Expressive Robot Motion Timing

02/05/2018 ∙ by Allan Zhou, et al. ∙ 0

Our goal is to enable robots to time their motion in a way that is purposefully expressive of their internal states, making them more transparent to people. We start by investigating what types of states motion timing is capable of expressing, focusing on robot manipulation and keeping the path constant while systematically varying the timing. We find that users naturally pick up on certain properties of the robot (like confidence), of the motion (like naturalness), or of the task (like the weight of the object that the robot is carrying). We then conduct a hypothesis-driven experiment to tease out the directions and magnitudes of these effects, and use our findings to develop candidate mathematical models for how users make these inferences from the timing. We find a strong correlation between the models and real user data, suggesting that robots can leverage these models to autonomously optimize the timing of their motion to be expressive.



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1 Introduction

Robot motion trajectories have two components. There is a kinematic component, which is the geometric path through the robot’s configurations space – a sequence of configurations that the robot will traverse. But there is also a timing component – a function that assigns a time stamp to each configuration along the path, dictating how the robot will traverse the configuration sequence.

Figure 1: Different timings of the same motion convey different things about the robot. We find effects on perceived confidence, naturalness, even the perceived weight of the object being manipulated. We develop mathematical models for these perceptions that correlate with user data and enable robots to optimize their timing for expressiveness.

Robotics motion planners for manipulation tend to focus on the path [18, 19, 33, 27], with few exceptions explicitly incorporating timing, for instance to improve efficiency or conservative obstacle avoidance [5, 4]. Most commonly, timing is an after-thought in robotics, left to the controller to assign post-hoc.

And yet, timing is crucial in HRI. Imagine seeing a robot arm carry a cup smoothly across the table, like in the top image in Fig.1. Now, imagine seeing a different arm pausing and restarting, slowing down and then speeding back up, like in the bottom image. The path might be the same, but the difference in timing might make us think very differently about the robots and about what they are doing. We might think that the second robot is less capable, or maybe that its task is more difficult. Perhaps it doesn’t have as much payload, perhaps the cup is heavier, or perhaps it does not know what to do:

The timing of a path affects how observers perceive the robot and the task that it is performing.

Studies have already shown that the average velocity and changes in velocity of motion affect perceptions of expressed emotion [7], intent [10], elation [2], animacy [32], arousal and dominance [25], and energy [3]. When it comes to robot motion, human observers will interpret the timing regardless of whether the robot is planning to express anything or not. Our goal is to give robots control over what their timing inadvertently expresses:

Robots should leverage timing to be more expressive of their internal states.

Techniques from animation can be useful in improving robot expressiveness [29, 24], and animated characters have long taken advantage of timing, both for making motion more natural (e.g. ease-in ease-out is one of the 12 animation principles [31]), and more expressive [23, 25]. This made timing a center of focus in the graphics community, developing automated tools for assigning timing to a path. Most tools still leave the animator in control of the timing, but simplify the assignment process (e.g. by allowing the animator to “act out” the timing of a motion with something like a pen and tablet [30]). Other tools align timing to a different trajectory or an external event like a beat [17, 14]. Others yet re-time a particular motion to satisfy new constraints (like finishing faster) while maintaining physical realism [21].

Overall, although realistic timing can be automated, even virtual characters still rely on an external expert when it comes to expressive timing – be it on an animator or on an artist’s trajectory. Robots, on the other hand, can’t afford to rely on experts for every motion they need to perform. They plan their motion autonomously, and have to autonomously decide on how to time it.

Our focus is on enabling robots to produce expressive timing. Two questions remain in this area. First, there is the question of what timing can express in the first place – prior work looked at effects on perceived emotional state, but are there also effects on function-related properties? Second, there is the synthesis question – how can we enable robots to autonomously generate timing from scratch that is purposefully expressive, rather than efficient or physically realistic. We take a step in this direction by analyzing motion timing during manipulation, from an open ended study, to hypothesis-driven experiments, to candidate mathematical models that capture human timing-based inferences.

We make three contributions:

Exploring the possible effects of timing. So far, studies focusing on timing mainly looked for effects on perceived emotional state. We designed and conducted a study to identify what types of variables timing influences more broadly. Rather than biasing users with questionnaires that already suggest how the timing should be interpreted, we used simple open-ended questions. We systematically manipulated timing across three axes inspired by prior work in a factorial design, and asked users to characterize the robot and the task. We clustered their responses to identify common interpretations, and uncovered robot competence, confidence, disposition, along with (unsurprisingly) motion naturalness, and a manipulation-specific characteristic of the task: the weight of the object being manipulated by the robot. This list of variables by no means comprise the entirety of timing effects, nor is it as specific as we would ultimately desire. It does, however, provide us with a rich set of dependent measures for more in-depth analysis.

Experiments that test these effects.

Only after identifying candidate dependent variables based on open-ended questions did we put these effects to the test. We conducted a hypothesis-driven experiment to understand the magnitude and directionality for each. Some of our findings support intuition, like the robot being perceived as less confident if it pauses during the motion. Others are quite surprising. For instance, when the robot is carrying an object, we found that people estimated that object to have approximately the same weight regardless of whether or not the timing had pauses or speed changes. Overall, pausing had a much stronger effect than speed.

Mathematical models and evaluation. Our experiment shows what effects timing has, but not why

. For robots to generate their timing autonomously in different situations, they need a mechanism for generalizing these findings. We attempt such a mechanism for three of the dependent variables. We introduce mathematical models for the inferences that humans make from motion timing. We take a Bayesian inference approach, in which the timing serves as an observation to the human about the states that they can’t observe, like the robot’s confidence or the object’s weight. We show strong correlations between these models and the real user data. The models are constructive, in the sense that robots can use them to optimize their timing to be expressive.

Overall, this paper shows how several timing features interact to affect perceptions of the robot and task, and uses these findings to introduce optimization criteria that correlate with the user data and that robots could use to autonomously time motion in a way that is expressive (e.g. of the robot’s confidence). We look forward to future work on further analysis and refinement of these criteria to ensure generality across settings, as well as further exploration of effects that are more difficult to model, such as how timing influences the robot’s perceived disposition.

Figure 2: Norm of the robot’s configuration space velocity for each way point configuration in each of our 20 conditions. Each column is a different speed change pattern, with the top row representing the conditions without a pause and the bottom representing the conditions with a pause, where we see the velocity go to 0. Each plot contains both the slow motion (lighter color) and its fast counterpart (darker color).

2 Notation

A trajectory in our experiment consists of two components: the sequence of of way point configurations that the robot moves through, and the time at which the robot reaches each way point. We use to represent the robot configuration. is the time that the robot reaches the configuration. We assume that trajectories begin at time 0. is the total duration of the trajectory (i.e., the time at which the robot reaches the final configuration).

We will usually be interested in the speed the robot travels over the course of its trajectory. We use

to represent this velocity (in radians per second). We use

where is the robot’s forward kinematics function, to represent the velocity of the end effector (in meters per second).

To summarize:

  • A sequence of robot configurations that represents the kinematic component (path) of a trajectory.

  • The robot configuration in the trajectory.

  • A sequence of time stamps that represents the timing component of a trajectory.

  • The time when the robot reaches .

  • The total time taken by the trajectory.

  • The velocity of the robot from to .

  • The end effector velocity of the robot from to .

3 Exploratory Study

We start with a study that builds on prior work to find what kinds of effects timing can have on what people infer about the robot during a manipulation task. Our goal with this study is to find the different dimensions of perception that timing affects, i.e. the dependent measures we should test – is it energy, elation, dominance, or something different? We need to avoid biasing the users towards a particular interpretation, so we ask the users open-ended questions and use their responses to form hypotheses for our next experiment.

3.1 Study Design

Robot Task. We used a Kinova 6DOF Mico arm (Fig.1) in our study. We chose one of the most common interactive manipulation tasks for the robot: a handover [6, 28, 20, 22]. The robot carried an object (a cup) from a table to a handover configuration (see Fig.1).

Figure 3: Words that participants in the exploratory study used to characterize the robot (left) and what it was carrying (right). The histograms cluster words into equivalence classes. We use these classes to devise the 5 dependent measures for our experiment, 4 of which are perceptions of the robot and its motion: competence, confidence, disposition, and naturalness. The other dependent measure is functional, the perceived weight of the object in the handover.

Manipulated Factors. The biggest challenge in studying the effects of timing on people’s perceptions is identifying which timing variations to test. A simple answer would be to randomly sample timings, which would uniformly cover the space of all timings. However, they would almost uniformly be interpreted in the same way – as erratic and unnatural. Instead, we decided to systematically generate timings by manipulating several factors.

Our first factor is overall speed. Previous studies found that overall robot speed has effects on perceptions [2, 25]. We use 2 levels for this factor: slow and fast.

Our second factor is change in speed. In studies on abstract characters and human motion, changes in speed have been shown to affect perceived animacy [32], emotion content [7], and energy [3]. Here, we considered 0, 1, and 2 changes, leading to a total of 5 levels for this factor: none (no change), StoF (starting to go faster), FtoS (slowing down), StoFtoS (faster, then slower), and FtoStoF (slower, than faster). Fig.2 (top) shows the magnitude of the velocity across the trajectory way points for each of these patterns, and for both the overall slow and the overall fast levels.

Finally, we also explore an edge case of change in speed: coming to a full stop. Our third factor is thus pause, with 2 levels: either the robot pauses or it does not. The pausing variants are at the bottom of Fig.2, and they differ in that the velocity goes to 0 for a portion of the trajectory.

We used a 2 by 5 by 2 factorial design, leading to a total of 20 conditions, each corresponding to a different timing for a path (shown in Fig.2).

Dependent Measures. We asked users to describe how the robot moved the cup, what adjectives they would use to characterize the robot, and what they think is in the cup.

Subject Allocation. We wanted a within-subjects design to enable users to see multiple possible timings and have bases for comparisons, as they would if they would interact with the robot on a longer term. However, we had 20 conditions, making within-subjects infeasible. We opted for a randomized assignment, where each participant evaluated 8 randomly sampled conditions. There were a total of 61 participants (63% male and 37% female, median age ) all from the United States and recruited through Amazon’s Mechanical Turk platform. All had a minimum approval rating of 95% on Mechanical Turk.

3.2 Analysis

We started by computing word counts for each question. Fig.3 shows the word cloud that this induced for the question of describing the robot, and identifying what the robot is carrying.111We quickly realized that when asked to describe how the robot moved the cup, users quite literally described what the robot did (e.g. “move the cup slowly away from the table”), and we are not including the analysis for that question. We then clustered the words into equivalence classes for easier analysis, ignoring words that appear fewer than 3 times in the data.

Adjectives. Two of the most common adjectives used to describe the robot were literal: slow and fast (or quick, speedy, rapid, efficient), giving rise to two of our clusters. But beyond those, many users described the motion as smooth, natural, predictable, or fluid, which formed the natural cluster with the highest word count (left histogram in Fig.3). The counterparts were also present (unnatural, jerky, robotic, mechanical, uneven, awkward), forming the unnatural cluster. And finally, users described the robot as careful, cautious, deliberate, hesitant, indecisive, calculated. We split these into two clusters: one that suggests low confidence but high ability, like deliberate, and one that suggests low confidence and low ability, like hesitant. The exact split of these is difficult to determine, so the relative counts for deliberate and hesitant should be taken with a grain of salt. The sum of the two, however, is important, and is larger than any of our other clusters, suggesting the importance of timing in perceptions of competence/confidence. Their counterpart, careless, is also present. The histogram plots clusters with more than 10 entries, and groups the remaining words into “other”.

Based on these clusters, we see that motion timing might affect perceived motion naturalness, but also two important other variables: perceived competence and perceived confidence. Confidence alone is not sufficient, because it doesn’t enable us to differentiate between hesitation and deliberation. But taken together, these two variables can help represent our clusters. Surprisingly, none of the descriptions directly related to arousal, dominance, energy (with the exception of the word “aggressive”). Previous studies found effects when directly measuring these, but they do not seem to specifically occur when users aren’t directly asked about them. However, the adjectives that participants used could be interpreted as suggesting higher or lower values for these variables, and we capture them with a broader term: perceived disposition (with positive or negative values).

Object. For what the robot was carrying, the most common words were “water” and “nothing”. While this is not surprising, their difference is important: the cup is heavier when it has something inside. Participants provided many other options, and we differentiated them between standard (e.g. something, water, soda, etc.), options that mentioned hot liquids (e.g. coffee), and solids that cannot be spilled, e.g. (jell-o, or even metal). Hot contents and solid contents were uncommon (right histogram) and not generalizable far beyond open containers, so we identify one variable here: perceived weight of the object that the robot is carrying.

4 Experiments on Timing Effects

We designed our experiments based on the findings from the exploratory study. The word counts already suggest certain effects, e.g. that slower motion tends to be more often described as careful or cautious or deliberate than fast motion (all three appear in top 15 words for slow and do not for fast).

However, we noticed that the difference remains the same when considering changes in speed and when not. Because of this and because changes in speed and pauses are related, we decided to separate into two experiments rather than testing all possible interactions: a first one focusing on speed and pauses, and a second focusing on the speed change patterns.

Figure 4: Our first hypothesis-driven experiment measured the effect of overall speed and pausing and their interaction. Speed significantly affected perceived confidence and weight. Pausing significantly affected perception of every property except weight.

4.1 Speed, Pauses, and Their Interaction

4.1.1 Experiment Design

Manipulated Factors. In this experiment, we manipulate the speed and pause factors as we did in Sec. 3.1.

Dependent Measures. We use the measures we identified in the previous section. We measure perceived competence, confidence, disposition, naturalness, and weight using 7-point scales.

For each dependent measure, the scales were labeled at either end and in the very middle. For example, the disposition scale was labeled “very negative”, “neither positive nor negative”, and “very positive” on the leftmost, middle, and rightmost options, respectively. We chose to label the scales in this fashion instead of having participants mark their agreement with a statement (as is typical in Likert scales) such as “The robot’s disposition is positive.” We did so because disagreeing with that statement does not necessarily mean the same thing as the robot having a “very negative” disposition: disagreeing with positive does not imply agreeing with negative. This is important because here, we are just as interested in whether timing can cause the perception of negative disposition as we are in whether timing can cause the perception of a positive one.

Subject Allocation. The experiment was within-subjects, every participant saw each of the 4 conditions. There were a total of 40 participants (80% male and 20% female, median age ). As in the exploratory study, all participants were from the United States and were recruited through Amazon’s Mechanical Turk, with a minimum approval rating of 95%.

Hypotheses. We state generic and intuitive hypotheses, motivated in part by prior work findings when it comes to disposition and weight, and extrapolating to confidence and competence. However, the devil is in the details, and as we will see in the analysis, not all factors will have their anticipated effects, nor the effect sizes will be the same. Our mathematical models will leverage these details.

We hypothesize that faster motion is more positively perceived (it has already been shown before to positively affect disposition-related perceptions [3], and this could extrapolate), and makes objects look lighter (known from animating dropping objects [23]):

H1. Increasing speed positively affects perceived competence, confidence, disposition, naturalness, and negatively affects perceived weight.

In contrast, pausing (incorporating infinitesimally slow motion) should have the opposite effect:

H2. Pausing negatively affects perceived competence, confidence, disposition, naturalness, and positively affects perceived weight.

4.1.2 Analysis

We first performed a multivariate analysis on the data, and found that the different items were not highly correlated (we computed item reliability, and found Cronbach’s

), so we proceeded with separate analyses for each.

We used a factorial repeated measures ANOVA with speed and pause as factors for each dependent measure. Fig.4 plots the results.

Competence. We found a significant main effect for pause (, ), and no other effects (main or interaction). Pausing made the robot seem less competent. Surprisingly, moving faster made the robot seem only ever-so-slightly more competent, suggesting that it is not overall efficiency that matters for perceived competence.

Confidence. Pausing made the robot seem significantly less confident (, ). But unlike for competence, higher speed made the robot seem significantly more confident (, ), but resulted in a smaller mean difference than pausing. The interaction effect was not significant.

Disposition. Pausing resulted in a more negative disposition (, ). Surprisingly, speed did not have a significant effect, though moving faster did result in a sightly more positive perception, in line with prior work [3].

Naturalness. Again, we found that pausing has a significant main negative effect (, ). Pausing made the motion less natural, intuitively because it is not as smooth, or because it does not match what a person would expect the robot to do. Speed had a very marginal positive effect (, ), though perhaps looking at other values for overall speed would lead to the motion becoming less natural.

Figure 5: Our second hypothesis-driven experiment measured the effect of speed changes on competence, confidence, disposition, weight, and naturalness. Fast-to-Slow was the highest rated pattern for every property except weight.

Weight. In the case of weight, it was speed that had a significant negative effect (, ), with moving faster resulting in the object being perceived as lighter. This is in line with animation advice for animating objects dropping, and physically it makes sense that objects that drop faster are lighter. But seeing this effect on a robot is important because the object is no longer free, but rather being moved by an agent. The robot does not need to move any different when the object is heavier, and yet people do make inferences on weight based on how the robot moves. Surprisingly, pausing did not affect weight, even though pausing did make the robot seem less confident and competent, which could suggest that it is carrying something heavier.

Summary. Overall, the effects we did find were intuitive: pausing negatively affected competence, confidence, disposition, and naturalness, while speed positively affects confidence and negatively affects weight. Participant comments suggested that pauses make the robot look like it is “planning” – it is uncertain about something or trying to locate something. We build on this uncertainty idea in our model in the next section.

It is the effects that we did not find that were surprising. For instance, speed did not seem to influence perceived competence, but influenced perceived confidence. Pausing did not seem to influence perceived weight. Of course, not finding an effect does not mean it is not there, but here the means suggest a small effect size, if at all. We dig deeper into these findings in our model section.

4.2 Speed Change Patterns

4.2.1 Experiment Design

Manipulated Factors. We manipulated speed changes as in Sec. 3.1, using the levels for 1 and 2 changes (previous experiment already evaluated 0 changes).

Dependent Measures. We used the same measures as in Sec. 4.1.

Subject Allocation. There were 40 participants (59% male and 41% female, median age ), selected and allocated as in 4.1.

Hypothesis. We hypothesize that changes in speed will make the robot seem more hesitant and have a negative disposition, but make the object look heavier:

H3. More changes in speed have a negative effect on perceived competence, confidence, disposition, naturalness, and a positive effect on perceived weight.

Which kind of changes (e.g. StoF vs FtoS) have which effect remains to be determined.

4.2.2 Analysis

Number of Speed Changes.

We first analyzed the effects that the number of speed changes has, combining data from this experiment with data from the former. A regression analysis shows, in line with our hypothesis, that having

more changes significantly decreases perceived competence (, ), confidence (,), disposition (, ), and naturalness (, ). It does not, however, significantly affect perceived weight, and in fact the slope on the linear fit is very close to , namely . This is consistent with our finding that pausing did not significantly affect perceived weight, but counter-intuitive nonetheless.

Speed Change Patterns. Aside from number of changes, the actual pattern is interesting as well – does it make a difference, for instance, if the robot starts slower and accelerates, or starts faster and decelerates? We ran a repeated measures ANOVA for each dependent measure, and found a significant effect for every case but perceived weight, so we followed up with Tukey HSD. The results are plotted in Fig.5.

For competence, we found that FtoS was the best option, significantly better than StoFtoS, the worst option (). This was similar for confidence, but here the worst option was FtoStoF. Disposition had the same result as confidence. For naturalness, FtoS was better than every other option, all with .

Summary. Overall, more speed changes negatively impacted all perceptions but weight. Slowing down was the most positively perceived speed change of all. At least for manipulation tasks, if the robot is going to change speed, slowing down will make it seem more competent, confident, and natural compared to speeding up or even speeding up and then slowing back down. This is somewhat surprising, but likely has to do with the notion of reaching a goal that the robot needs to do something with, like handing over the bottle or picking it up. Indeed, participants did often comment in this condition that the robot is changing speed to hand the object over more smoothly.

Surprisingly, speed changes had no effect (close to 0 slope) on perceived weight, even though intuitively the ability to change speed could indicate a lighter object, and the need to change speed could indicate a heavier object. Neither option seemed to be the case.

5 Candidate Mathematical
Models for Timing-Based
Human Inferences

Figure 6: Correlation between our model predictions and real user data. We varied the timing of the trajectories across 8 different conditions (described in Section 4

) and the plots above have one data point per condition. The x-coordinate of each data point is the best model’s prediction for that condition and the y-coordinate is the mean subject response, with a 95% confidence interval. We also show a 95% confidence interval on the regression. We see that higher probability for the robot being confident, the object being heavy, and the motion being natural usually does imply a higher rating along these criteria from the users. This suggests that the models are good candidates for capturing the inferences that people make, enabling robots to predict what their timing will convey. We also test how well the models can fit random data as opposed to real user data to check that they actually approximate the inferences that people make and not just overfitting.

Our findings inform us about the effects of motion timing, but they are only descriptive and not constructive: the robot can’t use them to time its motion automatically in order to express what it wants. All the robot can do now is compare the specific timings we explored and predict what users will infer based on them. But what it needs instead is to be able to predict how any timing will be interpreted.

We take an inferential approach to enabling generalization in this section. We construct models of the inferences that people make from robot motion timing based on our findings so far, and show how they correlate with the real user data. Armed with such models, the robot can simulate what a new timing would convey to a person, given a path, and even optimize its timing to purposefully convey something.

5.1 General Formulation

We start with a general approach, and then fill in the details for confidence, weight, and naturalness.

We will model people’s inference on some hidden robot or task state (e.g., the robot’s confidence) given timing and path as evidence. Thus, we model people as estimating via Bayesian inference from an observation model . If the robot can approximate the person’s , then it knows what conveys about .

To model , we suppose that the person expects the timing to be based on some criterion, with different hidden variables leading to different criteria. We use to represent the criterion for a timing given a value (e.g., a weight or a confidence value) and a path . Given and , the probability of a trajectory timing is


Such a formulation has been used for paths and general actions in an MDP in [8, 1, 11, 12].

In our experiments, the users observe timed trajectories and infer . To get this from our model, we apply Bayes’ rule to compute


Note that given this probability distribution, the robot can also search for a timing for its path that maximizes the probability of a particular

, e.g. .

Model Evaluation and Parameter Selection. Next, we consider what can be for confidence, weight, and naturalness. We will evaluate these models by measuring the correlation between the model prediction, for a given trajectory timing, with the mean subject response for that timing. Our models include some free parameters (e.g., in (1)) that we fit to the data by doing a grid search and selecting the parameters that correlate best with the data.

This means that there are two possible explanations for a high correlation: either the model actually explains people’s inferences, or it is complex enough that it can overfit to any data. Thus, we have a confound. To address it, we run the same procedure on randomly generated synthetic data: if we get a high correlation with random data, then it is likely that our model has overfit. On the other hand, a low correlation with random synthetic data suggests that our model does actually help explain the predictions that users made.

5.2 Confidence


We observed that high speed led to an increase in perceived confidence and pausing led to a decrease in perceived confidence. We propose that a mathematical model for confidence can be the precision (i.e., inverse variance) in the robot’s belief state.

We thus model the observer as assuming, for simplicity, that the robot’s belief state is a Gaussian with initial precision

where high corresponds to high confidence and vice versa.

The robot gets observations at a constant rate over the course of the trajectory. Our observer expects the robot to use a different timing depending on the confidence – intuitively, if it starts with low precision, it needs to get more observations than if it starts with high precision. More concretely, the timing that the observer will expect for an initial precision, , is related to the cost that the observer expects the robot to optimize when timing motion. We propose that should target high final precision , while trading off with being efficient on the task:


If the robot moves faster, it gets fewer observations so its final precision is lower. controls the relative importance of speed versus precision. If each of these observations has Gaussian noise with precision

, then the robot’s belief state updates with a Kalman filter 

[16]. The precision at the end of the trajectory is thus


where is the number of observations the robot gets during the trajectory.

This first model attempt explains the interaction between speed and perceived confidence, but can not explain the interaction with pauses; paused trajectories in our experiment still have the same overall duration, so the effect we found for pausing can not be explained by the current model.

To account for pauses, we further suppose that the quality of each observation depends on the robot’s velocity. If the robot is not moving, then it gets observations with precision . As the robot speeds up, the precision of its observations decreases. This gives us the following formula for :


Recall that is the time the robot reaches configuration and is the corresponding velocity. governs how quickly the observation precision falls off as the robot speeds up.

The inference task is to determine the value of , given a timed trajectory. We consider two possible values for : represents “high confidence” and represents “low confidence.”

Evaluation. We used grid search to fit and . For each parameter we consider 10 values between and , evenly distributed in log space. The best fit parameters were . The corresponding correlation is . The average best-fit correlation with random data was 0.3. Fig.6 (left) plots the confidence model’s output versus the mean student ratings for confidence.

5.3 Weight

Model. Previous work has shown that humans make inferences about the weights of objects based on their motions and that their perceptions can be modelled as Bayesian inference with a simplified physics model [26, 13]. However, this work focused on objects in free fall and collisions, while we are interested in objects being moved by a different entity, namely the robot. We found that human inferences about weight depend primarily on the speed of the object. A higher speed led users to infer that the held object was lighter.

It is tempting to apply a model where trajectories trade off between, e.g., energy for the robot (sum of squared torques on its joints) and the duration of the trajectory. This would lead to the appropriate inference with respect to speed; a higher weight means that the same torque results in a lower speed. However a sum of squared torques cost does not give a good explanation of the impact speed changes or pauses had on the inferred weight. A robot minimizing sum of squared torques will pay a higher penalty to pause with a heavy object, so pausing would change the inferred weight in this model.

As an alternative, we model the robot as attempting to control the overall momentum of the object it is holding. In this model, the robot is not minimizing the effort it expends to move the object, but rather it minimizes the amount of effort it would take to bring the object to a halt. The overall cost function trades off between duration and the sum of momentums across the trajectory:


where is the mass of the object being held, is the duration, and is the velocity of the end effector. In this model, the inferred mass of the object will depend on the average velocity of the object and does not have any dependence on speed changes that occur during the trajectory. This is in contrast to cost functions that minimize the sum of squared torques or the kinetic energy of the object.

The inference task is to determine , given a timed trajectory. We considered two physically plausible values of : kg represents a light mass and kg represents a heavy mass.

Evaluation. We used grid search to fit and . For each parameter we considered 10 values between and , evenly distributed in log space. The best fit parameters were . The corresponding correlation is . The average best-fit correlation with random data was 0.18. Fig.6 (center) plots the confidence model’s output versus the mean student ratings for confidence.

5.4 Naturalness

Model. Our model for naturalness is the simplest of the three. Natural human arm motions can be modeled as minimizing an objective function defined as the magnitude of jerk integrated over the motion [9]. In the case of robot-human handovers (the task used in our experiment), it has been shown that minimum jerk motions lead to faster reaction times from the human [15].

The cost function for naturalness inference is therefore a tradeoff between the duration of the trajectory (as before) and the sum of squared jerks along the trajectory:


where is the naturalness parameter that governs how natural the trajectory should be in comparison to its duration. is the jerk associated with the we can express this in terms of stepwise velocities as


The inference task is to determine given a trajectory timing. We suppose that can take on two possible value and that we fit to the data.

Evaluation. We used grid search to fit . We considered 10 values between and , evenly distributed in log space. During the grid search, we enforced the constraint that . The best fit was . The corresponding correlation was . The average best-fit correlation with random data was . Fig.6 (right) plots the naturalness model’s output versus the mean student ratings for naturalness.

6 Discussion

Summary. We already knew from prior work that timing is important, and expected to see effects on perceptions of non-functional properties of the robot, like disposition and naturalness. More exciting is that we have also found effects on perceptions of functional properties as well, like competence, capability, and carried object weight.

We introduced mathematical models for some of these perceptions, whose predictions strongly correlated with the perceptions of actual users. These contribute to enabling robots to anticipate what their timing will convey, as well as to optimize their timing, given a path, to purposefully convey that they are not confident, that they are handing the person over a heavy object, or to simply produce more natural or predictable motion.

Limitations and Future Work. Despite these promising results supporting the importance of timing and bringing us closer to autonomous expressive timing, we have just scratched the surface of this deep area of research. Timing is complex and multi-faceted, and we have only studied three factors that contribute to timing: speed, changes of speed (in particular ways), and pausing (at particular times).

Our models for weight, confidence, and naturalness help generalize to new timings outside of the conditions in our study. But more investigation is needed to put each model to the test with novel timing situations, new paths, new robots, and new tasks. Further, the fact that the current models correlate with the data we collected does not necessarily imply that they produce useful timings when optimized. Performing the timing generation and adjusting the models accordingly is our main direction of future work.

Finally, for each of our current models we defined a timing cost function based on some physical or informational quantity (e.g., momentum in the weight cost or precision in the confidence cost). Doing the analogous for effects like disposition is a significant future challenge, because such quantities are hard to directly relate to concrete physical properties.

7 Acknowledgments

This work is supported by The Center for Information Technology Research in the Interest of Society, Berkeley DeepDrive, and the Center for Human-Compatible AI.


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