To capture the control principle essential for the interlimb coordination mechanism, which works largely in a decentralized manner in insects' thoracic ganglia, it is important to determine a basic building block to be used for the distributed control system. From a control perspective, past studies have intensively argued mainly from the viewpoint of two distinct control paradigms: chains of reflexes (Cruse, 1983, 1990; Cruse et al., 1998; Dürr et al., 2004; Schilling et al., 2013) and CPGs (Pearson and Iles, 1973; Bässler and Wegner, 1983; Bässler, 1986, 1993; Ryckebusch and Laurent, 1993; Büschges et al., 1995, 2004; Bässler and Büschges, 1998; Büschges, 2005; Borgmann et al., 2009; Daun-Gruhn and Büschges, 2011; Marder and Bucher, 2011). In the chain-of-reflex approach, a control system is modeled by using many chained discontinuous reflexive events, in which locomotion can be generated purely from the interaction between sensory feedback signals and the body. However, the discontinuity in this approach may impede mathematical tractability (Daun-Gruhn and Büschges, 2011). In contrast, in the CPG approach, a control system is modeled by using directly coupled oscillators to generate feedforward motor commands, based on a continuous dynamical system, i.e., a set of differential equations, for the interlimb coordination. Considering the mathematical tractability stemming from a continuous model, we employ the CPG approach as a control paradigm. The CPG approach offers various ways to model a basic building block at different levels of abstraction (Ijspeert, 2008), ranging from detailed models using a single cell (Hodgkin and Huxley, 1952; Hellgren et al., 1992) to abstract oscillator models (Fitz-Hugh, 1969; Van der Pol, 1972; Kuramoto, 1984). Here we use a phase oscillator (Kuramoto, 1984) for each leg to build a minimal model of the interlimb coordination mechanism on the basis of a highly abstract model.

The time evolution of the oscillator phase is described by a differential equation as follows: (1)ϕ˙i=ω+fi, where ω is the intrinsic angular velocity; ϕ_i is the phase of the oscillator implemented into the ith leg; and f_i is a local sensory feedback term, which plays an essential role in the interlimb coordination. This equation is one of the abstract oscillator models, i.e., the Kuramoto model (Kuramoto, 1984) (a case without coupling between oscillators and with local sensory feedback f_i), which describes a one-dimensional, reduction model of oscillatory behaviors. Using the trigonometric functions (sin ϕ_i, cos ϕ_i, etc.) of oscillator phases enables us to generate a periodic motor command to control the legs of a robot. As an example of implementation, we describe the target angles θ~yaw,i and θ~roll,i for the proportional and derivative (PD) control of the motors (as explained in Section 2.4 and Figure 6 in detail) through the following equations: (2)θ˜yaw,i = −A cosϕi, (3)θ˜roll,i= {B sin ϕi,when 0≤ϕi<π,B′ sin ϕi,when π≤ϕi<2π, where A, B, and B′ are user-defined parameters, describing amplitudes in the yaw and roll direction for leg motion (see Section 2.4 and Table 1). Thus, the ith leg is actively controlled according to ϕ_i such that the ith leg is in the swing phase when 0 ≤ ϕ_i < π, i.e., sin ϕ_i > 0, and in the stance phase when π ≤ ϕ_i < 2π, i.e., sin ϕ_i < 0, as shown in Figure 1. Below, we explain how we design local sensory feedback f_i by introducing the concept of “Tegotae” in a systematic manner.

Here we explain the core concept Tegotae in detail. Tegotae is a novel concept describing the extent to which a perceived reaction matches an expectation (intention) of a controller. For ease of understanding, let us explain it metaphorically. Imagine you want to lean against a wall nearby. Note that what you want to do, i.e., leaning against the wall, is regarded as the intention of the controller, i.e., your nervous system. When you lean against the wall, if you feel that the reaction force from the wall is sufficient for supporting your body, we say “good” Tegotae is obtained. If the reaction force you receive is insufficient (imagine the wall were a curtain/screen for example), “bad” Tegotae is obtained. Notice that Tegotae stems not only from the reaction received from the environment, but also from the consistency between the perceived reaction and the intention/expectation of the controller, i.e., what the controller wants to do.

Now the question is how to quantify Tegotae. Of course, there are various ways to accomplish this. As the initial step of the investigation, we quantify Tegotae in the simplest mathematical form, i.e., a function based on the type of separation of variables as follows: (4)Ti(ϕi,N)=C(ϕi)S(N).

Hereafter, we refer to the function T_i as the “Tegotae function”—a function that quantitatively measures Tegotae. ϕ_i is a control variable (in this case the phase of the oscillator), and N is the sensory information obtained from multiple sensors embedded in the body. Note that, the Tegotae function T_i is expressed as the product of two functions C(ϕ_i) and S(N): the former is a function expressing the intention of the controller, and the latter denotes the reaction obtained from the environment. Here, we design T_i such that it becomes more positive when enhanced Tegotae is detected. Next, we explain how we can design the sensory feedback term f_i by using T_i.

Given that the Tegotae function is defined, the local sensory feedback term f_i is designed in such a way that the control system modulates ϕ_i in order to increase the amount of Tegotae received. Thus, because a continuous system is used, f_i is expressed simply as the partial derivative of the Tegotae function T_i with respect to the control variable ϕ_i, as follows: (5)fi=∂Ti(ϕi,N)∂ϕi.

Note that we can systematically design decentralized controllers by only designing the Tegotae functions required.

Now, the question is how to define T_i(ϕ_i, N) to satisfactorily reproduce the hexapedal interlimb coordination observed in insect locomotion. In this study, we define T_i(ϕ_i, N) as follows: (6)Ti(ϕi,N)=σ1Ti,1(ϕi,N)+σ2Ti,2(ϕi,N), (7)Ti,1(ϕi,N)=(−sin ϕi)NiV, (8)Ti,2(ϕi,N)=sin ϕi(1nL∑j∈L(i)nLkjNjV).

As Equation (6) indicates, T_i(ϕ_i, N) consists of two Tegotae functions, T_{i, 1}(ϕ_i, N) and T_{i, 2}(ϕ_i, N), both of which are linearly coupled via the positive constants σ₁ and σ₂. The suffix i denotes the leg number (i:1, 2, …, 6). Sensory information N consists of vertical ground reaction forces (GRFs) acting on each leg N=[N1V,N2V,…,N6V]T. L(i) denotes a set consisting of the legs neighboring the ith leg, and n_L is the number of elements in L(i) and k_j (k_a, k_p, k_c ≥ 0) denotes the weight for each GRF NjV, as shown in Figure 2. Further, we present a detailed explanation of the approach we followed when designing these two Tegotae functions.

T_{i, 1} quantifies Tegotae on the basis of the information that is only locally available at the corresponding leg; when the local controller intends to be in the stance leg (−sin ϕ_i > 0), and results in receiving a ground reaction force (NiV>0) (Figure 3, top), T_{i, 1} evaluates this situation as “good” Tegotae, and returns a positive value.

On the other hand, T_{i, 2} quantifies Tegotae on the basis of the relationship between the movements of the corresponding leg and its neighboring legs; when the local controller intends to be in the swing phase (sin ϕ_i > 0) and its neighboring legs offer good support to the body at that time (1nL∑j∈L(i)nLkjNjV>0) (Figure 3, bottom), T_{i, 2} evaluates that the corresponding leg adequately establishes a relationship with its neighboring legs and returns a positive value.

By substituting Equations (6–8) into Equations (1) and (5), we obtain our interlimb coordination mechanism as follows: (9)ϕ˙i=ω−σ1NiVcos ϕi+σ2(1nL∑j∈L(i)nLkjNjV)cos ϕi.

Introduction of the Tegotae-based approach enables us to easily design a minimal model for hexapedal interlimb coordination in a systematic manner.

Figure 4 shows the structure of our hexapod robot. The robot consists of six leg segments (Figure 5) and a body segment. The robot is 0.40 m long, 0.30 m wide, 0.20 m high, and weighs 2.4 kg. The leg and body consist of carbon fiber rods and acrylonitrile butadiene styrene (ABS) resin printed using a 3-D printer. For each leg, we used two servo motors (Futaba Corporation, Japan: RS405CB), which generate leg motion during the swing and stance phases according to the corresponding oscillator phase (Figure 5B). As shown in Figure 6, we describe the target angles θ~yaw,i and θ~roll,i for proportional and derivative (PD) control of the motors through the following equations: (10)θ˜yaw,i = −A cos ϕi, (11)θ˜roll,i = {B sin ϕi,when 0≤ϕi<π,B′ sin ϕi,when π≤ϕi<2π.

Based on this control scheme, we can generate periodic leg motion as shown in Figure 5B. From the viewpoint of neurophysiological findings for a locomotor CPG system in animals (Lafreniere-Roula and McCrea, 2005; Rybak et al., 2016), Equation (9) corresponds to the rhythm generator (RG) and Equations (10) and (11) correspond to a pattern formation (PF) network in the two-level CPG concept. For the robot, we choose parameter values A, B, B′ for the geometric path of the foot by tuning them through trial and error as shown in Table 1. We employ passive springs (MISUMI Corporation: WM8-20, 2.9 N/mm) in each leg for shock absorption. Furthermore, we use three-axis force sensors (OptoForce Ltd., Hungary: OMD-20-SE-40N) in the feet of the robot to detect ground reaction forces (GRFs), as shown in Figure 5A.

The body contains a main control board. We calculate the oscillator phase in each leg by using microcontrollers (mbed NXP LPC1768) on the main control board. We manipulated each servo motor installed in the legs using proportional-derivative (PD) control as explained above.

Figure 7 shows the results of measurements conducted when our robot was engaged in steady walking. Here, we set the parameter ω = 2.0 rad/s. Figure 7 shows the gait diagram (upper graph) and time evolution of the oscillator phases of the legs (lower graph, sin ϕ_i) for the period 0.0–20.0 s. In the gait diagram, the colored regions represent the stance phase, which is distinguished by using the threshold data value (1.5 N: less than 10% of the maximum force detected) from the force sensor. Hereafter, we use the gait diagrams and movies (i.e., Movies S1–S3) recorded by a video camera as a qualitative evaluation index and the average duty factors (the ratio of the stance phase to one period) as a quantitative evaluation index. For the quantitative analysis, the duty factors obtained by the gait diagrams reflect the direction of the robot motion (i.e., straightness) because the asymmetric duty factors in the left and right legs indicate turning in the locomotion. Moreover, the duty factors indirectly represent the foot point velocity during the locomotion because the leg trajectory of our robot is determined in response to oscillator phases (Figure 6). Thus, the data of the duty factors from the gait diagrams indirectly include physical information about the speed and the direction of the locomotion (see SM for more details). The gait pattern rapidly converges from the initial phase relationship to a tetrapod gait—the ipsilateral feet touch the ground in the order of hind, middle, and fore legs—within approximately two periods. Furthermore, we tested the effect of the variation in the initial oscillator phases on the gait patterns. The results confirmed that the initial patterns converged to the same gait patterns from any initial phase relationship (in 10 out of 10 trials: 100%).

We tested the ability of the proposed control scheme to change the gait patterns according to the locomotion speed by linearly changing the parameter ω from 2.0 to 4.0 rad/s during the time period 40.0 to 42.0 s. Figure 8A shows the gait diagram (upper graph) and the time evolution of oscillator phases of legs (lower graph, sin ϕ_i), during the time period 30.0–50.0 s in this experiment. After ω was chenged, the gait pattern spontaneously changed from that of a tetrapod to that of a tripod—the (L1, R2, L3) and (R1, L2, R3) feet alternately touch the ground in the anti-phase (Movie S1). Figure 8B shows the profile of vertical and horizontal GRFs (NiV and NiH) in the same experiment. In this figure, the upper, middle, and lower graphs show the GRF profile of the front (L1), middle (L2), and hind (L3) legs, respectively. Furthermore, we confirmed this result for the gait transition in all 10 trials (10/10: 100%). The results indicate that leg coordination is appropriately modified according to the locomotion speed via Tegotae-based control.

Here, we show the adaptability of our robot to changes in weight distribution by applying a load (500 g) to the hind portion of the body (upper photograph in Figure 9). The lower graphs in Figure 9 show the experimental result. Here, we changed the parameter ω from 2.0 to 4.0 rad/s during the period 40.0 to 42.0 s as in the previous gait transition experiments (Section 3.2). After changing ω, the gait pattern did not change to that of a tripod; instead, a tetrapod gait was maintained (Movie S2). We obtained the same results in 10 out of 10 trials (100%). Figure 10 compares the average duty factor of the front, middle, and hind legs without and with the load for 10 trials (ω = 4.0 rad/s). The duty factor, which is the ratio of the stance phase to one period, was calculated by using the gait patterns during six periods for each trial. This result indicates that the duty factor of the loaded hind legs is larger than that of legs that do not bear any load. This result demonstrates the adaptability of our proposed control scheme to changes in the weight distribution without requiring prior data about these changes.

Figure 11 shows the experimental results of the leg amputation test after both of the middle legs were amputated. In spite of the amputation, the robot was able to continue walking. Furthermore, the gait patterns converged to a trot or an L-S walk gait observed in quadrupeds—i. e. the (L1, R3) and (R1, L3) feet alternately touch the ground in nearly anti-phase, or more precisely, focusing on the timing of touch down, the feet touch the ground in the order from L1, R3, R1, L3 (Movie S3). Figure 12 compares the average duty factor of the front, middle, and hind legs for 10 trials of the leg amputation experiment. The duty factor of each leg was modulated according to the remaining number of legs, which mainly resulted in increasing the duty factor of the hind legs. Furthermore, we confirmed that the initial patterns converged to the same gait patterns from any initial phase relationship (in 10 out of 10 trials: 100%). These results also indicate that the proposed control scheme can achieve interlimb coordination according to the physical properties of the robot's body in a self-organizing manner, without any predefined gait patterns.

The usefulness of our proposed local sensory feedback was verified based on the Tegotae approach by conducting experiments with the following conditions: we set the parameters ω = 2.0, σ₁ = 0.2, σ₂ = 0, which is a model similar to our previous model for quadrupeds (Owaki et al., 2012; Owaki and Ishiguro, 2017) or Barikhan's model for hexapod models (Barikhan et al., 2014). We conducted 10 trials in this experiment using randomly selected initial phases. Figure 13 shows the experimental results obtained using these parameters. The gait patterns mostly did not converge to insect-like gaits, e.g., tetrapod/tripod gaits, but converged to other patterns under many initial conditions (in 7 out of 10 trials: 70%) in this model. In these gaits, the left legs touched the ground in the order L3, L2, and L1 (hind to fore), whereas the right legs touched in the order R1, R2, and R3 (fore to hind). This result indicates that the model with only the second term in Equation (9) (similar to Barikhan's model) sometimes reproduced a gait pattern similar to that of insects, but its robustness against the initial conditions was insufficient.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.