- Research Article
- Open Access

# Virtual-dynamics-based reference gait speed generator for limit-cycle-based bipedal gait

- Taisuke Kobayashi
^{1}Email authorView ORCID ID profile, - Kosuke Sekiyama
^{2}, - Yasuhisa Hasegawa
^{2}, - Tadayoshi Aoyama
^{2}and - Toshio Fukuda
^{3, 4}

**Received:**11 February 2018**Accepted:**17 August 2018**Published:**28 August 2018

## Abstract

This paper addresses a reference gait speed generator for limit-cycle-based bipedal gait of humanoid robots in order to achieve secure and efficient acceleration and deceleration without falling down at respective gait speeds. The reference gait speed generator is hard to be developed by analytical approaches due to the necessity of barren work for identifying the respective basins of attraction. We challenge this issue by designing virtual dynamics among a robot, a virtual leader point, and a goal, and adapting it according to a falling risk of the robot. The virtual dynamics, which has settling and acceleration times as design parameters, gives the reference speeds derived from states of the robot and the leader point to a gait speeds controller. In the dynamics, the robot’s mass is optimized virtually to maximize efficiency while ensuring stability stochastically by using a selection algorithm for locomotion. Even when there were obstacles or an up slope in traveling courses of simulations, the robot achieved the autonomous traveling from the start to the goal securely. Specific resistance was also kept small in comparison with local-stability-based walking. The proposed method makes the limit-cycle-based bipedal gait more practical and contributes toward replacing the major method that ensures stability of every step.

## Keywords

- 3-D bipedal gait
- Artificial potential field
- Leader-follower control
- Stochastic optimization

## Introduction

In the research fields of humanoid robots, bipedal gait control to travel in a variety of environments, such as disaster sites, human-living buildings, etc., is an essential issue. Limit-cycle-based bipedal gait [1–7] has excellent mobility in terms of energy efficiency from utilization of natural dynamics of robots, in contrast to major approach, which wastes the energy to ensure the stability of every step [8, 9]. This approach, however, makes footstep planning (i.e., the robot’s global position control) more challenging compared to the major approach to ensure the stability of every step [10, 11], although a few studies achieved footstep planning by combining several stable gait primitives [7].

Such energy-efficient but less-controllable gait is therefore suitable for traveling on large and long spaces where the footstep planning is not required. Instead of the footstep planning, gait speed control is easily achieved in the limit-cycle-based bipedal gait. Gait speed controllers consist of three types: forward speed control [2–4]; sideward speed control [3, 4]; and turning speed control [5, 6, 12]. Our approach, namely quasi-passive dynamic autonomous control (Q-PDAC) with adaptive speed controller (ASC) [4, 5], has achieved all types of speed control. In addition, an autonomous transition between walking and running has been implemented to expand capabilities of the bipedal gait [13].

Q-PDAC with ASC has solved the gait speed control adaptively, not analytically. This concept may cause falling down when the reference gait speeds are drastically changed, i.e., the adaptation cannot catch up their change. Nevertheless, analytical approaches are unfortunately infeasible to solve due to their complexity.

Therefore, a remaining issue to travel from a start to a goal autonomously is the way to generate secure but efficient reference speeds. We notice that asymptotic stability is expressed as a limit cycle formed by states of the robot. For this reason, the reference speeds should be designed based on the current states, not time, to ensure asymptotic stability.

However, this reference generator does not guarantee asymptotic stability. Alternatively, more rapid acceleration would be possible. To ensure asymptotic stability and keep high speed for a long time in pursuit of efficiency, the virtual dynamics can be optimized according to a falling risk of the robot by using a selection algorithm for locomotion (SAL) proposed in previous work [19]. SAL optimizes the robot’s mass virtually to maximize an evaluation function, in other words, efficiency within the allowable falling risk, called VM-SAL. Note that unlike previous use of SAL, the evaluation function is difficult to be maximized by the change of the virtual mass. This is because a dominant factor to determine the evaluation function is a set of interaction forces and torques, which are difficult to be optimized freely. To solve this problem, such interaction is regarded as stochastic variable [20], and instead of the evaluation function, its expected value is maximized.

To evaluate proposed VM-SAL, two types of simulations are carried out, while comparing the case without and with VM-SAL. In simulations, the robot achieved the secure and efficient traveling from the start to the goal autonomously. Namely, the robot could travel in the area with obstacles or on a 5° up slope. Specific resistance was fairly small in comparison with walking by a local-stability-based method conducted in ref. [21].

Our contribution in this paper is not only in system integration from the viewpoint of practicality, but also to expand applicable problems of SAL from the theoretical point of view. Specifically, SAL has solved the optimization problems for discrete variable using deterministic or stochastic objective [19, 22] and for continuous variables using deterministic objective [19, 23] so far. VM-SAL corresponds to the optimization for continuous variables using stochastic objective, which has not been solved yet. By enabling SAL to cover this problem, most of the optimization problems for locomotion can be handled with SAL.

## Prerequisites for limit-cycle-based bipedal gait and its speed control

### Passive dynamic autonomous control: PDAC

Let us introduce a method used as the limit-cycle-based bipedal gait, called passive dynamic autonomous control (PDAC) [21]. PDAC models the robot as an inverted telescopic pendulum model, which has a point mass *m* and three-dimensional half-spherical coordinates \((\theta , \phi , \ell )\), where \(\theta \in [0, \pi /2]\) means the inclination of the pendulum, \(\phi \in [-\pi , \pi ]\) is for the direction of the inclination, and \(\ell \in (0, \infty )\) gives the pendulum length. Here, \(\ell\) is constrained by a virtual holonomic constraint (VHC) for robots in accordance with ref. [24], where all joints of the robot interlock with single state variable: in this paper, it is given by \(\theta\) as \(\ell := \ell (\theta )\). This paper employs the dynamics-based VHC [13] specifically.

*g*is the gravitational acceleration. \(C_\theta\) and \(C_\phi\) are conserved quantities in each gait step, named PDAC constants. \(M(\theta )\), \(C(\theta )\), and \(L(\theta )\) are substituted for respective terms. In particular, \(L(\theta )\) can be solved under the condition that \(\ell ^3(\theta )\sin \theta\) is integrable. The sign ± in Eq. (2) is simply chosen by the sign of the initial angular velocity of \(\phi\), \(\dot{\phi }_0\). The sign \(\mp\) in Eq. (1) is inverted before and after an apex.

The bipedal gait exposed by PDAC is fully described by given VHC and two PDAC constants. In particular, the gait speeds are given by the combination of two PDAC constants, whose square roots have the same units of angular momentum.

### Adaptive speed controller (ASC)

ASC proposed in the previous work [4] controls the translational (forward and sideward) speeds. Let us introduce the principle of ASC. Now, the reference and actual gait speeds are defined as \(v_{x,y}^{\text {ref}} {\text{\,and\,}} {v_{x,y}}\), respectively, where *x* is the forward direction of the robot is facing and *y* is the sideward direction orthogonal to it. The origin of coordinate system of the swing-leg position \({\varvec{p}_L}\) is on a hip joint, where is geometrically connected to the vicinity of the center of gravity (COG) position.

#### Forward speed control

*m*, but in fact, the mass of the swing leg \(m_L\) is not lost as modeling error. The forward swing-up and touchdown positions, \(p_{Lx}^{su}\) and \(p_{Lx}^{td}\), are simply designed based on a capture point [25], while the term of the COG position in the capture point is ignored according to their coordinate system as follows:

#### Sideward speed control

### Quasi-passive dynamic autonomous control (Q-PDAC)

ASC is insufficient to reach arbitrary goal because its sideward speed control is limited depending on the forward speed. In that case, the turning speed control is additionally required like a vehicle. We have proposed Q-PDAC in previous work [5] to overcome this problem.

## Design of virtual dynamics

### Overview

The reference gait speeds, the translational speeds \(v_{x,y}^{\text{ref}}\) and the turning speed \(v_{w}^{\text{ref}}\), are generated only from virtual attractive (sometimes repulsive) dynamics with a virtual leader point. Note that this leader point is regarded as a ghost of the robot, which has the same physical parameters as the robot’s one and goes ahead. The leader point is towed by a goal, while it is repelled by obstacles, such as walls.

Such design of dynamics aims that the robot is indirectly towed by the goal and repelled by the obstacles. This indirect interaction yields secure acceleration and deceleration, which are suitable for the limit-cycle-based gait to maintain stability at respective gait speeds, although responsiveness is deteriorated reluctantly.

In this section, an attractive force (and torque) between the robot and the leader point, \(\varvec{f}_L\) (and \(\tau _L\)), and an attractive force between the leader point and the goal, \(\varvec{f}_G\) (and \(\tau _G\)), are designed. They entail the secure reference speeds without falling down at respective gait speeds.

In Appendices A, B, and C, a repulsive force from the obstacles, \(\varvec{f}_O\) (and \(\tau _O\)), and an explorative force to escape from stationary points, \(\varvec{f}_E\), are defined. Although they are not main focus of this study, they should be implemented as a practical matter, as can be seen from many types of previous work [14, 15].

### Connection among robot—leader-point—goal

*L*and

*G*. For instance, dampers and springs for their connections are given as \(c_{L,G}\) and \(k_{L,G}\) respectively. From these dampers and springs, attractive forces \(\varvec{f}_L\) and \(\varvec{f}_G\) are given as follows:

Given parameters for the connections, namely \(c_{L,G}\) and \(k_{L,G}\), are key parameters to generate the secure reference gait speeds. Hence, they are designed by giving following intuitive conditions.

#### Dynamics between leader point and goal

Firstly, the dynamics between the leader point and the goal is designed. To facilitate intuitive comprehension for design, this dynamics is represented by a damping ratio \(\zeta _G\) and a natural angular frequency \(\omega _G\), instead of \(c_G\) and \(k_G\).

Now, design criteria are given based on two kinds of time. One is a desired time to arrive at the goal in consideration with the robot’s locomotion ability, \(T_s\), and another is an acceleration time to converge on steady states, \(T_a\). These two are highly easy to be given by designer.

*m*is translated into

*I*, which means a moment of inertia, for rotational dynamics.

Now, we focus on the constraint, i.e., \(T_s \ge \ln {e_s}^{-1} T_a\), given at deriving \(\zeta _G\). To effectively and surely converge to the goal on time, \(T_s\) can be updated every gait step as \(T_s \leftarrow T_s - T_{\text{sup}}\), where \(T_{\text{sup}}\) means the elapsed time at *k*-th gait step. Such updating, however, reaches the limitation \(T_s=\ln {e_s}^{-1} T_a\). In that case, \(\zeta _G\) becomes 1, and therefore, the leader point is expected to converge on the goal without oscillation in accordance with the critical damping.

#### Dynamics between robot and leader point

Secondly, the dynamics between the robot and the leader point is designed. Now, not only \(c_L\) and \(k_L\) but also a damping ratio \(\zeta _L\) and a natural angular frequency \(\omega _L\) are used.

The robot should not become closer to the goal rather than the leader point due to risk of collision with the obstacles. \(\zeta _L\) is therefore designed to restrict an overshoot of the damped oscillation, in other words, to entail the critical damping, i.e., \(\zeta _L := 1\). From \(k_L\) and \(\zeta _L\), \(c_L\) is given. Now, as another point of view, the derivation of \(k_L\) and \(\zeta _L\) sets the settling time of the dynamics between the robot and the leader point to \(\sqrt{ R_{obs} / \max (R_{obs}, \Vert \Delta p_G \Vert _2) } T_s \le T_s\). This means that the robot will converge on the leader point faster than the time when the leader point converges on the goal, and eventually it will converge on the goal by tracking the leader point by \(T_s\).

#### Design of rotational dynamics between robot and leader point

The above connection models reveal the translational motions of the robot and the leader point. The rotational motions, however, should be treated because the robot cannot travel omni-directional without rotation, as mentioned above. Hence, the dampers and the springs, which are the same design for translational motions, are connected to rotate the leader point and the robot by generating attractive torques \(\tau _G\) and \(\tau _L\), respectively. Note that *m* is replaced with the moment of inertia *I* in rotation.

Unfortunately, \(\tau _L\) is insufficient to reach the leader point (the goal eventually) in most cases. This is because the sideward speed is absolutely dependent on the forward speed, i.e., the bipedal gait is a nonholonomic system similar to wheeled robots. To overcome this limitation of the sideward speed, an additional rotational torques \(\tau _S\) is required.

The leader point receives the reaction force \(-f_S\), which rotates it. Note that the point of load of \(-f_S\) is regarded as the rear of the leader point \(-t_R/2\), namely the rotational torques of the leader point is the same as \(\tau _S\) by canceling the sign. This reaction would result in that the robot moves only straight since the attitude error between the robot and the leader point becomes equal to 0 and the sideward error also becomes equal to 0.

### Update of reference gait speed

The attractive forces, \(\varvec{f}_L\) (precisely \(\hat{\varvec{f}}_L\), which was replaced into \(\varvec{f}_L\) for simplicity) and \(\varvec{f}_G\), and the rotational torques, \(\tau _L\), \(\tau _G\), and \(\tau _S\), are given in above sections. Besides, the force to avoid the obstacles, \(\varvec{f}_O\) (and \(\tau _O\)), and the force to escape from the stationary points, \(\varvec{f}_E\), are designed in Appendices A, B, and C. Accordingly, the reference gait speed, \(\varvec{v}^{\text{ref}} = (v_x^{\text{ref}}, v_y^{\text{ref}}, v_w^{\text{ref}})^\top\), and the leader point’s position, \(\varvec{p}^L = (p_x^L , p_y^L , p_w^L)^\top\), are updated under the influence of them.

*dt*is a control period at

*t*-th control step.

### Confirmation of convergence

*x*,

*y*,

*w*): two goals are in (10 m, 10 m, 0°) and (0 m, 10 m, 0°); the settling time is at 30 s. The gait speeds converge on the reference gait speeds as given immediately, and the references are updated at about 0.35 s intervals in accordance with the gait step time of the actual robot. \(\sigma _m\) is fixed to 1. Other parameters are the same as Table 1.

Parameters for robot model and proposed method

Symbol | Meaning | Value |
---|---|---|

DOF | Degree of freedoms | 22 |

| Total mass of robot | 22.545 kg |

\(m_L\) | Mass of each leg | 1.983 kg |

\(2d_L\) | Distance between legs | 0.21 m |

\(\ell _{\text{leg}}\) | Total length of leg | 0.595 m |

\(t_R\) | Thickness of robot | 0.1 m |

\(T_s\) | Arrival time | {45, 30} s |

\(T_a\) | Acceleration time | \(\sim\) 3.5 s |

\(R_{obs}\) | Observable range | 2.5 m |

\(\mu _S\) | Friction coefficient for rotation | 0.5 |

\(\gamma _O\) | Coefficient for obstacle avoidance | 0.9 |

\(\sigma _m\) | Virtual mass coefficient | [0.2, 5] |

\(\lambda _0\) | Maximum skewness | 5.0 |

\(\bar{v}\) | Expected average speed in traveling | 0.2 or 0.5 |

\(\sigma _0\) | Default variance | 1.0 |

\(G_\sigma\) | Gain to update \(\sigma _m\) | 0.5 |

#### One-dimensional simulations

As a remarkable point, the robot accelerated its speed again after the disturbance to catch up. This is the advantage of the virtual-dynamics-based (i.e., the state-based) reference speed generator.

#### Three-dimensional simulations

As a remarkable point, the distance between the robot and the leader point seemed to be so far, although it is in the observable range \(R_{obs}\). If this goes on, the robot would not avoid obstacles due to the tracking delay. This risk comes from no consideration of stability in the design of the virtual dynamics. Therefore, an additional design of \(\sigma _m\) to ensure asymptotic stability is proposed in the next section.

## Optimization of virtual mass by selection algorithm for locomotion (VM-SAL)

### Overview

We notice that \(\sigma _m\) changes the robot’s mass virtually so as to adjust trackability to the leader point. If \(\sigma _m\) is smaller than 1, the update of \(\varvec{v}^{\text{ref}}\) is restrained, and such a behavior facilitates the convergence on the limit cycles at current gait speeds; otherwise, it yields rapid acceleration or deceleration in pursuit of speed or stability. Both of these properties are required, but it is difficult to achieve them simultaneously. In this section, therefore, online optimization of \(\sigma _m\) by using SAL [19], called VM-SAL, is explained for secure and efficient bipedal gait.

An unconventional key point to apply SAL to the optimization of \(\sigma _m\) is to estimate an expected value of the evaluation function, which is maximized by optimizing \(\sigma _m\). This is because a main factor for the evaluation function is stochastically given by the virtual dynamics designed in above, and \(\sigma _m\) determines just a tendency of its dynamics.

### Selection algorithm for locomotion

In general, locomotion has a trade-off relation between stability and efficiency (e.g., energy efficiency and gait speed). Several approaches to select locomotion, therefore, have been proposed for switching the priority of stability and efficiency according to the situation [19, 26, 27]. SAL is the state-of-the-art algorithm among those approaches. SAL is divided into two phases: a recognition phase and a selection phase (see ref. [19] for more details).

In the recognition phase, the robot estimates the many uncertainties for locomotion from sensors: in this paper, zero moment point (ZMP) errors on x- and y-axes; a touchdown timing error; a swing-leg trajectory error; a step height; and a slope angle. They are integrated stochastically as a falling risk *S* using a Bayesian network. The structure of the Bayesian network and the connection strength between the nodes are obtained via offline and online learning.

*S*is proportional to the change of the gait speed \(\Delta v\). The gait-speed-based falling risk \(S^v\) is therefore defined as follows:

In the selection phase, the robot reveals the desired balance of stability and efficiency, and adjusts the balance toward the desired one by changing its variables of locomotion. Here, the desired balance of stability and efficiency is defined the maximum efficiency within allowable falling risk (in most cases, without falling down). The variable is given as \(\sigma _m\) in this paper (see the next section).

*R*is defined by product of a smooth threshold of the allowable falling risk and the efficiency given as a linear combination of the gait speed

*v*and a reciprocal of specific resistance (or cost of transport) \(1/C_{mt}\) [28].

*v*and \(C_{mt}\).

*t*is the current time step, and \(N = 5\) is the maximum time step used for predicting the falling risk. Further,

*r*designs the shape of the logistic function. \(\gamma _S = 0.8\) is the reliability of the prediction and \(\gamma _S^{\text {reg}}\) is a normalization term.

### Assumption for interaction forces as stochastic variables

Maximization of *R* yields the desired balance of stability and efficiency for locomotion. The virtual mass variable \(\sigma _m\) is therefore newly optimized for the purpose of maximization of *R*. Now, the role of \(\sigma _m\) has appeared in Eqs. (21) and (22): large/small \(\sigma _m\) strengthens/weakens the influences of given interaction forces and torques. In other words, \(\sigma _m\) can adjust \(\Delta v\) in SAL indirectly.

However, dominant factors for \(\Delta v\) are obviously the interaction forces and torques, \(\varvec{f}_L\), \(\tau _L\), and \(\tau _S\), and therefore, the effect of \(\sigma _m\) for *R* highly depends on them. This means that \(\sigma _m\) would oscillate if it is optimized by previous SAL, which decides optimal values deterministically.

To solve this problem, the interaction forces and torques are regarded as stochastic variables: i.e., \(\Delta v\) is also regarded as a stochastic variable. In general, when the gait speed *v* is under the average speed to travel from the start to the goal by \(T_s\), namely \(\bar{v}\), \(\Delta v\) tends to be positive to accelerate; otherwise, \(\Delta v\) tends to be negative to decelerate. Hence, \(\Delta v\) has a skewness depending on *v* in its distribution.

*x*for the sake of convenience. \(\phi\) is the standard normal probability density function with its cumulative distribution function \(\Phi\). This \(\mathcal {SN}\) has three parameters that should be given: a location \(\mu\); a scale \(\sigma\); and a shape \(\lambda\). From these three, its mean \(\mu _{\text {skew}}\), its variance \(\sigma _{\text {skew}}\), and its skewness \(\gamma _{\text {skew}}\) are derived.

With respect to \(\mu\), it is simply designed to be 0. In that case, \(\mu _{\text {skew}}\) is given to be \(\mu _{\text {skew}} = \sigma \delta\).

*v*.

### Maximization of expected value of locomotion reward

*R*, namely \(E_R\), can be derived as follows:

*R*is therefore approximated by third-order Maclaurin expansion, which gives an analytical solution according to properties of probability distribution by using \(\mu _{\text {skew}}\), \(\sigma _{\text {skew}}\), and \(\gamma _{\text {skew}}\). Here,

*η*

_{0–3}means 0–3-th order terms of

*R*regarding \(\sigma _m\).

- 1.
Stable state with low speed (the gradient is plus): the robot would accelerate rapidly to get high speed.

- 2.
Stable state with high speed (the gradient is minus): the robot would keep high speed for efficiency to reach the goal fast.

- 3.
Unstable state with low speed (the gradient is minus): the robot would keep low speed for stability not to fall down.

- 4.
Unstable state with high speed (the gradient is plus): the robot would decelerate rapidly to travel carefully.

- 5.
Highly unstable state (the gradient is almost zero): this region would be out of scope of VM-SAL.

## Simulation

### Simulation conditions

#### Robot details

Parameters for the robot model and proposed method are summarized in Table 1. Note that torque and angular velocity of all joints are almost limitless for simplicity. Now, the gait speed is represented by a dimensionless format, i.e., a Froude number \(F_{rd}\) calculated by \(v/\sqrt{g\ell _{\text {leg}}}\) (\(\ell _{\text {leg}}\) is a leg length).

#### Environment details

In the first type of simulation, the robot will go toward the goal: \((p_x, p_y, p_w)\) = (15 m, 15 m, 90°). In the middle of traveling, four pillars are arranged as obstacles to disturb traveling. This simulation is desired to be finished by \(T_s = 45\) s, namely, \(\bar{v}\) is derived to be about 0.2.

In the second type of simulation, the robot will go straight toward the goal: \((p_x, p_y, p_w)\) = (25 m, 0 m, 0°). The settling time is given as \(T_s = 30\) s, namely, \(\bar{v}\) is derived to be about 0.5. When the gait speed of our robot model is over 0.5, the gait will transit to running in pursuit of energy minimization [13, 31], although running is easy to break its balance [32]. A slope with 5° inclination is set on the way, and therefore, the robot should transit to walking again for secureness.

In both types, two cases, without and with VM-SAL, are compared to evaluate the performance of VM-SAL. In terms of secure traveling, a distance between the robot and the leader point or a phase of the gait formed by \((\theta , \dot{\theta }, \dot{\phi })\) is confirmed. In terms of efficient traveling, the specific resistance \(C_{mt}\) [28] is evaluated.

The above simulations are finished as successful cases when the robot steps into the radius of 0.5 m of the goal at a stopable gait speed (\(|F_{rd}| < 0.1\)). This stopable gait speed was given from experience of experiments and simulations so far. In failure cases, the COG trajectory cannot be generated by PDAC due to the infeasible initial parameters, and the robot would fall down eventually.

### Simulation results

All simulations were recorded in the attached video. In the following, effectiveness of VM-SAL is verified from the simulation results (Additional file 1).

#### Traveling while avoiding obstacles

As can be seen in these figures, in the case without VM-SAL, the robot went through the goal due to insufficient brake, while in the case with VM-SAL, the robot could stop near the goal on the first attempt. As a result, respective arrival times were highly different: 49.6 s in the case without VM-SAL; and 35.4 s in the case with VM-SAL.

#### Traveling on slope

## Conclusion

In this paper, we achieved the secure and efficient reference gait speed generator, i.e., the virtual dynamics with VM-SAL. The virtual dynamics was given among the robot—the leader point—the goal, and designed based on the desired time to arrive at the goal, \(T_s\), and the acceleration time, \(T_a\). Namely, this dynamics enabled the robot to reach the goal on time implicitly if no disturbances were added. The reference gait speeds were generated from this dynamics, however, they did not always maintain the limit cycle at respective gait speeds. Alternatively, more rapid acceleration and deceleration would be possible.

To ensure asymptotic stability and enhance efficiency, SAL optimizes the robot’s mass virtually depending on the gait speed and the falling risk, called VM-SAL. This VM-SAL aimed to maximize the locomotion reward stochastically. Namely, its expected value was maximized by updating the virtual mass variable \(\sigma _m\) in accordance with its gradient.

As a result, the robot traveled from the start to the goal in two types of environment: one is with the four pillars as obstacles; another is with the 5° up slope. When the robot traveled in the area with these obstacles, VM-SAL improved both trackability (i.e., secureness) and the specific resistance (i.e., efficiency) doubled in comparison with the case without VM-SAL. When the robot traveled on the slope, VM-SAL achieved rapid transition between walking and running according to the gait speed and prevented disturbance caused by unnecessary acceleration and deceleration. Such transition succeeded in traveling even though the case without VM-SAL failed to travel.

To regard \(\Delta v\) as the stochastic variable shown in Fig. 5 is a fairly rough assumption, which would cause unexpected behaviors. Future challenge of this research is therefore to reflect observed data into parameters of the stochastic variable for more effective optimization of \(\sigma _m\). Such reflection restrains the behavior contrary to expectation.

## Declarations

### Authors’ contributions

TF, YH, and KS designed and directed the project. TK and TA processed the experimental data, performed the analysis, drafted the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

This work was supported by JSPS KAKENHI, Grant-in-Aid for JSPS Fellows, Grant Number 16J05354.

### Competing interests

The authors declare that they have no competing interests.

### Publisher's Note

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## Authors’ Affiliations

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