Integrating Human Observer Inferences
into Robot Motion Planning
Anca Dragan and Siddhartha Srinivasa
The Robotics Institute, Carnegie Mellon University
{adragan,siddh}@cs.cmu.edu
Abstract —Our goal is to enable robots to produce mo-
tion that is suitable for human-robot collaboration and co-
existence. Most motion in robotics is purely functional ,
ideal when the robot is performing a task in isolation. In
collaboration, however, the robot’s motion has an observer ,
watching and interpreting the motion.
In this work, we move beyond functional motion, and
introduce the notion of an observer into motion planning,
so that robots can generate motion that is mindful of how it
will be interpreted by a human collaborator.
We formalize predictability and legibility as properties of
motion that naturally arise from the inferences in opposing
directions that the observer makes, drawing on action in-
terpretation theory in psychology. We propose models for
these inferences based on the principle of rational action,
and derive constrained functional trajectory optimization
techniques for planning motion that is predictable or legible.
Finally, we present experiments that test our work on
novice users, and discuss the remaining challenges in en-
abling robots to generate such motion online in complex
situations.1
I. Introduction
In this paper, we explore the problem where a robot
and a human are collaborating side by side to perform
a tightly coupled physical task together, like clearing a
table (a running example in our paper).
The task ampliﬁes the burden on the robot’s motion.
Most motion in robotics is purely functional : industrial
robots move to package parts, vacuuming robots move
to suck dust, and personal robots move to clean up a
dirty table. This type of motion is ideal when the robot
is performing a task in isolation.
Collaboration, however, does not happen in isolation.
In collaboration, the robot’s motion has an observer ,
watching and interpreting the motion.
In this work, we move beyond functional motion, and
introduce the notion of an observer and their inferences
into motion planning, so that robots can generate motion
that is mindful of how it will be interpreted by a human
collaborator.
When we collaborate, we make two inferences
(Fig.1,lower center) about our collaborator [12, 54, 56]:
1) we infer their goal based on their ongoing action
(action-to-goal), and 2) if we know their goal, we infer
their future action from it (goal-to-action). Our work
1This paper combines work from [15] and [18].
Fig. 1. Above: Predictable, day-to-day, expected handwriting vs. leg-
ible handwriting. Upper Center: A predictable and a legible trajectory
of a robot’s hand for the same task of grasping the green object. Lower
Center: Predictability and legibility stem from inferences in opposing
directions. Below: The legibility optimization process for a reaching
task. By exaggerating to the right, the robot is more clear about its
intent to grasp the object on the right .
formalizes the two properties of motion that enable these
two inferences: predictability and legibility .
Legibility stems from the ﬁrst inference, and is about
conveying intent – moving in a manner that makes the
robot’s goal clear to observer. Predictability stems from
the second inference, and is about matching the ob-
server’s expectation – matching the motion they predict
when they know the robot’s goal.
Predictable and legible motion can be correlated. For
example, in an unambiguous situation, where an actor’s
observed motion matches what is expected for a given
intent (i.e. is predictable), then this intent can be used
to explain the motion. If this is the only intent which
explains the motion, the observer can immediately infer
the actor’s intent, meaning that the motion is also legible.
This is why we tend to assume that predictability implies
legibility – that if the robot moves in an expected way,
then its intentions will automatically be clear [2, 6, 35].
The writing domain, however, clearly distinguishes
the two. The word legibility , traditionally an attribute
of written text [52], refers to the quality of being easy
to read. When we write legibly, we try consciously,
and with some effort, to make our writing clear and
readable to someone else. The word predictability , on the
other hand, refers to the quality of matching expectation.
When we write predictably, we fall back to old habits,
and write with minimal effort.
As a consequence, our legible and predictable writings
are different : our friends do not expect to open our
diary and see our legible writing style. They rightfully
assume the diary will be written for us, and expect our
usual, day-to-day style. In this paper, by formalizing
predictability and legibility as directly stemming from
the two inferences in opposing directions, goal-to-action
and action-to-goal, we show that the two are different in
motion as well:
Predictability and legibility are fundamentally dif-
ferent properties of motion, stemming from observer
inferences in opposing directions.
Ambiguous situations, occurring often in daily tasks,
make this opposition clear: more than one possible intent
can be used to explain the motion observed so far,
rendering the predictable motion illegible. Fig.1(center)
exempliﬁes the effect of this contradiction. The robot
hand’s motion on the left is predictable in that it matches
expected behavior. The hand reaches out directly to-
wards the target. But, it is not legible, failing to make
the intent of grasping the green object clear. In contrast,
the trajectory on the right is more legible, making it clear
that the target is the green object by deliberately bending
away from the red object. But it is less predictable, as
it does not match the expected behavior of reaching
directly. We will show in Sections IV and V how we
can quantify this effect with Bayesian inference, which
enables us to derive the online probabilites of the motion
reaching for either object, illustrated as bar graphs in
Fig.1.
Our work makes the following contributions:
1.We formalize legibility and predictability in the con-
text of goal-directed motion in Sec. III as stemming
from inferences in opposing directions. The formalism
emphasizes their difference, and directly relates to the
theory of action interpretation [12] and the concepts of
“action-to-goal” and “goal-to-action” inference.
2.Armed with mathematical deﬁnitions of legibility and
predictability, we propose a way in which a robot could
model these inferences in order to evaluate how legible
or predictable a motion is (Sections IV and V). The
models are based on cost optimization, resonate withthe principle of rational action [11, 23], and echo earlier
works on action understanding via inverse planning [5].
3.We derive methods for generating predictable and
legible motion. Although our model enables us to eval-
uate how predictable or legible a motion trajectory is,
it does not enable us to generate trajectories that are
predictable or legible. Going from evaluation to gen-
eration means going beyond modeling the observer’s
goal inference, to creating motion that results in the
correct goal being inferred, i.e. going from "I can tell
that you believe I am grasping this.", to "I know how
tomake you believe I am grasping this". We do so via
functional gradient optimization in the space of motion
trajectories, echoing earlier works in motion planning
[9, 28, 31, 43, 45, 53, 55, 61], now with optimization
criteria based on the observer’s inferences.
4.We propose a method for optimizing for legibility
while still maintaining predictability. The ability to op-
timize the legibility criterion leads us to a surprising
observation: that there are cases in which the trajectory
becomes too unpredictable . As our user studies show
(Sec. X-B), some unpredictability is often necessary to convey
intent — it is unpredictability beyond a threshold (like
the outermost trajectory in Fig.4) that confuses users
and lowers their conﬁdence in what the robot is doing.
We address this fundamental limitation by prohibiting
the optimizer to “travel to uncharted territory”, i.e. go
outside of the region in which its assumptions have
support — we call this a “trust region” of predictability.
5.We test our work in two experiments with novice
users, one focused on the models, and the other on
motion generation. We demonstrate legibility and pre-
dictability are contradictory not just in theory, but also in
practice, and follow-up with an analysis of the trade-off
between the two. We experiment with three characters
that differ in their complexity and anthropomorphism:
a simulated point robot, the bi-manual mobile manipu-
lator HERB[48], and a human (Sec. X).
Understanding and planning motion that is pre-
dictable or legible can greatly enhance human-robot
collaboration, but many challenges remain in generating
such motion online in complex situations, and in decid-
ing how to trade off between the two properties in each
situation. We discuss these in Sec. XI.
II. R elation to Prior Work
“Predictable” means “expected”, and is usually used
to refer to a desirable property of motion without need-
ing additional clariﬁcation [2, 6]. “Legible”, on the other
hand, is typically reserved for writing, and appears in
the robot motion literature accompanied by an expla-
nation: being legible means that observers are able to
recognize [6] , infer [39], or understand [3, 35] the inten-
tions of the robot, or that robot indicates the goal it will
reach [35] – overall, legibility is about expressing intent.
Building on this, we deﬁne predictability and legibility as
enabling inferences that the collaborator needs to make,
and propose trajectory optimization criteria for the two.
One exception to using legibility to mean expressing
intent is a deﬁnition of legibility as both being intent-
expressive and matching expectation [40]. Our work,
however, shows that matching expectation and express-
ing intent are fundamentally different properties that can
at times contradict.
Existing methods for generating legible motion fall
in three categories. (1) increasing legibility indirectly by
increasing predictability (e.g. via learning from demon-
stration) [6]; (2) increasing legibility indirectly by increas-
ing visibility to the observer [3, 29]; and (3) increasing
legibility directly by encoding animation principles like
anticipation in the motion [24, 49].
In contrast, our work explicitly formalizes intent-
expressiveness for goal-direct motion in the form of a
trajectory optimization criterion: rather than targeting a
related property, the robot directly maximizes the prob-
ability of the correct intent being inferred. The resulting
motion can be interpreted using animation principles,
but we do not encode these explicitly – instead, they
emerge out of the mathematics of legible motion.
III. F ormalizing Legibility
and Predictability
In common use, legible motion is intent-expressive,
and predictable motion matches what is expected. Here,
we formalize these deﬁnitions for the context of goal-
directed motion, where a human or robot is executing
a trajectory x:[0, 1]!Q , lying in a Hilbert space of
trajectories X.xstarts at a conﬁguration Sand ends at
a goal Gfrom a set of possible goals G, like in Fig.1. In
this context, Gis central to both properties:
Deﬁnition 1: Legible motion is motion that enables an
observer to quickly and conﬁdently infer the correct goal G.
Deﬁnition 2: Predictable motion is motion that matches
what an observer would expect, given the goal G.
A. Formalism
1) Legibility: Imagine an observer watching the orange
trajectory from Fig.1. As the robot’s hand departs the
starting conﬁguration and moves along the trajectory, the
observer is running an inference, predicting which of the
two goals it is reaching for. We denote this inference
function that maps (snippets of) trajectories from all
trajectories Xto goals as
IL:X!G
The bar graphs next to the hands in Fig.1 signify the
observer’s predictions of the two likely goals. At the very
beginning, the trajectory is confusing and the observerhas little conﬁdence in the inference — in what follows,
we model this conﬁdence based on the probability that
the observer assigns to the inferred goal. However, the
observer becomes conﬁdent very quickly – even from the
second conﬁguration of the hand along the trajectory,
it becomes clear that the green object is the target.
This quick and conﬁdent inference is the hallmark of
legibility.
We thus formalize legible motion as motion that en-
ables an observer to conﬁdently infer the correct goal
conﬁguration Gafter observing only a snippet of the
trajectory, xS!Q, from the start Sto the conﬁguration at
a time t,Q=x(t):
IL(xS!Q) =G
The quicker this happens (i.e. the smaller tis), the more
legible the trajectory is.
The deﬁnition in terms of ILis an interpretation
in the motion domain for terms like “readable” [49],
or “understandable”[3], and encourages “anticipatory”
motion [24] because it brings the relevant information for
goal prediction towards the beginning of the trajectory,
thus lowering t. The formalism can also generalize to
outcome-directed motion (e.g. gestures such as pointing
at, waving at, etc.) by replacing the notion of goal with
that of an outcome – here, legible motion becomes
motion that enables quick and conﬁdent inference of
the desired outcome. Our recent work illustrated this for
pointing gestures [20].
2) Predictability: Now imagine someone knowing that
the hand is reaching towards the green goal. Even before
the robot has moved, the observer creates an expectation,
making an inference on how the hand will move – for ex-
ample, that the hand will start turning towards the green
object as it is moving directly towards it. We denote this
inference function mapping goals to trajectories as
IP:G!X
We formalize predictable motion as motion for which
the trajectory xS!Gmatches this inference:
IP(G) =xS!G
The better the actual trajectory matches the inference,
measurable for example using a distance metric between
IP(G)and xS!G, the more predictable the trajectory is.
B. Connection to Action Interpretation in Psychology
A growing amount of research in psychology sug-
gests that humans interpret observed behaviors as goal-
directed actions [10, 12, 26, 42, 47, 58, 59], a result
supported by studies observing infants and how they
show surprise when exposed to inexplicable action-goal
pairings. Csibra and Gergeley [12] summarize two types
Human Inference Type Example Analogy in Motion Property of Motion
action7!goal ... pour beans in grinder 7!coffee xS!Q7!G legibility
goal7!action coffee 7!... pour beans in grinder ... G7!xS!G predictability
TABLE I
Legibility and predictability as enabling inferences from action interpretation in opposing direction .
(G)=argminξ∈ΞS→GC[ξ](ξS→Q)=argmaxG∈GP(G|ξS→Q)(G)ξS:QG1G2G
ξ*S→G=argminξ∈ΞS→GC(ξ)G*=argmaxG∈GP(G|ξS→Q)ξ*S→GξS→QG1G2
G
Fig. 2. In our models, the observer expects the robot’s motion to optimizea cost functionC(left), and uses that expectation to identity which goal ismost probable given the robot’s motion so far (right)B. Connection to PsychologyA growing amount of research in psychology suggests thathumans interpret observed behaviors as goal-directed actions[11], [15]–[19], a result stemming from studies observing in-fants and how they show surprise when exposed to inexplicableaction-goal pairings. [11] ] summarize two types of inferencestemming from the interpretation of actions as goal directed:“action-to-goal” and “goal-to-action”.“Action-to-goal” refers to an observer’s ability to infersomeone’s goal state from their ongoing actions (e.g. becausethey are pouring coffee beans into the grinder, the willeventually hold a cup of coffee). “Action-to-goal” inferenceanswers the question “What is the function of this action?”.“Goal-to-action” refers to an observer’s ability to predictthe actions that someone will take based on their goal (e.g.because they want to make coffee, they will will pour coffeebeans into the grinder). “Goal-to-action” inference answers thequestion “What action would achieve this goal?”.This has a natural connection to our formalism. In goal-directed motion, actions are trajectories and goals are goalconﬁgurations. Thus the inference occurring in legibility,from trajectory to goal,⇠S!Q7!G, relates naturally to“action-to-goal” inference. Likewise, the inference occurringin predictability, from goal to trajectory,G7!⇠S!G, relatesnaturally to “goal-to-action”.C. SummaryOur formalism emphasizes the difference between legibilityand predictability in theory: they stem from inferences inopposing directions(from trajectories to goals vs. from goalsto trajectories), with strong parallels in the theory of actioninterpretation. In what follows, we introduce one way for arobot to model these two inferences (summarize in Fig.2), andpresent an experiment that emphasizes the difference betweenthe two properties in practice.III. MODELINGPREDICTABLEMOTIONA. The Trajectory InferenceIPTo modelIPis to model the observer’s expectation. Oneway the robot could do so is by assuming that the humanobserver expects it to be a rational agent acting efﬁciently[11] or justiﬁably [13] to achieve a goal. This is known asthe principle of rational action [12], [13], and it has beenshown to apply to non-human agents, including robots [20].The robot could model this notion of “efﬁciency” via a costfunction deﬁning what it means to be efﬁcient. For example,if the observer expected the robot’s hand to move directlytowards the object it wants to grasp (as opposed to takingan unnecessarily long path to it), then “efﬁciency” would bedeﬁned by the cost function penalizing the trajectory’s length.Throughout this paper, we will refer to the cost functionmodeling the observer’s expectation asC:C:⌅!R+with lower costs signifying more “efﬁcient” trajectories.The most predictable trajectory is then the most “efﬁcient”:IP(G) = arg min⇠2⌅S!GC(⇠)(1)Crepresents what the observer expects the robot to opti-mize, and therefore encompasses every aspect of the observer’sexpectation, including (when available) body motion, handmotion, arm motion, and gaze.B. Evaluating and Generating PredictabilityPredictability can be evaluated based onC: the lowerthe cost, the more predictable (expected) the trajectory. Wepropose a predictabilityscorenormalized from 0 to 1:predictability(⇠)=expC(⇠)(2)Generating predictable motion means maximizing thisscore, or equivalently minimizing the cost functionC– asin (1). This presents two major challenges: learningC, andminimizingC.First, the robot needs access to the cost functionCthatcaptures how the human observer expects it to move. If thehuman observer expects human-like motion, animation (e.g.[21]) or biomechanics (e.g. [22], [23]) literature can serve toprovide approximations forC. Our experiment (Section V)uses trajectory length as a proxy for the realC, resulting inthe shortest path to goal – but this is merely one aspect ofexpected behavior. As our experiment will reveal, efﬁciencyof robot motion has different meanings for different observers.If the observer were willing to provide examples of what theyexpect, the robot could learn how to act via Learning fromDemonstration [24]–[26] or Inverse Reinforcement Learning[27]–[29]. Doing so in a high-dimensional space, however, isstill an active area of research.Second, the robot must ﬁnd a trajectory that minimizesC.This is tractable in low-dimensional spaces, or ifCis convex.While efﬁcient trajectory optimization techniques do exist forhigh-dimensional spaces and non-convex costs [30], they aresubject to local minima, and how to alleviate this issue inpractice remains an open research question [31], [32].away from the red object. But it is less predictable, as it doesnot match the expected behavior of reaching directly. We willshow in Sections III and IV how we can quantify this effectwith Bayesian inference, which allows us to derive, amongother things, the online probabilites of the motion reachingfor either object, illustrated as bar graphs in Fig.1.Our work makes the following three contributions:1.We formalize legibility and predictability in the contextof goal-directed motion in Section II as stemming frominferences inopposingdirections. The formalism emphasizestheir difference, and directly relates to the theory of actioninterpretation [11] and the concepts of “action-to-goal” and“goal-to-action” inference. Our formalism also uniﬁes pre-vious descriptions of legibility, quantifying readability andunderstandability, and encouraging anticipation as a directconsequence of our deﬁnitions.2.Armed with mathematical deﬁnitions of legibility andpredictability, we propose a way in which a robot could modelthese inferences in order to evaluate and generate motion thatis legible or predictable (Sections III and IV). The models arebased on cost optimization, and resonate with the principle ofrational action [12], [13].3.We demonstrate that legibility and predictability are contra-dictory not just in theory, but also in practice. We presentan extensive experiment for three characters that differ intheir complexity and anthropomorphism: a simulated pointrobot, the bi-manual mobile manipulator HERB [14], and ahuman (Section V). The experiment conﬁrms the contradictionbetween predictable and legible motion, and reveals interestingchallenges (Section VI). We found, for instance, that differentpeople expect a complex robot like HERB to act in differentways: for a robot to be predictable, it must adapt to theparticulars of the observer.The difference between legibility and predictability of mo-tion is crucial for human-robot interaction, in particular forcollaboration between humans and robots. Collaboration is adelicate dance of prediction and action, where agents mustpredict their collaborator’s intentions as well as make theirown intentions clear – they must act legibly. We are excitedto be taking an essential step towards better human-robotcollaboration: by emphasizing the difference between legibilityand predictability, we advocate for a different approach tomotion planning, in which robots decide between optimizingfor legibility and optimizing for predictability, depending onthe context they are in.II. FORMALIZINGLEGIBILITYANDPREDICTABILITYSo far, we have identiﬁed that legible motion is intent-expressive, and predictable motion matches what is expected.Here, we formalize these deﬁnitions for the context of goal-directed motion, where a human or robot is executing atrajectory towards one goalGfrom a set of possible goalsG,like in Fig.1. In this context,Gis central to both properties:the intent is reaching the goalG, and what is expected dependsonG:Deﬁnition 2.1:Legible motion is motion that enables anobserver to quickly and conﬁdently infer the goal.Deﬁnition 2.2:Predictable motion is motion that matcheswhat an observer would expect, given the goal.A. Formalism1) Legibility:Imagine someone observing the orange tra-jectory from Fig.1. As the robot’s hand departs the startingconﬁguration and moves along the trajectory, the observer isrunning an inference, predicting which of the two goals itis reaching for. We denote this inference function that maps(snippets of) trajectories from all trajectories⌅to goals asIL:⌅!GThe bar graphs next to the hands in Fig.1 signify the observer’spredictions of the two likely goals. At the very beginning, thetrajectory is confusing and the observer has littleconﬁdenceinthe inference. However, the observer becomes conﬁdent veryquickly– even from the second conﬁguration of the hand alongthe trajectory, it becomes clear that the green object is thetarget. This quick and conﬁdent inference is the hallmark oflegibility.We thus formalizelegiblemotion as motion that enables anobserver toconﬁdentlyinfer thecorrectgoal conﬁgurationGafter observing only a snippet of the trajectory,⇠S!Q, fromthe startSto the conﬁguration at a timet,Q=⇠(t):IL(⇠S!Q)=GThequickerthis happens (i.e. the smallertis), the morelegible the trajectory is.This formalizes terms like “readable” [4], or “understand-able” [6], and encourages “anticipatory” motion [5] because itbrings the relevant information for goal prediction towards thebeginning of the trajectory, thus loweringt. The formalism canalso generalize to outcome-directed motion (e.g. gestures suchas pointing at, waving at, etc.) by replacing the notion of goalwith that of an outcome – here, legible motion becomes motionthat enables quick and conﬁdent inference of the desiredoutcome.2) Predictability:Now imagine someone knowing that thehand is reaching towards the green goal. Even before the robotstarts moving, the observer creates an expectation, making aninference on how the hand will move – for example, that thehand will start turning towards the green object as it is movingdirectly towards it. We denote this inference function mappinggoals to trajectories asIP:G!⌅We formalizepredictablemotion as motion for which thetrajectory⇠S!Gmatches this inference:IP(G)=⇠S!GThe more the trajectory matches the inference, measurablefor example using a distance metric betweenIP(G)and⇠S!G, the more predictable the trajectory is.
ξ*S→G=argminξ∈ΞS→GC(ξ)G*=argmaxG∈GP(G|ξS→Q)ξ*S→GξS→QG1G2
G
Fig. 2. In our models, the observer expects the robot’s motion to optimizea cost functionC(left), and uses that expectation to identity which goal ismost probable given the robot’s motion so far (right)B. Connection to PsychologyA growing amount of research in psychology suggests thathumans interpret observed behaviors as goal-directed actions[11], [15]–[19], a result stemming from studies observing in-fants and how they show surprise when exposed to inexplicableaction-goal pairings. [11] ] summarize two types of inferencestemming from the interpretation of actions as goal directed:“action-to-goal” and “goal-to-action”.“Action-to-goal” refers to an observer’s ability to infersomeone’s goal state from their ongoing actions (e.g. becausethey are pouring coffee beans into the grinder, the willeventually hold a cup of coffee). “Action-to-goal” inferenceanswers the question “What is the function of this action?”.“Goal-to-action” refers to an observer’s ability to predictthe actions that someone will take based on their goal (e.g.because they want to make coffee, they will will pour coffeebeans into the grinder). “Goal-to-action” inference answers thequestion “What action would achieve this goal?”.This has a natural connection to our formalism. In goal-directed motion, actions are trajectories and goals are goalconﬁgurations. Thus the inference occurring in legibility,from trajectory to goal,⇠S!Q7!G, relates naturally to“action-to-goal” inference. Likewise, the inference occurringin predictability, from goal to trajectory,G7!⇠S!G, relatesnaturally to “goal-to-action”.C. SummaryOur formalism emphasizes the difference between legibilityand predictability in theory: they stem from inferences inopposing directions(from trajectories to goals vs. from goalsto trajectories), with strong parallels in the theory of actioninterpretation. In what follows, we introduce one way for arobot to model these two inferences (summarize in Fig.2), andpresent an experiment that emphasizes the difference betweenthe two properties in practice.III. MODELINGPREDICTABLEMOTIONA. The Trajectory InferenceIPTo modelIPis to model the observer’s expectation. Oneway the robot could do so is by assuming that the humanobserver expects it to be a rational agent acting efﬁciently[11] or justiﬁably [13] to achieve a goal. This is known asthe principle of rational action [12], [13], and it has beenshown to apply to non-human agents, including robots [20].The robot could model this notion of “efﬁciency” via a costfunction deﬁning what it means to be efﬁcient. For example,if the observer expected the robot’s hand to move directlytowards the object it wants to grasp (as opposed to takingan unnecessarily long path to it), then “efﬁciency” would bedeﬁned by the cost function penalizing the trajectory’s length.Throughout this paper, we will refer to the cost functionmodeling the observer’s expectation asC:C:⌅!R+with lower costs signifying more “efﬁcient” trajectories.The most predictable trajectory is then the most “efﬁcient”:IP(G) = arg min⇠2⌅S!GC(⇠)(1)Crepresents what the observer expects the robot to opti-mize, and therefore encompasses every aspect of the observer’sexpectation, including (when available) body motion, handmotion, arm motion, and gaze.B. Evaluating and Generating PredictabilityPredictability can be evaluated based onC: the lowerthe cost, the more predictable (expected) the trajectory. Wepropose a predictabilityscorenormalized from 0 to 1:predictability(⇠)=expC(⇠)(2)Generating predictable motion means maximizing thisscore, or equivalently minimizing the cost functionC– asin (1). This presents two major challenges: learningC, andminimizingC.First, the robot needs access to the cost functionCthatcaptures how the human observer expects it to move. If thehuman observer expects human-like motion, animation (e.g.[21]) or biomechanics (e.g. [22], [23]) literature can serve toprovide approximations forC. Our experiment (Section V)uses trajectory length as a proxy for the realC, resulting inthe shortest path to goal – but this is merely one aspect ofexpected behavior. As our experiment will reveal, efﬁciencyof robot motion has different meanings for different observers.If the observer were willing to provide examples of what theyexpect, the robot could learn how to act via Learning fromDemonstration [24]–[26] or Inverse Reinforcement Learning[27]–[29]. Doing so in a high-dimensional space, however, isstill an active area of research.Second, the robot must ﬁnd a trajectory that minimizesC.This is tractable in low-dimensional spaces, or ifCis convex.While efﬁcient trajectory optimization techniques do exist forhigh-dimensional spaces and non-convex costs [30], they aresubject to local minima, and how to alleviate this issue inpractice remains an open research question [31], [32].
Fig. 2. Our models for IPandIL: the observer expects the robot’s motion to optimize a cost function C(IP, left), and identiﬁes based on C
which goal is most probable given the robot’s motion so far ( IL, right).
of inference stemming from the interpretation of actions
as goal directed: “action-to-goal" and “goal-to-action".
“Action-to-goal” refers to an observer’s ability to infer
someone’s goal state from their ongoing actions (e.g.
because they are pouring coffee beans into the grinder,
the will eventually hold a cup of coffee). “Action-to-goal”
inference answers the question “What is the function of
this action?”.
“Goal-to-action” refers to an observer’s ability to pre-
dict the actions that someone will take based on their
goal (e.g. because they want to make coffee, they will
will pour coffee beans into the grinder). “Goal-to-action”
inference answers the question “What action would
achieve this goal?”.
This has a natural connection to our formalism. In
goal-directed motion, actions are trajectories and goals
are goal conﬁgurations. Thus the inference occurring
in legibility, from trajectory to goal, xS!Q7!G, re-
lates naturally to “action-to-goal” inference. Likewise,
the inference occurring in predictability, from goal to
trajectory, G7!xS!G, relates naturally to “goal-to-
action”.
C. Summary
Our formalism emphasizes the difference between
legibility and predictability in theory: they stem from
inferences in opposing directions (from trajectories to goals
vs. from goals to trajectories), with strong parallels in the
theory of action interpretation. Table I shows a summary.
In what follows, we introduce one way for a robot to
model these two inferences (summarized in Fig.2), de-
rive motion generating algorithms based on this model,
and present experiments that emphasize the difference
between the two properties in practice.IV . P redictable Motion
A. Modeling the Trajectory Inference IP
To modelIPis to model the observer’s expectation.
One way the robot could do so is by assuming that
the human observer expects it to be a rational agent
acting efﬁciently[12] or justiﬁably[11] to achieve a goal.
This is known as the principle of rational action[11, 23],
and it has been shown to apply to non-human agents,
including robots[33]. The robot can model this notion of
“efﬁciency” via a cost function deﬁning what it means
to be efﬁcient. For example, if the observer expected the
robot’s hand to move directly towards the object it wants
to grasp (as opposed to taking an unnecessarily long
path to it), then “efﬁciency” would be deﬁned by the
cost function penalizing the trajectory’s length.
Throughout this paper, we will refer to the cost function
modeling the observer’s expectation as C :
C:X!R+
with lower costs signifying more “efﬁcient” trajectories.
The principle of rational action suggests that the most
predictable trajectory is the most “efﬁcient”, for some
deﬁnition of efﬁciency C:
IP(G) =arg min
x2XS!GC[x] (1)
Crepresents what the observer expects the robot to
optimize, and therefore encompasses every aspect of the
observer’s expectation, including (when available) body
motion, hand motion, arm motion, and gaze.
B. Predictability Score
Predictability can be evaluated based on C: the lower
the cost, the more predictable (expected) the trajectory.
We propose a predictability score normalized from 0 to
1, where trajectories with lower cost are exponentially
more predictable (following the principle of maximum
entropy, which we detail in Sec. V-A):
Predictability (x) =expC[x]
(2)
C. Generating Predictable Motion
Generating predictable motion means maximizing the
predictability score, or equivalently minimizing the cost
function C– as in (1).
One way to do so is via functional gradient descent.
We start from an initial trajectory x0and iteratively
improve its score. At every iteration i, we maximize the
regularized ﬁrst order Taylor series approximation of C
about the current trajectory xi:
xi+1=arg min
x2XC[xi] +rCT
xi(xxi) +h
2jjxxijj2
M(3)
withh
2jjxxijj2
Ma regularizer restricting the norm of
the displacement xxiw.r.t. an M, as in [45].
By taking the functional gradient of (12) and setting it
to 0, we obtain the following update rule for xi+1:
xi+1=xi1
hM1¯rC (4)
Recent work has successfully applied this method to
motion planning, using a Cthat trades off between an
efﬁciency and an obstacle avoidance component [61]. The
non-convexity of this type of Cmakes trajectory opti-
mization prone to convergence to high-cost local minima,
which can be mediated by learning from experience
[13, 14].
A key challenge, which we discuss in Sec. XI, is ﬁnding
theCthat each observer expects the robot to minimize.
V . Legible Motion
A. Modeling the Goal Inference IL
To modelILis to model how the observer infers the
goal from a snippet of the trajectory xS!Q. One way to
do so is by assuming that the observer compares the
possible goals in the scene in terms of how probable each
is given xS!Q. This is supported by action interpretation:
Csibra and Gergeley [12] argue, based on the principle
of rational action, that humans assess which end state
would be most efﬁciently brought about by the observed
ongoing action.Taking trajectory length again as an ex-
ample for the observer’s expectation, this translates to
predicting a goal because xS!Qmoves directly toward
it and away from the other goals, making them less
probable.
One model forILis to compute the probability for
each goal candidate Gand to choose the most likely:
IL(xS!Q) =arg max
G2GP(GjxS!Q) (5)To compute this probability, we start with Bayes’ Rule:
P(GjxS!Q)µP(xS!QjG)P(G) (6)
where P(G)is a prior on the goals which can be uniform
in the absence of prior knowledge, and P(xS!QjG)is the
probability of seeing xS!Qwhen the robot targets goal
G. The is in line with the notion of action understanding
as inverse planning proposed by Baker et al.[5], here
P(xS!QjG)relating to the forward planning problem of
ﬁnding a trajectory given a goal.
We compute P(xS!QjG)as the ratio of all trajectories
from StoGthat pass through xS!Qtoalltrajectories
from StoG(Fig.3):
P(xS!QjG) =R
xQ!GP(xS!Q!G)
R
xS!GP(xS!G)(7)
Following [60], we assume trajectories are separable, i.e.
P(xX!Y!Z) =P(xX!Y)P(xY!Z), giving us:
P(xS!QjG) =P(xS!Q)R
xQ!GP(xQ!G)
R
xS!GP(xS!G)(8)
At this point, the robot needs a model of how probable
a trajectory xis in the eye of an observer. The observer
expects the trajectory of minimum cost under C. It is
unlikely, however, that they would be completely sur-
prised (i.e. assign 0 probability) by all other trajectories,
especially by one ever so slightly different. One way to
model this is to make suboptimality w.r.t. Cstill possible,
but exponentially less probable, i.e. P(x)µexpC[x]
,
adopting the principle of maximum entropy [60]. With
this, (8) becomes:
P(xS!QjG)µexpC[xS!Q]R
xQ!GexpC[xQ!G]
R
xS!GexpC[xS!G]
(9)
Computing the integrals is still challenging. In [19],
we derived a solution by approximating the probabil-
ities using Laplace’s method (also proposed indepen-
dently in [38]). If we approximate Cas a quadratic, its
Hessian is constant and according to Lapace’s method,R
xX!YexpC[xX!Y]kexpC[x
X!Y]
(with ka con-
stant and x
X!Ythe optimal trajectory from XtoYw.r.t.
C). Plugging this into (9) and using (6) we get:
P(GjxS!Q) =1
ZexpC[xS!Q]VG(Q)
expVG(S) P(GR)(10)
with Za normalizer across Gand VG(q) =
min x2Xq!GC[x]
Much like teleological reasoning suggests [12], this
evaluates how efﬁcient (w.r.t. C) going to a goal is
through the observed trajectory snippet xS!Qrelative
to the most efﬁcient (optimal) trajectory, x
S!G. In am-
biguous situations like the one in Fig.1, a large portion
SGQ
Fig. 3. xS!Qin black, examples of xQ!Gin blue, and further examples
ofxS!Gin purple. Trajectories more costly w.r.t. Care less probable.
SGR

GO

0!10!100!1000!10000!100000!
Fig. 4. The trajectory optimization process for legibility: the ﬁgure
shows the trajectories across iterations for a situation with a start and
two candidate goals.
ofx
S!Gis also optimal (or near-optimal) for a different
goal, making both goals almost equally likely along it.
This is why legibility does not also optimize C — rather than
matching expectation, it manipulates it to convey intent.
B. Legibility Score
A legible trajectory is one that enables quick and
conﬁdent predictions. A score for legibility therefore
tracks the probability assigned to the actual goal G
across the trajectory: trajectories are more legible if this
probability is higher, with more weight being given to
the earlier parts of the trajectory via a function f(t)(e.g.
f(t)=T-t, with T the duration of the trajectory):
legibility (x) =R
P(GjxS!x(t))f(t)dtR
f(t)dt(11)
with P(GjxS!x(t))computed using C, as in (10).
C. Generating Legible Motion
In order to maximize the L egibility functional, we
start from an initial trajectory x0and iteratively improveits score via functional gradient ascent (Fig.4). This pro-
cess is analogous to generating predictable motion, now
with a different optimization criterion:
xi+1=arg max
xLegibility [xi] +h¯rLegibility ,(xxi)i
h
2jjxxijj2
M (12)
By taking the functional gradient of (12) and setting it
to 0, we obtain the following update rule for xi+1:
xi+1=xi+1
hM1¯rLegibility (13)
To ﬁnd ¯rLegibility , letP(x(t),t) =
P(GRjxS!x(t))f(t)and K=1R
f(t)dt. The legibility
score is then
Legibility [x] =KZ
P(x(t),t)dt (14)
and
¯rLegibility =K¶P
¶xd
dt¶P
¶x0
(15)
Pis not a function of x0, thusd
dtdP
dx0=0.
dP
dx(x(t),t) =g0hh0g
h2P(GR)f(t) (16)
with g= exp
VGR(S)VGR(Q)
and h=
åGexp
VG(S)VG(Q)
, which after a few
simpliﬁcations becomes
¶P
¶x(x(t),t) =exp
VGR(S)VGR(x(t))

åGexp
VG(S)VG(x(t))2
å
G
expVG(x(t))
expVG(S)(V0
G(x(t))V0
GR(x(t)))!
P(GR)f(t)
(17)
Finally,
¯rLegibility (t) =K¶P
¶x(x(t),t) (18)
with¶P
¶x(x(t),t)from (17).
VI. E xample
Fig.4 portrays the functional optimization process for
legibility for a point robot moving from a start location
to one of two candidate goals.
A. Parameters
We detail our choice of parameters for creating this
example below.
oEfﬁciency cost C.In this example, we use sum squared
velocities as the cost functional Ccapturing the user’s
expectation:
C[x] =1
2Z
x0(t)2dt (19)
This cost, frequently used to encourage trajectory
smoothness [45], produces trajectories that reach directly
toward the goal — this is the predictable trajectory,
shown at iteration 0. Our experiments below show that
this trajectory geometry is accurate for 2D robots like
in Fig.4. This Calso allows for an analytical VGand its
gradient.
oTrajectory initialization. We set x0=arg min xC[x]: we
initialize with the most predictable trajectory, treating C
as a prior.
oTrajectory parametrization. We parametrize the trajec-
tory as a vector of waypoint conﬁgurations.
oNorms w.r.t. M.We use the Hessian of CforM. As
a result, the update rule in (4) and (13) propagates local
gradient changes linearly to the rest of the trajectory.
B. Interpretation
The predictable trajectory (iteration 0) is efﬁcient for
achieving the goal, but it is not the best at helping an
observer distinguish which goal the robot is targeting
from the beginning. By exaggerating the motion to the
right, the robot is increasing the probability of the goal
on the right relative to the one on the left.
Exaggeration is one of the 12 principles of animation
[51]. However, nowhere did we inform the robot of what
exaggeration is and how it might be useful for legibility.
The behavior elegantly emerged out of the optimization.
VII. T heUnpredictability of Legibility
The example leads to a surprising observation: in
some cases, by optimizing the Legibility functional, one
can become arbitrarily unpredictable .
Proof: Our gradient derivation in (17) enables us to
construct cases in which this occurs. In a two-goal case
like in Fig.4, with our example Cfrom (19), the gradient
for each trajectory conﬁguration points in the direction
GRGOand has positive magnitude everywhere but at
¥, where C[x] =¥. Fig.5 (red) plots Cacross iterations.
The reason for this peculiarity is that the model for
how observers make inferences in (5) fails to capture how
humans make inferences in highly unpredictable situations .
In reality, observers might get confused by the robot’s
behavior and stop reasoning about the robot’s possible
SGR

GO

!=160"!=80"!=40"!=20"!=10"
02004006008001000050100Iteration NumberC! Fig. 5. The predictable trajectory in gray, and the legible trajectories for
different trust region sizes in orange. On the right, the cost Cover the
iterations in the unconstrained case (red) and constrained case (green).
goals the way the model assumes they would — com-
paring the sub-optimality of its actions with respect to
each of them. Instead, they might start believing that the
robot is malfunctioning [46] or that it is not pursuing any
of the goals and doing something else entirely — this is
supported by our user study in Sec. X-B, which shows
that this belief signiﬁcantly increases at higher Ccosts.
This complexity of action interpretation in humans,
which is difﬁcult to capture in a goal prediction model,
can signiﬁcantly affect the legibility of the generated
trajectories in practice. Optimizing the legibility score
outside of a certain threshold for predictability can actu-
ally lower the legibility of the motion as measured with
real users (as it does in our study in Sec. X-B). Unpre-
dictability above a certain level can also be detrimental
to the collaboration process in general [2, 27, 41].
We propose to address these issues by only allowing
optimization of legibility where the model holds, i.e.
where predictability is sufﬁciently high. We call this
a “trust region” of predictability — a constraint that
bounds the domain of trajectories, but that does so w.r.t.
the cost functional C, resulting in C[x]b:
The legibility model can only be trusted inside this
trust region.
The parameter b, as our study will show, is identiﬁable
by its effect on legibility as measured with users —
the point at which further optimization of the legibility
functional makes the trajectory less legible in practice.
VIII. C onstrained Legibility Optimization
In order to prevent the legibility optimization from
producing motion that is too unpredictable, we deﬁne a
trust region of predictability, constraining the trajectory
to stay below a maximum cost in Cduring the optimiza-
tion in (12):
xi+1=arg max
xLegibility [xi] +¯rLegibilityT(xxi)
h
2jjxxijj2
M
s.t.C[x]b (20)
To solve this, we linearize the constraint, which now
becomes ¯rCT(xxi) +C[xi]b. The Lagrangian is
L[x,l] =Legibility [xi] +¯rLegibilityT(xxi)(21)
h
2jjxxijj2
M+l(b¯rCT(xxi)C[xi])
with the following KKT conditions:
¯rLegibilityhM(xxi)¯rCl=0 (22)
l(b¯rCT(xxi)C[xi]) = 0 (23)
l0 (24)
C[x]b (25)
Inactive constraint: l=0 and
xi+1=xi+1
hM1¯rLegibility (26)
Active constraint: The constraint becomes an equality
constraint on the trajectory. The derivation for xi+1is
analogous to [17], using the L egibility functional as
opposed to the classical cost used by the CHOMP motion
planer[45]. From (22)
xi+1=xi+1
hM1(¯rLegibilityl¯rC)|{z}
¯r(LegibilitylC)(27)
Note that this is the functional gradient of L egibility
with an additional (linear) regularizer lC penalizing un-
predictability . Substituting in (23) to get the value for l
and using (22) again, we obtain a new update rule:
xi+1=xi+1
hM1¯rLegibility
1
hM1¯rC(¯rCTM1¯rC)1¯rCTM1¯rLegibility
| {z }
projection on ¯rCT(xxi) =0
M1¯rC(¯rCTM1¯rC)1(C[xi]b)| {z }
offset correction to ¯rCT(xxi) +C[xi] =b(28)
Fig.5 shows the outcome of the optimization for vari-
ousbvalues. In our second experiment below, we ana-
lyze what effect bhas on the legibility of the trajectory
in practice, as measured through users observing the
robot’s motion.
(a) Multiple goals
 (b) Initialization
Fig. 8. (a) Legible trajectories for multiple goals. (b) Legibility is
dependent on initialization.
IX. U nderstanding Legible Trajectories
Armed with a legible motion generator, we investigate
legibility further, looking at factors that affect the ﬁnal
trajectories.
Ambiguity. Certain scenes are more ambiguous than
others, in that the legibility of the predictable trajectory
is lower. The more ambiguous a scene is, the greater
the need to depart from predictability and exaggerate
the motion. Fig.6(a) compares two scenes, the one on
the right being more ambiguous by having the candi-
date goals closer and thus making it more difﬁcult to
distinguish between them. This ambiguity is reﬂected
in its equivalent legible trajectory (both trajectories are
obtained after 1000 iterations). The ﬁgure uses the same
cost Cfrom Sec. VI.
Scale. The scale does affect legibility when the value
functions VGare affected by scale, as in our running
example. Here, reaching somewhere closer raises the de-
mand on legibility (Fig.6(b)). Intuitively, the robot could
still reach for GOand suffer little penalty compared to
a larger scale, which puts an extra burden on its motion
if it wants to institute the same conﬁdence in its intent.
Weighting in Time. The weighting function f(11) qual-
itatively affects the shape of the trajectory by placing the
emphasis (or exaggeration) earlier or later (Fig.6(c)).
Multiple Goals. Although for simplicity, our examples
so far were focused on discriminating between two goals,
legibility does apply in the context of multiple goals
(Fig.8(a)). Notice that for the goal in the middle, the
most legible trajectory coincides with the predictable
one: any exaggeration would lead an observer to predict
a different goal — legibility is limited by the complexity in
the scene .
Obstacle Avoidance. In the presence of obstacles in the
scene, a user would expect the robot to stay clear of
these obstacles, which makes Cmore complex. We plot
in Fig.7 an example using the cost functional from the
CHOMP motion planner[45], which trades off between
the sum-squared velocity cost we have been using thus
far, and a cost penalizing the robot from coming too
SSGR

GO

GR

GO

(a) Ambiguity
SGR

GO

SGO

GR

 (b) Scale
SGR

GO

f1f2 (c)f
Fig. 6. The effects of ambiguity, scale, and the weighting function fon legibility.
Fig. 7. Legibility given a Cthat accounts for obstacle avoidance. The gray trajectory is the predictable trajectory (minimizing C), and the orange
trajectories are obtained via legibility optimization for 10, 102, 103, 104, and 105iterations. Legibility purposefully pushes the trajectory closer
to the obstacle than expected in order to express the intent of reaching the goal on the right.
close to obstacles. Legibility in this case will move the
predictable trajectory much closer to the obstacle in
order to disambiguate between the two goals.
Local optima. There is no guarantee that L egibility
is concave. This is clear for the case of a non-convex
C, where we often see different initializations lead to
different local maxima, as in Fig.8(b).
In fact, even for quadratic VGs,P(GRjxS!Q)is – aside
from scalar variations – a ratio of sums of Gaussian
functions of the form expVG(x(t))
. Convergence to
local optima is thus possible even in this simple case.
As a side-effect, it is also possible that initializing the
optimizer with the most predictable trajectory leads to
convergence to a local maxima.
X. From Theory to Practice
Predictability and legibility are intrinsically properties
that depend on the observer: a real user. Here, we go
from the theory of the two properties to what happens
in practice, when novice users observe motion.
We present two studies. The ﬁrst tests our models for
predictability and legibility: it tests whether a trajectory
that is more legible but less predictable according to our
theoretical scores is also more legible but less predictable
to an observer.
The second study tests the motion generation: it tests
the notion of optimizing for legibility within a trustregion of predictability.
A. The Contradiction Between Predictability and Legibility
The mathematics of predictability and legibility imply
that being more legible can mean being less predictable
and vice-versa. We set out to verify that this is also true
in practice, when we expose subjects to robot motion.
We ran an experiment in which we evaluated two tra-
jectories – a theoretically more predictable one xPand
a theoretically more legible one xL– in terms of how
predictable and legible they are to novices.
Hypothesis. There exist two trajectories xLandxPfor the
same task such that xPis more predictable than xLandxLis
more legible than xP.
Task. We chose a task like the one in Fig.1: reaching
for one of two objects present in the scene. The objects
were close together in order to make this an ambiguous
task, in which we expect a larger difference between
predictable and legible motion.
Manipulated Variables.
oCharacter: We chose to use three characters for this
task – a simulated point robot, a bi-manual mobile
manipulator named HERB [48], and a human – because
we wanted to explore the difference between humans
and robots, and between complex and simple characters.
oTrajectory: We designed (and recorded videos of)
trajectories xPandxLfor each of the characters such that
Fig. 9. The trajectories for each character.
Predictability (xP)>Predictability (xL)according to
(2), but L egibility (xP)<Legibility (xL)according to
(11) (evaluated based on the parameters from Sec. VI-A.
We describe below several steps we took to eliminate
potential confounds and ensure that the effects we see
are actually due to the theoretical difference in the score.
With the HERB character, we controlled for effects
of timing, elbow location, hand aperture and ﬁnger
motion by ﬁxing them across both trajectories. For the
orientation of the wrist, we chose to rotate the wrist
according to a proﬁle that matches studies on natural
human motion [21, 36]), during which the wrist changes
angle more quickly in the beginning than it does at
the end of the trajectory. Fig.15 plots the end effector
trace for the HERB trajectories: the gray one has a larger
Fig. 10. The end effector trace for the HERB predictable (gray) and
legible (orange) trajectories.
predictability score (0.54 >0.42), while the orange one
has a higher legibility score (0.67 >0.63).
With the human character, we used a natural reach
for the predictable trajectory, and we used a reach that
exaggerates the hand position to the right for the legible
trajectory (much like with HERB or the point robot). We
cropped the human’s head from the videos to control for
gaze effects.
We slowed down the videos to control for timing
effects.
Dependent Measures.
oPredictability: Predictable trajectories match the ob-
server’s expectation. To measure how predictable a tra-
jectory is, we showed subjects the character in the initial
conﬁguration and asked them to imagine the trajectory
they expect the character will take to reach the goal. We
then showed them the video of the trajectory and asked
them to rate how much it matched the one they expected,
on a 1-7 Likert scale. To ensure that they take the time to
envision a trajectory, we also asked them to draw what
they imagined on a two-dimensional representation of
the scene before they saw the video. We further asked
them to draw the trajectory they saw in the video as an
additional comparison metric.
oLegibility: Legible trajectories enable quick and conﬁ-
dent goal prediction. To measure how legible a trajectory
is, we showed subjects the video of the trajectory and
told them to stop the video as soon as they knew the
goal of the character. We recorded the time taken and
the prediction.
Subject Allocation. We split the experiment into two
sub-experiments with different subjects: one about mea-
suring predictability, and the other about measuring
legibility.
For the predictability part, the character factor was
between-subjects because seeing or even being asked
about trajectories for one character can bias the expec-
tation for another. However, the trajectory factor was
within-subjects in order to enable relative comparisons
on how much each trajectory matched expectation. This
lead to three subject groups, one for each character. We
counter-balanced the order of the trajectories within a
group to avoid ordering effects.
For the legibility part, both factors were between-
subjects because the goal was the same (further, right)
in all conditions. This leads to six subject groups.
We recruited a total of 432 subjects (distributed ap-
proximately evenly between groups) through Amazon’s
Mechanical Turk, all from the United States and with
approval rates higher than 95%. To eliminate users that
do not pay attention to the task and provide random
answers, we added a control question, e.g. "What was
the color of the point robot?" and disregarded the users
who gave wrong answers from the data set.
Analysis.
oPredictability: In line with our hypothesis, a fac-
torial ANOVA revealed a signiﬁcant main effect for
the trajectory: subjects rated the predictable trajectory
xPas matching what they expected better than xL,
F(1, 310 ) =21.88, p<.001, with a difference in the mean
rating of 0.8, but small effect size ( h2=.02). The main
effect of the character was only marginally signiﬁcant,
F(2, 310 ) = 2, 91, p=.056. The interaction effect was
signiﬁcant however, with F(2, 310 ) = 10.24, p<.001.
The post-hoc analysis using Tukey corrections for mul-
tiple comparisons revealed, as Fig.11(a) shows, that our
hypothesis holds for the point robot (adjusted p<.001)
and for the human (adjusted p=0.28), but not for HERB.
The trajectories the subjects drew conﬁrm this (Fig.12):
while for the point robot and the human the trajectory
they expected is, much like the predictable one, a straight
line, for HERB the trajectory they expected splits be-
tween straight lines and trajectories looking more like
the legible one.
For HERB, xLwas just as (or even more) predictable
than xP. We conducted an exploratory follow-up study
with novice subjects from a local pool (with no technical
background) to help understand this phenomenon. We
asked them to describe the trajectory they would expect
HERB to take in the same scenario, and asked them
to motivate it. Surprisingly, all 5 subjects imagined a
different trajectory, motivating it with a different reason.
Two subjects thought HERB’s hand would reach from
the right side because of the other object: one thought
HERB’s hand is too big and would knock over the
other object, and the other thought the robot would be
more careful than a human. This brings up an interest-
ing possible correlation between legibility and obstacle
avoidance. However, as Fig.7 shows, a legible trajectory
still exaggerates motion away from the other candidate
objects even in if it means getting closer to a static
obstacle.
Another subject expected HERB to not be ﬂexible
enough to reach straight towards the goal in a natural
Expected!Predictable!Legible! Point Robot! HERB!Human!Fig. 12. The drawn trajectories for the expected motion, for xP
(predictable), and for xL(legible).
way, like a human would, and thought HERB would fol-
low a trajectory made out of two straight line segments
joining on a point on the right. She expected HERB
to move one joint at a time. We often saw this in the
drawn trajectories with the original set of subjects as well
(Fig.12, HERB, Expected).
The other subjects came up with interesting strategies:
one thought HERB would grasp the bottle from above
because that would work better for HERB’s hand, while
the other thought HERB would use the other object as a
prop and push against it in order to grasp the bottle.
Overall, that xPwas not more predictable than xL
despite what the theory suggested because the cost func-
tion we assumed did not correlate to the cost function
the subjects actually expected. What is more, every sub-
ject expected a different cost function, indicating that a
predictable robot would have to adapt to the particulars
of a human observer.
oLegibility: We collected from each subject the time at
which they stopped the trajectory and their guess of the
goal. Fig.11(b) (above) shows the cumulative percent of
the total number of subjects assigned to each condition
that made a correct prediction as a function of time along
the trajectory. With the legible trajectories, more of the
subjects tend to make correct predictions faster.
To compare the trajectories statistically, we uniﬁed
time and correctness into a typical score inspired by
the Guttman structure (e.g. [7]): guessing wrong gets a
score of 0, and guessing right gets a higher score if it
0!1!2!3!4!5!6!7!Human!HERB!Point Robot!Legible Traj!Predictable Traj!(a) Predictability Rating
Point Robot!HERB!Human!time (s)!Probability(correct inference)!Point Robot!HERB!Human!% of users that made an inference and are correct !
time (s)!0
  1.5
  3
  4.5
  6
  7.5
  9
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   (b) Legibility Measures
Fig. 11. (a) Ratings (on Likert 1-7) of how much the trajectory matched the one the subject expected. Error bars represent standard error on
the mean. (b) Cumulative number of users that responded and were correct (above) and the approximate probability of being correct (below).
happens earlier.A factorial ANOVA predicting this score
revealed, in line with our hypothesis, a signiﬁcant effect
for trajectory: the legible trajectory had a higher score
than the predictable one, F(1, 241 ) =5.62, p=.019. The
means were 6.75 and 5.73, much higher than a random
baseline of making a guess independent of the trajectory
at uniformly distributed time, which would result in a
mean of 2.5 – the subjects did not act randomly. The
effect size was small, h2=.02. No other effect in the
model was signiﬁcant.
Although a standard way to combine timing and
correctness information, this score rewards subjects that
gave an incorrect answer 0 reward. This is equivalent
to assuming that the subject would keep making the
incorrect prediction. However, we know this not to be
the case. We know that at the end (time T), every subject
would know the correct answer. We also know that at
time 0, subjects have a probability of 0.5 of guessing cor-
rectly. To account for that, we computed an approximate
probability of guessing correctly given the trajectory so
far as a function of time – see Fig.11(b)(below). Each
subject’s contribution propagates (linearly) to 0.5 at time
0 and 1 at time T. The result shows that indeed, the
probability of making a correct inference is higher for
the legible trajectory at all times.
This effect is strong for the point robot and for HERB,
and not as strong for the human character. We believe
that this might be a consequence of the strong bias hu-
mans have about human motion – when a human moves
even a little unpredictably, conﬁdence in goal prediction
drops. This is justiﬁed by the fact that subjects did havehigh accuracy when they responded, but responded later
compared to other conditions. Thus, legible human tra-
jectories would need a stronger emphasis on optimality
w.r.t. C(i.e. smaller trust region parameter bin (20)).
Interpretation. Overall, the results do come in partial
support of formalism: legible trajectories were more leg-
ible for all characters, and predictable trajectories were
more predictable for two out of the three characters (not
for HERB). However, the effect sizes were small, mainly
pointing to the challenge of ﬁnding the right cost Cfor
each observer.
B. The Trust Region
So far, we manipulated legibility using two levels. In
this section, we test our legibility motion planner, as well
as our theoretical notion of a trust region, by analyzing
the legibility in practice as the trajectory becomes more
and more legible according to our formalism. If our
assumptions are true, then by varying b2[bmin,bmax],
we expect to ﬁnd that an intermediate value bpro-
duces the most legible result: much lower than band
the trajectory does not depart predictability enough to
convey intent, much higher and the trajectory becomes
too unpredictable, confusing the users and thus actually
having a negative impact on legibility.
Hypotheses.
H1The size of the trust region, b, has a signiﬁcant effect
on legibility.
H2Legibility will signiﬁcantly increase with bat ﬁrst, but
start decreasing at some large enough b.
3!3.5!4!4.5!5!5.5!6!6.5!
0!40!320!Rating!! !Conﬁdence in Prediction!
0.8!0.85!0.9!0.95!1!
0!40!320!Success Rate!! !Success Rate!
1!2!3!4!5!6!
0!40!320!Rating!! !Belief in "Neither Goal"!
15!17!19!21!23!25!27!
0!10!20!40!80!160!320!Legibilty Score!!"Score w. Self-Chosen Times!
****Fig. 13. Left: The legibility score for all 7 conditions in our main experiment: as the trust region grows, the trajectory becomes more legible.
However, beyond a certain trust region size ( b=40), we see no added beneﬁt of legibility. Right: In a follow-up study, we showed users the
entire ﬁrst half of the trajectories, and asked them to predict the goal, rate their conﬁdence, as well as their belief that the robot is heading
towards neither goal. The results reinforce the need for a trust region.
Legibility Score Legibility Score Legibility Score Frequency
Frequency
Frequency  Histogram for β = 0 Histogram for β = 40 Histogram for β = 320
Fig. 14. The distribution of scores for three of the conditions. With a very large trust region, even though the legibility score does not signiﬁcantly
decrease, the users either infer the goal very quickly, or they wait until the end of the trajectory, suggesting a legibility issue with the middle
portion of the trajectory.
Manipulated Variables. We manipulated b, selecting
values that grow geometrically (with scalar 2) starting
at 10 and ending at 320, a value we considered high
enough to either support or contradict the expected
effect. We also tested b=min xC[x], which allows for no
additional legibility and thus produces the predictable
trajectory (we denote this as b=0 for simplicity). We
created optimal trajectories for each bfor the point robot
character.
Dependent Measures. We measured the legibility of the
seven trajectories in the same way as before, combining
the time and correctness into a Guttman score as in
the Analysis for the previous experiment. We used slow
videos (28s) to control for response time effects.
Subject Allocation. We chose a between-subjects de-
sign in order to not bias the users with trajectories
from previous conditions. We recruited 320 participants
through Amazon’s Mechanical Turk service, and took
several measures to ensure reliability of the results. All
participants were located in the USA to avoid language
barriers, and they all had an approval rate of over
95%. We asked all participants a control question that
tested their attention to the task, and eliminated data
associated with wrong answers to this question, as well
as incomplete data, resulting in a total of 297 samples.
Analysis. An ANOVA using bas a factor supportedH1, showing that the factor had a signiﬁcant effect on
legibility ( F(6, 290 ) =12.57, p<0.001), with a medium-
large effect size, h2=.2. Fig.13(left) shows the means
and standard errors for each condition.
An all-pairs post-hoc analysis with Tukey corrections
for multiple comparisons revealed that all trajectories
with b20 were signiﬁcantly more legible than the
predictable trajectory ( b=0), all with p<0.001, the
maximum being reached at b=40. This supports the
ﬁrst part of H2, that legibility signiﬁcantly increases
with bat ﬁrst: there is no practical need to become more
unpredictable beyond this point.
The post-hoc analysis also revealed that the trajectories
with b=20, 40, 80, or 320 were signiﬁcantly more legible
than the trajectory with b=10 ( p=.003, p<.001,
p=.004, and p=.002 respectively).
The maximum mean legibility was the trajectory with
b=40. Beyond this value, the mean legibility stopped
increasing. Contrary to our expectation, it did not signif-
icantly decrease. In fact, the difference in score between
b=40 and b=320 is in fact signiﬁcantly less than 2.81
(t(84) = 1.67, p=0.05). At a ﬁrst glance, the robot’s
overly unpredictable behavior seems to not have caused
any confusion as to what its intent was.
Analyzing the score histograms (Fig.14) for different b
values, we observed that for higher bs, the wide majority
of users stopped the trajectory in the beginning. The
consequence is that our legibility measure failed to capture
whether the mid-part of the trajectory becomes illegible : the
end of the trajectory conveys the goal because it reaches
it, but what happens in between the beginning and end?
Thus, we ran a follow-up study to verify that legibility in
this region does decrease at b=320 as compared to our
b=40, in which we explicitly measured the legibility
in the middle of the trajectory.
Follow-Up. Our follow-up study was designed to inves-
tigate legibility during the middle of the trajectories. The
setup was the same, but rather than allowing the users
to set the time at which they provide an answer, we ﬁxed
the time and instead asked them for a prediction and a
rating of their conﬁdence on a Likert scale from 1 to 7.
We hypothesize that in this case, the users’ conﬁdence
(aggregated with success rate such that a wrong pre-
diction with high conﬁdence is treated negatively) will
align with our H2: it will be higher for b=40 than for
b=320.
We conducted this study with 90 users. Fig.13 plots
the conﬁdences and success rates, showing that they are
higher for b=40 than they are for both of the extremes,
0 and 320. An ANOVA conﬁrmed that the conﬁdence
effect was signiﬁcant ( F(2, 84) = 3.64, p=0.03). The
post-hoc analysis conﬁrmed that b=40 had signiﬁcantly
higher conﬁdence t(57) =2.43, p=0.45.
We also asked the users to what extent they believed
that the robot was going for neither of the goals depicted
in the scene (also Fig.13). In an analogous analysis, we
found that users in the b=40 condition believed this
signiﬁcantly less than users in the b=320 condition
(t(57) =5.7, p<0.001).
Interpretation. Overall, the results suggest the existence
ofa trust region of expectation within which legibility op-
timization can make trajectories signiﬁcantly more legible to
novice users. Outside of this trust region, being more
legible w.r.t. L egibility is an impractical quest, because
it no longer improves legibility in practice. Furthermore,
the unpredictability of the trajectory can actually confuse
the observer enough that they can no longer accurately
and conﬁdently predict the goal, and perhaps even doubt
that they have the right understanding of how the robot
behaves. They start believing in a "neither goal" option
that is not present in the scene. Indeed, the legibility
formalism can only be trusted within this trust region .
XI. D iscussion
This paper studied motion planning in the presence
of an observer who is watching the motion and making
inferences about it.
We ﬁrst formalized predictability and legibility based
on the inferences that the observer makes, which have
opposing directionality. We then proposed mathematical
10!20!40!
Fig. 15. Legible trajectories on a robot manipulator assuming C,
computed by optimizing L egibility in the full dimensional space. The
ﬁgure shows trajectories after 0 (gray), 10, 20, and 40 iterations. Below,
a full-arm depiction of the trajectories at 0 and 20 iterations.
models for these inferences in order to arrive at pre-
dictability and legibility scores that a robot can evalu-
ate. Finally, we derived functional gradient optimization
methods for generating predictable or legible motion, as
well as a constrained optimization method for optimiz-
ing for legibility in a trust region of predictability.
Our studies on novice users provided some support
for our models — trajectories more predictable according
to the model were overall more predictable in practice,
and trajectories more legible according to the model
were overall more legible in practice while inside a trust
region of predictability. However, three main challenges
remain in planning such motions for complex cases (i.e.
high-dimensional spaces and non-convex C):
Challenge I: Finding C.If the human observer expects
human-like motion, cues from animation or biomechanics
[22, 25, 37, 57] can help provide good approximations for
C. However, our studies suggest that efﬁciency of robot
motion has different meanings for different observers
(see follow-up experiment in Sec. X-A).
A possibility is to learn from demonstrations provided
by the observer. Here, the robot can learn a Cthat
explains the demonstrations[4], using tools like Inverse
Optimal Control (IOC) [1, 44, 60]. However, extending
these tools to higher dimensions is an open problem [44],
and recent work focuses on learning costs that make the
demonstrations locally optimal [32, 38], or on restricting
the space of trajectories to one in which optimization is
tractable [30].
Aside from investigating the extension of IOC to high-
dimensional spaces, we also propose a second thread
of research: the idea of familiarizing users to robot
behavior. Can users be taught a particular Cover time?
Our preliminary results [16] suggest that familiarization
helps for the motion generated by the Cfrom (19), but
that it suffers from severe limitations, especially on less
natural choices of C. One possibility is that motion for
non-anthropomorphic arms is complex enough that we
cannot rely solely on the user to do all the learning, sug-
gesting that the two threads of research – familiarization
and learning from demonstration – are complementary.
Challenge II: Computing VG.Given a C, legibility op-
timization requires access to its value function for every
goal. In simple cases, like the one we focused on in this
paper, Vhas an analytical form. Legibility optimization
happens then in real-time even for high-dimensional
cases, as shown in Fig.15.
But this is not the case, for instance, for non-convex
functions that require obstacle avoidance, when the robot
has many degrees of freedom. In such cases, ﬁnd-
ing good approximations for Vbecomes crucial, and
many techniques value function approximation tech-
niques could be applied toward this goal [8].
What makes our problem special, however, is that
the quality of the approximation is deﬁned in terms of
its impact on legibility , and not on the original value
function itself. There could be approximations, such as
ignoring entire components of C, or only focusing on
some lower-dimensional aspects, which are very poor
approximations of Vitself, but might have little effect
on legibility in practice.
Challenge III: Finding b.The ﬁnal challenge is ﬁnding
how unpredictable the trajectory can become in a given
situation. This too can be learned based on the user,
or set based on the ambiguity level of the situation
(as measured by the legibility score of the predictable
trajectory).
Other Future Work Directions. Even though this pa-
per was about goal-directed motion, the formalism for
legibility can be applied more generally to transform
an efﬁciency cost Cinto a legible one. We are excited
to investigate this formalism with other channels of
communication and other robot morphologies. Recently,
we showed the formalism’s applicability to pointing
gestures [20], Tellex et al. [50] used the same under-
lying mathematics to generate legible natural language
requests, and Szaﬁr et al. [ ?] used our results to generate
legible quadrotor ﬂight.
Still for goal-directed motion, we are interested in the
concept of legibility when each goal could be achievedin multiple different robot conﬁgurations, as it happens
in most manipulation tasks [17]. In the example from
Fig.1, the added ﬂexibility of goal sets could enable the
robot to grasp the object from the side when legible, and
closer to the front when predictable.
Another avenue of further research is investigating
the role that perceived capability plays. Our follow-up
users in the ﬁrst experiment had different expectations
about what HERB is capable of, which shaped their
expectations about the motion. Here, familiarization to
the robot can potentially be useful in adjusting the
perceived capability to better match the real capability.
Furthermore, the perceived capability plays a role in
what goals the observer might attribute to the robot,
which can be captured in our formalism as the prior
P(G)over the candidate goals.
Limitations. Our work is limited in many ways. As the
previous section discussed, in generating predictable or
legible motion, we inherit the challenges of learning and
optimizing non-convex functions in high-dimensional
spaces. Furthermore, adding a trust region to the opti-
mization is a way to prevent the algorithm for traveling
on “uncharted territory” — from reaching trajectories
where the model’s assumptions stop holding. It does not,
however, ﬁx the model itself.
Despite its limitations and remaining challenges, this
work integrates the idea of an observer and the in-
ferences that he makes directly into motion planning,
paving the road to more seamless human-robot collabo-
rations.
Acknowledgements
We thank Geoff Gordon, Jodi Forlizzi, Hendrik Chris-
tiansen, Kenton Lee, Chris Dellin, Alberto Rodriguez,
and the members of the Personal Robotics Lab for fruit-
ful discussion and advice. This material is based upon
work supported by NSF-IIS-0916557, NSF-EEC-0540865,
ONR-YIP 2012, the Intel Embedded Computing ISTC,
and the Intel PhD Fellowship.
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