Gait Genesis Through Emergent Ordering of RBF Neurons on Central Pattern Generator for Hexapod Walking Robot

Abstract

The neurally based gait controllers for multilegged robots are designed to reproduce the plasticity observed in animal locomotion. In animals, gaits are regulated by Central Pattern Generator (CPG), a recurrent neural network producing rhythmical signals prescribing each leg's action timing, leading to coordinated motion of multiple legs. The biomimetic CPG-RBF architecture, where leg motion timing is encoded by Radial Basis Function (RBF) neurons coupled with CPG, is used in recent gait controllers. However, the RBF neurons coupling is usually parameterized by the supervisor. Therefore, the RBF parameters get outdated when the CPG signal's wave-form changes. We propose self-supervised dynamics for RBF parameters adapting to a given CPG and producing the required gait rhythm. The method orders the leg activity with respect to inter-leg coordination rules and maps the activity onto CPG states. The proposed dynamics produce rhythmic control for three different hexapod gaits and adapts to the CPG parametric changes.

1 Introduction

The biomimetic approach of the gait control is adopted in multi-legged robotics to imitate robustness and adaptability observed in animal locomotion [ 1 ]. The locomotion is driven by a neural network that continually controls and adapts to the environment during movement. In the context of the gait control, the essential part of the neural network is a Central Pattern Generator (CPG) [ 2 ], recurrently connected neurons generating rhythmical signals that drive the motion. The CPG is thus employed in many biomimetic multi-legged robot gait controllers [ 3 ].

In CPG-based controllers, the CPG drives the repetitive gait motion. During a regular motion, the gait can be described as a repeating sequence of leg movements, where each movement is performed at a certain motion phase. The motion phase is a hidden state that can be inferred from sensory feedback [ 4 ], or, as in our case, tracked by the CPG.

The CPG signal is periodic, and thus the state of the recurrent neural network representing the CPG creates a limit cycle, a closed trajectory in the state space. The CPG state then tracks the hidden motion phase of the gait [5,

Copyright ©2021 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). 6], where each leg movement corresponds to some CPG state. The method of mapping the leg movements onto the CPG state is determined by the selected architecture of the CPG-based controller.

In this paper, we focus on the architecture where the CPG is coupled with Radial Basis Function (RBF) neurons, where each RBF neuron fires at a particular CPG state [ 7, 8 ]. The RBF neuron activity dependes on the distance of the input point to the neuron’s parameter point, i.e., the activity dependes on the radius around the fixed parameter determining vicinity in which the input point is. Hence, radial basis function neurons.

In our method, the RBF neurons are parameterized by the centers placed into the CPG’s state space such that when the CPG state, representing the input, is near one of the centers, the corresponding RBF peaks in the activity. The RBF activations can then be used as motion phase encoding, motion primitive trigger, or couple multiple CPGs.

As far as we know, in all current CPG-RBF controllers, the RBF centers are set up by a supervisor. Such a prior parametrization assumes that the CPG properties remain unchanged during the locomotion. Due to the assumption of static properties, the CPG cannot be optimized (e.g., by Righetti’s learning rule [ 9 ]) nor the entraining waveform can be changed, which poses a limitation to the adaptability of the system.

In this work, we propose a dynamic rule for RBF centers self-organization that generates gaits. The proposed method decomposes the RBF centers organization into two tasks. First, the organization of leg movements in phase space that is consistent with Inter-leg Coordination Rules (ICRs) [ 10 ] and given phase offset of consecutive legs’ activity [ 11 ], providing phase relations within legs actions (see Fig. 1). The second task is the mapping of the organized leg movements onto CPG states. Both tasks are processed continually by the proposed dynamic rules and organize the RBF centers along the CPG’s limit cycle, so the resulting rhythm produces a corresponding gait.

The method is implemented on a hexapod walking robot in the simulated environment, where we show that the proposed solution generates multiple gaits consistent with the ICRs. We also demonstrate the adaptive capabilities during change of CPG properties, where the proposed solution adapts to the changes. 2

Related Work

CPG-based controllers can be found in wearable and legged robotics. The controllers can consist of two submodules: amplitude control, providing the magnitude of actuation, and phase control, providing the actuation timing [ 12, 13 ]. The CPG is involved in the phase control, where the CPG state represents the motion phase. One of the distinguishing features of the CPG-based controllers is how the CPG state is mapped into the movement phase. Three types of CPG-to-motion mapping can be distinguished: continuous, binary-phase switch, and its generalization multi-phase switch, characterized as follows.

Continuous mapping reshapes the CPG signal into the motion command with continuous function. In [ 14 ], where the CPG signal is empirically reshaped into joint angle command. The authors of [ 15 ] interpret the CPG output as a foot tip position that is transformed into joint angles by inverse kinematics. In [ 16 ], the CPG output is directly fed as an input of a PD controller that transforms the CPG output into angles of leg joints. Continuous maps depend on the wave-form of the CPG, which might limit the system adaptation, as the wave-form changes non-linearly with changing CPG parameters.

In contrast, using the CPG as a binary switch makes the system independent of the exact shape of the waveform and rather uses the CPG as a timing generator. The CPG can be used for switching between the stance and swing leg motion modes, where each mode has its control rules [ 12, 17 ]. The switching approach can be combined with a continuous mapping approach as in [ 18 ], where two different CPG output shapers are defined for the stance and swing motion modes, respectively. Using the CPG as a switch between stance and swing leg motions is straightforward; however, it forces the architecture to contain at least one CPG per leg as each leg needs its swing/stance timing. In multi CPG networks, the different gaits are implemented by learning the correct connectivity between CPGs, which might be difficult as the interaction between CPGs is generally non-linear. The networks of multiple CPGs can be avoided by generalizing the binaryphase into a multi-phase approach provided by CPG-RBF architecture.

The CPG-RBF architecture has been recently proposed in [ 19 ], and it provides a straightforward representation of the map between CPG states and motion phase using the RBF layer. The straightforward motion phase representation is utilized in [ 8 ], where the RBF output is used to learn the amplitude control with reinforcement learning mechanisms. The RBF neurons themselves can be trained to adapt general CPG with the periodic Grossberg rule. It is presented in [ 7 ], where the learning rule is used to couple two CPG layers specialized in sensory estimation and motor phase control.

In the context of gait generation, each of the three approaches has a different way of parameterizing the gait. While there are already proposed methods for gait learning for continuous and binary mapping approaches [ 14, 13, 12, 18 ], there is none for the CPG-RBF architecture. The aforementioned CPG-RBF controllers have RBF centers set by a supervisor, limiting the system to static CPG’s properties. Since we aim to increase the adaptability capabilities of the CPG-RBF controllers, we propose a selforganizing method for the RBF centers that generate the desired gait patterns for the hexapod walking robot. 3

Problem Statement

The gait phase controller provides the timing for each ith leg to coordinate the leg movement. We focus on the movement patterns that are consistent with three inter-leg coordination rules (ICRs) observed from hexapod insect gaits [ 10 ]: (i) while a leg is lifted-off, suppress the lift-off of the consecutive leg; (ii) if the leg touches the ground, initiate the lift-off of the consecutive leg; (iii) do not lift off the contralateral legs at the same time. The swing duration given by the phase offset Df [ 11 ] then determines the exact motion pattern (see Tab. 1) as it is depicted in Figs. 2 and 3, where the motion states of the feasible gaits are visualized with the color labeling as in Fig. 1.

For the CPG-RBF phase controller, we encode the coordinated timing by coupling RBF neurons to a single CPG, where each i-th RBF neuron drives the corresponding i-th leg. Generally, the CPG state evolution, y(t) 2 RD, can be modeled by the differential equation y˙ = f (y(t); c(t)), where the dot notation represents differential with respect to time, and the system f contains a limit cycle attractor, y0 RD: a looped trajectory to which all neighboring states converge. Thus, after the convergence, the CPG is T -periodic, y(t) = y(t + T ), and the CPG generates a periodic signal. The CPG state is an input for the RBF neuron activation j (y;w) = exp wk22 , where the ceny ky ter w 2 RD and hyperparameter y determines the timing and duration of activation, respectively. The RBF neuron peaks when the CPG state is close to the RBF center, y(t) w(t), generating periodic peaks. For each i-th leg, there is a corresponding RBF neuron with center wi, which peaks trigger the lift-off.

For a general CPG, the centers wi cannot be placed a priory, as the shape of the limit cycle y0 is not known. Moreover, the limit cycle can change its shape dynamically with changing parametrization of CPG dynamics f ( ) or different CPG input c(t). Thus the centers wi of RBF neurons need to be dynamically adjusted to drive the locomotion according to the coordination rules and given phase offset Df . 4

Method

We propose dynamic rules that form feasible gait patterns by organizing the RBF centers wi on the CPG’s limit cycle y0 while respecting the ICRs and maintaining the given phase offset Df . The task is decoupled into two subtasks: (i) order the lift-offs of each i-th leg into a sequence and (ii) map the sequence onto the CPG limit cycle.

The i-th RBF center dynamics are given by the periodic Grossberg rule [ 7 ] w˙ i = (y(t) wi(t)) pi(t); (1) that pushes the center wi towards the states y(t) at which the target signal pi(t) 2 [0; 1] is nonzero. In the previous work [ 7 ], the target signal is given by a supervisor. We introduce a new method for forming the target signal pi(t) = j fˆ (t); fi(t) (2) where fi 2 [0; 2p ) is the phase of the i-th leg lift-off that determines the sequence of lift-offs, and fˆ 2 [0; 2p ) that maps the [0; 2p ) phases onto the limit cycle y0. As the phase is within the circular space S1, we define the circle metric as jjf f 0jj = min(jf f 0j; 2p jf f 0j), which gives the closest distance between two phases. For both variables, fi and fˆ , we present their dynamics in the following sections. 4.1

Organizing Phase of Legs Activity

The motion start phases fi of each i-th leg must be ordered within the interval [0; 2p ), where the ordering has to be consistent with the ICRs, and phase offset Df . Both constraints can be defined as distances between phases fi: (i) front (hind) leg i is shifted by Df with respect to ipsilateral middle leg j, jjfi f jjj = Df ; (ii) contralateral legs i and j are in anti-phase jjfi f jjj = p , see Fig. 1.

We present the following dynamics to order randomly initialized phases fi

6 f˙i(t) = å ai j (Di j + sign (fi j f j) jjfi f jjj) ; (3) where Di j 2 [ p ; p ] parameterize the target offset between fi and f j, and ai j 2 IR+ weighs the relation influence. The relationships between contralateral legs are parameterized as D2;1 = D4;3 = D6;5 = p and D1;2 = D3;4 = D5;6 = p . The relation between ipsilateral legs are D2;6 = D1;5 = Df and D4;6 = D3;5 = Df . The above-defined relations have set ai j = 1 while the rest is turned off by a j0i0 = 0. The variable fi serves as a parameter of the target signal (2), the next input is the phase estimation. 4.2

Mapping the Legs Activity onto the Limit Cycle

Using Phase Estimation

The target signal (2) should have the same periodicity as the CPG; however, the periodicity of a general CPG nor its frequency cannot be obtained analytically. However, the frequency is needed to modulate the phase of the target signal fˆ(t), and thus the frequency must be learned. We propose to dynamically learn the CPG frequency by coupling the CPG with the pivot RBF neuron and measuring the RBF activity period to determine the CPG’s frequency.

First, the randomly initialized pivot RBF center wsig 2 IRD must get close to the limit cycle. The pivot center wsig(t) is attracted to the CPG state y(t) if the CPG state is within e-neighborhood of the center where clip(x) = min(1; max(0; x)) and hyperparameters s1 = 80; s2 = 2 are set empirically. As the pivot center wsig(t) converges to the limit cycle, the pivot RBF activation psig = j(y(t);wsig) produces T -periodic pulses.

From the pivot RBF activation psig, we extract the frequency of the CPG by adjusting descend of the variable s˙ = ((1

s)x a(t) psig(t) otherwise 1 ; (5) (6) (7) (8) (9) where x is large enough to reset s to 1 when the pivot activation peaks (psig(t) 1), and a 2 IR+ determines the slope of the descend. If the slope of a has such a value that descends from s(t1) = 1 to s(t1 + T ) = 0, then a is the CPG frequency. Thus slope a is adjusted as follows a(t+) := (a(t ) + ks(t ) a(t ) psig(t) 1 otherwise; which increases the frequency if s(t1 + T ) > 0 and decreases the frequency if s(t1 + T ) < 0. The hyperparameter u˙ = v; v˙ = z 1 u2 v u; where, z is a parameter indicating the strength of damping and the CPG’s state is y = (u; v) 2 IR2. 5.1

Adaptability Experiments

The system’s adaptability to different limit cycle shapes is demonstrated by learning the transition gait in four different scenarios.

1https://www.coppeliarobotics.com k is empirically set to 0:02. The obtained CPG frequency is used to estimate the CPG phase fˆ(t) = 2p (1 s(t)) : (10)

After the convergence of the CPG estimation fˆ(t) and lift-off ordering fi, the variables pi(t) of (2) produce the target signal that orders the RBF centers of each i-th leg to produce the gait pattern; which is demonstrated in the following section. 5

Results

The feasibility of the proposed method has been validated by experimental deployment in several scenarios to demonstrate the adaptability of the developed solution. We used Euler’s method to run the dynamic system consisting of the proposed equations, running for 20000 iterations with the step size 0:01. The correctness of the generated rhythm for different gaits is demonstrated using modified signals, obtained from the RBF neuron activations j(y;wi), to trigger the predefined swing movement, followed by a predefined stance movement for the robot’s legs in CoppeliaSim1 simulator. The methods have been implemented in Python 3 and a hexapod model PhantomX MK-III has been used to run the simulations. The system’s adaptability has been evaluated for two different CPG models.

The first CPG model is Matsuoka oscillator given by t1v˙1 = h(u1) v1; t1v˙2 = h(u2) v2; t2u˙1 = t2u˙2 = u1 u2 h(u2)b1 h(u1)b1 v1b2 + 1; v2b2 + 1; h(x) := max(x; 0); where the hyperparameters are set to t1 = 0:5; t2 = 0:25; b1 = b2 = 2:5, and the function h(x) represents the rectifier (i.e., ReLU function). For the Matsuoka oscillator the CPG’s state is y = (u1; u2; v1; v2) 2 IR4.

The second CPG is Van der Pol’s oscillator (VdP), given by (11) (12) (13) (14) (15) (16) (17) 1. Using unperturbed Matsuoka oscillator as the CPG model. 2. Using Matsuoka oscillator synchronized with four other coupled Matsuoka oscillators to demonstrate the method’s adaptability to small perturbations. 3. Using the VdP with z = 3 to demonstrate the usage on a different oscillator. 4. Using the VdP with the changed parameter z = 1 to show the adaptability to change of the oscillator parameter.

In scenario 2. the motion phases fi, their respective RBF centers wi, and the parameter a are initialized to the transition gait values, which were previously successfully learned with the unperturbed CPG. The RBF neurons provide the rhythmical input, producing the signal psig(t) based on the position of the center wsig with the dynamic vicinity.

The CPG’s phase is estimated as fˆ(t) to map the centers wi 2 IRD on the CPG’s limit cycle, by learning the frequency a, (9), and pivot center wsig, (4). A showcase of wsig center’s attraction to the CPG’s limit cycle together with the progress of its dynamic vicinity radius e, (6), is shown in Fig. 4 for both types of the oscillators with differing limit cycle shapes of a prior unknown shape.

The frequency a is learned based on the signal generated by the RBF neuron corresponding to the center wsig. The learning process of a is shown in Fig. 5. In Fig. 6, we provide a learning process of a for Matsuoka oscillator with differently initialized a. In all the cases, the frequency a successfully converges. The progression and convergence of s(t), estimating the CPG’s phase growth, and the modulated learning signal psig(t) are presented in Fig. 7, where the fact that a converges can be seen in convergence to zero of local minimum values, marked by the orange line. As mentioned in Section 4, if learned correctly, the value of s(t) declines from one to zero during each period, i.e., at the psig(t) signal pulse occurs, s(t) equals zero and it is reset to one.

Based on the estimated phase, the centers are organized around the CPGs’ limit cycles, as demonstrated by the transition gait for Matsuoka and VdP oscillators shown in Fig. 8, where centers are organized around the CPGs limit cycles of various a prior unknown shapes. 5.2

Different Gait Patterns Experiment

The ability to generate different gait patterns is demonstrated using Matsuoka neural oscillator. The motion phases fi 2 [0; 2p) interact with each other according to the given phase offset Df of two consecutive leg’s actions, and to ICRs, as shown in Fig. 9 for all three gait patterns. The correctness of the process can be observed by comparing the results with schema,2 describing the gait patterns in Fig. 3. The process of ordering the motion phases within the phase is independent of the used CPG model.

The successful mapping of the RBF centers’ for three different gait patterns with the use of Matsuoka oscillator is shown in Fig. 10. The signals produced via the centers’ RBF neurons producing gait pattern rhythm are visualized in Fig. 11.

The RBF signals are used in the CoppeliaSim simulator to make the hexapod robot walking, as shown by snapshots of one gait cycle for each of the given gait patterns in Fig. 12.

The experiments demonstrated that the proposed mechanism successfully produced rhythm for the desired gait patterns on both the CPG models with different dimensionality and different shape of their limit cycles.

2Note that important are the relative positions, i.e., ordering of the motion phases with correct distance (phase offset) between them. The current CPG-RBF controllers require setting the gaitpattern-determining RBF neurons parameters by a supervisor, assuming that the CPG produces a signal of unchanging wave-form and frequency. However, this assumption limits the architecture’s ability to adapt to changing CPG parameters, enabling higher frequency (thus faster movement) or adaptation after a change of the synchronizing signal. Our method enables the change of the CPG parameters, and therefore improves the adaptability to evolving conditions for the CPG-RBF architectures.

The gait pattern rhythm is generated by correctly ordered RBF centers wi 2 IRD (see Figs. 8 and 10), corresponding to legs actions, around the CPG’s limit cycle. The centers wi produce signals via their respective RBF neurons if the CPG’s state is close enough to the corresponding center. The produced signal provides a rhythm for the required gait pattern if the centers’ ordering along the limit cycle respects ICRs and the given phase offset Df of consecutive legs, determining the required gait pattern. As shown in Fig. 12, we successfully produced three desired gait patterns for the hexapod walking robot, tripod, transition, and wave gaits, described in Figs. 2 and 3.

The model learns parameters which have real-wolrd meaning. The phases fi represent the respective leg’s swing phase start within the walking cycle. The placing of the centers wi around the limit cycle represents the respective leg’s swing phase start within the repeating walking cycle. Slope a represents the frequency of the used CPG. The pivot center wsig marks the start of the walking cycle on the CPG limit cycle. Hence, the learning process is explainable, which is an advantage in comparison with black-box approaches.

The current method does not enable the generation of gait rhythm for different numbers of legs than six without further modifications. In our future work, we would like to explore the ICRs possibilities in automatically generating gait patterns for any number of legs. Robots with differing numbers of legs exist and malfunctions of the robot are also possible, requiring to learn to walk with damaged or missing limbs while deployed on a mission. 6

Conclusion

In this work, we propose and test self-supervised dynamics for organizing the RBF centers producing rhythm for required gait patterns. The method improves CPG-RBF controllers’ gait-generating adaptability towards a change of CPG properties. The method decouples the gait rhythm generating problem into two tasks, the legs activity ordering, and the CPG phase estimation, leading to mapping the legs’ activity ordering onto CPG’s states. The ordering of legs’ activity within the phase is driven by biomimetic inter-leg coordination rules and given phase offset of consecutive legs’ activity, determining the required gait pattern. The phase estimation is based on estimating the phase growth (phase angular velocity) from the signal with the period equal to the CPG’s period. Combining the proposed mechanisms enables mapping the RBF centers, corresponding to ordered actions, onto CPG’s limit cycle. The phase controller produces the rhythm for three desired gait patterns, tripod, transition, and wave gaits.

We demonstrate the correct functionality of the proposed method, including showcase from CoppeliaSim simulator, where the generated gait pattern rhythms are used to invoke the movement of the simulated hexapod walking robot. The results are demonstrated for two different CPG models, Matsuoka neural oscillator with/without a rhythmical input from other coupled CPGs, and Van der Pol’s oscillator with two different parameter settings.

In our future work, we aim to extend the model to generate gait patterns for robots with differing numbers of legs.

Acknowledgments – This work has been supported by the Czech Science Foundation (GA CˇR) under research project No. 21-33041J.

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