A stochastic algorithm for automatic hand pose and motion estimation

Abstract

In this paper, a novel, robust, and simple method for automatically estimating the hand pose is proposed and validated. The method uses a multi-camera optoelectronic system and a model-based stochastic algorithm. The approach is marker-based and relies on an Unscented Kalman Filter. A hand kinematic model is introduced for constraining relative marker’s positions and improving the algorithm robustness with respect to outliers and possible occlusions. The algorithm outputs are 3D coordinate measures of markers and hand joint angle values. To validate the proposed algorithm, a comparison with ground truths for angular and 3D coordinate measures is carried out. The comparative analysis shows the advantages of using the model-based stochastic algorithm with respect to standard processing software of optoelectronic cameras in terms of implementation simplicity, time consumption, and user effort. The accuracy is remarkable, with a difference of maximum 0.035r a d and 4m m with respect to angular and 3D Cartesian coordinates ground truths, respectively.

Appendix

The Denavit-Hartenberg (DH) parameters for the index finger and for the thumb are shown in Tables 1 and 2, respectively. The other long fingers have the same DH parameters of the index. DH parameters are evaluated in such a way as to obtain a generic algorithm valid for different hand sizes. Therefore, the algorithm envisages an initial calibration phase, where the 3D Cartesian coordinates of the markers center are detected manually in the first image acquired by the camera and the link lengths are measured, by means of the 3-dimensional information provided by the vision system.

Table 1 Denavit-Hartenberg parameters of the index finger

Table 2 Denavit-Hartenberg parameters of the thumb

In the Tables 1 and 2, L ^index and L ^thumb represent the link lengths of the index finger and of the thumb, respectively.

Once the DH parameters have been computed, the rotation matrices can be extracted. Given the symbolic form of a generic rotation matrix

$$ \textbf{R} = \left[ \begin{array}{ccc} r_{11} & r_{12} & r_{13}\\ r_{21} & r_{22} & r_{23}\\ r_{31} & r_{32} & r_{33} \end{array} \right] , $$

(19)

the corresponding Euler angles in configuration ZYX, under the assumption that r ₁₃≠0 and r ₂₃≠0, are

$$\begin{array}{@{}rcl@{}} &&\phi = atan2(r_{23}, r_{13})\\ &&\theta = atan2(\sqrt{r_{13}^{2} + r_{23}^{2}}, r_{33})\\ &&\psi = atan2(r_{32}, -r_{31}) \end{array} $$

(20)

where a t a n2(x,y) is the arctangent of two arguments, the choice of the positive sign for the term $\sqrt {r_{13}^{2} + r_{23}^{2}}$ limits the range of the feasible values of 𝜃 to (0,π). If 𝜃 is chosen in the range (−π, 0), Eq. 20 becames

$$\begin{array}{@{}rcl@{}} &&\phi = atan2(-r_{23}, -r_{13})\\ &&\theta = atan2(-\sqrt{r_{13}^{2} + r_{23}^{2}}, r_{33})\\ &&\psi = atan2(-r_{32}, r_{31}) \end{array} $$

(21)