Bidirectional invariant representation of rigid body motions and its application to gesture recognition and reproduction

Abstract

In this paper we propose a new bidirectional invariant motion descriptor of a rigid body. The proposed invariant representation is not affected by rotations, translations, time, linear and angular scaling. Invariant properties of the proposed representation enable to recognize gestures in realistic scenarios with unexpected variations (e.g., changes in user’s initial pose, execution time or an observation point), while Cartesian trajectories are sensitive to these changes. The proposed invariant representation also allows reconstruction of the original motion trajectory, which is useful for human-robot interaction applications where a robot recognizes human actions and executes robot’s proper behaviors using same descriptors. By removing the dependency on absolute pose and scaling factors of the Cartesian trajectories the proposed descriptor achieves flexibility to generate different motion instances from the same invariant representation. In order to illustrate the effectiveness of our proposed descriptor in motion recognition and generation, it is tested on three datasets and experiments on a NAO humanoid robot and a KUKA LWR IV$+$ manipulator and compared with other existing invariant representations.

Appendices

Appendix A: Rigid Body Motion Representation

To represent rigid body motions it is convenient to attach an orthogonal frame to the rigid body (body frame) and to describe the pose (position and orientation) of the body frame wrt a fixed frame (world frame). In each time instant the position of the rigid body is represented by the vector $\mathbf {p}$ connecting the origin of the body frame with the origin of the world frame. The axes of the body frame can be projected along the axes of the world frame by the means of the direction cosines. Hence, the orientation of the rigid body is described by collecting the direction cosines into a $3 \times 3$ rotation matrix $\mathbf {R}$. It is possible to show that a minimal representation of the orientation consists of 3 values (Siciliano et al. 2009). In this work, we use the rotation vector to represent the orientation.

The rotation vector $\mathbf {r} = \theta \hat{\mathbf{r}}$ is computed from $\mathbf {R}$ as:

$$\begin{aligned}&\theta = \text {arccos} \left( \frac{trace\left( \mathbf {R}\right) -1}{2}\right) ,\nonumber \\&\hat{\mathbf{r}} = \frac{1}{2\sin {\theta }} \begin{bmatrix} \mathbf{{R}}\left( 3,2\right) - \mathbf{{R}}\left( 2,3\right) \\ \mathbf{{R}}\left( 1,3\right) - \mathbf{{R}}\left( 3,1\right) \\ \mathbf{{R}}\left( 2,1\right) - \mathbf{{R}}\left( 1,2\right) \\\end{bmatrix} \end{aligned}$$

The rotation matrix $\mathbf {R}$ is computed from $\mathbf {r}$ as:

$$\begin{aligned} \mathbf {R} = \exp (\mathbf {r}) = \mathbf {I} + \frac{\mathbf {S}(\mathbf {r})}{\theta } \sin (\theta ) + \frac{\mathbf {S}^{2}(\mathbf {r})}{\theta ^{2}}(1 - \cos (\theta )) ~, \end{aligned}$$

where the skew-symmetric matrix $\mathbf {S}(\mathbf {r})$ is given by:

$$\begin{aligned} \mathbf {S}(\mathbf {r}) = \begin{bmatrix} 0&-r_z&r_y \\ r_z&0&-r_x \\ -r_y&r_x&0 \\ \end{bmatrix} ~. \end{aligned}$$

Appendix B: Proofs of the relationships in Sect. 7.1

1. $m_{\omega } = d_{\omega }^1$ derives from (20) and $d_{\omega }^1$ in Table 2.

2. $\theta _{\omega }^{1} \approx d_{\omega }^{2} \Delta t$. For $\Delta t \longrightarrow 0$, we can neglect the arc tangent in (28). Hence, we can rewrite $\theta _{\omega }^{1}$ in (23) as:

$$\begin{aligned} \begin{aligned} \theta _{\omega }^{1}&\approx \frac{\Vert {\varvec{\omega }}_t \times {\varvec{\omega }}_{t+1}\Vert }{{\varvec{\omega }}_{t} \cdot {\varvec{\omega }}_{t+1}} = \frac{\Vert {\varvec{\omega }}_t \times ({\varvec{\omega }}_{t} + \Delta {\varvec{\omega }}_{t})\Vert }{{\varvec{\omega }}_{t} \cdot ({\varvec{\omega }}_{t} + \Delta {\varvec{\omega }}_{t})}\\&\approx \frac{\Vert {\varvec{\omega }}_t \times \Delta {\varvec{\omega }}_{t}\Vert }{\Vert {\varvec{\omega }}_{t} \Vert ^{2}}\frac{\Delta t}{\Delta t} \approx \frac{\Vert {\varvec{\omega }}_t \times \dot{{\varvec{\omega }}}_{t}\Vert }{\Vert {\varvec{\omega }}_{t} \Vert ^{2}}\Delta t = d_{\omega }^{2} \Delta t~. \end{aligned} \end{aligned}$$

(43)

3. $\theta _{\omega }^{2} \approx d_{\omega }^{3}\Delta t$. Recall that $\mathbf {a} \times \mathbf {b} = -\mathbf {b} \times \mathbf {a}$ and that $\mathbf {a}\cdot (\mathbf {b} \times \mathbf {c}) = \mathbf {c}\cdot (\mathbf {a} \times \mathbf {b})$. $\theta _{\omega }^{2}$ in (23) can be re-written as:

$$\begin{aligned} \begin{aligned} \theta _{\omega }^{2}&= \arctan {\left( \frac{\Vert {\varvec{\omega }}_{t+1}\Vert {\varvec{\omega }}_{t+2} \cdot \left( {\varvec{\omega }}_{t+1}\times {\varvec{\omega }}_{t} \right) }{\left( {\varvec{\omega }}_{t+1} \times {\varvec{\omega }}_{t}\right) \cdot \left( {\varvec{\omega }}_{t+1} \times {\varvec{\omega }}_{t+2}\right) }\right) } \\&= \arctan {\left( \frac{\Vert {\varvec{\omega }}_{t+1}\Vert \left( {\varvec{\omega }}_{t}\times {\varvec{\omega }}_{t+1} \right) \cdot {\varvec{\omega }}_{t+2}}{\left( {\varvec{\omega }}_{t} \times {\varvec{\omega }}_{t+1}\right) \cdot \left( {\varvec{\omega }}_{t+1} \times {\varvec{\omega }}_{t+2}\right) }\right) } \end{aligned} \end{aligned}$$

(44)

The denominator of (44) can be re-written as:

$$\begin{aligned} \begin{aligned}&\left( {\varvec{\omega }}_{t} \times {\varvec{\omega }}_{t+1}\right) \cdot \left( {\varvec{\omega }}_{t+1} \times {\varvec{\omega }}_{t+2}\right) \\&\approx \left( {\varvec{\omega }}_{t} \times \Delta {\varvec{\omega }}_{t}\right) \cdot \left[ \left( {\varvec{\omega }}_{t} \times \Delta {\varvec{\omega }}_{t}\right) \times \left( {\varvec{\omega }}_{t} \times 2\Delta {\varvec{\omega }}_{t}\right) \right] \\&= \left( {\varvec{\omega }}_{t} \times \Delta {\varvec{\omega }}_{t}\right) \cdot \left[ 2({\varvec{\omega }}_{t} \times \Delta {\varvec{\omega }}_{t})-({\varvec{\omega }}_{t} \times \Delta {\varvec{\omega }}_{t})\right] \frac{\Delta {t}^{2}}{\Delta {t}^{2}} \\&\approx \left( {\varvec{\omega }}_{t} \times \dot{{\varvec{\omega }}}_{t}\right) \cdot \left( {\varvec{\omega }}_{t} \times \dot{{\varvec{\omega }}}_{t}\right) \Delta {t}^{2} = \Vert {\varvec{\omega }}_{t} \times \dot{{\varvec{\omega }}}_{t} \Vert ^{2}\Delta {t}^{2} \end{aligned} \end{aligned}$$

(45)

Considering that $\ddot{{{\varvec{a}}}}_t \approx ({{\varvec{a}}}_{t+2} + {{\varvec{a}}}_t)/\Delta {t}^2$, the numerator of (44) can be re-written as:

$$\begin{aligned} \begin{aligned}&\Vert {\varvec{\omega }}_{t+1}\Vert \left( {\varvec{\omega }}_{t}\times {\varvec{\omega }}_{t+1} \right) \cdot {\varvec{\omega }}_{t+2} \approx \Vert {\varvec{\omega }}_t \Vert \left( {\varvec{\omega }}_{t}\times \Delta {\varvec{\omega }}_{t} \right) \cdot \\&(\ddot{{\varvec{\omega }}}_{t} \Delta t^{2} - {\varvec{\omega }}_t) \approx \Vert {\varvec{\omega }}_t \Vert \left( {\varvec{\omega }}_{t}\times \dot{{\varvec{\omega }}}_{t} \right) \cdot \ddot{{\varvec{\omega }}}_{t} \Delta t^{3} \end{aligned} \end{aligned}$$

(46)

Finally, combining (45), (46) and (44), and neglecting the arc tangent, we obtain that $\theta _{\omega }^{2} \approx d_{\omega }^{3}\Delta t$ for $\Delta t \longrightarrow 0$.

Appenix C: Proofs of the relationships in Sect. 7.2

1. $m_{v} = e_{v}^1$ derives from (19) and $e_{v}^1$ in Table 2. $m_{\omega } = e_{\omega }^1$ derives from (20) and $e_{\omega }^1$ in Table 2.

2. $\theta _{\omega }^{1} \approx e_{\omega }^2 \Delta t$ derives from (43) recalling that $e_{\omega }^2 = d_{\omega }^2$. $\theta _{v}^{1} \approx e_{v}^2 \Delta t$ can be proven by following similar steps as in (43) and considering $e_{v}^2$ in Table 2.

3. $\theta _{v}^{2} \approx e_{v}^3 \Delta t$ and $\theta _{\omega }^{2} \approx e_{\omega }^3 \Delta t$. Following similar steps as in (45) and (46), and recalling that $(\mathbf {a} \times \mathbf {b})\times (\mathbf {a} \times \mathbf {c}) = \left[ \mathbf {a}\cdot (\mathbf {b} \times \mathbf {c})\right] \mathbf {a}$, it is possible to prove that $\theta _{v}^{2} \approx e_{v}^3 \Delta t$ and $\theta _{\omega }^{2} \approx e_{\omega }^3 \Delta t$.