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1. From Bipedal Walking to Quadrupedal Locomotion: Full-Body Dynamics Decomposition for Rapid Gait Generation

   https://doi.org/10.1109/ICRA40945.2020.9196841  

   Ma, Wen-Loong; Ames, Aaron D. (January 2020, IEEE International Conference on Robotics and Automation (ICRA))

   null (Ed.)

   This paper systematically decomposes a quadrupedal robot into bipeds to rapidly generate walking gaits and then recomposes these gaits to obtain quadrupedal locomotion. We begin by decomposing the full-order, nonlinear and hybrid dynamics of a three-dimensional quadrupedal robot, including its continuous and discrete dynamics, into two bipedal systems that are subject to external forces. Using the hybrid zero dynamics (HZD) framework, gaits for these bipedal robots can be rapidly generated (on the order of seconds) along with corresponding controllers. The decomposition is achieved in such a way that the bipedal walking gaits and controllers can be composed to yield dynamic walking gaits for the original quadrupedal robot - the result is the rapid generation of dynamic quadruped gaits utilizing the full-order dynamics. This methodology is demonstrated through the rapid generation (3.96 seconds on average) of four stepping-in-place gaits and one diagonally symmetric ambling gait at 0.35 m/s on a quadrupedal robot - the Vision 60, with 36 state variables and 12 control inputs - both in simulation and through outdoor experiments. This suggested a new approach for fast quadrupedal trajectory planning using full-body dynamics, without the need for empirical model simplification, wherein methods from dynamic bipedal walking can be directly applied to quadrupeds. 

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2. Using Energy Shaping and Regulation for Limit Cycle Stabilization, Generation, and Transition in Simple Locomotive Systems

   https://doi.org/10.1115/1.4051589  

   Yeatman, Mark; Gregg, Robert D. (September 2021, Journal of Computational and Nonlinear Dynamics)

   Abstract This paper explores new ways to use energy shaping and regulation methods in walking systems to generate new passive-like gaits and dynamically transition between them. We recapitulate a control framework for Lagrangian hybrid systems, and show that regulating a state varying energy function is equivalent to applying energy shaping and regulating the system to a constant energy value. We then consider a simple one-dimensional hopping robot and show how energy shaping and regulation control can be used to generate and transition between nearly globally stable hopping limit cycles. The principles from this example are then applied on two canonical walking models, the spring loaded inverted pendulum (SLIP) and compass gait biped, to generate and transition between locomotive gaits. These examples show that piecewise jumps in control parameters can be used to achieve stable changes in desired gait characteristics dynamically/online. 

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3. Inferring Distributions of Parameterized Controllers for Efficient Sampling-Based Locomotion of Underactuated Robots

   Chavali, Raghu A; Kent, Nathan; Napoli, Michael E; Howard, Thomas M; Travers, Matthew (August 2019, Proceedings of the ... American Control Conference)

   Sampling-based motion planning algorithms provide a means to adapt the behaviors of autonomous robots to changing or unknown a priori environmental conditions. However, as the size of the space over which a sampling-based approach needs to search is increased (perhaps due to considering robots with many degree of freedom) the computational limits necessary for real-time operation are quickly exceeded. To address this issue, this paper presents a novel sampling-based approach to locomotion planning for highly-articulated robots wherein the parameters associated with a class of locomotive behaviors (e.g., inter-leg coordination, stride length, etc.) are inferred in real-time using a sample-efficient algorithm. More specifically, this work presents a data-based approach wherein offline-learned optimal behaviors, represented using central pattern generators (CPGs), are used to train a class of probabilistic graphical models (PGMs). The trained PGMs are then used to inform a sampling distribution of inferred walking gaits for legged hexapod robots. Simulated as well as hardware results are presented to demonstrate the successful application of the online inference algorithm. 

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4. Inferring Distributions of Parameterized Controllers for Efficient Sampling-Based Locomotion of Underactuated Robots

   Chavali, Raghu Aditya; Kent, Nathan; Napoli, Michael E.; Howard, Thomas M.; Travers, Matthew (August 2019, American Control Conference)

   Sampling-based motion planning algorithms provide a means to adapt the behaviors of autonomous robots to changing or unknown a priori environmental conditions. However, as the size of the space over which a sampling-based approach needs to search is increased (perhaps due to considering robots with many degree of freedom) the computational limits necessary for real-time operation are quickly exceeded. To address this issue, this paper presents a novel sampling-based approach to locomotion planning for highly-articulated robots wherein the parameters associated with a class of locomotive behaviors (e.g., inter-leg coordination, stride length, etc.) are inferred in real-time using a sample-efficient algorithm. More specifically, this work presents a data-based approach wherein offline-learned optimal behaviors, represented using central pattern generators (CPGs), are used to train a class of probabilistic graphical models (PGMs). The trained PGMs are then used to inform a sampling distribution of inferred walking gaits for legged hexapod robots. Simulated as well as hardware results are presented to demonstrate the successful application of the online inference algorithm. 

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5. Natural variability can increase human walking metabolic costs and its implications to simulation-based metabolic estimation

   https://doi.org/10.1098/rsif.2025.0602  

   Alwan, Aya; Srinivasan, Manoj (November 2025, Journal of The Royal Society Interface)

   Human walking contains variability due to small intrinsic perturbations arising from sensory or motor noise, or to promote motor learning. We hypothesize that such stride-to-stride variability may increase the metabolic cost of walking over and above a perfectly periodic motion, and that neglecting such variability in simulations may mis-estimate the metabolic cost. Here, we quantify the metabolic estimation errors accrued by neglecting the stride-to-stride variability using human data and a musculoskeletal model by comparing the cost of multiple strides of walking and the cost of a perfectly periodic stride with averaged kinematics and kinetics. We find that using an averaged stride underestimates the cost by approximately 2.5%, whereas using a random stride may mis-estimate the cost positively or negatively by up to 15%, ignoring the contribution of measurement errors to the observed stride-to-stride variability. As a further illustration of the cost increase in a simpler dynamical context, we use a feedback-controlled inverted pendulum walking model to show that increasing the sensory or motor noise increases the overall metabolic cost, as well as the variability of stride-to-stride metabolic costs, as seen with the musculoskeletal simulations. Our work establishes the importance of accounting for stride-to-stride variability when estimating metabolic costs from motion. 

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