# Adaptation for Validation of a Consolidated Control Barrier Function based
Control Synthesis

Mitchell Black1 Dimitra Panagou2 The authors would like to acknowledge the
support of the National Science Foundation award number 1931982.1Dept. of
Aerospace Engineering, Univ. of Michigan, 1320 Beal Ave, Ann Arbor, MI 48109,
USA<EMAIL_ADDRESS>of Robotics and Dept. of Aerospace Engineering,
Univ. of Michigan, Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>
###### Abstract

We develop a novel adaptation-based technique for safe control design in the
presence of multiple control barrier function (CBF) constraints. Specifically,
we introduce an approach for synthesizing any number of candidate CBFs into
one consolidated CBF candidate, and propose a parameter adaptation law for the
weights of its constituents such that the controllable dynamics of the
consolidated CBF are non-vanishing. We then prove that the use of our
adaptation law serves to certify the consolidated CBF candidate as valid for a
class of nonlinear, control-affine, multi-agent systems, which permits its use
in a quadratic program based control law. We highlight the success of our
approach in simulation on a multi-robot goal-reaching problem in a crowded
warehouse environment, and further demonstrate its efficacy experimentally in
the laboratory via AION ground rovers operating amongst other vehicles
behaving both aggressively and conservatively.

##  I Introduction

Since the arrival of control barrier functions (CBFs) to the field of safety-
critical systems [1], much attention has been devoted to the development of
their viability for safe control design [2, 3, 4]. As a set-theoretic approach
founded on the notion of forward invariance, CBFs encode safety in that they
ensure that any state beginning in a safe set remains so for all future time.
In the context of control design, CBF conditions are often used as constraints
in quadratic program (QP) based control laws, either as safety filters [5] or
in conjunction with stability constraints (e.g. control Lyapunov functions)
[6]. Their utility has been successfully demonstrated for a variety of safety-
critical applications, including mobile robots [7, 8], unmanned aerial
vehicles (UAVs) [9, 10], and autonomous driving [11, 12]. But while it is now
well-established that CBFs for controlled dynamical systems serve as
certificates of safety, the verification of candidate CBFs as valid is in
general a challenging problem.

Though for a single CBF there exist guarantees of validity under certain
conditions for systems with either unbounded [2] or bounded control authority
[13, 14], these results do not generally extend to control systems seeking to
satisfy multiple candidate CBF constraints. Recent approaches to control
design in the presence of multiple CBF constraints have mainly circumvented
this challenge by considering only one such constraint at a given time
instance, either by assumption [15] or construction in a non-smooth manner
[16, 17]. In contrast, the authors of [18] and [19] each propose smoothly
synthesizing one candidate CBF for the joint satisfaction of multiple
constraints, but make no attempt to validate their candidate function. The
problem of safe control design under a multitude of constraints is especially
relevant in practical applications involving autonomous mobile robots, where
the main challenge is in the robot completing its nominal objective while
satisfying constraints related to collision avoidance with respect to
obstacles both static and dynamic.

Figure 1: Parameter adaptation for our C-CBF leads to a gain-dependent (and
time-varying) controlled-invariant set $C(k)\subset S=\bigcap_{i=1}^{c}S_{i}$.
$C(k)$ is shown here with a dotted white boundary for gains $k_{0}$ at time
$t_{0}$ and $k_{1}$ at $t_{1}$.

It is with this problem in mind that we propose a consolidated CBF (C-CBF)
based approach to control design for multi-agent systems in the presence of
both non-communicative and non-responsive (though non-adversarial) agents.
Constructed by smoothly synthesizing any arbitrary number of candidate CBFs
into one, our C-CBF defines a new super-level set that can under-approximate
the intersection of its constituent sets arbitrarily closely (see Figure 1).
We further propose a parameter adaptation law for the weighting of the
constituent functions, and prove that its use renders our C-CBF valid and the
super-level set controlled invariant for the class of nonlinear, control-
affine, multi-agent systems under consideration. And while various works have
utilized parameter adaptation in the context of control for safety-critical
systems, usually in an attempt to either learn [20, 21] or compensate for [22]
unknown parameters in the system dynamics, our proposed adaptation law is the
first to our knowledge to be used for the simultaneous verified satisfaction
of multiple CBF constraints. To show the effectiveness of our proposed control
formulation, we study a decentralized multi-robot goal-reaching problem in a
crowded warehouse environment amongst non-responsive agents. As a practical
demonstration, we tested our controller experimentally on a collection of
ground rovers in the laboratory setting and found that it succeeded in safely
driving the rovers to their goal locations amongst non-responsive agents
behaving both aggressively and conservatively.

The paper is organized as follows. Section II introduces some preliminaries,
including set invariance, optimization based control, and our first problem
statement. In Section III, we introduce the form of our C-CBF and propose a
parameter adaptation law for rendering it valid. Sections IV and V contain the
results of our simulated and experimental case studies respectively, and in
Section VI we conclude with final remarks and directions for future work.

##  II Mathematical Preliminaries

We use the following notation throughout the paper. $\mathbb{R}$ denotes the
set of real numbers. The set of integers between $i$ and $j$ (inclusive) is
$[i..j]$. $\|\cdot\|$ represents the Euclidean norm. A function
$\alpha:\operatorname{\mathbb{R}}\rightarrow\operatorname{\mathbb{R}}$ is said
to belong to class $\mathcal{K}_{\infty}$ if $\alpha(0)=0$ and $\alpha$ is
increasing on the interval $(-\infty,\infty)$, A function
$\phi:\operatorname{\mathbb{R}}\times\operatorname{\mathbb{R}}\rightarrow\operatorname{\mathbb{R}}$
is said to belong to class $\mathcal{L}\mathcal{L}$ if for each fixed $r$
(resp. $s$), the function $\phi(r,s)$ is decreasing with respect to $s$ (resp.
$r$) and is such that $\phi(r,s)\rightarrow 0$ for $s\rightarrow\infty$ (resp.
$r\rightarrow\infty$). The Lie derivative of a function
$V:\mathbb{R}^{n}\rightarrow\mathbb{R}$ along a vector field
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ at a point $x\in\mathbb{R}^{n}$ is
denoted $L_{f}V(x)\triangleq\frac{\partial V}{\partial x}f(x)$.

In this paper we consider a multi-agent system, each of whose $A$ constituent
agents may be modelled by the following class of nonlinear, control-affine
dynamical systems:

$\dot{\bm{x}}_{i}=f_{i}(\bm{x}_{i}(t))+g_{i}(\bm{x}_{i}(t))\bm{u}_{i}(t),\quad\bm{x}_{i}(0)=\bm{x}_{i0}$
(1)

where $\bm{x}_{i}\in\operatorname{\mathbb{R}}^{n}$ and
$\bm{u}_{i}\in\mathcal{U}_{i}\subseteq\operatorname{\mathbb{R}}^{m}$ are the
state and control input vectors for the ith agent, with $\mathcal{U}_{i}$ the
input constraint set, and where
$f_{i}:\operatorname{\mathbb{R}}^{n}\rightarrow\operatorname{\mathbb{R}}^{n}$
and $g_{i}:\operatorname{\mathbb{R}}^{n\times
m}\rightarrow\operatorname{\mathbb{R}}^{n}$ are known, locally Lipschitz, and
not necessarily homogeneous $\forall i\in\mathcal{A}=[1..A]$. We denote the
concatenated state vector as
$\bm{x}=[\bm{x}_{1},\ldots,\bm{x}_{A}]^{T}\in\operatorname{\mathbb{R}}^{N}$,
the concatenated control input vector as
$\bm{u}=[\bm{u}_{1},\ldots,\bm{u}_{A}]^{T}\in\mathcal{U}\subseteq\operatorname{\mathbb{R}}^{M}$,
and as such express the full system dynamics as

$\dot{\bm{x}}=F(\bm{x}(t))+G(\bm{x}(t))\bm{u}(t),\quad\bm{x}(0)=\bm{x}_{0},$
(2)

where
$F=[f_{1},\ldots,f_{A}]^{T}:\operatorname{\mathbb{R}}^{N}\rightarrow\operatorname{\mathbb{R}}^{N}$
and $G=\textrm{diag}([g_{1},\ldots,g_{A}]):\operatorname{\mathbb{R}}^{M\times
N}\rightarrow\operatorname{\mathbb{R}}^{N}$. We assume that a (possibly empty)
subset of the agents are communicative, denoted
$j\in\mathcal{A}_{c}=[1..A_{c}]$, in the sense that they share information
(e.g. states, control objectives, etc.) with one another, and that the
remaining agents are non-communicative, denoted
$k\in\mathcal{A}_{n}=[(A_{c}+1)..A]$, in that they do not share information,
where $A_{c}\geq 0$ and $A_{n}=A-A_{c}\geq 0$ are the number of communicative
and non-communicative agents respectively. We further assume that all agents
are non-adversarial in that they do not seek to damage or otherwise deceive
others, though there may be non-communicative agents which are non-responsive
($l\in\mathcal{A}_{n,n}\subseteq\mathcal{A}_{n}$) in that they do not actively
avoid unsafe situations.

###  II-A Safety and Forward Invariance

Consider a set of safe states $S$ defined implicitly by a continuously
differentiable function
$h:\operatorname{\mathbb{R}}^{N}\rightarrow\operatorname{\mathbb{R}}$, as
follows:

$S=\\{\bm{x}\in\operatorname{\mathbb{R}}^{N}\;|\;h(\bm{x})\geq 0\\},$ (3)

where the boundary and interior of $S$ are denoted as $\partial
S=\\{\bm{x}\in\operatorname{\mathbb{R}}^{N}\;|\;h(\bm{x})=0\\}$ and
$\textrm{int}(S)=\\{\bm{x}\in\operatorname{\mathbb{R}}^{N}\;|\;h(\bm{x})>0\\}$
respectively. In many works (e.g. [23, 24]), the set $S$ defined by (3) is
referred to as safe if it is forward-invariant, i.e. if $\bm{x}(0)\in
S\implies\bm{x}(t)\in S$, $\forall t\geq 0$. Nagumo’s Theorem provides a
necessary and sufficient condition for rendering the set $S$ forward-invariant
for the system (2).

######  Lemma 1 (Nagumo’s Theorem[25]).

Suppose that there exists $\bm{u}(t)\in\mathcal{U}$ such that (2) admits a
globally unique solution for each $\bm{x}_{0}\in S$. Then, the set $S$ is
forward-invariant for the controlled system (2) if and only if

$L_{F}h(\bm{x})+L_{G}h(\bm{x})\bm{u}\geq 0,\;\forall\bm{x}\in\partial S.$ (4)

One way to render a set $S$ forward-invariant is to use CBFs in the control
design.

######  Definition 1.

[2, Definition 5] Given a set $S\subset\operatorname{\mathbb{R}}^{N}$ defined
by (3) for a continuously differentiable function
$h:\operatorname{\mathbb{R}}^{N}\rightarrow\operatorname{\mathbb{R}}$, the
function $h$ is a control barrier function (CBF) defined on a set $D\supseteq
S$ if there exists a Lipschitz continuous class $\mathcal{K}_{\infty}$
function
$\alpha:\operatorname{\mathbb{R}}\rightarrow\operatorname{\mathbb{R}}$ such
that, for all $\bm{x}\in D$,

$\sup_{\bm{u}\in\mathcal{U}}\left[L_{F}h(\bm{x})+L_{G}h(\bm{x})\bm{u}\right]\geq-\alpha(h(\bm{x})).$
(5)

In this paper, we assume that $\frac{\partial h}{\partial\bm{x}}$ is Lipschitz
continuous so that $L_{F}h(\bm{x})$ and $L_{G}h(\bm{x})$ are likewise. In
other works (e.g. [26]), the function $h$ responsible for defining $S$ is a
CBF if there exists a class $\mathcal{K}_{\infty}$ function $\alpha$
satisfying

$L_{G}h(\bm{x})=\mathbf{0}_{1\times M}\implies
L_{F}h(\bm{x})+\alpha(h(\bm{x}))>0.$ (6)

We note, however, that with unbounded control authority (i.e.
$\mathcal{U}=\operatorname{\mathbb{R}}^{M}$) a sufficient condition for the
existence of some $\alpha\in\mathcal{K}_{\infty}$ satisfying (5), and thus for
$h$ to be a CBF, is $L_{G}h(\bm{x})\neq\mathbf{0}_{1\times M}$,
$\forall\bm{x}\in S$, though this does not generally hold for a system with
multiple CBF constraints.

###  II-B Control Design using CBFs

Decentralized controllers, in which agents compute inputs based on local
information, have found empirical success as a control strategy for multi-
agent systems of the form (2) [27, 28]. The following is an example of one
such controller for an agent $i\in\mathcal{A}$ with safety constraints encoded
via $c>1$ candidate CBFs:

$\displaystyle\bm{u}_{i}^{*}=\operatorname*{arg\,min}_{\bm{u}_{i}\in\mathcal{U}_{i}}$
$\displaystyle\frac{1}{2}\|\bm{u}_{i}-\bm{u}_{i}^{0}\|^{2}$ (7a) s.t.
$\displaystyle\forall s\in[1..c]$ $\displaystyle
a_{s,i}+\bm{b}_{s,i}\bm{u}_{i}$ $\displaystyle\geq 0,$ (7b)

where (7a) seeks to produce a control solution $\bm{u}_{i}^{*}$ that deviates
minimally from some nominal input $\bm{u}_{i}^{0}$, and (7b) encodes $c$
safety constraints of the form (5) via candidate CBFs $h_{s}$, where
$a_{s,i}=L_{f_{i}}h_{s}+\alpha_{s}(h_{s})$ and $\bm{b}_{s,i}=L_{g_{i}}h_{s}$.
Notably, for many classes of systems (7) is neither guaranteed to be feasible
nor to preserve safety between agents [8]. When some subset of agents are able
to communicate with one another, i.e. agents $j\in\mathcal{A}_{c}$ share
information, their control inputs
$\bm{u}_{\mathcal{A}_{c}}=[\bm{u}_{1},\ldots,\bm{u}_{A_{c}}]^{T}$ may be
computed in a centralized fashion as follows:

$\displaystyle\bm{u}_{\mathcal{A}_{c}}^{*}=\operatorname*{arg\,min}_{\bm{u}_{\mathcal{A}_{c}}\in\mathcal{U}_{\mathcal{A}_{c}}}$
$\displaystyle\frac{1}{2}\|\bm{u}_{\mathcal{A}_{c}}-\bm{u}_{\mathcal{A}_{c}}^{0}\|^{2}$
(8a) $\displaystyle\textrm{s.t.}\quad\forall j,k$
$\displaystyle\in\mathcal{A}_{c},\;k\neq j$ $\displaystyle
a_{s,j}+\bm{b}_{s,j}\bm{u}_{j}$ $\displaystyle\geq 0,\;\forall
s\in[1..c_{I}],$ (8b) $\displaystyle
a_{s,jk}+\bm{b}_{s,j}\bm{u}_{j}+\bm{b}_{s,k}\bm{u}_{k}$ $\displaystyle\geq
0,\;\forall s\in[c_{I}+1..c]$ (8c)

where
$\bm{u}_{\mathcal{A}_{c}}^{0}=[\bm{u}_{1}^{0},\ldots,\bm{u}_{A_{c}}^{0}]^{T}$
is the nominal input vector shared amongst communicative agents,
$\mathcal{U}_{\mathcal{A}_{c}}=\bigoplus_{j=1}^{A_{c}}\mathcal{U}_{j}$ is the
Minkowski sum of their input constraint sets, (8b) denotes the $c_{I}\geq 0$
individual CBF constraints for agent $j$ (e.g. speed), and (8c) represents
combinations of safety constraints between agents (e.g. collision avoidance),
where $a_{s,jk}=L_{f_{j}}h_{s}+L_{f_{k}}h_{s}+\alpha_{s}(h_{s})$,
$\bm{b}_{s,j}=L_{g_{j}}h_{s}$, and $\bm{b}_{s,k}=L_{g_{k}}h_{s}$. When all
agents are communicative, (8) is guaranteed to be safe provided that it is
feasible.

A challenge when it comes to both (7) and (8) is in satisfying all of the
safety constraints simultaneously, especially when it comes to the design of
$\alpha_{s}$. In some recent works, authors have proposed setting
$\alpha_{s}(h_{s})=p_{s}h_{s}$ and including the parameters $p_{s}$ as
decision variables in the QP [29] (and thus an additional term
$\sum_{s=1}^{c}\frac{1}{2}q_{s}p_{s}^{2}$ for $q_{s}>0$ in the objective
function), but the performance of these approaches are still heavily dependent
on the gains $q_{s}$. Other techniques have avoided the issue of multiple
candidate CBFs by assuming that only one constraint is in need of satisfaction
at once [15] or by synthesizing a single non-smooth candidate CBF [16, 17],
both of which may lead to undesirable chattering behavior or the loss of
existence and uniqueness of solutions. We seek to address this open problem,
and require the following assumption to do so.

######  Assumption 1.

The intersection of the safe sets $S_{s}$ for all $s\in[1..c]$ is non-empty,
i.e. $S=\bigcap_{s=1}^{c}S_{s}\neq\emptyset$.

######  Problem 1.

Given that Assumption 1 holds for a collection of $c>1$ candidate control
barrier functions $h_{s}$ corresponding to safe sets $S_{s}$, design a
consolidated control barrier function candidate
$H:\operatorname{\mathbb{R}}^{N}\times\operatorname{\mathbb{R}}_{+}^{c}\rightarrow\operatorname{\mathbb{R}}$
with constituent gains
$\bm{k}=[k_{1},\ldots,k_{c}]^{T}\in\operatorname{\mathbb{R}}_{+}^{c}$ for the
zero super-level set
$C(\bm{k})=\\{\bm{x}\in\operatorname{\mathbb{R}}^{N}\;|\;H(\bm{x},\bm{k})\geq
0\\}$ such that $C(\bm{k})\subseteq S$ for all $\bm{k}$ satisfying
$0<k_{s}<\infty$, $\forall s\in[1..c]$.

##  III Consolidated CBF based Control

In this section, we first introduce our proposed solution to Problem 1, a
consolidated control barrier function (C-CBF) candidate that smoothly
synthesizes multiple candidate CBFs into one, and then design a parameter
adaptation law which renders the candidate C-CBF valid for safe control
design.

###  III-A Consolidated CBFs

Let the vector of $c>1$ candidate CBFs evaluated at a given state $\bm{x}$ be
denoted
$\bm{h}(\bm{x})=[h_{1}(\bm{x})\;\ldots\;h_{c}(\bm{x})]^{T}\in\operatorname{\mathbb{R}}^{c}$,
and define a gain vector as
$\bm{k}=[k_{1}\;\ldots\;k_{c}]^{T}\in\operatorname{\mathbb{R}}^{c}$, where
$0<k_{s}<\infty$ for all $s\in[1..c]$. Our C-CBF candidate
$H:\operatorname{\mathbb{R}}^{N}\times\operatorname{\mathbb{R}}_{+}^{c}\rightarrow\operatorname{\mathbb{R}}$
is the following:

$H(\bm{x},\bm{k})=1-\sum_{s=1}^{c}\phi\Big{(}h_{s}(\bm{x}),k_{s}\Big{)},$ (9)

where $\phi:\operatorname{\mathbb{R}}_{\geq
0}\times\operatorname{\mathbb{R}}_{\geq
0}\rightarrow\operatorname{\mathbb{R}}_{+}$ belongs to class
$\mathcal{L}\mathcal{L}$, is continuously differentiable, and satisfies
$\phi(h_{s},0)=\phi(0,k_{s})=\phi(0,0)=1$. For example, the decaying
exponential function, i.e. $\phi(h_{s},k_{s})=e^{-h_{s}k_{s}}$, satisfies
these requirements over the domain $\operatorname{\mathbb{R}}_{\geq
0}\times\operatorname{\mathbb{R}}_{\geq 0}$. With $\phi$ possessing these
properties, it follows then that the new zero super level-set
$C(\bm{k})=\\{\bm{x}\in\operatorname{\mathbb{R}}^{N}\;|\;H(\bm{x},\bm{k})\geq
0\\}$ is a subset of $S$ (i.e. $C(\bm{k}\subset S$), where the level of
closeness of $C(\bm{k})$ to $S$ depends on the choices of gains $\bm{k}$. This
may be confirmed by observing that if any $h_{s}(\bm{x})=0$ then
$H(\bm{x})\leq 1-1-\sum_{j=1,j\neq s}^{c}\phi(h_{j}(\bm{x}),k_{j})<0$, and
thus for $H(\bm{x})\geq 0$ it must hold that $h_{s}(\bm{x})>0$, for all
$s\in[1..c]$.

As such, $H$ defined by (9) is a solution to Problem 1, i.e. $H$ is a C-CBF
candidate. This implies via Lemma 1 that if $H$ is valid over the set
$C(\bm{k})$, then $C(\bm{k})$ is controlled invariant and thus the
trajectories of (2) remain safe with respect to each constituent safe set
$S_{s}$, $\forall s\in[1..c]$. By Definition 1, for a static gain vector (i.e.
$\dot{\bm{k}}=\mathbf{0}_{c\times 1}$) the function $H$ is a CBF on the set
$S$ if there exists $\alpha_{H}\in\mathcal{K}_{\infty}$ such that the
following condition holds for all $\bm{x}\in S\supset C(\bm{k})$:

$L_{F}H(\bm{x},\bm{k})+L_{G}H(\bm{x},\bm{k})\bm{u}\geq-\alpha_{H}(H(\bm{x},\bm{k})),$
(10)

where from (9) it follows that

$\displaystyle L_{F}H(\bm{x})$
$\displaystyle=-\sum_{s=1}^{c}\frac{\partial\phi}{\partial
h_{s}}L_{F}h_{c}(\bm{x}),$ (11) $\displaystyle L_{G}H(\bm{x})$
$\displaystyle=-\sum_{s=1}^{c}\frac{\partial\phi}{\partial
h_{s}}L_{G}h_{c}(\bm{x}).$ (12)

Again taking $\phi(h_{s},k_{s})=e^{-h_{s}k_{s}}$ as an example, we obtain that
$\frac{\partial\phi}{\partial h_{s}}=-k_{s}e^{-h_{s}k_{s}}$, in which case it
is evident that the role of the gain vector $\bm{k}$ is to weight the
constituent candidate CBFs $h_{s}$ and their derivative terms $L_{F}h_{s}$ and
$L_{G}h_{s}$ in the CBF condition (10). Thus, a higher value $k_{s}$ indicates
a weaker weight in the CBF dynamics, as the exponential decay overpowers the
linear growth. Due to the combinatorial nature of these gains, for an
arbitrary $\bm{k}$ there may exist some $\bm{x}\in C(\bm{k})$ such that
$L_{G}H(\bm{x})=\mathbf{0}_{1\times M}$, which may violate (6) and lead to the
state exiting $C(\bm{k})$ (and potentially $S$ as a result). Using online
adaptation of $\bm{k}$, however, it may be possible to achieve
$L_{G}H(\bm{x})\neq\mathbf{0}_{1\times M}$ for all $t\geq 0$, which motivates
the following problem.

######  Problem 2.

Given a C-CBF candidate
$H:\operatorname{\mathbb{R}}^{N}\times\operatorname{\mathbb{R}}_{+}^{c}\rightarrow\operatorname{\mathbb{R}}$
defined by (9) and associated with the set $C(\bm{k})$, design an adaptation
law $\dot{\bm{k}}=\kappa(\bm{x},\bm{k})$ such that
$L_{G}H\neq\mathbf{0}_{1\times M}$ for all $t\geq 0$.

###  III-B Adaptation for Control Synthesis

Before proceeding with our main result, we require the following assumption.

######  Assumption 2.

The matrix of controlled candidate CBF dynamics
$\bm{L}_{g}\in\operatorname{\mathbb{R}}^{c\times M}$ is not all zero, i.e.

$\bm{L}_{g}=\begin{bmatrix}L_{g}h_{1}\\\ \vdots\\\
L_{g}h_{c}\end{bmatrix}\neq\mathbf{0}_{c\times M}.$ (13)

We now present our main result, an adaptation law that solves Problem 2 and
thus renders $H$ a valid CBF for the set $C(\bm{k}(t))$, for all $t\geq 0$.

######  Theorem 1.

Suppose that there exist $c>1$ candidate CBFs
$h_{s}:\operatorname{\mathbb{R}}^{N}\rightarrow\operatorname{\mathbb{R}}$
defining sets
$S_{s}=\\{\bm{x}\in\operatorname{\mathbb{R}}^{N}\;|\;h_{s}(\bm{x})\geq 0\\}$,
$\forall s\in[1..c]$, and that it is known that
$\mathcal{U}=\operatorname{\mathbb{R}}^{M}$. If $\bm{k}(0)$ is such that
$L_{G}H\neq\mathbf{0}_{1\times M}$ at $t=0$, then, under the ensuing
adaptation law,

$\displaystyle\kappa(\bm{x},\bm{k})=\operatorname*{arg\,min}_{\bm{\mu}\in\operatorname{\mathbb{R}}^{c}}\;\frac{1}{2}(\bm{\mu}-\bm{\mu}_{0})^{T}\bm{P}$
$\displaystyle(\bm{\mu}-\bm{\mu}_{0})$ (14a)
$\displaystyle\mathrm{s.t.}\quad\quad\quad$
$\displaystyle\bm{\mu}+\alpha_{k}(\bm{k}-\bm{k}_{min})$ $\displaystyle\geq 0,$
(14b)
$\displaystyle\bm{p}^{T}\bm{Q}\dot{\bm{p}}+\bm{p}^{T}\dot{\bm{Q}}\bm{p}+\alpha_{p}(h_{p})$
$\displaystyle\geq 0,$ (14c)

the controlled CBF dynamics $L_{G}H\neq\mathbf{0}_{1\times M}$ for all $t\geq
0$, and thus the function $H$ defined by (9) is a valid CBF for the set
$C(\bm{k}(t))=\\{\bm{x}\in\operatorname{\mathbb{R}}^{N}\;|\;H(\bm{x},\bm{k})\geq
0\\}$, for all $t\geq 0$, where $\bm{P}\in\operatorname{\mathbb{R}}^{c\times
c}$ is a positive-definite gain matrix,
$\alpha_{k},\alpha_{p}\in\mathcal{K}_{\infty}$, $\bm{\mu}_{0}$ is the nominal
$\dot{\bm{k}}$, $\bm{k}_{min}=[k_{1,min},\ldots,k_{c,min}]^{T}$ is the vector
of minimum allowable values $k_{s,min}>0$, and

$\displaystyle\bm{p}$ $\displaystyle=\left[\frac{\partial\phi}{\partial
h_{1}}\;\ldots\;\frac{\partial\phi}{\partial h_{c}}\right]^{T},$ (15)
$\displaystyle\bm{Q}$
$\displaystyle=\bm{I}-(\bm{N}\bm{N}^{T})^{T}-\bm{N}\bm{N}^{T}-(\bm{N}\bm{N}^{T})^{T}\bm{N}\bm{N}^{T}$
(16)

with $h_{p}=\frac{1}{2}\bm{p}^{T}\bm{Q}\bm{p}-\varepsilon$, $\varepsilon>0$,
and

$\bm{N}=[\bm{n}_{1}\;\ldots\;\bm{n}_{r}],$ (17)

such that $\\{\bm{n}_{1},\ldots,\bm{n}_{r}\\}$ constitutes a basis for the
null space of $\bm{L}_{g}^{T}$, i.e.
$\mathcal{N}(\bm{L}_{g}^{T})=\mathrm{span}\\{\bm{n}_{1},\ldots,\bm{n}_{r}\\}$,
where $\bm{L}_{g}$ is given by (13).

###### Proof.

First, given (9), we have that

$\displaystyle\dot{H}$
$\displaystyle=-\sum_{s=1}^{c}\left(\frac{\partial\phi}{\partial
h_{s}}\dot{h}_{s}+\frac{\partial\phi}{\partial k_{s}}\dot{k}_{c}\right)$
$\displaystyle=\bm{p}^{T}\dot{\bm{h}}+\bm{q}^{T}\dot{\bm{k}}$
$\displaystyle=\bm{p}^{T}(\bm{L}_{f}+\bm{L}_{g}\bm{u})+\bm{q}^{T}\dot{\bm{k}}$

where $\bm{p}$ is given by (15), $\bm{L}_{g}$ by (13),
$\bm{L}_{f}=[L_{F}h_{1}\;\ldots\;L_{F}h_{c}]^{T}$, and
$\bm{q}=[\frac{\partial\phi}{\partial
k_{1}}\;\ldots\;\frac{\partial\phi}{\partial k_{c}}]^{T}$. As such,
$L_{F}H=\bm{p}^{T}\bm{L}_{f}+\bm{q}^{T}\dot{\bm{k}}$ and
$L_{G}H=\bm{p}^{T}\bm{L}_{g}$. With
$\mathcal{U}=\operatorname{\mathbb{R}}^{M}$, it follows that as long as
$L_{G}H\neq\mathbf{0}_{1\times M}$ it is possible to choose $\bm{u}$ such that
$\dot{H}(\bm{x},\bm{u})\geq-\alpha_{H}(H)$. We will now show that with
$\dot{\bm{k}}=\kappa(\bm{x},\bm{k})$ given by (14) it holds that
$L_{G}H\neq\mathbf{0}_{1\times M}$ and thus $H$ is a CBF for $C(\bm{k}(t))$,
for all $t\geq 0$.

Since $L_{G}H=\bm{p}^{T}\bm{L}_{g}$, the problem of showing that
$L_{G}H\neq\mathbf{0}_{1\times M}$ is equivalent to proving that
$\bm{p}\notin\mathcal{N}(\bm{L}_{g}^{T})=\mathrm{span}\\{\bm{n}_{1},\ldots,\bm{n}_{r}\\}$.
Since the vector $\bm{p}$ can be expressed as a sum of vectors perpendicular
to and parallel to $\mathcal{N}(\bm{L}_{g}^{T})$ (respectively
$\bm{p}^{\perp}$ and $\bm{p}^{\parallel}$), it follows that
$\bm{p}\notin\mathcal{N}(\bm{L}_{g}^{T})$ as long as $\|\bm{p}^{\perp}\|>0$,
where $\bm{p}^{\perp}=\left(\bm{I}-\bm{N}\bm{N}^{T}\right)\bm{p}$ by vector
projection, and $\bm{N}$ is given by (17). Thus, a sufficient condition for
$\bm{p}\notin\mathcal{N}(\bm{L}_{g}^{T})$ is that

$\frac{1}{2}\|(\bm{I}-\bm{N}\bm{N}^{T})\bm{p}\|^{2}=\frac{1}{2}\bm{p}^{T}\bm{Q}\bm{p}>\varepsilon$
(18)

for some $\varepsilon>0$, where $\bm{Q}$ is given by (16). Then, by defining a
function $h_{p}=\frac{1}{2}\bm{p}^{T}\bm{Q}\bm{p}-\varepsilon$, it follows
from (5) that when (18) is true at $t=0$, it is true $\forall t\geq 0$ as long
as (14c) holds.

Therefore, gains $\bm{k}$ adapted according to the law (14) are guaranteed to
result in $L_{G}H\neq\mathbf{0}_{1\times M}$. Thus, $H$ is a CBF for the set
$C(\bm{k}(t))$, for all $t\geq 0$. This completes the proof. ∎

######  Remark 1.

With $\bm{Q}$ depending on basis vectors spanning
$\mathcal{N}(\bm{L}_{g}^{T})$, it is not immediately obvious under what
conditions $\dot{\bm{Q}}$ is continuous (or even well-defined). Prior results
show that if the rank of $\mathcal{N}(\bm{L}_{g}^{T})$ is constant then
$\dot{\bm{Q}}$ varies continuously $\forall\bm{x}\in B_{\epsilon}(\bm{x})$
[30], but analytical derivations of $\dot{\bm{Q}}$ are not available to the
best of our knowledge. In practice, we observe that the rank of
$\mathcal{N}(\bm{L}_{g}^{T})$ is indeed constant, and we approximate
$\dot{\bm{Q}}$ numerically using finite-difference methods.

With $H$ consolidating the many constituent constraints into one CBF
condition, we can then replace the centralized CBF-QP controller (8) with the
following:

$\displaystyle\bm{u}_{\mathcal{A}_{c}}^{*}$
$\displaystyle=\operatorname*{arg\,min}_{\bm{u}_{\mathcal{A}_{c}}\in\mathcal{U}_{\mathcal{A}_{c}}}\frac{1}{2}\|\bm{u}_{\mathcal{A}_{c}}-\bm{u}_{\mathcal{A}_{c}}^{0}\|^{2}$
(19a) s.t. $\displaystyle a$ $\displaystyle+\bm{b}\bm{u}_{\mathcal{A}_{c}}\geq
0,$ (19b)

where $a=L_{F}H+\alpha_{H}(H)$ and $\bm{b}=L_{G}H_{[i\in\mathcal{A}_{c}]}$. If
all agents are communicative, i.e. $\mathcal{A}_{c}=\mathcal{A}$, then since
$H$ is a CBF for the set $C(\bm{k}(t))\subset S$, for all $t\geq 0$, the
system trajectories are guaranteed to stay within $C(\bm{k}(t))\subset S$ and
thus remain safe. In the presence of non-communicative agents, we replace the
decentralized CBF-QP controller (7) with

$\displaystyle\bm{u}_{i}^{*}$
$\displaystyle=\operatorname*{arg\,min}_{\bm{u}_{i}\in\mathcal{U}_{i}}\frac{1}{2}\|\bm{u}_{i}-\bm{u}_{i}^{0}\|^{2}$
(20a) s.t. $\displaystyle a$ $\displaystyle+\bm{b}_{i}\bm{u}_{i}\geq d,$ (20b)

where $\bm{b}_{i}=L_{G}H_{[mi:m(i+1)]}$, i.e. the portion of the dynamics of
$H$ that agent $i$ controls, and
$d=e^{-rH}\max_{\bm{u}\in\mathcal{U}}\sum_{j=1,j\neq
i}^{A}L_{G}H_{[jm:j(m+1)]}u_{j}$, where $r>0$. While for the case of unbounded
control authority $d$ is similarly unbounded, in practice it is reasonable to
assume that agents have limited control authority and thus to use (20)
assuming some bounded $\mathcal{U}$.

##  IV Multi-Robot Numerical Study

In this section, we demonstrate our C-CBF controller on a decentralized multi-
robot goal-reaching problem.

Consider a collection of 3 non-communicative, but responsive robots
($i\in\mathcal{A}_{n}\setminus\mathcal{A}_{n,n}$) in a warehouse environment
seeking to traverse a narrow corridor intersected by a passageway occupied
with 6 non-responsive agents ($j\in\mathcal{A}_{n,n}$). The non-responsive
agents may be e.g. humans walking or some other dynamic obstacles. Let
$\mathcal{F}$ be an inertial frame with a point $s_{0}$ denoting its origin,
and assume that each robot may be modeled according to a dynamic extension of
the kinematic bicycle model described by [31, Ch. 2], provided here for
completeness:

$\displaystyle\dot{x}_{i}$
$\displaystyle=v_{i}\left(\cos{\psi_{i}}-\sin{\psi_{i}}\tan{\beta_{i}}\right)$
(21a) $\displaystyle\dot{y}_{i}$
$\displaystyle=v_{i}\left(\sin{\psi_{i}}+\cos{\psi_{i}}\tan{\beta_{i}}\right)$
(21b) $\displaystyle\dot{\psi}_{i}$
$\displaystyle=\frac{v_{i}}{l_{r}}\tan{\beta_{i}}$ (21c)
$\displaystyle\dot{\beta}_{i}$ $\displaystyle=\omega_{i}$ (21d)
$\displaystyle\dot{v}_{i}$ $\displaystyle=a_{i},$ (21e)

where $x_{i}$ and $y_{i}$ denote the position (in m) of the center of gravity
(c.g.) of the ith robot with respect to $s_{0}$, $\psi_{i}$ is the orientation
(in rad) of its body-fixed frame, $\mathcal{B}_{i}$, with respect to
$\mathcal{F}$, $\beta_{i}$ is the slip angle111$\beta_{i}$ is related to the
steering angle $\delta_{i}$ via
$\tan{\beta_{i}}=\frac{l_{r}}{l_{r}+l_{f}}\tan{\delta_{i}}$, where
$l_{f}+l_{r}$ is the wheelbase with $l_{f}$ (resp. $l_{r}$) the distance from
the c.g. to the center of the front (resp. rear) wheel. (in rad) of the c.g.
of the vehicle relative to $\mathcal{B}_{i}$ (assume
$|\beta_{i}|<\frac{\pi}{2}$), and $v_{i}$ is the velocity of the rear wheel
with respect to $\mathcal{F}$. The state of robot $i$ is denoted
$\bm{z}_{i}=[x_{i}\;y_{i}\;\psi_{i}\;\beta_{i}\;v_{i}]^{T}$, and its control
input is $\bm{u}_{i}=[a_{i}\;\omega_{i}]^{T}$, where $a_{i}$ is the
acceleration of the rear wheel (in m/s2), and $\omega_{i}$ is the angular
velocity (in rad/s) of $\beta_{i}$.

The challenges of this scenario relate to preserving safety despite multiple
non-communicative and non-responsive agents present in a constrained
environment. A robot is safe if it 1) obeys the speed restriction, 2) remains
inside the corridor area, and 3) avoids collisions with all other robots.
Speed is addressed with the following candidate CBF:

$\displaystyle h_{v}(\bm{z}_{i})$ $\displaystyle=s_{M}-v_{i},$ (22)

where $s_{M}>0$, while for corridor safety and collision avoidance we used
forms of the relaxed future-focused CBF introduced in [11] for roadway
intersections, namely

$\displaystyle\begin{split}h_{c}(\bm{z}_{i})&=(m_{L}(x_{i}+\dot{x}_{i})+b_{L}-(y_{i}+\dot{y}_{i}))\cdot\\\
&\quad\;(m_{R}(x_{i}+\dot{x}_{i})+b_{R}-(y_{i}+\dot{y}_{i}))\end{split}$ (23)
$\displaystyle\begin{split}h_{r}(\bm{z}_{i},\bm{z}_{j})&=D(\bm{z}_{i},\bm{z}_{j},t+\hat{\tau})^{2}\\\
&\quad+\epsilon
D(\bm{z}_{i},\bm{z}_{j},t)^{2}-(1+\epsilon)(2R)^{2},\end{split}$ (24)

where (23) prevents collisions with the corridor walls (defined as lines in
the $xy$-plane via $m_{L},b_{L},m_{R},b_{R}\in\operatorname{\mathbb{R}}$), and
(24) prevents inter-robot collisions and is defined $\forall
i\in\mathcal{A}_{n}\setminus\mathcal{A}_{n,n}$, $\forall j\in\mathcal{A}_{n}$,
where $\epsilon>0$, $D(\bm{z}_{i},\bm{z}_{j},t_{a})$ is the Euclidean distance
between agents $i$ and $j$ at arbitrary time $t_{a}$, and $\hat{\tau}$ denotes
the time in the interval $[0,T]$ at which the minimum inter-agent distance
will occur under constant velocity future trajectories. For a more detailed
discussion on future-focused CBFs, see [11]. As such, (22), (23), and (24)
define the sets

$\displaystyle S_{v,i}$
$\displaystyle=\\{\bm{z}_{i}\in\operatorname{\mathbb{R}}^{n}\;|\;h_{v}(\bm{z}_{i})\geq
0\\},$ $\displaystyle S_{c,i}$
$\displaystyle=\\{\bm{z}_{i}\in\operatorname{\mathbb{R}}^{n}\;|\;h_{c}(\bm{z}_{i})\geq
0\\},$ $\displaystyle S_{r,i}$ $\displaystyle=\bigcap\limits_{j=1,j\neq
i}^{A}\\{\bm{z}\in\operatorname{\mathbb{R}}^{N}\;|\;h_{r}(\bm{z}_{i},\bm{z}_{j})\geq
0\\},$

the intersection of which constitutes the safe set for agents $i$, i.e.
$S_{i}(t)=S_{v,i}\cap S_{c,i}\cap S_{r,i}$.

We control robots $i\in\mathcal{A}_{n}\setminus\mathcal{A}_{n,n}$ using a
C-CBF based decentralized controller of the form (20) with constituent
functions $h_{c}$, $h_{s}$, $h_{r}$, an LQR based nominal control input (see
[11, Appendix 1]), and initial gains $\bm{k}(0)=\mathbf{1}_{10\times 1}$. The
non-responsive agents used a similar LQR controller to move through the
passageway in pairs of two, with the first two pairs passing through the
intersection without stopping and the last pair stopping at the intersection
before proceeding.

As shown in Figure 2, the non-communicative robots traverse both the narrow
corridor and the busy intersection to reach their goal locations safely. The
trajectories of the gains $\bm{k}$ for each warehouse robot are shown in
Figure 3, while their control inputs are depicted in Figure 4. The CBF time
histories for the constituent and consolidated functions are highlighted in
Figures 5 and 6 respectively, and show that the C-CBF controllers maintained
safety at all times.

Figure 2: XY paths for the warehouse robots (blue) and non-responsive agents
(red) in the warehouse control problem. Figure 3: Gains $\bm{k}$ for the C-CBF
controllers in the warehouse study. Robot 1 denoted with solid lines, dotted
for robot 2, dash-dots for robot 3. AgentA and AgentB denote the other two
non-communicative robots from the perspective of one (e.g. AgentA=Agent1 and
AgentB=Agent3 for robot 2).  Figure 4: Warehouse robot controls: accel. ($a$)
and slip angle rate ($\omega$). Figure 5: Evolution of warehouse robot
constituent CBF candidates, $h_{s}$ $\forall s\in[1..c]$, synthesized to
construct C-CBF. Figure 6: Evolution of C-CBF $H$ for warehouse robots 1, 2,
and 3.

##  V Experimental Case Study

For experimental validation of our approach, we used an AION R1 UGV ground
rover as an ego vehicle in the laboratory setting and required it to reach a
goal location in the presence of two non-responsive rovers: one static and one
dynamic. We modeled the rovers as bicycles using (21), and sent angular rate
$\omega_{i}$ and velocity $v_{i}$ (numerically integrated based on the
controller’s acceleration output) commands to the rovers’ on-board PID
controllers. The ego rover used our proposed C-CBF (20) with constituent
candidate CBFs (22) (with $s_{M}=1$ m/s) and the rff-CBF defined in (24) for
collision avoidance. The nominal input to the C-CBF controller was the LQR law
from the warehouse robot example, as was the controller used by the dynamic
non-responsive rover. A Vicon motion capture system was used for position
feedback, and the state estimation was performed by extended Kalman filter via
the on-board PX4.

Figure 7: A rover avoids a static and dynamic rover using our proposed C-CBF
controller en route to a target in the laboratory setting.

For the setup, the static rover was placed directly between the ego rover and
its goal, while the dynamic rover was stationary until suddenly moving across
the ego’s path as it approached its target. As highlighted in Figure 7, the
ego rover first headed away from the static rover and then decelerated and
swerved to avoid a collision with the second rover before correcting course
and reaching its goal. Videos and code for both this experiment and the
simulation in Section IV is available on Github222Link to Github repo:
github.com/6lackmitchell/CCBF-Control.

##  VI Conclusion

In this paper, we addressed the problem of safe control under multiple state
constraints via a C-CBF based control design. To ensure that the synthesized
C-CBF is valid, we introduced a parameter adaptation law on the weights of the
C-CBF constituent functions and proved that the resulting controller is safe.
We then demonstrated the success of our approach on a multi-robot control
problem in a crowded warehouse environment, and further validated our work on
a ground rover experiment in the lab.

In the future, we plan to explore conditions under which the C-CBF approach
may preserve guarantees in the presence of input constraints, including
whether alternative adaptation laws for the weights assist in guarantees of
liveness in addition to safety.

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