# Universal scaling and criticality of extremes in random matrix theory

Sina Saber  Department of Physics, University of Tehran, P. O. Box 14395-547,
Tehran, Iran  Abbas Ali Saberi  (corresponding author<EMAIL_ADDRESS>Department of Physics, University of Tehran, P. O. Box 14395-547, Tehran, Iran
Institut für Theoretische Physik, Universität zu Köln, Zülpicher Str. 77,
50937 Köln, Germany

###### Abstract

We present a random-matrix realization of a two-dimensional percolation model
with the occupation probability $p$. We find that the behavior of the model is
governed by the two first extreme eigenvalues. While the second extreme
eigenvalue resides on the moving edge of the semicircle bulk distribution with
an additional semicircle functionality on $p$, the first extreme exhibits a
disjoint isolated Gaussian statistics which is responsible for the emergence
of a rich finite-size scaling and criticality. Our extensive numerical
simulations along with analytical arguments unravel the power-law divergences
due to the coalescence of the first two extreme eigenvalues in the
thermodynamic limit. We develop a scaling law that provides a universal
framework in terms of a set of scaling exponents uncovering the full finite-
size scaling behavior of the extreme eigenvalue’s fluctuation. Our study may
provide a simple practical approach to capture the criticality in complex
systems and their inverse problems with a possible extension to the
interacting systems.

Random matrix theory (RMT) and percolation theory are considered as two
cornerstones of probability theory with numerous applications in statistical
physics and description of various complex systems r1 ; r2 ; r3 ; r4 ; r5 ; r6
; r7 . There have been also simultaneous contributions of both theories in the
past to describe diverse physical problems ranging from the integer quantum
Hall effects r9  to wireless communications r10 . RMT and percolation offer
their own distinct universal classification schemes characterized by universal
distribution functions of their extremes and a set of critical exponents,
respectively. The emergence of extreme value theory in the dynamics of
percolation models has just recently been reported r11 . However, to our best
knowledge, there has been no report if these two theories are mixed. Here we
aim to fill this gap by presenting a random matrix realization of a two-
dimensional (2D) percolation model, and discuss how a universal scaling
framework emerges near the criticality.

We directly map a percolation problem with occupation probability $p$ on a
square lattice of linear size $L$, to a $L\times L$ random matrix
$\mathcal{M}^{\prime}$ where every element $\mathcal{M}^{\prime}_{ij}$ is
equal to $+1$ with probability $p$ or $-1$ with probability $1-p$. A bond
percolation model of this kind is known r6 ; r7 ; r8  to exhibit a continuous
phase transition at the critical threshold $p_{c}=1/2$ with a step-like
percolation probability function about $p_{c}$ in the thermodynamic limit, and
power-law divergent correlation length $\xi\propto|p-p_{c}|^{-\nu}$ and mean
cluster size $\chi\propto|p-p_{c}|^{-\gamma}$ with a bunch of characteristic
universal exponents $\nu$, $\gamma$, etc. In this letter, we report on a
manifestation of such critical concepts in the random matrix realization of a
percolation problem. We shall indeed refer to the matrix and percolation
ensembles interchangeably.

In our mapping, the matrix elements $\mathcal{M}^{\prime}_{ij}$ are chosen to
be either $-1$ or $+1$ (instead of being $0$ or $1$), to satisfy the mean zero
condition at $p_{c}=1/2$ which typically appears in the random matrix
literature, and the additional advantage of symmetry
$\mathcal{M}^{\prime}_{ij}\leftrightarrow-\mathcal{M}^{\prime}_{ij}$ when
$p\leftrightarrow 1-p$. Here, the statistics of extreme eigenvalues is of
interest. In order to have real eigenvalues, we use symmetrization
$\mathcal{M}=(\mathcal{M}^{\prime}+\mathcal{M}^{\prime T})/2$, where
$(\cdot)^{T}$ denotes the transpose of the matrix. Thus, the real symmetric
matrix $\mathcal{M}$ has $\frac{1}{2}L(L+1)$ independent elements
$\mathcal{M}_{ij}$ which can take values $-1$, $0$ and $+1$ with probabilities
$(1-p)^{2}$, $2p(1-p)$ and $p^{2}$, respectively, for $i>j$. The diagonal
elements $\mathcal{M}_{ii}$ can be either $-1$ or $+1$ with probabilities
$(1-p)$ or $p$, respectively. Every element has the average
$\mu=\langle\mathcal{M}_{ij}\rangle=2p-1$. The zero mean condition thus only
holds for $p_{c}=1/2$.

The limiting distribution of eigenvalues of various random symmetric matrices
is given by Wigner’s semicircle law as the size of the matrix approaches
infinity, i.e., $L\rightarrow\infty$ r12 ; r13 . This law was originally
stated for matrices with entries taking the values $\pm 1$ each with
probability $1/2$ r12 , and then it has been generalized to matrices with
entries distributed symmetrically about zero and having finite moments of all
orders r13 .
Figure 1 presents the probability distribution function of the eigenvalues for
an ensemble of rescaled symmetric random matrices $\\{\mathcal{M}\\}$ as a
realization of a percolation model on a square lattice with occupation
probabilities $p=0.2,0.5$ and $0.8$. The matrices for all $p$’s are rescaled
by a factor of $1/\sqrt{L}$, so that the edges of the semi-circle distribution
function of eigenvalues would lie in the interval $[-\sqrt{2},\sqrt{2}]$. As
shown in Figure 1, since the zero-mean condition holds for $p=0.5$, the
distribution of eigenvalues (the red dashed line) coincides with the Wigner’s
semicircle law with edges exactly located at $\pm\sqrt{2}$. However, the
distribution of eigenvalues for $p=0.2$ and $0.8$ consists of two parts: (i) A
semicircle part for the bulk eigenvalues whose edges approach zero as
$|p-p_{c}|\rightarrow 1/2$, and due to the symmetry $p\leftrightarrow 1-p$,
both collapse on the top of each other for $p=0.2$ and $0.8$, and (ii) A
disjoint isolated distribution of the extremes including the largest
(smallest) eigenvalues for $p=0.8$ ($0.2$) with a perfect Gaussian
distribution (shown in the Insets of Fig. 1).

Figure 1:  Eigenvalue distribution of an ensemble of rescaled symmetric random
matrices as a realization of a percolation model on a square lattice of size
$L=2^{10}$ with occupation probabilities $p=0.2,0.5$ and $0.8$. For $p=0.5$
(the red dashed line) which satisfies the zero mean condition, reproduces the
Wigner’s semicircle law with edges exactly located at $\pm\sqrt{2}$. The
distributions for $p=0.2$ and $0.8$ have two parts: (i) A semicircle law for
the bulk eigenvalues whose edges approach zero, and due to the symmetry
$p\leftrightarrow 1-p$, both collapse on the top of each other for $p=0.2$ and
$0.8$, and (ii) A disjoint isolated Gaussian distribution of the extremes
(Insets).

For real symmetric (orthogonal) random matrices, it is known r14  that the
limiting distribution of the largest eigenvalue sitting at the spectral edge
is given by the Gaussian Orthogonal Ensemble (GOE) Tracy-Widom distribution
r15  with the characteristic skewness $\sim 0.29346$ and excess kurtosis $\sim
0.16524$. In Figure 2, we report our estimates for the skewness and the excess
kurtosis of the largest eigenvalue $\lambda_{\text{max}}$ as a function of
$p$, compared with the corresponding predictions for the GOE and Gaussian
distributions (shown by the horizontal dashed lines). Our results suggest the
following: (i) The isolated Gaussian distribution for $p>p_{c}$ ($p<p_{c}$)
(see Fig. 1) only includes the first largest (smallest) eigenvalues, and, the
second largest (smallest) eigenvalues are sitting at the spectral edges. (ii)
The step-like behavior in Fig. 2, is suggestive of a sharp transition very
close to the critical threshold i.e., $p\rightarrow p_{c}$, when the two
Gaussian and GOE distributions merge r15-1 . We noted that a similar
observation has been recently reported in a different context r16  for the
extremal eigenvalues of sparse Erdős-Rényi graphs. As we will see in the
following, this merging in our model is accompanied by some power-law
divergences in the statistics of the first largest/smallest extremes which can
be described within a remarkable universal scaling framework.

Figure 2: The skewness and the excess kurtosis of the largest eigenvalues
$\lambda_{\text{max}}$ estimated from an ensemble of $10^{5}$ independent
percolation realizations on a square lattice of size $L=2^{11}$ as a function
of $p$. The horizontal dashed lines indicate the corresponding predictions for
the GOE and Gaussian distributions.  Figure 3: The average absolute value of
the second extremes $\langle|\lambda_{\text{2nd}}|\rangle$ in the limit of the
infinite system size $1/L\rightarrow 0$ as a function of $p$. The solid line
shows a semi-ellipse functionality which well describes our data.

Let us now investigate how the location of the semicircle edges in Fig. 1
behaves as a function of $p$. To this end, we have measured the average
absolute value of the second extremes $\langle|\lambda_{\text{2nd}}|\rangle$
in the limit of the infinite system size $L\rightarrow\infty$ for various
choices of $p$. Our computed data are shown by symbols in Fig. 3. The best fit
to our data suggests a perfect agreement with a semi-ellipse functionality
shown by the solid line, i.e.,

$\Big{[}\frac{\langle|\lambda_{\text{2nd}}|\rangle}{\sqrt{2}}\Big{]}^{2}+\Big{[}\frac{(p-1/2)}{1/2}\Big{]}^{2}=1.$
(1)

This can be used for the rescaling of the matrix $\mathcal{M}$ by an
additional factor $1/\sqrt{1-\langle\mathcal{M}_{ij}\rangle^{2}}$, to enforce
the edges of the semicircles to stay at $\pm\sqrt{2}$ for all $p$’s.

In the rest of the Letter, we shall focus our attention on the statistical and
scaling properties of the isolated extreme eigenvalue and its finite size
effects. It is shown in r17  that for a real symmetric matrix with nonzero
mean, there always exists an eigenvalue whose magnitude is greater than or
equal to the mean of the sums of the elements in a row of the matrix. This
means that the absolute value of the largest eigenvalue in $\mathcal{M}$ is in
the order of $|\langle\mathcal{M}_{ij}\rangle|L$, i.e.,
$\langle|\tilde{\lambda}_{\text{max}}|\rangle\sim|2p-1|L$ where the tilde
denotes for the eigenvalues of ’non-rescaled’ matrices (Note that the largest
eigenvalue of a deterministic matrix $\mathcal{M}_{ij}=\mu$ for $i\geq j$, is
equal to $\mu L$, and all the other eigenvalues are equal to zero. Thus the
fluctuation of the entries of the matrix each with the variance $\sigma^{2}$
for $i>j$, changes $\tilde{\lambda}_{\text{max}}$ only with a normal random
variable of expectation $\sigma^{2}/\mu$ and variance $2\sigma^{2}$ not
growing with the size $L$—see Fig. 1.). The quantity of interest is indeed the
average fluctuation of the extreme eigenvalue i.e.,
$\langle|\tilde{\lambda}_{\text{max}}|\rangle_{c}:=\langle|\tilde{\lambda}_{\text{max}}|\rangle-|2p-1|L$
in the asymptotic limit $L\rightarrow\infty$.
We have carried out extensive Monte Carlo simulations of extremely large
system sizes $100\leq L\leq 146400$ of our model in the vicinity of the
critical occupancy $p_{c}=1/2$ involving strong finite size effects, to
estimate $\langle|\tilde{\lambda}_{\text{max}}|\rangle_{c}$ for every $p$ in
the limit $L\rightarrow\infty$. As shown in the left Inset of Figure 4, the
finite-size effects necessarily become essential near the critical point and
the convergence of $\langle|\tilde{\lambda}_{\text{max}}|\rangle_{c}$ occurs
at much larger system size when $p\rightarrow p_{c}$. Our estimates for the
asymptotic $\langle|\tilde{\lambda}_{\text{max}}|\rangle_{c}$ as a function of
$p$ in the vicinity of $p_{c}$ are shown in the main panel of Fig. 4. The best
fit to our data suggests the following power-law behavior:

$\big{\langle}|\tilde{\lambda}_{\text{max}}|\big{\rangle}_{c}=\frac{1}{4}|p-p_{c}|^{-1}.$
(2)

This finding confirms our above mentioned expectation that the average
fluctuations about $\mu L$ should asymptotically be given by
$\sigma^{2}/\mu=2p(1-p)/(2p-1)$, which gives rise to the scaling relation
given in Eq. (2) as $p(1-p)|p-p_{c}|^{-1}$ with the identification that the
prefactor $p(1-p)\rightarrow 1/4$ when $p\rightarrow p_{c}=1/2$.

Figure 4: The scaling law developed for the finite-size behavior of the
fluctuations of the first extreme eigenvalues. Main: Asymptotic estimates of
the average fluctuation of the largest absolute eigenvalue versus $p-p_{c}$.
The solid line is the power-law with exponent $-1$ which best describes our
data. Left Inset: The average fluctuation of the largest eigenvalue for five
occupation probabilities with $p>0.5$ as a function of inverse system size.
The average fluctuation increases with system size up to $L^{*}$ after which
it saturates to the asymptotic value. Right Inset: All data that are shown in
the left Inset collapse onto a single universal curve when they are suitably
normalized according to the scaling formalism proposed in Eq. 4 with
$\alpha=1$, $\beta=1/2$ and $z=2$.

Although Eq. (2) is asymptotically true in the infinite size limit
$L\rightarrow\infty$, the left Inset in Figure 4 suggests that the extreme
eigenvalue’s fluctuation generally depends on the length scale of observation:
At a given $p$, the fluctuation grows with system size until a characteristic
length scale $L^{*}$ above which, it saturates to a size-independent value. We
are interested in if, in analogy with the roughness growth in non-equilibrium
stochastic surface-growth phenomena, a ”Family-Vicsek”-like scaling law
provides a universal framework for our observations summarized in Figure 4. We
have examined our hypothesis, and the answer was surprisingly consistent. We
find that the following spatio-occupational scale-invariant behavior governs
the fluctuations of the first extreme eigenvalue

$\big{\langle}|\tilde{\lambda}_{\text{max}}|\big{\rangle}_{c}(p,L)=|p-p_{c}|^{-\alpha}\mathpzc{F}\bigg{(}\Big{(}\frac{1}{L}\Big{)}|p-p_{c}|^{-z}\bigg{)},$
(3)

where the universal function $\mathpzc{F}(x)\rightarrow 1/4$ as $x\rightarrow
0$ (see Eq. (2)) and $\mathpzc{F}(x)\sim x^{-\alpha/z}$ as
$x\rightarrow\infty$, so that
$\langle|\tilde{\lambda}_{\text{max}}|\rangle_{c}$ grows with size like
$L^{\alpha/z}$ until it saturates to $(1/4)|p-p_{c}|^{-\alpha}$ when
$L\sim|p-p_{c}|^{-z}$. More specifically,

$\big{\langle}|\tilde{\lambda}_{\text{max}}|\big{\rangle}_{c}(p,L)\propto\begin{cases}|p-p_{c}|^{-\alpha}&\text{if
$L^{*}\ll L$}\\\ (1/L)^{-\beta}&\text{if $L\ll L^{*}$},\\\ \end{cases}$ (4)

with $L^{*}$ being a saturation length proportional to $|p-p_{c}|^{-z}$, and
$\alpha$, $\beta$ and $z=\alpha/\beta$ are three positive scaling exponents
featuring the universality of the extreme eigenvalue’s fluctuation.
As shown in the right Inset of Fig. 4, we find that a perfect collapse onto a
universal function $\mathcal{F}$ is achieved when the data is appropriately
rescaled by the following examined (conjectured) exponents

$\alpha=1,\hskip 22.76228pt\beta=1/2,\hskip 22.76228ptz=2.$

Within the standard Family-Vicsek scaling law in surface-roughness growth, one
may note that the same corresponding set of exponents describes the
universality of the linear Edward-Wilkinson (EW) equation r18  (with exactly
known exponents $\alpha=(2-d)/2$, $\beta=(2-d)/4$ and $z=2$) in $0+1$
dimensions.

Figure 5: Main: The effective exponent $\zeta_{\text{eff}}$ defined in Eq. (5)
as a function of $1/L$ for different values of occupation probability $p$ in
the vicinity of the critical point $p_{c}$. All data behave linearly with
$1/L$ with a negative slope $\xi^{*}(p)$ and clear extrapolations to $1/2$ in
the infinite-size limit (solid lines). Inset: Power-law divergence of
$\xi^{*}(p)$ measured from the data presented in the Main panel. The solid
line shows the power-law $\frac{1}{8}|p-p_{c}|^{-2}$ which best describes our
data.

Until now, we have focused on the scaling properties of the ’non-rescaled’
average extreme eigenvalue fluctuations around the mean value $|2p-1|L\propto
L^{1}$. This means that the leading size contribution in the ’rescaled’
average extreme eigenvalue is proportional to $L^{\zeta}$ with $\zeta=1/2$. We
are interested in realizing to what extent the exponent $\zeta=1/2$ for the
average extreme eigenvalue $\langle|\lambda_{\text{max}}|\rangle$ remains
valid when $p\rightarrow p_{c}$. Without loss of generality, in the following
we discuss the supercritical regime $p>p_{c}$with a straightforward extension
to $p<p_{c}$ regime due to the symmetry $p\leftrightarrow 1-p$. Our attempt to
carefully validate the exponent $\zeta=1/2$, has led to capture the power-law
behavior we have already observed for $L^{*}$ in more detail.
In order to delicately estimate the exponent $\zeta$ characterizing the
dominant size dependence of $\langle\lambda_{\text{max}}\rangle$, we define an
effective exponent $\zeta_{\text{eff}}$ which allows to easily monitor the
approach to the true asymptotic behavior. For a given occupation probability
$p$, $\zeta_{\text{eff}}(L_{k})$ is defined by the successive slopes of the
line segments connecting two neighboring points of
$\big{(}L_{k-1},\langle\lambda_{\text{max}}\rangle(L_{k-1})\big{)}$ and
$\big{(}L_{k},\langle\lambda_{\text{max}}\rangle(L_{k})\big{)}$ in which
$\langle\lambda_{\text{max}}\rangle(L_{k})$ stands for the average largest
eigenvalue for a system size $L_{k}$,

$\zeta_{\text{eff}}(L_{k})=\frac{\ln\big{[}\langle\lambda_{\text{max}}\rangle(L_{k})/\langle\lambda_{\text{max}}\rangle(L_{k-1})\big{]}}{\ln[L_{k}/L_{k-1}]}.$
(5)

It is plotted against $1/L$ in Figure 5 for various choices of $p$ to capture
their extrapolations in the infinite-size limit. We observe that the effective
exponent increases almost linearly with $1/L$ for all $p>p_{c}$ with a clear
extrapolation to $\zeta=1/2$ in the limit $1/L\rightarrow 0$ (the solid
lines). However, our remarkable observation from Fig. 5 is that in the
sufficiently large system-size regime where the effective exponent
$\zeta_{\text{eff}}$ increases linearly with the inverse system size,

$\zeta_{\text{eff}}=\frac{1}{2}-\xi^{*}(p)\bigg{(}\frac{1}{L}\bigg{)},$ (6)

the corresponding slope $\xi^{*}(p)$ exhibits a power law divergence as the
critical threshold $p_{c}$ is approached (see the Inset of Fig. 5). Our best
fit to our data, provides the following conjecture in perfect agreement with
our data

$\xi^{*}(p)=\frac{1}{8}|p-p_{c}|^{-2}.$ (7)

The length scale $\xi^{*}(p)$ should be identical with
$L^{*}\propto|p-p_{c}|^{-z}$ with $z=2$, as the only relevant length scale in
our study. This is independently verified when we have marked
$(1/L^{*})|p-p_{c}|^{-2}$ with the vertical dotted line in the right Inset of
Fig. 4 where crosses the horizontal axis at $8$. For every $p>p_{c}$,
$\xi^{*}(p)$ remains finite and the exponent $\zeta=1/2$ is always achieved in
the asymptotic limit $L\rightarrow\infty$. Exactly at the critical occupancy
$p_{c}$, the extreme eigenvalues reside on the spectral edges in the finite
interval $[-\sqrt{2},\sqrt{2}]$ which necessarily needs $\zeta=0$.

Our study can be appropriately generalized to construct a random-matrix
representation of a 2D bond percolation model described by a sum of two
independent random real matrices $\mathcal{M^{\prime}}^{h}$ and
$\mathcal{M^{\prime}}^{v}$, one for the horizontal and one for the vertical
bonds:
$\mathcal{M^{\prime}}=\mathcal{M^{\prime}}^{h}+i\mathcal{M^{\prime}}^{v}$, in
which $\mathcal{M^{\prime}}^{h}_{ij}$ and $\mathcal{M^{\prime}}^{v}_{ij}$
denote as the horizontal and vertical bonds to the right resp. above site
($i,j$). This associates to each open/closed horizontal bond a value $\pm 1$,
and to each open/closed vertical bond a complex value $\pm i$. Therefore, each
bond configuration corresponds to a matrix $\mathcal{M^{\prime}}$ with
elements $\pm 1\pm i$. Thus 2D bond percolation on the square lattice is
associated with an ensemble of asymmetric complex matrices.
We have carried out new simulations to analyze the eigenvalue spectrum of an
ensemble of such matrices after symmetrization
$\mathcal{M}=(\mathcal{M}^{\prime}+\mathcal{M}^{\prime t})/2$, where
$(\cdot)^{t}$ denotes the conjugate transpose of the matrix. We find that the
same scaling laws proposed in Eqs. (3) and (4) with the same scaling exponents
$\alpha=1,\beta=1/2$ and $z=2$ well describe the 2D bond percolation model
with the only difference that here the corresponding prefactors in Eqs. (2)
and (7) take on exactly twice values, i.e.,
$\big{\langle}|\tilde{\lambda}_{\text{max}}|\big{\rangle}_{c}=\frac{1}{2}|p-p_{c}|^{-1}$,
and $\xi^{*}(p)=\frac{1}{4}|p-p_{c}|^{-2}$.

To conclude, we have presented a random matrix realization of a 2D percolation
model which offers a novel formalism to study complex systems in a rather
simple approach that is in all aspects consistent with the concepts in
critical phenomena. We find that the behavior of the model is encoded in the
fluctuations of the extreme isolated eigenvalues whose divergence signals the
criticality in the vicinity of the critical point. We have established a
scaling law, which provides a universal framework characterizing the finite-
size effects in terms of a set of critical exponents. Self-consistency
conditions along with analytical arguments let us making exact conjectures for
the exponents and prefactors in perfect agreement with our computations which,
however, calls for future mathematical attempts for their rigorous proofs. The
random matrix realization of interacting and/or correlated statistical models
may provide an inroad for progress towards the understanding of elusive models
in two dimensions.

Acknowledgment. We would like to thank S.N. Majumdar, H. Spohn, and especially
P.L. Ferrari for useful discussions. A.A.S. acknowledges the supports from the
Alexander von Humboldt Foundation (DE) and the research council of the
University of Tehran. We also thank the High-Performance Computing (HPC)
center at the University of Cologne, Germany, where a part of computations has
been carried out.

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# Benchmarking local motion planners
for navigation of mobile manipulators

Sevag Tafnakaji† Hadi Hajieghrary† Quentin Teixeira† Yasemin Bekiroglu†,∗
†Department of Electrical Engineering, Automatic Control Research Group,
Chalmers University of Technology, Göteborg, Sweden. ∗University College
London, Statistical Machine Learning Group, AI Center, London, UK. Email:
<EMAIL_ADDRESS>
###### Abstract

There are various trajectory planners for mobile manipulators. It is often
challenging to compare their performance under similar circumstances due to
differences in hardware, dissimilarity of tasks and objectives, as well as
uncertainties in measurements and operating environments. In this paper, we
propose a simulation framework to evaluate the performance of the local
trajectory planners to generate smooth, and dynamically and kinematically
feasible trajectories for mobile manipulators in the same environment. We
focus on local planners as they are key components that provide smooth
trajectories while carrying a load, react to dynamic obstacles, and avoid
collisions. We evaluate two prominent local trajectory planners, Dynamic-
Window Approach (DWA) and Time Elastic Band (TEB) using the metrics that we
introduce. Moreover, our software solution is applicable to any other local
planners used in the Robot Operating System (ROS) framework, without
additional programming effort.

##  I INTRODUCTION

Motion planning is a key component for mobile robotic applications [1, 2, 3,
4]. In order to reduce the complexity, motion planning is often divided into
two logical parts: global and local planning [5]. A global planner searches
for the overall path toward the goal on the static map of the environment. The
local planner uses the sensory information of the robot to perceive the
current state of the robot’s immediate surroundings and generates feasible
trajectories to be followed by the robot. Avoiding any dynamic or static
obstacles that may or may not have been included in the given global map is
the responsibility of the local planner. The local planner fuses data from
sensors such as laser scanners, ultra-sound sensors, and various cameras to
predict the state of the immediate surroundings of itself, and plan a
dynamically and kinematically feasible trajectory for the mobile robot to
follow. Therefore, the performance of the local planner will directly affect
the overall success outcome of a mobile robot’s task. In the case of a mobile
manipulator, this matter is of paramount importance, since any rugged motion
of the mobile base may translate into catastrophic trembling of the payload
grasped by the manipulator.

Figure 1: (Left) A mobile manipulator, including a Mobile Industrial Robots
MiR mobile base, a Universal Robot UR10 robotic arm, and OnRobot RG6 gripper.
(Right) The corresponding robot model shown in Rviz (3D visualization tool
from ROS).

In this paper, we present a framework to evaluate the performance of a local
planner and use the framework to benchmark two predominant methods, the latest
implementation of Dynamic-Window Approach (DWA), called DWB, [6], and Time
Elastic Band (TEB) [7, 8]. We evaluate their performance in terms of
generating smooth feasible trajectories under various conditions.
Nevertheless, many of the presented methodologies and provided machinery can
be easily used to benchmark other local planners. The result of the benchmark
presented in this document aims to determine how effectively a planner
navigates an industrial mobile manipulator moving a heavy load. We introduce
an extensive set of metrics, and hypothesize that these measures will paint a
clear picture about the performance of a mobile manipulator. Then we test our
hypothesis by inspecting the performance of a realistically simulated
industrial mobile manipulator in different environments.

The Dynamic-Window Approach to planning the trajectory for a mobile robot
constructs the velocity space around the robot, i.e., a space which is
accessible with a span of the possible linear and angular velocities. Within
this space a small window around the current pose of the robot is chosen. Any
of the velocity profiles which may lead the robot to collision with an
obstacle is removed. The remaining set of velocities that are deemed safe is
sampled, and the samples are simulated to determine where they would drive the
robot if they were applied. The velocity commands that offer the best result
are chosen as the control signal and then sent to the robot. The performance
is usually measured based on the advance the robot will make along the global
path.

The DWB is a sampling-based algorithm. In contrast, the Time Elastic Band
approach (TEB) is a potential-field optimization-based algorithm. The TEB
approach is designed to deform and optimize the global trajectory, which is
initially coarsely planned and is purely geometric. The deformation of the
global trajectory is guided by the presence of potential fields which are to
design to shorten the path and obtain the shortest path, while maintaining a
separation between the path and obstacles [9].

The primary task of a local planner is to turn a path generated by the global
planner into a dynamically feasible trajectory, which can be followed by the
robot. The similarly important task of the local planner is to re-evaluate the
path generated by the global planner, and determine if that path is still
feasible. The initially proposed path might no longer be feasible due to
several factors: the map uploaded to the robot may not be valid anymore, since
some changes may have occurred in the environment; or a unexpected obstacle is
found on the path of the robot. In both cases, it is the responsibility of the
local planner to generate a feasible and safe trajectory for the robot to
follow. In this paper, we are testing the performance of the local planner.
However, it would be quite easy to deploy this framework to benchmark any
dynamic obstacle avoidance algorithm in the Robot Operating System (ROS)
Framework.

The proposed benchmarking framework is implemented in ROS. We run it for a
simulated mobile manipulator (see Fig.1) inside the Gazebo simulation. In this
paper, we present the results of the benchmarking of DWB and TEB in guiding a
mobile manipulator using three Gazebo worlds: playground, warehouse, and
office environment, see Fig. 2. In each case, we benchmark the performance of
the local planner for two scenarios: first, the complete map of the
environment is available for the robot with high fidelity. Second, some
objects in the environment are not included in the map the robot possesses. In
all of these tests, the performance of the local planners is measured using
the metrics we have defined to quantify the smoothness of the planned path,
the smoothness of the motion of the manipulator’s end-effector, and the
discrepancy between the planned global path and the actual one. Also, we
compare the algorithms based on overall performance metrics such as total
time, distance traveled, and final position accuracy.

In [10] and [11], motion planning algorithms such as DWB, TEB, and E-band are
tested using small mobile robots in both simulation and real experiments. The
general consensus is that DWB performs the fastest, E-band is the most
accurate local planner, and TEB plans motions that lead the robot to the goal
slightly faster than the other two algorithms. The authors also report that
TEB would react the fastest to a dynamic obstacle, while DWB sometimes stops
to calculate its subsequent route. In [12] and [13], similar results are
reported with a heavy differential-driven robot carrying a Baxter manipulator.
These results indicate that in general, TEB performs better than the baseline
methods at obstacle avoidance in simulation and real experiments.

In order to improve navigation benchmarking efforts, different standards are
proposed, [14], [15]. In [14], the authors introduce the standard of worlds
with different types of obstacles. They also propose adding variations in how
the dynamic obstacles are set up (from simpler to more complicated
environments) and perform experiments in real and simulation environments.
There are also studies, [15] focusing more on crafting standard metrics that
can meaningfully benchmark the navigation algorithms using standard
worlds/environments.

Metrics used in evaluations are key to analyze the performance of the
algorithms. In [15], the authors have found that TEB generates much smoother
paths when using a metric they introduced. This metric constitutes the sum of
the difference between two position measurements when the mobile base is
traveling along a path. However, this method may fail to discern how smooth
the generated path is. In [16], the authors simulate a mobile robot in
different scenarios to measure the generated path’s smoothness. They analyze
energy consumption in a real robot for trajectories using different smoothness
factors. They also analyze social acceptance for different smoothness factors
by presenting different simulated situations to different people. Their
results indicate that in general smoother paths decrease energy consumption
and increase acceptability, as long as important factors such as distance to
people are fulfilled.

Our proposed framework is different from the previous approaches in the
following aspects: we propose a framework to evaluate the performance of local
trajectory planners to generate dynamically and kinematically feasible
trajectories for a mobile manipulator carrying a load; and, we concentrate on
the smoothness of the motion for the load carried by the manipulator. In
addition, our benchmarking measures indicate, in return, how the motion of the
manipulator affect the maneuver of the mobile base.

Studying the stability of the local planners in this context reveals the
necessary precautions that should be taken while designing a mobile
manipulator to transport sensitive material, as mobile manipulators are more
complex and have more noise sources than the standalone mobile robot. There
have been many benchmarks and evaluations on local planners for using mobile
robots, but not many that focus on applications of mobile manipulators. This
paper focuses on the performance of local planners for the specific use cases
of a mobile manipulator rather than a more simple mobile robot. With the added
complexity of the arm attached to the base, there are more components and
aspects that the local planner can affect. One of these, which is the paper’s
primary focus, is how smoothly the mobile manipulator can carry loads. The
ability of the robot to transport a load can be hindered due to the poor
performance of the local planner. If the abilities of a mobile manipulator is
not quantified systematically, the user will limit its use due to, mainly,
safety reasons, and leave much of its economic potential unrealized. The
proposed framework to benchmark a local planner in particular, and a mobile
manipulator in general, is intended to help devise an optimal navigation stack
to fully exploit a mobile manipulator’s functionalities in an industrial
setting.

##  II PROPOSED METHOD

In this paper, we benchmark the local planning algorithms for mobile
manipulators in terms of 1) the total time taken to traverse from the origin
to the goal; 2) the total distance traveled to reach the goal ; 3) the
accuracy of the position of the end-effector; 4) the capability of the
algorithm to avoid the obstacles; and 5) the smoothness of the planned path.

(a) Playground

(b) Office

(c) Warehouse

Figure 2: Simulated Gazebo worlds with/without dynamic obstacles, used to
benchmark the TEB and DWB algorithms.

The procedure is repeatable, i.e., it is iterated for the desired number of
times using the same test. The user can choose and modify different parameters
of the planner, as well as the simulated system. During each trial, all the
necessary measurements are collected to create the set of proposed metrics to
analyze the performance of the local planner:

  1. A)

Path smoothness: As the robot travels along the global path, a measurement of
its pose is sampled and stored. With enough measurements, i.e., $N$, the taken
path is retraced by simply treating the pose as vectors and finding how
smoothly the robot travels between measurements, i.e.,
$\Delta\mathbf{x}_{i}=x_{i}-x_{i-1},i=2,\dots,N-1$, where $x_{i}$ is the
measurement of the position of the robot in Cartesian coordinates. We can find
the angle of a path segment as the angle between two succeeding positions and
present the measure of path smoothness, $p_{s}$, as:

$p_{s}=\sum\alpha_{i},$ (1)

where, $\alpha_{i}$ is the angle between the vectors $\mathbf{x}_{i}$ and
$\mathbf{x}_{i+1}$. The desired smooth path will be the one that there is only
a slight angle between each two consequent vectors; preferably, the angle
would be zero; hence, the optimal value for this metric would be as small as
possible. Another approach for this metric could be to take the inverse of the
sum of all angles, which would make it desirable to maximize this value.

  2. B)

End-effector velocity smoothness: This measures the discrepancy between the
actual configuration of the robot - more specifically, the pose of the end-
effector, with the expected pose of the end-effector based on the designed
trajectory for the mobile base. We define the difference between the expected
position of the end-effector, $\mathbf{p}_{exp}$, and the measured position of
the end-effector,$\mathbf{p}_{act}$, as:

$\mathbf{e}=\mathbf{p}_{exp}-\mathbf{p}_{act}.$ (2)

Once the error is calculated, we integrate the absolute value of the error
over the entire traveled path. This results in a quantity which quantifies the
stability of the load being transported by the mobile manipulator:

$\mathbf{p}_{e}=\begin{bmatrix}\ p_{e_{x}}\ \\\ \ p_{e_{y}}\ \\\ \
p_{e_{z}}\end{bmatrix}=\int_{T}\ \left|\mathbf{e}\ \right|dt=\begin{bmatrix}\
\int_{T}\ \left|e_{x}\right|\ dt\ \\\ \ \int_{T}\ \left|e_{y}\right|\ dt\ \\\
\ \int_{T}\ \left|e_{z}\right|\ dt\ \end{bmatrix}.$ (3)

$T\in[t_{start},t_{final}]$ is the time over which we integrate and $t_{f}$ is
the time it takes for the robot to complete its travel along the path. The
expected path of the end-effector can be found in many ways, depending on
whether the arm is moving or not. If the arm is moving, then by using the
joint angles and the corresponding forward kinematic equations one can find
the expected pose. If the arm is not moving and is in a known pose, then by
using the simulation software’s own physics engine measurements for the base
and applying a simple translation, we can find the expected pose. The actual
pose would be measured directly from the simulation. In a real scenario, the
actual pose may be measured with an external sensor or with a camera mounted
on the manipulator.

  3. C)

Distance travelled: This metric is the sum of Euclidean distances between the
sampled position of the end-effector:

$d_{travelled}=\sum^{N-1}_{i=2}||\Delta\mathbf{x}_{i}||$ (4)

  4. D)

Integrated difference between global and local path: As the travelled path
deviates from the global path, the area between them can be approximated by
taking the distance between the paths:

$d_{between}=\sum^{\text{min}(N,n)}_{i=1}||\mathbf{x}_{i}-\mathbf{X}_{i}||^{2}$

where $n$ is the number of way-points on the planned path,
$\Delta\mathbf{X}_{i}$ is the position of the robot at the planned path (taken
from the global planner), and $\Delta\mathbf{x}_{i}$ is the actual position of
the robot (taken from the local planner). The area between the global and
local paths can be approximated as:

$A_{\text{between}}=\dfrac{d_{between}\cdot d_{travelled}}{\text{min}(N,n)}$
(5)

  5. E)

Final position accuracy: The final position accuracy is measured when the
navigation is done, and it is simply the Euclidean distance between the goal
position, $\mathbf{p}_{goal}$, and the final position of the robot,
$\mathbf{p}_{final}$:

$p_{acc}=||\mathbf{p}_{final}-\mathbf{p}_{goal}||^{2}$ (6)

It is important to note that this metric does not account for the orientation
difference between the final pose and the goal pose of the robot.

  6. F)

total time taken: Once the ROS navigation stack receives a goal position, a
timer is started at time $t_{start}$ and stops once the robot reaches the goal
at time $t_{final}$:

$T_{taken}=t_{start}-t_{end}$ (7)

The total time consists of two essential parts: the time taken for
computations and the time taken to travel along the computed path.

It is important to note that $x_{i}$ is measured from the centre of the mobile
base MiR, whereas $\mathbf{p}_{exp}$ and $\mathbf{p}_{act}$ is the position of
the end-effector (expected and actual, respectively).

##  III RESULTS

(a) Static Playground

(b) Dynamic Playground

(c) Static Office

(d) Dynamic Office

(e) Static Warehouse

(f) Dynamic Warehouse

Figure 3: Stability metric, $\mathbf{p}_{e}$, indicating the difference
between the expected trajectory of the end-effector and its real trajectory
while the mobile manipulator is navigating through the environments using TEB
and DWB local planners. TEB results in more smooth motions for the end-
effector compared to DWB, particularly, in dynamic environments. In the
dynamic warehouse DWB was not able to plan any trajectory to avoid the
obstacle.

(a) Static - DWB

(b) Dynamic - DWB

(c) Static - TEB

(d) Dynamic - TEB

Figure 4: Plot of the travelled path (x,y) along with the smoothness at that point as the z-axis, showing the angle between the current direction and the new direction. This example shows the results from one trial using the playground world, and how the addition of obstacles affects the smoothness. The added obstacle blocks the path and hence a new one needs to be followed (hence the large peaks and change in direction in the figures when comparing static and dynamic worlds). | $p_{s}$ | $p_{e,x}$ | $p_{e,y}$ | $p_{e,z}$ | $d_{\text{travelled}}$ | $A_{\text{between}}$ | $p_{\text{acc}}$ | $T_{\text{taken}}$
---|---|---|---|---|---|---|---|---
DWB | TEB | DWB | TEB | DWB | TEB | DWB | TEB | DWB | TEB | DWB | TEB | DWB | TEB | DWB | TEB
A | $11.82$ | $16.67$ | $5.36$ | $6.86$ | $7.15$ | $9.38$ | $0.24$ | $0.32$ | $24.82$ | $25.06$ | $1.27$ | $2.68$ | $0.25$ | $0.14$ | $33.88$ | $44.9$
$\text{A}^{\prime}$ | $88.91$ | $74.14$ | $9.12$ | $10.82$ | $10.88$ | $13.61$ | $0.40$ | $0.50$ | $36.64$ | $34.91$ | $1.86$ | $3.72$ | $0.30$ | $0.14$ | $54.96$ | $68.9$
B | $28.98$ | $21.37$ | $16.35$ | $33.51$ | $11.61$ | $22.68$ | $0.36$ | $0.50$ | $44.06$ | $44.72$ | $2.73$ | $5.41$ | $0.09$ | $0.05$ | $58.97$ | $80.9$
$\text{B}^{\prime}$ | $18.81$ | $31.79$ | $13.00$ | $16.99$ | $9.70$ | $13.00$ | $0.39$ | $0.52$ | $44.97$ | $45.95$ | $1.98$ | $5.21$ | $0.13$ | $0.09$ | $62.14$ | $83.5$
$\text{B}^{\prime\prime}$ | $17.85$ | $17.54$ | $9.70$ | $11.77$ | $9.02$ | $11.81$ | $15.91$ | $18.04$ | $44.93$ | $44.51$ | $1.75$ | $4.86$ | $0.15$ | $0.10$ | $62.01$ | $78.7$
C | $9.47$ | $8.69$ | $3.17$ | $4.35$ | $4.29$ | $5.79$ | $0.84$ | $1.15$ | $15.00$ | $15.62$ | $1.62$ | $1.54$ | $0.11$ | $0.18$ | $20.54$ | $28.1$
$\text{C}^{\prime}$ | - | $84.61$ | - | $9.85$ | - | $13.08$ | - | $2.7$ | - | $34.46$ | - | $6.23$ | - | $0.19$ | - | $70.5$
Table I: Summary of the data collected in the simulations. The letters
represent specific worlds (A for the playground, B for the office, and C for
the warehouse), and primes represent their dynamic versions. The
$\text{B}^{\prime\prime}$ row is the data from the simulation where the arm
was moved while the base was moving. For $\text{C}^{\prime}$, the algorithm
DWB could not avoid the added obstacle and hence data could not be collected.

The metric that quantifies the load’s stability ($\mathbf{p}_{e}$) can be seen
in Fig. 3. Data on the average performance of algorithms can be found in Table
I. We can see that, on average, DWB leads to more stable paths at the end-
effector, suggesting that it provides more stable load carrying capabilities.

It is also worth noting how a change in the environment affects the
performance of local planners in this regard. When dynamic obstacles are
introduced to the simulated environment, the performance of both planners
visibly degrades, and the smoothness of the motion of the end-effector
suffers. This may be seen by comparing dynamic and static figures in 3 or
simply by checking the average smoothness of the end-effector, as in Table I.
One reason for this drawback is the fact that the robot would have to make
sudden changes to its path to avoid an obstacle; in some cases, it comes to a
complete stop before planning a new path. These sudden movements and brakes
lead to unplanned and unexpected movement from the robot. This is further
amplified by the fact that the local planner only accounts for the
acceleration of the mobile base and does not consider the forces applied to
the arm of the manipulator. The drivers of the arm motors would work to oppose
these movements, leading to higher energy usage than expected. While DWB has a
more stable performance than the performance of TEB in dynamic environments,
TEB is more effective at avoiding unexpected obstacles than DWB (TEB has less
of an increase than DWB when comparing static and dynamic world smoothness
metric values.) This can also be seen in the moving arm test scenario
($D^{\prime\prime}$ in table I), where the smoothness values (both path at the
base and end-effector) were improved, possibly due to the fact that the arm
was moved to a more compact and robust configuration, yet the improvement in
smoothness for TEB was still higher than that of DWB. A possible explanation
for the increase in the z-axis could be due to the measurement timing mismatch
leading to incorrect data being compared.

Most non-smooth segments in the path are at segments of the path where the
robot needs to make large turns, which for static worlds are at the path’s
beginning or the end. At those points, the robot needs to make significant
turns; hence, the angle between each path segment increases by a sizable
amount. In some cases, however, that is not true. Even in a static world, the
robot can find itself too close to an obstacle and must re-adapt its approach.
Such a maneuver will significantly increase the angles between the trajectory
segments. This issue would be quite prevalent in more narrow pathways. As it
is possible that there may be obstacles in the planned path in a dynamic
environment, the robot must make significant turns to follow a newly generated
trajectory to avoid the obstacle. This decreases the smoothness of the path
and forces the robot to approach the desired end pose from an unfavorable
direction in some cases, which can also decrease the overall smoothness of the
path (see how the addition of obstacles in Fig. 4 creates a large peak in the
angles between directions.)

Figure 5: Trajectories of the mobile manipulator in the playground environment
with a single obstacle in the path. The global planner has devised a path
which the robot initially is following until it encounters the obstacle. This
figure demonstrates how both local planners react in such situations. As it
can be seen, TEB reacts more radically at the beginning, but soon stabilizes
on a straight trajectory, while DWB tries to get the robot back to the
globally devised trajectory.

The results for accuracy and total time in general can be summarized as
follows: DWB has faster performance than TEB, and TEB has a more accurate
final position than DWB. We can also see that TEB tends to drive further,
except when there are obstacles in the way, in which case TEB would travel
less.

We can see that TEB, on average, deviates more from the global path than DWB,
as suggested by the $A_{\text{between}}$ values. DWB more easily follows the
global path. However, the increase in the area between the global paths and
the local trajectory caused by unforeseen obstacles is smaller for TEB than
for DWB ($\approx 39\%$ increase for TEB and $\approx 47\%$ increase for DWB),
see Fig. 5.

It is important to mention scenarios in which specific local planners failed
to find/execute a path to the goal. In the world of wide mazes, there is a
tight corridor in the map through which the robot would have to travel to
reach the goal. The scenario in which DWB failed was dynamic warehouse. The
hypothesised reason for this failure is attributed to the size of the added
obstacles. As the sampling window which the DWB algorithm uses to find viable
paths is small, an obstacle that covers the window would lead to the robot
being unable to find a path. In most cases, this happens due to the poor
reaction of the algorithm where the robot drives too close to the obstacle,
leading to the covering of sample window. If this problem is noticed soon
enough, it can be avoided by calling the global planner and re-planning a new,
viable plan around the new obstacle. Finally, seeing how for most metrics, the
increase in value was less for TEB than for DWB, we can conclude that while
TEB was affected less by the obstacles, DWB still has a more reliable and
smooth performance in both static and dynamic environments. The data and code
to validate these conclusions can be found in our 111https://github.com/sevag-
tafnakaji/local_planner_benchmarkingGitHub page.

##  IV CONCLUSION

In this paper, we present a set of measures to benchmark the performance of
local motion planners for mobile manipulator navigation and a software
framework to implement these benchmarks conveniently. The stability of the
manipulator is one of the most critical considerations in planning the
trajectory of a mobile manipulator. The robot should move without any sudden
stop, violent twist, or pull of the load. In this paper, we quantify the
motion’s smoothness for the load being carried by a mobile manipulator. We
benchmark two prominent local planners, DWB and TEB, to demonstrate which one
delivers the most smooth trajectory. The findings of this paper indicate that,
on average, DWB results in a trajectory that causes less disturbance to the
manipulator in comparison to TEB. Nonetheless, when an unforeseen obstacle
appears in the scope of the robot’s sensors on its way to the goal, TEB can
handle the transition to the exogenous path around the obstacle more
efficiently than its DWB counterpart, however DWB proved to still provide
smoother paths (both for the end-effector and the mobile base). We plan to
extend this work by utilizing these measures to find the safest trajectories
for a real mobile manipulator in order to achieve various tasks successfully.

## ACKNOWLEDGMENT

This work was supported by by Chalmers AI Research Center (CHAIR), Chalmers
Gender Initiative for Excellence (Genie), and the project AIMCoR - AI-enhanced
Mobile Manipulation Robot for Core Industrial Applications.

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