# Understanding URDF: A Survey Based on User Experience

Daniella Tola and Peter Corke D. Tola is with the Department of Electrical and
Computer Engineering, Aarhus University, Aarhus, Denmark (e-mail:
dt@ece.au.dk).P. Corke is with the Queensland University of Technology (QUT)
Centre for Robotics, QUT, Brisbane, Australia (e-mail:
peter.corke@qut.edu.au).This work was supported by the Innovation Foundation
Denmark through the MADE FAST project. We would like to thank our colleagues
for the detailed feedback on the survey, and particularly Cláudio Gomes for
helping with both revision of the questions and distribution of the survey.
The study was approved by the Institutional Review Board at Aarhus University
with the approval number TECH-2022-013.

###### Abstract

With the increasing complexity of robot systems, it is necessary to simulate
them before deployment. To do this, a model of the robot’s kinematics or
dynamics is required. One of the most commonly used formats for modeling
robots is the Unified Robot Description Format (URDF). The goal of this
article is to understand how URDF is currently used, what challenges people
face when working with it, and how the community sees the future of URDF. The
outcome can potentially be used to guide future research.

This article presents the results from a survey based on 510 anonymous
responses from robotic developers of different backgrounds and levels of
experience. We find that 96.8% of the participants have simulated robots
before, and of them 95.5% had used URDF. We identify a number of challenges
and limitations that complicate the use of URDF, such as the inability to
model parallel linkages and closed-chain systems, no real standard, lack of
documentation, and a limited number of dynamic parameters to model the robot.
Future perspectives for URDF are also determined, where 53.5% believe URDF
will be more commonly used in the future, 12.2% believe other standards or
tools will make URDF obsolete, and 34.4% are not sure what the future of URDF
will be. Most participants agree that there is a need for better tooling to
ensure URDF’s future use.

##  I Introduction

Simulating robots has become an integrated part of robot system development
[1, 2], as it reduces the cost by allowing experimentation with parameters and
environments in advance of committing to the physical hardware. A commonly
used method to represent a robot’s geometry and physical appearance is the
Unified Robot Description Format (URDF) [3]. It was introduced with the Robot
Operating System (ROS) in 2009 as a format to describe the kinematics,
geometries, and dynamics of robots in a universal manner [4].

A URDF file is an XML-based file with an extension of _.urdf_ , that can be
imported and exported by different tools for visualization or simulation
purposes. It describes robot links (rigid bodies) and the joints that connect
the links using a tree structure.

Figure 1: Google trend (trends.google.com/trends) for the search term ‘URDF’
since 2009. Note that the data is an approximation of the number of Google
searches. The data points where the search is zero percent are due to
insufficient data.

The use of URDFs has increased over the years, as evidenced by the trend in
Google queries for the term ‘URDF’, see Fig. 1. Although URDF is widely used,
it has inherent issues which have been made clear by multiple roboticists
through online forums such as ROS Discourse [5] (posts from 2016-2022), GitHub
repositories [6] (posted in 2015), and Google groups [7] (posted in 2015).
Additionally, the first ROSCast (in 2016) at MetroRobots.com discussed some of
the underlying issues of URDF and potential future directions [8]. At the time
of writing (February 2023) many of the challenges with URDF, that were
described since 2015, still persist. Over time, see Fig. 2, other open-source
object description formats have been developed such as the Simulation
Description Format (SDF) [9], Universal Scene Description (USD) [10], and
MuJoCo Modeling XML File (MJCF) [11].

Figure 2: Timeline presenting the year of the first commit to the default
branch of the GitHub repository of each description format.

With various formats emerging, it is necessary to understand the current state
of URDF and its potential future directions. We conducted an anonymous survey
on the experience of roboticists with URDF, where we asked about the
development of URDF, its limitations, and desired improvements. The results of
this survey can be used to direct future research on URDF and robot
description formats.

Our contributions in this article include:

  * •

a study with over 500 robotics developers,

  * •

an analysis of challenges and possible improvements,

  * •

presentation of quantitative and qualitative data on the current use of URDF
and perspectives for its future use,

  * •

experienced users’ knowledge of tools to create URDFs,

  * •

and the survey materials and results allowing other roboticists to build on
top of this work.

Section II provides a brief introduction to URDF, followed by the methodology
of the survey in Section III, and the results in Section IV. An overall
summary and discussion of the survey results is presented in Section V. We
conclude in Section VI with our main findings and directions for future work.

##  II Background

A URDF file is a human-readable XML file describing the kinematic structure,
dynamics, and visual representation of a robot. The visual representation can
be defined using basic 3D shapes such as boxes and cylinders or using 3D
polygon meshes that typically consist of connected triangles defining the
shape of the object. Figure 3 shows two examples of robotic arms modeled with
URDF, the left one using basic 3D shapes, and the right one using 3D polygon
meshes.

Figure 3: Example of a URDF with basic shapes (left) and a URDF with polygon
meshes (right).

###  II-A URDF File

URDF is a stand-alone file format that includes relevant modeling details of a
robot. It was initially introduced with ROS but has since been adopted by
tools both within and outside of the ROS ecosystem. Listing LABEL:lst:urdf2dof
illustrates how a link and a joint of a robot can be modeled using URDF.

⬇

1 <link name="base link">

2 <visual>

3 <origin xyz="0 0 0.25"/>

4 <geometry>

5 <cylinder length="0.05" radius="0.1"/>

6 </geometry>

7 </visual>

8 </link>

9...

10 <joint name="joint 1" type="continuous">

11 <parent link="base link" />

12 <child link="link 1" />

13 <axis xyz="0 0 1" />

14 <origin xyz="0 0 0.05"/>

15 </joint>

Listing 1: Parts of a URDF file for the robotic arm shown on the left in Fig.
3.

####  II-A1 Links

are rigid bodies of the robot that can be connected using joints. The base
link, on lines 1-8 in Listing LABEL:lst:urdf2dof, represents the fixed base of
the robot which is visualized as a cylinder, the upright gray cylinder shown
in Fig. 3 (left robot). Further information on link properties can be found in
the ROS Wiki page [12]. Note, that only links of rigid bodies can be
represented using URDF, deformable bodies cannot.

####  II-A2 Joints

connect two links, a parent and child link. The parent link is closer to the
robot base, and the child link is closer to the tool tip. The joint, on lines
10-15 in Listing LABEL:lst:urdf2dof, connects the base link with another link,
link 1. These two links can be seen on the left robot in Fig. 3, in which the
bottom light gray cylindrical link is the parent link, and the long vertical
dark gray cylindrical link is the child link. This joint is continuous,
meaning it is revolute with no motion limits. Additional joint properties can
be found in the ROS Wiki page [13].

###  II-B Xacro

Xacro is a macro language for XML that is used to construct URDF files [14]
and the xacro preprocessor is part of the ROS ecosystem. It provides tags that
can be used to configure URDF files based on the application, to reduce
redundancy and improve the maintainability of the models. These tags are
defined in a URDF-based xacro file with the extension _(.xacro)_. The xacro
preprocessor is an executable used to generate URDF files by interpreting the
xacro tags and using input values. The preprocessor allows combining multiple
xacro files, which is especially beneficial when dealing with complex robotic
structures [15].

##  III Methodology

###  III-A Motivation

The motivation for creating the survey is to understand the current position
of URDF in the robotics community. We use the survey to answer the following
questions:

  * Q1: Who is using URDF?

  * Q2: What is URDF being used for?

  * Q3: What are the limitations of URDF?

  * Q4: What is the future of URDF?

We ask these questions to understand how common the format is, its perceived
complexity by the robotics community, and how users envision its future. These
questions are important to guide the direction of future research in the area
of robot description formats.

###  III-B Survey Design

An online survey among robotics developers was conducted in December 2022 to
answer our research questions. The survey contained in total 22 questions, of
them 3 are demographic, 16 are multiple choice where 8 of them also had a text
option, 2 are rating questions, and 1 is open-ended. The demographic questions
were used to understand the experience and background of the participants.
This structure of starting with demographic questions was adapted from another
survey on simulation for testing robots [2]. We used the online tool
SurveyXact [16] to conduct the survey and collect data. To avoid asking
irrelevant questions, the survey was structured with different paths depending
on the participant responses, resulting in varying numbers of responses for
different questions. A list of the questions, their ID, and the number of
responses is presented in the appendices in Table V. We will refer to the
questions by their ID. Ethical approval of the survey was granted by the
Institutional Review Board at Aarhus University with the approval number
TECH-2022-013.

###  III-C Recruitment

To collect responses we distributed the survey via email to the robotics
community, through LinkedIn posts (by both authors, nearly 40.000
impressions), a Twitter post, an email to robotics-worldwide, a ROS Discourse
post, and posting in robotics Reddit communities. In total, 689 participants
took the survey, where 515 of them fully completed the survey and the
remaining partially completed it. Out of the 515 responses, 5 did not give
their consent therefore ending the survey, leaving 510 completed responses
which this article is based on. In our recruitment emails and posts, we
described that we are conducting a survey on robot simulation and URDF. As
URDF was directly stated in the recruitment description, we can assume that
most participants have experience with or knowledge of the format.

The experience of the participants and the organizations where they have
simulated robots or worked with URDF are presented in Table I, where 13/510
participants had no experience with this. In total 16/510 participants had
never simulated a robot before (based on answers to D1 and S1). The majority
of the participants have heard of URDF (477/497) and used it (472/477). The
experience of the participants on URDF is presented in Table II. Participants
reported different levels of experience with simulation and URDFs, within
different sizes and types of organizations. We can claim that the sample size
(510) and the participants’ background is diverse ensuring that the study is
not limited to one type of population, but there is a potential bias towards
people in academia (82%) compared to industry (52%).

###  III-D Analysis

The survey consisted of both quantitative and qualitative questions, allowing
the participants to express their own opinion on the challenges they found,
desired improvements, and future perspectives on URDF. We used descriptive
coding to analyze the open-ended questions, which entails summarizing
responses into words or phrases that describe a main topic in the data.
Specifically, we used in vivo coding as the phrases defining the topics are
taken directly from terms used by the participants themselves [17]. The
classification of responses into the different topics was performed based on
our subjective opinion.

###  III-E Threats to Validity

To mitigate the risk of bias in the questions, we followed survey best
practices [18] such as allowing open-ended answers for participants to explain
their choice, and iteratively pre-testing with volunteers.

Typical biases that can occur in multiple-choice questions are position bias
and forced-choice bias. Position bias can occur depending on the order of the
choices, and to reduce this we have ordered the choices alphabetically.
Forced-choice bias occurs when participants do not agree with any of the
choices, but are forced to choose one of the options to continue the survey.
To reduce this, we have in many questions allowed participants to select
“Other” and describe their answer using text, allowing them to provide
information based on their subjective experience/opinion. A bias may also
occur when allowing participants to answer both via multiple-choice and text,
as many participants want to limit the time spent on a question, and therefore
may choose to skip providing additional information using the text option.

Adding further comments or notes was possible at the end of the survey. Some
of these comments can be found in the appendix in Appendix B, and pointed out
potential biases in the formulation of the questions, where:

  * •

Three reported that S10 is limiting and induces a bias (participants A, B, and
C). Question 10 asks the participants how they have used URDF, with the
possibility to choose only one of the choices: ‘Always in combination with
ROS’, ‘Always in combination with ROS, but would prefer not to require ROS’,
or ‘Without ROS’. We agree with the participants’ statements and therefore
chose to limit the conclusions drawn from these results.

  * •

One reported S16 is leading by using the word painful and makes it clear that
we are looking for a specific result (participant D). The question is open-
ended and goes as follows ‘What would you say is the most painful part of
creating/modifying a URDF model?’. One of our motivations for creating this
survey is a result of our experience with the workflow of creating URDFs,
which we believe is painful, and is unfortunately made clear in the question
potentially inducing a bias.

  * •

One reported S18 as biased, as it seemed we were asking which of the following
improvements should we implement first (participant A). This does not reflect
the truth behind the question and its choices. The improvements we listed in
the multiple-choice question were found on community posts and were internally
discussed and agreed upon as what evidently seemed to be desired by people.
Naturally, such a question will be biased as the majority will choose answers
from the list instead of writing their own opinion.

External validity describes how well the results can be generalized [19]. The
threat to external validity in this survey is reduced with a large number of
participants ($>$500), their various organizational backgrounds, and differing
years of experience, see Table I and Table II.

Construct validity is related to generalization of results with focus on
correctly labeling results by making inferences based on observations. As we
have categorized the responses to open-ended questions ourselves, there is a
potential threat to the construct validity in the results. Conclusion validity
concerns defining the relationship between variables in the data and whether
or not the conclusion is reasonable. As we have compared variables between
responses from different questions, there is a potential that the conclusions
we have made are not entirely reasonable. To mitigate these threats, we have
shared the preliminary findings of the survey with people from the community
through the same media used for recruitment.

To promote further research we share our recruitment materials, questionnaire,
and additional results at the URL111github.com/Daniella1/urdf_survey_material.

##  IV Results and Analysis

###  IV-A General Results

The participants’ experience with simulation and/or URDF, and the organization
types where they have used this, are shown in Table I. The participants had
last simulated a robot (S1, 497 responses) within the last month (70%), within
the last year (24%), in the last 5 years (4%), and 1% more than 5 years ago.
The remaining 1% had never simulated a robot.

TABLE I: Demographics of the participants’ experience with robot simulation and/or URDF. Examples of Unaffiliated groups are hobby or school clubs. Note that the tables are positioned independently of each other. Years of experience | % |  Organization type | % |  #Organization employees | %
---|---|---|---|---|---
$<$1 | 10.0 |  Academia | 82.3 |  1–10 | 26.0
1–5 | 61.2 |  Industry | 52.5 |  11–50 | 23.7
6–10 | 16.7 |  Government | 4.0 |  51–100 | 10.3
$>$10 | 9.6 |  Unaffiliated groups | 14.7 |  100+ | 40.0
None | 2.5 |  Individual | 25.8 |  |
|  |  Other | 0.4 |  |
510 responses | 497 responses | 493 responses

Table II shows the participants’ experience with URDF. 497/510 participants
had simulated a robot or worked with URDF before, where 477 had heard of URDF
and 472 had used it. One of the participants that had not previously used URDF
stated it was too difficult, while another stated it was due to the
limitations of representing deformable bodies.

TABLE II: Participants’ experience with URDF. The last column represents how experienced the participants rate themselves with URDF. Note that the tables are positioned independently of each other. Years of experience | % |  Last used | % |  Experience | %
---|---|---|---|---|---
$<$1 | 16.1 |  Last month | 66.5 |  Beginner | 21.8
1–5 | 61.8 |  Last year | 27.3 |  Intermediate | 57.4
6–10 | 15.7 |  Last 5 years | 5.3 |  Expert | 20.8
$>$ 10 | 5.2 |  $>$5 years ago | 0.8 |  |
None | 1.0 |  |  |  |
477 responses | 472 responses | 472 responses

The software tools used by participants to simulate robots (S2, 494 responses)
are shown in Fig. 4, where Gazebo was the most used tool (83%), followed by
RViz (78%). RViz is part of the ROS ecosystem which is provided by the Open
Source Robotics Foundation (OSRF) [20]. The foundation also provides Gazebo,
which is easily integrated to work with the ROS ecosystem. The least-used tool
is FlexSim which less than 1% of the participants have used. Of the
participants, 21% stated they use other tools for simulating robots, and the
most popular answers among them were PyBullet (17%), NVIDIA Omniverse
consisting of Isaac Sim and Gym (14%), MuJoCo (11%), manufacturer-specific
tools such as ABB studio or URSim (9%), and Drake (7%). Manufacturer-specific
tools do not typically support URDF, as they provide models of their own
robots, making it difficult to simulate a complete application with components
from other manufacturers or custom robots.

Figure 4: The tools that the participants have used to simulate robots with 3D
visualizations (S2, 494 responses).

###  IV-B Use of URDF

Application domains (S7, 472 responses): URDF has been used in a variety of
application domains, including Manufacturing (46%), Transportation (24%),
Agriculture (14%), Medical (12%), Cleaning (8%), Defense (6%), and Marine
(6%). Additionally, 34% of respondents stated they used URDF in other domains,
with 28% of the 34% using them in Academia, 7% in Service robots (e.g. for
household, retail, healthcare), 6% in Logistics or Warehouses, 6% in
Construction robots, and 5% use URDF in Mining.

Manufacturers (S8, 472 responses): Of the robots most commonly used with URDF,
the manufacturers are Universal Robots (54%), Kuka (37%), Franka Emika (26%),
Robotiq (24%), ABB (21%), Clearpath Robotics (21%), Kinova Robotics (15%),
Willow Garage (14%), Fanuc (10%), Boston Dynamics (9%), and Yaskawa Motoman
Robotics (6%). In addition, 40% of the participants stated they use URDFs of
other manufacturers, with 31% stating they have used URDF for self-made/custom
robots, and 8% stating the robot URDFs they use are not of any manufacturer.
The 8% that did not belong to a manufacturer can also be classified as custom
robots, resulting in a total of 39%.

Robot types (S9, 472 responses): The types of robots modeled in URDF are shown
in Fig. 5. Of these, single robotic arms were the most commonly modeled (81%),
while delta robots were the least modeled (4%). Other types of robots were
used by 6% of the participants, 14% of whom modeled underwater vehicles. The
low percentage of delta robots modeled with URDF can potentially be related to
the limitations of URDF not supporting parallel linkages or closed-chain
systems.

Figure 5: Robot types modeled in URDF (S9, 472 responses).

Responses to multiple questions provide consistent information. The fact that
URDFs from Universal Robots (54%) and Kuka (37%) are most commonly used, and
the main types being single robotic arms (81%) and mobile robots (64%),
implies that these robots are commonly used in the dominant application for
URDF which is manufacturing (46%).

URDF and ROS (S10, 472 responses): Looking at the use of URDF and ROS, 66%
stated always using URDF in combination with ROS, 18% always with ROS but
would prefer not to require ROS, and 16% without ROS. As described earlier in
Section III, this question was biased as the users could not choose with and
without ROS. We can only conclude that URDF is commonly used with ROS, by the
large number of responses to this (66%), but also by looking at the high usage
of simulation tools compatible with ROS (Gazebo and RViz). We can also state
that a number of the participants prefer URDF’s independence from ROS, but we
cannot rely on the percentage here, as the question is leading and could
affect the responses.

###  IV-C Creating URDFs

Tools to create URDF (S11, 441 responses): Tools the respondents were aware of
are xacro (82%), SolidWorks URDF exporter (57%), Fusion2URDF (17%), OnShape to
URDF exporter (12%), the Blender extension Phobos (12%), and PTC Creo to URDF
exporter (4%). Additionally, 90% have used xacro to generate URDFs (S12, 361
responses).

URDF and 3D polygon meshes (S13, 472 responses): URDF can be used with basic
shapes, such as boxes and cylinders, or with 3D polygon meshes that visualize
the robot, see Fig. 3. Most of the participants (93%) have used URDF with 3D
polygon meshes.

Methods to obtain URDF (S14, 439 responses): Of the participants that had used
URDF with 3D geometrical meshes, 67% stated they developed the URDF by hand,
61% obtained the URDF from a CAD tool, 59% from a ROS package, 47% from the
website of the robot manufacturer, and 20% from a simulation tool. It is
important to note that this question was asked with the possibility to answer
multiple options, and therefore if a participant has developed a URDF by hand,
they may have also used a CAD tool for aligning or working with the meshes.
Furthermore, it is possible that some participants have obtained a URDF and
subsequently modified it, therefore choosing the option “developing the URDF
by hand”.

Most robots in ROS packages are described using xacro. The results from the
survey showed that 261 participants have used ROS packages, 313 have used
xacro, and in total 203 have both obtained URDFs from ROS packages and at some
point in their work used xacro. As most ROS packages contain xacro files that
need to be processed with the xacro preprocessor, we would assume that the
number of participants that have obtained URDFs from ROS packages would be
similar to the number of participants that have used xacro before. The fact
that only 203/261 have obtained URDFs from ROS packages and also used xacro
before shows a discrepancy in our expectations. This may indicate that either
many ROS packages contain URDF files that do not need to be preprocessed with
xacro, or a misunderstanding by the participants of what a ROS package is.
Another interesting fact about the results is the excessive use of xacro,
where 313/361 respondents stated they have used xacro before. This indicates
that it is a useful tool, but similarly to URDF, it has a number of issues
[15].

Difficulty rating of creating/modifying URDFs (S15, 439 responses): The
participants that have created/modified URDFs rated the difficulty of this, of
whom the majority (57%) rated it as medium, 32% experienced it as difficult,
7% experienced it as easy, 1% answered “Do not know”, and 3% stated they have
no experience with this.

Furthermore, one of the participants stated We have a coffee mug in the lab
that says, ‘DO NOT CHANGE THE URDF!’, anecdotally illustrating the perceived
complexity of modifying URDFs.

Challenges with creating/modifying URDFs (S16, 278 responses): Of the
respondents, that have created/modified URDFs, the most painful parts of the
process were setting up the poses and meshes of the robot (21.2%), the lack of
tooling making it difficult to debug the URDF (18.6%), and adjusting dynamic
and kinematic parameters to be accurate (9.0%). The full overview of the
categories is shown in Table VI in the appendix.

###  IV-D Challenges, Improvements, and Future Directions

Challenges and limitations (S17, 440 responses): The challenges experienced by
the participants are shown in Fig. 6. In addition, 10% of the participants
experienced other challenges presented in Table VII in the appendix.

The most common challenges found in the survey from responses to various
questions are summarized in Table III.

Figure 6: Challenges encountered by the participants (S17, 440 responses). TABLE III: Summarized most common challenges of URDF derived from responses to S16, S17, S18, and S19. Challenges |  Details
---|---
Format |  XML syntax and boilerplate code.
Multiple xacro files and very long URDF files make it difficult to find
components and keep an overview.
Does not support parallel linkages or closed-chain systems.
No real standard and no versioning, making it difficult to know which features
of URDF a simulator supports.
Only supports solid bodies, meaning it is not possible to model soft or
deformable objects.
There are not enough dynamic parameters, e.g. it is not possible to define the
elasticity of objects.
Does not support nonlinear mimic joints.
Not possible to define limits to higher-order variables of joints, e.g.
acceleration, jerk, etc.
Cannot change parameters online in simulation.
Creating URDFs |  Meshes: difficult to display colors, file paths for meshes, matching visualization and collision meshes, setting the origins.
Setting up frames and ensuring they are accurate.
Tedious workflow: debugging is complicated as you need to reload the URDF into
a visualization tool every time you make a change, and you need to use
multiple tools (CAD, XML, and visualization) to develop one URDF.
Adjusting parameters to be accurate, e.g. the dynamics or kinematic parameters
of inertia, joint limits, and positions need to be accurately defined to
represent the real world.
Surroundings |  Lack of documentation makes the learning curve steep.
Lack of tooling: to create robots out-of-the-box, for debugging, for
validating URDFs, intellisense or linter, one framework or tool to be able to
create/modify/validate URDFs, or a light preview tool for live editing, and
tools for converting between URDF and other formats.
More manufacturers should provide accurate kinematic and dynamic parameters.
Difficult to build applications with URDF.

Improvements (S18, 461 responses): The main improvements chosen by the
participants are better tools for combining and manipulating URDF files (70%),
easier methods or tools to create URDF files (69%), better tools for
validating URDF files (65%), better documentation (56%), a versioned standard
(i.e., URDF v1.0, URDF v1.1, etc.) (35%), and standardized metadata like
originator, date, version, etc. A number of the participants suggested
improvements which are categorized in Table VIII in the appendix, with the key
points being a clear demand for better tooling for creating, combining and
manipulating, and validating URDFs.

Future directions (S19, 477 responses): Of the respondents, 53% believed that
URDF will be more commonly used in the future with the main arguments being
that URDF is the de-facto standard in ROS, and as a lot of tools in ROS rely
on it, it would be too difficult to replace URDF with a new format. 34% were
not sure of URDF’s future use, and 12% did not believe it would be used in the
future. Their main arguments were that URDF is too limited and instead other
formats that tackle these limitations will take over. More details on these
elaborations are presented in Table IX, Table X, and Table XI in the appendix.

The main benefits of URDF perceived by the respondents are summarized in Table
IV.

TABLE IV: Summarized benefits and advantages of URDF derived from responses to S19. Benefits |  Details
---|---
Open-source |  Open-source robotics is becoming more common in the industry, and URDF is supported by some of the main ROS tools e.g. RViz and Moveit which are used in open-source robotics.
Interoperable |  As URDF is not dependent on one specific simulation tool, but is a portable, manufacturer-independent model, it allows users to simulate in any tool that supports URDF and exchange models within the community.
Accessible |  It is simple enough (compared to some other formats) that many non-experts can use it and follow the structure of the robot, also considering it is a human-readable format.
Custom models |  It allows developers to create custom models of their robots.

###  IV-E Analysis

In this section we analyze responses from multiple questions and check if
there are correlations in the results.

To validate the participants’ self-rating of their URDF competences, we
compare the years of experience with the participants’ rating, see Fig. 7. It
is clear that most participants that rate themselves as intermediate have
between 1-5 years of experience with URDF, whilst very few with less than a
year of experience rate themselves as experts, suggesting the competence self-
ratings are reasonable.

Figure 7: The participants’ self-rated URDF competences based on their years
of experience with URDF.

Figure 8 shows the participants’ future predictions based on self-rated
competences with URDF. The results imply there is no significant correlation
between the competences of the participants and their future predictions for
URDF.

Figure 8: The future predictions of the participants based on their self-rated
competences with URDF.

Figure 9 shows the participants’ perceived difficulty of creating/modifying
URDFs based on their self-rated competences with URDF. The results imply that
the majority of the participants experienced creating/modifying URDFs as of
medium difficulty, regardless of their self-rated competence with URDF. There
is a clear pattern within the beginner and experts groups. There are fewest
beginners in the easy column, with more in medium, and the most in difficult,
whilst for the experts the rating is the opposite, i.e., there are fewest
experts in the difficult column. This may indicate, as some of the
participants have mentioned themselves, that the learning curve of working
with URDF is steep, and with more experience the easier it becomes to work
with URDF.

Figure 9: The difficulty of creating/modifying URDFs based on the
participants’ self-rated competences.

Of the participants that found it difficult to create/modify URDFs, 74% found
it difficult to create a URDF (as expected), and 60% had issues importing
URDFs into simulation tools, see Fig. 10. Issues with URDF not supporting
parallel linkages were experienced by 50% of the participants that rated the
process of creating/modifying URDF as easy, whilst only 3% of them had issues
with creating URDFs. These results can be associated with the results in Fig.
9, where the majority of the URDF experts rated the process of
creating/modifying the file as easy, indicating the learning curve of working
with URDF is steep. This could potentially be a symptom of a lack of
documentation.

Figure 10: The participants that have chosen a specific difficulty rating for
creating/modifying URDFs, and the percentage of the total that have
encountered the different challenges.

Of the participants that do not believe URDF will be used in the future, 45%
of them have had difficulties importing the URDF into simulation tools, and
40% have had difficulties creating a URDF, as shown in Fig. 11. Although those
numbers are high, there is also a high percentage of participants that believe
URDF will be used in the future and have encountered the same challenges,
indicating there is no specific challenge that has affected the opinion of the
participants.

Figure 11: The participants that have provided future predictions of URDF, and
the percentage of them that have encountered different challenges.

##  V Summary and Discussion

In this section, we answer the questions asked in Section III based on the
survey results. Furthermore, discussions on the future focus of URDF are
presented.

###  V-A Summary

####  V-A1 Q1: Who is using URDF?

URDF has mainly been used in academic (82%) and industrial (53%)
organizations. The majority of the users (62%) have 1-5 years of experience,
and have used URDF within the last month (67%) of taking the survey (conducted
December 2022). More than 66% of the participants have used URDF in
combination with ROS while others have used it without ROS, showing the format
is used both within and outside of the ROS ecosystem.

####  V-A2 Q2: What is URDF being used for?

Manufacturing (46%) and Transportation (24%) are the dominating application
domains for URDF’s use. The most common robot types modeled with URDF are
single robotic arms (81%) and mobile robots (64%), which are also the main
types of robots used in manufacturing. Most of the participants use URDF with
3D visualizations (93%).

####  V-A3 Q3: What are the limitations of URDF?

Most of the respondents had created URDFs by hand, and this is where the main
limitations were found. When creating a URDF multiple tools are needed, such
as CAD software for modifying the meshes, a program to write the XML
structure, and additional software to visualize and test the URDF.
Participants described that the debugging process was tedious; every time a
parameter in the URDF file was tweaked, the file was loaded into the
visualization software (again) and tested. Additionally, difficulties when
importing URDFs into simulation tools, and the inability to model parallel
linkages or closed-chain systems were reported.

The main improvements for URDF, desired by the participants, are better tools
for creating, manipulating, and validating URDFs. Improved documentation to
decrease the steep learning curve of URDF was also high on the list of desired
improvements for URDF.

####  V-A4 Q4: What is the future of URDF?

The majority (53%) of the participants believed that URDF will be used in the
future. Their main argument being that URDF is the de-facto standard in ROS
where many tools rely on it to function, making it too difficult to replace
URDF with another format. The remaining participants were either not sure of
or did not believe in URDF’s future use. Their reasoning was primarily based
on the current limitations of URDF. They suspect that another format, which
addresses these limitations, will take over in the future.

###  V-B Discussion

####  V-B1 Other formats

Many of the participants mentioned other formats such as SDF [9], USD [10],
and MJCF [11]. A few also mentioned they are working on new formats that they
believe will replace URDF. With the existing and emerging formats that can be
used for modeling robots, it can be overwhelming for newcomers in robot
modeling to determine which format to use. It may be beneficial to create an
overview of the capabilities and limitations of the different formats, to
allow users to easily determine which format is most suitable for their needs.

####  V-B2 XML

Contrasting opinions were revealed on using XML for describing URDF files.
Some saw an advantage in its human-readability and simplicity, whilst others
believed that XML is outdated and the current structure of the format contains
a lot of boiler-plate code.

####  V-B3 Tackling challenges

One of the main challenges is having difficulties with importing URDFs into
simulation tools. Although, we have not analyzed the causes of this, we
suggest creating guidelines for tools to support URDF. These guidelines would
potentially make it easier for simulation tools to implement support for URDF,
but also provide a consistent user interface across simulation tools,
especially if the same function names are used.

Another interesting challenge is that participants could not find URDFs of the
specific versions of the robots they were using. This could potentially be
tackled by creating a URDF database in which it would be possible for users to
share URDFs, rate each other’s URDFs, describe issues/limitations, and update
their URDF with improved versions.

##  VI Conclusion

In this article, we conducted a survey of 510 robotics developers to determine
the use of URDF, its current challenges, desired improvements, and future
outlooks of URDF by the robotics community. We found that most of the
participants that have simulated a robot before have also used URDF.
Additionally, the majority had used URDF within the last month when taking the
survey (conducted December 2022). The majority of the participants believed
that URDF will be more commonly used in the future, especially if it is
improved. We identified challenges and limitations of the format, that the
participants would like to be tackled or improved. Furthermore, we analyzed
and discussed results and provided suggestions on how some challenges can be
tackled. The results of this survey can be used as guidelines for future
research.

## References

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The appendices contain the raw results from the survey in the form of
diagrams, and the categorized results from open-ended questions.

##  Appendix A Questions

TABLE V: Overview of question IDs, questions, and number of responses in the survey. ID |  Question | #Responses
---|---|---
D1 |  For how long have you used simulation tool(s) to simulate a robot and/or used URDF files? | 510
D2 |  In which of these types of environments have you used simulation tool(s) to simulate a robot and/or used URDF files? | 497
D3 |  If you have used simulation tool(s) to simulate a robot and/or used URDF files at any organization, how large was the most recent organization | 493
S1 |  If you have simulated a robot before, when was the last time you did this? | 497
S2 |  If you have simulated a robot with 3D visualizations, which tools have you used to do this? | 494
S3 |  Have you heard of (Unified Robot Description Format) URDF before? | 497
S4 |  If you have used URDFs before, for how long have you been using them? | 477
S5 |  When was the last time you used a URDF? | 472
S6 |  How experienced do you feel you are with URDFs? | 472
S7 |  In which application domains have you used URDFs? | 472
S8 |  Which manufacturers of robots or end effectors have you used URDFs of? | 472
S9 |  Which types of robots have you used URDFs of? | 472
S10 |  How have you used URDFs? | 472
S11 |  Are you aware of any of the following or other tools for creating URDFs? | 441
S12 |  Have you ever used the tool ‘xacro’ to generate URDFs? | 361
S13 |  Have you ever used URDF models (xml file + meshes) with geometrical meshes of the robot (i.e., not just boxes or cylinders, etc.)? | 472
S14 |  How did you obtain the URDF models? Please mention the specific tools. | 439
S15 |  If you have earlier created/modified a URDF model, what level of difficulty would you rate the process? | 439
S16 |  What would you say is the most painful part of creating/modifying a URDF model? (Optional) | 278
S17 |  Have you faced any challenges or limitations with URDFs? | 440
S18 |  Are there any improvements you would like for URDF? | 461
S19 |  Do you think that URDF will be more commonly used in the future? Please justify if possible. | 477

##  Appendix B Results

Important comments on the survey questions:

  * •

“The question about if the participant uses URDFs in conjunction with ROS is
very limiting and induces a problematic bias.” - Participant A

  * •

“The section about improvements the participant would like to see in URDFs
makes it very clear what the researchers would like to see the URDFs, and do
not necessarily reflect what the participants want, it reads more like ’which
of the things we are already doing should we do first’.” - Participant A

  * •

“Note for Q10: I use URDFs both with and without ROS” - Participant B

  * •

“Note for question 10. I have used URDFs both with and without ROS. No
preference on the use - it is situationally dependent.” - Participant C

  * •

“(Side note: ’painful’ is a leading word in this question, it makes it clear
you are looking for a specific result. ’difficult’ or ’challenging’ would be
more fair)” - Participant D (part of answer to question 16)

TABLE VI: The most painful parts of creating/modifying URDFs described by the participants (S16). Most painful part of creating/modifying URDF | % |  Description
---|---|---
Poses and meshes | 21.2 |  Would be nice if the manufacturers provided separate meshes for each link and its origin etc., so they easily could be put together in a URDF file. Ensuring correct file paths to meshes can be annoying. Ensuring the axes are accurate. Difficult to get colors to work in the meshes. And also matching the collision and visual geometries.
Lack of tools and difficult to debug | 18.6 |  We need intellisense, linters, single tools to develop, debug, and visualize URDFs while in the making with e.g. a preview tool. A tool for validating the URDF or XML file would be beneficial. Would be nice with tools for live editing.
Adjusting the parameters to be correct (e.g. dynamics, physics) | 9.0 |  Setting the values of the dynamics, kinematics correctly, e.g. joint limits or position, inertia, etc.
Workflow | 8.8 |  The complete workflow of creating a URDF.
Syntax | 8.8 |  The syntax is complicated, and XML is a disadvantage. Includes a lot of boilerplate code that is just repeated, e.g. if a box is created for a link, you need to copy it to other links.
Limitations of parallel links and closed chains | 7.1 |  The limitation of being able to model robots with parallel links or closed chains.
No real standard | 3.7 |  Changing standards and versions without real versioning, parsers have different ways of interpreting frames e.g. Y axis up or Z-axis up.
Lack of documentation | 3.4 |  Not enough documentation or not good enough, i.e. not intuitive.
Multiple xacro files or long URDF files | 2.8 |  Following the chain of xacro files for more complicated workcells or robots is tedious. Long files make it difficult to keep an overview or find specific components.
Using multiple software tools to generate a URDF is tedious | 1.1 |  Using CAD software and other programs is time-consuming and difficult.
TABLE VII: Other challenges described by the participants. The percentage shown in the table is from the 10% that answered ’Other’ to S17 in the survey. Challenges/limitations | % |  Description
---|---|---
Issues with dynamics | 10.4 |  Difficulty creating correct inertial parameters, lack of dynamic parameters.
Complex linkages | 8.3 |  Difficult to model conveyor belts, difficulty modeling more complex linkages even for serial robots.
Not possible to define limits of higher-order variables to joints | 8.3 |  Higher-order variable limits cannot be specified such as acceleration and jerk, can also not define coupled joints.
Lack of tooling | 8.3 |  Need tools to create robots out of the box, lack of intellisense, need tools to verify URDF parameters for application.
URDF only supports solid bodies | 6.3 |  Lack of support for deformable or elastic links, i.e. limitations modeling non-rigid structures.
Issues with mimic joints | 6.3 |  Only linear joint mimic relationships.
No .xsd (standard) | 4.2 |  No working/accepted XSD, incomplete format.
Challenges when mixing with SDF | 4.2 |  Turning URDF to SDF is difficult.
Inadequate documentation | 4.2 |  Lack of documentation, and need to read through xacro before being able to configure robot.
Issues with origins | 4.2 |  Discrepancy between link origin and geometric mesh origin, missing frame support.
Maintainability | 2.1 |  Difficult to maintain.
TABLE VIII: Other improvements described by the participants. The percentage shown in the table is from the 11% that answered ’Other’ to S18. Improvements | % |  Description
---|---|---
Better tools | 19.6 |  Syntax checking, linter, online combining and manipulation of URDFs, tools for building and visualizing URDF.
Closed chain support | 14.3 |  Support for closed chain robots such as delta robots.
Support for more complex and parallel linkages | 12.5 |  Support for complex robots and parallel links.
Improve towards SDF | 8.9 |  Better tools to describe scenes, deprecate URDF or change URDF towards SDF.
Standardization | 5.4 |  Standardization, standard method to describe extensions to the format (e.g. new joint types), versioning.
Support for deformable bodies | 3.6 |  Deformable links.
Better Gazebo integration | 3.6 |  Better integration using Gazebo plugins.
More dynamic parameters | 3.6 |  Additional dynamic parameters such as elasticity.
Better conversion support to other formats | 1.8 |  Better conversion to other formats such as URDF to USD.
More accurate kinematics/dynamics | 1.8 |  More manufacturers should provide accurate kinematic and dynamic parameters.
TABLE IX: The summarized opinions of the participants that answered YES to URDF being more commonly used in the future (S19). Future development thoughts (YES) | % |  Description
---|---|---
Commonly used/have experience with it already | 19.2 |  It is a widely accepted format and many tools are built around it (especially ROS tools).
De-facto ROS standard | 14.7 |  It is the de-facto standard in ROS, and as ROS is a growing community, URDF and its use will grow with it.
Good option | 13.6 |  Good option (best in some opinions) as it is simple to use, enables interoperability, and there seems to be no other alternative.
Good basis, but needs improvement | 10.7 |  It’s a great tool but needs usability and workflow improvements and more tools around it. It should also support closed chained and nonlinear mimic joints.
Not too difficult to use/intuitive | 6.8 |  It is accessible for non-experts, easy to use, human-readable format, and most developers can use it without being lost in complexity.
Such an interchangeable standard is needed | 6.8 |  URDF provides interoperability for robot models, allows people to create models of custom robots, is manufacturer-independent, and there should always be a standard for this.
SOTA robotic simulation tools support URDF | 5.1 |  Many software tools and especially robotics software tools are using it.
Increase in use of simulators, open source, and robots overall | 4.0 |  More people are working with Open Source Robotics and more companies are valuing simulation before deployment, and with this growth, more people will be using URDF.
Don’t know other good formats | 3.4 |  It’s the best current option for visualization when combined with ROS/moveit/Rviz and there does not seem to be an alternative.
TABLE X: The summarized opinions of the participants that answered NO to URDF being more commonly used in the future (S19). Future development thoughts (NO) | % |  Description
---|---|---
New standard or other software tools will take over, e.g. SDF, USD, MJCF | 37.1 |  A new format or software tool will take over, which will be more robust and easier to use. The format or tool could be SDF, Omniverse, MuJoCo, or USD.
URDF is limited | 12.9 |  URDF is limited in terms of support for mechanisms such as closed chain systems, description of actuators and sensors, and has poor abstractions. XML is limiting.
Working with URDF is difficult | 5.7 |  It is widely adopted, but people don’t seem to like it as it can be formatted in a cleaner way, there is no standardization and few updates.
URDF is not supported by enough software tools | 2.9 |  Several simulators do not use URDF.
TABLE XI: The summarized opinions of the participants that answered DON’T KNOW to URDF being more commonly used in the future (S19). Future development thoughts (DON’T KNOW) | % |  Description
---|---|---
Other formats may emerge/take over | 24.7 |  There are other formats such as SDF that overcome some of the disadvantages of URDF and might be replacing URDF in the future.
URDF is limited | 8.2 |  URDF has limitations with regards to modeling systems with more intricate dynamics, not allowing parallel linkages, and has a steep learning curve.
URDF tooling needs to improve | 7.1 |  There is a need for tools to address the current limitations and better tools for creating URDFs.
URDF needs improvement to be more commonly used | 7.1 |  For URDF to be more commonly used it needs to be improved with a special focus on the creation of URDFs of robots and also building applications using URDFs.
URDF is currently commonly used | 5.9 |  URDF is commonly used in the ROS community and is used between different simulation tools.
URDF is difficult to use | 5.9 |  URDF has a steep learning curve and limited resources, making it difficult to use and create customized robots.
Such a format is useful | 5.9 |  It is important to have a format that is as simple and portable as the URDF.
Depending on the use of ROS | 3.5 |  If ROS becomes more widely used then URDF will most likely be as well.

Figure 12: Responses to D1. Figure 13: Responses to D2. Figure 14: Responses
to D3. Figure 15: Responses to S1. Figure 16: Responses to S2. Figure 17:
Responses to S3. Figure 18: Responses to S4. Figure 19: Responses to S5.
Figure 20: Responses to S6. Figure 21: Responses to S7. Figure 22: Responses
to S8. Figure 23: Responses to S9. Figure 24: Responses to S10. Figure 25:
Responses to S11. Figure 26: Responses to S12. Figure 27: Responses to S13.
Figure 28: Responses to S14. Figure 29: Responses to S15. Figure 30: Responses
to S17. Figure 31: Responses to S18. Figure 32: Responses to S19.


# A Characterization of concave mappings

V. Bravo and R. Hernández and O. Venegas  Facultad de Ingeniería y Ciencias
Universidad Adolfo Ibáñez
Av. Padre Hurtado 750, Viña del Mar, Chile<EMAIL_ADDRESS>Facultad de
Ingeniería y Ciencias
Universidad Adolfo Ibáñez
Av. Padre Hurtado 750, Viña del Mar, Chile<EMAIL_ADDRESS>Departamento de Ciencias Matemáticas y Físicas. Facultad de Ingeniería
Universidad Católica de Temuco<EMAIL_ADDRESS>
###### Abstract.

This study focuses on Concave mappings, a class of univalent functions that
exhibit a unique property: they map the unit disk onto a domain whose
complement is convex. The main objective of this work is to characterize these
mappings in terms of the real part of the expression
$1+zf^{\prime\prime}(z)/f^{\prime}(z)$, considering scenarios where the
omitted convex domain is either bounded or unbounded. In the case of a bounded
convex domain, we investigate the pivotal role played by the Schwarzian
derivative and the order of the functions in understanding the behavior and
properties of these mappings.

The authors were partially supported by Fondecyt Grants # 1190756. Key words:
Concave mappings, Schwarzian derivative, Caratheodory class. 2020 AMS Subject
Classification. Primary: 30C45, 30C55; Secondary: 31A10.

##  1\. Introduction

A complex function defined in the unit disk is said to be concave as opposed
to a convex function, that is, those functions that map the unit disk into the
complement of a convex domain. In other words, the set of points it omits
forms a convex domain in the complex plane. From this definition, we can
conclude that these functions, or more precisely, the image domain
$f(\mathbb{D})$ cannot be a bounded set and therefore, either inside or on the
boundary, $f$ has a pole. Since we consider these functions to be univalent,
this pole must be simple. This type of functions are the subject of study in
this manuscript.

The two scenarios mentioned above, one with the pole inside the disk, and the
other, when the pole is on the boundary, are treated differently and
characterized ad-hoc. In the first case, we consider conformal functions that
omit an unbounded convex domain and have the pole at $z=1$. This set of
functions is denoted as $Co$. While it is true that characterizations of this
set of functions exist in terms of the real part of
$1+zf^{\prime\prime}(z)/f^{\prime}(z)$, a particular case of these types of
functions are those that have an angle at infinity less than or equal to
$\pi\alpha$, and they are referred to in the literature as $Co(\alpha)$. Here,
we propose a new characterization of functions in $Co$ using the well-known
differential operators, Schwarzian derivative and the order operator, much
like what happens with the extensively studied convex functions defined in the
unit disk. The geometric significance of the Schwarzian derivative as a tool
in the context of the geometric theory of functions is well known, and in this
case, it plays a role equivalent to what it does for convex functions, and
this is where we see the specific value of these notes. Section 2 is dedicated
to this task, particularly focusing on the extreme case of equality in the
obtained characterization. Subsection 2.2 deals with the aforementioned
functions in the class $Co(\alpha)$ and their characterization in terms of the
real part, which extends the known results up to now.

On the other hand, when the (simple) pole is in the interior, the omitted
convex domain is bounded. Several authors have addressed this type of
functions by considering the pole to be at $z=p$, where $p\in[0,1)$, and
referring to this set as $Co(p)$. A particular case of study is when
considering $p=0$, and it has indeed been characterized equivalently to the
case of convex functions, except that in this case, the real part of
$1+zf^{\prime\prime}(z)/f^{\prime}(z)$ is negative. Note that this operator is
well-defined when $f$ has a simple pole at the origin. In Section 3, we study
this class of concave functions, provide a characterization that includes the
aforementioned case, and calculate the value of the Schwarzian derivative for
any function within this family. The section concludes with the case of
equality in this characterization.

In this manuscript, we study some characterizations of this two classes of
concave mappings, $Co(p)$ with $p\in[0,1)$ and $Co$, and the Schwarzian
derivative for this mappings.

##  2\. On the analytic concave mappings: The class $Co$.

Let $\mathbb{D}$ denote the open unit disk in the complex plane, and let
$\mathbb{C}$ be the complex plane. Consider analytic mappings
$f:\mathbb{D}\to\mathbb{C}$ into the complex plane, such that the image of $f$
lies in the complement of a convex domain. Furthermore, let $f(1)=\infty$ be a
normalization condition for these mappings. The set of all such functions is
denoted by $Co$. The class called $Co(\alpha)$ consists of analytic concave
mappings that are normalized such that $f(0)=0$, $f^{\prime}(0)=1$, and
$f(1)=\infty$. The opening angle of $f(\mathbb{D})$ at infinity is then
bounded by $\pi\alpha$, where $\alpha\in(1,2]$. For further details on these
classes, refer to [2] as well as [5, 8].

In [1] the authors shows that if $f\in Co$ then the function $\phi$ defined by
the formula

(1) $\phi(z)=z+\dfrac{2f^{\prime}(z)}{f^{\prime\prime}(z)}$

is a holomorphic mapping defined in ${\mathbb{D}}$ and such that
$|\phi(z)|\leq 1$, $z=1$ is the unique fixed point, and
$\phi^{\prime}(1)\in[0,1/3]$.

The Schwarzian derivative for locally univalent meromorphic functions $f$, is
a well-known differential operator given by

$Sf=\left(\dfrac{f^{\prime\prime}}{f^{\prime}}\right)^{\prime}-\frac{1}{2}\left(\dfrac{f^{\prime\prime}}{f^{\prime}}\right)^{2}.$

Is a mathematical tool to measure the distortion caused by a complex mapping.
It provides a quantitative measure of how a mapping deforms angles and shapes
locally and captures the second-order behavior of $f(z)$. In fact, $f$ maps
conformally the unit disk onto a convex domain if and only if

$|Sf(z)|(1-|z|^{2})^{2}+2\left|\dfrac{\varphi(z)-\overline{z}}{1-z\varphi(z)}\right|^{2}\leq
2,$

when

$\varphi(z)=\dfrac{f^{\prime\prime}(z)/f^{\prime}(z)}{2+zf^{\prime\prime}(z)/f^{\prime}(z)}.$

The reader can be found this characterization in [9].

Let $f:{\mathbb{D}}\to{\mathbb{C}}$ be a conformal mapping. In [11], the
authors introduce the definition of inferior order of $f$ as

$\inf_{z\in{\mathbb{D}}}|A_{f}(z)|=\inf_{z\in{\mathbb{D}}}\frac{1}{2}\left|(1-|z|^{2})\frac{f^{\prime\prime}}{f^{\prime}}(z)-2\overline{z}\right|.$

In the same paper, they proved that $f$, with the normalization that
$f(1)=\infty$, is concave if and only if $\inf|A_{f}|=1$.The reader can see
this result as equivalent to the well-known fact that a function is convex if
and only if $\sup|A_{f}|=1$. Likewise, this is also equivalent to

$(1-|z|^{2})^{2}|Sf(z)|+2|A_{f}(z)|\leq 2,$

see [9, eq. (9)]. The following results, is in this sense, can be understood
as the generalization of this inequality.

######  Theorem 1.

Let $f:{\mathbb{D}}\to{\mathbb{C}}$ be a conformal mapping with $f(1)=\infty$,
then $f\in Co$ if and only if

(2) $|Sf(z)|(1-|z|^{2})^{2}+2\leq 2|A_{f}(z)|^{2}.$

###### Proof.

Since $f\in Co$ we have that there exists a holomorphic mapping
$\phi:{\mathbb{D}}\to\overline{{\mathbb{D}}}$ such that the equation (1) holds
(see [1]). Thus, $f^{\prime\prime}/f^{\prime}(z)=2/(\phi(z)-z)$, which implies
that $Sf(z)=-2\phi^{\prime}(z)/(\phi(z)-z)^{2}.$ From where, and using the
Schwarz’s Lemma, we have that

$|Sf(z)|(1-|z|^{2})^{2}\leq\frac{2(1-|\phi(z)|^{2})(1-|z|^{2})}{|\phi(z)-z|^{2}}=2\left(\frac{|1-\overline{z}\phi(z)|^{2}}{|\phi(z)-z|^{2}}-1\right).$

By equation (1), a direct computation shows that

$\frac{1-\overline{z}\phi(z)}{\phi(z)-z}=\frac{1}{2}\left((1-|z|^{2})\frac{f^{\prime\prime}}{f^{\prime}}(z)-2\overline{z}\right)=A_{f}(z),$

which completes the proof of the necessity part. Now, suppose that
$|Sf(z)|(1-|z|^{2})^{2}+2\leq 2|A_{f}(z)|^{2}$, then $|A_{f}(z)|\geq 1$ which
is equivalent with the function $f$ is concave (see [11, Thm. 4 part (ii)]).
Then the proof is completed. ∎

###  2.1. The case of equality in Theorem 1

The equality in (2) for some $z\in{\mathbb{D}}$, implies that $\phi$ satisfies
the equality case in the Schwarz’s Lemma, therefore $\phi$ is an automorphism
of the unit disk given by $\phi(z)=\nu(a-z)/(1-\overline{a}z)$ for some
$a\in{\mathbb{D}}$ and $|\nu|=1$. Since $\phi$ has a fixed point at $z=1$ we
have that $\nu=-(1-\overline{a})/(1-a)$, and
$\phi^{\prime}(1)=(1-|a|^{2})/|1-a|^{2}\in[0,1/3]$. Moreover,

$\dfrac{f^{\prime\prime}}{f^{\prime}}(z)=\dfrac{2(1-\overline{a}z)}{\nu
a-(\nu+1)z+\overline{a}z^{2}}.$

Observe that if $a=0$ then $\phi(z)=z$ which says that all points are fixed
point, which is a contradiction with Theorem 2 part (iii) in [1]. Since $a\neq
0$, $\nu a/\overline{a}\neq 1$, in other case, $\mbox{Re}\\{a\\}=|a|^{2}$
which implies that $\phi^{\prime}(1)=1$. Therefore, we have that

$\dfrac{f^{\prime\prime}}{f^{\prime}}(z)=\dfrac{2(1-\overline{a}z)}{\overline{a}(z-1)(z-\lambda)}=\dfrac{2(1-\overline{a})}{\overline{a}(1-\lambda)}\cdot\dfrac{1}{z-1}+\dfrac{2(\nu
a-1)}{\overline{a}(1-\lambda)}\cdot\dfrac{1}{z-\lambda},\quad\lambda=\nu
a/\overline{a}.$

Then

$f^{\prime}(z)=A(z-1)^{-b-2}(z-\lambda)^{b},$

for some complex constant $A$ and $b=2(\nu a-1)/(\overline{a}-\nu
a)=(1-|a|^{2})/(|a|^{2}-\mbox{Re}\\{a\\})$. A straightforward computation and
the fact that $\phi^{\prime}(1)\in[0,1/3]$, shows that
$b=2\phi^{\prime}(1)/(1-\phi^{\prime}(1))\in[0,1]$. In this case,

$f(z)=A\left(\dfrac{z-\lambda}{z-1}\right)^{1+b}+B,$

for some constant $B$. Observed that $f$ is an affine maps of the unit disk
into a concave domain. This domain is defined as the complement of an angle
less than or equal to $\pi$ centered at the origin.

###  2.2. The class of $Co(\alpha)$.

In their seminal paper [1], the authors introduced a noteworthy class of
mappings, characterized by a specific normalization condition. This condition
ensures that the angle at infinity in $f(\mathbb{D})$ is at most $\alpha\pi$,
where $\alpha$ lies in the interval $[1,2]$. Notably, when $\alpha=1$ the
mapping reduces to the well-known half-plane mapping $\ell(z)=z/(1-z)$ or
affine transformation of this mapping. These functions are commonly referred
to as $Co(\alpha)$ mappings. For a more comprehensive understanding of this
family, interested readers can refer to the following references: [2, 5, 6, 7,
8].

In particular, the author in [8] demonstrates that $(1-|z|^{2})^{2}|Sf(z)|\leq
2(\alpha^{2}-1)$, which can be derived using our last theorem and the fact
that $\sup|A_{f}|=\alpha$ (see [11]) for any mapping belonging to this class.
In [2] the authors proved (see Theorem 2) that, for $\alpha\in(1,2]$, $f\in
Co(\alpha)$ if and only if

(3)
$\text{Re}\left\\{\dfrac{\alpha+1}{2}\cdot\dfrac{1+z}{1-z}-1-z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)\right\\}>0.$

Similarly to the case of convex functions, this last inequality can be
improved in the sense that not only is the real part of this expression
positive, but it is also greater than or equal to a positive expression
involving the same elements. This is the goal of the next result.

######  Theorem 2.

$f\in Co(\alpha)$ with $\alpha\in(1,2]$ if and only if

$\textup{Re}\left\\{\dfrac{\alpha+1}{2}\cdot\dfrac{1+z}{1-z}-1-z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)\right\\}\geq\dfrac{1}{2(\alpha-1)}\left|\dfrac{f^{\prime\prime}}{f^{\prime}}-\dfrac{\alpha+1}{1-z}\right|^{2}(1-|z|^{2}).$

###### Proof.

By inequality (3), there exists a Schwarz’s map $\omega$ such that

$\dfrac{\alpha+1}{2}\cdot\dfrac{1+z}{1-z}-1-z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)=\frac{\alpha-1}{2}\cdot\frac{1+\omega}{1-\omega}.$

Since $\omega(z)=z\varphi(z)$ for some holomorphic mapping
$\varphi:{\mathbb{D}}\to{\mathbb{D}}$, we have that

$E:=\frac{1}{\alpha-1}\left(\frac{\alpha+1}{1-z}-\frac{f^{\prime\prime}}{f^{\prime}}(z)\right)=\frac{\varphi(z)}{1-z\varphi(z)},$

which is equivalent to

$\varphi(z)=\frac{E}{1+zE}.$

Since $|\varphi(z)|\leq 1$ it follows that

$1+2\text{Re}\\{zE\\}\geq|E|^{2}(1-|z^{2}|).$

But,

$\dfrac{\alpha+1}{2}\cdot\dfrac{1+z}{1-z}-1-z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)=\frac{\alpha-1}{2}\left(1+2zE\right),$

using the last inequality, we conclude as desire. ∎

##  3\. On the meromorphic concave mappings: The class $Co(p)$.

When this pole is located within $\mathbb{D}$, a well-known class of such
mappings arises, known as $Co(p)$, which is normalized with respect to the
pole at $p\in[0,1)$. These classes have been extensively studied in [1, 2, 3,
4]. Moreover, in [10], the authors provide a characterization of concave
functions with a pole at the origin, normalized as follows:
$f(z)=1/z+b_{0}+b_{1}z+\cdots$. Now, in general, let $f$ be a concave mapping
with a simple pole at $p\in[0,1)$ normalized to

(4) $f(z)=\dfrac{1}{z-p}+b_{0}+b_{1}(z-p)+\cdots,\quad z\in{\mathbb{D}}.$

This class of mappings are called $Co(p)$. This functions satisfies that

$\mbox{Re}\left\\{1+z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}\right\\}<0.$

The reader can found more information in [3, 6, 10, 12]. In particular, the
authors in [10] showed that when the pole at the origin, this functions can be
characterized as

$\mbox{Re}\left\\{1+z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)\right\\}<0.$

Furthermore, within the context of the same paper, it is demonstrated that
this inequality can be further refined through, $f\in Co(0)$ if and only if

(5)
$\mbox{Re}\left\\{1+z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)\right\\}\leq-\dfrac{1}{4}(1-|z|^{4})\left|z\frac{f^{\prime\prime}}{f^{\prime}}(z)\right|^{2}.$

The equality at a specific point is attained if and only if the dilation
function $\omega(z)=\lambda z^{2}$, where $\lambda$ is an arbitrary unimodular
constant. Considering the normalization that the residue at the pole located
at $z=0$ admits only one possibility, namely $\lambda=1$. Consequently, in
this particular scenario, equality at a point distinct from the origin arises
solely when the function $f$ is defined as follows:

$f(z)=\frac{1}{z}+a_{0}+z.$

The subsequent theorem is presented in [10]; however, it is not explicitly
delineated. The proof outlined herein significantly diverges from the
rationale presented in that publication.

######  Theorem 3.

$f\in Co(0)$ if and only if
$|Sf(z)|(1-|z|^{2})^{2}+2(2|\varphi|+1)|\Phi|^{2}\leq 2(2|\varphi|+1)$, where

$\varphi(z)=\frac{z+2f^{\prime}(z)}{z^{3}f^{\prime\prime}(z)},\quad\quad\mbox{and}\quad\quad\Phi(z)=\frac{\overline{z}-z\varphi(z)}{1-z^{2}\varphi(z)}.$

###### Proof.

Let $f$ be a normalized function in $Co(0)$ given by
$f(z)=\dfrac{1}{z}+b_{0}+b_{1}z+b_{2}z^{2}+b_{3}z^{3}+\cdots$. Then
$z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)=-2-2b_{1}z^{2}-2b_{1}z^{3}-6b_{2}z^{3}-\cdots$
hence $z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)=\dfrac{2}{\omega-1}$,
therefore

$z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)+2=\dfrac{2\omega}{\omega-1}.$

From $z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)=\dfrac{2}{\omega-1}$ we have
that $\dfrac{f^{\prime\prime}}{f^{\prime}}(z)=\dfrac{2}{z(\omega-1)}$, and
also

$Sf(z)=\left(\dfrac{2}{z(\omega-1)}\right)^{\prime}-\dfrac{1}{2}\left(\dfrac{2}{z(\omega-1)}\right)^{2}=\dfrac{-2(\omega+z\omega^{\prime})}{z^{2}(\omega-1)^{2}}.$

There exists a holomorphic mapping $\varphi:{\mathbb{D}}\to{\mathbb{D}}$ such
that $\omega(z)=z^{2}\varphi(z)$ from where we obtain that
$\omega^{\prime}=2z\varphi+z^{2}\varphi^{\prime}$ and therefore

$Sf(z)=\dfrac{-6\varphi-2z\varphi^{\prime}}{(1-z^{2}\varphi)^{2}}.$

Consider $\phi$ such that $\phi(z)=z\varphi(z)$ then
$\phi^{\prime}(z)=\varphi(z)+z\varphi^{\prime}(z)$. Thus

$\displaystyle|Sf(z)|(1-|z|^{2})^{2}$ $\displaystyle=$
$\displaystyle\dfrac{2(1-|z|^{2})^{2}}{|1-z\phi|^{2}}|2\varphi+\phi^{\prime}|$
$\displaystyle\leq$ $\displaystyle
2\left(\dfrac{|2\varphi|(1-|z|^{2})^{2}}{|1-z\phi|^{2}}+\dfrac{|\phi^{\prime}|(1-|z|^{2})^{2}}{|1-z\phi|^{2}}\right)$
$\displaystyle\leq$ $\displaystyle
2\left(\dfrac{|2\varphi|(1-|z|^{2})^{2}}{|1-z\phi|^{2}}+\dfrac{(1-|\phi|^{2})(1-|z|^{2})}{|1-z\phi|^{2}}\right).$

Define $\Phi:{\mathbb{D}}\to{\mathbb{C}}$ by
$\Phi(z)=\dfrac{\overline{z}-\phi(z)}{1-z\phi(z)}$ hence $|\Phi(z)|^{2}<1$ and
$1-|\Phi(z)|^{2}=\dfrac{(1-|\phi|^{2})(1-|z|^{2})}{|1-z\phi|^{2}}$. Then

$\displaystyle|Sf(z)|(1-|z|^{2})^{2}$ $\displaystyle\leq$ $\displaystyle
2\left(\dfrac{|2\varphi|(1-|z|^{2})^{2}}{|1-z\phi|^{2}}+1-|\Phi|^{2}\right)$
$\displaystyle=$ $\displaystyle
2\left(\dfrac{2|\varphi|(1-|z|^{2})}{1-|\phi|^{2}}\right)(1-|\Phi|^{2})$
$\displaystyle\leq$ $\displaystyle 2(2|\varphi|+1)(1-|\Phi|^{2}),$

which is equivalent to

$|Sf(z)|(1-|z|^{2})^{2}+2(2|\varphi|+1)|\Phi|^{2}\leq 2(2|\varphi|+1).$

∎

This result, can be understood as an extension to equation (9) in [9].

######  Corollary 1.

$f\in Co(0)$ then $|Sf(z)|(1-|z|^{2})^{2}\leq 6$.

###### Proof.

Since $\varphi:{\mathbb{D}}\to{\mathbb{D}}$ we have that the right side is
less than or equal to 6. ∎

######  Theorem 4.

Let $f:{\mathbb{D}}\to{\mathbb{C}}$ satisfying equation (4). Then $f\in
Co(p)$, $p\in[0,1)$ if and only if

$\textup{Re}\left\\{1+z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)+q\right\\}\leq-\dfrac{(1-|z|^{2})(1+2a_{p}|z|+|z|^{2})}{4(1+a_{p}|z|)^{2}}\left|z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)+q\right|^{2},$

where $a_{p}=\dfrac{2b_{1}p^{2}+O(p^{3})}{-1+b_{1}p^{2}+O(p^{3})}$ and
$q=\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}$.

###### Proof.

We known that $f\in Co(p)$ if and only if the real part of

$M(z)=1+z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}$

is negative. Since $z+2f^{\prime}(z)/f^{\prime\prime}(z)$ has a unique fixed
point at $p$ (see [1, Thm. 2]), we have that
$1+zf^{\prime\prime}(z)/f^{\prime}(z)$ is equal to $1$ when $z=0$, thus
$M(0)=-1$. Therefore, exists a Schwarz’s mapping $\omega$ such that

$1+z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}=\dfrac{\omega+1}{\omega-1}.$

Then $\omega=(M+1)/(M-1)$, and $\omega(z)=z\varphi(z)$ where
$\varphi:{\mathbb{D}}\to{\mathbb{D}}$ given by

(6)
$\varphi(z)=\dfrac{\dfrac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2}{z-p}-\dfrac{2p}{1-pz}}{z\dfrac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}}=\dfrac{(z-p)\dfrac{f^{\prime\prime}}{f^{\prime}}(z)+2-\dfrac{2p(z-p)}{1-pz}}{z(z-p)\dfrac{f^{\prime\prime}}{f^{\prime}}(z)+2p-\dfrac{2pz(z-p)}{1-pz}}.$

A straightforward calculations shows that

$\varphi(0)=\dfrac{2b_{1}p^{2}+O(p^{3})}{-1+b_{1}p^{2}+O(p^{3})}=a_{p},$

which tends to 0 when $p$ goes to 0. Now,

$|M+1|^{2}\leq|z|^{2}|\varphi(z)|^{2}|M-1|^{2},$

which is equivalent to

$\left|2+z\frac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}\right|^{2}\leq|z|^{2}|\varphi(z)|^{2}\left|z\frac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}\right|^{2},$

or

$4+4\text{Re}\left\\{z\frac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}\right\\}\leq\left|z\frac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}\right|^{2}(|z\varphi(z)|^{2}-1),$

from where

$\text{Re}\left\\{1+z\frac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}\right\\}\leq-\frac{1-|z|^{2}|\varphi(z)|^{2}}{4}\left|z\frac{f^{\prime\prime}}{f^{\prime}}(z)+\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}\right|^{2}.$

Then, using that the generalization of Schwarz’s Lemma, we have that

$\text{Re}\left\\{1+z\frac{f^{\prime\prime}}{f^{\prime}}(z)+q\right\\}\leq-\frac{1-|z|^{2}\left(\dfrac{|a_{p}|+|z|}{1+|a_{p}||z|}\right)^{2}}{4}\left|z\frac{f^{\prime\prime}}{f^{\prime}}(z)+q\right|^{2},$

where $q=\dfrac{2p}{z-p}-\dfrac{2pz}{1-pz}$. Thus, the proof is completed. ∎

Observed that the inequality (5) is a particular case of this Theorem
considering $p=0$, in which case $a_{p}=0$, or equivalently $\varphi(0)=0$.
Since $\varphi(z)$ is given by equation (6), we have that
$\varphi(0)=-a_{2}-\dfrac{1}{p}-p$, then $|a_{2}|\leq 1/p+p$, and the equality
case attained by $f(z)=e^{i\theta}k_{p}(e^{-i\theta}z)$, where

$k_{p}(z)=\dfrac{z}{(1-z/p)(1-pz)},$

which maps the unit disk into a hole the complex plane minus the real segment
$\left[\dfrac{1}{2-1/p-p},\dfrac{-1}{2+1/p+p}\right]$.

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