State-Space Characterization of Balance Capabilities in Biped Systems with Segmented Feet

Abstract

The human ability of keeping balance during various locomotion tasks is attributed to our capability of withstanding complex interactions with the environment and coordinating whole-body movements. Despite this, several stability analysis methods are limited by the use of overly simplified biped and foot structures and corresponding contact models. As a result, existing stability criteria tend to be overly restrictive and do not represent the full balance capabilities of complex biped systems. The proposed methodology allows for the characterization of the balance capabilities of general biped models (ranging from reduced-order to whole-body) with segmented feet. Limits of dynamic balance are evaluated by the Boundary of Balance (BoB) and the associated novel balance indicators, both formulated in the Center of Mass (COM) state space. Intermittent heel, flat, and toe contacts are enabled by a contact model that maps discrete contact modes into corresponding center of pressure constraints. For demonstration purposes, the BoB and balance indicators are evaluated for a whole-body biped model with segmented feet representative of the human-like standing posture in the sagittal plane. The BoB is numerically constructed as the set of maximum allowable COM perturbations that the biped can sustain along a prescribed direction. For each point of the BoB, a constrained trajectory optimization algorithm generates the biped’s whole-body trajectory as it recovers from extreme COM velocity perturbations in the anterior–posterior direction. Balance capabilities for the cases of flat and segmented feet are compared, demonstrating the functional role the foot model plays in the limits of postural balance. The state-space evaluation of the BoB and balance indicators allows for a direct comparison between the proposed balance benchmark and existing stability criteria based on reduced-order models [e.g., Linear Inverted Pendulum (LIP)] and their associated stability metrics [e.g., Margin of Stability (MOS)]. The proposed characterization of balance capabilities provides an important benchmarking framework for the stability of general biped/foot systems.

A Computational Framework Towards the Tele-Rehabilitation of Balance Control Skills

“…The anthropometric parameters and joint angle/torque limits of the reference subject are from the literature ( Winter 2005 ; Mummolo and Vicentini, 2020 ). The LIP and the 4-DOF models have a rigid foot link with no metatarsal joint, which is assumed to maintain a fixed contact with the ground at all times; the 5-DOF model includes a two-link foot, where a metatarsal joint and a multimodal foot-ground interaction model ( Mummolo et al, 2021 ) allow the foot to rotate about its heel and metatarsal. All models have a total foot length fl = 0.23 m.…”

“…The COM velocity extrema are calculated along the anterior (+ X ) and posterior (− X ) direction, by sampling the COM initial positions at a constant height = 1.12 m, corresponding to the subject’s COM Y -coordinate in the upright standing configuration. The BoB of the LIP model can be found analytically using the linear inequalities that limit the position of the extrapolated center of mass ( ) within the base of support [0, fl ] ( Mummolo et al, 2021 ), i.e., , where and are the COM position and velocity, respectively, and are the pendulum’s length and natural frequency.…”

“…Details of the numerical optimization algorithm and its solution via sequential quadratic programming can be found in previous work ( Mummolo et al, 2018b ). The construction method of the balance threshold has been demonstrated for the study of gait and posture stability of human, robot, and exoskeleton systems ( Mummolo et al, 2017 ; Mummolo et al, 2018a ; Mummolo et al, 2018b ; Mummolo and Vicentini, 2020 ; Mummolo et al, 2021 ).…”

“…Depending on how this distance is measured in the state space, these indicators can include position, velocity, or mixed margins. In this study, the extended Margin of Stability (eMOS) ( Mummolo et al, 2021 ) is used as a position margin that quantifies the distance from a given state to the BoB along the position coordinate of the state space ( Figure 1B ). The eMOS is equivalent to the Margin of Stability (MOS) for a linear inverted pendulum (LIP) model ( Hof et al, 2005 ; Mummolo et al, 2021 ), but it can be applied to any generic biped system.…”

“…In this study, the extended Margin of Stability (eMOS) ( Mummolo et al, 2021 ) is used as a position margin that quantifies the distance from a given state to the BoB along the position coordinate of the state space ( Figure 1B ). The eMOS is equivalent to the Margin of Stability (MOS) for a linear inverted pendulum (LIP) model ( Hof et al, 2005 ; Mummolo et al, 2021 ), but it can be applied to any generic biped system. This indicator, unlike the boundary margin, is specific to the motor task performed by the subject in a given stance, allowing for the continuous evaluation of the COM state of balance.…”