Variant and invariant patterns embedded in human locomotion through whole body kinematic coordination

Abstract

Step length, cadence and joint flexion all increase in response to increases in gradient and walking speed. However, the tuning strategy leading to these changes has not been elucidated. One characteristic of joint variation that occurs during walking is the close relationship among the joints. This property reduces the number of degrees of freedom and seems to be a key issue in discussing the tuning strategy. This correlation has been analyzed for the lower limbs, but the relation between the trunk and lower body is generally ignored. Two questions about posture during walking are discussed in this paper: (1) whether there is a low-dimensional restriction that determines walking posture, which depends not just on the lower limbs but on the whole body, including the trunk and (2) whether some simple rules appear in different walking conditions. To investigate the correlation, singular value decomposition was applied to a measured walking pattern. This showed that the whole movement can be described by a closed loop on a two-dimensional plane in joint space. Furthermore, by investigating the effect of the walking condition on the decomposed patterns, the position and the tilt of the constraint plane was found to change significantly, while the loop pattern on the constraint plane was shown to be robust. This result indicates that humans select only certain kinematic characteristics for adapting to various walking conditions.

Appendix: Description of walking motion on joint spaces

In order to describe the motion of DS and SS in the same space, the time series of the motion R(t) and the constraint planes $\left( {z_{1}^{\text{DS}} ,\,z_{2}^{\text{DS}} } \right),\,\left( {z_{1}^{\text{SS}} ,\,z_{2}^{\text{SS}} } \right)$ are embedded in the $(\mathop {\tilde{z}}\nolimits_{1} ,\mathop {\tilde{z}}\nolimits_{2} ,\mathop {\tilde{z}}\nolimits_{3} )$ space built by the movement when DS and SS phases are considered together. The posture at time t:R(t) in this space is

$$ \widetilde{R}_{i} (t) = (R(t) - \widetilde{R}_{0} ) \cdot \mathop {\tilde{z}}\nolimits_{i} ,\,(i = 1 \cdots 3) $$

(5)

where $\widetilde{R}_{0}$ is the mean posture over both DS and SS phases.

The spaces $\mathop {\tilde{z}}\nolimits^{\text{DS}}$ and $\mathop {\tilde{z}}\nolimits^{\text{SS}}$ are embedded with origins $\mathop {\tilde{z}}\nolimits^{{0{\text{DS}}}}$and $\mathop {\tilde{z}}\nolimits^{{0{\text{SS}}}}$ determined from the mean postures $R_{0}^{\text{DS}}$and $R_{0}^{\text{SS}}$ by

$$ \left\{ \begin{aligned} \mathop {\tilde{z}}\nolimits_{i}^{{0{\text{DS}}}} & = \left( {R_{0}^{\text{DS}} - \widetilde{R}_{0} } \right) \cdot \mathop {\tilde{z}}\nolimits_{i} \\ \mathop {\tilde{z}}\nolimits_{i}^{{0{\text{SS}}}} & = \left( {R_{0}^{\text{SS}} - \widetilde{R}_{0} } \right) \cdot \mathop {\tilde{z}}\nolimits_{i} \\ \end{aligned} \right.,\,\left( {i = 1 \cdots 3} \right) . $$

(6)

The axes of the intersegmental coordination $\left( {\mathop {\tilde{z}}\nolimits_{i}^{jDS} ,\,\mathop {\tilde{z}}\nolimits_{i}^{jSS} } \right)$ are determined from $z_{j}^{\text{DS}} ,\,z_{j}^{\text{SS}} \,\left( {j = 1, 2} \right)$ when DS and SS are considered separately:

$$ \left\{ \begin{aligned} \mathop {\tilde{z}}\nolimits_{i}^{jDS} & = z_{j}^{DS} \cdot \mathop {\tilde{z}}\nolimits_{i} \\ \mathop {\tilde{z}}\nolimits_{i}^{jSS} & = z_{j}^{SS} \cdot \mathop {\tilde{z}}\nolimits_{i} \\ \end{aligned} \right.,\left( {i = 1 \cdots 3,\;j = 1 \cdots 2} \right) . $$

(7)