Research

Fundamental gait models are to legged systems what model organisms are to genetics or linear time-invariant systems are to control theory: they provide the theoretical foundation for systematic understanding and principled design. In humanoid robotics, for instance, these models form the bedrock of locomotion planning and control, directly linking a robot's stability, robustness, and dexterity to the properties of the underlying gait models.

Foundational Contributions. Working primarily with the spring-mass model, our early works made several foundational contributions. We discovered self-stable running dynamics and pioneered time-embedded deadbeat control strategies capable of rejecting large ground disturbances without prior terrain knowledge. Furthermore, we derived approximate solutions that enabled theoretical insights into the parametric dependencies of model behavior. Finally, we unified walking and running within a single framework—the bipedal spring-mass model. This work revealed walking and running to not represent distinct locomotion mechanics but belong to a range of compliant gaits governed by speed and system energy.

Current Directions. Our current research explores new dimensions of complexity within these models. We are investigating turning dynamics and control of legged systems for highly maneuverable steering, and we are exploring active spring-mass formulations that negotiate multi-terrain environments with biomimetic ground reaction forces. We aim to use these models as predictors, assessing human intent as well as fall risk during assistive robot control.

• Self-stable running dynamics: Seyfarth, Geyer, Guenther & Blickhan (2002), [DOI]
• Deadbeat control strategy: Seyfarth, Geyer & Herr (2003), [DOI], Wu & Geyer (2013) [DOI]
• Approximate analytical solutions: Geyer, Seyfarth & Blickhan (2005), [DOI]
• Bipedal robot demonstration of deadbeat control running: Martin, Wu & Geyer (2017), [DOI], Albert Wu's PhD thesis (2017) [DOI], William Martin's PhD thesis (2019) [DOI]
• Unifying walking and running: Geyer, Seyfarth & Blickhan (2006), [DOI]
• Bipedal robot control inspired by unified model: Hubicki et al. (2018) [DOI], William Martin's PhD thesis (2019) [DOI]
• Multi-terrain locomotion: Kumar & Geyer (2025), [DOI]

As a research community, we tend to start investigating the control of a complex motion system such as a legged system by reducing it to a point mass, and later on, a slightly more complex centroidal body. But once we exhaust understanding at this level, we lack good tools for deciding how control should expand hierarchically when incorporating more structure such as segmented limbs and their dynamics in legged systems. Although hierarchical control has a long history in control theory, the methods developed for finding control hierarchies focus on decision criteria—interaction measures, passivity, controllability and observability, time-scale separation—that typically do not take the objective of the original control problem into account, which can lead to poor closed-loop performance. Nor do these methods provide insights into how close the performance of resulting hierarchical controls is to the best possible, optimal control of the system (typically intractable to compute).

Core Mechanism. Over the past several years we developed policy decomposition, a novel framework for identifying computationally tractable, hierarchical control strategies that give up little in closed-loop performance. This work comprised establishing the theoretical foundation of policy decomposition including a priori estimates of control performance, identifying system state and input transformations that favor finding good decompositions, and overcoming the vast combinatorics of all possible hierarchies of a general complex system by adapting genetic algorithm search to policy decomposition.

Example. An example outcome that combines all three of these elements is shown in figure 2. Within 20 minutes of search over the realm of 120 million possible control hierarchies for quadcopter control (a system with 12 states and 4 inputs), the adapted genetic algorithm identifies a hierarchical control that outperforms heuristic control hierarchies identified by scientists as well as control identified with popular policy optimization algorithms. 

Application Domains. We have demonstrated policy decomposition on models of bipedal systems, manipulators, and quadcopters, achieving substantially lower trajectory costs than state-of-the-art reinforcement learning while providing explicit sub-optimality estimates. More generally, this new framework allows to systematically study the effects of changes in the controller structure on control performance.

• Policy decomposition framework: Khadke & Geyer (2021) [DOI]
• Sparsity-inducing representations: Khadke & Geyer (2022) [DOI]
• Automated control policy discovery: Khadke & Geyer (2024) [DOI]
• Ashwin Khadke's PhD Thesis (2024) [DOI]
• Policy Decomposition Code Repository: [here]

Foundation (2003-2010). Our entry into this research area established that neural control could be understood through the lens of mechanical principles rather than purely from anatomical connectivity. We pioneered encoding biomechanical principles of locomotion in neural control via spinal reflex circuits, which resulted in a neuromuscular gait model that predicts detailed activation patterns of major leg muscles observed in human walking.

Spinal Control Architecture (2010-2018). Building on this foundation, we expanded the spinal control architecture and demonstrated a 3-D neuromuscular model capable of diverse steady and transitional locomotion behaviors including walking and running, acceleration and deceleration, slope and stair negotiation, turning, and deliberate obstacle avoidance. In addition, we provided evidence for the model's validity by demonstrating that it responds to a range of electrical and mechanical gait disturbances in ways remarkably similar to humans, and that it can predict the effects of aging on human walking performance.

Neural Learning Mechanisms (2020-present). As neuromuscular models grow in complexity, over-parameterization makes it increasingly difficult to distinguish essential biological control pathways from arbitrary parameter choices. In response, we pivoted our approach from attempting to directly identify the structure of the spinal controller to understanding how it may be learned through a transfer of control from the brain to the spinal cord. This idea builds on the observation that locomotor skills like skiing require effort and attention at first but become "second nature" with practice. Brain imaging studies have corroborated that such learning involves a transfer of activity from resource-intensive attentional networks to more subconscious sensorimotor networks in the brain. Since experimental studies have repeatedly shown that spinal output can be modified over time by inputs from the brain and afferent projections, we speculated that the transfer extends to spinal cord networks.

In initial work, we demonstrated with a neuromuscular model that the spinal controller of the essential leg rebound motion may be automatically learned through hetero-synaptic modulation of spinal reflex gains. After learning, the controller contained individual spinal reflexes well known from physiological experiments but previously thought to serve separate purposes, suggesting a more holistic interpretation of how individual sensory projections function within spinal networks.

In recent work, we tested the model's key prediction that spinal reflex gains keep adapting, which should be especially visible when the human system experiences unnatural locomotion environments. Performing H-reflex measurements during split-belt treadmill locomotion, we could confirm such a dynamic adaptation of spinal reflex gains and provide evidence for their causal role in producing the gait asymmetries observed in split-belt locomotion. We currently seek to extend this understanding to clinical applications.

• Embedding spring-like leg behavior with spinal reflexes: Geyer et al. (2003) [DOI]
• Spinal reflex walking: Geyer & Herr (2010) [DOI]
• 3-D neuromuscular model: Song & Geyer (2015) [DOI]
• Evaluation of model vs human responses in disturbance experiments: Song & Geyer (2017) [DOI]
• Predicting effects of aging on human walking performance: Song & Geyer (2018) [DOI]
• Seungmoon Song's PhD Thesis (2017) [DOI]
• Learning spinal cord control through transfer from the brain: Sar & Geyer (2020) [DOI]
• Evidence of spinal control adaptation during split-belt locomotion: Refy et al. (2023) [DOI]
• Causal relation between spinal control adaptation and step length asymmetry in split-belt locomotion: Refy et al. (2025) [DOI]
• Omar Refy's PhD Thesis (2024) [DOI]

Human-in-the-Loop Challenge. Compared to autonomous legged robots, prostheses and exoskeletons present a distinct human-in-the-loop challenge. These devices connect to the human body and require control strategies that co-operate rather than conflict with human motor control of locomotion. Direct neural interfacing offers one solution to this problem; however, the associated surgeries remain inaccessible to the vast majority of users. The alternative is decentralized control strategies that integrate predictions about the human user's intent. Our research progresses along this alternative direction.

Virtual Neuromuscular Control (2010–2019). Human motor control of locomotion itself is decentralized, with the spinal cord controlling basic functions of the lower limbs. Our neuromuscular models capture this distributed control and demonstrate how spinal reflexes create stable and adaptive human-like locomotion (see Neuromuscular Control & Learning). We have translated this knowledge into decentralized control algorithms for robotic limbs. Early contributions pioneered the use of reflex-like control strategies in robotic ankle-foot prostheses that adapt to ground slope variations and stairs in a manner similar to the biological ankle joint. This approach was patented and later commercialized (eventually reaching the market as the Ottobock EmPower prosthesis). Subsequently, we generalized virtual neuromuscular control to powered knee-ankle prostheses, which enables adaptive, compliant limb behavior in stance and improves balance recovery through prosthesis foot placement in swing.

Online Control Tuning (2016–2018). As part of the research on virtual neuromuscular control, we addressed the individualization of lower-limb assistive robots to user behavior and preferences, exploring and validating methods for online optimization of these device controls characterized by high dimensional parameter spaces.

Human Intent Prediction and Environment Perception (2019–Present). Our current work focuses on integrating virtual neuromuscular control with environment perception and user intent prediction to enable seamless mobility with assistive lower-limb robots across challenging terrain. Our contributions include a comprehensive predictor for steady and transitional motions of the human leg in swing, prosthesis state estimation based on algorithms developed for pedestrian dead reckoning (inertial navigation), and the combination of these elements in real-time control of powered knee-ankle prostheses for seamless navigation on, over and off objects as intended by the human user.

• Reflex-like ankle-foot prosthesis control: Eilenberg et al (2010) [DOI]
• Virtual neuromuscular control: Schepelmann et al (2015) [DOI], Thatte & Geyer (2015) [DOI], Thatte et al (2019a) [DOI]
• Alexander Schepelmann's PhD Thesis (2016) [DOI], Nitish Thatte's PhD Thesis (2019) [DOI]
• Online Control Tuning: Thatte et al (2017) [DOI] and (2018) [DOI]
• Prosthesis state estimation: Thatte et al (2019a) [DOI], 
• Human intent prediction and environment perception: Thatte et al (2019b) [DOI], Gordon et al (2019) [DOI], Xing et al (2022) [DOI], Xing et al (2025) [DOI]
• Haosen Xing's PhD Thesis (2025) [DOI]