A human inspired handover policy using Gaussian Mixture Models and haptic cues

Abstract

A handover strategy is proposed that aims at natural and fluent robot to human object handovers. For the approaching phase, a globally asymptotically stable dynamical system (DS) is utilized, trained from human demonstrations and exploiting the existence of mirroring in the human wrist motion. The DS operates in the robot task space thus achieving independence with respect to the robot platform, encapsulating the position and orientation of the human wrist within a single DS. It is proven that the motion generated by such a DS, having as target the current wrist pose of the receiver’s hand, is bounded and converges to the previously unknown handover location. Haptic cues based on load estimates at the robot giver ensure full object load transfer before grip release. The proposed strategy is validated with simulations and experiments in real settings.

Appendix: Unit quaternion preliminaries

A four-parameter representation of the orientation is the unit quaternion defined as $Q=\begin{bmatrix} \eta&\epsilon ^T\end{bmatrix}^T$ with $\eta \in \mathfrak {R}$ being the scalar part and $\epsilon \in \mathfrak {R}^3$ the vector part and which are related with the angle-axis representation of orientation, $r \in \mathfrak {R}^3, \ \theta \in \mathfrak {R}$ as follows:

$$\begin{aligned} \eta= & {} \cos \frac{\theta }{2} \end{aligned}$$

(21)

$$\begin{aligned} \epsilon= & {} r\sin \frac{\theta }{2} \end{aligned}$$

(22)

Notice that utilizing the angle-axis representation, one specific rotation can be expressed with two different ways: a rotation by $-\theta$ around $-r$ or a rotation by $\theta$ around r. On the contrary, utilizing the quaternion representation there is a unique expression for each rotation (Siciliano et al. 2010). The following properties hold for unit quaternions:

$$\begin{aligned} Q^TQ= & {} 1 \end{aligned}$$

(23)

$$\begin{aligned} Q^T\dot{Q}= & {} 0 \end{aligned}$$

(24)

$$\begin{aligned} Q^{-1}= & {} \begin{bmatrix}\eta \\ -\epsilon \end{bmatrix} \end{aligned}$$

(25)

Composition of unit quaternions $Q_1 = \begin{bmatrix}\eta _1&\epsilon _1^T\end{bmatrix}^T$ and $Q_2 = \begin{bmatrix}\eta _2&\epsilon _2^T\end{bmatrix}^T$, denoted by the operator $*$, yields the unit quaternion corresponding to the respective rotation matrix product $R_1R_2$ where $R_1, R_2 \in SO(3)$:

$$\begin{aligned} Q_1*Q_2 = \begin{bmatrix} \eta _1\eta _2-\epsilon _1^T\epsilon _2 \\ \eta _1\epsilon _2+\eta _2\epsilon _1+\epsilon _1\times \epsilon _2 \end{bmatrix} \end{aligned}$$

(26)

Given the current and desired rotation matrices $R \in SO(3)$, $R_d \in SO(3)$ as well as the respective quaternions Q, $Q_d$, the relative rotation $RR_d^T$, can be defined in terms of the quaternion $\varDelta Q=\begin{bmatrix}\varDelta \eta&\varDelta \epsilon ^T\end{bmatrix}^T$ as follows:

$$\begin{aligned} \varDelta Q = Q*Q_d^{-1} \end{aligned}$$

(27)

Notice that $\varDelta Q = \begin{bmatrix}1&0_{1\times 3}\end{bmatrix}^T$ if and only if $R_d = R$.

Unit quaternion time derivatives are related to the angular velocity $\omega$ expressed in the inertia frame as follows (equations (3.94)-(3.95) from Siciliano et al. (2010)):

$$\begin{aligned} \begin{aligned} \dot{\eta }&= -\frac{1}{2}\epsilon ^T\omega \\ \dot{\epsilon }&= \frac{1}{2}\left( \eta I_3 - \hat{\epsilon }\right) \omega \end{aligned} \end{aligned}$$

(28)

which can be written compactly as:

$$\begin{aligned} \dot{Q} = \frac{1}{2}J_Q\omega \end{aligned}$$

(29)

where

$$\begin{aligned} J_Q = \left[ \begin{array}{c} -\epsilon ^T \\ \eta I_3 - \hat{\epsilon } \end{array} \right] \in \mathfrak {R}^{4\times 3}. \end{aligned}$$

(30)

Given (26)–(27), a minimal representation of the orientation error $e_o \in \mathfrak {R}^3$ can be defined via the vector part of the quaternion error as follows:

$$\begin{aligned} e_o \triangleq \varDelta \epsilon = \eta _d\epsilon - \eta \epsilon _d - \hat{\epsilon }\epsilon _d \end{aligned}$$

(31)

with $\hat{\epsilon }$ denoting the skew symmetric matrix of vector $\epsilon$. Notice that (31) can be written as:

$$\begin{aligned} e_o = -J_Q^T Q_d \end{aligned}$$

(32)

Substituting (29) in (24) yields $J_Q^T Q = 0$; thus, $e_o$ can be written also in the following form:

$$\begin{aligned} e_o = J_Q^T\left( Q-Q_d\right) . \end{aligned}$$

(33)