From Motions to Emotions: Can the Fundamental Emotions be Expressed in a Robot Swarm?

Abstract

This paper explores the expressive capabilities of a swarm of miniature mobile robots within the context of inter-robot interactions and their mapping to the so-called fundamental emotions. In particular, we investigate how motion asnd shape descriptors that are psychologically associated with different emotions can be incorporated into different swarm behaviors for the purpose of artistic expositions. Based on these characterizations from social psychology, a set of swarm behaviors is created, where each behavior corresponds to a fundamental emotion. The effectiveness of these behaviors is evaluated in a survey in which the participants are asked to associate different swarm behaviors with the fundamental emotions. The results of the survey show that most of the research participants assigned to each video the emotion intended to be portrayed by design. These results confirm that abstract descriptors associated with the different fundamental emotions in social psychology provide useful motion characterizations that can be effectively transformed into expressive behaviors for a swarm of simple ground mobile robots.

Appendices

A Swarm Behaviors

In Sect. 3.1, a series of swarm behaviors were designed based on the movement and shape attributes associated with the different fundamental emotions. This appendix includes the mathematical expressions of the control laws used to produce the different swarm behaviors. Note that all the control laws included here treat each robot in the swarm as a point that can move omnidirectionally according to single integrator dynamics as in (1). The transformation from single integrator dynamics to unicycle dynamics is discussed in detail in Appendix B.

Fig. 16

Density functions associated to represent the emotions of anger (a), disgust (b) and fear (c). The higher the density (darker color), the higher the concentration of robots will be in that area. The red circles represent the position of the agents once the control law in (13) has converged

1.1 A.1 Happiness

The swarm movement selected for the happiness behavior consists of the robots following the contour of a circle with a superimposed sinusoid. This shape is illustrated in Fig. 15a and can be parameterized as

$$\begin{aligned} \begin{aligned} x_{h}(\theta )&= (R + A\sin (f\theta )) \cos \theta ,\\ y_{h}(\theta )&= (R + A\sin (f\theta )) \sin \theta , \end{aligned} \quad \theta \in [0, 2\pi ), \end{aligned}$$

(3)

where R is the radius of the main circle and A and f are the amplitude and frequency of the superposed sinusoid, respectively. For the shape in Fig. 15a, the frequency of the superimposed sinusoid is $f=6$.

If we have a swarm of N robots, we can initially position Robot i according to

$$\begin{aligned} p_i(0) = [x_h(\theta _i(0)), ~y_h(\theta _i(0))]^T,\quad i=1,\dots , N, \end{aligned}$$

(4)

with

$$\begin{aligned} \theta _i(0) = 2\pi i/N. \end{aligned}$$

(5)

Then the team will depict the desired shape if each robot follows a point evolving along the contour in (3),

$$\begin{aligned} \dot{p}_i = [x_h(\theta _i(t)), y_h(\theta _i(t))]^T - p_i, \end{aligned}$$

(6)

with $\theta _i$ a function of time $t\in \mathbb {R}_+$,

$$\begin{aligned} \theta _i(t) = \text {atan2}(\sin (t + \theta _i(0)), \cos (t + \theta _i(0))). \end{aligned}$$

(7)

1.2 A.2 Surprise

In the case of the surprise emotion, each robot follows a point moving along a circle with expanding radius, as in Fig. 15b. Such shape can be parameterized as,

$$\begin{aligned} \begin{aligned} x_{sur}(\theta , t)&= R(t) \cos \theta ,\\ y_{sur}(\theta , t)&= R(t) \sin \theta , \end{aligned} \quad \theta \in [0, 2\pi ), \end{aligned}$$

(8)

with

$$\begin{aligned} R(t) = \text {mod}(t, R_{max}-R_{min})+R_{min},\quad t\in \mathbb {R}_+, \end{aligned}$$

(9)

to create a radius that expands from $R_{min}$ to $R_{max}$.

Analogously to the procedure described in Appendix A.1, in this case the robots can be initially located at

$$\begin{aligned} p_i(0) = [x_{sur}(\theta _i(0), 0), y_{sur}(\theta _i(0), 0)]^T,\quad i=1,\dots , N, \end{aligned}$$

(10)

with $\theta _i(0)$ given by (5). The controller for each robot is then given by,

$$\begin{aligned} \dot{p}_i = [x_{sur}(\theta _i(t), 0), y_{sur}(\theta _i(t), 0)]^T - p_i, \end{aligned}$$

(11)

with $\theta _i(t)$ as in (7).

1.3 A.3 Sadness

For the case of the sadness emotion, the robots move along a circle of small dimension as compared to the domain. The strategy is analogous to the ones in (6) and (11), with the parameterization of the contour given by,

$$\begin{aligned} \begin{aligned} x_{sad}(\theta )&= R \cos \theta ,\\ y_{sad}(\theta )&= R \sin \theta , \end{aligned} \quad \quad \theta \in [0, 2\pi ), \quad R>0. \end{aligned}$$

(12)

1.4 A.4 Anger, Fear and Disgust

For the remaining emotions—anger, disgust and fear—the swarm coordination is based on the coverage control strategy, which allows the user to define which areas the robots should concentrate around.

If we denote by D the domain of the robots, the areas where we want to position the robots can be specified by defining a density function, $\phi :D\rightarrow [0,\infty )$, that assigns higher values to those areas where we desire the robots to concentrate around. We can make the robots distribute themselves according to this density function by implementing a standard coverage controller such as [13], where

$$\begin{aligned} \dot{p}_i = \kappa (c_i(p) - p_i), \end{aligned}$$

(13)

where $p = [p_1^T,\dots ,p_N^T]^N$ denotes the aggregate positions of the robots and $\kappa >0$ is a proportional gain. In the controller in (13), $c_i(p)$ denotes the center of mass of the Voronoi cell of Robot i,

$$\begin{aligned} c_i(p) = \frac{\int _{V_i(p)}q\phi (q)dq}{\int _{V_i(p)}\phi (q)dq}, \end{aligned}$$

(14)

with the Voronoi cell being characterized as,

$$\begin{aligned} V_i(p) = \{q\in D ~|~\Vert q-p_i\Vert \le \Vert q-p_j\Vert , j\ne i \}. \end{aligned}$$

(15)

Figure 16 shows the densities selected for each of the emotions, where the red circles represent the positions of the robots in the domain upon convergence, achieved by running the controller in (13).

B Individual Robot Control

The swarm behaviors described in Appendix A assume that each robot in the swarm can move omnidirectionally according to

$$\begin{aligned} \dot{p}_i = u_i, \end{aligned}$$

(16)

with $p_i=(x_i, y_i)^T\in \mathbb {R}^2$ the Cartesian position of Robot i in the plane and $u_i=(u_{ix}, u_{iy})^T\in \mathbb {R}^2$ the desired velocity. However, the GRITSBot (Fig. 1) has a differential-drive configuration and cannot move omnidirectionally as its motion is constrained in the direction perpendicular to its wheels. Instead, its motion can be expressed as unicycle dynamics,

$$\begin{aligned} \dot{x}_i&= v_i \cos \theta _i,\nonumber \\ \dot{y}_i&= v_i \sin \theta _i,\nonumber \\ \dot{\theta }_i&= \omega _i, \end{aligned}$$

(17)

with $\theta _i$ the orientation of Robot i and $(v_i, \omega _i)^T$ the linear and angular velocities executable by the robot, as shown in Fig. 17.

In this paper, the single integrator dynamics in (16) are converted into unicycle dynamics, as in (17), using a near-identity diffeomorphism [41],

$$\begin{aligned} \begin{pmatrix} v_i\\ \omega _i \end{pmatrix} = K \begin{pmatrix} \cos \theta _i &{} \sin \theta _i\\ -\dfrac{\sin \theta _i}{l} &{} \dfrac{\cos \theta _i}{l} \end{pmatrix} \begin{pmatrix} u_x\\ u_y \end{pmatrix}, \quad K, l>0. \end{aligned}$$

(18)

Fig. 17

Parameters involved in the near-identity diffeomorphism in (18), used to transform the single integrator dynamics in (16) into unicycle dynamics (17), executable by the GRITSBots. The pose of the robot is determined by its position, $p=(x,y)^T$, and its orientation, $\theta$. The single integrator control, u, is applied to a point $\tilde{p}$ located at a distance l in front of the robot. The linear and angular velocities, v and $\omega$, that allow the robot to track $\tilde{p}$ are obtained applying the near-identity diffeomorphism in (18)

A graphical representation of this transformation is included in Fig. 17: the input $u = (u_x, u_y)^T$ is applied to a point located at a distance of l in front of the robot, $\tilde{p}$, which can move according to the single integrator dynamics in (16). The effect of this parameter in the movement of the robot is illustrated in Fig. 9. The parameter K acts as a proportional gain.