- Research article
- Open Access

# Motion-tracking skill assist and power assist for sinusoidal motions with a semi-active assist mechanism using energy control

- Takashi Kusaka†
^{1}Email author, - Takayuki Tanaka†
^{1}and - Hidekazu Kajiwara†
^{2}

**Received: **1 January 2016

**Accepted: **6 December 2016

**Published: **3 January 2017

## Abstract

This paper describes a skill assist method for sinusoidal motions using a semi-active assist mechanism through an energy control method. In a previous study, we had developed and verified the effectiveness of a power assist device with a semi-active assist mechanism and a control method for reducing loads during periodic motion. Here, we have developed a skill assist method as an extension of our power assist device for periodic motion. The skill of performing sinusoidal motions is defined in this study as an operator’s ability to track such motions. Therefore, our skill assist method attempts to improve the operator’s tracking ability. The proposed skill assist method is implemented using our previous power assist device; therefore, the device provides not only a power assist effect but also a skill assist effect to correct the motion. Hence, an operator obtains both the power and the skill assist effects simultaneously.

## Keywords

## Background

We have developed semi-active assist mechanisms to facilitate power assist and motion correction [1, 2]. The semi-active assist mechanisms use elastic materials as the source of the assist force and as actuators for controlling the elastic force. Elastic materials such as springs retain equilibrium because their material generates a restoring force towards the natural length. This characteristic is effective in assisting movement that requires a restoration of posture, such as bending and stretching of the human limbs [3]. Moreover, the actuator of a semi-active assist mechanism can control the restoring force by adjusting its equilibrium point.

The semi-active assist mechanism consists of elastic materials and actuators, and therefore, it is a series of elastic actuators. This series of elastic actuators can use the energy of the actuators efficiently by utilizing the elastic materials for energy storage. Uemura et al. [4] suggested an assist mechanism for sinusoidal motion that uses the stiffness control of an elastic material in a such way that the mechanism’s actuator resonates with the operator’s sinusoidal motions. Moreover, they stated that “in power assisting, it is important that cooperation exists among human, mechanical elastic elements, and actuators from the viewpoint of energy efficiency.” Walsh et al. [5] also developed a highly energy efficient device that utilizes elastic materials to assist in human ankle extension using the energy stored when the knee is in flexion.

In this paper, we propose a new assist method for a semi-active assist mechanism by employing the concept of “skill assist” that is defined as motion correction or motion induction. “Skill” is defined as tracking ability in this study. We realize such motion assistance by impedance control of the man-machine systems. There are some studies pertaining to the adjustment of impedance in man-machine systems, such as self-impedance matching (SIM) [6], impedance training [7], and subliminal calibration [8]. SIM is an adaptive algorithm to improve the maneuverability of man-machine systems by estimation of their optimal impedance parameters. Its effectiveness in energy consumption has been confirmed by experiments. Impedance training is a method of training the human hand to adjust its impedance, for rehabilitation purposes. This training needs to measure many parameters for determining the impedance of the human body, such as electromyogram (EMG), forces, and posture. Subliminal calibration allows for an impedance change in the man-machine system without the user’s notice. It has two characteristics: the operator keeps the control initiative, and it does not disturb the operator learning the motion. These studies have realized impedance adjustment and have confirmed its effectiveness. However, they have not improved and corrected the motion because they have focused on controlling the maneuverability only.

Therefore, we aim to realize motion correction by using the phase difference between the forces as a method of controlling the impedance of the man-machine systems, limiting the target to periodic motion only. In periodic motion, impedance control is equivalent to motion control; moreover, many motions in daily life and work are periodic, such as walking and repeatable tasks.

Specifically, we have defined the phase difference between the operator’s force and the actuator’s force, and the phase difference generates the induction force automatically. The force transducing characteristic of the semi-active assist mechanism can realize such control.

*KEIROKA*(fatigue reduction) assist that we had proposed as part of a Japan Science and Technology Agency project [11]. Through these studies, we had confirmed that there were many periodic motions in those tasks. Therefore, developing an assist method for periodic tasks is beneficial and the method we have developed has the ability to assist walking, running, and other daily tasks.

## Methods

### Energy control for power assist and skill assist

#### Semi-active assist mechanism

*F*, the user’s force \(F_{h}\), and the assist force \(F_s\) generated by the semi-active assist mechanism can be expressed as follows:

*k*is the spring constant, \(x_1\) is the handle position, \(x_2\) is the actuator position controlled by actuator, \(l_0\) is the natural length of the spring, and \(l_g\) is the elongation of the spring by gravity. \(l_g\) is calculated as \(l_g = -mg/k\), where

*m*is the total mass of the handle and the weight and

*g*is the gravitational constant. \(F_D\) is the friction of the handle, however we have considered \(c=0\) for easy analysis. If we need to consider friction, our control method described later, that cancels the term can be used. Because the frictional and dissipative forces are sufficiently small, we can ignore them in our analysis. We thus restrict ourselves to considering an ideal system.

To realize a power assist, the device must decrease the operator’s force \(F_h\) by appropriately controlling \(F_s\) according to Eq. (1). We can control the assist force by adjusting \(x_2\), and if \(F_s\) is synchronized with \(F_h\), then the user’s force is decreased by the provision of a power assist.

Conversely, if \(F_s = 0\) then \(F = F_h\). However, \(F_s\) can also be applied to alter the user’s motion. Namely, the assist force can adjust the impedance of the man-machine system, especially in a periodic motion. If the user cannot perform a desired motion, we can correct the user’s motion by adding an appropriate \(F_s\) to provide a skill assist.

Therefore, not only power assist but also skill assist can also be provided by using the elastic material as a power source. The elastic element constrains motion only if it is infinitely stiff; otherwise it does not constrain motion at all, but provides forces that influence motion.

#### Concept of skill assisting during power assisting

*mg*, the operator’s force \(F_h\), and the assist force \(F_s\). Therefore, the system under consideration assists a part of the load using the gravitational force.

*T*. The skill assist effect is thus described by this equation, because it represents the energy that is used to change the operator’s motion. The effect of changing the motion owing to the work done by the phase difference was also analyzed as the impedance adjustment effect in our previous study [16].

Finally, we can summarize the power assist effect described by \(\cos \psi\) and the skill assist effect described by \(\sin \psi\) through the phase difference \(\psi\), as shown in Fig. 2. Specifically, for the phase difference \(\psi\), the power assist effect- Eq. (10) is changed by \(\cos \psi\), and the skill assist effect- Eq. (11) is changed by \(\sin \psi\). In this paper, we do not discuss the direction of the radius, in order to consider the effect of the phase only. Further, the axis of the skill assist effect indicates both motion excitation (\(0<\psi <\pi /2\)) and motion inhibition (\(-\pi /2<\psi <0\)). Therefore, the skill assistance effect increases with \(\sin \psi\). In the \(\psi >\pi /2\) and \(\psi <-\pi /2\) regions, the effect can be understood as a motion disturbance through Eqs. (10) and (11). When \(\psi = 0\), the controller performs solely as a power assist. Note that the controller cannot perform only as a skill assist, because it is an extension of the power assist that uses \(\psi\). Therefore, if the operator can perform a tracking task without skill assist, the controller automatically performs strictly as a power assist.

### Energy control by periodic input control

#### Energy control for skill assist

*M*and

*K*are imaginary inertia and stiffness satisfying \(\omega = \sqrt{K/M}\)· \(x_1d\) is the desired value of the human’s motion and \(x_1\) is the measured human’s motion. \(r_E\) is the ratio between the energy required for a motion state change and the available energy of the assist system:

When \(r_E>1\), however, the assist system cannot supply the total energy required for the motion state change; therefore, \(|E_s|\) supplies only a portion of the energy. Hence, as described above, the control method can move the motion state closer to the desired state by using \(\psi\).

#### Synchronization of sinusoidal motion using a hybrid oscillator

To employ energy control, the assist force must be synchronized with the operator’s force possessing an arbitrary phase difference. Therefore, we apply the periodic input control to our system by utilizing a hybrid oscillator.

Periodic input control is a control method that can synchronize an input and an output with an arbitrary phase difference.

#### Frequency entrainment of the hybrid oscillator

*u*by the frequency entrainment phenomena.

*a*(

*t*) and \(\phi (t)\) are unknown functions that change slowly. Then, the differential equation can be rewritten by using Eq. (19) and its derivative:

#### Periodic input control

*u*and \(\xi\) are given by simple sinusoidal motions \(u=\alpha \sin \omega t\) and \(\xi =a \sin \omega t\), we can obtain the synchronized output \(x_{pic}\) with a phase difference of \(\varphi (\psi )\), by using \(\xi\) and its differential solution \(\dot{\xi }\) as shown below:

Figure 3 shows the controller performance. These results were obtained by a simulation under the condition \(\psi =0\) and an ideal input which has constant frequencies. The frequencies of the input are 1.0 Hz during 0–5 s, 1.2 Hz during 5–10 s, and 0.7 Hz during 10–15 s. We confirmed the ability of the frequency entrainment of the developed controller. According to the results, the controller can synchronize \(x_2\)–\(x_1\) by adapting to the frequency change in the input. Our controller can perform without phase delay, adapting the frequency automatically to the change in the input frequency, because of frequency entrainment by the controller.

## Experiment on the developed skill assist

### Experimental set-up and conditions

Experimental conditions

Set-up conditions | |

Weight (handle mass) | 3.2 kg |

Spring constant | 72.5 N/m |

\(\tilde{x}_{1d}\) | 200 mm |

\(\omega _c\) | \(2\pi\)rad/s (1 HZ) |

\(\Delta \omega\) | \(\pi\)rad/s (0.5 HZ) |

Sampling time | 10 ms |

Experimental conditions | |

Subjects | 5 subjects |

Trials | 5 times each pattern |

Trial time | 30 s |

(60 s resting interval between trials) | |

Conditions | 4 conditions |

1 : No assist without weight | |

2 : No assist with weight | |

3 : Only power assist | |

\((\psi =0)\) | |

4 : Power assist with skill assist | |

\((\psi =\psi (\Delta E))\) |

Conditions for each case

Attaching weight | Applying power assist | Applying skill assist | |
---|---|---|---|

Condition 1 | \(\times\) | \(\times\) | \(\times\) |

Condition 2 | \(\checkmark\) | \(\times\) | \(\times\) |

Condition 3 | \(\checkmark\) | \(\checkmark\) | \(\times\) |

Condition 4 | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) |

Next, we investigate the effects of the power assist and the skill assist through a visual tracking experiment by controlling the phase difference of the synchronized assist force. The effect of controller is already analyzed and confirmed by simulation with ideal input in our previous study [16]. We have confirmed that the control method can control operator’s movement by adjusting the phase difference of assist force. Therefore, the assist system should be able to improve the motion of subjects in this experiment.

Figure 4 shows an overview of the experiment. Here, in order to validate the motions of subjects, we controlled the experimental conditions as much as possible. For example, the standing position and the posture of grasping handle were decided strictly like Fig. 1. The handle’s height is measured by a position sensor. The operator moves the handle (the bar in Fig. 4) to follow a tracking target (the circle in Fig. 4) that is displayed on a monitor. In this experiment, the target is a periodic motion; thus, if the operator becomes accustomed to the experiment, they may fall into a natural rhythmic motion instead of visually tracking the target. To investigate the effect of visual tracking only, we use frequency modulation to prevent motion prediction. The operator performs the tracking task watching the target \(x_{1d}\) and the handle position \(x_1\). In this study, the graph plotting of the motion history is not displayed to the operator, to prevent predicting the next motion. Furthermore, in order to observe the motion correction effect clearly, we decided that the trial time and the resting time are 30 and 60 s, respectively, because we confirmed that the repeated trials with over 30 s trial time provide a large fatigue to the subjects and such fatigues might affect the result of the skill by preliminary experiments.

RMSEs of the position tracking (\(\mu \pm SD\)) and *p* values between the cases applying power assist only and power assist with skill assist

Subject A | B | C | D | E | Average of the normalized error | |
---|---|---|---|---|---|---|

RMSE of position tracking (mm) | ||||||

No assist without weight | 30.9 ± 3.09 | 28.5 ± 2.21 | 47.0 ± 5.11 | 20.3 ± 7.13 | 24.5 ± 3.43 | 1.00 ± 0.186 |

No assist with weight | 30.2 ± 1.43 | 28.1 ± 2.79 | 40.8 ± 10.0 | 32.4 ± 8.24 | 33.5 ± 4.73 | 1.16 ± 0.356 |

Only power assist | 30.2 ± 2.49 | 28.4 ± 2.25 | 49.3 ± 6.08 | 35.9 ± 8.59 | 34.6 ± 7.20 | 1.24 ± 0.391 |

Power assist with skill assist | 25.9 ± 1.57 | 26.8 ± 1.13 | 47.7 ± 5.72 | 25.3 ± 4.72 | 25.3 ± 4.72 | 1.01 ± 0.199 |

p value of t test | 0.012 | 0.121 | 0.353 | 0.035 | 0.034 | 0.008 |

Statistical power (\(\alpha = 0.05\)) | 0.86 | 0.20 | 0.08 | 0.74 | 0.76 | 0.83 |

Standard deviation of the phase difference \(\sigma _\psi\)

Subject A | B | C | D | E | |
---|---|---|---|---|---|

Standard deviation of the phase difference (rad) | |||||

No assist without weight | 0.213 | 0.269 | 0.466 | 0.262 | 0.332 |

No assist with weight | 0.229 | 0.243 | 0.356 | 0.264 | 0.330 |

Only power assist | 0.222 | 0.263 | 0.598 | 0.305 | 0.376 |

Power assist with skill assist | 0.200 | 0.230 | 0.448 | 0.291 | 0.303 |

Appropriate excitation and inhibition of the motion improves the operability of the motion correction. Here we investigate the effectiveness of the motion correction through energy control of the semi-active mechanism. Energy control is performed by adjusting the phase difference as the situation demands, through Eq. (12).

Thus, to apply energy control to the actuator by periodic input control, the phase difference must always be equal to \(\psi (\Delta E)\) (see Eq. (12)). To provide skill assist, the control supplies the required energy to move the energy towards the desired state. In this experiment, using the system represented in the block diagram in Fig. 5, the actuator control is made to depend upon the operator’s motion.

### Results of the skill assist experiment

We analyze the results of the experiments. In this analysis, we use only the latter half of the measured data for analysis because the target of our controller is only steady-state of periodical motions.

In the case of power assist only, the motion of the subjects was disturbed when the frequency was decelerated by frequency modulation. However, applying the skill assist improved the tracking errors.

*e*defined as the root mean square error (RMSE) is as follows:

*e*compared to all the other cases (including using the power assist method only) for all the subjects.

According to the results, the motions of the subjects A, D, and E were improved by a statistically significant amount. The results of a t test for subjects A–E were \(p = 0.012\) \((p<0.05\), \(d= 1.90, 95\%\, CI \,[1.26,7.33])\), \(p =0.121\) \((d= 0.55,\) \([-1.00, 4.19]\), \(p = 0.353\) \((d= 0.17,\) \([-7.00,10.20])\), \(p = 0.035\) \((p<0.05,\) \(d= 1.59,\) [0.49, 20.70]) and \(p = 0.034\) \((p<0.05,\) \(d= 1.64,\) [0.42, 18.17]), respectively. In general, if the statistical power \((1-\beta )\) is more than 0.8, then the statistical analysis has enough samples. Therefore, we confirmed the statistical power by the post-hoc power analysis [23]. The statistical powers for those tests were 86, 20, 8, 74 and \(76\%\), respectively. The subject A is enough and the subjects D and E are almost enough statistical power for statistical analysis.

Next, we must verify that the power assist effects of our system are retained, because the skill assist method is an extension of the power assist method. Hence, the skill assist method should have a power assist effect related to Fig. 2. The power assist effect is evaluated by measuring the operational force with a force gauge on the handle. The detailed analysis of the power assist effects will be discussed in another paper; here, we merely establish that the results of the skill assist method demonstrate a significant load reduction effect similar to the power assist method (Figs. 6, 7). The force data shown in Fig. 7 was measured in the experiment. We confirmed that the tracking error was reduced by applying phase control as a skill-assist. It can be discerned that our system also works simultaneously as a power assist system.

In the experiment, the load reduction effect can be explained as the average phase difference tending to zero at all times. Under these circumstances, our skill assist method provides a local motion correction, and provides a load reduction equal to that given by the power assist method (i.e., \(\psi =0\)) globally. If the average phase difference is not zero, then we find that the power assist effect is less for the extended method than for the power assist method. This possibility may exist in cases where continuously increasing energy is needed during motion acceleration.

## Discussion

By examining the results for each subject in detail, we see that a skill assist effect is clearly manifested for subject A because application of the skill assist decreased *e* compared to the other three cases. The value of *e* is almost the same for all cases. The mean of *e* for subject B was decreased by the skill assist, but the decrease was not statistically significant. Comparing this outcome with the results from subject A, we can see that subject B already has sufficient ability to perform the task and is able to perform the required motion. Subject B therefore does not need a skill assist to accomplish the task. The mean of *e* for subject C is also not statistically significant. Moreover, *e* was particularly insignificant for subject C in the no assist case with a weight, as compared to the case without a weight; the weight enables the subject to slow down the motion and determine the handle position. In the case where only the power assist is applied, the motion is disturbed by the force from the assist device. The skill assist effect is significant for subject D. Subject E also clearly shows the effect. The motions of subjects D and E were disturbed by the assist device when applying the power assist alone, but were improved by simultaneously applying the skill assist.

In summary, we confirmed the applicable range of the developed system from the heterogeneity across subjects, especially our assist system has the ability to correct the operator’s motion when the skill becomes worse by applying power assist.

We also investigated the total effect for all the subjects collectively, to confirm some system-wide characteristics, irrespective of the differences between the subjects. To compare the errors between the subjects, the errors were normalized by the average error of the condition “no assist without weight.” Figure 6f depicts the normalized error that has been calculated using the data from all the subjects. The skill assist effect is significant at \(p = 0.008\, (p<0.01)\) with a statistical power of \(83\%\). Therefore, we could confirm that this total analysis has enough meaning statistically. Hence, we conclude that the system we have developed improves the operators’ motion, and at a minimum, does not increase the error.

## Conclusion

We have proposed a skill assist method by using a semi-active assist mechanism through energy control. This method was realized using the phase difference that is a parameter of the control system (Fig. 2), and has enabled us to employ our previously developed semi-active assist device.

We have considered details of the skill assist effect based upon the phase difference resulting from periodic input control. According to experimental results, the motion correction effect of the proposed method is validated because the average tracking error has decreased by 16.1% at the 5% significance level. We have also established from Fig. 6 that the power assist effect of the extended method remains high. The effectiveness of the proposed method is similar to the case in which only the power assist is applied, namely, the phase difference is approximately zero. We have concluded that the reason for this result was that the distribution of changes in the phase difference was around zero, in the experiment.

We have applied our skill assist method to the semi-active assist mechanism in this study, but it can be applied to any force-control system having a periodic motion. Therefore, our method can be applied to any area with periodic systems.

In future, we will develop a system that has the ability to control the assist ratio by designing \(x_{20}\) and \(\tilde{x}_{2}\), extending the skill assist to more complex periodic motions, for practical applications such as assistance for walking. Moreover, because the semi-active assist mechanism is implemented using elastic materials, the risk of harm to the human body in the case of erroneous operation of the system is lower than that for other assist systems that do not incorporate an SEA mechanism. In the design of human-robot systems such as the one proposed here, human safety is of paramount importance [24, 25], and assistive robot systems must have the autonomy to ensure the safety of the humans operating them. Therefore, we will continue the pursuit for safer and more convenient autonomous systems that can be applied to real-world tasks.

## Notes

## Declarations

### Authors' contributions

TK took the lead in experimentation, implemented our developed cotrol method, and wrote this paper as corresponding author. TT devised the basic concept of the skill assist method for sinusoidal motions by using semi-active assist mechanisms and the periodic input control. HK improved the theoretical analisys of the effect of the periodic input control. All authors read and approved the final manuscript.

### Acknowledgements

The authors thank Prof. Kaneko of the Graduate School of Information Science and Technology, Hokkaido University for his advice on the experimental methods and its statistical analysis. This work was supported by JSPS KAKENHI Grant Number 11J01391.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Tanaka T, Satoh Y, Kaneko S, Suzuki Y, Sakamoto SN (2008) Seki: soft power suit with semi-active assist mechanism -prototype for supporting waist and knee joint. In: Proceedings of ICROS-SICE international joint conference ICCAS-SICEGoogle Scholar
- Kusaka T, Tanaka T, Kaneko S, Suzuki Y, Saito M, Kajiwara H (2009) Assist force control of smart suit for horse trainer considering motion synchronization. Int J Autom Technol 3(6):723–730View ArticleGoogle Scholar
- Rosendo A, Tanaka T, Kaneko S (2012) A yank-based variable coefficient method for a low-powered semi-active power assist system. J Robot Mechatron 24(2):291–297View ArticleGoogle Scholar
- Uemura M, Kanaoka, K, Kawamura S (2006) Power assist system for sinusoidal motion by passive element and impedance control. In: Proceedings of the IEEE international conference on robotics and automation, pp 3935–3940Google Scholar
- Walsh CJ, Pasch K, Herr H (2006) An autonomous, underactuated exoskeleton for load carrying augmentation. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, pp 3935–3940Google Scholar
- Morishita S, Tanaka T, Yamafuji K, Kanamori N (1999) Improvement of maneuverability of man-machine system for wearable nursing robots. J Robot Mechatron 11(6):461–467View ArticleGoogle Scholar
- Takeda Y, Tanaka Y, Tsuji T (2003) A virtual training sports system for measuring human hand impedance. In: Proceedings of the IEEE/ASME international conference on advanced intelligent mechatoronics, pp 914–919Google Scholar
- Igarashi H (2012) Subliminal calibration for machine operation. J Adv Comput Intell Intell Inform 16(1):108–116Google Scholar
- Kusaka T, Tanaka T, Nara H (2014) Smart suit for dairy working-a wearable power assistive device based on semi-active assist mechanism. In: Proceedings of the 2nd international conference of universal village, robotics and future universal village, pp 1–5Google Scholar
- Imamura Y, Tanaka T, Kaneko S, Yamanaka M, Suzuki Y, Saitoh M (2009) Design of smart suit for snow shoveling based on motion based assist method. Proc JSME Mech Eng Congr 2009:287–288 (in Japanese)Google Scholar
- Nara H, Tanaka T, Imamura Y, Yoshinari S, Nakajima Y, Kaneko S (2011)Fundamental study on evaluation of keiroka (fatigue-reduction) technology in using ud shovel for removing snow by musculo-skeletal dynamics simulator. In: Proceedings of the IEEE international conference on robotics and biomimetics, pp 1579–1584Google Scholar
- Kazarooni H (1993) Extender: a case study for human-robot interaction via transfer of power and information signals. In: Proceedings of the IEEE international workshop on robot and human communication, pp 10–20Google Scholar
- Pratt JE, Pratt GA (1999) Exploiting natural dynamics in the control of 3D bipedal walking simulation. In: Proceedings of the international conference on climbing and walking robots, pp 974–981Google Scholar
- Kajita S, Yamaura T, Kobayashi A (1999) Dynamic walking control of a biped robot along a potential energy conserving orbit. IEEE Trans Robot Autom 8(4):431–438View ArticleGoogle Scholar
- Kusaka T, Tanaka T, Kaneko S, Kajiwara H (2013) Power assist and skill assist for periodical motions by using semi-active assist mechanism—control of power assist ratio under use of skill assist. In: Proceedings of the 18th robotics symposia, pp 9–14. (in Japanese)Google Scholar
- Kusaka T, Tanaka T, Kaneko S, Kajiwara H (2013) Power assist and skill assist for sinusoidal motions by using semi-active assist mechanisms-analysis of impedance adjustment effect of phase difference control-. Proc JSME ROBOMEC 2013:9–14 (in Japanese)Google Scholar
- Kajiwara H, Hashimoto Y, Tsuchiya T (2001) Motion control of a pendulum via periodic input. Trans Jpn Soc Mech Eng 67(663):84–90Google Scholar
- Umegae M, Kajiwara H, Tanaka T, Kaneko S (2009) A robust walk of quasi passive walking robot controlled by periodic input. Trans Jpn Soc Mech Eng 42(8):974–981 (in Japanese)Google Scholar
- Hachiya M, Suzuki S (2009) Stabilization of a biped quasi passive walking robot via periodic input. Trans Jpn Soc Mech Eng 33(1):57–63 (in Japanese)Google Scholar
- Huys R, Jirsa VK (2010) Nonlinear dynamics in human behavior. Springer, BerlinGoogle Scholar
- Zwillinger D (1997) Handbook of differential equations, 3rd edn. Academic Press, CambridgeMATHGoogle Scholar
- Susuki Y, Yokoi Y, Hikihara T (2007) Energy-based analysis of frequency entrainment described by van der pol and phase-locked loop equations. CHAOS Interdiscip J Nonlinear Sci 17:023108View ArticleMATHGoogle Scholar
- Faul F, Erdfelder E, Buchener A, Lang A (2009) Statistical power analysis using g*power 3.1: tests correlation and regression analysis. Behav Res Methods 41(4):1149–1160View ArticleGoogle Scholar
- Yamada Y, Hirasawa Y, Huang S, Umetani Y, Suita K (1997) Human–robot contact in the safeguarding space. IEEE ASME Trans Mechatron 2(4):230–236View ArticleGoogle Scholar
- Suita K, Ikeda H (1995) A failure-to-safety “kyozon” system with simple contact detection and stop capabilities for safe human–autonomous robot coexistence. In: Proceedings of the IEEE international conference on robotics and automation, pp 3089–3096Google Scholar