Reactive Navigation in Crowds for Non-Holonomic Robots With Convex Bounding Shape

Abstract

This paper describes a novel method for non-holonomic robots of convex shape to avoid imminent collisions with moving obstacles. The method's purpose is to assist navigation in crowds by correcting steering from the robot's path planner or driver. We evaluate its performance using a custom simulator which replicates real crowd movements and corresponding metrics which quantify the agent's efficiency and the robot's impact on the crowd and count collisions. We implement and evaluate the method on the standing wheelchair Qolo. In our experiments, it drives in autonomous mode using on-board sensing (LiDAR, RGB-D camera and a system to track pedestrians) and avoids collisions with up to five pedestrians and passes through a door.

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and Automation Letters
IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED FEBRUARY, 2021 1
Reactive Navigation in Crowds for Non-holonomic
Robots with Convex Bounding Shape
David J. Gonon, Diego Paez-Granados, and Aude Billard
Abstract—This paper describes a novel method for non-
holonomic robots of convex shape to avoid imminent colli-
sions with moving obstacles. The method’s purpose is to assist
navigation in crowds by correcting steering from the robot’s
path planner or driver. We evaluate its performance using a
custom simulator which replicates real crowd movements and
corresponding metrics which quantify the agent’s efficiency and
the robot’s impact on the crowd and count collisions. We
implement and evaluate the method on the standing wheelchair
Qolo. In our experiments, it drives in autonomous mode using
on-board sensing (LiDAR, RGB-D camera and a system to track
pedestrians) and avoids collisions with up to five pedestrians and
passes through a door.
Index Terms—Collision Avoidance; Reactive and Sensor-Based
Planning; Human-Aware Motion Planning; Velocity Obstacle
I. INT ROD UC TI ON
THIS work considers robots that need to navigate within
crowds to reach their goal (as in Fig. 1), such as e.g.
electric wheelchairs and delivery robots (see also Fig. 2). This
is challenging because individual pedestrians’ decisions are
uncertain and also, pedestrians expect cooperative behaviour,
as they anticipate and leave space for each other’s future mo-
tion. Thus, robots need to coordinate with pedestrians but also
to adapt quickly to surprising behaviour and avoid imminent
collisions that can endanger humans. When evaluating such a
robot’s controller, one needs to take into account its impact
on pedestrians to quantify the robot’s social performance.
This paper is about avoiding imminent collisions between
pedestrians and a mobile robot that is non-holonomic (e.g.
having wheels preventing sideways motion) and non-circular
(e.g. being of elongated shape). We propose the method
Reactive Driving Support (RDS) for such robots to correct
nominal commands (from the driver or high-level planner)
as far as necessary for avoiding previously unanticipated
yet imminent collisions. RDS employs Velocity Obstacles
(VO) [1], [2], whose basic concept we describe in detail
in Sec. II-A. RDS constructs VO between each obstacle
and the robot’s closest subpart and constrains its velocity
accordingly to avoid collisions locally. This constitutes a novel
Manuscript received: October 15, 2020; Revised January 23, 2021; Ac-
cepted February 20, 2021.
This paper was recommended for publication by Editor Nancy M. Amato
upon evaluation of the Associate Editor and Reviewers’ comments. This work
was supported by the EU H2020 project “Crowdbot” (779942). (Correspond-
ing author: David J. Gonon)
The authors are with LASA, School of Engineering, Swiss
Federal School of Technology in Lausanne - EPFL, Switzer-
land (e-mail: david.gonon@epfl.ch; dfpg@ieee.org;
aude.billard@epfl.ch)
Digital Object Identifier (DOI): see top of this page.
Fig. 1: The robot Qolo is passing between pedestrians using
the proposed reactive controller (from right to left).
Fig. 2: Qolo (left) and Starship’s delivery robot (right) have
footprints which capsules can bound well tightly (red) or more
conservatively (green) and still yield a smaller width than the
tightest bounding circle (blue).
way to extend VO to non-holonomic robots of non-circular
shape. This paper formulates RDS for a capsule, which is
a generic shape that fits many delivery robots and robotic
wheelchairs (see Fig. 2). RDS does not require pre-processing
which merges overlapping obstacles (in contrast to e.g. [3],
[4]) and is computationally lightweight itself even for very
many obstacles. Its implementation is publicly available at
https://github.com/epfl-lasa/rds/.
A. Related Work
There haven been diverse research efforts about robotic
navigation in crowds recently. They have mostly modeled
robots as circles that can move omni-directionally, thereby
idealizing the shape and kinematics to ease investigating
the specific properties and challenges which crowds create.
Particularly, they have explored navigating cooperatively and
according to social norms [5]–[8], predicting the surrounding
crowd’s future motion [9], planning the robot’s motion be-
yond the interactions with its immediate neighbours [10], and
conservative collision avoidance under incomplete knowledge
about obstacles’ position and behaviour [11].

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and Automation Letters
2 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED FEBRUARY, 2021
Several approaches exist for circular [12] or even non-
convex [13] robots to avoid static obstacles by optimizing
over discrete candidate trajectories under dynamic and kine-
matic constraints. Vector Field Histogram Plus [14] is another
optimization-based method, which identifies narrow passages
for the robot, but it also assumes static environments. The
method in [11] guarantees that the robot is at rest when
a collision happens for any behaviour of obstacles, which
can be prohibitive in crowds. Another method [15] aims at
increasing the feasible command space by conditioning on
surrounding agents’ most probable maneuvers, but it also
assumes a circular robot.
Methods using VO are suitable for short-term planning in
dynamic environments such as crowds. The classical VO [1]
and its derivatives, e.g. Optimal Reciprocal Collision Avoid-
ance (ORCA) [2], assume circular holonomic agents. Some
frameworks enable their application to non-holonomic robots,
including [16] and a simple approach that shifts the center
(which we discuss in Sec. II-B). However, they artificially
increase the robot’s radius and thus reduce the capability
to plan through narrow passages. An extension of the VO
concept to non-linear motion control models [17] can di-
rectly treat non-holonomic circular vehicles. However, this
relies on forward-integration to yield trajectory candidates
from a discretized solution space, which is computationally
expensive. Other methods [18], [19] treat non-circular robots
with holonomic kinematics, where they separate the step
for computing rotations from the step applying VO to find
translations. Such an approach is not applicable to robots with
non-holonomic constraints, where translation and rotation are
coupled. [20] introduce a velocity-continuous formulation of
VO which is applicable to non-holonomic but only circular
robots.
Crowd simulations have been common as tools to evaluate
and study methods for navigation in crowds [6], [8], [11],
[21]. Many such evaluations assign a fixed goal to each
agent and evaluate the ability to find an efficient trajectory
to the goal by suitable metrics. Very common metrics include
agents’ path length and time for traveling from their starting
to their goal positions for quantifying efficiency, the rate of
success (reaching the goal without collision), and the number
of collisions or minimum separation of agents for quantifying
safety (e.g. in [5], [6], [8], [21]–[23]). The evaluation in
[24] has the most similar perspective to ours, as it measures
the robot’s deviations from the high-level planner’s (time-
independent) reference path by the squared deviation integral.
B. Contributions
Our method RDS extends VO to non-holonomic non-
circular robots for reactive control in crowds. This paper
presents RDS and its evaluation tailored to this context, where
we employ a custom simulator and four novel metrics for
quantitative evaluation. The simulator combines original time-
dependent trajectories of video-tracked pedestrians (from [25])
with local collision avoidance such that agents incorporate
high-level planning, social coordination, and local collision
avoidance. By using empirical reference trajectories, we aim
Fig. 3: The capsule approximation (solid green) fits through
gaps which are not accessible with an abstracting circle (dotted
green) centered ahead of the wheel axle (yellow).
at making agents’ arrangements and motion patterns rep-
resentative of the original crowd. Corresponding to these
time-varying goals, we introduce the robot’s and the crowd’s
average deviation from their reference trajectories as metrics to
quantify how efficient a reactive controller’s corrections are at
avoiding collisions while continuing tracking of the reference
motions. Thus, our first two metrics reflect the task’s focus on
complementing high-level motion plans. Further, we introduce
two metrics that directly measure how a robot’s presence
and its controller impact other agents’ velocities. While it is
common to consider velocities, measuring one specific agent’s
impact on others is far less common. Both of these metrics
relate to the compatibility between a robot’s and pedestrians’
different ways of navigation.
Complementing this evaluation by experiments with the
standing wheelchair Qolo [26] (in Fig. 1, 2, 3), on which we
have implemented our method, we demonstrate the method to
be effective at avoiding collisions and feasible in practice. The
comparison with another method shows that our approach is
more advantageous in crowds due to its ability to lead through
narrow gaps between obstacles. The next section (Sec. II)
provides the concepts underlying this paper’s techniques. The
method’s description (Sec. III), its evaluation in simulation
(Sec. IV) and with the robot Qolo (Sec. V) follow. Finally,
Sec. VI concludes this work.
II. BACKGROU ND
This section reviews important technical prerequisites.
A. Velocity Obstacles for Circular Holonomic Robots
The original VO [1] describes for each circular obstacle
the corresponding cone-shaped set of constant velocities for
the circular robot that will eventually lead to a collision (as
Fig. 4 shows). ORCA [2] uses VO for multi-agent navigation
as follows (see also Fig. 4). It disregards collisions after the
time horizon τand thus spherically truncates the VO cone.
Further, ORCA conservatively approximates the relative VO
by the halfplane ¯
Hrel whose boundary touches the relative VO
boundary at its point closest to the previous relative velocity
v−
rel. Then, it obtains the reciprocal absolute VO for the first
agent Aby shifting the relative VO by the second agent B’s
previous velocity plus its reciprocal contribution cB∈R2,
which accounts for the second agent’s contribution to collision

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and Automation Letters
GONON et al.: REACTIVE NAVIGATION IN CROWDS FOR NON-HOLONOMIC ROBOTS WITH CONVEX BOUNDING SHAPE 3
Fig. 4: In relative velocity space (left), ORCA truncates and
linearizes the relative VO (dark gray) between the robot A
(blue) and the obstacle B(gold) around the previous relative
velocity v−
rel. Shifting the resulting halfplane ¯
Hrel by the
obstacle’s previous velocity v−
Byields the halfplane ¯
HAof
avoiding velocities in the robot’s velocity space (right), if B
keeps its velocity. Shifting by B’s reciprocal contribution cB
yields the halfplane HAof reciprocally avoiding velocities.
avoidance, and vice-versa for the second agent. ORCA thus
generates a linear constraint for an agent’s velocity due to
each other agent, where each constraint is characterized by its
normal nand offset b. Finally, it solves a quadratic program
to find the velocity closest to the preferred one while adhering
to the constraints.
B. Abstraction from the Robot’s Shape and Kinematics
We define for comparison with our method later on the
baseline method as using ORCA [2] (assuming no reciprocity,
i.e. cB=0) for the given non-holonomic non-circular robot by
the following abstracting technique (from [27]). We consider
a generic robot’s kinematic model which is conceptually
equivalent to an axle with two wheels that rotate independently
around it and roll on a plane without slip. The model views the
robot as a rigid body which moves in the plane and is subject to
the non-holonomic constraint which requires its instantaneous
center of rotation to be on the infinite line that contains the
wheel axle.
For abstracting from such non-holonomic kinematics and
the robot’s possibly non-circular shape, [27] defines the control
point as a point on the robot’s body that does not lie on the
wheel axle. The control point’s cartesian velocity has degree of
freedom two and thus it serves to receive velocity commands
that address a holonomic robot. Further, a circle whose center
is the control point and which contains the robot serves to
mask its true shape such that one can apply conservatively
to this virtual circle a method that avoids collisions for a
circular robot. We note that the control point needs to be ahead
of the wheel axle in the robot’s preferred direction of travel
(if it exists) to yield proper signs of angular velocities when
avoiding obstacles, i.e. rotating clockwise/counter-clockwise
when passing on the right/left, respectively. Consequentially,
the virtual circle for this abstraction is particularly conservative
when the robot is longer in the rearward than in the forward
direction from the wheel axle (as Fig. 3 shows for Qolo).
Fig. 5: The method constructs the velocity obstacles (light
grey) for obstacles (gold) and the robot’s closest incircle (dark
blue), linearizes them as Hi(red) and re-maps them as ˜
Hi
(dark red) to the reference point’s velocity space.
III. MET HO D
The method avoids collisions by constructing constraints for
the velocity of a particular robot-fixed point, which we refer
to as the reference point, according to Fig. 5. Each obstacle is
taken into account by determining the robot’s incircle which is
closest to it and constructing the VO that the obstacle induces
for the incircle. A linear constraint is derived for the incircle
center’s velocity and transformed into the equivalent constraint
for the reference point’s velocity via the robot’s kinematic
relation between different points’ velocities.
A. Definitions
We assume the robot to have non-holonomic kinematics ac-
cording to Sec. II-B. The robot’s fixed right-handed coordinate
system is defined such that the y-axis separates the wheels
symmetrically and points forward and the x-axis coincides
with the wheel axle’s line. Any cartesian vector components
in the method’s description refer to this coordinate system.
The method’s description here assumes the robot’s shape as a
capsule which is symmetric in the y-axis and corresponds to
sweeping a circle of radius rwith its center on x= 0 from
the rear end yrear <0to the front end yf ront >0. An incircle
is any circle with its center on the line between the endpoints
and radius r.
The command vector u= [v, ω]Tdefines the robot’s
linear and angular velocity command vand ωin such a way
that positive values yield forward translations and counter-
clockwise rotations around the origin, respectively. Given any
point (x, y), its cartesian velocity vcorresponding to ucan
be expressed via the Jacobian J(x, y)as v=J(x, y)u. One
can show that
J(x, y) = 0−y
1x,J−1(x, y) = x/y 1
−1/y 0,(1)
where J−1exists for any point (x, y)with y6= 0. Finally,
(xref , yref )defines the reference point with yref 6= 0. Alter-
natively to u, its velocity vref can describe via the inverse of
its Jacobian Jref the robot’s motion, as u=J−1
ref vref .

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B. Velocity Constraints for Local Incircles
Assuming circular obstacles (with known radii, positions
and velocities), the method constructs for each obstacle Oi
the VO which it induces for the respective closest incircle,
located at (0, yi). The VO’s construction, truncation by the
time horizon τ, and linearization around the incircle center’s
and the obstacle’s previous relative velocity are identical to
the approach in [2] (as in Sec. II-A) apart from reciprocity,
i.e. the method assumes the obstacle to maintain its velocity
(corresponding to cB=0in Fig. 4). Thus, each obstacle
Oicreates for the velocity viof the point (0, yi)the linear
constraint
nT
ivi≤bi,(2)
which describes the feasible halfplane Hi(ni, bi)with the
outwards unit normal niand the offset from the origin bi.
C. Optimization Problem
The method aims at optimizing the command vector ufor
the robot to execute such that it minimally deviates from
the nominal command ¯
u= [¯v, ¯ω]Te.g. by the driver. The
optimization problem is formulated in the reference point’s
velocity space, wherein the nominal command is mapped as
¯
vref =Jref ¯
u. Further, the method maps in this space the
constraints for the incircles’ velocities due to Nobstacles by
expressing the local velocity in each constraint (2) as vi=
J(0, yi)J−1
ref vref using (1). We incorporate these constraints
and the objective in the quadratic program
v∗
ref = arg min
vref
|vref −¯
vref |2(3)
s.t. nT
iJ(0, yi)J−1
ref vref ≤bi∀i∈1, ..., N (4)
nT
v,j vref ≤bv,j ∀j∈1, ..., 4(5)
nT
a,kvr ef ≤ba,k ∀k∈1, ..., 4(6)
where the additional constraints in (5), (6) represent the robot’s
velocity and acceleration limits, respectively. We assume that
they result respectively from four fixed constraints for uand
from the four box constraints around the previous command
[v−, ω−]Twhich encode |v−v−| ≤ ˆa∆tand |ω−ω−| ≤ ˆα∆t,
with ˆaand ˆαdenoting the maximum absolute linear and angu-
lar acceleration, respectively, and the control cycle time ∆t.
These constraints’ normals are multiplied by J−T
ref and their
offsets are adopted to yield nv,j , bv,j ,na,k , ba,k in (5), (6).
However, one can employ any given number and arrangement
of constraints instead.
D. Solution and Command Computation
The method employs an incremental algorithm very sim-
ilar to [28], [29] to solve the quadratic program (3)-(6) or
determine that its constraints are infeasible. Importantly, the
maximum number of obstacles bounds a priori the number of
iterations which the algorithm requires. If the constraints are
feasible, the solution defines the velocity command according
to u∗=J−1
ref v∗
ref . Otherwise, the pre-defined maximum linear
and angular braking decelerations ˆa > 0and ˆα > 0define the
command according to v∗=h(v−,ˆa)and ω∗=h(ω−,ˆα),
where h(u, m):=u−sign(u) min(|u|,∆t m).
E. Discussion and Generalizations
Two simplifying approximations underlie the method. First,
it reduces the robot’s shape to the closest incircle for each
obstacle. Second, it approximates the incircles’ motions over
the planning horizon as straight with individual constant
velocities such that they only initially verify a rigid body’s
velocity distribution. Consequentially, the method requires a
high control frequency to prevent collisions, i.e. it must update
the robot’s velocity command not just as the time horizon
elapses but early enough such that both approximations remain
valid. However, this is also necessary as obstacles may change
their velocities faster than the horizon.
The choice (xref , yref )defines the relative costs for deviat-
ing from ¯
ualong different axes in the command space. When
viewing uas the optimization variable in (3) by plugging
in vref =Jref u, the objective becomes the quadratic form
defined by JT
ref Jref , whose principal axes and eigenvalues
can be tuned via (xref , yref ).
The expression (4) maps each halfplane Hi(ni, bi)(con-
straining the local velocity vi) to the corresponding constraint
for the reference point’s velocity vref . With M(x, y):=
J−T
ref J(x, y)T, the latter constraint can be geometrically in-
terpreted, if M(0, yi)ni6=0, as the transformed fea-
sible halfplane ˜
Hi(˜
ni,˜
bi)with the unit normal ˜
ni=
M(0, yi)ni/||M(0, yi)ni|| and the offset ˜
b=b/||M(0, yi)ni||
(see also Fig. 5). In the case M(0, yi)ni=0, the constraint
reads 0≤bi, which occurs if and only if yi=ni,y = 0 (e.g.
when an obstacle approaches the static robot along the line
y= 0). There, the local VO constrains vi,x independently (of
vi,y) while kinematically any command must yield vi,x = 0.
Thus, vref does not enter the constraint, which makes the
optimization infeasible and triggers braking if bi<0. This
effect limits the ability to escape close and fast obstacles
approaching along y≈0.
The method’s formulation here assumes a capsule-shaped
robot. However, any convex shape which a convex polygon and
a sweeping circle generate together can replace the capsule.
This only requires a routine to compute for a given obstacle
the robot’s corresponding subpart as the closest instance of the
sweeping circle.
IV. EXP ER IM EN TS I N SIM UL ATIO N
The simulations in this section compare our method, RDS,
to the baseline method (from Sec. II-B). The comparison in
Sec. IV-C additionally includes the method “Blank”, which
is defined as outputting directly its input, such that the robot
executes the nominal command, i.e. setting u∗=¯
u. With the
method “Blank”, only the nominal command and other agents
contribute to cooperative navigation, allowing to estimate a
compared reactive controller’s additional contribution.
We define the control point for all methods and the reference
point for RDS to coincide (for simplicity), where the control
point (whose velocity the baseline method acts on) additionally
represents the robot’s position for tracking the reference tra-
jectory given by the experiment. We choose the robot’s param-
eters as xref =0, yrear =-0.5 m, yfront =yref =0.18 m, r=0.45m
(corresponding to Qolo’s conservative capsule in Fig. 2), and
τ=1.5 s, ˆa=2m/s2,ˆα=3 rad/s2.

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and Automation Letters
GONON et al.: REACTIVE NAVIGATION IN CROWDS FOR NON-HOLONOMIC ROBOTS WITH CONVEX BOUNDING SHAPE 5
Algorithm 1 In each simulation update, pedestrians compute
their velocities via ORCA, and the robot reacts via RDS.
for i= 1, ..., Npdo
vp
i,t =ORC A ¯
vp
i,t,xp
i,t,{xj,t ,vj,t−∆t}Np+Nc
j=16=i
end for
ur
t=RDS ¯
ur
t,ur
t−∆t,xr
t, ϕr
t,xp
j,t,vp
j,tNp
j=1
xp
i,t+∆t=xp
i,t + ∆tvp
i,t
(xr
t+∆t, ϕr
t+∆t)=(xr
t+ ∆tvr(ur
t, ϕr
t), ϕr
t+ ∆t ωr(ur
t))
The following Sec. IV-A introduces the simulation frame-
work and its specific performance metrics. A qualitative com-
parison follows in Sec. IV-B. A test series for quantitative
comparison follows in Sec. IV-C, whose source code is avail-
able in the provided repository.
A. Simulation Framework and Performance Metrics
This section describes our framework for simulating how the
robot navigates in environments with pedestrians. We represent
them by circular agents (of radius 0.3m) which track individual
reference trajectories while avoiding collisions with each other
and the robot by applying ORCA (assuming reciprocity, and
τ=1.5 s). The robot also tracks a reference trajectory and in
turn uses the method RDS (or baseline) to avoid collisions
with the pedestrians, reacting to their current position and
updated velocities (or not, with the method “Blank”). The
simulation’s update scheme (for the case with RDS) is given
in Algorithm 1, with ∆t=0.05 s being its time step and also
the cycle time for RDS and other agents’ controllers. The
following paragraphs introduce the notation and explain the
simulation framework.
For each pedestrian i, let xp
i,t,vp
i,t ∈R2denote respectively
the global position and velocity at time t. Let xr
t∈R2and ϕr
t
denote respectively the robot’s position (i.e. where its control
point is) and orientation at time t. Accordingly, let vr
t∈R2
and ωr
tdenote respectively the robot’s global cartesian velocity
(of its control point) and angular velocity at time t. Let ur
t
denote the robot’s command vector [v∗, ω∗]Tat time t, whose
components are respectively the forward and angular velocity
which result from the method for collision avoidance. They
prescribe the robot’s global velocities, which are thus functions
vr
t=vr(ur
t, ϕr
t)and ωr
t=ωr(ur
t).
Pedestrians perceive the robot as a collection of Ncvirtual
circular agents, which are attached to the robot, covering its
actual capsule. When simulating with the baseline method
for collision avoidance, the collection includes the enlarged
bounding circle, otherwise it contains only several tightly
fitting circles. The virtual agents adopt the position and ve-
locity of their respective point of attachment on the robot. Let
xj,t,vj,t ∈R2denote the position and velocity for a generic
circular agent (i.e. a pedestrian or a virtual agent).
The robot’s reference trajectory ¯
xr
t:R→R2defines the
robot’s reference position at each time tand prescribes the
nominal velocity ¯
vr
tfor the robot’s control point according to
¯
vr
t=d¯
xr
t
dt+k(¯
xr
t−xr
t).(7)
Therein, the reference trajectory’s derivative forms a feed-
forward term and the tracking error is added as a feedback
term (with the gain k > 0). For pedestrians, the reference
trajectories ¯
xp
i,t :R→R2prescribe the corresponding
nominal velocities ¯
vp
i,t in analogy to (7). Both terms in (7)
together achieve a vanishing tracking error over time such that
agents converge to their (moving) reference position even after
perturbations. A set n¯
xr
t,¯
xp
1,t, ..., ¯
xp
Np,towhich contains the
robot’s and Nppedestrians’ reference trajectories over a time
window [t1, t2]defines a particular simulation configuration.
Our metrics rely on the following definitions. The static
area of evaluation Adefines where relevant interactions are
expected (Sec. IV-C). Let the indicator function I{B}equal 1
if Bis true or 0otherwise. Further, let h·i =Rt2
t1(·)dt/(t2−t1)
and hh·ii =PNp
i=1h·i/Npdenote respectively averaging over
the sample’s time window and averaging over both the time
window and the crowd. We use the following metrics.
•The robot’s mean tracking error Er=h|¯
xr
t−xr
t|i.
•The pedestrians’ mean tracking error Ep=hh|¯
xp
i,t −
xp
i,t|wi,t ii, with wi,t ∝I{¯
xp
i,t∈A}and hhwi,t ii = 1.
•The crowd’s velocity reduction due to the robot Vc=
Vc,0/Vc,r, where Vc,r =hh|vp
i,t|ˆwiii denotes the crowd’s
weighted average velocity for the case with the robot, and
Vc,0denotes the analogous quantity for the case without a
robot. Herein, the pedestrians’ weights ˆwi∝ hI{xp
i,t∈A}i
are proportional to their actual time in Aand normalized
as hh ˆwiii = 1.
•The neighbours-to-crowd velocity ratio Vn=Vn,r/Vc,r ,
where Vn,r =hh|vp
i,t|˘wiii and ˘wi∝ hI{|xp
i,t−xr
t|<D}iare
weights proportional to pedestrians’ time D-close to the
robot (D= 3m) and normalized as hh ˘wiii = 1.
•Crcounting collisions with the robot’s capsule.
The metric Vccompares the crowd’s speed when the robot is
not present (leaving its place to a regular agent instead) to
the crowd’s speed when the robot is present. Thus, evaluating
Vcfor a given simulation configuration and method (e.g.
RDS) requires to execute one simulation without and one with
the robot. The metric Vncompares the speed of the robot’s
neighbours to the entire crowd’s speed. If the robot tends to
slow down pedestrians, we expect that Vc>1and Vn<1.
B. Crossing with Variable Head Start
The following experiment lets the robot and a pedestrian
cross while both contribute to collision avoidance according
to our simulation framework (Sec. IV-A). Their reference tra-
jectories move at the same speed 1.3 m/s and their paths cross
orthogonally, however, the pedestrian starts from a variable
distance to the crossing point. The pedestrian’s head start tp
hs
denotes the time difference between the moment when the
pedestrian’s reference trajectory reaches the crossing point and
the moment when the robot’s reference trajectory reaches it.
Fig. 6 shows both agent’s trajectories that result respectively
for different values of the pedestrian’s head start in the range
tp
hs±1.5s. The result shows for both methods how the crossing
order changes around tp
hs = 0. Over the test series, the pedes-
trian’s mean tracking error is similar with both methods to

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and Automation Letters
6 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED FEBRUARY, 2021
Fig. 6: The robot (moving to the right) and a pedestrian
(moving upwards) cross with variable relative head starts,
resulting in the corresponding (color-coded) trajectories and
the particular crossing order. The robot uses either RDS (left)
or the baseline method (right) for collision avoidance.
Fig. 7: The robot uses RDS (left) or the baseline method
(right) to traverse the dynamic crowd in this empirically based
simulation example. For three sequential time windows, the
initial state of the robot (green capsule) and crowd (black
circles) and their future motion (yellow to red) and future
nominal motion (dashed lines) are shown.
control the robot (Erds
p= 0.10±0.12m, Eb.l.
p= 0.09±0.06m),
whereas the robot’s tracking error is clearly lower with RDS
(Erds
r= 0.20 ±0.05m, Eb.l.
r= 0.35 ±0.14m).
C. Navigating in a Sparse Crowd
For quantitative comparison, this experiment series evalu-
ates RDS, the baseline method, and the trivial method “Blank”
(that adopts nominal commands) in simulations that are driven
by original crowd movements (from a pedestrian intersection
on a campus) which are available in the “Crowds-by-Example”
dataset [25] as timed waypoint sequences. We use these orig-
inal trajectories to generate 430 different sample simulation
TABLE I: The metrics’ mean and standard deviation (or for
Cr, the sum) is shown over the sparse crowd simulations
for the three methods. Between the baseline method and
RDS, superior mean values are marked in bold and significant
differences by asterisks (except for Cr).
Method Er[m] Ep[m] Vc[-] Vn[-] ΣCr[-]
RDS 0.8∗±0.9 0.20∗±0.09 0.999∗±0.007 1.07±0.24 0
Baseline 2.2∗±2.0 0.21∗±0.09 0.994∗±0.010 1.08±0.25 0
“Blank” 0.0±0.0 0.20±0.09 0.996±0.009 1.07±0.23 6
configurations by replacing a different original pedestrian by
the robot and simulating the remaining original pedestrians
via regular agents. For each agent (including the robot), we
define its reference trajectory as the two cubic splines fitting
the respective original pedestrian’s 2D-waypoints over time.
The area of evaluation Ais chosen as a tight bounding box of
all the waypoints. The time window [t1, t2]for a given sample
configuration’s simulations matches the time window of the
waypoints for the robot’s original pedestrian. Other agents’
trajectories outside their original waypoints’ time windows are
linear extrapolations (which are mostly outside A). For each
sample configuration, we evaluate the metrics from Sec. IV-A
for each method (simulating once without a robot). Fig. 7
shows for an exemplary sample configuration the robot’s
and the surrounding crowd’s motion during sequential time
windows for RDS and the baseline method, respectively. These
motion snippets exemplify how the robot can often follow
closely its nominal motion with RDS, whereas the baseline
method leads it on detours around dense groups.
Table I reports the metrics’ sample averages and standard
deviations over the 430 sample configurations for the three
methods (or for Cr, the sum over all configurations). For
all the metrics except Crwe compare their distributions for
RDS against the baseline using a two-sample t-test with a
significance level α= 0.05. We find p<α, i.e. significant
differences in the mean values, for all the metrics except Vn.
In comparison to the baseline method, we attribute RDS’
significantly lower tracking error for the robot and the pedes-
trians (i.e. Erds
r< Eb.l
r,Erds
p< Eb.l
p) to the tighter shape
representation. It allows the robot to maneuver through narrow
gaps between pedestrians and requires less deviations from
them. On the other hand, the circular shape representation with
the baseline method encourages agents to maneuver around the
robot with increased velocity (as typical for ORCA), whereas
the multi-circle shape representation they perceive for the
robot with RDS often traps them between two such circles,
thus Vb.l.
n> V r ds
n. While the assumption that other agents
perceive the enlarged bounding circle for the robot with the
baseline method is not realistic, we observe that it is actually
favourable for the baseline method’s performance, since other-
wise the robot frequently experiences virtual collisions (with
its bounding circle), and while trying to resolve them, it is
prone to colliding truly, as it does not represent the robot’s
true capsule shape.
With RDS or the baseline method, collisions do not occur
for the robot throughout the simulations. This is due to
contributions from both the robot’s reactive controller and the

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and Automation Letters
GONON et al.: REACTIVE NAVIGATION IN CROWDS FOR NON-HOLONOMIC ROBOTS WITH CONVEX BOUNDING SHAPE 7
pedestrians’ cooperative controller. The robot’s contribution is
still necessary sometimes to avoid collisions, as the method
“Blank” (which does not contribute) leads to a few collisions
(ΣCr>0).
Comparing Ep,Vc, and Vnbetween RDS and the method
“Blank”, we find that RDS does not facilitate other agents
to follow their references and generally reduces the crowd’s
velocity in our simulation framework. However, also with the
method “Blank” the robot already receives high-level guidance
via the reference trajectory (which avoids other agents original
positions) and therefore, this result mainly shows that the
smaller holonomic pedestrians can resolve efficiently the slight
collisions due to the robotic agent’s larger shape even when
the robot does not contribute.
In summary, these results show that RDS successfully
corrects the robot’s motion to account for its capsule shape
(whose potential collisions the reference motion does not
avoid) while at the same time achieving a low tracking error
for the robot and the crowd.
V. EX PE RI ME NT S WI TH T HE RO BO T QOLO
The robot Qolo [26] is an electrically powered standing
wheelchair (in Fig. 1, 2, 3). In this section’s experiments,
Qolo is driven by RDS which receives a constant forward-
pointing command (i.e. with vanishing nominal angular ve-
locity) simulating a driver’s primitive input. RDS uses the
same parameters’ values given already for the experiments in
simulation (Sec. IV), except for ˆa=1.5m/s2,ˆα=1.5 rad/s2. The
experiments’ videos are available in the paper’s supplementary
material.
A. Implementation
The robot’s sensors include a front and a rear LiDAR and
a front RGB-D camera. They inform the modules for SLAM,
person detection tracking and collision avoidance.
1) SLAM: The robot estimates its own trajectory by match-
ing scans from the rear LiDAR using the ROS package
hector slam [30], which allows the tracker to transform the
sensors’ spatial data into a static fixed reference frame and to
estimate obstacles’ absolute velocities.
2) Person detection tracking: The module tracks persons’
positions from both LiDARs and the camera’s RGB images
(using the pipeline in [31]).
3) Collision avoidance: The module implements RDS or
the baseline method (as in Sec. IV). It treats every scanpoint
from both LiDARs as a separate circular obstacle with a
small radius and zero velocity. Further, every person track is
perceived as a circular obstacle with 0.3 m radius and with
the track’s estimated velocity.
B. Test in a Static Environment
The test (in Fig. 8, top) compares how RDS and the baseline
method can assist passing through a door. As the nominal
command would drive the robot forward into the door frame,
correction is necessary to avoid the collision and ideally lead
through the door. The trajectories for the baseline method and
Fig. 8: The robot Qolo uses RDS (left) or the baseline method
(right) to pass through a door (top), overtake three pedestri-
ans (middle) or a surrounding crowd (bottom). Its trajectory
and the tracker’s estimates of persons are shown (blobs and
triangles, respectively, encoding time in yellow-red). Also, the
robot’s footprint (green capsule), tracked persons’ footprints
(circles) and LiDAR scanpoints (blobs) are shown at the
beginning (in cyan) and at the end (in blue).
RDS in Fig. 8 right and left, respectively, show that among
both methods, only RDS can lead the robot through the door
due to the tighter capsule shape representation.
C. Tests with Pedestrians
The following two tests evaluate the robot’s ability to
overtake pedestrians that move in the same direction but are
distributed ahead of and around the robot.
1) Row of pedestrians: The experiment (in Fig. 8, middle
row) involves three pedestrians walking next to each other,
forming a line with a larger gap between two of them such that
the robot could pass in between while respecting a comfortable
distance.
With the baseline method (Fig. 8, right), the robot ap-
proaches the moving pedestrians and then alternates between
different angles while attempting to pass through the gap,
which is due to the fact that the perceived orientation of the
gap oscillates as the foot patterns and relative advancement of
the pedestrians vary slightly over time. Using RDS (Fig. 8,
left), the robot adjusts its orientation early towards the gap in
order to avoid colliding with the middle pedestrian, and then
it moves straight forward and passes through the gap.
2) Unidirectional crowd: In the experiment (in Fig. 8,
bottom), there are five pedestrians surrounding the robot. With
the baseline method, the robot’s motion is heavily constrained
and it moves always towards small free areas created randomly
by small irregularities in the crowd motion. With RDS, the

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and Automation Letters
8 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED FEBRUARY, 2021
robot adjusts its orientation early to avoid colliding with one
pedestrian ahead, and subsequently it converges to a collision-
free course and overtakes the surrounding crowd.
VI. CO NC LU SI ON
We have developed a method to apply the Velocity Obstacle
(VO) to non-holonomic capsule-shaped robots and highlighted
its effectiveness at avoiding collisions with static obstacles
and interacting pedestrians, both in simulation and physical
experiments with the robot Qolo. The comparison with another
method using VO has demonstrated our method’s advantage
due to allowing maneuvers through narrow gaps. Our simu-
lations of agents tracking real crowds’ motions show that the
method avoids collisions efficiently such that the robot and
pedestrians remain close to their references. We have described
and evaluated four novel metrics to support this analysis.
ACK NOW LE DG EM EN T
We thank Dan Jia and Sabarinath Mahadevan for providing
and supporting us with the person tracker. We thank Fabien
Grzeskowiak and Julien Pettr´
e for proposing the metrics that
quantify the robot’s impact on the crowd’s speed.
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