Three-dimensional kinematics and dynamics of the foot during walking: a model of central control mechanisms

Abstract

The foot is a critical interface between the body and supporting surface during walking, but there is no coherent framework on which to model the dynamics of the stance and swing phases. To establish this framework, we studied the rotational and translational dynamics of foot movement in three dimensions with a motion detection system (OPTOTRAK), while subjects walked on a treadmill. Positions, velocities, and durations were normalized to leg-length and gravity. Foot position and rotation at toe-off were closely related to walking velocity. Foot pitch at toe clearance increased with walking velocity, but the medial–lateral and vertical toe positions were unaltered. Phase–plane trajectories along the fore-aft direction, i.e., plots of toe velocity versus position, were circular during the swing phases, with radii proportional to walking velocity. Peak forward, lateral, and upward velocities were linearly related to corresponding excursions, forming main sequences. A second order model predicted the changes in toe position and velocity, and the approximately hyperbolic decrements in duration as a function of walking velocity. The model indicates that the foot is controlled in an overdamped manner during the stance phase and as a feedback-controlled undamped pendulum during the swing. The data and model suggest that the state of the foot at toe-off, set by walking velocity during the stance phase, determines the dynamics of the swing phase. Thus, in addition to determining locomotion kinematics, walking velocity plays a critical role in determining the phase–plane trajectories and main sequence relationships of foot movements during the swing phases.

Appendix

Measurement coordinate system

The lower left LED was the origin (Fig. 1a, LED 4). The line connecting LEDs 4 and 3 was the Z _S-axis (Fig. 1a). It was aligned with the spatial vertical and directed upward. The line connecting LEDs 4 and 1 was the X _S direction. It was aligned with the spatial horizontal in the direction of walking and was orthogonal to the Z _S-axis. The line normal to the LED plate and positive to the left side of the walking direction was taken as the Y _S-axis. These vectors formed a right-handed spatial coordinate frame (Fig. 1a).

Rigid bodies for the foot and shank consisted of four markers embedded 48 mm apart on a plastic frame (Fig. 1b). Five additional markers were placed on the right leg to form shank and foot coordinate frames (Fig. 1b). The foot coordinate markers were placed at the lateral malleolus, the lateral calcaneus (heel) and the fifth metatarsal (toe) (Fig. 1b). The foot rigid body, which was fixed on the shoe, was placed at the proximal part of the foot, approximately on the cuboid bone and lateral cuneiform. The positive Z _f-axis of the foot coordinate frame was upward and spatially vertical when standing erect. The X _f-axis was a cross product of the Z _f-axis and a vector normal to a plane that contained the lateral malleolus, the lateral calcaneus and the fifth metatarsal (positive forward). The Y _f-axis was a cross product of the Z _f- and X _f-axes (positive left).

Shank coordinate markers were placed at the medial and lateral epicondyles of the knee and lateral malleolus of the ankle (Fig. 1b). The shank rigid body was located in the middle of the knee and the ankle on the front-lateral side. The positive Z _sh-axis of the shank coordinate frame was upward and spatially vertical when standing erect. The Y _sh-axis was a cross product of the Z _sh-axis and a vector normal to a plane that contained the medial and lateral epicondyles and the lateral malleolus (positive left). The X _sh-axis was a cross product of the Z _sh- and Y _sh-axes (positive forward).

All rigid bodies were computed dynamically using Toolbench (Northern Digital, Inc) as part of the calibration procedure. Calibration was done by asking the subject to stand erect in a natural hip-wide, fully-extended knee position on a platform (Fig. 1b) and not to move one body part relative to another while holding onto a bar connected to the rotator. The platform was then rotated slowly by hand over a 30 s time period. This calibration procedure allowed the OPTOTRAK sensors to sample all LED coordinates several times so that the coordinate frames and rigid bodies could be computed accurately. After the data were accessed, a program built the coordinate frames and rigid bodies. This procedure took from 20 s to 1 min on a dual processor Opteron (AMD)-based computer.

A yaw angle of 0° of the foot indicated that X _f-axis, which is the projection of the line between the lateral calcaneus and the fifth metatarsal (lateral border of the foot) onto the X–Y plane, was in parallel with the spatial X _S-axis. During calibration, the averaged yaw angle between the X _h- and X _f-axes across subjects was −9°. This indicates that the coordinate frame of the foot was rotated laterally relative to the head during calibration, because the X _f-axis was directed laterally relative to the line between the calcaneus and big toe. As the subjects were instructed to fixate a visual target placed in the middle of the trajectory, the X-axis of the head (X _h) during locomotion was almost equal to the X _S-axis, the walking direction. Thus, the yaw angle of the foot relative to space was ≈−9° in the calibrated position, which we denote as primary yaw positions of the foot.

Computation of orientation and translation of body part relative to space

Data were processed by programs written in MATLAB (Mathworks, Inc., Natick, MA, USA). The rotation and translation of the foot and shank were obtained as follows: At each orientation, a plane was fitted to the LED coordinates and a unit normal vector, ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{1} ,$ was determined (Fig. 1c). A vector was then determined as the average vector connecting the midpoint between LEDs 1 and 2 and the midpoint of LEDs 3 and 4 $({\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{{\mathbf{d}}} )$ (Fig. 1c, dashed vector). The vectors, ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{1}$ and ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{{\mathbf{d}}} ,$ were then averaged over 11 samples and a normalized cross product between the averaged vector, ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{{\mathbf{d}}} ,$ and the averaged normal-to-plane vector, ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{1} ,$ was computed as a unit vector, ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{2} .$ The vector ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{2}$ was orthogonal to the averaged vector ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{1}$ and lay in the average plane that was fitted to the LED coordinates. A third vector, ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{3} ,$ was determined as the cross product of the averaged ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{1}$ and ${\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{n}}}_{2} ,$ which formed the coordinate triple for the rigid body. A matrix whose columns were the three coordinate vectors was the rotation matrix for the rigid body relative to space over the observation window. The rotation of the rigid body in space was obtained by multiplying the rotation matrix by the inverse of the rotation matrix obtained during calibration. The translation of the rigid body was obtained by subtracting the coordinates of the LEDs after these had been corrected for their rotation and averaging over the four LEDs. Thus, the relationship of the rigid body on the any body part was determined relative to the coordinate frame of that body part, using the markers on the anatomical landmarks. The rotations and translations of the rigid body could then be converted to rotations and translations of the coordinate frame and in turn related to anatomical markers. The toe movement was separately obtained from an LED at the fifth metatarsal.

The rotation and translation of the head were determined from four visible markers on the headband. Since head direction is the most reliable measure of heading during straight walking (Imai et al. 2001), it was used to compute the medial–lateral position of the subject on the treadmill and to obtain the yaw orientation of the foot. Angular rotations of the foot about the ankle were calculated from the rotation matrices of the shank and foot. Although somewhat different from anatomical descriptions of the ankle joint, this relative rotation was defined as the ‘joint angle’ for simplicity.

Definition of the stance and swing phases

Components of the stride cycle (stance and swing phases) were determined from Heel contact (HC) and Toe-Off (TO) which were the points where the velocity of the X component of the heel were equal to zero. Our method for determining HC and TO from the heel movement is approximately equivalent to the determinations of HC and TO in terms of force exerted on a force plate during locomotion (Borghese et al. 1996; Grasso et al. 1998; Ivanenko et al. 2002, 2003). In these studies, a virtual line connecting the greater trochanter (GT) and the lateral malleolus (LM) projected into the sagittal plane is defined as the limb axis. The time of maximum (forward) and minimum (backward) elevation of the limb axis is taken as the heel contact and toe-off respectively. Their estimated times of HC and TO were within 2% of the gait cycle determined by the data from the force plate (2 N/cm² as threshold) (Borghese et al. 1996; Grasso et al. 1998; Ivanenko et al. 2002, 2003). Since the elevation angle was given by Tan⁻¹[(LMx-GTx)/(LMz-GTz)], maximum and minimum values of this parameter occur when LM reaches its forward and backward limit and when the X velocity of the ankle becomes zero. It is close to the time when heel X velocity becomes 0, which is how HC and TO were determined in the present study. Waveforms representing forward movement obtained during treadmill walking at 1.5 m/s (0.52 Froude) had consistent linear backward stance movements and sigmoidal swing phases (Fig. 1d). Waveforms at other velocities of walking had similar characteristics.