Derivation and Application of a Conserved Orbital Energy for the Inverted Pendulum Bipedal Walking Model

We present an analysis of a point mass, point foot, planar inverted pendulum model for bipedal walking. Using this model, we derive expressions for a conserved quantity, the “Orbital Energy”, given a smooth Center of Mass trajectory. Given a closed form Center of Mass Trajectory, the equation for the Orbital Energy is a closed form expression except for an integral term, which we show to be the first moment of area under the Center of Mass path. Hence, given a Center of Mass trajectory, it is straightforward and computationally simple to compute phase portraits for the system. In fact, for many classes of trajectories, such as those in which height is a polynomial function of Center of Mass horizontal displacement, the Orbital Energy can be solved in closed form. Given expressions for the Orbital Energy, we can compute where the foot should be placed or how the Center of Mass trajectory should be modified in order to achieve a desired velocity on the next step. We demonstrate our results using a planar biped simulation with light legs and point mass body. We parameterize the Center of Mass trajectory with a fifth order polynomial function. We demonstrate how the parameters of this polynomial and step length can be changed in order to achieve a desired next step velocity

Derivation and Application of a Conserved Orbital Energy for the Inverted Pendulum Bipedal Walking Model

Abstract—We present an analysis of a point mass, point foot, planar inverted pendulum model for bipedal walking. Using this model, we derive expressions for a conserved quantity, the “Orbital Energy”, given a smooth Center of Mass trajectory. Given a closed form Center of Mass Trajectory, the equation for the Orbital Energy is a closed form expression except for an integral term, which we show to be the first moment of area under the Center of Mass path. Hence, given a Center of Mass trajectory, it is straightforward and computationally simple to compute phase portraits for the system. In fact, for many classes of trajectories, such as those in which height is a polynomial function of Center of Mass horizontal displacement, the Orbital Energy can be solved in closed form. Given expressions for the Orbital Energy, we can compute where the foot should be placed or how the Center of Mass trajectory should be modified in order to achieve a desired velocity on the next step. We demonstrate our results using a planar biped simulation with light legs and point mass body. We parameterize the Center of Mass trajectory with a fifth order polynomial function. We demonstrate how the parameters of this polynomial and step length can be changed in order to achieve a desired next step velocity. I

Dynamic Walking: Toward Agile and Efficient Bipedal Robots

Dynamic walking on bipedal robots has evolved from an idea in science fiction to a practical reality. This is due to continued progress in three key areas: a mathematical understanding of locomotion, the computational ability to encode this mathematics through optimization, and the hardware capable of realizing this understanding in practice. In this context, this review article outlines the end-to-end process of methods which have proven effective in the literature for achieving dynamic walking on bipedal robots. We begin by introducing mathematical models of locomotion, from reduced order models that capture essential walking behaviors to hybrid dynamical systems that encode the full order continuous dynamics along with discrete footstrike dynamics. These models form the basis for gait generation via (nonlinear) optimization problems. Finally, models and their generated gaits merge in the context of real-time control, wherein walking behaviors are translated to hardware. The concepts presented are illustrated throughout in simulation, and experimental instantiation on multiple walking platforms are highlighted to demonstrate the ability to realize dynamic walking on bipedal robots that is agile and efficient

Mechanics and Control of Human Balance

A predictive, forward-dynamic model and computer simulation of human gait has important medical and research applications. Most human simulation work has focused on inverse dynamics studies to quantify bone on bone forces and muscle loads. Inverse dynamics is not predictive - it works backwards from experimentally measured motions in an effort to find the forces that caused the motion. In contrast, forward dynamics determines how a mechanism will move without the need for experimentation. Most of the forward dynamic gait simulations reported consider only one step, foot contact is not modeled, and balance controllers are not used. This thesis addresses a few of the shortcomings of current human gait simulations by contributing an experimentally validated foot contact model, a model-based stance controller, and an experimentally validated model of the relationship between foot placement location and balance. The goal of a predictive human gait simulation is to determine how a human would walk under a condition of interest, such as walking across a slippery floor, using a new lower limb prosthesis, or with reduced leg strength. To achieve this goal, often many different gaits are simulated and the one that is the most human-like is chosen as the prediction for how a person would move. Thus it is necessary to quantify how `human-like' a candidate gait is. Human walking is very efficient, and so, the metabolic efficiency of the candidate gait is most often used to measure the performance of a candidate gait. Muscles consume metabolic energy as a function of the tension they develop and the rate at which they are contracting. Muscle tension is developed, and contractions are made in an effort generate torques about joints in order to make them move. To predict human gait, it is necessary for the simulation to be able to walk in such a way that the simulated leg joints use similar joint torques and kinematics as a human leg does, all while balancing the body. The joint torques that the legs must develop to propel the body forward, and balance it, are heavily influenced by the ground reaction forces developed between the simulated foot and the ground. A predictive gait simulation must be able to control the model so that it can walk, and remain balanced while generating ground reaction force profiles that are similar to experimentally observed human ground reaction force profiles. Ground reaction forces are shaped by the way the foot interacts with the ground, making it very important to model the human foot accurately. Most continuous foot contact models present in the literature have been experimentally validated using pendulum impact methods that have since been shown to produce inaccurate results. The planar foot contact model developed as part of this research was validated in-vivo using conventional force plates and optical tracking markers. The experimental data was also highly useful for developing a computationally efficient foot model by identifying the dominant contact properties of a real foot (during walking), without the complexity of modelling the 26 bones, 33 joints, over 90 ligaments, and the network of muscles that are in a real foot. Both ground reaction forces and the balance of the model are heavily influenced by the way the stance limb is controlled. Anthropomorphic multibody models typically have a fragile sense of balance, and ground reaction force profiles that do not look similar to experimentally measured human ground reaction force profiles. In contrast, the simple point-mass spring-loaded-inverted-pendulum (SLIP) can be made to walk or run in a balanced manner with center-of-mass (COM) kinematics and ground reaction force profiles that could be mistaken for the equivalent human data. A stance limb controller is proposed that uses a planar SLIP to compute a reference trajectory for a planar anthropomorphic multibody gait model. The torso of the anthropomorphic model is made to track the computed trajectory of the SLIP using a control system. The aim of this partitioned approach to gait simulation is to endow the anthropomorphic model with the human-like gait of the simpler SLIP model. Although the SLIP model-based stance-controller allows an anthropomorphic gait model to walk in more human-like manner, it also inherits the short comings of the SLIP model. The SLIP can walk or run like a human, but only at a fixed velocity. It cannot initiate or terminate gait. Fall preventing movements, such as gait termination and compensatory stepping, are of particular relevance to kinesiologists and health care professionals. Kinesiologists have known for nearly a decade that humans restore their balance primarily by systematically altering their foot placement location. This thesis presents a human experimental validation of a planar foot placement algorithm that was originally designed to restore the balance of bipedal robots. A three-dimensional (3D) theoretical extension to the planar foot placement algorithm is also presented along with preliminary human experimental results. These models of foot placement can be used in the future to improve the capabilities of gait simulations by giving simple models human-like compensatory stepping abilities. The theoretical extension also provides some insight into how instability and balance performance can be quantified. The instability and balance performance measures have important applications for diagnosing and rehabilitating balance problems. Despite all of the progress that has been made, there is still much work to be done. Work needs to be continued to find methods that allow the anthropomorphic model to emulate the SLIP model more faithfully. Experimental work needs to be completed to realize the potential diagnostic and rehabilitation applications of the foot placement models. With continued effort, a predictive, balanced, multi-step gait simulation can be developed that will give researchers the time-saving capability of computerized hypothesis testing, and medical professionals improved diagnostic and rehabilitation methods