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Identifying essential factors for energy-efficient walking control across a wide range of velocities in reflex-based musculoskeletal systems [1]

['Shunsuke Koseki', 'Department Of Robotics', 'Graduate School Of Engineering', 'Tohoku University', 'Sendai', 'Mitsuhiro Hayashibe', 'Dai Owaki']

Date: 2024-02

Abstract Humans can generate and sustain a wide range of walking velocities while optimizing their energy efficiency. Understanding the intricate mechanisms governing human walking will contribute to the engineering applications such as energy-efficient biped robots and walking assistive devices. Reflex-based control mechanisms, which generate motor patterns in response to sensory feedback, have shown promise in generating human-like walking in musculoskeletal models. However, the precise regulation of velocity remains a major challenge. This limitation makes it difficult to identify the essential reflex circuits for energy-efficient walking. To explore the reflex control mechanism and gain a better understanding of its energy-efficient maintenance mechanism, we extend the reflex-based control system to enable controlled walking velocities based on target speeds. We developed a novel performance-weighted least squares (PWLS) method to design a parameter modulator that optimizes walking efficiency while maintaining target velocity for the reflex-based bipedal system. We have successfully generated walking gaits from 0.7 to 1.6 m/s in a two-dimensional musculoskeletal model based on an input target velocity in the simulation environment. Our detailed analysis of the parameter modulator in a reflex-based system revealed two key reflex circuits that have a significant impact on energy efficiency. Furthermore, this finding was confirmed to be not influenced by setting parameters, i.e., leg length, sensory time delay, and weight coefficients in the objective cost function. These findings provide a powerful tool for exploring the neural bases of locomotion control while shedding light on the intricate mechanisms underlying human walking and hold significant potential for practical engineering applications.

Author summary Previous reflex-based control systems face significant limitations in accurately regulating walking velocity owing to the vast number of control parameters involved. This hinders identifying essential reflex circuits that have a significant impact on energy-efficient walking across a wide range of speeds. Our research tackles this challenge by developing a reflex-based control framework that precisely regulates the velocities of the bipedal model through the performance-weighted least squares (PWLS) method that optimizes the control parameter values while considering energy efficiency. Through a detailed analysis, we identify two key reflex circuits essential for generating energy-efficient walking over a wide range of velocities. We hope that our research will inspire future investigations into reflex mechanisms and facilitate the development of advanced walking control systems for practical applications, such as gait-assisted exoskeletons and prosthetic legs, and robot control.

Citation: Koseki S, Hayashibe M, Owaki D (2024) Identifying essential factors for energy-efficient walking control across a wide range of velocities in reflex-based musculoskeletal systems. PLoS Comput Biol 20(1): e1011771. https://doi.org/10.1371/journal.pcbi.1011771 Editor: Jian Liu, JHU, UNITED STATES Received: April 12, 2023; Accepted: December 18, 2023; Published: January 19, 2024 Copyright: © 2024 Koseki et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All experimental data and programming code are publicly available on GitHub, at https://github.com/Shunsuke-KK/reflex_plos_revision. Funding: This work was supported by the JSPS KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas Hyper-Adaptability Project (JP20H05458 and JP22H04764, M.H.) and Scientific Research (A) (JP23H00481, D.O.). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

Introduction Walking is a fundamental mode of locomotion in our daily lives. The neurological and biochemical control mechanisms that support it comprise one of the most complex autonomous control systems in the human body [1–6]. Modeling and replicating the underlying walking mechanisms is expected to contribute to engineering applications including energy-efficient bipedal robots [7–11], gait-assisted exoskeletons [12–15], smart prosthetic legs [16, 17]. One of the most crucial aspects of human walking is the energy-efficient maintenance of our controlled velocity in the range of 1.0–1.6 m/s [18]. Central pattern generators in the central nervous system engage rhythmic neural circuits that generate basic leg movements [19–23]. Equally indispensable are reflex control mechanisms, which provide rapid adjustments to external stimuli from the sensory organs [19–21, 24], even for unexpected balance loss or unpredicted ground variations during walking, thereby maintaining velocity. Notably, humans modulate the reflex response depending on the task in locomotion [25, 26]. These findings indicate that reflex mechanisms have a crucial impact on achieving stable and energy-efficient walking. A physical simulation is a powerful tool for exploring the neural basis of walking, a complex phenomenon generated by the interaction of versatile mechanisms [27]. Previous studies have shown that reflex-based control systems can generate human-like walking in terms of muscle activities, joint angles, and torque patterns [28–37]. Notwithstanding the great advancements thereof, these systems have difficulty regulating velocity, owing to the large number of control parameters that must be properly tuned [38], e.g., Geyer [28] used 36 control parameters to generate a steady gait in a two-dimensional musculoskeletal model. Previous studies have attempted to regulate velocities within the reflex-based control frameworks [30, 32], but their methods were limited to transitions between predetermined velocities and did not provide precise velocity controls. Furthermore, the transitions between these predetermined velocities were accomplished through the utilization of distinct control parameters designated for these transitions. Given that walking speed assumes continuous values, achieving precise control over walking speed through such means would theoretically require infinite parameters for transitions. Because “transition” differs from “control” in terms of its adaptability, it is essential to extend reflex-based systems to affect precise velocity controls while improving energy efficiency to explore reflex control mechanisms in walking. Furthermore, the development of energy-efficient control over a wide range of velocities in the reflex-based system will lead to improving the performance of the controller for exoskeletons [39, 40] and prosthetic leg [17, 41] by adjusting their control parameters according to user walking velocity. The purpose of this study is to extend the reflex-based control system to enable controlled walking velocities based on target speeds and gain a better understanding of its modulation and energy-efficient maintenance mechanism across a wide range of walking velocities. To achieve this, we developed a novel performance-weighted least squares (PWLS) method to design a parameter modulator that coordinates a vast number of control parameters for an input target velocity while maintaining energy efficiency. In short, the reflex-based control system with the parameter modulator optimized via the PWLS successfully and energetic-efficiently maintains the desired velocity from 0.7 to 1.6 m/s in a two-dimensional musculoskeletal model. Subsequently, the detailed analysis of the parameter modulator in the reflex-based system identified two key reflex circuits affecting energy efficiency across a wide range of walking velocities. The contributions of this work include (i) extending a reflex-based control system to include velocity control, (ii) providing an adaptive polynomial regression method that uses performance indices to control performance, and (iii) identifying the key reflex circuits related to energy efficiency across a wide range of walking velocities.

Discussion This study aimed to extend the reflex-based control system including velocity control and identify key reflex circuits that play a significant role in energy-efficient walking. We demonstrated that a musculoskeletal model driven by reflex-based control can achieve controlled speeds based on the input target velocity. Subsequently, by utilizing the generated gaits of varying energy efficiencies, we identified two key reflex circuits having a significant impact on CoT values. We found that the proposed PWLS regression method optimizes the parameter modulator for the reflex-based controller to reproduce more energy-efficient walking across a wide range of velocities. We verified that the ∫CoT values were decreased by giving the bias to the high-performing data in calculating regression curves as shown in Fig 10. These results demonstrate that the PWLS fitting method is more adaptive in evaluating each data point as a weight for performance. However, we also found that the walking generated at an excessively large A value resulted in less stability, resulting in a fall at specific target velocities. This can be attributed to insufficiently fast leg swing. Putting the swing leg fast enough in front of the stance leg is essential to prevent falling down [52]. As illustrated in Fig 17, the stimulation applied to the HFL during the swing phase decreased with larger A. Consequently, this led to a slow hip flexion resulting in less stability. As shown in Fig 14, we identified that and significantly contributed to improve the energy efficiency of the gait generated through reflex-based control. Furthermore, this conclusion has been robustly validated across various settings, including a different body structure, different neural systems, and different cost functions, as shown in Fig 18. Our findings elucidated that the modulation of these reflex circuits resulted in a reduction of stimulation to the HFL during the swing phase, subsequently leading to reduced energy consumption by the HFL, GLU, and HAM (Fig 15). HFL expended less energy due to reduced stimulation. GLU and HAM were mainly activated during the stance phase (Fig 16), and these two muscles are used to maintain the torso balance in the stance phase [28]. Reduced HFL activity during the swing phase leads to less energy consumption at the contralateral GLU and HAM because the work to compensate for torso acceleration caused by HFL is decreased. Therefore, it can be concluded that minimizing the stimulation to the HFL during the swing phase while ensuring sufficient hip swing to prevent falls is essential to improve energy efficiency in reflex-based walking control. This not only reduces HFL activity but also the effort of the GLU and HAM, which compensates for torso acceleration caused by HFL. The generated walking has several characteristics that demonstrate its biomechanical validity in comparison to the human gait. (i) the R values of the hip and knee kinematics were close to 1 (R > 0.94 in all cases); (ii) The trajectory of the hip segment height from the ground exhibited a sinusoidal pattern (see Fig. S7 in S1 Appendix) [53]; (iii) we observed both heel strike and toe-off events [54] (see attached movie file); and (iv) a quadratic relationship between walking velocities and CoT [50] (Fig 10). These features are similar to those of human walking from a biomechanical perspective, proving that our simulations do not compromise the dynamically reasonable properties of human walking. Although we agree that reducing the number of control parameters would be essential and helpful for understanding the mechanism underlying walking, we used Wang’s extended model rather than Geyer’s original model to robustly generate walking without losing biomechanical explanatory ability. The results of this paper have some limitations concerning similarity to human walking. First, overshoots and undershoots were observed in the measured GRFs (Fig 8) and time evolution of the walking velocities (Fig 9). In the GRF profiles, we found two undershoots in GRF x and two overshoots in GRF z . The first occurred at heel strike, and the second occurred at toe-off of the contralateral leg. These overshoots and undershoots can be suppressed by using a lower-impedance ground. However, employing a lower-impedance ground resulted in foot penetration into the ground (refer to S1 Appendix). While the extreme GRF peaks are not biomechanically meaningful, we strongly believe that it is difficult to accurately model contact in a simulation environment [55, 56]. Moreover, we did not observe abrupt or significant changes in joint kinematics attributable to these extreme GRF peaks. Thus, given the difficulty of modeling contact in the simulation environment, the extreme peaks in GRF do not significantly affect kinematics. We conclude that the effects of the extreme peaks of GRF on walking in the musculoskeletal model do not negatively affect the biomechanical meaning of the findings obtained in this study. Second, the cross-correlation values of ankle dorsiflexion were close to 0 compared to hip and knee joints, as shown in Fig 8. At knee joints, the model straightened the knee earlier in the stance phase compared to humans. This strategy is known to generate more efficient solutions in gait optimization [32]. The previous study also showed low cross-correlation values at ankle joints when optimized to minimize energetic cost [32]. Third, we designed the cost function employed in this study to have minimal task terms to generate a human-like gait with more weight on the energy efficiency-related term. This approach optimized the control parameters to improve energy efficiency, consequently allowing the identification of factors that are significant to the energy efficiency of the generated gait over a wide range of walking velocities. More complex cost functions [29, 34, 36] may be required to generate human-like ankle joint kinematics. Finally, although this study successfully demonstrated the implementation of velocity control in a reflex-based control system, it took 20 s to transit walking speed by 1.0 m/s without falling. On the other hand, humans can adjust their walking speed by 1.0 m/s in less than 2 s [57]. Consequently, our control framework lacks the component to change walking speeds rapidly while ensuring stability. Extending the proposed framework is one of the future works to achieve velocity control as fast as humans. Human locomotion involves complex interactions between descending supraspinal commands, interconnected spinal circuits involving reflexes and CPGs, and the musculoskeletal system. This study primarily focuses on reflexes in the spinal cord. Therefore, we must add other control components for fast walking velocity control. Moreover, we constrained the bipedal model motion into the sagittal plane. Hence, the extension of the motion into three dimensions is another essential future work to apply the controller to engineering applications.

Conclusion Reflex mechanisms contribute significantly to the generation of stable and energy-efficient walking. However, a major limitation of generating gaits in musculoskeletal models through reflex-based control is the difficulty in precisely regulating velocity due to the large number of control parameters that need to be properly tuned. Extending reflex-based systems to affect velocity controls is essential to explore the reflex modulation mechanism and to understand its energy-efficient maintenance mechanism across a wide range of velocities. Furthermore, the development of energy-efficient control over a wide range of velocities in the reflex-based system will facilitate advanced engineering applications. Therefore, we developed a reflex-based control framework that enables the regulation of walking velocity over a wide range of velocities. Our parameter modulation method using PWLS that calculates the control parameters in response to a target velocity while optimizing efficiency successfully demonstrates generating walking gaits from 0.7 to 1.6 m/s. Furthermore, after a detailed analysis of the parameter modulator in a reflex-based system, we identified that the modulations of two reflex circuits, and , improve energy efficiency of the gait. The coordinated activity in the swing phase between the HFL and HAM reduced the stimulation applied to HFL during the swing phase, which not only caused the reduction of HFL activity but also alleviated the effort of GLU and HAM that compensates for the torso acceleration induced by the HFL. This research will inspire future investigations into reflex mechanisms and facilitate the development of advanced walking control systems for practical applications, such as gait-assisted exoskeletons and prosthetic legs, and robot control.

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[1] Url: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1011771

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