Simultaneous stabilizing feedback control of linear and angular momentum in human walking
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Stabilizing bipedal gait is mechanically challenging. Previous studies have used the assumption that the linear momentum of the body center of mass (CoM) is controlled to analyze how gait is stabilized. In these studies, the position of the center of pressure (CoP) or of the foot relative to the COM is often used as an indicator of corrections of the CoM state. Mechanically, neither of these variables is directly related to changes in linear momentum, whereas they do directly affect whole-body angular momentum (WBAM), which has also been suggested to be a controlled variable. We show that, in human walking, linear and angular momentum follow quasi-periodic functions with similar periodicity and phase. Combining the equations of linear and rotational motion for a system of linked rigid segments shows that, in this case, the horizontal distance between CoP and CoM is a good predictor of horizontal forces in the corresponding direction. This suggests that linear and angular momentum are simultaneously controlled to follow similar quasi-periodic functions and may explain the success of preceding studies that correlated CoM states to CoP or foot locations. We developed feedback models to predict the ground reaction force and its moments along the sagittal and transverse axes from the preceding CoM state and WBAM respectively. These models were fitted to experimental data of participants walking at normal and slow speeds. The consistent, good fit of both models supports that linear and angular momentum are controlled simultaneously in human walking.
Nomenclature
All kinematic variables are expressed with respect to an inertial reference frame affixed to the world. All moments of inertia are with respect to the center of mass of the respective segment/body. Dot notations denote derivatives with respect to time.
a CoM’ = linear acceleration of the projection of body center of mass on the support surface.
CoM = whole-body center of mass.
CoM’ = projection of the whole-body center of mass on the support surface.
CoP = center of pressure, or point of application of the ground reaction force.
F gr = ground reaction force vector.
g = gravitational constant.
H CoM = angular momentum of the whole body around its center of mass.
I i = inertia tensor of segment i. m = body mass.
m i = mass of segment i .
p CoM = linear momentum of the whole body.
r CoM = position vector of the body center of mass.
r CoM’ = position vector of the projection of the body center of mass on the support surface.
r CoP = position vector of the point of application of the ground reaction force or center of pressure.
r i = position vector of the center of mass of segment i relative to the body center of mass.
v CoM = velocity of the body center of mass.
v i = velocity of segment i relative to the body center of mass.
ω i = angular velocity of segment i .
