The ability to efficiently move in complex environments is a key property for animals and their survival. This implies that many aspects of their morphology and central nervous system are shaped by constraints related to their locomotor skills. Animal locomotion is not generated merely from neural systems; instead, it is generated from the close interaction between neural systems, musculoskeletal systems, and the real-world environment (Pfeifer and Bongard, 2006; Pfeifer et al., 2007). Thus, it is essential to elucidate the locomotion generation mechanism by analysing the interaction dynamics among these three systems and by analysing the neural systems themselves. Understanding these mechanisms is expected to result in contributions to biology and robotics by facilitating the design of durable and resilient robots that are energy-efficient.

Central pattern generators (CPGs) are neural circuits that are found in invertebrate (Pearson and Iles, 1973; Bässler and Wegner, 1983; Bässler, 1986) and vertebrate animals (Shik et al., 1966; Grillner, 1975; Grillner, 1985). CPGs can produce rhythmic patterns of neural activity without receiving any rhythmic inputs. The term central indicates that the sensory feedback from the peripheral nervous system is not needed for generating the rhythms (Marder and Bucher, 2001; Ijspeert, 2008). Biological CPGs underlie many fundamental rhythmic activities such as chewing, breathing, and digesting. In addition, they also serve as the fundamental building blocks for locomotor neural circuits. From the perspective of control, they have several interesting characteristics such as a distributed control, the ability to deal with redundancies, the presence of fast control loops, and the ability to modulate the locomotion by using simple control signals. Owing to these properties, CPGs are considered to be transferred mathematical models. In addition, CPGs serve as the building blocks of robotic locomotion controllers and are being increasingly used in the robotics community (Ijspeert, 2008). To enable biologically inspired robotic control, the architecture of CPGs has been extensively adopted to generate periodic patterns for locomotion control owing to the properties of nonlinear oscillators (Kimura et al., 1999; Fukuoka et al., 2003; Tsujita et al., 2003; Aoi and Tsuchiya, 2005; Buchli et al., 2006; Kimura et al., 2007; Righetti and Ijspeert, 2008; Wang et al., 2011).

Although sensory feedback in CPGs is not necessary for generating rhythmic patterns, it plays a central role in guaranteeing adaptivity to the environmental conditions (Ijspeert, 2008).

Sensory feedback in CPGs for animal locomotion was first studied in the pioneering work on bipedal walking conducted by Taga et al. (1991), Taga (1994), Taga (1995). In these studies, sensory information from the environment was fed back into the nervous system model to generate a walking pattern from the interaction among the nervous system model, musculoskeletal model, and environment (“Global Entrainment”). Kimura et al. (1999); Fukuoka et al. (2003) proposed a model by integrating CPG and reflex mechanisms to realise uneven terrain quadruped walking. Aoi and Tsuchiya (2005), Aoi and Tsuchiya (2006) focused on “phase resetting” (Schomburg et al., 1998), a feedback mechanism found in animals, to include gait stabilisation in CPG-based control models. Aoi’s group also applied the phase resetting feedback in CPGs to human-like musculoskeletal models of bipedal walking (Aoi et al., 2010), quadrupedal gait transitions (Aoi et al., 2011; Aoi et al., 2013), and a hexapod walking model (Ambe et al., 2018). Steingrube et al. (2010); Manoonpong et al. (2010) proposed a modular neural control with bio-inspired CPG-based network and sensory feedback, demonstrating environmental adaptability, such as walking on uneven terrain and avoiding unknown obstacles, and then extended the models by introducing forward models (Manoonpong et al., 2013; Dasgupta et al., 2015), visual feedback (Goldschmidt et al., 2014; Grinke et al., 2015), muscle models (Xiong et al., 2014; Xiong et al., 2015), and so on. Buchli et al. (2006); Nachstedt et al. (2017) proposed an adaptive frequency oscillator that could learn motion frequency adaptively and verified the generation of gait according to body characteristics. Furthermore, an interlimb coordination model that employed load information as sensory information and generated adaptive and diverse quadruped walking patterns was proposed (Maufroy et al., 2010; Fukuoka et al., 2015; Owaki and Ishiguro, 2017a). Sensory feedback inclusion significantly modifies the dynamics of the CPG’s architecture, which often leads to bifurcations and other dynamic phenomena (Aoi et al., 2011; Wang et al., 2011; Aoi et al., 2013).

To establish a systematic design principle of the sensory feedback in the CPGs to achieve biologically inspired robotic locomotion, a novel concept called “Tegotae” is proposed. Tegotae is a Japanese concept that describes the extent to which a perceived reaction matches the intended motor command. The potential of the Tegotae approach in reproducing animals’ locomotion and understanding the underlying mechanism has been previously demonstrated based on synthetic approaches. The Tegotae approach was first used by Owaki et al. (2017) to develop a minimal model for interlimb coordination on hexapod robot locomotion with CPG-based control. Kano et al. (2018) demonstrated gait transition between the concertina and scaffold-based locomotion in a snake model simulation with reflex-based control. Kano et al. (2019) proposed the detailed design of the Tegotae function, particularly for motor commands, using the genetic algorithm (GA) to simulate a simple 1-D earthworm model with CPG-based control. Owaki et al. (2021) demonstrated adaptive walking control on a biped model with CPG and reflex-based controllers.

The main contribution of this study is the construction of a specific proprioceptive feedback law through the so-called Tegotae approach (Owaki et al., 2017). Together with a specific control policy, i.e. reflex-like actuation, it exploits it fruitfully based on the concept of embodied intelligence (Pfeifer and Bongard, 2006; Pfeifer et al., 2007). Then, the feedback is applied to certain mechanical systems, i.e. hopping systems; is first considered for the simplest case of one leg, and is then extended to two legs. In such circumstances, the sensory feedback plays an important role in shaping the rhythmic patterns and ensuring coordination between the CPGs and body movements. This study demonstrates the adaptation processes as well as the acquirement of the different gait. In addition, it compares the analytical solution for the single-leg case with an optimal controller solution that is based on direct methods such as the multiple shooting methods (Bock and Plitt, 1984; Diehl et al., 2005; Fagiano, 2019). This confirms the intuitions for the energy efficiency of the control policy. Finally, we extensively analyse the approach in relation with the considerations for learning and energy efficiency (Hayashibe and Shimoda, 2014).

The following section presents the materials and methods used in this study. First, we briefly describe the Tegotae approach. Second, we present the mathematical model for the Tegotae-based control. Third, we discuss the Tegotae approach based on the learning framework by comparing it with tacit learning as described in Hayashibe and Shimoda (2014). Then, we present the simulation results to validate the Tegotae controller and then evaluate the energy efficiency. Finally, in Section 5, we discuss the results and future work.

The Tegotae approach has certain interesting similarities with other learning frameworks, which motivates some of the intuitions for its energy efficiency. The adaptivity in the learning processes is typically defined for the parameters/weights of the controller/learning agent. In the Tegotae framework, although a further adaptation of the feedback coefficients σ may be included, the main adaptation is induced by modifying the dynamics of the oscillators. This factor is taken into account in the comparison, since the eventual adaptation of the parameters is straightforward.

First, it is interesting to note how the Tegotae approach shares some similarities with the tacit learning, which is a learning framework that was introduced in Berenz et al. (2014); Berenz et al. (2015). In tacit learning, the control law consists of an extension for the PD controller with a tacit learner block with the time frame (Lt). By using the scalar case for simplicity, the following expression can be obtained. u = k x c T + q q = ∫ f ( e ) d t   ( Lt ) (23) where u, x _c , k, and e are respectively the control, the state variable that is expressed in the control space, the proportional and derivative gain, and any type of quantity that needs to be minimised. The learning process is obtained in the (Lt) block by accumulating the integral over the time of the quantity that needs to be minimised. On this basis, we neglect the proportional and derivative terms in this study. u = q q = ∫ f ( e ) d t   ( Lt ) (24)

The function f(e) is recommended to have the form f(e) = p(ξ)a(e) ^T . In the one-dimensional case, a(e) can be a simple linear transformation a(e) = ae and p(ξ) is a periodic function of ξ. Both of these additional terms are selected to guarantee the following. p ( ξ )   x e | | p ( ξ ) | |   | | x e | | = 1   if   α = 0 p ( ξ )   x e | | p ( ξ ) | |   | | x e | | = − 1   if   α = π (25)

In Eq. (25), x _e represents the state variable that is expressed in the task space, in which the error e is minimized. In contrast, α is generically defined as the angle between e ˙ and D(e); the latter is the direction toward which e is minimized. In the one-dimensional case, α = 0 ∧ π . This formulation guarantees that min (f(e)) = min(e). Now, let us consider the Tegotae framework. The objective is to construct feedback and not a feedforward controller. To do this, let us consider the factor that needs to be minimized that corresponds to e = −F _k , the virtual variable ξ to the physical variable ϕ, and the error function a(e) = σe. By neglecting the constant terms due to the integration, the feedback over the oscillator results in the following expressions. u = q e = − F k p ( ϕ ) = − sin ( ϕ ) q = ∫ f ( e ) d ϕ = ∫ σ sin ( ϕ ) F k d ϕ = − σ cos ( ϕ ( t ) ) F k   ( L ϕ ) (26)

In the Tegotae framework, x _e = Δl represents the elongation speed of the spring length. This variable points towards the direction of the minimisation of the value of e = −F _k . Thus, the following expression is obtained. − sin ( ϕ )   x e | | sin ( ϕ ) | |   | | x e | | = 1   if   α = 0 − sin ( ϕ )   x e | | sin ( ϕ ) | |   | | x e | | = − 1   if   α = π (27)

This shows how the Tegotae approach is de facto obtaining a tacit learning feedback (Lt) as previously described. Nevertheless, this is achieved by accumulating the quantity that needs to be minimised for the integral of the state space variable that is directly from (Lϕ). The integration over the state space frame ϕ is coherent with the CPGs’ framework. The role of the oscillators is to provide a different time frame to the dynamics, which is reproduced by the linear transformation ϕ = ωt. Thus, in the CPGs’ framework, the integration/derivation over the state variable of the oscillator ϕ is conceptually equivalent to the integration over the time. Interestingly, it has been demonstrated in Hayashibe and Shimoda (2014) that this controller can guarantee energy efficiency during the task realisation in case the quantity that needs to be minimised is the actuation torque.

The main contribution of this study is to propose a control policy with a reflex-like actuation (Eq. (11)) for the Tegotae-based feedback law in the CPG in such a way that the controller fruitfully exploits the embodiment (Pfeifer and Bongard, 2006; Pfeifer et al., 2007). For the validation of the proposed method, we first demonstrated the energy efficiency of the monopod model as well as its robustness and adaptability using the controller. Then, we demonstrated the gait transition for the bipedal model with its robustness and adaptability. Based on the optimal control theory, we designed an optimal controller and then compared it with the Tegotae-based control input. The results indicate the Tegotae-based feedback with reflex-like actuation results for optimal and energy-efficient motion. This suggests the first evidence concerning the optimal energy efficiency for the Tegotae approach.

This study is the first attempt to analyse the optimal energy efficiency along with the adaptivity of the Tegotae approach. Previous studies (Owaki et al., 2017) have mainly focused on the temporal (timing/phase) modulation in the oscillators by the Tegotae feedback on GPG-based models. The proposed reflex-like actuation can modulate the “amplitude” of the actuation via F _a function (Eq. (11)), depending on sensory feedback F _k . As presented in Table 1, in comparison with the previous methods, the introduction of the Tegotae feedback f ₁ was optimal in terms of the transient period for synchronisation and energy efficiency. The reflex-like pathway (Figure 1A) resulted in a rapid response (fast control loop) on motion generation, leading to the first convergent time in Table 1. Furthermore, the proposed reflex-like actuation (Eq. (11)) induced by the Tegotae feedback in the CPG could generate an input (Figures 7, 9) identical to that of the optimally designed controller, resulting in energy-efficient motion, as presented in Table 1. As discussed in Section. 3, the Tegotae approach has similarities (Eq. (26)) with the tacit learning frameworks in Hayashibe and Shimoda (2014). Energy efficiency is also achieved by the accumulation of a quantity that needs to be minimised when directly integrating the state variable. These facts suggest that our control policy, i.e. reflex-like actuation with the Tegotae-based proprioceptive feedback in the CPG, accomplishes optimal energy-efficient motion through the dynamical learning process along with the interaction between the controller, body, and environments (Pfeifer and Bongard, 2006; Pfeifer et al., 2007).

The reflex-based leg coordination models (Ekeberg and Pearson, 2005; Manoonpong et al., 2007; Lewinger and Quinn, 2011; Schilling et al., 2013; Dürr et al., 2019) and reflex-like feedback integration into CPG (Ajallooeian et al., 2013; Dzeladini et al., 2014; Li et al., 2014) have been studied in the past two decades. Pioneering research on “event-driven” reflex models in cats (Ekeberg and Pearson, 2005) and insects (Lewinger and Quinn, 2011; Schilling et al., 2013; Dürr et al., 2019) has been conducted, successfully reproducing various aspects of animal inter- and intra-leg coordination during locomotion. Manoonpong et al. (2007) demonstrated that a reflex-based neural controller could achieve stable and fast bipedal walking. Following the pioneering work integrating a CPG with reflex models (Kimura et al., 1999), similar approaches have been proposed. Ajallooeian et al. (2013); Li et al. (2014) also proposed to integrate a CPG with “event-driven” reflex models for adaptability against perturbations and environmental changes; One of characteristic approaches in this line, Dzeladini et al. (2014) introduced CPG as feed-forward components in reflex-based neuromuscular models for human walking, confirming the idea of using CPGs as feedback predictors (Kuo (2002)) from the viewpoint of gait modulation. In our work, the CPG oscillator is not a feedback predictor, but can be considered as a representation of the movement (phase ϕ _i ), that is, an internal model. In the Tegotae approcah, the Tegotae function T _i (ϕ _i ,F _k ) is defined as the product of the function of intended motor command C(ϕ _i ) and sensory information S(F _k ); hence, our reflex-like actuation always modulates the motion based on the Tegotae feedback f _i , which increases the value of the Tegotae function T _i (ϕ _i ,F _k ), leading to its adaptability and optimal energy efficiency, as mentioned in previous paragraph.

Past studies that have used the Tegotae approach (Owaki et al., 2012; Owaki and Ishiguro, 2017b; Owaki et al., 2017) have demonstrated adaptability and behavioural diversity for reproducing animal-like legged locomotion. For quadruped locomotion, the simple and local sensory feedback law in the CPG reproduced the adaptability against the change in mass distribution, which resulted in horse-like or primate-like walking patterns, and a spontaneous gait transition, from walking to trotting and galloping, in response to the locomotion speed. These studies for quadruped robots provide a basis for establishing a design scheme based on the Tegotae approach. For hexapod locomotion, Owaki et al. (2017) designed a minimal model for the inter-limb coordination in a systematic manner based on the Tegotae concept, successfully reproducing the various aspects of the insect locomotion patterns, which includes adaptability to changes in the body properties, e.g. leg amputation. In line with these studies, this investigation also successfully reproduces the adaptability (Figures 3, 6), and behavioural diversity (Figures 4, 5) as well as the energy efficiency. As discussed in previous studies, in the Tegotae approach, the main aim of designing the Tegotae function is to consider the physical consistency of the action and reaction for the desired motion, and to design the Tegotae function such that its value increases in such cases. Once such a Tegotae function is designed, it is possible to modify the control variables in a situation-dependent manner by increasing the value of the Tegotae function as a feedback term ∂ T ( x , S ) / ∂ x . Therefore, the Tegotae approach enables the design of an autonomous decentralised controller in a systematic manner, by designing the Tegotae function in line with the desired motions.

This study proposes a reflex-like actuation for the Tegotae-based feedback law in the CPG. This is a significant contribution for the actuation and sensory feedback on the adaptation process to the environment and the optimisation process for energy efficiency. However, one of the limitations of this study is that we did not test the applicability of the Tegotae approach to the real-world environment with a physical robot. In addition, it is extremely difficult to perfectly model the dynamics in the real-world environment. One of the key aspects based on the Tegotae approach is the verification in the real world as shown in Owaki et al. (2012); Owaki and Ishiguro (2017b); Owaki et al. (2017). Instead, we analysed the Tegotae control by using the optimal control theory and provided evidence concerning the optimal control input. Regarding the energy efficiency of tacit learning in the real-world environment, it has been verified by achieving a task with a redundant arm in Hayashibe and Shimoda (2018). One potential future direction is to apply our control policy to a robot with more degrees of freedom that performs more complicated tasks. Our control policy is compatible with the force/torque-based control of a physical robot, which is a promising direction of study for future research.