1 Introduction

Modeling control mechanisms underlying speed control of bipedal locomotion is a fundamental problem in neuromechanics. This is because most methods are based on muscle-reflex controllers (MRCs) with no feed-forward control mechanisms (e.g., central pattern generators (CPGs)) [1]. The MRCs produce walking that is not only robust to perturbations, but also characterized with muscle activations and joint angles that are surprisingly comparable to those observed in humans. However, the feedback-based MRCs are not robust to changes in the walking speed; typically, they are optimized for a single speed. This is unlike animals and humans, which regulate speed via descending signals that originate from cortical and spinal regions of the central nervous system (CNS). Inspired with this, we here propose a patterned muscle-reflex controller (PMRC) combined with a forward dynamics musculoskeletal modeling formulation driven by muscle primitives [2].

This results in a novel closed-loop modeling formulation that synthesizes the neural integration of afferent/efferent commands for the stable control of muscle-actuated simulations. We demonstrate this formulation in a 2D biped actuated by 16 Hill-type muscle-tendon units (MTUs).

Fig. 1
figure 1

Patterned muscle-reflex control (PMRC) for bipedal locomotion. Via the speed input S, feed-forward control outputs the pre-stimulations \(p_{i}\) (\(i = 1,2,\ldots , 15,16\)) to muscle-reflex control of 16 Hill-type MTUs actuating a biped

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2 Materials and Methods

Figure 1 shows our proposed PMRC modeling formulation. In this, the pattern generator and formation components produce baseline MTU activations for a given target locomotion condition, i.e. target speed and ground elevation. Baseline activation are further modulated by the MRC to achieve a new activation set that ensures locomotion stability.

2.1 Pattern Generator

The pattern generator is modeled as a Matsuoka neural oscillator consisting of four mutually inhibiting neurons. The oscillator is given by:

$$\begin{aligned} S \frac{dx_j}{dt} + x_j = - \displaystyle \sum _{h=1}^{4} (a_{jh} c_h) + I_j - b f_j, \end{aligned}$$
(1)
$$\begin{aligned} c_j = max(0,x_j), \end{aligned}$$
(2)
$$\begin{aligned} S G_{j} \frac{df_j}{dt} + f_j = c_j, \end{aligned}$$
(3)

where \(a_{jh}\) is a weight of inhibitory synaptic connection from the h-th neuron to the j-th neuron (\(j= h = 1,2,3,4\)), which is set as: \(a_{jj} = 0\), \(a_{13} = a_{24} = a_{32} = a_{41} = 1.5\), and other \(a_{jh} = 2.0\). \(I_j\) and \(G_{j}\) are the constant input and gain to the j-th neuron, which are set as: \(I_j = G_{j} = 1.6\). S and \(c_j\) are the input and output of the pattern generator. Adjusting the speed input S results in the outputs \(c_j\) with different oscillations (see Fig. 1), which leads to a simple strategy of speed control of bipedal locomotion.

Fig. 2
figure 2

Outputs \(c_{1-4}\) of the Matsuoka neural oscillator (see Fig. 1) with respect to the speed input S

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Fig. 3
figure 3

Basic activation patterns \(c_{1-4}\) of the Hill-type MTUs for coordinating the left and right legs of the biped (\(S= 4.7\))

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Fig. 4
figure 4

Adaptive weights \(W_{1-4,1}\) and \(W_{1-4,3}\) for the GMAX and HAMS muscles with respect to different desired speeds [km/h]. The associated activation patterns of the adaptive weights can be seen at Fig. 2

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2.2 Pattern Formation

The pattern formation is to transform the basic activation patterns \(c_{1-4}\) (see Fig. 2) to the pre-stimulations \(p_i\) (\(i = 1,2,\ldots ,15,16\)) for 16 Hill-type MTUs actuating the biped. First, the phase difference of the basic activation patterns \(c_{1-4}\) for the Hill-type MTUs actuating its left and right legs is set to one-half of the cycle (see Fig. 3). Second, the pre-stimulation matrix \(P_{2\times 8}\) is the product of the pattern matrix \(C_{2\times 4}\) and the weight matrix \(W_{4\times 8}\) (see Fig. 1),

$$\begin{aligned} P_{2\times 8} = C_{2\times 4}W_{4\times 8}, p_{1-8} = P_{1,1-8}, p_{9-16} = P_{2,1-8}, \end{aligned}$$
(4)

where the first and second rows of \(C_{2\times 4}\) represent the basic activation patterns for the 16 Hill-type MTUs actuating the left and right legs (see Fig. 3), respectively. The eight columns of \(W_{4\times 8}\) denote the eight sets of weights for the eight Hill-type MTUs actuating each leg of the biped (e.g., see Fig. 4). The considered MTUs are GMAX (gluteus maximus), ILPSO (iliopsoas), HAMS (biarticular hamstrings), RF (rectus femoris), VAS (vasti), GAS (gastrocnemius), SOL (soleus), TA (tibialis anterior). The calculation of \(W_{4\times 8}\) is based on a predictive regression model where the relationship between muscle weights and walking speeds is generalized [3].

3 Results

The PMRC (see Fig. 1) enables the biped to easily regulate speed via a simple strategy where the neural oscillator and weight prediction are integrated into feed-forward control [2], and further adjusted via muscle reflex rules. This is because feed-forward control of the PMRC can easily produce activation oscillations (see Fig. 2) and leg coordination (see Fig. 3) while producing adaptive weights for their associated activation patterns with respect to speeds (Fig. 4). As a result, bipedal locomotion does not need to be re-optimized in order to walk at a different speed. The video of speed control of the biped under the PMRC can be seen at https://www.youtube.com/watch?v=z_iU89xYKX0&feature=youtu.be.

4 Discussions and Conclusions

The feed-forward neural and feedback-driven muscle-reflex control is integrated to facilitate speed control of bipedal locomotion. The method provides a way forward to understand bipedal locomotion and develop intelligent assistive devices. This approach enables synthesizing important neuromechanical processes underlying human locomotion. Future work will extend this method by synthesizing the neuro-mechanical processes underlying modulation of joint viscoelasticity [4]. It is expected this will enable achieving adaptive locomotion not only to speed but also across a variety of terrains.