Learning inverse dynamics for human locomotion analysis

Abstract

In this work, learning-based inverse dynamics algorithms are proposed for the analysis of human motion. Immeasurable joint torques and exterior contact forces are directly estimated from motions by machine learning techniques including deep neural networks, random forests and Ridge regression. A multistage subclass approach is introduced. The method recovers occluded motion data and generates meaningful features, as well as gait phase labels to restrict and facilitate the regression of forces and moments. In contrast to the state-of-the-art inverse dynamics optimization, the learning-based methods are independent of ground reaction force measurements and the global position and orientation of the human body. These properties make the application to reconstructed poses from videos or inertial measurements possible, creating fast and simple access to the underlying dynamics of recorded human motions. The performance of the proposed methods is evaluated on a self-recorded data set including walking and running motions and on a publicly available gait data set by Fukuchi et al. (PeerJ 6:e4640, 2018). Furthermore, the applicability to reconstructed gait sequences taken from the well-known CMU database (Human motion capture database, 2014. http://mocap.cs.cmu.edu/) is investigated. Finally, the method is tested as a tool to detect abnormal torque distributions in gait, based on a reconstructed 3D motion of a limping subject.

Appendix 1: Implementation details concerning [20]

We compare the performance of our learning-based inverse dynamics methods to a data-driven maximum a posteriori approach by [20]. For this purpose, the referenced method is implemented with a few modifications to allow a fair comparison between the two methods. These modifications were made to achieve a better operation using our specific data and physical model.

To find z(t) at each frame, the following weighted sum of energy terms is minimized:

$$\begin{aligned} E({\varvec{z}}(t)) = \lambda _1E_{\mathrm {physical}} + \lambda _2E_{\mathrm {prior}} + \lambda _3E_{\mathrm {data}} + \lambda _4E_{\mathrm {smooth}}, \end{aligned}$$

(18)

with the weights $(\lambda _1,\lambda _2,\lambda _3,\lambda _4) = (2,2,100,1)$. In consistence with our physical model, the friction term $E_{\mathrm {friction}}$ is left out, assuming a negligible effect for the considered movements. The state and control parameters are

$$\begin{aligned} {\varvec{z}}(t) = ({\varvec{q}}(t), \dot{{\varvec{q}}}(t), {\varvec{F}}_c(t), \varvec{\tau }(t)), \end{aligned}$$

(19)

where ${\varvec{F}}_c$ consists of the GRF and the GRM. In contrast to [20], the center of pressure and the torsional torque are replaced by the resulting GRM.

Similar to [20], principle component analysis is used to linearize the local parameter space at each frame. The local environment is built of the 200 next neighbors of ${\varvec{z}}(t)$. We only optimize the scores ${\varvec{s}}$ of the first n principle components stacked in the matrix ${\varvec{K}}$. These n components constitute 95 % of the overall variability of the local data. This way the number of optimization variables is drastically reduced. The optimization problem becomes

$$\begin{aligned} \underset{{\varvec{s}}}{\min }\{E(\varvec{\mu }+ {\varvec{K}}{\varvec{s}}))\}, \end{aligned}$$

(20)

with the mean $\varvec{\mu }$ of the neighbouring parameter vectors.

As indicated before, the state vector ${\varvec{z}}$ is adapted to fit the applied physical model, which has an immediate effect on the physical term $E_{\mathrm {phys}}$. This term describes the deviation of the kinematic state $({\varvec{q}}(t), \dot{{\varvec{q}}}(t), \ddot{{\varvec{q}}}(t))$ given by ${\varvec{z}}(t)$ from the kinematics arising from the acting forces and torques via the EOM. Based on Eq. (4), the energy term is defined as

$$\begin{aligned} E_{\mathrm {physical}} = \Vert \varvec{{\mathcal {M}}}\ddot{{\varvec{q}}} - \varvec{{\mathcal {F}}}({\varvec{q}}, \dot{{\varvec{q}}}, {\varvec{F}}_c, \varvec{\tau })\Vert ^2. \end{aligned}$$

(21)

The changed physical model further necessitates a slight modification of the smoothness term:

$$\begin{aligned} E_{\mathrm {smooth}} = \Vert {\varvec{F}}_c(t-1) - 2{\varvec{F}}_c(t) + {\varvec{F}}_c(t+1)\Vert ^2. \end{aligned}$$

(22)

The remaining energy terms $E_{\mathrm {prior}}$ and $E_{\mathrm {data}}$ can be employed without adaptation.