# Synchronizing Full-Body Avatar Transforms with WebRTC DataChannel on
Educational Metaverse

Yong-Hao Hu  Virtual Reality Educational Research Center
The University of Tokyo
Tokyo, Japan
<EMAIL_ADDRESS>Kenichiro Ito  Information Technology
Center
The University of Tokyo
Tokyo, Japan
<EMAIL_ADDRESS>Ayumi Igarashi  Graduate School of Medicine
The University of Tokyo
Tokyo, Japan
<EMAIL_ADDRESS>
###### Abstract

Full-body avatars are suggested to be beneficial for communication in virtual
environments, and consistency between users’ voices and gestures is considered
essential to ensure communication quality. This paper propose extending the
functionality of a web-based VR platform to support the use of full-body
avatars and delegated avatar transforms synchronization to WebRTC DataChannel
to enhance the consistency between voices and gestures. Finally, we conducted
a preliminary validation to confirm the consistency.

###### Index Terms:

Metaverse, Real-time Communication, Web User Interface

##  I Introduction

’Metaverse’ is commonly defined as 3D virtual environments where interactions
among users occur and are accessible via computers or Virtual Reality (VR)
devices, and it has found utility across diverse areas, including education.
We have started developing an educational platform in metaverse based on
Mozilla Hubs, an open-source web-based VR platform [1].

’Avatar’ refers to a character that represents and is controlled by a user in
a virtual environment. Currently, Mozilla Hubs only supports avatars having a
simplified upper body (Fig. 3(a)), which may reduce computation costs on the
client side, ensuring Mozilla Hubs’ high accessibility even on low-end
devices. On the other hand, it limits the conveyance of non-verbal information
through body gestures. We believed that it is also essential for effective
communication, and the use of full-body avatars that possess complete body
parts like a real human (Fig. 3(b)) can compensate this function.

In fact, previous studies have shown how full-body avatars benefit
communication in virtual environments [2, 3]. Similarly, full-body avatars can
improve sense of presence [4], which in turn play an important role in
learning [5]. Following these findings, we decided to integrate full-body
avatars to the Mozilla-Hubs-based platform we are developing.

In addition, Mozilla Hubs currently synchronizes avatar transforms (changes in
the position, orientation, or size of an avatar’s bones) through WebSocket on
a mesh network, while we considered WebRTC DataChannel more suitable to
synchronize avatar transforms due to security concerns and consistency between
users’ voices and gestures. In this study, we aimed to expand Mozilla Hubs’
implementation to enable the use of full-body avatars and to have the full-
body avatar transforms synchronized by WebRTC DataChannel.

##  II Proof of Concept

###  II-A Implementation

####  II-A1 Accommodate Full-body Avatars in Mozilla Hubs

In the original implementation in Mozilla Hubs, avatars are hard-coded to
contain limited bone names and hierarchy
111https://github.com/MozillaReality/hubs-avatar-pipelines, and other avatars
with different skeletons, including full-body avatars, are usually neither
operable nor rendered properly. To address this, we prepared a bone mapping
function that maps the bones by checking the similarity of their name with
their corresponding body parts (e.g. LowerArm.R is mapped to right elbow).
Then we implemented Cyclic Coordinate Descent Inverse Kinematics [6] so that a
full-body avatar can still reflect its user’s poses naturally with limited
inputs.

####  II-A2 Synchronization of Full-body Avatar Transforms by WebRTC
DataChannel

WebRTC (Web Real-Time Communication) 222https://webrtc.org/ enables real-time
data transmission between web browsers without passing through a central
server. WebRTC DataChannel is capable of transmitting text or binary data,
implemented on top of UDP while remaining reliability similar to TCP, and
incorporating DTLS to encode the data transmission.

In our previous study [1], we introduced an alternative WebRTC SFU solution
into Mozilla Hubs to enhance its audio transmission, and we also delegated the
transmission of avatar transforms to DataChannel. In this current study, we
continued to employ this delegation and extended it to encompass full-body
avatars for the following reasons.

Firstly, latency on the synchronization of voices and gestures may cause
inconsistency between verbal and non-verbal cues, impacting the communication
quality and effectiveness. We argued that real-time alignment between verbal
and non-verbal cues with minimal latency should be prioritized, and
DataChannel holds the potential to address this concern.

Moreover, within the metaverse, avatar transforms can be regarded as sensitive
personal data, especially when using full-body avatars which reproduce a
user’s real body poses, revealing more about the user’s identity[7].
Leveraging WebRTC DataChannel allows us to encode avatar transforms
transmission and avoid routing through central servers, thereby enhancing
higher security.

###  II-B Preliminary Validation in Data Transmission Latency

A preliminary validation was conducted to measure the transmission latency
between audio and avatar transforms within both the original Mozilla Hubs
architecture and our proposed one. We hypothesized that our implementation
brings lower latency of avatar transforms from audio, resulting in higher
consistency between audio and avatar transforms.

One Apple MacBook serving as the Sender and one Windows 10 Desktop Computer
serving as the Observer, two devices were connected to the same room on self-
hosted Mozilla Hubs (Fig.1 and 2). For 5 minutes, the Sender repeatedly played
a 3 second audio clip and synchronized circular movements of its avatar’s left
hand. The avatar’s left hand positions were transmitted to the Observer, while
the audio clip was captured by the Sender’s microphone and also transmitted.
Timestamps were recorded whenever the Observer received the audio or observed
changes in the Sender’s avatar.

Figure 1: Validation Settings with Original Implementation.

Figure 2: Validation Settings with Proposed Implementation.

(a) Simplified avatar supported by original implementation

(b) Full-body avatar used in our proposed implementation

Figure 3: Screenshots by the Observer during validation.

Figure 4: Latency between audio and avatar transforms over time.

Subsequently, the transmission latency was calculated between avatar
transforms and audio using the recorded timestamps, and found that avatar
transforms were transmitted faster than audio in both conditions (Fig. 4). In
our proposed implementation, the average latency was 257.64 ms (SD = 16.10
ms), while in the original implementation, it was 184.04 ms (SD = 49.81 ms).
This suggests that, compared to the original implementation, our approach
results in higher dispersion between audio and avatar transforms due to either
slower audio transmission or faster synchronization of avatar transforms.
Further investigation is needed to clarify this.

Higher variation was also observed in latency in the original implementation,
as evident from the graph and the larger standard deviation, indicating
greater stability in our proposed implementation, which can also contribute to
higher consistency between audio and avatar transforms.

Regarding the limitations of this validation, it is worth noting that the
avatar’s movement was triggered when the audio clip was played, not precisely
when the Sender started the audio data transmission. This discrepancy might
had contributed to slower audio data arrivals at the Observer side.

Lastly, it is essential to acknowledge that this preliminary validation
involved only two devices in the same room. In rooms accommodating more users,
latency in both audio transmission and avatar transforms synchronization may
become more obvious, allowing for more meaningful comparisons.

##  III Conclusion

We extended the functionality of Mozilla Hubs to support the use of full-body
avatars and delegated full-body avatar transforms synchronization to WebRTC
DataChannel. The result of a preliminary validation failed to demonstrate a
more accurate synchronization but indicated more consistent time differentials
between audio and avatar transforms in our implementation. To gain a clearer
understanding of the latency improvement, further investigation is required
under higher client load, and we are also planning an usability assessment for
our implementation within an educational context.

## Acknowledgment

This work was partially supported by the following grants: JST Grant Number
JPMJPF2202.

## References

  * [1] Y.-H. Hu, K. Ito, and A. Igarashi, “Improving real-time communication for educational metaverse by alternative webrtc sfu and delegating transmission of avatar transform,” in _Proceedings of International Conference on Consumer Electronics-Taiwan 2023 (ICCETW)_ , July 2023.
  * [2] J. Cassell and H. H. Vilhjálmsson, “Fully embodied conversational avatars: Making communicative behaviors autonomous,” _Autonomous Agents and Multi-Agent Systems_ , vol. 2, pp. 45–64, 1999.
  * [3] T.-Y. Wang, Y. Sato, M. Otsuki, H. Kuzuoka, and Y. Suzuki, “Effect of full body avatar in augmented reality remote collaboration,” _2019 IEEE Conference on Virtual Reality and 3D User Interfaces (VR)_ , pp. 1221–1222, 2019.
  * [4] P. Heidicker, E. Langbehn, and F. Steinicke, “Influence of avatar appearance on presence in social vr,” _2017 IEEE Symposium on 3D User Interfaces (3DUI)_ , pp. 233–234, 2017.
  * [5] F. Mantovani and G. Castelnuovo, “The sense of presence in virtual training: Enhancing skills acquisition and transfer of knowledge through learning experience in virtual environments,” in _Being there: Concepts, effects and measurements of user presence in synthetic environments._ , G. Riva, F. Davide, and W. A. IJsselsteijn, Eds. IOS Press, 01 2003, ch. 11, pp. 167–181.
  * [6] B. Kenwright, “Inverse kinematics – cyclic coordinate descent (ccd),” _Journal of Graphics Tools_ , vol. 16, no. 4, pp. 177–217, 2012.
  * [7] V. Nair, W. Guo, J. Mattern, R. Wang, J. F. O’Brien, L. Rosenberg, and D. Song, “Unique identification of 50,000+ virtual reality users from head & hand motion data,” in _32nd USENIX Security Symposium (USENIX Security 23)_ , Anaheim, CA, Aug. 2023, pp. 895–910.


# A simplified counterexample to the integral representation
of the relaxation of double integrals

Andrea Braides
SISSA, via Bonomea 265, Trieste, Italy

###### Abstract

We show that the lower-semicontinuous envelope of a non-convex double integral
may not admit a representation as a double integral. By taking an integrand
with value $+\infty$ except at three points (say $-1$, $0$ and $1$) we give a
simple proof and an explicit formula for the relaxation that hopefully may
shed some light on this type of problems. This is a simplified version of
examples by Mora-Corral and Tellini, and Kreisbeck and Zappale, who
characterize the lower-semicontinuous envelope via Young measures.

Double-integral functionals defined in $L^{p}$ spaces of the form

$F(u):=\int_{\Omega\times\Omega}f(u(x)-u(y))\,dxdy$ (1)

can be treated using the direct methods of the Calculus of Variations. To that
end, necessary and sufficient conditions for the lower semicontinuity of $F$
with respect to weak $L^{p}$ topologies turn out to be the convexity and lower
semicontinuity of $f$, exactly as in the case of single-integral functionals
(see e.g. [1, 5]). In the case of non-convex $f$ the parallel is lost. Indeed,
in [4] it is shown that the lower-semicontinuous envelope of $F$ cannot be
represented as a double integral of the same form when the function $f$ is a
simple double-well potential. The proof in [4] relies on the representation of
the relaxed functional in terms of Young measures and on the study of the
optimality conditions satisfied by such measure-valued minimizers. We now give
a simple explanation of the non-representability of the relaxed functional
when $f$ is a double-well potential (or rather a “triple-well” potential with
wells in $-1$, $0$ and $1$) with “infinite depth”; namely,

$f(z)=\begin{cases}0&\hbox{if }z\in\\{-1,1\\}\cr 1&\hbox{if
}z=0\cr+\infty&\hbox{otherwise.}\end{cases}$ (2)

For simplicity we chose $\Omega=(0,1)$. We remark that we can extend this
example to everywhere finite integrands $f$ by approximation.

We note that other examples are shown in [3] when the integrand is of the form
$f(u(x),u(y))$. In that case the functionals are not invariant by
translations, so the parallel with local functionals would be with integrands
depending on $(u(x),\nabla u(x))$, for which lower-semicontinuity conditions
are more complex [2].

We now turn to the analysis of the counterexample.

Characterization of the lower-semicontinuous envelope. Note preliminarily that
a lower bound for the lower-semicontinuous envelope $\overline{F}$ of $F$ with
respect to the weak $L^{1}$-convergence is

$F_{0}(u):=\int_{\Omega\times\Omega}f^{**}(u(x)-u(y))\,dxdy,$ (3)

where the lower-semicontinuous convex envelope of $f$ is

$f^{**}(z)=\begin{cases}0&\hbox{if
}z\in[-1,1]\cr+\infty&\hbox{otherwise;}\end{cases}$ (4)

that is,

$F_{0}(u)=\begin{cases}0&\hbox{if }{\rm ess\hbox{-}sup}\,u-{\rm
ess\hbox{-}inf}\,u\leq 1\cr+\infty&\hbox{otherwise.}\end{cases}$ (5)

This lower bound implies that $\overline{F}$ is finite at most on functions
$u\in L^{\infty}(0,1)$ such that

${\rm ess\hbox{-}sup}\,u-{\rm ess\hbox{-}inf}\,u\leq 1.$ (6)

Let $u\in L^{\infty}(0,1)$ satisfy (6), and let $u_{j}$ be a sequence weakly
converging to $u$ and such that $F(u_{j})<+\infty$ for all $j$. Note that for
fixed $j$ the function $u_{j}$ can take at most two values almost everywhere
and these values are at distance $1$. Indeed by Fubini’s theorem for almost
all $y\in(0,1)$ we have $u_{j}(x)\in\\{u_{j}(y),u_{j}(y)-1,u_{j}(y)+1\\}$ for
almost every $x\in(0,1)$. Hence, there exists $z^{j}$ such that
$u_{j}(x)\in\\{z^{j},z^{j}-1,z^{j}+1\\}$ for almost every $x\in(0,1)$. If both
values $z^{j}-1$ and $z^{j}+1$ were taken on sets of positive measure, then we
would have $F(u_{j})=+\infty$, and a contradiction. Hence, we can suppose that
there exist $z^{j}$ such that $u_{j}(x)\in\\{z^{j},z^{j}+1\\}$ almost
everywhere. We can assume, up to subsequences, that $z_{j}\to z$, so that

$z\leq{\rm ess\hbox{-}inf}\,u\quad\hbox{ and }\quad{\rm ess\hbox{-}sup}\,u\leq
z+1,$ (7)

and that, if we let $A^{j}:=\\{x:u_{j}(x)=z+1\\}$, there exists $t\in[0,1]$
such that $\lim\limits_{j\to+\infty}|A^{j}|=t$. Hence, we obtain

$\displaystyle\lim_{j\to+\infty}F(u_{j})=\lim_{j\to+\infty}(|A^{j}|^{2}+(1-|A^{j}|)^{2})=t^{2}+(t-1)^{2}=2t^{2}-2t+1.$
(8)

Note that the minimum of $t^{2}+(1-t)^{2}$ is $1\over 2$ so that (8) implies
that $\overline{F}(u)\geq{1\over 2}$ for all $u$.

Since by the convergence of $\int_{(0,1)}u_{j}\,dx$ to $\int_{(0,1)}u\,dx$ we
have

$t=\int_{(0,1)}u\,dx-z,$ (9)

the limit of $F(u_{j})$ can be described in terms of $\int_{(0,1)}u\,dx$ and
$z$ only, and is independent of the particular sequence $u_{j}$.

Note conversely that if $u$ and $z$ are such that (7) holds, then there exist
$u_{j}$ with $u_{j}\in\\{z,z+1\\}$ and weakly converging to $u$, so that the
value $2t^{2}+2t+1$ is achieved on this sequence with $t$ given by (9). By
optimizing in $z$ we then have a description of $\overline{F}(u)$ as

$\displaystyle\overline{F}(u)$ $\displaystyle=$
$\displaystyle\min\biggl{\\{}\Bigl{(}\int_{(0,1)}u\,dx-z\Bigr{)}^{2}+\Bigl{(}\int_{(0,1)}u\,dx-z-1\Bigr{)}^{2}$
(10) $\displaystyle\hskip 113.81102pt:z\leq{\rm ess\hbox{-}inf}\,u,\ {\rm
ess\hbox{-}sup}\,u\leq z+1\Bigl{\\}}$ $\displaystyle=$
$\displaystyle\min\biggl{\\{}2\Bigl{(}\int_{(0,1)}u\,dx\Bigr{)}^{2}-2(2z+1)\Bigl{(}\int_{(0,1)}u\,dx\Bigr{)}+2z^{2}+2z+1$
$\displaystyle\hskip 113.81102pt:z\leq{\rm ess\hbox{-}inf}\,u,\ {\rm
ess\hbox{-}sup}\,u\leq z+1\Bigl{\\}}.$

We can make this formula more symmetric by the change of variables
$w=z+{1\over 2}$, so that

$\displaystyle\overline{F}(u)$ $\displaystyle=$
$\displaystyle\min\biggl{\\{}2\Bigl{(}\int_{(0,1)}u\,dx\Bigr{)}^{2}-4w\Bigl{(}\int_{(0,1)}u\,dx\Bigr{)}+2w^{2}+{1\over
2}$ (11) $\displaystyle\hskip 113.81102pt:{\rm ess\hbox{-}sup}\,u-{1\over
2}\leq w\leq{\rm ess\hbox{-}inf}\,u+{1\over 2}\Bigl{\\}}.$

Furthermore, noting that the functionals are invariant if we add a constant to
$u$, replacing $u$ by $u-\int_{(0,1)}u\,dx=0$ we also have

$\overline{F}(u)=\min\biggl{\\{}2w^{2}+{1\over 2}:{\rm
ess\hbox{-}sup}\,u-\int_{(0,1)}u\,dx-{1\over 2}\leq w\leq{\rm
ess\hbox{-}inf}\,u-\int_{(0,1)}u\,dx+{1\over 2}\Bigl{\\}}.$ (12)

Non representability of the lower-semicontinuous envelope. We now prove that
there exists no $g$ such that

$\overline{F}(u)=\int_{\Omega\times\Omega}g(u(x)-u(y))\,dxdy.$ (13)

Note that $g$ can be assumed to be even, up to replacing $g(z)$ with ${1\over
2}(g(z)+g(-z))$.

We first describe $\overline{F}(u)$ more precisely in some ‘extreme’ cases. In
the first one the minimization does not involve constraint (7), so that
$\overline{F}(u)={1\over 2}$. To get this, we note that if

${\rm ess\hbox{-}sup}\,u-{\rm ess\hbox{-}inf}\,u\leq{1\over 2}$ (14)

then we can take

$z=\int_{(0,1)}u\,dx-{1\over 2},$

and by (14) we have

${\rm ess\hbox{-}sup}\,u\leq{\rm ess\hbox{-}inf}\,u+{1\over 2}\leq
z+1\quad\hbox{ and }\quad z\leq{\rm ess\hbox{-}sup}\,u-{1\over 2}\leq{\rm
ess\hbox{-}inf}\,u,$

and $\overline{F}(u)={1\over 2}$ by formula (10). As a particular case of a
function satisfying (14) we can take $u$ a constant. In this case (13) would
give

$g(0)={1\over 2}\,.$ (15)

The other ‘extreme’ case is when only one $z$ is involved in the minimization
in (10); which is the case when ${\rm ess\hbox{-}sup}\,u-{\rm
ess\hbox{-}inf}\,u=1$, so that $z={\rm ess\hbox{-}inf}\,u$ and $z+1={\rm
ess\hbox{-}sup}\,u$. The value of $\overline{F}(u)$ is then simply

$\overline{F}(u)=\Bigl{(}\int_{(0,1)}u\,dx-{\rm
ess\hbox{-}inf}\,u\Bigr{)}^{2}+\Bigl{(}\int_{(0,1)}u\,dx-{\rm
ess\hbox{-}sup}\,u\Bigr{)}^{2}\,.$

This can be applied, for fixed $t\in(0,1)$, with $u$ given by

$u(x)=\begin{cases}1&\hbox{ if }x\leq t\cr 0&\hbox{ if }x>t,\end{cases}$

for which $\overline{F}(u)=2t^{2}-2t+1$. If (13) held true then by (15) we
would also have

$\overline{F}(u)=(2t^{2}-2t+1)g(0)+2t(1-t)g(1)={1\over
2}(2t^{2}-2t+1)+2t(1-t)g(1),$

which would give

$g(1)={2t^{2}-2t+1\over 4t(1-t)}={1\over 4}\Bigl{(}{t\over 1-t}+{1-t\over
t}\Bigr{)}.$

Taking different values for $t\in(0,1)$ we get different values for $g(1)$,
which is a contradiction.

Conclusions and remarks. Formula (10) shows that $\overline{F}(u)$ is obtained
by functions $u_{j}$ weakly converging to $u$ and oscillating between two
values $z$ and $z+1$ maximizing the measure of the subset of points
$(x,y)\in\Omega\times\Omega$ such that $u_{j}(x)=z$ and $u_{j}(y)=z+1$. This
operation depends only on $z$, which satisfies some constraints due to the
convergence of $u_{j}$ to $u$; minimizing the outcome in $z$ gives the optimal
choice of $u_{j}$. Minimization in $z$ is unconstrained if ${\rm
ess\hbox{-}sup}\,u-{\rm ess\hbox{-}inf}\,u\leq{1\over 2}$, while it is limited
to a single $z$ when ${\rm ess\hbox{-}sup}\,u-{\rm ess\hbox{-}inf}\,u=1$. The
dependence on the quantity ${\rm ess\hbox{-}sup}\,u-{\rm ess\hbox{-}inf}\,u$
highlights the nonlocality of the recovery sequences. An example of this fact
is obtained by considering constants $u=c$, for which we have minimizing
sequences oscillating between $c-{1\over 2}$ and $c+{1\over 2}$, while this is
not true for piecewise-constant functions: if $u$ takes only two values at
distance $1$ then a recovery sequence is $u$ itself, without oscillations.

We remark that from this example we also obtain examples with finite
integrand. Indeed, if $f_{n}$ is a sequence of functions increasingly
converging to $f$ given by (2) and

$F_{n}(u):=\int_{\Omega\times\Omega}f_{n}(u(x)-u(y))\,dxdy,$ (16)

then the lower-semicontinuous envelopes $\overline{F}_{n}$ converge to
$\overline{F}$. If there existed (convex) functions $g_{n}$ such that

$\overline{F}_{n}(u):=\int_{\Omega\times\Omega}g_{n}(u(x)-u(y))\,dxdy,$ (17)

then this would hold also for $\overline{F}$.

Acknowledgmnents. The content of this work is a lesson of the course “Local
and nonlocal variational problems in Sobolev spaces” held at SISSA in the
Winter Semester 2022-2023. The author gratefully acknowledges valuable
comments by Carolin Kreisbeck.

## References

  * [1] Bellido, J.C., Mora-Corral, C. Lower semicontinuity and relaxation via Young measures for nonlocal variational problems and applications to peridynamics. SIAM J. Math. Anal. 50 (2018), 779–809
  * [2] De Giorgi, E., Buttazzo, G., Dal Maso, G. On the lower semicontinuity of certain integral functionals Atti Accad. Naz. Lincei. Classe Sci. Fis. Mat. Natur. 74 (1983), 274–282
  * [3] Kreisbeck, C., Zappale, E. Loss of double-integral character during relaxation. SIAM J. Math. Anal. 53 (2021), 351–385.
  * [4] Mora-Corral, C., Tellini, A. Relaxation of a scalar nonlocal variational problem with a double-well potential. Calc. Var. Partial Differential Equations 59 (2020), 67.
  * [5] Pedregal, P. Weak lower semicontinuity and relaxation for a class of non-local functionals. Rev. Mat. Complut. 29 (2016), 485–495.


# Model-Free Safety-Critical Control for Robotic Systems

Tamas G. Molnar, Ryan K. Cosner, Andrew W. Singletary, Wyatt Ubellacker, and
Aaron D. Ames *This research is supported in part by the National Science
Foundation, CPS Award #1932091, Dow (#227027AT) and Aerovironment.The Authors
are with the Control and Dynamical Systems and the Department of Mechanical
and Civil Engineering, California Institute of Technology, Pasadena, CA 91125,
USA. {tmolnar, rkcosner, asinglet, wubellac<EMAIL_ADDRESS>
###### Abstract

This paper presents a framework for the safety-critical control of robotic
systems, when safety is defined on safe regions in the configuration space. To
maintain safety, we synthesize a safe velocity based on control barrier
function theory without relying on a – potentially complicated – high-fidelity
dynamical model of the robot. Then, we track the safe velocity with a tracking
controller. This culminates in model-free safety critical control. We prove
theoretical safety guarantees for the proposed method. Finally, we demonstrate
that this approach is application-agnostic. We execute an obstacle avoidance
task with a Segway in high-fidelity simulation, as well as with a Drone and a
Quadruped in hardware experiments.

##  I INTRODUCTION

Safety is a fundamental requirement in the control of many robotic systems,
including legged [1], flying [2] and wheeled robots [3]. Provable safety
guarantees and safety-critical control for robotics have therefore attracted
significant attention. Synthesizing safety-critical controllers, however,
typically relies on high-fidelity dynamical models describing the robots,
which are often complicated and high-dimensional. The underlying control laws,
therefore, are nontrivial to synthesize and implement [4, 5]. For example,
control barrier functions (CBFs) [6] are a popular tool to achieve provable
safety guarantees, although designing CBFs and calculating the corresponding
safe control inputs may be nontrivial if the dynamics are complicated.

To tackle this, [7] proposed model-free barrier functions by a data-driven
approach, while [8, 9] used robust CBFs to overcome the effects of unmodeled
dynamics. Furthermore, many works rely on reduced-order models for planning
and control [10]. These include single integrator models for multi-robot
applications [11, 12] or unicycle models for wheeled robots [13, 14], which
have proven to be extremely useful models despite being overly simplistic.
Here we draw inspiration from these models and approaches.

In this paper, we rely on CBFs to synthesize safe controllers for robotic
systems in which safe regions are defined in the configuration space. We treat
the safety-critical aspect of this problem in a model-free fashion, without
relying on the full-order dynamics of the robot. We follow the approach of
[15, 16], where a safe velocity was designed based on reduced-order kinematics
– i.e., without the full dynamical model – and this safe velocity was tracked
by a velocity tracking controller. This approach is agnostic to the
application domain, although the underlying tracking controllers depend on the
system and their synthesis or tuning may require knowledge about the full
model. Velocity tracking, however, is well-established in robotics [17] and
controllers executing stable tracking are available for many robots. Once
velocity tracking is established, enforcing safety does not require further
consideration of the high-fidelity model — we refer to this as model-free
safety-critical control.

While the idea behind this control method was established in [15], the present
paper formalizes and generalizes this approach via two main contributions.
First, we provide a theoretical proof of the safe behavior for robotic systems
executing the proposed control approach. Second, we demonstrate the
applicability of this method on wheeled, flying and legged robots: a Segway
(in simulation), a Drone and a Quadruped (in hardware experiments). This
justifies that the method is agnostic to the application domain.

Figure 1:  The proposed control method and its execution on hardware. While
the safety-critical controller does not rely on the full dynamical model of
the robot, it controls the motion in a provably safe manner.

The paper is organized as follows. Section II revisits control Lyapunov and
control barrier functions to achieve stability and safety. Section III
outlines the proposed control method, states and proves the safety guarantees
thereof. Section IV discusses robotic applications through simulations and
hardware experiments. Section V concludes the paper.

##  II PRELIMINARIES

Our approach relies on stable tracking of a safe velocity to achieve safety
for robotic systems. Thus, first we introduce the notions of stability and
safety, and the guarantees thereof provided by control Lyapunov functions
(CLFs) and control barrier functions (CBFs). CLFs and CBFs are illustrated in
Fig. 2 together with a stable and a safe trajectory.

Consider control-affine systems with state space ${X\subseteq\mathbb{R}^{n}}$,
state ${x\in X}$, set of admissible inputs ${U\subseteq\mathbb{R}^{m}}$, and
control input ${u\in U}$:

$\dot{x}=f(x)+g(x)u.$ (1)

Let ${f:X\to\mathbb{R}^{n}}$ and ${g:X\to\mathbb{R}^{n\times m}}$ be Lipschitz
continuous. For an initial condition ${x(0)=x_{0}\in X}$ and a Lipschitz
continuous controller ${k:X\to U}$, ${u=k(x)}$, the system has a unique
solution ${x(t)}$ which we assume to exist for all ${t\geq 0}$. We also assume
that ${x(t)\equiv 0}$ is an equilibrium of (1) if ${u(t)\equiv 0}$ (i.e.,
${f(0)=0}$) and $X$ is an open and connected neighborhood of ${x=0}$.

Throughout the paper we use the following notation. ${\|.\|}$ is Euclidean
norm and ${\|.\|_{\infty}}$ is maximum norm. We say that a continuous function
${\gamma:[0,b)\to\mathbb{R}_{\geq 0}}$, ${b\in\mathbb{R}_{>0}}$ is of
class-$\mathcal{K}$ (or ${\gamma:(-a,b)\to\mathbb{R}}$,
${a,b\in\mathbb{R}_{>0}}$ is of extended class-$\mathcal{K}$) if $\gamma$ is
strictly monotonically increasing and ${\gamma(0)=0}$.

###  II-A Stability and Control Lyapunov Functions

Hereinafter, we rely on the notion of exponential stability.

######  Definition 1.

The equilibrium ${x=0}$ of system (1) is exponentially stable if there exist
${a,M,\beta\in\mathbb{R}_{>0}}$ such that ${\|x_{0}\|\leq
a\Rightarrow\|x(t)\|\leq M{\rm e}^{-\beta t}\|x_{0}\|}$, ${\forall t\geq 0}$.

An efficient technique to achieve exponential stability is control synthesis
via control Lyapunov functions (CLFs) [18], as stated formally below.

######  Definition 2.

A continuously differentiable function ${V:X\to\mathbb{R}_{\geq 0}}$ is a
control Lyapunov function (CLF) for (1) if there exists
${c,k_{1},k_{2},\lambda\in\mathbb{R}_{>0}}$ such that ${\forall x\in X}$:

$\displaystyle\begin{split}k_{1}\|x\|^{c}\leq V(x)\leq k_{2}\|x\|^{c}\\\
\inf_{u\in U}\dot{V}(x,u)\leq-\lambda V(x),\end{split}$ (2)

where

$\dot{V}(x,u)=\nabla V(x)(f(x)+g(x)u)$ (3)

is the derivative of $V$ along system (1).

######  Theorem 1 ([18]).

If $V$ is a CLF for (1), then any locally Lipschitz continuous controller
${u=k(x)}$ satisfying

$\dot{V}(x,k(x))\leq-\lambda V(x),$ (4)

${\forall x\in X}$ renders ${x=0}$ exponentially stable.

Theorem 1 establishes that synthesizing a control input $u$ while enforcing
condition (4) achieves exponential stability.

Figure 2:  (a) A CLF and a stable trajectory. (b) A CBF and a safe trajectory.
While $V$ is nonnegative, $h$ may take any real value. (c) The stability
condition (4) and a stable trajectory (green), the ISS condition (10) and an
input-to-state stable trajectory (purple). For ISS the trajectory converges to
a neighborhood of ${x=0}$. (d) The safety condition (8) and a safe trajectory
(green), the ISSf condition (12) and an input-to-state safe trajectory
(purple). For ISSf a superset $S_{d}$ of $S$ is forward invariant.

###  II-B Safety and Control Barrier Functions

We consider system (1) safe if its state $x(t)$ is contained in a safe set
${S\subset X}$ for all time, as stated below.

######  Definition 3.

System (1) is safe w.r.t. $S$ if $S$ is forward invariant under (1), that is,
${x_{0}\in S\Rightarrow x(t)\in S}$, ${\forall t\geq 0}$.

The choice of the safe set is application-driven, e.g., it may represent
positions where a robot does not collide with obstacles. Here, we define the
safe set $S$ as the 0-superlevel set of a continuously differentiable function
${h:X\to\mathbb{R}}$:

$S=\\{x\in X:h(x)\geq 0\\}.$ (5)

Then, control barrier functions (CBFs) can be used as tools to synthesize
provably safe controllers in a similar fashion to how CLFs achieve stability.

######  Definition 4.

A continuously differentiable function ${h:X\to\mathbb{R}}$ is a control
barrier function (CBF) for (1) if there exists ${\alpha\in\mathbb{R}_{>0}}$
such that ${\forall x\in S}$: 111In general, $\alpha$ can be chosen as an
extended class-$\mathcal{K}$ function, while here we use a constant for
simplicity.

$\sup_{u\in U}\dot{h}(x,u)\geq-\alpha h(x),$ (6)

where

$\dot{h}(x,u)=\nabla h(x)(f(x)+g(x)u)$ (7)

is the derivative of $h$ along system (1).

######  Theorem 2 ([6]).

If $h$ is a CBF for (1), then any locally Lipschitz continuous controller
${u=k(x)}$ satisfying

$\dot{h}(x,k(x))\geq-\alpha h(x),$ (8)

${\forall x\in S}$ renders (1) safe w.r.t. $S$.

Theorem 2 establishes safety-critical controller synthesis by condition (8).
For example, a desired but not necessarily safe controller $k_{\rm d}(x)$ can
be modified in a minimally invasive way to a safe controller by solving the
quadratic program:

$\displaystyle\begin{split}k(x)={\rm arg}\\!\min_{u\in U}&\;(u-k_{\rm
d}(x))^{\top}(u-k_{\rm d}(x))\\\ \text{s.t.}\ &\;\dot{h}(x,u)\geq-\alpha
h(x).\end{split}$ (9)

The Lipschitz continuity of this controller is discussed in [6].

###  II-C Effect of Disturbances

In practice, robotic systems are often subject to unknown disturbances that
may compromise stability or safety. For example, a bounded disturbance
${d\in\mathbb{R}^{m}}$ added to the input $u$ leads to the system
${\dot{x}=f(x)+g(x)(u+d)}$.

To address disturbances, the notion of exponential stability can be extended
to exponential input-to-state stability (ISS) by modifying Definition 1.
Namely, we require that there exists a class-$\mathcal{K}$ function $\mu$ such
that $\|x_{0}\|\leq a\Rightarrow{\|x(t)\|\leq M{\rm e}^{-\beta
t}\|x_{0}\|+\mu(\|d\|_{\infty})}$, ${\forall t\geq 0}$. That is, solutions
converge to a neighborhood of the origin which depends on the size of the
disturbance. [19, 20] showed that exponential ISS is achieved by strengthening
(4) in Theorem 1 to:

$\dot{V}(x,u,d)\leq-\lambda V(x)+\iota(\|d\|_{\infty}),$ (10)

for some class-$\mathcal{K}$ function $\iota$.

Similarly, safety can be extended to input-to-state safety (ISSf) by requiring
that the system stays within a neighborhood ${S_{d}\supseteq S}$ of the safe
set $S$ which depends on the size of the disturbance: ${x_{0}\in
S_{d}\Rightarrow x(t)\in S_{d}}$, ${\forall t\geq 0}$. We define this
neighborhood as a 0-superlevel set:

$S_{d}=\\{x\in X:h(x)+\gamma(\|d\|_{\infty})\geq 0\\},$ (11)

with some class-$\mathcal{K}$ function $\gamma$. It was established in [21]
that ISSf is guaranteed by replacing (8) in Theorem 2 with:

$\dot{h}(x,u,d)\geq-\alpha h(x)-\iota(\|d\|_{\infty}),$ (12)

for some class-$\mathcal{K}$ function $\iota$.

##  III MODEL-FREE SAFETY-CRITICAL CONTROL

Now consider robotic systems with configuration space
${Q\subseteq\mathbb{R}^{n}}$, configuration coordinates ${q\in Q}$, set of
admissible inputs ${U\subseteq\mathbb{R}^{m}}$, control input ${u\in U}$, and
dynamics:

$D(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=Bu,$ (13)

where ${D(q)\in\mathbb{R}^{n\times n}}$ is the inertia matrix,
${C(q,\dot{q})\in\mathbb{R}^{n\times n}}$ contains centrifugal and Coriolis
forces, ${G(q)\in\mathbb{R}^{n}}$ involves gravity terms and
${B\in\mathbb{R}^{n\times m}}$ is the input matrix. $D(q)$ is symmetric,
positive definite, ${\dot{D}(q,\dot{q})-2C(q,\dot{q})}$ is skew-symmetric. We
consider control laws ${k:Q\times\mathbb{R}^{n}\to\mathbb{R}^{m}}$,
${u=k(q,\dot{q})}$, initial conditions ${q(0)=q_{0}}$,
${\dot{q}(0)=\dot{q}_{0}}$, and assume that a unique solution $q(t)$ exists
for all ${t\geq 0}$.

We consider the robotic system safe if its configuration $q$ lies within a
safe set $S$ for all time: ${q(t)\in S}$, ${t\geq 0}$.

######  Assumption 1.

The safe set is defined as the 0-superlevel set of a continuously
differentiable function ${h:Q\to\mathbb{R}}$:

$S=\\{q\in Q:h(q)\geq 0\\},$ (14)

where the gradient of $h$ is finite: ${\exists C_{h}\in\mathbb{R}_{>0}}$ such
that ${\|\nabla h(q)\|\leq C_{h}}$, ${\forall q\in S}$. That is, safety
depends on the configuration $q$ only and $h$ is independent of $\dot{q}$.

######  Problem Statement.

For the robotic system (13), design a controller ${u=k(q,\dot{q})}$ that
achieves safety with respect to set $S$ in (14), i.e., ${q(t)\in S}$,
${\forall t\geq 0}$ given certain initial conditions ${q_{0}\in Q}$ and
${\dot{q}_{0}\in\mathbb{R}^{n}}$.

###  III-A Control Method

Following [15, 16], we seek to maintain safety by synthesizing and tracking a
safe velocity. This reduces the complexity of safety-critical control
significantly, while velocity tracking controllers are widely used [17]. The
approach allows safety-critical control in a model-free fashion.

We synthesize the safe velocity ${\dot{q}_{\rm s}\in\mathbb{R}^{n}}$ so that
it satisfies:

$\nabla h(q)\dot{q}_{\rm s}\geq-\alpha h(q),$ (15)

cf. (8), for some ${\alpha\in\mathbb{R}_{>0}}$ to be selected. The safe
velocity $\dot{q}_{\rm s}$ depends on the configuration $q$. Note that (15) is
a kinematic condition that does not depend on the full dynamics (13).

To track the safe velocity, we define the tracking error:

$\dot{e}=\dot{q}-\dot{q}_{\rm s}.$ (16)

and use a velocity tracking controller ${u=k(q,\dot{q})}$. First, we consider
the scenario that $u$ is able to drive the error $\dot{e}$ to zero
exponentially, then we address the effect of disturbances.

######  Assumption 2.

The velocity tracking controller ${u=k(q,\dot{q})}$ achieves exponentially
stable tracking: ${\|\dot{e}(t)\|\leq M\|\dot{e}_{0}\|{\rm e}^{-\lambda t}}$
for some ${M,\lambda\in\mathbb{R}_{>0}}$. That is, if $\dot{e}$ is
differentiable ($\ddot{e}$, $\ddot{q}_{\rm s}$ exist), there exists a
continuously differentiable Lyapunov function
${V:Q\times\mathbb{R}^{n}\to\mathbb{R}_{\geq 0}}$ such that
${\forall(q,\dot{e})\in Q\times\mathbb{R}^{n}}$:

$k_{1}\|\dot{e}\|\leq V(q,\dot{e})\leq k_{2}\|\dot{e}\|,$ (17)

for some ${k_{1},k_{2}\in\mathbb{R}_{>0}}$, and there exists
${\lambda\in\mathbb{R}_{>0}}$ such that
${\forall(q,\dot{e},\dot{q},\ddot{q}_{\rm s})\in
Q\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}}$ $u$ satisfies:

$\dot{V}(q,\dot{e},\dot{q},\ddot{q}_{\rm s},u)\leq-\lambda V(q,\dot{e}),$ (18)

cf. (4). For exposition’s sake, below we assume $\ddot{q}_{\rm s}$ exists and
we use (18). This assumption is relaxed later in Remark 4.

Before discussing its safety guarantees, we demonstrate the applicability of
this method on a motivating example.

######  Example 1 (Double integrator system).

Here we revisit the example in [15]. As the simplest instantiation of (13),
consider a double integrator system in two dimensions:

$\ddot{q}=u,$ (19)

where ${q\in\mathbb{R}^{2}}$ is the planar position of the robot and
${u\in\mathbb{R}^{2}}$. Our goal is to navigate the system from a start
position $q_{0}$ to a goal $q_{\rm g}$ while avoiding obstacles. A simple
solution is to realize the desired velocity ${\dot{q}_{\rm d}=-K_{\rm
P}(q-q_{\rm g})}$ that is based on a proportional controller with gain
${K_{\rm P}\in\mathbb{R}_{>0}}$.

We can avoid an obstacle of radius $r$ centered at $q_{\rm o}$ by the help of
the distance ${d=\|q-q_{\rm o}\|}$ and the CBF:

$h(q)=d-r,$ (20)

with gradient ${\nabla h(q)=(q-q_{\rm o})^{\top}/\|q-q_{\rm o}\|=n_{\rm
o}^{\top}}$ equal to the unit vector $n_{\rm o}$ pointing from the obstacle to
the robot. Then, the safe velocity can be found by using condition (15).
Specifically, we modify the desired velocity $\dot{q}_{\rm d}$ in a minimally
invasive fashion by solving the quadratic program:

$\displaystyle\begin{split}{\rm arg}\\!\\!\min_{\dot{q}_{\rm
s}\in\mathbb{R}^{2}}&\;(\dot{q}_{\rm s}-\dot{q}_{\rm d})^{\top}(\dot{q}_{\rm
s}-\dot{q}_{\rm d})\\\ \text{s.t.}\ &\;n_{\rm o}^{\top}\dot{q}_{\rm
s}\geq-\alpha(d-r),\end{split}$ (21)

cf. (9). Based on the KKT conditions [22], it has the solution:

$\dot{q}_{\rm s}=\dot{q}_{\rm d}+\max\\{-n_{\rm o}^{\top}\dot{q}_{\rm
d}-\alpha(d-r),0\\}n_{\rm o}.$ (22)

The safe velocity can be tracked for example by the controller ${u=-K_{\rm
D}(\dot{q}-\dot{q}_{\rm s})}$ with gain ${K_{\rm D}\in\mathbb{R}_{>0}}$.

Figure 3:  Numerical simulation of the double integrator system (19) tracking
the safe velocity (22). The controller is able to keep the system safe if
parameter $\alpha$ is selected to be small enough.

Fig. 3 shows four simulation results for avoiding two obstacles with
parameters ${K_{\rm P}=0.2\,{\rm s^{-1}}}$, ${K_{\rm D}=1\,{\rm s^{-1}}}$ and
${\alpha=0.1,0.2,0.5}$ and ${1\,{\rm s^{-1}}}$, respectively. With the
proposed approach, the double integrator system avoids the obstacles, although
the second-order dynamics was not directly taken into account during the CBF
and control design. The condition for safety, however, is picking a small
enough $\alpha$ value (e.g. 0.1 or 0.2), while safety is violated for larger
$\alpha$ (e.g. 0.5 or 1). We remark that for multiple obstacles we considered
the closest one at each time. This results in a nonsmooth CBF which has been
analyzed in [23]. Accordingly, the safe velocity $\dot{q}_{\rm s}$ is only
piecewise differentiable; for simplicity, our constructions are restricted to
the differentiable segments.

###  III-B Main Result

In what follows, our main result proves that tracking the safe velocity
achieves safety for the full dynamics if parameter $\alpha$ is selected to be
small enough. Specifically, for tracking controllers satisfying Assumption 2
stability translates into safety for the full system (13) if
${\lambda>\alpha}$. As this result is agnostic to the application domain, this
culminates in model-free safety-critical control. Realizing velocity tracking
controllers, however, depends on the application. Later we give examples for
such controllers and corresponding CLFs.

The following theorem summarizes the safety guarantees provided by tracking
the safe velocity.

######  Theorem 3.

Consider system (13), safe set (14), safe velocity satisfying (15), and
velocity tracking controller satisfying (18). If ${\lambda>\alpha}$, safety is
achieved such that ${(q_{0},\dot{e}_{0})\in S_{V}\Rightarrow q(t)\in S}$,
${\forall t\geq 0}$, where:

$\displaystyle\begin{split}&S_{V}=\\{(q,\dot{e})\in
Q\times\mathbb{R}^{n}:h_{V}(q,\dot{e})\geq 0\\},\\\
&h_{V}(q,\dot{e})=-V(q,\dot{e})+\alpha_{\rm e}h(q),\end{split}$ (23)

with ${\alpha_{\rm e}=(\lambda-\alpha)k_{1}/C_{h}>0}$ and $C_{h}$, $k_{1}$
defined at (14, 17).

###### Proof.

Since ${V(q,\dot{e})\geq 0}$, the implication ${h_{V}(q,\dot{e})\geq
0}\Rightarrow{h(q)\geq 0}$ holds. Thus, ${h_{V}(q(t),\dot{e}(t))\geq 0}$,
${\forall t\geq 0}$ is sufficient to prove. We prove this by noticing that the
initial conditions satisfy ${h_{V}(q_{0},\dot{e}_{0})\geq 0}$ and we also
have:

$\displaystyle\begin{split}\dot{h}_{V}&(q,\dot{e},\dot{q},\ddot{q}_{\rm
s},u)=-\dot{V}(q,\dot{e},\dot{q},\ddot{q}_{\rm s},u)+\alpha_{\rm e}\nabla
h(q)\dot{q}\\\ &\geq\lambda V(q,\dot{e})+\alpha_{\rm e}\nabla h(q)\dot{q}_{\rm
s}+\alpha_{\rm e}\nabla h(q)\dot{e}\\\ &\geq\lambda V(q,\dot{e})-\alpha_{\rm
e}\alpha h(q)+\alpha_{\rm e}\nabla h(q)\dot{e}\\\
&\geq(\lambda-\alpha)V(q,\dot{e})-\alpha_{\rm e}\|\nabla
h(q)\|\|\dot{e}\|-\alpha h_{V}(q,\dot{e})\\\
&\geq(\lambda-\alpha)k_{1}\|\dot{e}\|-\alpha_{\rm e}C_{h}\|\dot{e}\|-\alpha
h_{V}(q,\dot{e})\\\ &\geq-\alpha h_{V}(q,\dot{e}).\end{split}$ (24)

Here we used the following properties in the 6 lines of the inequality: (i)
definition (23) of $h_{V}$, (ii) stability condition (18) and definition (16)
of $\dot{e}$, (iii) condition (15) on the safe velocity, (iv) definition (23)
of $h_{V}$ and the Cauchy-Schwartz inequality, (v) lower bound of $V$ in (17)
and upper bound $C_{h}$ of $\|\nabla h(q)\|$, (vi) definition of $\alpha_{\rm
e}$. This guarantees ${h_{V}(q(t),\dot{e}(t))\geq 0}$, ${\forall t\geq 0}$ by
Theorem 2. ∎

######  Remark 1.

Condition ${\lambda>\alpha}$ means the controller tracks the safe velocity
fast enough (characterized by $\lambda$) compared to how fast the boundary of
the safe set may be approached (characterized by $\alpha$). In practice, one
can pick a small enough $\alpha$ for a given velocity tracking controller, for
example, by gradually increasing $\alpha$ from $0$. The existence of such
$\alpha$ is guaranteed by the Theorem. Note that there is a trade-off: for
smaller $\alpha$ the system may become more conservative, evolving farther
from the boundary of the safe set.

######  Remark 2.

Condition (15) is equivalent to designing a safe control input $\dot{q}_{\rm
s}$ for the single integrator system $\dot{q}=\dot{q}_{\rm s}$. Thus, this
approach is a manifestation of control based on reduced-order models. While
$h$ is a CBF for the reduced-order model, $h_{V}$ is a CBF for the full system
(13) as a dynamic extension of $h$, similar to the energy-based extension in
[16]. Other reduced-order models of the form ${\dot{q}=A(q)\mu_{\rm s}}$ with
control input ${\mu_{\rm s}\in\mathbb{R}^{k}}$ and transformation
${A(q)\in\mathbb{R}^{n\times k}}$ can also be used. This, for example,
includes the unicycle model for wheeled robots with
${q=(x,y,\psi)\in\mathbb{R}^{3}}$ containing Cartesian positions and yaw angle
and ${\mu_{\rm s}=(v_{\rm s},\omega_{\rm s})\in\mathbb{R}^{2}}$ containing
forward velocity and yaw rate:

$\begin{bmatrix}\dot{x}\\\ \dot{y}\\\
\dot{\psi}\end{bmatrix}=\begin{bmatrix}\cos\psi&0\\\ \sin\psi&0\\\
0&1\end{bmatrix}\begin{bmatrix}v_{\rm s}\\\ \omega_{\rm s}\end{bmatrix}.$ (25)

The safe velocity $\mu_{\rm s}$ is given by ${\nabla h(q)A(q)\mu_{\rm
s}\geq-\alpha h(q)}$ based on (15), and the proof of Theorem 3 holds with
substitution ${\dot{q}_{\rm s}=A(q)\mu_{\rm s}}$. The tracking controller $u$,
however, must provide property (18) with respect to
${\dot{e}=\dot{q}-A(q)\mu_{\rm s}}$.

######  Remark 3.

Theorem 3 requires initial conditions to satisfy ${(q_{0},\dot{e}_{0})\in
S_{V}\iff h(q_{0})\geq V(q_{0},\dot{e}_{0})/\alpha_{\rm e}}$. This is a
stricter condition than ${q_{0}\in S\iff h(q_{0})\geq 0}$ that is usually
required in safety-critical control (cf. Definition 3). The additional
conservatism is reduced when the initial tracking error $\dot{e}_{0}$ is
smaller (since ${V(q_{0},\dot{e}_{0})}$ is smaller) and when the tracking is
faster, i.e., ${\lambda-\alpha}$ is larger (since $\alpha_{\rm e}$ is larger).

######  Remark 4.

The error $\dot{e}$ is assumed to be differentiable in Assumption 2 only for
exposition’s sake. Theorem 3 can be extended to non-differentiable signals
satisfying ${\|\dot{e}(t)\|\leq M\|\dot{e}_{0}\|{\rm e}^{-\lambda t}}$. The
proof relies on the fact that ${\dot{h}(q,\dot{q})\geq-\alpha
h(q)-C_{h}M\|\dot{e}_{0}\|{\rm e}^{-\lambda t}}$ holds, and by the comparison
lemma with ${\dot{y}(t)=-\alpha y(t)-C_{h}M\|\dot{e}_{0}\|{\rm e}^{-\lambda
t}}$, ${y(0)=h(q_{0})}$ one can show that ${h(q(t))\geq y(t)\geq 0}$.

###  III-C Effect of Disturbances

Now consider that ideal exponential tracking of the safe velocity is not
possible. This can be captured via a bounded input disturbance $d$, that
represents the effect of imperfect tracking controllers, time delays or
modeling errors. Then, instead of safety, one shall guarantee input-to-state
safety (ISSf), i.e., the invariance of the larger set ${S_{d}\supseteq S}$:

$\displaystyle\begin{split}&S_{d}=\\{q\in Q:h_{d}(q)\geq 0\\},\\\
&h_{d}(q)=h(q)+\gamma(\|d\|_{\infty}),\end{split}$ (26)

where $\gamma$ is a class-$\mathcal{K}$ function to be specified. We also
introduce the dynamic extension ${S_{Vd}\supseteq S_{V}}$ of set $S_{d}$:

$\displaystyle\begin{split}&S_{Vd}=\\{(q,\dot{e})\in
Q\times\mathbb{R}^{n}:h_{Vd}(q,\dot{e})\geq 0\\},\\\
&h_{Vd}(q,\dot{e})=h_{V}(q,\dot{e})+\gamma(\|d\|_{\infty}).\end{split}$ (27)

We show that ISSf is guaranteed by input-to-state stable (ISS) tracking:
${\|\dot{e}(t)\|\leq M\|\dot{e}_{0}\|{\rm e}^{-\lambda
t}+\mu(\|d\|_{\infty})}$. Note that exponential ISS is our strongest
assumption. When the tracking is poor, $\mu(\|d\|_{\infty})$ dominates this
bound. If the error does not decay (${M=0}$), the bound reduces to
${\|\dot{e}(t)\|\leq\|\dot{e}\|_{\infty}}$ and we recover the traditional ISSf
guarantees in [21]. For ISS, instead of (18) the tracking controller shall
satisfy:

$\dot{V}(q,\dot{e},\dot{q},\ddot{q}_{\rm s},u,d)\leq-\lambda
V(q,\dot{e})+\iota(\|d\|_{\infty}),$ (28)

for some class-$\mathcal{K}$ function $\iota$. The connection between ISS and
ISSf is summarized in the following Corollary of Theorem 3.

######  Corollary 1.

Consider system (13), sets $S_{d}$ and $S_{Vd}$ in (26) and (27), safe
velocity satisfying (15), and velocity tracking controller satisfying (28). If
${\lambda>\alpha}$, input-to-state safety is achieved such that
${(q_{0},\dot{e}_{0})\in S_{Vd}\Rightarrow q(t)\in S_{d}}$, ${\forall t\geq
0}$, where $\alpha_{\rm e}$ is given in Theorem 3 and
${\gamma(\|d\|_{\infty})=\iota(\|d\|_{\infty})/\alpha}$.

The proof follows the same steps as those in the proof of Theorem 3, by
replacing $h$ and $h_{V}$ with $h_{d}$ and $h_{Vd}$. Corollary 1 concludes
that input-to-state stable tracking of a safe velocity implies input-to-state
safety for the full system, i.e., robust velocity tracking implies robust
safety guarantees.

###  III-D Velocity Tracking Controllers

Figure 4:  High-fidelity simulation of a Ninebot E+ Segway platform. (a)
Planar dynamical model (13, 39) with the model-free safety-critical controller
(41, 42). (b) Spatial dynamical model with the model-free controller (44, 45).
The controllers keeps the system safe (the CBF $h$ is positive for all time).

Finally, we consider examples of velocity tracking controllers that provide
stability by (18) or ISS by (28). As the simplest choice, we consider a model-
free D controller:

$u=-K_{\rm D}\dot{e},$ (29)

where ${K_{\rm D}\in\mathbb{R}^{m\times n}}$ is selected so that ${K=BK_{\rm
D}}$ is positive definite. Furthermore, when model-dependent terms are well-
known, they can also be included in the control law. If ${n=m}$ and $B$ is
invertible (i.e., the system is fully actuated), one may use a D controller
with gravity compensation:

$u=B^{-1}(G(q)-K\dot{e}),$ (30)

with a positive definite gain ${K\in\mathbb{R}^{n\times n}}$. Moreover, one
can also use a heavily model-dependent extension:

$u=B^{-1}(D(q)\ddot{q}_{\rm s}+C(q,\dot{q})\dot{q}_{\rm s}+G(q)-K\dot{e}).$
(31)

While this controller may achieve better tracking, it requires $D(q)$ and
$C(q,\dot{q})$ which may have complicated expressions and may be expensive to
compute in practice.

We characterize these controllers by the constant
${\lambda\in\mathbb{R}_{>0}}$:

$\lambda=\frac{\sigma_{\rm min}(K)}{\displaystyle\sup_{q\in Q}\sigma_{\rm
max}(D(q))},$ (32)

where $\sigma_{\rm min}$ and $\sigma_{\rm max}$ denote the smallest and
largest eigenvalue. The eigenvalues are positive real numbers due to the
positive definiteness of $D(q)$ and $K$. Accordingly, $\lambda$ represents the
smallest gain divided by the largest inertia, hence characterizes how fast
controllers may track. We associate the controllers with the Lyapunov function
candidate:

$V(q,\dot{e})=\sqrt{\frac{1}{2}\dot{e}^{\top}D(q)\dot{e}},$ (33)

that has the bound (17) with ${k_{1}=\inf_{q\in Q}\sqrt{\sigma_{\rm
min}(D(q))/2}}$ and ${k_{2}=\sup_{q\in Q}\sqrt{\sigma_{\rm max}(D(q))/2}}$. We
also define the linear class-$\mathcal{K}$ function
${\iota(\|d\|_{\infty})=\|d\|_{\infty}/(2k_{1})}$.

With the above controllers, the parameters to be selected during control
design are $\alpha$ and $K_{\rm D}$ or $K$. Now we state that these
controllers satisfy the required stability properties.

######  Proposition 1.

Consider system (13), Lyapunov function $V$ defined by (33), constant
$\lambda$ given by (32) and ${\dot{e}\neq 0}$.

  * (i)

Controller (29) satisfies the ISS condition (28) with respect to
${d=-D(q)\ddot{q}_{\rm s}-C(q,\dot{q})\dot{q}_{\rm s}-G(q)}$.

  * (ii)

Controller (30) satisfies the ISS condition (28) with respect to
${d=-D(q)\ddot{q}_{\rm s}-C(q,\dot{q})\dot{q}_{\rm s}}$ when ${\dot{q}_{\rm
s}\not\equiv 0}$ and the stability condition (18) when ${\dot{q}_{\rm s}\equiv
0}$.

  * (iii)

Controller (31) satisfies the stability condition (18).

###### Proof.

The proof follows that in Section 8.2 of [17]. Here we prove case (i) only.
The proof of case (ii) is the same when ${\dot{q}_{\rm s}\not\equiv 0}$,
whereas the proofs of case (ii) when ${\dot{q}_{\rm s}\equiv 0}$ and case
(iii) can be obtained by substituting ${d\equiv 0}$.

We differentiate $V$ given by (33):

$\dot{V}(q,\dot{e},\dot{q},\ddot{q}_{\rm
s},u,d)=\frac{1}{2V(q,\dot{e})}\left(\frac{1}{2}\dot{e}^{\top}\dot{D}(q,\dot{q})\dot{e}+\dot{e}^{\top}D(q)\ddot{e}\right),$
(34)

and substitute the error dynamics corresponding to (13, 16):

$D(q)\ddot{e}=-C(q,\dot{q})\dot{e}-D(q)\ddot{q}_{\rm
s}-C(q,\dot{q})\dot{q}_{\rm s}-G(q)+Bu.$ (35)

For controller (29) this leads to:

$\dot{V}(q,\dot{e},\dot{q},\ddot{q}_{\rm
s},u,d)=\frac{-\dot{e}^{\top}K\dot{e}+\dot{e}^{\top}d}{2V(q,\dot{e})},$ (36)

where the term ${\dot{e}^{\top}(\dot{D}(q,\dot{q})-2C(q,\dot{q}))\dot{e}}$
dropped since ${\dot{D}(q,\dot{q})-2C(q,\dot{q})}$ is skew-symmetric.

Based on (36), now we show (28) holds. Since (32) implies
${\dot{e}^{\top}K\dot{e}-\lambda\dot{e}^{\top}D(q)\dot{e}\geq 0}$, the
definition (33) of $V$ leads to:

$\frac{-\dot{e}^{\top}K\dot{e}}{2V(q,\dot{e})}\leq-\lambda V(q,\dot{e}),$ (37)

${\forall q\in Q,\dot{e}\in\mathbb{R}^{n}}$. Furthermore, the Cauchy-Schwartz
inequality, the bound (17) on $V$ and the definition of $\iota$ yield:

$\frac{\dot{e}^{\top}d}{2V(q,\dot{e})}\leq\frac{\|\dot{e}\|\|d\|_{\infty}}{2k_{1}\|\dot{e}\|}=\iota(\|d\|_{\infty}),$
(38)

where $\|\dot{e}\|$ drops, making the right-hand side independent of time.
Substituting (37, 38) into (36) yields (28). ∎

##  IV APPLICATIONS TO WHEELED, FLYING AND LEGGED ROBOTS

Figure 5:  Hardware experiments using the proposed model-free safety-critical
control method. An obstacle avoidance task is accomplished by two
fundamentally different robots: a custom-made racing Drone (top) and a Unitree
A1 Quadruped (bottom). (a) The Drone is tracking a safe velocity determined
based on single integrator model. (b) The Quadruped is tracking a safe
velocity based on single integrator model via side-stepping and (c) based on
unicycle model via turning. Both robots executed the task with guaranteed
safety. A video of the experiments can be found at [24].

Now we apply the proposed control method to robotic platforms, including high-
fidelity simulations of a Segway and hardware experiments on a Drone and a
Quadruped.

###  IV-A Numerical Simulation of Segway

We consider a Ninebot E+ Segway platform with its planar and spatial high-
fidelity dynamical models described in [25].

######  Example 2 (Segway in plane).

Consider the two-degrees of freedom planar Segway model in Fig. 4(a) with
configuration
${q=[p,\,\varphi]^{\top}\in\mathcal{Q}=\mathbb{R}\times[0,2\pi]}$ including
the position $p$ and pitch angle $\varphi$. The dynamics are in form (13),
where:

$\displaystyle\begin{split}D(q)&\\!=\\!\begin{bmatrix}m_{0}&mL\cos\varphi\\\
mL\cos\varphi&J_{0}\end{bmatrix}\\!,\;\;G(q)\\!=\\!\begin{bmatrix}0\\\
-mgL\sin\varphi\end{bmatrix}\\!,\\\ C(q,\dot{q})&\\!=\\!\begin{bmatrix}b_{\rm
t}/R&-b_{\rm t}-mL\dot{\varphi}\sin\varphi\\\ -b_{\rm t}&b_{\rm
t}R\end{bmatrix}\\!,\;\;B\\!=\\!\begin{bmatrix}K_{\rm m}/R\\\ -K_{\rm
m}\end{bmatrix}\\!,\end{split}$ (39)

with parameters given in Table I and ${u\in U=[-20,20]\,{\rm V}}$.

TABLE I: Parameters of the Segway Model Description | Parameter | Value | Unit
---|---|---|---
gravitational acceleration | $g$ | 9.81 | m/s2
radius of wheels | $R$ | 0.195 | m
mass of wheels | $M$ | 2$\times$2.485 | kg
mass moment of inertia of wheels | $J_{C}$ | 2$\times$0.0559 | kgm2
distance of wheel center to frame CoM | $L$ | 0.169 | m
mass of frame | $m$ | 44.798 | kg
mass moment of inertia of frame | $J_{G}$ | 3.836 | kgm2
lumped mass ${m_{0}=m+M+J_{C}/R^{2}}$ | $m_{0}$ | 52.710 | kg
lumped inertia ${J_{0}=mL^{2}+J_{G}}$ | $J_{0}$ | 5.108 | kgm2
torque constant of motors | $K_{\rm m}$ | 2$\times$1.262 | Nm/V
damping constant of motors | $b_{\rm t}$ | 2$\times$1.225 | Ns

Our goal is to realize a desired forward velocity $\dot{p}_{\rm d}$ until
reaching a wall at position $p_{\rm max}$, then stop automatically and safely
in front of the wall. This is captured by the CBF:

$h(q)=p_{\rm max}-p,$ (40)

which, by condition (15), leads to the safe forward velocity:

$\dot{p}_{\rm s}=\min\\{\dot{p}_{\rm d},\alpha(p_{\rm max}-p)\\},$ (41)

similar to (22). This safe velocity is tracked by the controller:

$u=K_{\dot{p}}(\dot{p}-\dot{p}_{\rm
s})+K_{\varphi}\varphi+K_{\dot{\varphi}}\dot{\varphi}$ (42)

with ${K_{\dot{p}}=50\,{\rm Vs/m}}$, ${K_{\varphi}=150\,{\rm V/rad}}$,
${K_{\dot{\varphi}}=40\,{\rm Vs/rad}}$, which also stabilizes the Segway to
the upright position.

Fig. 4(a) shows simulation results where the Segway executes the task starting
from ${p_{0}=0}$, ${\varphi_{0}=-0.138\,{\rm rad}}$ (where its frame is
vertical), ${\dot{p}_{0}=0}$, ${\dot{\varphi}_{0}=0}$, for ${\dot{p}_{\rm
d}=1\,{\rm m/s}}$, ${p_{\rm max}=2\,{\rm m}}$ and ${\alpha=0.5\,{\rm
s^{-1}}}$. Notice that controller (41, 42) is model-free, it does not rely on
the full dynamics (13, 39). The gains $K_{\dot{p}}$, $K_{\varphi}$ and
$K_{\dot{\varphi}}$, however, are tuned so that the full dynamics achieves
stable velocity tracking. These gains were tuned based on linearization and
LQR in [25] and they determine the tracking performance with the associated
$\lambda$.

######  Example 3 (Segway in space).

Consider the spatial model of the Segway in Fig. 4(b) with 7-dimensional state
space and 2 control inputs. The task is to navigate it from a start point to a
goal (left panel) while avoiding obstacles of radius 0.5 m (solid black),
similar to Example 1. The obstacle radius is buffered by the size of the
Segway (dashed black) and the Segway’s center must be kept outside this zone.

This task is accomplished by tracking a safe velocity obtained for the
unicycle model (25); cf. Remark 2. We set the desired forward velocity and yaw
rate ${\mu_{\rm d}=(v_{\rm d},\omega_{\rm d})}$ based on the distance ${d_{\rm
g}=\|(x_{\rm g}-x,y_{\rm g}-y)\|}$ to the goal as ${v_{\rm d}=K_{v}d_{\rm g}}$
and ${\omega_{\rm d}=-K_{\omega}(\sin\psi-(y_{\rm g}-y)/d_{\rm g})}$. To avoid
obstacles, we use a CBF that includes the heading direction:

$h(q)=d-r-\delta\cos(\psi-\theta),$ (43)

where ${d=\|(x_{\rm o}-x,y_{\rm o}-y)\|}$ is the distance from the obstacle,
${\theta=\arctan((y_{\rm o}-y)/(x_{\rm o}-x))}$ is the angle towards the
obstacle, and ${\delta\in\mathbb{R}_{>0}}$ is a tunable parameter.

This CBF is incorporated into the quadratic program:

$\displaystyle\begin{split}{\rm arg}\\!\\!\\!\min_{\mu_{\rm
s}\in\mathbb{R}^{2}}&\;(\mu_{\rm s}-\mu_{\rm d})^{\top}\Gamma(\mu_{\rm
s}-\mu_{\rm d})\\\ \text{s.t.}\ &\;\nabla h(q)A(q)\mu_{\rm s}\geq-\alpha
h(q),\end{split}$ (44)

cf. (21), where ${\Gamma={\rm diag}\\{1,R^{2}\\}}$ is a weight between forward
velocity and yaw rate with parameter ${R\in\mathbb{R}_{>0}}$. The resulting
safe velocity ${\mu_{\rm s}=(v_{\rm s},\omega_{\rm s})}$ is tracked by the
controller:

$u_{1,2}=K_{\dot{p}}(\dot{p}-v_{\rm
s})+K_{\varphi}\varphi+K_{\dot{\varphi}}\dot{\varphi}\pm
K_{\dot{\psi}}(\dot{\psi}-\omega_{\rm s})$ (45)

used at the two wheels with the same gains as in Example 2 and a gain
${K_{\dot{\psi}}=10\,{\rm Vs/rad}}$ on the yaw rate $\dot{\psi}$.

With this approach, the Segway is able to move to the goal safely, while its
controller (44, 45) is model-free. Fig. 4(b) shows the safe motion for
${K_{v}=0.16\,{\rm s^{-1}}}$, ${K_{\omega}=0.8\,{\rm s^{-1}}}$,
${\alpha=0.2\,{\rm s^{-1}}}$, ${\delta=0.5\,{\rm m}}$ and ${R=0.25\,{\rm m}}$.

###  IV-B Hardware Experiments on Drone and Quadruped

We executed the obstacle avoidance task of Example 3 on two fundamentally
different hardware platforms: a Drone and a Quadruped; see Fig. 5. The
obstacle locations were known to the robots, sensory information was used to
determine the robots’ position only. We performed two classes of experiments:
by synthesizing safe velocities based on the single integrator and unicycle
models, respectively; cf. Remark 2. A video of the experiments can be found at
[24].

First, we considered the single integrator model, and we tracked the
associated safe velocity with the Drone and the Quadruped by platform-specific
tracking controllers. We used CBF (20) and safe velocity (22). The desired
velocity was ${\dot{q}_{\rm d}=-K_{\rm P}(q-q_{\rm g})}$ with saturation; cf.
Example 1.

The Drone was a custom-built racing drone [26], shown in Fig. 5(a). It has 6
degrees of freedom and 4 actuators. The state of the Drone (position,
orientation and corresponding velocities) were measured by IMU and an
OptiTrack motion capture system. State estimation and control action
computation ran at 400 Hz. The safe velocity was commanded to the drone
wireless from a desktop computer, while velocity tracking was done using an
on-board betaflight flight controller. The safe velocity was calculated with
${K_{\rm P}=0.7\,{\rm s^{-1}}}$ and ${\alpha=0.2\,{\rm s^{-1}}}$. Fig. 5(a)
shows the Drone reaching the goal safely, as guaranteed by Theorem 3 since
$\alpha$ was selected small enough for the available tracking performance. The
value of $\alpha$ was chosen based on the simulated response of the single
integrator. $\alpha$ was not tuned for optimal performance, and it could
potentially be increased for less conservatism.

The Quadruped was a Unitree A1 quadrupedal robot, shown in Fig. 5(b), which
has 18 degrees of freedom and 12 actuators. Its position was measured based on
odometry assuming the feet do not slip, while joint states were available via
built-in encoders. An ID-QP walking controller was realized at 1 kHz loop rate
on this robot to track a stable walking gait with prescribed forward and
lateral velocities and yaw rate, designed using the concepts in [27].
Individual commands were tracked via a motion primitive framework described in
[28]. In the single integrator experiments, the yaw rate was set to zero,
while the safe velocity (22) with ${K_{\rm P}=0.1\,{\rm s^{-1}}}$ and
${\alpha=0.2\,{\rm s^{-1}}}$ was tracked by forward- and side-stepping. The
Quadruped executed the task safely similar to the Drone (see Fig. 5(b)),
although it has fundamentally different dynamic behavior. This indicates the
application-agnostic nature of our model-free approach.

Finally, we used the unicycle model (25) and CBF (43) to achieve safety on the
Quadruped. The safe forward velocity and yaw rate in (44) were tracked by the
same ID-QP walking controller. Fig. 5(c) shows the Quadruped traversing the
obstacle course with ${K_{v}=0.08\,{\rm s^{-1}}}$, ${K_{\omega}=0.4\,{\rm
s^{-1}}}$, ${\alpha=0.2\,{\rm s^{-1}}}$, ${\delta=0.5\,{\rm m}}$ and
${R=0.5\,{\rm m}}$. While safety is maintained, the Quadruped performs the
task with different behavior than in the previous experiment: it walks forward
and turns instead of forward- and side-stepping. Still, safety is provably
guaranteed — and in a model-free fashion.

##  V CONCLUSIONS

We considered safety-critical control for robotic systems in a model-free
fashion following [15]. Our control method relies on a synthesizing a safe
velocity using control barrier functions and tracking this velocity. We stated
and proved theoretical guarantees for the safety of our method. Namely, safety
is achieved when the safe velocity is tracked faster than how fast the
corresponding safe motion may approach the boundary of the safe set. Due to
its model-free nature, our approach is application-agnostic. By simulation and
hardware experiments we demonstrated that it works for various robots such as
a Segway, a Drone and a Quadruped.

While our method does not rely on the full dynamical model of the robot to
achieve safety, it relies on kinematic models such as the single integrator or
unicycle models. Our future work includes further exploration of safety-
critical control based on reduced-order models beyond simple kinematic ones.
We also plan to study how to relax the assumption on the performance of the
velocity tracking controller.

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Internal connection between the field theory equations.

Fundamentals of the field theory

L.I. Petrova

Moscow State University, Department of Computational Mathematics and

Cybernetics, Russia, e-mail<EMAIL_ADDRESS>
###### Abstract

It is shown that there is a correspondence between field theory equations such
as the Dirac, Shrődinger, Maxwell, Einstein equations and closed exterior
forms of a certain degree.

In this case, the Dirac and Shrődinger equations for the wave function
correspond to closed exterior forms of zero degree. The Shrődinger equation
for the state functional corresponds to closed exterior forms of the first
degree. The Maxwell’s equations based on exterior forms of second degree.
Einstein’s equation for the gravitational field consists of covariant tensors,
which correspond to closed exterior forms of the second degree. However, the
covariant tensors of the Einstein equation are derived from the covariant
tensors, which correspond to closed exterior forms of third degree.

Such a correspondence between the field theory equations and closed exterior
forms of a certain degree reveals the internal connection between the field
theory equations.

At the same time, it was shown that closed exterior forms, on which the field
theory equations for physical fields are based, are associated with the
equations of mathematical physics for material media, such as thermodynamic,
gas-dynamic, electromagnetic, cosmological equations, etc.

This indicates the connection between the field theory equations and the
mathematical physics equations and reveals the foundations of field theory.

Keywords: the field theory equations; skew-symmetric differential forms; a
unified field theory; the mathematical physics equations; quantum nature of
the field theory equations.

##  1 Introduction

As is known, field theory equations describing physical fields, such as the
equations of Dirac, Shrődinger, Maxwell, Einstein, are based on the invariance
and covariance properties.

It turns out that there is a mathematical formalism that possesses these
properties. This is the theory of closed skew-symmetric differential forms
[1,2,3].

Closed exterior forms are defined on integrable manifolds and satisfy the
condition: $d\omega\,=0$. They are differentials, and therefore invariants,
and correspond to the conservation laws for physical fields: the presence of
conservative quantities or structures. (The example is Noether’s theorem.)
Moreover, the dual forms corresponding to closed inexact (defined only on
integrable structures) exterior forms have covariant properties.

It is shown that there is a correspondence between the field theory equations
and closed exterior forms. This reveals an internal connection between
theories describing different types of physical fields.

In addition, it is shown that closed exterior forms, on which the field theory
equations for physical fields are based, are connected with the mathematical
physics equations for material media, such as thermodynamic, gas-dynamic,
electromagnetic, cosmological equations, etc.

This reveals the basis of field theory equations and the nature of physical
fields.

##  2 Correspondence between field theory equations and closed exterior forms
of a certain degree

The study of field theory equations by closed external skew-symmetric forms
showed that there is a correspondence between the field theory equations and
closed external skew-symmetric forms. It can be seen that the field theory
equations are associated with closed exterior forms of a certain degree.

Closed exterior forms of zero degree correspond to quantum mechanics
equations.

The Dirac bra- and cket- vectors form a closed exterior form of zero degree.

The Shrődinger equation for the wave function:

$\Huge i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi$

consists of operators corresponding to closed exterior forms of zero degree.

The Heisenberg equation for quantum mechanics is associated with a dual form
corresponding to a closed exterior form of zero degree.

The Hamiltonian formalism is based on the properties of a closed exterior form
of the first degree and the corresponding dual form. The Shrődinger equation
for the action functional is such an equation:

$h{{\partial\psi}\over{\partial t}}\,=\,H\psi$

Here the Hamilton function $H$ is a closed skew-symmetric form of the first
degree.

The properties of closed exterior forms of the second degree and their dual
forms underlie the electromagnetic field equations. Covariant electromagnetic
field tensor $F_{\mu\nu}$ in Maxwell’s equations obeys the identical relations
$d\theta^{2}=0$, $d^{*}\theta^{2}=0$, where
$\theta^{2}=\frac{1}{2}F_{\mu\nu}dx^{\mu}dx^{\nu}$ is a closed exterior form
of the second degree, and ${}^{*}\theta^{2}$ is the corresponding dual form.

Einstein’s equation

$G_{\mu\nu}={{8{\pi}G}\over{c^{4}}}T_{\mu\nu}$

(where $G_{\mu\nu}=R_{\mu\nu}-1/2g_{\mu\nu}R$ is the Einstein tensor) is a
relation in the covariant tensors, which are related to closed exterior and
dual forms of the second degree. In this case, the Einstein equation is
obtained from the relation with covariant tensors, which are analogues of
closed exterior forms of the third degree.

### Approach to the unified field theory.

It can be seen that the field theory equations are associated with closed
exterior forms of a certain degree. This shows that there is an internal
connection between field theory equations that describe different types of
physical fields.

Obviously, that the degree of closed exterior forms is a parameter that unites
field theories into the unified field theory.

In addition, it can be shown that the degree of closed exterior forms is a
parameter that unifies interactions. If we denote the degree of closed
exterior forms by $k$, then $k=0$ corresponds to the strong interaction, $k=1$
corresponds to weak interaction, $k=2$ corresponds to electromagnetic, and
$k=3$ corresponds to the gravitational interaction [3, Appendix 5].

One more general property of field theory equations can be noted here. They
are relations for functionals such as the wave function, the action
functional, the Einstein tensor. For the electromagnetic field such a
functional is the Poynting vector [4].

Thus, one can see that the field theory equations are relations for
functionals and are based on the properties of closed exterior forms.

##  3 What is the physical meaning of closed exterior forms and how are they
being obtained?

It turns out that closed exterior forms, on which the field theory equations
that describe physical fields are based, are connected with the equations of
mathematical physics that describe material media, such as thermodynamic, gas-
dynamic, electromagnetic, cosmological, etc.

And this is due to the conservation laws.

The conservation laws have a feature.

As is known, conservation laws for physical fields are the presence of
conservative quantities or structures. Closed exterior forms describe such
conservative quantities or structures (as already noted, Noether’s theorem is
the example).

And the conservation laws for material media are the conservation laws of
energy, momentum, angular momentum and mass. These are differential
conservation laws. They are described by differential equations. Equations of
mathematical physics describing material media, as is known, consist of such
equations.

From the equations of conservation laws for material media, skew-symmetric
forms also are obtained. But these are skew-symmetric forms that are
evolutionary, since they are defined, in contrast to exterior forms, on a
nonintegrable manifold.

It turns out that such evolutionary skew-symmetric forms, which are obtained
from the mathematical physics equations, generate closed exterior forms that
underlie the field theory equations.

Such results were obtained in the study of the consistency of the conservation
laws equations for material media. (It should be noted that the quantum
properties of the mathematical physics equations and the field theory
equations expressly depend on the consistency of the conservation laws
equations.)

###  3.1 Investigation of the conservation laws equations consistency for
material media.

Some features of the study should be noted.

Conservation law equations were expressed in terms of state functionals [4],
such as wave function, entropy, action functional, etc., which are also
functionals of the field theory equations.

In addition, the conservation laws equations were written in frame of
reference, associated with the studied material medium. This is an
accompanying frame of reference is connected with the manifold constructed of
the trajectories of the material medium elements. (Lagrange coordinate system
is example of such frame of reference.)

Let us investigate the energy and momentum conservation laws equations
consistency.

In accompanying frame of reference, the energy conservation law equation is
written as:

${{\partial\psi}\over{\partial\xi^{1}}}\,=\,A_{1}$ $None$

Here $\xi^{1}$ \- coordinate along the trajectory, $A_{1}$ is a quantity that
depends on the material medium properties and on external energy actions onto
the medium.

Similarly, in accompanying frame of reference, the momentum conservation law
equation reduces to the equation:

${{\partial\psi}\over{\partial\xi^{\nu}}}\,=\,A_{\nu},\quad\nu\,=\,2,\,...$
$None$

where $\xi^{\nu}$ are coordinates in the normal direction to the trajectory,
$A_{\nu}$ \- quantities that depend on the medium properties and external
force actions.

Equations (1) and (2) can be reduced to the relation:

$d\psi\,=\,\omega$ $None$

where $\omega\,=\,A_{\mu}\,d\xi^{\mu}$ is a skew-symmetric differential form
of the first degree. (The summation is carried out over the repeating index.)

In the general case (for the conservation equations of energy, momentum,
angular momentum and mass), this relation will have the form:

$d\psi\,=\,\omega^{p}$ $None$

where $\omega^{p}$ is a skew-symmetric differential form of the degree
$p=0,1,2,3$.

Since the conservation laws equations are evolutionary, the skew-symmetric
form and relations (3) and (4) are also evolutionary.

###  3.2 Features of the evolutionary skew-symmetric form and evolutionary
relation.

The evolutionary form, in contrast to the exterior form, is defined on an
deformable, nonintegrable manifold (constructed of the trajectories of the
material medium elements). The differential of a form whose basis is a
nonintegrable manifold is not equal to zero.

This can be shown on the example of the evolutionary form of the first degree
$\omega\,=\,A_{\alpha}\,d\xi^{\alpha}$.

Differential $d\omega$ of the evolutionary form
$\omega\,=\,A_{\alpha}\,d\xi^{\alpha}$ can be written as
$d\omega=K_{\alpha\beta}d\xi^{\alpha}d\xi^{\beta}$, where
$K_{\alpha\beta}=A_{\beta;\alpha}-A_{\alpha;\beta}$ is a commutator of the
evolutionary form $\omega$, and $A_{\beta;\alpha}$ , $A_{\alpha;\beta}$ are
covariant derivatives. If we express the covariant derivatives in terms of
connectedness, then the form $\omega$ commutator can be written as:

$K_{\alpha\beta}=\left(\frac{\partial
A_{\beta}}{\partial\xi^{\alpha}}-\frac{\partial
A_{\alpha}}{\partial\xi^{\beta}}\right)+(\Gamma^{\sigma}_{\beta\alpha}-\Gamma^{\sigma}_{\alpha\beta})A_{\sigma}$
$None$

The evolutionary form commutator is not equal to zero, since it includes
connectedness of the metric form of a nonintegrable manifold that are not
symmetric, i.e. the expression
$(\Gamma^{\sigma}_{\beta\alpha}-\Gamma^{\sigma}_{\alpha\beta})$ not equal to
zero [5].

Since the commutator is not equal to zero, then, consequently, the
differential of the evolutionary form is also not equal to zero. And this
means that the evolutionary form, unlike the exterior form, cannot be a closed
form, i.e. a differential.

It can be shown that the evolutionary form $\omega^{p}$ is not a differential.

(The example of such an evolutionary form is the Lagrangian.)

Since on the right side of the evolutionary relation $d\psi\,=\,\omega^{p}$
there is an evolutionary form that is not a differential, this means that the
evolutionary relation turns out to be a nonidentical relation (there is a
differential on the left, and an evolutionary form that is not a differential
on the right).

(An example of a nonidentical evolutionary relation is the first law of
thermodynamics.)

[It can be noted that the nonidentity of the evolutionary relation, which was
obtained in the study of the conservation laws equations consistency,
indicates that on the original coordinate space the conservation laws
equations for material media turn out to be inconsistent, i.e., the
conservation laws are noncommutative. The hidden properties of the
mathematical physics equations [6] are connected precisely with the
noncommutativity of the conservation laws.]

##  4 Correspondence between field theory equations and evolutionary relation

As noted above, the field theory equations are relations for the $\psi$
functionals.

The resulting evolutionary relation $d\psi\,=\,\omega^{p}$ is a relation for
the same functionals.

However, these are different relations.

The field theory equations are identical relations, since the right side of
the relations contains closed exterior forms, i.e. differentials, as in the
left side.

And the evolutionary relation is a nonidentical relation, since the
evolutionary form on the right side of the evolutionary relation is not a
closed exterior form, i.e., a differential.

But it turns out that identical relations, which are analogous to the field
theory equations, are obtained from the evolutionary relation.

###  4.1 Realization of the identity relation.

An identity relation can be realized from a nonidentical evolutionary relation
if a closed exterior form, which is a differential, is obtained from an
evolutionary form that is not closed.

But this can only happen under a degenerate transformation, i.e., under a
transformation that does not preserve the differential, since the differential
of the evolutionary form is not equal to zero, while the differential of a
closed exterior form is equal to zero.

### The unique meaning of the degenerate transformation.

In field theory, as is known, nondegenerate transformations are used, i.e.,
transformations that preserve the differential. They make transitions between
integrable structures.

A degenerate transformation is a transformation that does not preserve the
differential. It has a unique property: under a degenerate transformation, an
integrable structure arises and a transition occurs from a nonintegrable space
to structures of an integrable manifold.

An example of a degenerate transformation is the Legendre transformation. It
is the Legendre transformation that makes the transition from a Lagrangian,
nonintegrable manifold to an Hamiltonian structures of integrable manifold.

Degenerate transformations occur discretely, arise spontaneously when any
degrees of freedom are realized. This corresponds to turning to zero of such
functional expressions as Jacobians, determinants, Poisson brackets, residues,
etc. Such conditions describe a realized integrable structures.

### Degenerate transformation. Realization of a closed external inexact skew-
symmetric form.

A degenerate transformation is a transformation of skew-symmetric differential
forms. It is described by the transition from the evolutionary form
differential, which is not equal to zero, to the differentials of the closed
inexact exterior form and the corresponding dual form, which are equal to
zero.

This can be written as:

$d\omega^{p}\neq 0\to$ (degenerate transformation) $\to
d_{\pi}{}^{*}\omega^{p}=0$, $d_{\pi}\omega^{p}=0$

The $d_{\pi}{}^{*}\omega^{p}=0$ condition indicates the implementation of a
closed dual form ${}^{*}\omega^{p}$, which describes some realized integrable
structure $\pi$ (pseudostructure by its metric properties).

And the $d_{\pi}\omega^{p}=0$ condition indicates that the closed inexact
exterior form $\omega^{p}_{\pi}$ is realized on the integrable structure
$\pi$.

### Identical relation.

Realization of a closed inexact exterior form $\omega^{p}_{\pi}$ and dual form
${}^{*}\omega^{p}$ leads to the fact that on the pseudostructure $\pi$ from
the nonidentical evolutionary relation $d\psi\,=\,\omega^{p}$ the relation

$d_{\pi}{\psi}=\omega^{p}_{\pi}$ $None$

which is the identical relation, is realized, since the right side has a
closed form, which is a differential, like the left side.

##  5 Identical relations of the mathematical physics equations as an analogue
of the field theory equations

As has been shown, the field theory equations describing physical fields are
relations for field theory functionals and are based on the closed exterior
forms properties.

Identical relations $d_{\pi}{\psi}=\omega^{p}_{\pi}$ derived from the
mathematical physics equations, describing material media, are relations for
the same functionals and contain closed exterior forms.

It turns out that there is a connection between the closed exterior forms of
the field theory equations and the closed exterior forms of the identical
relations obtained from the mathematical physics equations. The right parts of
the field theory equations are obtained from the mathematical physics
equations under a degenerate transformation.

Thus, the Hamilton function in the Hamiltonian formalism is obtained from the
Euler equation for the Lagrange function under the Legendre transformation,
which is a degenerate transformation [7].

The covariant tensors in the Maxwell’s equations are connected to the Poynting
vector, which describes the energy flow of electromagnetic media.

The energy-momentum tensor in the Einstein equation follows from the analysis
of the equations of the conservation laws of energy, momentum and the equation
of state [8], i.e. also from the mathematical physics equations. [Here it
should be noted that when deriving the energy-momentum tensor, Einstein
imposed a number of assumptions on the conservation laws equations in order to
prove the covariance of the energy-momentum tensor [8]. However the covariance
of the energy-momentum tensor is satisfied under a degenerate transformation,
in which the consistency of the energy and momentum equations is realized.]

The connection of the field theory equations with the identical relations of
the mathematical physics equations reveals the foundations of the field theory
and can serve as an approach to the general field theory.

### Quantum nature of the field theory equations.

The correspondence between the field theory equations and the identical
relations of the mathematical physics equations reveals the quantum character
of the field theory equations.

As has been shown, the identical relations of mathematical physics are
realized discretely under a degenerate transformation.

The connection of the field theory equations with the identical relations of
mathematical physics equations indicates that the field theory equations also
realize discretely. In this case, a structure appears (which is described by a
dual form) with a conservative quantity (a closed exterior form), i.e., an
invariant object, a quantum, arises.

This reveals the quantum character of the field theory equations.

In the quantum mechanics, this is taken into account by the discrete
Hamiltonian spectrum [7].

In the quantum physics, this is connected to the discrete properties of the
Hamilton function, which is defined on sections of the bundle of the cotangent
space [7].

The correspondence between the field theory equations and the identical
relations of the mathematical physics equations also substantiates the
Einstein equation discreteness. When deriving the Einstein equation [8], it
was assumed that the transitions from the curvature tensor to Einstein’s
tensor (at which the Bianchi identity is fulfilled, the connectedness
coefficients are symmetric ones, i.e., the connectedness coefficients are the
Christoffel symbols, and there exists a transformation under which the
connectedness coefficient becomes zero [9]) are nondegenerate transformations
(transitions were made from the tensors upper indices to the lower indices,
which is true only for tensors defined on integrable manifolds). However, on
spaces with curvature, which are non-integrable manifolds, such
transformations are nondegenerate, and are realized only discretely, when any
degrees of freedom are realized.

It turns out that the Einstein equation can only be satisfied discretely [9].
In this case, it should be taken into account that quantum effects arise in
discrete transitions. (In the material media, they appear as observable
formations, such as waves, vortices, pulsations, etc.)

Here you can pay attention to the following. The covariance conditions, which
the field theory equations must satisfy, are satisfied only for symmetric
connections. Symmetric connections are obtained by a degenerate transformation
from nonsymmetric connections.

Einstein, as shown in the book [10], in the last years of his life tried to
obtain symmetric connections from asymmetric ones in order to derive the
general field theory equation. He, as noted by Pauli [11], tried to find a
nondegenerate transformation that would make it possible to transform
nonsymmetric connections into symmetric ones. But such attempts did not lead
to success, because the transformation that transforms nonsymmetric
connections into symmetric ones is a degenerate transformation.

##  6 Conclution

It has been shown that there is a mathematical formalism that reveals the
properties of the field theory equations. These are skew-symmetric
differential forms.

It is shown that field theory equations are based on the properties of closed
exterior forms. At the same time, field theory equations describing physical
fields of various types are associated with closed exterior forms of a
certain, corresponding degree.

This reveals the internal connection of field theory equations, unifies field
theories, and can serve as an approach to a unified field theory.

It turns out that from the mathematical physics equations describing material
media, evolutionary skew-symmetric forms are obtained, which generate closed
exterior forms that underlie the field theory equations.

This points to the connection of the field theory equations with the
mathematical physics equations and reveals the foundations and nature of the
field theory equations.

## References

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Geometriques. -Paris, Hermann, 1945.

[2] Petrova L.I., Exterior and evolutionary differential forms in mathematical
physics: Theory and Applications, -Lulu.com,157, 2008.

[3] Petrova L.I., Role of skew-symmetric differential forms in mathematics,
2010. https://arxiv.org/pdf/1007.4757.pdf

[4] L.I. Petrova, Physical meaning and a duality of concepts of wave function,
action functional, entropy, the Pointing vector, the Einstein tensor.
https://arxiv.org/pdf/1001.1710.pdf

[5] Tonnelat M.-A., Les principles de la theorie electromagnetique et la
relativite, Masson, Paris, 1959.

[6] Petrova Ludmila, Discrete Quantum Transitions, Duality: Emergence of
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[7] Petrova L., Qualitative Investigation of Hamiltonian Systems by
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Switzerland), Vol 13, No.1, 1-7, 2021.

[8] A. Einstein, The Meaning of Relativity. Princeton, 1953.

[9] Petrova L., Evolutionary Relation of Mathematical Physics Equations.
Evolutionary Relation as Foundation of Field Theory. Interpretation of the
Einstein Equation. //Axioms, MDPI (Basel, Switzerland), Vol 10(2), No. 46,
1-10, 2021. https://doi.org/10.3390/axioms10020046

[10] Abraham PAIS, THE SCIENCE AND THE LIFE OF Albert EINSTEIN, Oxford
univerity New York Toronto Melbourno,1982.

[11] W. Pauli, Theory of Relativity, Pergamon Press, 1958.