Anticipatory Robot Navigation:
Incorporating Estimated Obstacle Behaviors with
the Social Force Model
Fengkai Liu∗ , Yuichi Ohsita† , Kenji Kashima‡ , Shinya Yasuda§ , Taichi Kumagai§ ,Hiroshi Yoshida§ , Dai Kanetomo§ , Masayuki Murata∗∗ Graduate School of Information Science and Technology, Osaka University, Osaka, Japan 565-0871

Email: {f-liu, murata}@ist.osaka-u.ac.jp† Cybermedia Center, Osaka University, Osaka, Japan 560-0043

Email: yuichi.ohsita.cmc@osaka-u.ac.jp‡ Graduate School of Informatics, Kyoto University, Kyoto, Japan 606-8317

Email: kk@i.kyoto-u.ac.jp§ Visual Intelligence Research Laboratories, NEC Corporation, Kanagawa, Japan 211-8666

Email: {shinya-yasuda, t kumagai, yoshida, d-kanetomo}@nec.com

Abstract—In environments populated by various dynamic
obstacles, such as people and robots, it is essential for each robot
to operate efﬁciently while avoiding these obstacles. Therefore, inthis paper, we propose a method for navigating a robot to avoid
obstacles and ensure smooth and efﬁcient movement in dynamic
environments. Moving obstacles may also actively attempt to
avoid the robots. Therefore, our navigation method should
consider such interactions to achieve an efﬁcient movement. To
achieve this, our method integrates principles from the Social
Force Model and models the movement of nearby obstacles by
estimating the parameter of the Social Force Model. Then, ourmethod controls the robot to ensure a balance between efﬁciency
and safety based on the model. We evaluate our method bysimulation and demonstrate that it achieves high efﬁciency andsafety by utilizing a model of nearby obstacles.
Index Terms—Obstacle Avoidance, Obstacle Behavior, Social
Force Model

I. I NTRODUCTION
With the continuous growth of the e-commerce and retailsectors, there’s an increasing demand for automation within
warehouses and logistic centers. Being central to this automation transition, the robot, speciﬁcally the Automated Guided
Vehicle (AGV), offers efﬁciency for warehouse operations,especially in tasks like material transport. However, the warehouse environment is unique, with robots interacting withmoving obstacles, e.g., humans or robots operating in othersystems, presenting distinctive challenges for robot navigation.
Particularly, given the efﬁciency demands of warehouse operations, the movement efﬁciency of the robot and safety become
paramount.
Traditionally, in situations with a possibility of collision,actions like stopping or signiﬁcantly limiting speed have beentaken to ensure human safety [1]. Several approaches havebeen proposed in the past, for example, Feder et al. [2]presented a method for real-time path planning in dynamic
environments using harmonic potentials. Arslan et al. [3]proposed a robot obstacle avoidance method based on power
diagrams. Missura et al. [4] enhanced the Dynamic Window
Approach and incorporated a dynamic collision model thatpredicts future collisions. However, neither of these methods
considers the potential impact of the robot’s movement on the
trajectory of obstacles, indicating that they lack interactionawareness. Such neglect could lead the robot to perceive agreater intrusion by obstacles into its trajectory, prompting it
to adopt a more conservative control strategy. As a result,these methods may introduce inefﬁciencies and, in extreme
scenarios, give rise to the Freezing Robot Problem [5].
Therefore, several interaction-aware methods have been
proposed. Helbing et al. [6], [7] introduced the Social ForceModel, a simple yet effective framework for describing pedestrian movement adaptable to a myriad of contexts, which was
also used in robot navigation. Ferrer et al. [8] presented ahuman-aware navigation approach based on the Social ForceModel. Shiomi et al. [9] improved the Social Force Model
based approach by introducing the concept of “social distance” and respecting personal spaces, aiming for human-likecollision avoidance in robots. Kamezaki et al. [10] enhanced
the Social Force Model by adding an inducible feature tohandle the Freezing Robot Problem. However, a common
limitation across the methods in the approaches presented in[8]–[10] is the use of ﬁxed parameter values. This assumes
uniform repulsive forces across all objects, implying that all
obstacles exhibit the same behavioral pattern. However, eachobstacle has its own behavior. Some obstacles may change
their trajectories to avoid the robot, while other obstacles may
not change their trajectories. If an obstacle does not avoid therobot but the robot assumes it will, this mismatch can resultin a collision.
An interaction-aware method based on reinforcement learning has also been proposed [11]. This method trains a model
to control a robot without violating social norms by using a

trajectory dataset. However, this method also does not consider
the variation in the behavior of the obstacles.
Hence, in this paper, we propose a method for navigating
a robot to avoid obstacles and ensure smooth and efﬁcient
movement by considering the interaction between the robot
and obstacles. In our method, we use the Social Force Model tomodel the behavior of obstacles. By estimating the parameterof the Social Force Model based on the observations ofobstacles, our method reﬂects the varied behaviors exhibitedby different obstacles. While we apply the Social Force Modelto prevent collisions, its parameters are adjusted in our system
to achieve a balance between safety and efﬁciency, factoring in
the behaviors of surrounding obstacles. By repeatedly updating
the models of the behaviors of nearby obstacles and updating
the parameters for controlling robots, our method achieves
safety and efﬁciency in dynamic environments.
Our primary contributions in this study are:1) We propose a method to model the behaviors of eachobstacle that avoids collisions with the robot by using
observations of obstacles.
2) We also propose a method to reﬁne the robot’s control
input, ensuring a balance between efﬁciency and safety,
based on the model of the behaviors of nearby obstacles.
In the following sections, we elaborate on our methodology.
Section II delves into the Social Force Model. Section III
outlines our approach for calculating the control input, while
Section IV-B presents the experimental setup, results, anddiscussions related to our evaluation.
II. S OCIAL F ORCE M ODEL
Social Force Model is a simple model designed to describe
pedestrian motion, and takes into account individual preferences, local intentions with neighbors, and the surrounding
environment. In this model, interactions with other objects inﬂuence the moving velocity. This inﬂuence is deﬁned in termsof repulsive forces. One object also experiences a simulated
attractive force originating from destinations or places they
intend to go to, guiding them towards their target. Figure 1
shows the overview of the Social Force Model. By utilizing
the principles of the Social Force Model, we can model the
behavior of the robot and the obstacles within a shared space,especially in contexts where their paths might intersect orcome into proximity.
In the Social Force Model, we can use (1) to determine theposition of object i for the subsequent time step t + ∆t.
1
ri (t + ∆t) = ri (t) + vi (t) · ∆t + ai (t) · ∆2 t,(1)
2
where ri (t) and vi (t) are the position and velocity of object
i at time step t, respectively. In the Social Force Model, theacceleration ai (t) is obtained by (2).
X rep
ai (t) =Fi,j (t) + Fiatt (t),
(2)
j̸=i
rep
where Fi,j(t) represents the repulsive force exerted between
att
object i and any distinct obstacle j, and Fi,j(t) denotes

w
r

rep
Fr,o
rep
Fr,w

o
Fratt
gr

Fig. 1. An example for Social Force Model. In this scenario, a robot rnavigates in a space with a moving obstacle o and a wall w, which serves
as a static obstacle. Therefore, there are three forces acting on robot r: two
rep
rep
repulsive forces exerted by moving obstacle o, Fr,oand wall w, Fr,w
, andthe attractive force Fratt directed towards its goal gr .

the attractive force directing object towards its designatedrepdestination. We accumulated all the repulsive forces Fi,j(t)
that arise between object i and any obstacles.
repFi,j(t) is the repulsive force to act the interaction, calculated by (3), which is a simpliﬁed version of the Social ForceModel from [7], often referred to as the circular speciﬁcation.
repFi,j(t) = Ai · e−||di,j (t)||/B

di,j (t),||di,j (t)||(3)

where di,j (t) means the distances between the object i andobject j, calculated by di,j (t) = ri (t) − rj (t). The parameter
Ai determines the strength of repulsive force exerted on object
i, which can be used to ﬂexibly depict the behavior of object i.
The parameter B determines the range of the repulsive force.
In this paper, by setting Ai for each object, we capture its
tendency to avoid obstacles.
Fiatt (t), the attractive force, is calculated by (4), using a
speciﬁc variable vi0 (t) that represents the desired velocity.
Fiatt (t) =
1 0v (t) − vi (t) ,
τ i

(4)

where τ represents the relaxation time. The desired velocity
vi0 (t) can be calculated by (5) from the concept in [6].
vi0 (t) = vimax (t) = vimax ·gi − ri (t),||gi − ri (t)||(5)

where the notion vimax represents the maximum velocity, while
gi denotes the goal of the object i.
For static obstacles, we only employ (3) to compute therepulsive force between the object and the static obstacle anddo not use other equations to calculate the displacement ofstatic obstacles.
III. C ONTROL OF THE ROBOTIn this section, we explain our assumption of the environment. Then, we explain the overview of our method.
Following that, we delve into the mechanisms by which therobot processes information about obstacles to ascertain theappropriate control input.

SensorInformation Update

Parameter A
Aoo
Estmation

Robot

Control Input
Calculation

0
Desired Velocity v
AroCalculation

Fig. 2. The procedure of control.

A. Assumed Environment
In our setup, we assume robot r can observe the positions ofobstacle o as reo (t) by sensors installed within the environmentor sensors mounted on the robot. Similarly, the obstacle canobserve the position of the robot as rr (t). We also assume
that the robot’s observation of the obstacle’s position reo (t)contains noises, potentially due to sensor precision. That is,reo (t) is represented byreo (t) = ro (t) + e(t),(6)

where ro (t) is the actual location of o at time t, and e(t) is errors. We assume that the error term e(t) follows a multivariate
normal distribution where the components in each dimension
are independent. To simplify our analysis, we consider theerrors included only in the observations by the robot and do
not consider the errors included in the observations by the
obstacles.
B. Method Overview
Our method employs the model introduced in Section II to
compute interaction and the robot’s control input. Using the
movement model, we assume both the robot r and the obstacleo have their own goals, denoted as gr and go . respectively. We
assume that we can use the goal for the obstacle estimated by
the methods proposed in [12], [13]. We also assume that both
the robot and the obstacle will make every effort to movetowards their destinations, without exhibiting unreasonablebehaviors like getting lost.
Our method operates based on a distinct workﬂow, comprising two cycles, as shown in Figure 2. The ﬁrst cycle focuses ondetermining the avoidance behavior and is characterized by its
shorter duration. This cycle involves obtaining the observationfrom sensors, as well as calculating the control input for the
robot based on the Social Force Model. The second, longer
cycle primarily deals with determining the parameters needed
to compute the robot’s control input. This not only includes
the parameter value Ao that reﬂects the tendency of obstacles
to avoid the robot, but also the desired velocity vr0 used to
calculate the robot’s attraction. The desired velocity vr0 affects
the attractive force from the goal. That is, by setting the desired
velocity, we can strike a balance between the attractive force
and the repulsive force from obstacles.
We will delve into further details in the subsequent Section
III-C and Section III-D.
C. Avoidance Behavior based on the Social Force ModelAt each time step t, the system ﬁrst employs the Kalmanﬁlter to reﬁne the observed position of an obstacle, and thencalculates the control input of the robot by integrating forcesderived from the Social Force Model.
1) Estimation for the position of obstacle: At a discrete
time step t, upon observing the position of the obstacle, we
use the Kalman ﬁlter to minimize the impact of observation
errors. The ﬁlter reﬁnes this prediction by (7), using the newly
observed position, reo (t), to produce an estimation of the true
value, rbo (t).
hirbo (t) = rbo′ (t) + K(t) · reo (t) − rbo′ (t) ,(7)where rbo′ (t) is the predicted value from the previous time step,
while K(t) is the Kalman gain, calculated by (8).
K(t) =P (t),P (t) + R

(8)

where P and R are pre-deﬁned covariances matrix. After thisstep, we update the covariance matrix P in preparation for the
next prediction by (9).
P (t + ∆t) = (I − K(t))P (t),(9)

We then predict the obstacle’s position for the subsequent
time step rbo′ (t + ∆t). In the prediction period, we use themodel we introduced in Section II, but use (10) to estimate
the obstacle’s velocity.
bo (t) =vb o (∆t) · ∆2 t
rbo (t) − rbo (t − ∆t) + 1/2 · a
∆t

(10)

This prediction is aimed at reﬁning the subsequent time step’s
estimation at t + ∆t, as we showed in (7).
2) Behavior based on Social Force Model: Using the estimated position of the obstacle rbo (t), and the model from Section II, we can utilize (3) to compute the repulsive forces and(4) to calculate the attractive force acting on the robot. Onceboth forces are obtained, the actual control input vr (t + ∆t)
is computed using (11).
vr (t + ∆t) = vr (t) + ar (t)∆t,(11)

where the desired velocity used in (4) is determined using themethod detailed in Section III-D2.
D. Control Considering the Avoidance Behavior of Obstacle
Within this discussion, we delve into a control strategy
that addresses the evasive actions of dynamic obstacles. This
includes estimating the tendency of nearby obstacles to avoid
the robot and the calculation of the desired velocity of therobot. First, based on the observed obstacle position, we infer
the parameter value Ao of the repulsive force acting on thebo ,obstacle, in (3). Then, using the estimated parameter value A
0
we compute the desired velocity vr , ensuring the formulationof a harmonious path. In the estimation step, we use theobserved obstacle position instead of the optimized outcomesfrom the Kalman ﬁlter, aiming to minimize the inﬂuence ofthe Kalman ﬁlter’s behavior on the estimation results.

1) Parameter Estimation for the Model: Using Bayesianestimation (12), we estimate the value of the parameter Ao .
ro (t), reo (t − ∆t)) =P (Aio |e
L reo (t)|Aio , reo (t − ∆t) · P (Aio )P,eo (t)|Aio , reo (t − ∆t) · P (Aio )iL rJt ([vr0 ]) =(12)||gr − rr (t + H∆t, [vr0 ])||
V vr (t + H∆t, [vr0 ]), gr , rr (t + H∆t, [vr0 ]) + ε

H 
X

+ ∆t ·
I gr , rbr (t + i∆t) ,
(16)
i=1

where Aio is a value sourced from the prior probability
distribution. This prior is derived from the posterior probabilityof the previous time step. We set the initial distribution of Aio
to a normal distribution N (0, σ 2 ), assuming that the obstaclemight not change its trajectory. L reo (t)|Aio , reo (t − ∆t) is thevalue of the likelihood function deﬁned by
L reo (t)|Aio , reo (t − ∆t) = f (ero (t); reo,i (t), Σ),

We deﬁne the function Jt ([vr0 ]) by(13)

where f is the probability density functionof the normalbo (t−
distribution of reo (t), given by reo (t)∼N(er(t−∆t)+vo
b o,i (t−∆t)·∆2 t), Σ , with the error terms e(t) is∆t)·∆t+ 21 aassumed to follow a same normal distribution N (0, Σ). Σ isthe covariance matrix of the error term’s components in eachdimension. reo,i (t) determined based on the sampled Aio by
reo,i (t) = reo (t − ∆t) − e(t − ∆t) +1bo (t − ∆t) · ∆t + ab o,i (t − ∆t) · ∆2 t − e(t), (14)v
2where rr (t + H∆t, [vr0 ]) and rr (t + H∆t, [vr0 ]) denote theposition and velocity of robot. Both of them are generated
based on the model introduced in Section III-B. I(g, r) is 1
if [g ̸= r] , otherwise 0. The function V(v, g, r) calculates
the projection of the velocity v on the vector formed by thedestination g and the position r,g−r
V(v, g, r) = v ·,(17)||g − r|| + ε
where ε is a small constant added to prevent division by zero.
To summarize the function Jt ([vr0 ]), the ﬁrst term of (16)calculate the required travel time to the destination for the
robot r after a time horizon of H∆t, and the second term
is added for the case if the robot arriving at the destination
within the time horizon H∆t.
However, merely minimizing the function Jt might lead toa local optimum solution. Therefore, we add a penalty function
related to the robot’s movement velocity, Pv ([vr0 ]), deﬁned by
Pv ([vr0 ]) =

t+H∆t
X h

i
vrmax − V vr (i, [vr0 ]), gr , rr (i, [vr0 ]) .

i=t+∆tb o,i (t − ∆t) is calculated by (2) using Aio .
where aAfter the calculation on time step t, we determine thebo from the expectation of the posterior probestimated value A
ability distribution, and we then treat the posterior probability
distribution as the prior probability distribution on the nexttime step t + ∆t.
2) Calculation of the Desired Velocity: Given the positionof the robot rr (t) and the Kalman ﬁlter corrected positionof the obstacle rbo (t) at current time step t, we calculate thedesired velocity used to calculate the robot’s control input by
considering their future interactions. The desired velocity vr0
for robot, used in (4), is computed by minimize objectivet+(H−1)∆t) in (15), where the independentfunction J ([vr0 ]tt+(H−1)∆tdenotes the sequence of desired vevariable [vr0 ]tlocity vectors from time t to t + (H − 1)∆t. For brevity, we
denote the array of the desired velocity as [vr0 ] in subsequent
equations. In our objective function, we aim to minimize boththe time the robot takes to reach its destination and the riskof collision with the obstacle.
minimize

J ([vr0 ]) =Jt ([vr0 ]) + Pv ([vr0 ]) + Pd ([vr0 ])
(15)

(18)
The addition of a penalty function related to the robot’s
movement velocity aims to ensure the velocity is maximized.
For safety, we also add a penalty function related to the
distance between both subjects, Pd ([vr0 ]) deﬁned by

t+H∆t
X
0
a · ekd dmin −||dr,o (t,[vr ])|| ,Pd ([vr0 ]) =(19)i=t+∆twhere ||dr,o (t, [vr0 ])|| denotes the distance between the robot
and the obstacle, calculated by on the model in Section II.
a and kd are parameters to determine the behavior of thefunction, and dmin is the parameter to decide the minimum
distance between the robot and the obstacle. If the distance
between both subjects becomes less than dmin , the value ofthe function rises sharply. By adding a distance-related penalty
function, it’s possible to avoid the computational issues ofproviding the minimum distance between both subjects as aconstraint condition up to the time horizon H∆t.
The values of the two penalty functions, Pv ([vr0 ]) andPd ([vr0 ]), will be repeatedly calculated until the time horizon
H∆t.
IV. E VALUATIONThis objective function consists of a function Jt ([vr0 ]) that
calculates the required time to the destination for the robot, apenalty function Pv ([vr0 ]) to ensure movement efﬁciency, anda penalty function Pd ([vr0 ]) to ensure safety.

In this research, we conduct a simulation using the MATLAB Optimization Toolbox to validate the effectiveness of the
proposed method. In the simulation, we simulate a robot anda moving obstacle. Subsequently, we analyze the minimumTABLE IS ETTINGS FOR THE ROBOT AND THE O BSTACLE
Initial position r(0)Initial velocity v(0)Goal gMaximum speed vmax

Obstacle o[1.5, 0.8]
[0, 1][1.5, 2.2]
1.5m/sRobot r[1.5, 2.2]
[0, -1][1.5, 0.8]
1.5m/sTABLE IIS ETTING OF THE PARAMETER VALUES
Parameter
Time slot ∆tParameter to control the strength of
repulsive force on robot Ar
Parameter to control the inﬂuence range ofthe repulsive force B
Relaxation time τ
Observation error covariance ΣTime horizon H∆ta in (19)
kd in (19)Minimum distance dmin

Value
0.1s
20
2
0.1s
0.1 · I
15∆t
20
20
1m

distance between the robot and the moving obstacle during
their interaction to demonstrate the safety of the proposed
method. Additionally, we evaluate the robot’s movement time
to validate the movement efﬁciency of our approach.
A. Experimental Setup
We set a ﬂat area of size 3m × 3m as the evaluation
environment, without static obstacles. The settings for the
robot and the obstacle are presented in Table I. We ensurethat the trajectories of the robot and the obstacle inevitablyintersect, leading to an interaction between the two. We
generated the obstacles to behave following the Social ForceModel, introduced in Section II. In this evaluation, we simulate
two cases; the case that the obstacle avoids the robot (thisinvolves assuming the parameter Ao , used in (3) to decide thetendency to avoid the robot, is set to 20) and the case that the
obstacle maintains its original trajectory without any deviation(this involves assuming the parameter Ao = 0). We generate
the observation errors of the obstacle’s position as follow anormal distribution N (0, 0.12 ). In this evaluation, we set theparameters as shown in Table II. We ran the simulation 100
times.
In this evaluation, we compare our method with a methodbothat calculates the desired velocity using a ﬁxed estimation A
to evaluate the signiﬁcance of estimating Ao .
B. Results
1) Estimation of the obstacles behavior: Before comparing
with the cases of ﬁxed parameters, we investigate the parameter estimation result produced by our method. Figures 3 and4 show examples of the estimated parameter at each time
bo is estimated to be a small
slot. Figure 3 indicates that A
value in the early time slots even when the actual value of
Ao = 20. That is, our method considers a case where the
obstacle may not change its trajectory. However, as the time

bo when the Fig. 4. Estimation result A
bo when theFig. 3. Estimation result A
actual value of Ao is set to 20.
actual value of Ao is set to 0.

bo is estimated to have a large value. This
slot continues, Achange is attributed to our method detecting that the obstacleis actively avoiding the robots. Consequently, our approach
can predict the behavior of obstacles that maneuver to avoid
robots. On the other hand, Figures 4 indicates that our method
bo to have a small value when the actual value of
estimates A
Ao = 0. As a result, our method can handle the obstacle as
the obstacle that does not change its trajectory.
2) Result when the obstacle will avoid the robot (Ao =20): Figure 5 presents the results for the scenario where the
obstacle will avoid the robot, means that the actual value of
Ao is set to 20. In this scenario, obstacles avoided the robot.
By accurately predicting the behavior of the obstacles (in thebo = 20), wecase of the method with a ﬁxed parameter A
can set an optimal strategy for controlling the robot to avoid
collisions with obstacles and achieve efﬁcient movement. In
contrast, a method that assumes the obstacle does not change
its trajectory, implying that the estimation of the parameterbo is ﬁxed at 0, requires more substantial evasive actions.
A
As a result, more time is required to reach its goal. Actually,
Figure 5 shows that the minimum distance from the obstacleis slightly far and the movement time is large in the case ofbo = 0. Figure 5 also
the method with the ﬁxed parameter A
indicates that our method achieves similar movement time tobo = 20. This is because our method
the method with a ﬁxed Abestimated Ao as the values indicating that the obstacle avoids
the robot.
3) Result when the obstacle will not avoid the robot (Ao =0): Figure 6 presents the results for the scenario where the
obstacle will not avoid the robot, means that the actual valueof Ao is set to 0. In this scenario, the obstacle did not change
its trajectory. In contrast, a method that assumes the obstacle
will avoid the robot, implying that the estimation of thebo is ﬁxed at 20, results in ineffective robot control.
parameter A
Consequently, the minimum distance between the robot andthe obstacle becomes less than 1, which violates our behavioral
constraint, as shown in Figure 6(a). On the other hand, thebo = 0 keeps the minimum
method with the correct estimation A
distance larger than 1 in most cases, though the results for our
method did exhibit a few outliers. Figure 6 also indicates that
our method achieves a result similar to that of the method withbo = 0. This is because our method
the correct estimation A
can estimate Ao accurately even when the obstacle does notFig. 5. Comparison with the scenario without parameter estimation. The Fig. 6. Comparison with the scenario without parameter estimation. The
results for the case where the actual value of Ao = 20, indicate that the results for the case where the actual value of Ao = 0, indicate that the
obstacle will attempt to avoid the robot.
obstacle will not attempt to avoid the robot.

change its trajectory.
4) Discussion: From our simulations, it is evident that byestimating the obstacle’s intention to avoid the robot, ourmethod consistently calculates an appropriate desired velocity
even if the observation includes errors. This ensures that the
robot not only moves efﬁciently towards its target, but also
maintains a safe distance from moving obstacles, even in
the presence of observational errors. The robustness of ourapproach was demonstrated across 100 iterations with varying
noise levels. Although inevitable path intersections test thecapabilities of our method, it is worth noting that potential
prediction inaccuracies can compromise either the safety or
efﬁciency of our method. Our method demonstrated a strong
balance between these two crucial aspects in most situations.
V. C ONCLUSION
In this paper, we utilized the Social Force Model to understand and predict obstacle avoidance behaviors, then appliedthese insights to calibrate the robot’s control input, striking
a balance between efﬁciency and safety. In the future, wewill delve deeper into the practical application of our method,
encompassing deployment on real robots and testing in realworld environments. Additionally, we will consider the volume
of obstacles and extend the obstacles to humans, individuals
who might exhibit irrational behavior.
ACKNOWLEDGMENT
These research results were partly obtained from the commissioned research (JPJ012368C00701) by National Institute of Information and Communications Technology (NICT),Japan.
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