A cardioid oscillator with asymmetric time ratio for establishing CPG models

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.

Design and Development of a Rayleigh Oscillator-Based Reference Angle Generator for Motion Control of Smart Prosthetic Knees

“…The RMSEs, the maximum absolute errors, and the cumulative errors between the gait cycle time of the thigh reference angle (θ t_R ) and that of the subject's thigh's swing angle (θ t_a ) in two walking processes are shown in Table 1. In Figures 11, 13 and Table 1, the traditional method [20]- [23], [26] that uses the subject's previous gait cycle time as its subsequent gait cycle time is added for comparison. In Table 1, in the first walking process, the RMSE between the gait cycle time of the thigh reference angle (θ t_R ) and that of the subject's thigh's swing angle (θ t_a ) is 0.014 s, the maximum absolute error is 0.030 s, TABLE 1.…”

“…Torrealba et al [19] used sines and cosines as the prosthesis' CPG model. We [20] also proposed a cardioid oscillator to model rhythmic motions with asymmetric time ratios in the CPG model. As a classic oscillator, Rayleigh oscillator is also used as CPG model.…”

Cardioid oscillator-based pattern generator for imitating the time-ratio-asymmetrical behavior of the lower limb exoskeleton

“…On the contrary, the reference trajectory generated by the central pattern generators (CPGs), a biological neural circuit that generates rhythmic behaviors in animals, is periodical and self-excited. It is important to note that these trajectories possess the locked phase relationships and can be adjusted according to environment changes, thereby attracting significant research attention (Conradt, 2003 ; Acebrón et al, 2005 ; de Pina Filho et al, 2005 ; Morimoto et al, 2008 ; Saito et al, 2009 ; Katayama, 2012 ; Mora et al, 2012 ; Dingguo et al, 2017 ; Ferrario et al, 2018 ; Fu et al, 2018 ; Payam et al, 2018 ; Xie et al, 2019 ; Mokhtari et al, 2020 ; Pasandi et al, 2022 ; Wei et al, 2022 ).…”

“…Recently, CPG models, such as neuron-based CPG model (Saito et al, 2009 ; Katayama, 2012 ; Dingguo et al, 2017 ; Ferrario et al, 2018 ; Payam et al, 2018 ; Xie et al, 2019 ; Mokhtari et al, 2020 ; Pasandi et al, 2022 ; Wei et al, 2022 ) and oscillator-based CPG model (Conradt, 2003 ; Acebrón et al, 2005 ; de Pina Filho et al, 2005 ; Morimoto et al, 2008 ; Mora et al, 2012 ; Fu et al, 2018 ), are used for imitating the swing angles of human lower limbs. The former utilizes an oscillator to imitate the functions of neural cells, while the latter utilizes an oscillator to imitate periodic motions/torques.…”

Central pattern generator–based coupling control method for synchronously controlling the two-degrees-of-freedom robot

“…6 Currently, the widely applied control methods for multi-joint robots include sliding mode control, 7 fuzzy control, 8 crossing-coupling control and contour error coupled control, [9][10][11] and central pattern generator (CPG) control. [12][13][14][15][16][17][18][19] Kamal et al 7 controlled TDFR's two joints to follow the two desired trajectories by applying the sliding mode control method. Combining proportional-integral-derivative (PID) control and fuzzy control, Mohan et al 8 controlled a TDFR to achieve the pre-appointed motions.…”