Individual and Team Trust Preferences for Robotic Swarm Behaviors

In this work we consider a swarm composed of ground-based robotic vehicles [8]. Each agent’s motion can be represented using unicycle dynamics commanded by a multi-agent controller.

To elicit human trust responses, we have implemented five swarm behaviors using the robotic platform: 1) cyclic pursuit, in which agents traverse a circle [9], 2) herding, in which agents move from one location to another while maintaining collective cohesion [10], 3) leader following, in which a leader moves while trailed by all other agents [11], 4) square formation, in which agents relocate to the vertices of a square shape [11], 5) line formation, in which the agents relocate to form a line [11]. The respective trajectories are depicted in Figure 1.

Given a set of $n_{b}$ swarm behaviors $B$, we wish to collect a set of $m_{b}$ pairwise comparison preferences $\mathcal{E}_{B}=\{(v_{i}^{1},v_{i}^{2})\,|\,v_{i}^{1},v_{i}^{2}\in B,\,i\in\{1,...,m_{b}\}\}$ that answer the general question “Comparing these two swarm behaviors, which do you trust more?”, presupposing an individual’s intuitive definition of trust. For the $i^{th}$ comparison, we record the more trusted behavior as $v_{i}^{1}$ (the first element in the pair) and the less trusted behavior as $v_{i}^{2}$ (the second element).

We may visualize an individual’s pairwise comparison preferences using a preference graph. Consistent preferences may be depicted as acyclic graphs, yielding a partial order over preferences, while inconsistent preferences generate a cyclic graph. For the $k^{th}$ individual, the directed preference graph $\mathcal{G}_{k}=(V_{k},\mathcal{E}_{k})$ is defined by a vertex set $V_{k}\subseteq B$ containing the compared behaviors, and an ordered edge set $\mathcal{E}_{k}\subseteq V_{k}\times V_{k}\subseteq\mathcal{E}_{B}$ indicating preferences among pairs of vertices. Here, a directed edge $(v_{i},v_{j})\in\mathcal{E}_{k}$ from vertex $v_{i}$ to vertex $v_{j}$ indicates behavior $v_{i}$ is preferred to behavior $v_{j}$. Note there is at most one edge between each vertex in a pair, i.e. we do not consider self-contradictions.

We assume that each preference is labeled as belonging to a given individual when constructing the individual’s directed preference graph. In a wider population for which demographic information cannot be collected, we instead collect anonymized unlabeled trust preferences such that we cannot distinguish between individuals. In this case we may define a population preference graph $\bar{\mathcal{G}}=(\bar{V},\bar{\mathcal{E}},W)$, with the vertex set $\bar{V}=\bigcup_{k=1}^{p}V_{k}$, the ordered set of edges $\bar{\mathcal{E}}\subseteq\mathcal{E}_{B}$, and the associated edge weight set $W$. The edge and weight set are formed by enumerating each pairwise preference as $a_{ij}=\sum_{k=1}^{p}\mathbb{I}_{\mathcal{E}_{k}}\left((v_{i},v_{j})\right)$ for all $v_{i},v_{j}\in\overline{V}$. The edge set $\bar{\mathcal{E}}$ contains the edge $(v_{i},v_{j})$ if $a_{ij}-a_{ji}>0$ (i.e. $v_{i}$ is more preferred to $v_{j}$) or $a_{ij}=a_{ji}\neq 0$ and $i<j$ (i.e. $v_{i}$ are equally preferred). The associated edge weight element of $W$ is $w_{k}=a_{ij}/(a_{ij}+a_{ji})$ for the $k^{th}$ preference pair $(v_{k}^{1},v_{k}^{2})=(v_{i},v_{j})$; these weights capture preference variability among individuals in the population.

We interpret $\bar{\mathcal{G}}$ as proportional to the average preference of the population, with each vertex in the graph mapped to a preferred preference. The directed weighted edges between vertices indicate the likelihood of the preference transition by the majority of individuals. A partially ordered set of instance preferences can be abstracted from the preference graph and is amenable to an ordinal optimization problem [12]. We instead locate the preference instances within a feature vector space and optimize within this space.

As we cannot reason about preferences towards behaviors not already in the graph, we may consider mapping each behavior to a point $x$ in a feature space $\mathcal{X}$. In $\S$V we present individuals with videos of swarm behaviors. We seek to work directly with these stimuli, to capture information about the swarm’s visual appearance and trajectory evolution encoded therein. Manual feature vector extraction from stimuli has been demonstrated for preference learning in [13], however doing so for videos is impractical and entails discretionary judgements regarding which features to select. Automatic feature vector extraction processes can address these two issues by removing human discretion in feature identification. One may consider dimensionality-reduction techniques for individual video frames (e.g. principle component analysis [14]), however temporal dynamics between frames will be neglected. To overcome this issue, we have adopted a similar approach to [15] by developing a neural network-based variational auto-encoder (Valma) that extracts a feature vector for a video of swarm behavior. As depicted in Figure 2, the model contains a pre-processing component (extracting frame features using the VGGNet-A computer vision model [16]) and two additional recurrent neural network components: an encoder and a decoder. The recurrent neural networks use an LSTM architecture [17] in order to learn temporal relationships between different data features. Between the two components there is a data bottleneck, with the encoded input projected into a lower-dimensional latent space.¹¹1The reader is referred to our repository for further details about the implementation and training of the model. By training the model to reproduce the encoder input at the decoder output, we can learn a mapping from the input space to a latent space in an unsupervised manner. and thus extract a compact feature vector $x\in\mathcal{X}$ automatically. We describe the mapping as $h:B\to\mathcal{X}$ from the behavior set to a $q$-dimensional feature space $\mathcal{X}\subseteq\mathbb{R}^{q}$.

The feature vectors for the trust preference pair $(v^{1}_{i},v^{2}_{i})\in\mathcal{E}_{B}$ are given by $x^{1}_{i}=h(v^{1}_{i})$ and $x^{1}_{i}=h(v^{1}_{i})$, respectively. Pairwise trust preferences imply the existence of an underlying quadratic trust function $f_{k}:\mathcal{X}\to\mathbb{R}$ such that

Consider the quadratic trust function

where $\bar{x}_{k}$ is a vector in $\mathcal{X}$ corresponding to optimal trust for individual $k$. We may estimate this function by considering the preference set as analogous to a set of affine classifications. In feature vector space, this set corresponds to a set of hyperplanes separating the pairs of behavior points with maximal distance to each point.

Given the $i^{th}$ preference pair $(x_{i}^{1},x_{i}^{2})\in\mathcal{X}\times\mathcal{X}$, (2) is equivalent to the halfspace

where $a_{i}=x_{i}^{2}-x_{i}^{1}$ and $b_{i}=a_{i}^{T}(x_{i}^{1}+x_{i}^{2})/2$. The closed halfspace indicates that $g_{i}(x)\leq 0$ (the region of the feature space containing preferred behaviors) is convex but not affine.

The $k^{th}$ individual’s preference set can be described by a set of halfspaces (4) in feature vector space, with the $i^{th}$ hyperspace associated with the preference $(v^{1}_{i},v^{2}_{i})\in\mathcal{E}_{k}$. Each preference instance reduces the halfspace of $\mathcal{X}$, further constraining $x$. The intersection of the closed halfspaces define a preference polytope, a region of feature vector space associated with greatest preference. The intersection satisfies the system of linear inequalities created by the preferences, and can be represented by the polytope

An example the intersection of eight preference’s corresponding halfspaces is given by the shaded interior region in Figure 3. The preference polytope can be unbounded for small $|\mathcal{E}_{k}|$ and poorly distributed preference pairs. The preference polytope can also be empty for cyclic preference graphs and poorly selected embeddings.

The closed region of the polytope $P_{k}$ can be used to determine preferred swarm behaviors; this process is often termed preference learning [18]. The polytope $P_{k}$ can be built iteratively, with new pairwise comparisons presented to the individual over time. Given the polytope $P_{k}(t)$ at sample time $t$, the addition of the $i^{th}$ preference at time $t+1$ forms the new polytope $P_{k}(t+1)=P_{k}(t)\bigcap\{x|g_{i}(x)\leq 0\}$. The strategic presentation of pairwise comparisons to the participant can rapidly reduce the volume of $P_{k}(t)$ over time.

As we have insufficient information within a bounded $P_{k}$ to find $\bar{x}_{k}$ in (3), we may substitute an alternative point in $\mathcal{X}$. We believe that the Chebyshev center of $P_{k}$ is a suitable candidate [6]. The Chebyshev center of $P_{k}$ is the center of the largest inscribed ball in $P_{k}$, also referred to as the in-center point. A visual interpretation of this is depicted in Figure 3. Let the Chebyshev center $x_{c}$ lie at the center of the largest possible ball $\mathcal{B}=\{x_{c}+u\,|\,\lVert u\rVert_{2}\leq r\}$ inside $P_{k}$. We may obtain $\mathcal{B}$ by maximising $r$. For a weaker constraint, let $\mathcal{B}$ lie in the half-space

The corresponding largest possible ball is given by $\text{sup}\{a_{i}^{T}u|\lVert u\rVert_{2}\leq r\}=r\lVert a_{i}\rVert_{2}.$ Hence the Chebyshev center lies within the halfspace $\mathcal{B}\subset P_{k}$ if and only if $a_{i}^{T}x_{\text{c}}+r\lVert a_{i}\rVert_{2}\leq b_{i},$ for all $(v_{i}^{1},v_{i}^{2})\in\mathcal{E}_{k}$. Given the ball radius $r\geq 0$, $x_{\text{c}}$ can be found by solving the optimization

The optimization is a linear program with many algorithms that can reliably and efficiently solve the problem [19]. The resulting $x_{c}$ can then be used as a proxy to compare individuals’ trust.

Given multiple individuals’ respective preferences, we may devise a measure of the individuals’ distinctiveness. The optimal trust value for each individual $k$ is denoted as $x+z^{k}$ where $x$ is assumed to be a global reference trust measure and $z^{k}$ the perturbation of individual $k$ away from this reference. If the magnitude of $z^{k}$ were small for all of the team, the group would exhibit similar trust behaviors. Individual with large $z^{k}$ would, in turn, have distinctive trust behaviors compared to the group as a whole. Examining the preferences across each individual $k$, the selection problem for their $i^{th}$ preference selection will generate the hyperspace $\left(a_{i}^{k}\right)^{T}\left(x+z^{k}\right)\leq b_{i}^{k}$. An optimization problem can then be posed to find $x$ with small $\left\|z^{k}\right\|_{1}$ across all members using

The accumulative 1-norm is used here to minimize $z$ due to its sparsifying properties as it promotes sparse solutions with small or even zero $\left\|z^{k}\right\|_{1}$ [19]. When $\left\|z^{k}\right\|_{1}=0$, the reference trust measure $x$ will satisfy all of the selection preferences for individual $k$. This subset of individuals will share a non-trivial intersection of their trust polytopes and an additional Chebyshev center selection could be performed to select a trust reference $x_{c}$ with the characteristics described in $\S$III-C.

We now examine the case where a preference may be expressed multiple times with contradictory responses. Consider the selection of the perturbation $z$ for each individual from a normal distribution $Z\sim\mathcal{N}(0,\Sigma)$, where $\Sigma$ is a symmetric positive definite matrix. From (4), the $i^{th}$ selection problem can subsequently be posed as a preferential selection of one choice over the other when the random variable $X_{i}=g_{i}(x+z)=a_{i}^{T}\left(x+z\right)-b_{i}$ is non-positive; the corresponding distribution is $\mathcal{N}(\mu=a_{i}^{T}x-b_{i},\sigma^{2}=a_{i}^{T}\Sigma a_{i})$. By applying the inverse distribution mapping (1), the probability of a positive preference selection $p_{i}=p(X_{i}\leq 0)$ is then $-\Phi^{-1}(p_{i})\sigma=\mu.$ Assuming that the covariance of $Z$ is bounded as $0\preceq\Sigma\preceq\alpha^{2}I$, then $\sigma\leq\alpha\left\|a_{i}\right\|_{2}$. For $p_{i}\geq 0.5$ then $\Phi^{-1}(p_{i})\geq 0$ and

Similarly, for $p_{i}<0.5$ then

such that each preference constrains $x$ to lie in what can be interpreted geometrically as a slab, i.e., a set of the form $\{x\in\mathbb{R}^{q}|\alpha\leq a^{T}x\leq\beta\}$ for scalars $\alpha\leq\beta$. With the data sampled from a finite population, the probability $p_{i}$ is calculated based on a confidence interval $\left[p_{i}-\delta,p_{i}+\delta\right]$ projected onto the unit interval $\left[0,1\right]$ as $\Delta=\left[\max(0,p_{i}-\delta),\min(1,p_{i}+\delta)\right])$. Here, $2\delta$ is the width of the confidence interval band and is based on the margin of error calculated from a number of samples. Applying the central limit theorem for the binomial distribution is one approach to calculate the width with $\delta=Z\sqrt{1/4n_{s}}$ where $Z$ is the $Z$-score associated with a confidence interval and $n_{s}$ is the number of samples of the $i^{th}$ preference [20].

We assume that $g_{i}(x)$ is constructed so that $X_{i}\leq 0$ for most of the population (i.e., $p_{i}\geq 0.5$), for example as per $\S$II-B for $\bar{\mathcal{G}}=(\bar{V},\bar{\mathcal{E}},W)$ with $p_{i}=w_{i}\in W$. Using the constraints (9) and (10) over $\Delta$, we may find the average trust measure $\overline{x}$ and minimum covariance bound $\overline{\alpha}$ for the population as

For the $i^{th}$ preference, the (positive) upper bound on the covariance $\alpha$ constrains the width of the slab containing $x$. Similarly, the closer $p_{i}$ is to $0.5$ (i.e. a split decision on the $i^{th}$ preference among individuals), the narrower the slab.

We pursued an online survey methodology involving filmed videos of the swarm behaviors from $\S$II-A. All participants were over 18 years of age and no demographic information was requested. Forty-three participants responded to the survey, and were not remunerated or otherwise rewarded for survey completion. In the first part, we presented the participant with a video of a swarm executing a leader following behavior (see $\S$II-A). The fourteen questions relating to trust in the swarm are a modified version of the Trust Perception Scale-HRI (TPS-HRI) [3], substituting the word ‘swarm’ for ‘robot’. In the second part, we presented the participant with pairs of swarm behavior videos and collected the participant’s pairwise trust preferences, asking ‘Comparing the two swarms, which do you trust more?’. To avoid priming the participants we did not specify the notion of trust further. The first six pairs of videos covered each combination among the first four of the five behaviors, shown to each participant in the same order. Each participant could repeat the process with four more video pairs, but this was optional to avoid survey fatigue impacting response quality.

We proceed to analyze data gathered from the online survey, focusing on the distinctiveness and cohesion of the cohort’s trust preferences. For each participant we have created a preference graph (exemplified by Figure 7) to determine the respective preference polytope. For the subset of participants with bounded preference polytopes, their individual Chebyshev centres could be found. In Figure 4 the aggregated population trust optimum $\bar{x}$ from (8) is compared with the Chebyshev centers for the preference polytopes belonging to a subset of participants.²²2For visualization purposes only two dimensions of $\mathcal{X}$ are depicted. For the same subset of participants, we also compare their distinctiveness $\left\|z{{}^{k}}\right\|_{1}$ with corresponding TPS-HRI trust scores in Figure 6. We observe that participants with distinctiveness $\left\|z^{k}\right\|_{1}\leq 0.035$ and a trust score in the range $[42\%,56.5\%]$ express preferences compatible with the population’s preferences. In contrast, participants with distinctiveness $\left\|z^{k}\right\|_{1}>0.035$ have preferences differing from the population. In Figure 5 we extrapolate the notion of distinctiveness and trust bounds to the whole population. Participants with trust values in the range $[42\%,56.5\%]$ are associated with low distinctiveness from the population’s preferences in Figure 6, hence have similar trust preferences to an average participant in the population. In this way low distinctiveness can become a criterion for selecting teams of participants.

In Figure 7 a partial ordering over the swarm behaviors is generated from an aggregation of unlabeled preferences from all participants, and represented as a preference graph $\bar{\mathcal{G}}=(\bar{V},\bar{\mathcal{E}},W)$ as depicted. We may then use the edge-weighted population preference graph to derive an average trust measure $\mu=\bar{x}$ and minimum covariance bound $\sigma\leq\bar{\alpha}$ from (11). In Table I we compare $\mu$ with the aggregated Chebyshev center $\bar{x}_{c}$ of the unweighted preference graph $(\bar{V},\bar{\mathcal{E}})$. We observe in Table II that $54.1\%$ of participant’s individual Chebyshev centers lie within the upper bound of one standard deviation $\alpha$ of the mean and $100\%$ lie within $2\alpha$. This matches well the theoretical bounds $p(-s<\left\|X-\mu\right\|_{2}/\sigma<s)=\Phi(s)-\Phi(-s)$ for $s\in\{1,2\}$, hence the aggregated unlabeled preference data is consistent with the model presented in Figure 7. The relatively small distance between the mean and the population’s Chebyshev center, $\left\|\mu-\bar{x}_{c}\right\|_{2}\leq 0.1\bar{\alpha}$, shows that the population preference graph $\bar{\mathcal{G}}$ and optimal solution of (11) is representative of the true value of $\bar{x}_{k}$. This suggests that we may analyze preference similarity to evaluate a population’s cohesiveness.