown that the gait pattern generated by a passive walking robot diverges in an experiment conducted on a real robot ³⁷.

In addition, bifurcation phenomena have been observed in areas other than walking motion. The advanced turning maneuvers of a multilegged centipede robot were demonstrated using the bifurcation phenomenon as a parameter of the flexibility of its body segments ³⁸. In a study of bifurcated gait transitions, a semi-autonomous control system with two bifurcated attractors was proposed ³⁹. These robot motion transitions exhibit an example of successfully using the stabilization and destabilization of a nonlinear dynamical system and are expected to realize highly efficient motion control because the system as a whole actively moves toward a stable state (entrainment phenomenon).

In several related works, during gait, the GRF generated by the contact between the foot and the ground is extremely important information in controlling gait stability and rhythm. Previous studies have proposed methods to generate and modify gait patterns of robots and musculoskeletal models using feedback of GRFs obtained from foot-mounted sensors. For example, Owaki et al. studied a method for gait control based on a CPG model, using the GRF feedback from the sole as input. In this method, spontaneous gait generation and its transition are achieved through sensory feedback of GRF ¹⁹. The GRF may contribute to adaptive gait generation to terrain and disturbances, suggesting that it can be used to adjust gait parameters according to the environment ⁴⁰. Real-time force information from the foot sensor enables optimization of weight shift and landing timing during gait, resulting in stable gait generation. From the above, the gait generation using GRF is designed considering the mechanical input from the environment because the gait generation is realized by the mechanical interaction with the environment. In this sense, it differs from conventional robot control methodologies that control a self-enclosed system.

On the other hand, gait generation based on mechanical stabilization is an approach that takes full advantage of the robot’s inherent passive dynamics and structural characteristics to achieve an energy efficiency and stable gait. A prime example is McGeer’s Passive Dynamic Walking, where in 1990 McGeer showed that gravity and mechanical dynamics alone can produce a natural gait rhythm without the use of actuators or complex controls ^25,26,27. This work introduced the concepts of “fixed point generation” and “entrainment phenomena” in gait generation and has been the basis for much subsequent research. Other approaches have also been proposed to incorporate stability constraints into gait generation using model predictive control (MPC) to guarantee theoretical stability based on the relationship between center of gravity (CoG) and ZMP ⁴¹. Furthermore, it has been reported that in 3D bipedal walking, a mechanism for autonomous synchronization and self-synchronization and self-stabilization of forward/backward and left/right motions during gait has been demonstrated, resulting in complex 3D gait patterns ⁴². Gait generation based on mechanical stabilization can be described as similar to gait generation based on GRFs in that it controls an open system. However, it incorporates a simpler robot body design, such as an equal distribution of driving forces, rather than only forces such as GRFs, and the model is designed for energetic considerations.

2.2. Stabilization of Locomotion by Equalizing the Ground Reaction Force

The GRF is an important mechanical element for gait control because it is an external force associated with walking, along with gravity and inertia. The CPG realizes stabilization and transition of motion in response to external inputs by means of stability points and bifurcation points intrinsic to the neural mechanism. Passive walking arbitrarily creates stable points inherent in the body’s dynamic system, including the environment, through body design, and realizes stable motion, but motion transitions are difficult to achieve.

Meanwhile, this study suggests a design theory that realizes motion transition as well as creating a stable point by body design. For bipedal walking, when the GRF transmitted to the legs is not equal when both legs are in the stance phase, internal forces are generated in the body, causing the animal to fall (Fig. 1(a)). In other words, the animal has a mechanism to equalize the GRF acting on both legs within the body, and this mechanism is assumed to generate a stable gait pattern without falling. Focusing on the mechanism for maintaining postural stability, we developed a robot equipped with a differential gear to achieve the equalization of the GRF ⁴³. In studies that employ CPGs or sensory feedback, each leg is typically equipped with one or more actuators. Under such arrangements, variations in ground reaction forces and individual motor characteristics make it difficult to keep the applied torque strictly constant unless a complete electro-mechanical model of each motor is available. In our work, the differential gear ensures equal torque distribution, enabling us to investigate the problem with a considerably simpler robot than those used in previous research. Note that our research does not claim originality in the mere incorporation of a differential mechanism into a robot, while geared differential design allow to build an extremely minimal robot for studying rhythmic locomotion in multi-legged systems that lack any electronic control circuitry. Figs. 1(b)–(e) shows the relationship between the differential device to eliminate the internal rotational forces and the generation of a rhythmic motion in a leg-equipped robot.

The red arrows represent the flow of force and torque until the input torque from the motor slips on the ground. Conversely, the blue arrows represent the reaction force or counter-torque that the legs receive from the ground. The shafts that power each leg are coupled by a motor and differential mechanism that can absorb each speed difference while achieving equal rotational force distribution.

The gait generated by the leg-equipped robot developed here was not reproducible and did not reach the point of motion stabilization. In addition, no theoretical analysis was conducted at all, and it is necessary to clarify the normative principles by attempting to formulate the phenomenon. This differential gear eliminates angular velocity differences and equalizes the driving force ⁴⁴, and it is based on a device mainly used in automobiles. Experiments using the leg-equipped robot confirmed that the robot generates a stable rhythmic motion in response to the input torque.

Our robot allows only the difference in angular velocity (or angle) between the hind legs. Even with this simpler constraint, stability has been demonstrated through actual experiments, enabling rhythmic motion generation. In this regard, the experimental setup differs from those based on GRF for gait generation.

3.1. Scheme of Robot Experiments

We developed a leg-equipped robot (Fig. 2), conducted walking experiments, measured its behavior, and compared it with a mathematical model to explore the proposed rhythmic motion mechanism (Fig. 3). Walking experiments were conducted using the developed leg-equipped robot. The torque input from the motor is transmitted to the left and right legs via a differential gear (Fig. 3(a)). Owing to the geometrical structure of the legs, a swing phase and a stance phase appear with each rotation of the drive shaft. The GRF generated by the contact between the left and right toes and the ground is transmitted to the motor via the differential gear. The generated GRF was measured using a pressure-sensitive sensor, and the change in the resistance of the pressure-sensitive sensor was converted to conductance. The data measured as conductance contained considerable noise owing to the flapping of the foot. Here, wavelet transform, fast Fourier transform (FFT), and a low-pass filter were applied to reduce the noise (Fig. 3(b)). The time between the adjacent peaks of the obtained waveforms was one period. The difference between the peak timing of the left and right waveforms was calculated based on the waveform of the data obtained from the right leg, and the value obtained by multiplying the ratio of the time difference to one cycle by 2\(\pi\) was expressed as the phase difference between the legs.

3.2. Robot Design

Our robot has the following features. I) A differential gear was used to transmit the driving force from a single motor to the left and right legs equally. II) The Theo–Janssen mechanism was used for both legs. This allows the trajectory of the toes to be adjusted by setting the length of each link arbitrarily.

Our robot weighs 2.452 \(\mathrm{kg}\) and has a degree of freedom in the yaw direction, in addition to the rotational axes of both legs. It is equipped with a mechanism that allows passive rotation in response to the movement of its legs. By providing degrees of freedom at the connection between the front wheels and the main body, the mechanism encourages the front wheels and the main body to move independently, preventing the friction of the front wheels from affecting the main body.

The robot is equipped with a single brushless DC motor. The motor was rated at 24 \(\mathrm{V}\) and operates at a maximum no-load speed of 320 \(\mathrm{rpm}\) and a maximum continuous torque of 1.2 \(\mathrm{Nm}\). The motor controls the input torque by controlling voltage via pulse width modulation (PWM) in the range of 0\(\mathrm{\%}\)–100\(\mathrm{\%}\) using controller area network (CAN) communication with 1 Mbps speed. The current flow through the motor and the rotation speed can be obtained as feedback information. The motor has back-drivability, which means that it can receive the reaction force from the ground as a single force input (Figs. 1(b)–(e)).

Pressure-sensitive sensors were attached to the robot’s soles to measure the GRF. The phase difference was obtained from the time variation of the measured GRF to determine whether the left or right leg was in a swing or stance phase.

3.3. Differential Gear

Differential gears are mechanical parts installed in vehicles such as automobiles and are useful when a speed difference occurs at the end of left and right shafts, such as when turning. As a vehicle turns, a speed difference arises between the inner and outer wheels. The differential gear absorbs this speed difference while simultaneously transmitting torque equally from the drive source to both the inner and outer wheels, thereby facilitating smooth driving ⁴⁴.

Differential gears have various types, including shaft-driven and belt-driven types ⁴⁵. Shaft-driven gears are used in automobiles because of their superior power transmission efficiency and strength. The belt-driven type is used in radio-controlled vehicles because of its simple structure, ability to transmit power between distant shafts, and superior quietness. Therefore, a belt-driven differential gear is used in this study.

The belt-driven differential gear comprises mechanical elements, such as a differential pulley integrated with the differential case, side gears, pinion gears, and cross shaft. The pinion gears mesh with the side gears on both sides, allowing them to rotate on their own axes around the cross shaft and revolve around the axes of the side gears. The gear ratio between the pinion gear and the side gear used in this study is \(5:9\). Moreover, to minimize damping effects, a low-friction strategy is pursued by avoiding the injection of oil into the gear case. Torque is transmitted from the timing belt to the differential pulley, the differential pulley and differential case rotate as a single unit, and torque is transmitted to side gears by the pinion gear’s revolution. When the angular velocities of the left- and right-side gears, \({\omega_{\rm{L}}}\) and \({\omega_{\rm{R}}}\), are different and a differential drive is performed (Fig. 4), the following equation holds:

3.4. Phase Analysis

We attached a pressure-sensitive sensor to the soles of bothblegs without performing calibration to acquire leg phase data and measured the leg phase from the timing of the leg ground contact (Fig. 3(a)). The pressure-sensitive range was 0.3–9.8 \(\mathrm{N}\), the pressure-sensitive area was \(39.6~\mathrm{mm} \times 39.6~\mathrm{mm}\), and the basic resistance was 20 M\(\Omega\) or higher. The resistance of the pressure-sensitive sensor was obtained using a voltage divider circuit, and the pressure-sensitive pressure value was output as the conductance \(G\) [S].

Since the conductance data obtained using this approach contain a lot of noise (Fig. 5), a wavelet transform was first performed (see Appendix A.2). This allows the waveforms that involve multiple peaks due to noise to be identified as a single peak. Furthermore, FFT analysis was performed to visualize the frequency domain of the leg rotation axis and the high-frequency domain due to noise (see Appendix A.3), and a low-pass filter was used to remove the high-frequency domain, including the noise (see Appendix A.1).

Using the waveforms of the left and right leg data with the reduced noise, the phase difference was calculated from the ratio of the time difference in the peak timing of the waveforms included in one cycle (see Appendix A.4).

3.5. Mathematical Model

In this section, rather than directly analyzing periodic motions, we focus on identifying bifurcation phenomena that arise as system parameters vary and on characterizing the types of these bifurcations. A simple mechanical model was developed for the robot (Fig. 6). Here, only the differential gear necessary for the system to operate, the left and right legs, and the axes necessary for the transmission of the driving force were retained. The left and right legs were cylinders (hereafter referred to as wheels) to avoid complications in the calculations, and various characters were defined for the differential gear and wheels \(\rm{L}\) and \(\rm{R}\) (Fig. 6). The subscripts \(\rm{c}\), \(\rm{L}\), and \(\rm{R}\) denote the differential gear, left wheel, and right wheel, respectively. The subscript \(\rm{LR}\) represents a common term. Additionally, Table 1 summarizes the various parameters and variables defined in this study.

The stability of walking motion is often formulated as a hybrid dynamical system. However, once such a formulation is adopted, conventional analytical methods developed for continuous functions can no longer be directly applied, making the analysis extremely difficult. Therefore, in this study, we intentionally minimized mathematical rigor and focused on whether bifurcation phenomena could emerge, which constitutes the main feature of our theoretical analysis. In gait stability analysis, Poincaré maps are commonly used to evaluate stability within a single gait cycle. From the perspective of gait pattern formation, however, it can also be valuable to investigate the continuous phase relationship between the legs rather than focusing solely on one cycle. We also aimed to include the non-differentiable transition points between the stance and swing phases in the analysis. To achieve this, we extracted only the simplest periodic component, which can be regarded as the first term of a Fourier series expansion, and approximated it as a differentiable continuous function. This approach enabled us to derive and analyze a self-consistent equation of motion^46,47. This approximation was determined from the anteroposterior GRF \(F_{y}\) shown in Fig. 7, which was measured at the sole of the foot during human walking. The frictional force acting on the left and right wheels is defined by the following continuous function:

The difference between the rotation angles of the left and right wheels, \(\theta_{\rm{L}}\) and \(\theta_{\rm{R}}\), is newly defined as the phase difference \(\phi\).

In our mathematical model, the equation of motion for the robot’s rhythmic motion is described by the equation of state, described as follows:

4.1. Rhythm Generation

The developed robot was used in a walking experiment on a treadmill. Because the hardware design made it impossible to fix the initial conditions, we randomly assigned the initial phase difference by hand for all experiments in this study. This is because the initial posture and position of the robot are not able to explicitly predefined. First, the differential gear attached to the robot allows the shaft angles—and hence angular velocities—to transmit torque to both legs to vary according to Eq. (2). Second, owing to the back-drivability of the single brushless motor that serves as the sole power source, reverse torques can further alter these angles. Consequently, the robot always begins from a fully relaxed, load-free state; the front wheels may spin freely, and the initial posture is determined only by the robot’s own weight as it settles into static balance. In the walking experiment, the output torque \(\tau_{\rm{c}}\) of the motor was adjusted by arbitrarily changing the voltage value, which is the only parameter. When the load applied to the motor is not constant, the motor torque cannot remain fixed because the current fluctuates. Therefore, we adopt the voltage value as the variable so that the equilibrium state is uniquely defined. This allows the robot’s moving speed to vary with the voltage. We generally kept the treadmill stationary; however, when the robot’s locomotion speed became high, we operated it at a constant speed to prevent the robot from falling. The experiments were conducted 10 times for each voltage, from 7 \(\mathrm{V}\) to 15 \(\mathrm{V}\) in 0.5 \(\mathrm{V}\) increments, and the phase difference was quantitatively obtained.

Stable walking was observed at 12 \(\mathrm{V}\) as the voltage to the motor changed (Fig. 8(a)). Conversely, at 8 \(\mathrm{V}\), the phase difference showed a walk close to zero, resulting in difference from the behavior at 12 \(\mathrm{V}\). In addition, the measured raw data show different grounding timings (Fig. 5), which is considered to be a result of bifurcation in the dynamics consisting of the robot and the ground, with input torque as the bifurcation parameter.

These hardware experiments demonstrated that walking gait patterns can be generated without using control laws, such as the CPG, owing to spontaneous gait generation that takes advantage of environmental characteristics and body structure, as in the case of a passive walking robot. This walking can regulate gait generation via a single control input, with only one area where it is generated by increasing or decreasing the input torque from the motor under adjustable control and one area where it is not (disorderly or different gait is generated).

When the driving force from the motor was transmitted via a single shaft, the left and right legs maintained their initial phase difference, and no pattern was generated. Alternatively, in the case of the differential gear used in this study, the phase difference can change from moment to moment because the mechanism can tolerate changes in the phase of the left and right legs. In fact, in the experimental results, we obtained a region where the phase difference was not constant at an input value of 11.5 \(\mathrm{V}\). However, at 12 \(\mathrm{V}\), the robot continued to walk with a phase difference of \(\pi\), suggesting the existence of an attractor for a stable rhythmic motion.

4.2. Energy Efficiency

To calculate the energy efficiency, we measured the distance traveled by the robot and the current value for each constant voltage value during a measurement time of 10 \(\mathrm{s}\). This allows us to calculate cost of transport (CoT) ^51,52, which is expressed by the following equation:

4.3. Mathematical Analysis

Mathematical analysis was performed to gain a more detailed understanding of motion stability and when walking is generated.

Here, the fixed points of the equation of state are described as follows:

The Jacobi matrices and their eigenvalues were calculated around each fixed point to determine the stability of the fixed points (see Appendix C.1). The eigenequations of each Jacobi matrix are described as follows:

The fixed point \(\boldsymbol{x}_{1}^{*}\) is unstable because it has eigenvalues with positive real parts. From the coefficients \(f_{\mathbf{A}_{3}}(\xi_{3})\) and \(f_{\mathbf{A}_{4}}(\xi_{4})\), Hurwitz’s stable discriminant method shows that the fixed points \(\boldsymbol{x}_{3}^{*}\) and \(\boldsymbol{x}_{4}^{*}\) are also always unstable. On the other hand, for \(\boldsymbol{x}_{2}^{*}\), the stability discriminant is determined by the positive and negative values of the real parts of the eigenvalues, where \(\cos(\sin^{-1}({\tau_{\rm{c}}}/({2Fr_{\rm{LR}}}))) = \sqrt{1 - ({\tau_{\rm{c}}}/({2Fr_{\rm{LR}}}))^{2}}\).

Assuming \(i(\xi_{2}) = 0,\ j(\xi_{2})=0\) based on Eq. (14), the following four eigenvalues are obtained:

The eigenvectors corresponding to each eigenvalue are described as follows:

These eigenvectors represent the direction of convergence and divergence of the fixed point \(\boldsymbol{x}_{2}^{*}\) on the phase diagram, and the magnitude of the eigenvalue is the system gain that indicates the degree of its decay or amplification. Thus, we obtain \(\mathrm{Re}(\xi_{22}) < 0\) and \(\mathrm{Re}(\xi_{24}) < 0\) for the real part of each eigenvalue, whereas \(\mathrm{Re}(\xi_{21})\) and \(\mathrm{Re}(\xi_{23})\) are positive or negative depending on the value of \(\tau_{\rm{c}}\). Therefore, the real part of the eigenvalues \(\mathrm{Re}(\xi_{21})\), where \(c_{\rm{LR}} = I_{\rm{LR}} = F = r_{\rm{LR}} = 1,\) for the input torque \(\tau_{\rm{c}}\), was plotted using numerical analysis (Fig. 9(a)). The Jacobian matrix \(\rm{\bf{A}}_{1}\) is a block diagonal matrix, and the stability with respect to the phase difference is determined by \(\mathrm{Re}(\xi_{21})\) and \(\mathrm{Re}(\xi_{22})\). Therefore, only the results for \(\mathrm{Re}(\xi_{21})\) are shown here, but \(\mathrm{Re}(\xi_{23})\) exhibits similar properties to \(\mathrm{Re}(\xi_{21})\). As a result, when the input torque \(\tau_{\rm{c}}\) is \(0\leq \tau_{\rm{c}}<2\), the fixed point \(\boldsymbol{x}_{2}^{*}\) is stable, and the phase difference is 0. The red line represents the mathematically calculated bifurcation point. Therefore, it shows a stable motion with a phase difference of 0. This result is consistent with the motion observed in the walking experiment for the voltage range of 7–11 \(\mathrm{V}\). If the value of \(\tau_{\rm{c}}\) is \(2 < \tau_{\rm{c}}\), the real parts of the eigenvalues are negative, indicating destabilization.

In addition, the results of the eigenvalue analysis of the nondimensionalized mathematical model are plotted with \(\bar{\tau}_{\rm{c}}\) on the horizontal axis and \(\bar{F}\) on the vertical axis (Fig. 9(b)). Assuming the real part of the eigenvalues to be 0 (that is, at the bifurcation boundary indicated by the red line), we obtain \(\bar{\tau}_{\rm{c}}/\bar{F} = 0,2\). \(\bar{\tau}_{\rm{c}}/\bar{F} = 2\) indicates that at the moment \(\bar{\tau}_{\rm{c}}/\bar{F} = 2\), the system diverges from a stable state (zero phase difference) to an unstable state. However, \(\bar{\tau}_{\rm{c}}/\bar{F} = 0\) indicates that \(\bar{\tau}_{\rm{c}} = 0\), and no input torque is applied; therefore, it was not included in the analysis.

4.4. Comparison of the Experimental and Mathematical Results

The results of the mathematical analysis were compared with those of the actual experiment conducted using nondimensionalization. To compare the results with continuous mathematical values, it is desirable to use experimental data with more subdivided input voltage values. Therefore, the median phase difference at each voltage, obtained from walking experiments conducted 10 times per voltage, was used to compare with the mathematical analysis. The values of the real part of the eigenvalues (black line) and the phase difference (green star) versus the ratio of the dimensionless quantity \(\bar{\tau}_{\rm{c}}\) and \(\bar{F}\) are plotted in Fig. 10. The black line represents the predicted change in stability based on the non-dimensionalized Eq. (18). The red square represents the data point at 12 \(\mathrm{V}\) in the actual machine experiments, where a phase difference of \(\pi\) was found to be the most stable. Comparative examination of Figs. 10(a) and (b) demonstrates that the qualitative state transitions associated with bifurcation exhibit equivalent characteristics.

4.5. Visualization of Frequency Distribution

We visualized the time-series data of the motor’s angular velocity for each voltage value obtained in Section 4.1, specifically the angular velocity of the differential gear’s input shaft and the GRFs data of both legs. The heatmap in Fig. 11 displays frequency on the vertical axis and voltage values on the horizontal axis, with color intensity representing the amplitude magnitude obtained from the FFT. However, amplitudes below 0.2 \(\mathrm{Hz}\) exhibit significant noise and were therefore excluded beforehand. Additionally, Figs. 11(b), (d), and (f) present the lines obtained by applying the least squares method to data sampled from Figs. 11(a), (c), and (e), respectively. These sampled data represent the maximum amplitude for each voltage value, and the sampling ranges differ for each corresponding figure. The heatmap was created based on a \(200 \times 200\) grid, resulting in approximately 0.04 \(\mathrm{V}\) increments. In Fig. 11(a), two datasets were created, defined as:

Figure 11(a) shows that the frequency of angular velocity change increases discretely at a voltage value of approximately 12–13.5 \(\mathrm{V}\). Because the motor’s angular velocity varies such that it oscillates periodically (Fig. 11(g)), it can be expressed by the following equation.

Focusing on the amplitude ratio, we derive:

A.1. Waveform After Low-Pass Filter

The experimental data are shown when the input voltage to the motor was 8 V and 12 V. In each Fig. 12, the blue line represents the experimental data for the left leg and the yellow line, for the right leg. The frequency at which the one-sided spectrum of the FFT reached its maximum was considered as the ground motion from the swing leg to the support leg, and the higher frequency components were removed.

A.2. Wavelet Transform

Experimental data are shown for motor input torques of 8 \(\mathrm{V}\) and 12 \(\mathrm{V}\). Fig. 13 shows the data of the right leg. Since the highlighted areas are the main frequency components of the acquired data, they are extracted and used for the phase difference analysis. The Morlet function expressed as a wavelet was used: \(mw(x) = {\exp}^{-({x^{2}}/{2})}\cos5x\), where \(x = {(t-\beta)}/{\alpha}\). \(t\): time of experimental data, \(\alpha\) and \(\beta\): parameters of the wavelet. Specifically, in this study, \(\alpha\) ranges from 0.000 to 1.000 and \(\beta\) ranges from 0.1 to 10.0. Furthermore, the data to be extracted was obtained by selecting \(\alpha\) that satisfies the following:

A.3. Fast Fourier Transform (FFT)

Experimental data are shown for motor input torques of 8 \(\mathrm{V}\) and 12 \(\mathrm{V}\) (Fig. 14). In the data extracted by the wavelet transform, a larger input voltage value results in increased noise. The main frequencies are visualized using toolbox on MATLAB fft function to reduce noise. The sampling frequency for the FFT was set to 10 \(\mathrm{Hz}\) because the experimental data were measured every 0.1 seconds. The red dashed line shows the frequency when the correction value is multiplied by 1.2 times the frequency when the one-sided spectrum is at its maximum. Note that these cutoff frequencies can take different values in each data.

A.4. Calculation of the Phase Difference

The phase difference was calculated by focusing only on the maximum value (“\(+\)" in Fig. 12) of each waveform. First, the average period \(T_{\rm{L}\_\mathit{ave}}\) and \(T_{\rm{R}\_\mathit{ave}}\) of each waveform obtained is calculated by the following equation.

From the above, the phase difference is obtained from the ratio of the average time difference of each maximum value to the average of the average period of the waveforms obtained from the left and right legs.

The above equation calculates the phase difference by multiplying the ratio of the phase difference in one period by the phase of one period, \(2\pi\).

C.1. Jacobi Matrix

The linearization of Eq. (7) is shown below using the Jacobi matrix obtained around fixed point \(\boldsymbol{x}^{*} = [\dot{\phi}^{*}, \phi^{*},\dot{\theta}_{\rm{c}}^{*}, \theta_{\rm{c}}^{*}]^{\rm{T}}\):

C.2. Nondimensionalization

Equation (7) is nondimensionalized with representative scales as \({M} = I_{\textrm{LR}}/r_{\textrm{LR}}^{2}, {L} = r_{\textrm{LR}}\), and \({T} = I_{\textrm{LR}}/c_{\textrm{LR}}\). Table 3 shows the parameters obtained upon nondimensionalization. After the nondimensionalization of Eq. (7), the following equation is obtained: