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Biological Cybernetics

, Volume 112, Issue 3, pp 227–235 | Cite as

A cardioid oscillator with asymmetric time ratio for establishing CPG models

Original Article
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Abstract

Nonlinear oscillators are usually utilized by bionic scientists for establishing central pattern generator models for imitating rhythmic motions by bionic scientists. In the natural word, many rhythmic motions possess asymmetric time ratios, which means that the forward and the backward motions of an oscillating process sustain different times within one period. In order to model rhythmic motions with asymmetric time ratios, nonlinear oscillators with asymmetric forward and backward trajectories within one period should be studied. In this paper, based on the property of the invariant set, a method to design the closed curve in the phase plane of a dynamic system as its limit cycle is proposed. Utilizing the proposed method and considering that a cardioid curve is a kind of asymmetrical closed curves, a cardioid oscillator with asymmetric time ratios is proposed and realized. Through making the derivation of the closed curve in the phase plane of a dynamic system equal to zero, the closed curve is designed as its limit cycle. Utilizing the proposed limit cycle design method and according to the global invariant set theory, a cardioid oscillator applying a cardioid curve as its limit cycle is achieved. On these bases, the numerical simulations are conducted for analyzing the behaviors of the cardioid oscillator. The example utilizing the established cardioid oscillator to simulate rhythmic motions of the hip joint of a human body in the sagittal plane is presented. The results of the numerical simulations indicate that, whatever the initial condition is and without any outside input, the proposed cardioid oscillator possesses the following properties: (1) The proposed cardioid oscillator is able to generate a series of periodic and anti-interference self-exciting trajectories, (2) the generated trajectories possess an asymmetric time ratio, and (3) the time ratio can be regulated by adjusting the oscillator’s parameters. Furthermore, the comparison between the simulated trajectories by the established cardioid oscillator and the measured angle trajectories of the hip angle of a human body show that the proposed cardioid oscillator is fit for imitating the rhythmic motions of the hip of a human body with asymmetric time ratios.

Keywords

Central pattern generator (CPG) Cardioid oscillator Limit cycle Asymmetrical time ratio Global invariant set theory 

1 Introduction

A lot of motions of creations, such as walking, crawling, and swing, are rhythmic behaviors, which are controlled by central pattern generators (CPGs) located in spines Grillner (1985). In order to imitate these rhythmic behaviors, CPG models described by dynamic equations are established by researchers. Generally speaking, the dynamic equations for establishing CPG models include neuron models Hodgkin and Huxley (1952); Pinto and Santos (2011); Matsuoka (1987); Fukuoka et al. (2003) and nonlinear oscillator models Buchli et al. (2005); Righetti et al. (2006); Pina Filho et al. (2005); Nandi et al. (2009); Wieczorek (2009); Hu et al. (2014); Rayleigh and Lindsay (1976); Wu et al. (2004); Acebron et al. (2005). The neuron models, such as the Voltage-gated channel model Hodgkin and Huxley (1952), the Wilson–Cowan model Pinto and Santos (2011), the Matsuoka model Matsuoka (1987), and the Kimura model Fukuoka et al. (2003), have the definite biological meaning and can be used to imitating many rhythmic behaviors, including those with asymmetric time ratios. However, the neuron models have structures with a lot of parameters that should be determined by training, so they are difficult to be analyzed. A nonlinear oscillator is a nonlinear dynamic system with at least one stable limit cycle. As the invariant set of a nonlinear dynamic system with nonzero states, the stable limit cycle makes the trajectory of the dynamic system converge to a limit cycle and stay on it. Thus, the nonlinear oscillator can be self-excited to generate periodic motions. Because of the characteristics that the nonlinear oscillator’s self-excited motions are periodic Awrejcewicz and Pyryev (2005), nonlinear oscillators are used for establishing CPG models for imitating rhythmic motions in the nature Bhuiyan et al. (2015); Buchli et al. (2005); Righetti et al. (2006); Pina Filho et al. (2005); Nandi et al. (2009). Recently, the most widely used nonlinear oscillators include the Hopf oscillator Wieczorek (2009); Hu et al. (2014), Rayleigh oscillator Rayleigh and Lindsay (1976); Wu et al. (2004), Kuramoto oscillator Acebron et al. (2005); Tsimring et al. (2005), and van der Pol (VDP) oscillator Pol and Mark (1928); Bi (2004). For example, Buchli et al. Buchli et al. (2005) designed the CPG model of a bipedal robot based on the Hopf oscillator for imitating human’s walking. On this basis, Righetti et al. Righetti et al. (2006) developed the CPG model of a bipedal robot based on the modified Hopf oscillator. de Pina Filho et al. Pina Filho et al. (2005) and Nandi et al. Nandi et al. (2009) separately developed the CPG models of a bipedal robot based on the Rayleigh oscillator. Furthermore, Nandi applied his developed CPG model to providing reference trajectories for the motion control of an above-knee prosthesis. Additionally, a phase oscillator with a variable amplitude based on the Kuramoto oscillator was designed by Ijspeert et al. Ijspeert et al. (2007), and the CPG models of a snake-like robot and fish-like robot were established based on the designed oscillator. Zielinska Zielinska (1996) developed the CPG model of a bipedal robot based on a VDP oscillator for imitating the motion of the hip joint and knee joint of human walking.

Although the above-mentioned nonlinear oscillators can be used to establish CPG models for imitating rhythmic motions, their drawbacks are apparent. According to the works conducted by Buchli et al. Buchli et al. (2006) and Chatterjee and Dey Chatterjee and Dey (2013), the normalized phase planes and the trajectories of a Hopf oscillator, Rayleigh oscillator, and VDP oscillator are shown in Fig. 1. Observing Fig. 1, these oscillators possess the trajectories with symmetric time ratios and the centrosymmetrical limit cycles in their phase planes, which are same as a sine curve or a quasi-sine curve. Nevertheless, the most of rhythmic motions in the nature possess asymmetrical time ratios, which means that the proportion of the forward process of rhythmic motions is different from the proportion of the backward process in one period Suchorsky and Rand (2009); Murray et al. (1964). The CPG models with sine curves or quasi-sine curves can imitate simple rhythmic motions like swimming and crawling; however, they are unsuited to complex rhythmic motions like walking Wu et al. (2009). Presently, the rhythmic motions with asymmetric time ratios are approximately modeled with conventional oscillators. For example, de Pina Filho et al. Pina Filho et al. (2005) and Nandi et al. Nandi et al. (2009) have approximately modeled the joint movements by applying a Rayleigh oscillator. The another way to model the sustained oscillating processes with asymmetric time ratios is realized by utilizing two Hopf oscillators with symmetric time ratios, and adjusting their frequency to model the forward motion and the backward motion of a trajectory, respectively Righetti and Ijspeert (2008). However, the former method can only approximately model rhythmic motions with large error, and the latter method will complicate the model. Therefore, it is significant for establishing CPG models that can generate special outputs and extend its engineering applications how to design the nonlinear oscillators with asymmetrical time ratios Wu et al. (2009); Xu et al. (2016); Fu et al. (2017).
Fig. 1

Nonlinear oscillators: a the phase plane and b the trajectories

Considering that a cardioid curve is a kind of closed curve with central asymmetry, and by designing a cardioid curve as the limit cycle of a cardioid oscillator, the cardioid oscillator will have an asymmetric limit cycle, which results in the oscillating trajectories with asymmetric time ratios. In order to design a nonlinear oscillator with an asymmetric time ratio, in this paper, the method to design the closed curve of a dynamic system as its limit cycle is proposed based on the property of an invariant set. Based on the proposed limit cycle design method and according to a global invariant set theory, the cardioid oscillator with an asymmetric time ratio is realized. Additionally, the affection of the oscillator parameters on the resulting trajectories, the anti-interference performance, and the convergence rate are numerically simulated and analyzed. And the example that utilizing the established cardioid oscillator to simulate the rhythmic motion of the hip joint of a human body in the sagittal plane is presented.

2 Designing method of a limit cycle

The limit cycle of a dynamic system is its invariant set with nonzero states. When a nonzero set could be designed as the invariant set of a dynamic system, the set would be its limit cycle.

Defining a dynamic system as
$$\begin{aligned} \bigg \{\begin{array}{l} \dot{x}_{1}=g_{1}(x_{1},x_{2}) \\ \dot{x}_{2}=g_{2}(x_{1},x_{2}) \\ \end{array} \end{aligned}$$
(1)
where \(x_1 \) and \(x_2 \) are the states of a dynamic system,\(g_1 \left( {x_1 ,x_2 } \right) \) and \(g_2 \left( {x_1 ,x_2 } \right) \) are the nonlinear functions of the states.
Defining the closed curve of the dynamic system given by Eq. (1) in its phase plane as
$$\begin{aligned} C:F\left( {x_1 ,x_2 } \right) =0, \end{aligned}$$
(2)
then the set of the points on the closed curve C is given by
$$\begin{aligned} \varOmega =\left\{ {\left( {x_1 ,x_2 } \right) \hbox {|}F\left( {x_1 ,x_2 } \right) =0} \right\} \end{aligned}$$
(3)
Considering that the derivation of the invariant set is equal to zero, and assuming that \(\varOmega \) is the invariant set of the dynamic system given by Eq. (1), its derivation is equal to 0, which yields
$$\begin{aligned} \left\{ {\dot{F}\left( {x_{1} ,x_{2} } \right) = 0{\text {|}}\left( {x_{1} ,x_{2} } \right) \in \varOmega } \right\} \end{aligned}$$
(4)
According to Eq. (4), we have
$$\begin{aligned} \dot{F}\left( {x_{1} ,x_{2} } \right) = \frac{{\partial F}}{{\partial x_{1} }}\dot{x}_{1} + \frac{{\partial F}}{{\partial x_{2} }}\dot{x}_{2} = 0. \end{aligned}$$
(5)
Substituting Eq. (1) into (5) leads to
$$\begin{aligned} \frac{\partial F}{\partial x_1 }g_1 \left( {x_1 ,x_2 } \right) +\frac{\partial F}{\partial x_2 }g_2 \left( {x_1 ,x_2 } \right) =0. \end{aligned}$$
(6)
For the invariant set, its derivation is equal to zero. If the states of the dynamic system given by Eq. (1) satisfy Eq. (6), \(\varOmega \) is its invariant set. Additionally, the states of the dynamic system given by Eq. (1) are not equal to 0 when they belong to \(\varOmega \). Consequently, \(\varOmega \) is the limit cycle of the dynamic system given by Eq. (1).

3 Cardioid oscillator with asymmetric time ratios

Utilizing the proposed method to design the limit cycle in Sect. 2, an oscillator with an asymmetric limit cycle and asymmetric trajectories can be designed. In this section, the cardioid curve, which is an asymmetric closed curve, is designed as the limit cycle of the oscillator.

Defining the cardioid curve C in Eq. (2) as
$$\begin{aligned} F\left( {x_1 ,x_2 } \right) =\left( {x_1 +a} \right) ^{2}+x_2 ^{2}+bx_2 +c\left( {x_1 ^{2}+x_2 ^{2}} \right) ^{2} \end{aligned}$$
(7)
where a, b, and c are the nonzero parameters of the curve C.

Figure 2 shows the modified cardioid curve given by Eq. (7) with \(a=5\), \(b=200\), and \(c=-4\). Observing Fig. 2, the curve C is a non-centrosymmetrical closed curve.

Substituting Eq. (7) into (3) leads to
$$\begin{aligned} \varOmega= & {} \Bigg \{ \left( {x_1 ,x_2 } \right) \hbox {|}F\left( {x_1 ,x_2 } \right) =\left( {x_1 +a} \right) ^{2}\nonumber \\&+\, x_2 ^{2}+bx_2 +c\left( {x_1 ^{2}+x_2 ^{2}} \right) ^{2}=0 \Bigg \} \end{aligned}$$
(8)
According to Eq. (8), defining \(g_1 \left( {x_1 ,x_2 } \right) \) and \(g_2 \left( {x_1 ,x_2 } \right) \) in Eq. (1) as
$$\begin{aligned} \left\{ {{\begin{array}{l} {g_1 \left( {x_1 ,x_2 } \right) =g_{11} +\gamma g_{12} F\left( {x_1 ,x_2 } \right) } \\ {g_2 \left( {x_1 ,x_2 } \right) =g_{21} +\gamma g_{22} F\left( {x_1 ,x_2 } \right) } \\ \end{array} }} \right. \end{aligned}$$
(9)
where \(g_{11} , g_{12} , g_{21} \), and \(g_{22} \) are the nonlinear functions of \(x_1 \) and \(x_2 \), \(\gamma \) is a nonzero real number.
Substituting Eq. (9) into (6) gives
$$\begin{aligned} \dot{F}\left( {x_{1} ,x_{2} } \right)= & {} \left( {\frac{{\partial F}}{{\partial x_{1} }}g_{{11}} + \frac{{\partial F}}{{\partial x_{2} }}g_{{21}} } \right) + \left( {\frac{{\partial F}}{{\partial x_{1} }}g_{{12}} + \frac{{\partial F}}{{\partial x_{2} }}g_{{22}} } \right) \nonumber \\ F\left( {x_{1} ,x_{2} } \right)= & {} 0 \end{aligned}$$
(10)
Because \(F\left( {x_1 ,x_2 } \right) \) is equal to 0 when \(\left( {x_1 ,x_2 } \right) \in \varOmega \), the necessary and sufficient condition for satisfying Eq. (10) is given by
$$\begin{aligned} \frac{\partial F}{\partial x_1 }g_{11} +\frac{\partial F}{\partial x_2 }g_{21} =0 \end{aligned}$$
(11)
Defining the Lyapunov function of the nonlinear system defined by Eq. (9) as
$$\begin{aligned} V=\frac{1}{2}\left[ {\left( {x_1 +a} \right) ^{2}+x_2 ^{2}+bx_2 +c\left( {x_1 ^{2}+x_2 ^{2}} \right) ^{2}} \right] ^{2} \end{aligned}$$
(12)
According to Eq. (9), the time derivative of Eq. (12) is given by
$$\begin{aligned} \dot{V}= & {} \left( {\frac{{\partial F}}{{\partial x_{1} }}g_{{11}} + \frac{{\partial F}}{{\partial x_{2} }}g_{{21}} } \right) F\left( {x_{1} ,x_{2} } \right) \nonumber \\&+ \left( {\frac{{\partial F}}{{\partial x_{1} }}g_{{12}} + \frac{{\partial F}}{{\partial x_{2} }}g_{{22}} } \right) F^{2} \left( {x_{1} ,x_{2} } \right) \end{aligned}$$
(13)
According to Eq. (11), we have
$$\begin{aligned} \dot{V}=\gamma \left( {\frac{{\partial F}}{{\partial x_{1} }}g_{{12}} + \frac{{\partial F}}{{\partial x_{2} }}g_{{22}} } \right) F^{2} \left( {x_{1} ,x_{2} } \right) \end{aligned}$$
(14)
Observing Eq. (12), \(\left( {x_1 ,x_2 } \right) \rightarrow \infty \), \(V\rightarrow \infty \). According to the global invariant set theory, when
$$\begin{aligned} \dot{V} \le 0 \end{aligned}$$
(15)
\(\left( {x_1 ,x_2 } \right) \rightarrow \left\{ {\left( {x_1 ,x_2 } \right) \hbox {|}\dot{V}=0} \right\} \hbox { as } t\rightarrow \infty \).
Fig. 2

Cardioid curve with \(a=5, b=200\), and \(c=-4\)

Fig. 3

Trajectories and phase planes of the cardioid oscillator with different ratios of a to b: a the trajectories and b the phase planes

Fig. 4

Phase planes of the cardioid oscillator with different initial conditions

Fig. 5

Phase planes and trajectories of the disturbed cardioid oscillator: a the phase planes and b the trajectories

Fig. 6

Convergence rate of the cardioid oscillator with \(\gamma \) of 0.05, 0.2, and 0.5, respectively: a in the whole process and b in the converging process

For \(F^{2}\left( {x_1 ,x_2 } \right) \ge 0\) and \(\gamma >0\), the necessary and sufficient condition for satisfying Eq. (15) is given by
$$\begin{aligned} \frac{\partial F}{\partial x_1 }g_{12} +\frac{\partial F}{\partial x_2 }g_{22} \le 0 \end{aligned}$$
(16)
In order to satisfy the condition given by Eqs. (11) and (17), we define \(g_{11} \), \(g_{12} \), \(g_{21} \), and \(g_{22} \) as
$$\begin{aligned} \left\{ {{\begin{array}{l} {{\begin{array}{l} {g_{11} =-\frac{\partial F}{\partial x_2 }} \\ {g_{12} =-\frac{\partial F}{\partial x_1 }} \\ \end{array} }} \\ {{\begin{array}{l} {g_{21} =\frac{\partial F}{\partial x_1 }} \\ {g_{22} =-\frac{\partial F}{\partial x_2 }} \\ \end{array} }} \\ \end{array} }} \right. \end{aligned}$$
(17)
According to Eq. (7), substituting Eq. (17) into (9) leads to
$$\begin{aligned} \left\{ \begin{array}{{l}} \dot{x}_{1} = - \left[ 2x_{2} + b + 2cx_{2} \left( {x_{1} ^{2} + x_{2} ^{2} } \right) \right] \\ \quad \qquad - \gamma \left[ {2x_{1} + 2a + 4cx_{1} \left( x_{1} ^{2} + x_{2} ^{2} \right) } \right] F\left( {x_{1} ,x_{2} } \right) \\ \dot{x}_{2} = \left[ 2x_{1} + 2a + 4cx_{1} \left( {x_{1} ^{2} + x_{2} ^{2} } \right) \right] \\ \quad \qquad - \gamma \left[ 2x_{2} + b + 2cx_{2} \left( {x_{1} ^{2} + x_{2} ^{2} } \right) \right] F\left( {x_{1} ,x_{2} } \right) \\ \end{array} \right. \end{aligned}$$
(18)
Consequently, because the dynamic system given by Eq. (18) satisfies the condition given by Eq. (10) and the condition of the global invariant set theory given by Eq. (17), \(\varOmega \) given by Eq. (8) is its limit cycle. That is to say, any trajectory starting in the phase space converges to this limit cycle.

4 Coupled cardioid oscillators

In engineering fields, multiple objects are always controlled to move coordinately. Therefore, the nonlinear oscillators for the individual objects need to exhibit the phase-locked behavior. By coupling multiple cardioid oscillators, the cardioid oscillators can exhibit phase-locked behavior.
Fig. 7

State variants \(x_1 ^{1}\) and \(x_1 ^{2}\) of the two coupled cardioid oscillators with the initial conditions of (14.145, 0) and (0, 0): a \(\sigma ^{1}=\sigma ^{2}=0.5\), b \(\sigma ^{1}=\sigma ^{2}=1\), c \(\sigma ^{1}=\sigma ^{2}=2\), and d \(\sigma ^{1}=\sigma ^{2}=3\)

The coupled cardioid oscillators can be expressed as
$$\begin{aligned} \left\{ {\begin{array}{l} {{\dot{x}}_{1} ^{{i}} = {g}_{1} \left( {{x}_{1} ^{{i}} ,{~x}_{2} ^{{i}} } \right) + {\sigma }^{{i}} \mathop \sum \nolimits _{{{j} = 1}}^{{N}} {L}_{1} ^{{{ij}}} \left( {{x}_{1} ^{{i}} ,{~x}_{2} ^{{i}} ,{x}_{1} ^{{j}} ,{~x}_{2} ^{{j}} } \right) } \\ {{\dot{x}}_{2} ^{{i}} = {g}_{2} \left( {{x}_{1} ^{{i}} ,{~x}_{2} ^{{i}} } \right) - {\sigma }^{{i}} \mathop \sum \nolimits _{{{j} = 1}}^{{N}} {L}_{2} ^{{{ij}}} \left( {{x}_{1} ^{{i}} ,{~x}_{2} ^{{i}} ,{x}_{1} ^{{j}} ,{~x}_{2} ^{{j}} } \right) } \\ \end{array} } \right. \end{aligned}$$
(19)
where \(\left( {x_1 ^{i},x_2 ^{i}} \right) \) and \(\left( {x_1 ^{j},x_2 ^{j}} \right) \) are the states of the ith and jth cardioid oscillators; N in the number of the cardioid oscillators that coupled with the ith cardioid oscillator; \(\sigma ^{i}\) is the coupling parameter of the ith cardioid oscillator; \(L_1 ^{ij}\left( {x_1 ^{i},x_2 ^{i},x_1 ^{j},x_2 ^{j}} \right) \) is the coupling factor between the state \(x_1 ^{i}\)and state \(x_1 ^{j}\), \(L_2 ^{ij}\left( {x_1 ^{i},x_2 ^{i},x_1 ^{j},x_2 ^{j}} \right) \) is the coupling factor between the state \(x_2 ^{i}\) and state \(x_2 ^{j}\). \(L_1 ^{ij}\left( {x_1 ^{i},x_2 ^{i},x_1 ^{j},x_2 ^{j}} \right) \) and \(L_2 ^{ij}\left( {x_1 ^{i},x_2 ^{i},x_1 ^{j},x_2 ^{j}} \right) \) are given by
$$\begin{aligned} \left\{ {{\begin{array}{l} {L_1 ^{ij}\left( {x_1 ^{i},x_2 ^{i},x_1 ^{j},x_2 ^{j}} \right) =\left( {x_1 ^{j}+x_2 ^{j}} \right) x_2 ^{i}x_2 ^{i}} \\ {L_2 ^{ij}\left( {x_1 ^{i},x_2 ^{i},x_1 ^{j},x_2 ^{j}} \right) =\left( {x_1 ^{j}+x_2 ^{j}} \right) x_1 ^{i}x_2 ^{i}} \\ \end{array} }} \right. \end{aligned}$$
(20)
Fig. 8

State variants \(x_1 ^{1}\) and \(x_1 ^{2}\) of the two coupled cardioid oscillators with the initial condition of (0, 0): a \(\sigma ^{1}=-\sigma ^{2}=0.1\), b \(\sigma ^{1}=-\sigma ^{2}=0.25\), c \(\sigma ^{1}=-\sigma ^{2}=0.5\), and d \(\sigma ^{1}=-\sigma ^{2}=1\)

5 Numerical simulations and analyses

For analyzing the properties of the proposed cardioid oscillator, verifying that the cardioid oscillator possesses the trajectories with asymmetric time ratio, and analyzing the affection of its parameters on the resulting trajectories, numerical simulations on the cardioid oscillator given by Eq. (18) are conducted in this section.

5.1 Asymmetry of the limit cycle and the time ratio of the cardioid oscillator

The time ratio of the trajectories and the limit cycle of the cardioid oscillator can be adjusted by altering the ratio between a and b. Figure 3 shows the trajectories and the phase planes of the cardioid oscillator when \(a=5\) and \(c=-4\), and the ratio between a and b is \(-\) 10, \(-\) 1, 1, and 10, respectively. As Fig. 3a shows, the time ratio of the forward motion of the trajectory increases from 38 to 62% approximately with increasing the ratio between a and b from \(-\) 10 to 10. Figure 3b shows that the asymmetry of the limit cycle is more and more obvious with increasing the absolute value of the ratio between a and b. When the ratio between a and b is further increased, the bifurcation is occurred. The reasons for the bifurcation are waiting for the further research.

5.2 Self-excited oscillations and anti-interference

Figure 4 shows the phase planes of the cardioid oscillators with initial conditions of (\(-\) 14, 152), (10, 90), and (19, \(-\) 120) when \(a=5\), \(b=200\), and \(c=-\,4\), respectively. Observing Fig. 4, it is clear that the trajectories of the cardioid oscillators are able to converge to the limit cycle without any outside inputs when the initial conditions of the states are not on the limit cycle.

A cardioid oscillator disturbed by a noise can be expressed as
$$\begin{aligned} \left\{ {\begin{array}{l} \dot{x}_{1} = - \left[ {2x_{2} + b + 2cx_{2} \left( {x_{1} ^{2} + x_{2} ^{2} } \right) } \right] \\ \qquad \quad - \gamma \left[ {2x_{1} + 2a + 4cx_{1} \left( {x_{1} ^{2} + x_{2} ^{2} } \right) } \right] F + \delta \\ \dot{x}_{2} = \left[ {2x_{1} + 2a + 4cx_{1} \left( {x_{1} ^{2} + x_{2} ^{2} } \right) } \right] \\ \qquad \quad - \gamma \left[ {2x_{2} + b + 2cx_{2} \left( {x_{1} ^{2} + x_{2} ^{2} } \right) } \right] F \\ \end{array} } \right. \end{aligned}$$
(21)
where \(\delta \) is the noise signal. Figure 5 shows the phase planes and the trajectories of the cardioid oscillators disturbed by a pulse signal with amplitude of 8, frequency of 1 Hz, delay of 0.5 s, and duty ratio of 1%. Figure 5 shows that the states of the disturbed cardioid oscillators are able to back to the limit cycle, which indicates that the proposed oscillator has strong anti-interference performance.

5.3 Convergence rate of the cardioid oscillator

Figure 6 shows the trajectories of the cardioid oscillators with initial condition of (30, 0), \(a=5\), \(b=200\), \(c=-4\), and \(\gamma \) of 0.005, 0.02, and 0.05 in the whole process and the converging process, respectively. Observing Fig. 6b, the trajectories of the cardioid oscillator with the initial state of (30, 0) converges from the initial state to the limit cycle regardless of \(\gamma \). When \(\gamma \) is equal to 0.005, the converging process that \(x_1 \) with initial condition of 30 converges to the limit cycle where \(x_1 \) is equal to 20 takes 15 ms. When \(\gamma \) is equal to 0.02, the converging process takes 5 ms. And when \(\gamma \) is equal to 0.05, the converging process takes 2 ms. It is clearly seen that the convergence rate of the oscillator increases with increasing \(\gamma \).

5.4 Behaviors of two coupled cardioid oscillators

According to Eqs. (19) and (20), Fig. 7 shows the trajectories of the states \(x_1 ^{1}\) and \(x_1 ^{2}\) of two coupled cardioid oscillators with the initiate conditions of (14.14, 0) and (0, 0) when their coupling coefficients \(\sigma ^{1}\)and \(\sigma ^{2}\)are equal to 0.5, 1, 2, and 3. As shown in Fig. 7, the trajectories of the states \(x_1 ^{1}\) and \(x_1 ^{2}\) of the two coupled cardioid oscillators with the different initiate conditions will be synchronous without the phase difference after several periods. When the coupling coefficients \(\sigma ^{1}\) and \(\sigma ^{2}\) were equal to 0.5, 1, 2, and 3, the coupling times \(T_\mathrm{c}\) are 3.6, 3, 1.5, and 1 s. Therefore, the coupling strength of two coupled cardioid oscillators is increasing with increasing the coupling coefficients \(\sigma ^{1} \)and \(\sigma ^{2}\).
Fig. 9

Simulating the trajectory of the hip joint of a human body with the proposed cardioid oscillator: a the trajectory and b the error

According to Eqs. (19) and (20), Fig. 8 shows the trajectories of the states \(x_1 ^{1}\) and \(x_1 ^{2}\) of the two coupled cardioid oscillators with the initiate condition of (0, 0) when the coupling coefficient \(\sigma ^{1}\) is 0.1, 0.25, 0.5, and 1, and the coupling coefficient \(\sigma ^{2}\) is \(-\) 0.1, \(-\) 0.25, \(-\) 0.5, and \(-\) 1. As shown in Fig. 8, the trajectories of the states \(x_1 ^{1}\) and \(x_1 ^{2}\) of these two coupled cardioid oscillators detach to each other and finally maintain a phase difference after several periods. When \(\sigma ^{1}\) and \(\sigma ^{2}\) are equal to 0.1 and \(-\) 0.1, the coupling time \(T_\mathrm{c}\) is 6 s; when \(\sigma ^{1}\) and \( \sigma ^{2}\) are equal to 0.25 and \(-\) 0.25, the coupling time \(T_\mathrm{c}\) is 3.8 s; when \(\sigma ^{1}\) and \(\sigma ^{2}\) are equal to 0.5 and \(-\) 0.5, the coupling time \(T_\mathrm{c}\) is 2.2 s; when \(\sigma ^{1}\) and \(\sigma ^{2}\) are equal to 1 and \(-\) 1, the coupling time \(T_\mathrm{c}\) is 1.2 s. Therefore, the coupling strength of two coupled cardioid oscillators is increasing with increasing the absolute value of the coupling coefficients \(\sigma ^{1}\) and \(\sigma ^{2}\).

6 Simulating the rhythmic motion of the hip joint of a human body with a cardioid oscillator

Figure 9a shows the simulated trajectory of the cardioid oscillator with \(a=5\), \(b=200\), \(c=-4\), and \(\gamma =0.02\). For comparison, Fig. 9a also shows the measured trajectory of the hip joint of a human body, which is measured on the test platform for the above-knee prosthesis developed by the authors Xu et al. (2016). Figure 9b shows the angle error between the simulated trajectory of the cardioid oscillator and the measured trajectory of the hip joint of the human body. Observing Fig. 9a, the time ratio of the forward motion of the simulated trajectory by the cardioid oscillator to the one period is about 60%, which is the same as that of the measured trajectory of the hip joint of the human body. Observing Fig. 9b, the maximum absolute error between the simulated trajectory by the cardioid oscillator and the measured trajectory of the hip joint is less than 6\(^{\circ }\), which means that the established cardioid oscillator is fit for simulating the rhythmic motions of the hip joint of the human body.

7 Conclusion

In this paper, a cardioid oscillator with asymmetric time ratios is designed and realized through designing a cardioid curve as its limit cycle. The numerical simulation results indicate that the proposed cardioid oscillator possesses the following properties: (1) the proposed cardioid oscillator is able to generate a series of periodic and anti-interference self-exciting trajectories, (2) the generated trajectories possess the asymmetric time ratios, and (3) the time ratios can be regulated by adjusting the cardioid oscillator’s parameters. Additionally, the maximum absolute error between the trajectory of the rhythmic motion of the hip joint of the human body simulated by the cardioid oscillator and the trajectory measured from the hip joint of the human body is less than 6\(^{\circ }\). In addition, the simulated trajectory by the cardioid oscillator has the asymmetric time ratio, which is the same as the trajectory measured from the hip joint of the human body.

On these bases, the cardioid oscillator with asymmetric time ratio proposed in this paper can be used to establish the CPG models for modeling the rhythmic motions with asymmetric time ratios.

Notes

Acknowledgements

This work has been partialy supported by the LPMT, CAEP (Grant No. 2015-01-001), the National Natural Science Foundation of China (Grant No. 51675070), and the Fundamental Research Funds for the Central Universities (Project No. CDJZR12 12 00 05)

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of Optoelectronic Technology and Systems of the Ministry of Education of ChinaChongqing UniversityChongqingChina
  2. 2.Precision and Intelligence Laboratory, Department of Optoelectronic EngineeringChongqing UniversityChongqingChina

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