# Using Circulation to Mitigate Spurious Equilibria in Control Barrier
Function

Vinicius Mariano Gonçalves1, Prashanth Krishnamurthy1,3, Anthony Tzes1,2, and
Farshad Khorrami1,3 1Center for Artificial Intelligence and Robotics (CAIR),
New York University Abu Dhabi, United Arab Emirates.2New York University Abu
Dhabi, Electrical Engineering, Abu Dhabi 129188, United Arab Emirates.3New
York University, Electrical & Computer Engineering Department, Brooklyn, NY
11201, USA.

###### Abstract

Control Barrier Functions and Quadratic Programming are increasingly used for
designing controllers that consider critical safety constraints. However, like
Artificial Potential Fields, they can suffer from the stable spurious
equilibrium point problem, which can result in the controller failing to reach
the goal. To address this issue, we propose introducing circulation
inequalities as a constraint. These inequalities force the system to
explicitly circulate the obstacle region in configuration space, thus avoiding
undesirable equilibria. We conduct a theoretical analysis of the proposed
framework and demonstrate its efficacy through simulation studies. By
mitigating spurious equilibria, our approach enhances the reliability of CBF-
based controllers, making them more suitable for real-world applications.

IEEE Publication Technology, This paper was produced by the IEEE Publication
Technology Group. They are in Piscataway, NJ.Manuscript received April 19,
2021; revised August 16, 2021.

##  I Introduction

Control barrier functions [1] (CBFs) have attracted much interest in recent
years in the field of automatic control, since they provide a convenient
incorporation of safety constraints as linear inequalities in the control
input [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. This inequality can be inserted into a
Quadratic Programming (QP) optimization problem, that is formulated based on
two objectives: making the resulting control action as close as possible to a
_nominal controller_ , that solves the task (e.g., navigating towards a goal
location) if we disregard all the constraints; ensuring that all the safety
constraints are satisfied. Solving this QP optimization problem provides a
control input that is able to preserve safety while driving the robot towards
the goal.

In the QP formulation, the constraints are linear in the control input
provided that we assume an affine dynamics for the system. This linearity
appears since we use _local information_ of the problem, like distance
function and Lyapunov function gradients. This locality of the approach
induces _spurious equilibrium points_ into the system, and so the closed loop
system can become stuck in such a point. Even worse, these spurious
equilibrium points can even be _stable_ [12].

This problem also appears in other approaches that use local information to
navigate. A quintessential example is using _artificial potential functions_
(APF) [13], where a combination of gradients of attractive and repulsive
potentials are employed for navigation. The QP-CBF formulation is
qualitatively similar to the APF. In fact, the Karush-Kuhn-Tucker conditions
(KKT), the fact that many of the CBF inequalities are generated using
gradients of “obstacle functions,” and the fact that the nominal controller is
usually the negative gradient of a potential function, implies that the CBF-QP
controller can be seen as a sum of attractive and repulsive vectors with a
choice of weights that comes from the optimization procedure.

Considering this local equilibrium issue, there are works that aim to modify
the traditional QP-CBFs/APF approaches to circumvent it. For example, [12]
introduces a new CBF inequality into the original CBF formulation that can, in
the case that there is a single convex obstacle, remove the equilibrium points
in the boundary of the obstacle. In the proposed approach in this paper, the
spurious equilibrium point problem is mitigated by introducing an inequality,
which we call _a circulation inequality_. This inequality forces the
configuration to circulate the obstacle when near it. This is motivated by the
observation that QP-CBF approaches often fail in large/complex obstacles,
specially when the nominal controller is defined in terms of the gradient of a
simple potential function as, for example, the squared distance between the
configuration and the goal in the configuration space. This is because the
controller seeks to decrease this potential function, and for large obstacles
this may be impossible to do in all situations: sometimes the configuration
must move around (i.e., circulate) the obstacle, resulting in temporary
increase of this function so as to enable eventual reaching of the goal at a
later time.

The idea of inducing circulation of obstacles appears in other works as well.
Early approaches for circulating obstacles appear in the _bug algorithm_ [14].
The proposed inequality can be seen as a low-level implementation of this
circulation aspect, although the proposed controller does not realize exactly
the bug algorithm or any of its variants. In order to avoid obstacles, [15]
proposes the addition of a circulation term, called “gyroscopic force”, and
[16, 17] propose the addition of a circulation term based on magnetic fields.
Furthermore, in [18], the authors propose a modification of the classic
attractive/repulsive potential fields approach so as to improve its
“efficiency” while retaining the Lyapunov stability analysis from traditional
potential fields. The approach is motivated by the observation that if the
attractive and repulsive vectors conflict with each other - i.e., if the
cosine of the angle between them is less than 0 - then, within a certain range
of angles, the repulsive vector can be replaced by a different vector that is
orthogonal to the attractive vector without violating the Lyapunov
inequalities for the closed loop system. This modification of the
attractive/repulsive potential fields approach allows the robot to move faster
towards the goal since the impact of the repulsive potential is reduced in the
system. This modification of making (in some circumstances) the repulsive and
attractive fields orthogonal can be seen as a way to generate a circulation
behavior in the controller. However, although this technique can remove some
spurious equilibrium points in some situations, it cannot completely solve the
problem. The “modification of the repulsive/attractive vectors” philosophy is
also employed in this work, since the proposed optimization formulation is
conceptually similar to APF approaches. The difference is that the
modification is introduced in this paper by adding a new constraint into the
optimization problem. This proposed strategy allows us to remove the spurious
equilibrium points in some circumstances, as shown in Figure 1.

(a) Without circulation (b) With circulation

Figure 1: Vector field generated by traditional CBF-QP formulations (III-A)
(left) and our formulation (III-A) (right), with a nominal controller that
points towards the goal. Note that in the original formulation, there is a set
of points (at the left of the obstacle) such that if we start from this set,
we converge to spurious equilibria near the boundary of the obstacle instead
of converging towards the target.

The problem considered in [19], which is to track a path while avoiding
obstacles, is similar to the problem addressed in this paper. To implement
obstacle avoidance, a “circulation mode”, based on the controller proposed in
[20], is initiated when the robot is near the obstacle until it can return to
navigating towards the target. This switching between circulation and goal-
directed navigation modes is done by monitoring the sign of a variable and can
produce discontinuities in the vector field. Our proposal is similar in the
sense that we also enforce circulation when near an obstacle, but the
enforcing of this behavior is achieved by imposing a circulation constraint,
which gradually increases in its effect as the robot approaches the obstacle,
instead of switching the vector fields in a discontinuous fashion. Since our
formulation is based on a strictly convex optimization problem with continuous
data, our proposed construction can be shown to have continuity properties
using classical theorems on QPs.

In this paper, we extend the QP-CBF formulations with a circulation
constraint, providing certain theoretical results (as characterization of the
equilibrium points, feasibility, and continuity). The major challenge is
related to how to _define_ circulation in higher dimensions. In order to
implement the constraint, we need to specify precisely _how_ to circulate the
obstacles. Clockwise or counter-clockwise can be used in planar situations,
but when we consider higher dimensions the situation is more complex, in which
topology may forbid us to defining circulation in a continuous way.

Paper organization: The notations are summarized in Section II. The proposed
CCBF-QP (Circulation Control Barrier Function Quadratic Program) approach and
its properties are discussed in Section III. Practical aspects in application
of the proposed approach are discussed in Section IV. Simulation studies of
the approach are presented in Section V. Finally, concluding remarks are
summarized in Section VI.

##  II Mathematical Notation

We denote by $\mathbb{R}^{+}~{}(\mathbb{R}^{-})$ the set of nonnegative
(nonpositive) reals. All vectors are column vectors unless stated otherwise,
and for a vector $v$, $v^{T}$ denotes its transpose. If
$f:\mathbb{R}^{n}\mapsto\mathbb{R}$, $\nabla f$ is its gradient, written as a
column vector. For a closed set $\mathcal{C}$, we denote its boundary by
$\partial\mathcal{C}$ and
$\textsl{Int}(\mathcal{C})\triangleq\mathcal{C}-\partial\mathcal{C}$ its
interior. We denote the identity matrix of appropriate order by $I$. A square
matrix is _skew-symmetric_ if $\Omega^{T}=-\Omega$ and _orthonormal_ if
$\Omega^{T}\Omega=I$.

##  III The CCBF-QP formulation and its properties

###  III-A The CCBF-QP formulation

We are interested in defining a controller that reaches a goal in the
configuration space while avoiding obstacles. For this purpose, we will state
certain assumptions (and related definitions) that will allow us to describe
the problem and our proposed solution formally.

###### Assumption 1

We assume the following:

  1. 1.

Our configuration space $\mathcal{Q}$ is an _even-dimensional_ Euclidean
space, that is, $\mathcal{Q}\subset\mathbb{R}^{n}$, $n$ even. Note that
$\mathcal{Q}$ can include forbidden configurations.

  2. 2.

We consider a single $n$-dimensional obstacle
$\mathcal{C}\subset\mathbb{R}^{n}$ in the configuration space that is modeled
as a closed set. Furthermore, we assume a function
$D:\mathbb{R}^{n}\mapsto\mathbb{R}^{+}$ that computes a metric of “closeness”
from the point $q\in\mathcal{Q}$ and $\mathcal{C}$. $D$ is not required to be
the Euclidean distance between $q$ and $\mathcal{C}$, but

$D(q)>0~{}\Rightarrow q\not\in\mathcal{C}.$ (1)

Furthermore, $D$ is assumed to be differentiable (i.e., $\nabla D(q)$ is
continuous) everywhere in $\mathbb{R}^{n}-\textsl{Int}(\mathcal{C})$ and
$\nabla D(q)\not=0$ almost everywhere. Define the set
$\mathcal{W}\triangleq\\{q\in\mathbb{R}^{n}\ |\
q\in\mathbb{R}^{n}-\textsl{Int}(\mathcal{C})\ ,\ \nabla D(q)\not=0\\}$. For
$q\in\mathcal{W}$, we also define the _normal vector_ $N(q)\triangleq\nabla
D(q)/\|\nabla D(q)\|$ .

  3. 3.

A skew-symmetric orthonormal matrix is defined such that
$\Omega\in\mathbb{R}^{n\times n}$. If $q\in\mathcal{W}$, we define the
_tangent vector_ as $T(q)\triangleq\Omega N(q)$.

  4. 4.

We aim to achieve a target configuration $q_{g}\in\mathcal{W}$, and that we
have a _nominal_ controller $u_{d}:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ that
achieves this task, as the negative gradient of a positive definite
differentiable function $V(q)$ such that $V$ and $\nabla V$ vanish only at
$q=q_{g}$. Furthermore, we assume that $\|\nabla V(q)\|$ is upper bounded by
all $q$, i.e., $\displaystyle\max_{q\in\mathbb{R}^{n}}\|\nabla V(q)\|$ is
finite.

  5. 5.

We have a set of configuration-dependent constraints to our control input,
written as $\mathcal{U}(q)\triangleq\\{\mu\ |\ A(q)\mu\geq b(q)\\}$, in which
$A:\mathbb{R}^{n}\mapsto\mathbb{R}^{l\times n}$ and
$b:\mathbb{R}^{n}\mapsto\mathbb{R}^{l}$ are continuous functions of $q$.
Furthermore, $\forall q$, let the set $\mathcal{U}(q)$ contain a ball of
radius $r>0$ centered at the origin, i.e., if $\|\mu\|\leq r$, then $A\mu\geq
b$.

  6. 6.

We have a continuous function $\alpha:\mathbb{R}^{+}\mapsto\mathbb{R}^{-}$
such that $\alpha(0)=0$ and $\alpha(D)<0$ for $D<0$ (e.g., $\alpha(D)=-\eta D$
for a constant $\eta>0$) .

  7. 7.

We have a continuous function $\beta:\mathbb{R}^{+}\mapsto\mathbb{R}$ with the
following properties: $\beta$ is decreasing, $\beta(0)>0$, and
$\beta(D)\rightarrow-\infty$ as $D\rightarrow\infty$ (e.g., $\beta(D)=a-bD$
for positive constants $a,b$). Furthermore, $\beta(D(q_{g}))<0$ and
$\beta(0)<r$.

  8. 8.

Our system’s dynamics is $\dot{q}=u$.

The most striking assumption is related to the even-dimensional assumption in
Assumption 1-1. It turns out that even-dimensional spaces are more amenable
for defining the circulation of obstacles than odd-dimensional spaces. This is
a topological issue: the _hairy ball theorem_ [21] establishes that there is
no way to define a continuous and non-vanishing circulation field (i.e.,
tangent to the surface) on the surfaces of odd-dimensional balls. Thus, since
$N$ is in the surface of a $n$-dimensional ball (thanks to $\|N\|=1$ and
$N\in\mathbb{R}^{n}$), creating a continuous non-vanishing function that maps
every possible $N$ to a vector orthogonal to it would be tantamount to
generating a continuous non-vanishing field at the surface of the $n$-sphere,
which is forbidden by the hairy ball theorem when $n$ is odd. As a consequence
of this result, there is no such matrix $\Omega$ as defined in Assumption 1-3
if $n$ is odd111Another way to see it without the hairy ball theorem is that
if $n$ is odd and $\Omega$ is skew-symmetric, $\det(\Omega^{T})=\det(-\Omega)$
implies $\det(\Omega)=(-1)^{n}\det(\Omega)$ and thus $\det(\Omega)=0$. Thus,
odd-dimensional skew-symmetric matrices are non-invertible and consequently
cannot be orthonormal., because $F(N)=\Omega N$ would define a continuous and
non-vanishing tangential field in the surface of an odd-dimensional sphere. In
Subsection IV-C, we discuss a workaround for the case when $n$ is odd.

If we want to reach the target $q_{g}$ while avoiding the obstacle, a common
approach is to formulate a _minimally invasive QP using CBFs_ and compute the
control input $u$ as:

$\displaystyle u(q)=$ $\displaystyle\arg\min_{\mu}\ \|\mu-u_{d}(q)\|^{2}$ such
that $\displaystyle N(q)^{T}\mu\geq\alpha(D(q))$ (2) $\displaystyle
A(q)\mu\geq b(q)$

for $q\in\mathcal{W}$ (we leave $u(q)$ undefined if $q\not\in\mathcal{W}$).
The QP tries to minimally modify the nominal controller $u_{d}(q)$ while
imposing the CBF constraint $\dot{D}=\nabla D(q)^{T}\mu\geq\|\nabla
D(q)\|\alpha(D(q))\geq 0$ that guarantees that the system will not enter the
forbidden set $\mathcal{C}$.

Unfortunately, the formulation in (III-A) is prone to have stable equilibrium
points besides the only desired one $q=q_{g}$. The result of (III-A) is
essentially a sum of potential fields, which is known for having spurious
equilibrium points, dependent on the repulsive potential $N(q)$. Indeed, if we
disregard the constraints $A(q)u\geq b(q)$, (III-A) can be solved
analytically:

$u(q)=-\nabla V(q)+N(q)\lambda(q)$

in which $\lambda(q)=\max\big{(}0,N(q)^{T}\nabla V(q)+\alpha(D(q))\big{)}$,
the dual variable for the normal constraint, is a nonnegative configuration-
dependent weight for the repulsive potential.

These spurious equilibrium points appear near the surface of the obstacle when
the controller cannot force the configuration to circulate the obstacle in
order to escape from it. Therefore, in this paper, we augment (III-A) with an
inequality that forces the configuration to circulate the obstacle when in its
neighborhood by including the inequality:

$T(q)^{T}u\geq\beta(D(q))$

(in which $\beta$ was defined in Assumption 1 \- 7) that forces the following
behavior. When near obstacles (that is, when $\beta>0$), we have a positive
projection into the tangent vector of the obstacle, which means that we should
circulate it. If $D$ is large enough, $\beta$ becomes very negative and thus
the constraint is trivialized. This is specially true for $q=q_{g}$: we do not
want to force any circulation when the goal is achieved and this is why
Assumption 1 \- 7 has $\beta(D(q_{g}))<0$. Nevertheless, after the inclusion
of this inequality into (III-A), we have the CCBF-QP (Circulation Control
Barrier Function Quadratic Program), including the validity of all terms in
Assumption 1:

$\displaystyle u(q)=$ $\displaystyle\arg\min_{\mu}\ \|\mu-u_{d}(q)\|^{2}$ such
that $\displaystyle N(q)^{T}\mu\geq\alpha(D(q))$ (3) $\displaystyle
T(q)^{T}\mu\geq\beta(D(q))$ $\displaystyle A(q)\mu\geq b(q).$

Figure 1 makes the comparison in the computed vector fields between (III-A)
and (III-A), disregarding the constraints $A(q)\mu\geq b(q)$, for a smooth
rectangle-like obstacle and goal $p_{\textsl{goal}}=[1\ 0]^{T}$ (green dot in
the two pictures). We can see that without the circulation term, there is a
spurious line of attractive points (in cyan), with a very large attractive
basin to the left of the obstacle. If we include the circulation terms, there
are no spurious attractive points anymore.

Next, we establish some properties for this formulation.

###  III-B Feasibility of CCBF-QP

One important question is whether the CCBF-QP formulation always admits a
feasible solution. The following result answers this question positively.

###### Theorem 1

Let $K$ be such that $\beta(0)<K<r$ (such $K$ always exists, since
$\beta(0)<r$, see Assumption 1-7). Then, if $q\in\mathcal{W}$, the CCBF-QP
formulation always has a feasible solution: $\mu_{f}\triangleq KT(q)$.
Furthermore, this particular solution lies in the interior of the feasible
set, achieving all inequalities without equality.

###### Proof:

We shall verify all of the three constraints in (III-A), and emphasize that
the inequalities are achieved without equality.

  * •

For $N(q)^{T}\mu_{f}\geq\alpha(D(q))$, we note that $N(q)^{T}T(q)=0$ and
$\alpha(D(q))<0$ if $q\not\in\mathcal{C}$.

  * •

For $T(q)^{T}\mu_{f}\geq\beta(D(q))$, we note that
$T(q)^{T}T(q)=\|T(q)\|^{2}=1$ and that $K>\beta(D(q))$ since $\beta$ being
decreasing (Assumption 1-7) implies that the maximum value for $\beta$ occurs
at $D=0$ and thus $K>\beta(0)\geq\beta(D(q))$.

  * •

For the constraint $A\mu\geq b$, we use the fact that $\|\mu\|\leq r$ is a
sufficient condition for this inequality to hold (Assumption 1-5). Now
$\|KT(q)\|=K<r$. The inequality being strict means that $\mu_{f}$ lies
strictly inside the set $A\mu\geq b$, i.e., $A\mu_{f}>b$.

∎

This implies that CCBF-QP allows as a feasible action for the system to move
tangent to its current contour surface of $D(q)$ (in the direction specified
by $T(q)$).

###  III-C Continuity of CCBF-QP

We now prove the continuity of $u(q)$.

###### Theorem 2

If $u(q)$ comes from a CCBF-QP, it is a continuous function of $q$ for any
$q\in\mathcal{W}$.

###### Proof:

This is a consequence of a result [22] that states the following: the single
minimizer of a strictly convex quadratic program
$\displaystyle\min_{\mu}\mu^{T}H\mu/2+f^{T}\mu\ \mbox{such that}\ G\mu\geq g$
is continuous on the parameters $\\{H,f,G,g\\}$ as long as the constraint
$G\mu\geq g$ is _regular_ , which means that there exists $\mu_{f}$ such that
$G\mu_{f}>g$ (i.e., the interior is non-empty).

In (III-A), $H=2I$, $f=2\nabla V(q)$, $G=[N(q)\ T(q)\ A(q)^{T}]^{T}$, and
$g(q)=[\alpha(D(q))\ \beta(D(q))\ b(q)^{T}]^{T}$. The regularity is a
corollary of Theorem 1, that states that $\mu_{f}$ (defined in that theorem)
satisfies the inequalities with all inequalities being strict. Now, the set of
Assumptions 1 guarantees that $f(q),G(q)$, and $g(q)$ are continuous functions
of $q$ for $q\in\mathcal{W}$, and the proof is complete. ∎

###  III-D CCBF-QP Equilibrium Points

We will characterize the subset of $\mathcal{W}$ in which the output of CCBF-
QP is the null vector, that is, the _equilibrium points_ of the dynamical
system $\dot{q}=u(q)$.

###### Theorem 3

Considering the set $q\in\mathcal{W}$, in the CCBF-QP formulation, $u(q)=0$
occurs only in the two mutually exclusive conditions:

  1. 1.

$q=q_{g}$;

  2. 2.

$\beta(D(q))=0$ and $-u_{d}(q)$ is positive parallel to $T(q)$, that is, there
exists a positive scalar $\lambda_{T}$ for which $u_{d}+\lambda_{T}T=0$.

###### Proof:

If $\mu=0$ is optimal, the KKT conditions, for the dual variables
$\lambda_{N},\lambda_{T}\in\mathbb{R}$ and $\lambda_{A}\in\mathbb{R}^{l}$, are
[23]:

$\displaystyle(i):\ u_{d}+N\lambda_{N}+T\lambda_{T}+A^{T}\lambda_{A}=0;$
$\displaystyle(ii):\ \lambda_{N}\alpha=0\ ,\ (iii):\ \lambda_{T}\beta=0\ ,\
(iv):\ \lambda_{A}^{T}b=0;$ $\displaystyle(v):\ \lambda_{N}\geq 0\ ,\ (vi):\
\lambda_{T}\geq 0\ ,\ (vii):\ \lambda_{A}\geq 0;$ $\displaystyle(viii):\
0\geq\alpha\ ,(ix):\ 0\geq\beta\ ,\ (x):0\geq b.$ (4)

The first important fact is that it is not possible for these conditions to
hold if $q\in\mathcal{C}$. Indeed, in this case, $\beta(D(q))>0$, which
contradicts ($ix$).

Thus, if $q\not\in\mathcal{C}$, we have that $\alpha\not=0$ (Assumption 1-6).
Consequently, we get from ($ii$) that $\lambda_{N}=0$. Furthermore, since the
polytope $A\mu\geq b$ contains a ball of non-null radius $r$ (Assumption 1-5),
$b$ is a strictly negative vector (since $\mu=0$ should be feasible and in the
interior of the polytope), and then, from ($iv$) and ($vii$), we have that
$\lambda_{A}=0$. Hence,

$(i):\ u_{d}+T\lambda_{T}=0\ ,\ (iii):\ \lambda_{T}\beta=0\ ,\ (vi):\
\lambda_{T}\geq 0.$ (5)

From ($iii$), either $\lambda_{T}=0$ or $\beta=0$. If $\lambda_{T}=0$, we have
$u_{d}(q)=0$. Due to Assumption 1-4, this implies $q=q_{g}$. If $\beta=0$, we
have $u_{d}+T\lambda_{T}=0$ with $\lambda_{T}\geq 0$. Note that since
$\beta(D(q_{g}))\not=0$ (Assumption 1-7), the two sets of vanishing points are
mutually exclusive. ∎

##  IV Practical aspects

###  IV-A The function D

The scalar function $D$ must encompass all the system constraints, codified as
an obstacle $\mathcal{C}$ in the configuration space. However, in practice, it
is easier to describe these constraints using $m$ scalar functions as
$F_{i}(q)\geq 0$, that is, $\mathcal{C}=\\{q\in\mathbb{R}^{n}\ |\ F_{i}(q)\geq
0\ i=1,2,..,m\\}$. In this case, one could use as function $D$ the minimum of
these $F_{i}$, since clearly $D>0$ implies that $q\not\in\mathcal{C}$.
However, even if $F_{i}$ is everywhere differentiable, the function $D$
defined in this way can have many non-differentiable points. More
specifically, whenever we have a $q$ in which the minimum is achieved for at
least two different functions $F_{i}$, $D$ is non-differentiable unless the
respective gradients $\nabla F_{i}$ are equal at this $q$, which is a very
strong condition. Being non-differentiable brings several practical problems
because the controller will be discontinuous at that point.

One idea to remove this problem is to use an approximate version of the
minimum function. The _softmin_ function [24] is a good candidate, defined
(with an averaging term) as
$\min^{h}(g_{1},g_{2},...,g_{m})\triangleq-h\ln(\frac{1}{m}\sum_{k}\exp(-g_{k}/h))$.
This function is always differentiable in the arguments $g_{i}$ and
approximates the minimum when $h$ is positive and close to $0$. However, we
can show that
$\min^{h}(g_{1},g_{2},...,g_{m})\geq\min(g_{1},g_{2},...,g_{m})$, and thus if
we choose $D(q)=\min^{h}_{i}F_{i}(q)$, $D>0$ may not imply $F_{i}(q)>0$ for
all $i$. In this case, we introduce a margin $\delta>0$ and choose
$D(q)\triangleq\min^{h}_{i}F_{i}(q)-\delta$. Another important aspect is that
a naive implementation of the softmin function can easily generate numerical
precision problems, since some of the terms $\exp(-g_{i}/h)$ can be too large
(so we have a $\ln(\infty)$ situation) or all of them too small (so we have a
$\ln(0)$ situation). In order to avoid this, we can use the fact that
$\min^{h}_{i}g_{i}=\min_{i}g_{i}+\min^{h}_{i}(g_{i}-\min_{i}g_{i})$. The terms
$\hat{g}_{i}\triangleq g_{i}-\min_{i}g_{i}$ are always positive and at least
one of them is $0$, which implies that the terms $\exp(-\hat{g}_{i}/h)$ will
be always less than or equal to one and at least one of them is $1$. This
eliminates this numerical issue.

###  IV-B Generating matrices $\Omega$

In order to implement the algorithm, we need skew-symmetric orthonormal
matrices, i.e, $\Omega\in\mathbb{R}^{n\times n}$ such that
$\Omega^{T}=-\Omega=\Omega^{-1}$. As mentioned before, they exist only in
even-dimensional spaces. A different choice of $\Omega$ implies a different
way to circulate the obstacle.

If $n=2$, there are only two such matrices:

$\Omega_{1}=\left[\begin{array}[]{cc}0&-1\\\ 1&\ 0\end{array}\right]\ ,\
\Omega_{2}=-\Omega_{1}=\left[\begin{array}[]{cc}\ 0&1\\\
-1&0\end{array}\right]$ (6)

corresponding to counter-clockwise and clockwise rotations, respectively. But
for $n>2$ (and even), infinite such matrices exist. A simple way to create
these matrices is to divide the set $\\{1,2,...,n\\}$ into $n/2$ pairs
$(i_{k},j_{k})$, $k=1,2,...,n/2$. We then start from the $n\times n$ identity
matrix, and multiply each $i_{k}^{th}$ row by $-1$. We then swap the
$i_{k}^{th}$ row with the $j_{k}^{th}$ row, and we can check by inspection
that the resulting matrix is skew-symmetric and orthonormal. For example, for
$n=4$, we can have the pairs $(1,3)$ and $(2,4)$ and then:

$\Omega=\left[\begin{array}[]{cccc}\ 0&\ 0&1&0\\\ \ 0&\ 0&0&1\\\ -1&\ 0&0&0\\\
\ 0&-1&0&0\end{array}\right].$ (7)

We can create $n!/(n/2)!$ such matrices, but for even numbers $n>2$, there are
other ways. One simple way is as follows: it is easy to show that if
$\Omega\in\mathbb{R}^{n\times n}$ is skew-symmetric orthonormal and
$Q\in\mathbb{R}^{n\times n}$ is an orthonormal matrix, then
$\hat{\Omega}=Q\Omega Q^{T}$ is also a skew-symmetric orthonormal matrix. We
can use this fact to generate new skew-symmetric orthogonal matrices by
constructing $\Omega$ by the simple pairing index procedure described above,
constructing an orthonormal matrix $Q$ and then calculating
$\hat{\Omega}=Q\Omega Q^{T}$. To generate orthonormal matrices in
$\mathbb{R}^{n\times n}$, we use the fact that if $A$ is skew-symmetric and
$I$ is the $n\times n$ identity matrix, $Q=(I-A)^{-1}(I+A)$ is orthonormal. We
can generate a skew-symmetric matrix by using any matrix $B\in\mathbb{R}^{n}$
and computing $A=B-B^{T}$.

###  IV-C Odd dimension extension

In Assumption 1-1, we consider even $n$. If $n$ is odd, there is no matrix
$\Omega\in\mathbb{R}^{n\times n}$ that is skew-symmetric and orthonormal. One
alternative for handling the odd-dimensional case is to drop one of these
properties. On the one hand, being skew-symmetric is important for
guaranteeing that $T=\Omega N$ is orthogonal to $N$ (for any $N$), which
guarantees that $T$ indeed induces circulations. On the other hand, being
orthonormal, is not as important as $\Omega$ being invertible, which is also
incompatible with skew-symmetry when $n$ is odd. Invertibility is important
because it guarantees that $T=\Omega N$ only vanishes when $N$ vanishes, i.e.,
when $q\not\in\mathcal{W}$. If $T$ vanishes when $\beta>0$, the circulation
constraint $T^{T}\mu\geq\beta$ cannot be satisfied and the QP is infeasible,
rendering the controller undefined even in some points inside $\mathcal{W}$.

This constraint in $\Omega$ for $n$ odd can be relaxed if we assume that
$\Omega$ is skew-symmetric but is orthonormal only in an $n-1$ dimensional
subspace of $\mathbb{R}^{n}$, being singular in the complementary
$1$-dimensional space. Such matrices can be created using a similar
construction as the one described in Subsection IV-B. However, if $n$ is odd,
the index $i$ that has no pair must have its respective row in the identity
matrix set to zero.

##  V Simulation Studies

###  V-A 4-DoF planar manipulator

In this simulation, we have a planar 4DOF revolute manipulator and rectangle-
shaped links of size $0.70$m$\times 0.15$m. The configuration
$q\in\mathbb{R}^{4}$ corresponds to the four joint positions. It must go from
an initial horizontal configuration to a vertical configuration while avoiding
a circular obstacle with radius 0.5m, centered at $x$=1.5m and $y=1.3$m.

Figure 2: Manipulator trajectory snapshots without (with) the circulation
constraint at left (right) column.

We introduce functions $F_{i}$ to model the constraints and obstacles in
configuration space, as described in Subsection IV-A. The first constraint is
the collision avoidance between the links and the obstacle (i.e., circle). We
consider functions $F_{i}(q)$, $i=1,2,3,4$ that compute the squared distance
(measured in meters square) between each rectangular link and the circle. We
also consider a joint limit of $\pm 360^{o}$ for the first joint and $\pm
120^{o}$ for all the other joints. These constraints also generate obstacles
in the configuration space, which are modeled using functions
$F_{i}(q)=0.02(q_{i}-q_{\min,i})$ and $F_{i}(q)=0.02(q_{\max,i}-q_{i})$
($q_{i}$ measured in degrees) for minimum and maximum joint limits,
respectively. We thus have another 8 functions $F_{i}$. For the softmin, we
used $h=0.04$ and a margin of $\delta=0.15$. We used $\alpha(D)=-0.5D$,
$\beta(D)=0.6-2.25D$, and $V(q)=\sum_{i=1}^{4}0.4|q_{i}-q_{g,i}|^{1.5}$ . We
also use the constraints of the form $A\mu\geq b$ to impose a maximum absolute
value of $1$ to each control input. The employed $\Omega$ matrix is

$\Omega=\left[\begin{array}[]{cccc}\ 0&\
\frac{\sqrt{3}}{3}&-\frac{\sqrt{3}}{3}&\frac{\sqrt{3}}{3}\\\
-\frac{\sqrt{3}}{3}&\ 0&\ \frac{\sqrt{3}}{3}&\frac{\sqrt{3}}{3}\\\ \
\frac{\sqrt{3}}{3}&-\frac{\sqrt{3}}{3}&\ 0&\frac{\sqrt{3}}{3}\\\
-\frac{\sqrt{3}}{3}&-\frac{\sqrt{3}}{3}&-\frac{\sqrt{3}}{3}&0\\\
\end{array}\right].$ (8)

Figure 3: Snapshots of the trajectory of the manipulator in the _configuration
space_.

Figure 2-(Top-Right) shows the path made by the manipulator in six different
timesteps, Figure 2-(Bottom-Right) shows the path made by the junction points
of the links/tip of the manipulator, and Figure 4 shows the evolution of the
distance function, which is always greater than 0. It is important to note
that without the circulation constraint, the system converges to a spurious
equilibrium point when the manipulator touches the circle from below. The
snapshots for the simulation without the circulation constraint can be seen in
the two figures at the Left column of Figure 2. It is also important to
mention that, although the obstacle in the 2D space is very simple (a circle),
the resulting obstacle $\mathcal{C}$ in the configuration space is very
complex. To illustrate this claim, Figure 3 shows snapshots of the obstacle in
the _configuration space_ during the same six timesteps as Figure 2. Since the
configuration space is 4-dimensional, we show 2-dimensional slices of this
space by keeping the same configurations $q_{1}$, $q_{3}$ but changing $q_{2}$
and $q_{4}$. The obstacle region is in gray.

Figure 4: Evolution of the distance function $D$.

###  V-B Multiple robot 3D-motion ($n=15$ case)

In this simulation, we have 5 flying robots for which we can control their
linear velocities. The configuration $q$ is the vector with all their $3D$
positions stacked, and thus $n=15$. They must navigate in a 3D environment,
avoiding collision with each other and the obstacles, and reach their
designated setpoints. The robots are represented by colored spheres, their
setpoints by small cubes, and the obstacles are in black. The robots must
navigate to their respective setpoints while avoiding collisions between
themselves and the environment, as shown in Figure 5 for the starting time.
The obstacles (three tree-shaped obstacles and a wall-shaped obstacle) are
modeled using primitives shapes: spheres, boxes, and cylinders. Overall, the
obstacles are modeled using $3$ spheres, $3$ cylinders, and $4$ boxes,
resulting in $10$ primitive obstacles.

Figure 5: Snapshot of the multi-robot simulation.

Figure 6: Snapshots for the multi-robot simulation.

Obstacles are modeled as spheres with radius $r=0.25m$, whereas for inter-
agent collision, we model them as cylinders with this same radius and infinite
height, with the cylinder axis normal to the ground. In this case, having one
of them going below another is undesirable because the downwash from the top
drone will push the bottom drone downwards. We then have $5\times 10$
functions $F_{i}(q)$ that compute the distance between each robot and each
primitive obstacle, $(5\times 4)/2$ functions $F_{i}(q)$ that compute the
distance between the robots, and finally $6\times 5$ functions $F_{i}(q)$ that
implement limits for the positions of the robots, since they must lie in a
6m$\times$6m$\times$3m box. Overall, we have $90$ functions $F_{i}$ that we
condense into a single function $D$ using the method described in Subsection
IV-A with $h=0.01$ and $\delta=0.13$.

Figure 7: Snapshot of the multi-robot simulation without the circulation
inequality. Figure 8: Evolution of the distance function $D$.

Moreover, $\alpha(D)=-0.5D$, $\beta(D)=0.9-6D$, and
$V(q)=\frac{1}{2}\sum_{i=1}^{5}\|p_{i}-p_{g,i}\|^{2}$, in which
$p_{i}\in\mathbb{R}^{3}$ is the position of the robots and
$p_{g,i}\in\mathbb{R}^{3}$ their respective target positions. There are no
constraints of the form $A\mu\geq b$. Since $n=15$ is odd-dimensional, we must
implement the workaround discussed in Subsection IV-C, where a skew-symmetric,
not orthonormal/invertible matrix $\Omega\in\mathbb{R}^{15\times 15}$ is
employed, such that $\Omega=[N_{2}\ \ {-}N_{1}\ \ N_{4}\ \ {-}N_{3}\ \ldots\
N_{14}\ \ {-}N_{13}\ 0]^{T}$.

Figure 6 shows snapshots of the simulations. In the first row, 5 snapshots are
shown, in sequential timeframes, using a top view of the path of each robots.
In the second row, we show the same timeframes as the first row, but with a
different view. We can see circulation-like behaviors that are essential to
accomplish the task and that would not be possible only with the CBF
constraint. For example, the yellow robot in the fourth snapshot circulates
the cross-shaped obstacle in order to reach its setpoint.

Indeed, Figure 7 shows the result of applying the controller without the
circulation inequality, in which we can see that the robots get stuck into a
spurious equilibrium point. Figure 8 shows the evolution of the positive
function $D$.

##  VI Conclusion

An optimization framework was proposed for controlling a system with
integrator dynamics to a goal in the configuration space while considering
obstacle avoidance in addition to other constraints. The novelty of the
framework is the introduction of a circulation inequality that forces the
system to circulate an obstacle - in the configuration space - when
sufficiently near it. The motivation for the approach is that often CBF-QP
formulations are prone to reach spurious equilibrium points instead of the
desired goal. The introduced inequality allows us to extend the range of
situations in which the approach is successful with a negligible increase in
the complexity of the formulation. We showed several formal results for the
controller including continuity, feasibility, and classification of the
equilibrium points, and also demonstrated the efficacy of the approach using
simulation studies.

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##  Appendix A Appendix A: Earlier Versions of XAIR

We provide the initial versions of the framework that were used at the
beginning of the three iterative workshops (from Fig. 10 to Fig. 12). These
examples show how XAIR improved throughout the workshops.

Figure 10. Version 1 before The 1rd Iterative Expert Workshop (Study 2)

Figure 11. Version 2 before The 2rd Iterative Expert Workshop (Study 2). Main
updates from Version 1: (When) Add dimensions and update the connection
between the key factors and the dimensions. (What) Add User Goal and User
Profile into the content type table. (How) Reorganize according to the
dimensions and simply the structure.

Figure 12. Version 3 before The 3rd Iterative Expert Workshop (Study 2). Main
updates from Version 2: (When) Update the connection between the key factors
and the dimensions. (What) Simplify the structure and provide default option
as guidance. (How) Remove the “location” dimension and improve the visual
design.

##  Appendix B Appendix B: Details of Application Scenarios

###  B.1. Explanation Details for The Two Applications

We present a more structured summary of the two scenarios in Sec. 6, together
with examples of all explanation content type (Tab. 3).

|  |
---|---|---
Scenario Info |  Scenario 1: Route Suggestion |  Scenario 2: Plant Fertilization Reminder
Scenario |  Nancy (AI expert, high AI literacy) is jogging in the morning on a quiet trail. Since it is the cherry-blossom season and Nancy loves cherries, her AR glasses display a map beside her and recommend a detour. Nancy is surprised since this route is different from her regular one, but she is happy to explore it. She is also curious to know the reason this new route was recommended. |  Sarah (general end-user, low AI literacy) was chatting with her neighbor about gardening. After she returned home and sat on the sofa, her AR glasses recommended instructions about plant fertilization by showing a care icon on the plant. Sarah is concerned about technology invading her privacy, and wants to know the reason behind the recommendation.
Platform- Agnostic Key Factors | System Goal |  User Intent Discovery (new route) |  Trust Building (clarification)
User Goal |  Resolve Surprise |  Privacy Awareness
User Profile |  User Preference: Cherry blossom tree lover; History: Regular jogging in the morning; AI Literacy: Expert, high |  User Preference: Plant enthusiast; History: Did not care of the plant for a while; AI Literacy: General end-user, low
AR-Specific Key Factors | Contextual Info |  Location: Outdoor; Time: Morning; Environment: Trails, streets |  Location: Home; Time: Afternoon; Environment: Living room furniture, the plant
User State |  Activity: Jogging; Cognitive Load: Low |  Activity: Sitting on the sofa; Cognitive Load: Low
Explanation Content Type Examples | Input/Output |  This route is recommended based on seasons, your routine and preferences. |  The system checks the plant’s current status by visually scanning the plant.
Why/Why-Not |  The route has cherry blossom trees that you can enjoy. The length of the route is appropriate and fits your morning schedule. |  The plant has abnormal spots on the leaves, which indicates fungi or bacteria infection.
How |  This algorithm finds and ranks possible routes based your location and other people who share similar preferences to you. |  The system checks the plant’s visual appearance, then searches online to find ways to cure it.
Certainty |  Match rate between this route’s condition and your preference: 93% |  The chance of the plant having disease is high (94%).
Example |  These photos captured memories about jogging during cherry blossom season. |  These are some images of other plants with similar symptoms.
What-If |  The recommended route will be a shorter one if you jog in the evening. |  N/A
How-To |  Disable the “season option” to return to the old route. |  N/A

Table 3. Details of The Two Application Scenarios in Sec. 6. “N/A” indicates
that this particular explanation type is not applicable for this case. The
same below.

###  B.2. Additional Application Scenarios

We further applied XAIR to additional everyday AR scenarios to illustrate the
practicability of XAIR. The four scenarios cover extra indoor & outdoor
recommendations (Tab. 4), as well as AR-based intelligent instructions and
automation aside from recommendations (Tab. 5).

|  |
---|---|---
Scenario Info |  Scenario 3: Food Rec for A Movie Night |  Scenario 4: Podcast Rec while Driving
Scenario |  Emma (general end-user, low AI literacy) has a few friends over for a small party. They decide to watch a Bollywood movie and now they are about to order food. The AR glasses recommends ordering from an Indian restaurant. Mary never heard of this restaurant before, but she loves this idea. She is also curious about the reason of this recommendation. |  Jeff (general end-user, low AI literacy) is about to drive to work. The AR glasses recommends a new podcast “TEDx Shorts” that Jeff is unfamiliar with. However, the topic is interesting and Jeff wants to give it a try. Meanwhile, Jeff is curious to know the reason for this recommendation.
Platform- Agnostic Key Factors | System Goal |  User Intent Assistance (find good food) |  User Intent Discovery (new podcast)
User Goal |  Reliability, Informativeness |  Informativeness
User Profile |  User Preference: Everyone’s food preferences; History: Just decided to watch a Bollywood movie; AI Literacy: General end-user, low |  User Preference: Topic interests; History: Morning driving routine; AI Literacy: General end-user, low
AR-Specific Key Factors | Contextual Info |  Location: Indoor; Time: Evening; Environment: Home with a group of friends |  Location: Outdoor; Time: Morning; Environment: Street conditions
User State |  Activity: Hanging out with friends; Cognitive Load: Low to Medium |  Activity: About to Start Driving to Work; Cognitive Load: High
Explanation Content Type Examples | Input/Output |  This restaurant is recommended based on everyone’s food preference and movie genre. |  The recommendation takes your playlist history and driving routine into account.
Why/Why-Not |  The food fits everyone’s food preferences and matches the genre of the movie you are watching. |  This podcast’s topic is in line with your interest, and its length fits your expected driving time.
How |  The algorithm filters the restaurants by food preferences and then finds the best match between the food and the related activity. |  The algorithm detects that it’s morning and you are driving to work, then recommends the new podcast whose topic may be of interest to you.
Certainty |  Match score between the restaurant and the food preference and the movie: 90% |  The podcast was liked by 85% of people with similar interests as you.
Example |  Last time, everyone enjoyed Chinese food while watching a Chinese movie. |  “The Daily” and “Fresh Air” are other appropriate examples when you drove to work
What-If |  If movie genre is disabled, other cuisines would be recommended. |  If the commute is longer, there are other episodes that may be of interest to you.
How-To |  N/A |  To listen to previous podcasts, you can set history as the main recommendation factor.
XAI Designs in AR: When |  Availability (G1) |  Always available |  Always available
Delivery (G2) |  Auto-trigger as both conditions is met (enough capacity and the user is not familiar with the recommendation). |  Manual-trigger (high cognitive load during driving).
XAI Designs in AR: What |  Content (G3) |  Input/Output & Why/Why-Not |  Input/Output & Why/Why-Not
Detail - Concise (G4) |  The Why part of the explanation examples. |  The Why part of the explanation examples.
Detail - Detailed (G5) |  A list of the two explanation content types, plus images of the movie and food to support the Why part. |  A list of the two explanation types.
XAI Designs in AR: How |  Modality (G6) |  Visual modality. |  Audio modality
Paradigm - Format (G7) |  Textual format, plus graphical format in the detailed explanations. |  N/A
Paradigm - Pattern (G8) |  Explicit pattern, presenting texts in the same window as the recommendations. |  N/A

Table 4. Additional Application Examples of XAIR on Two Recommendation
Scenarios.

|  |
---|---|---
Scenario Info |  Scenario 5: Cooking Instructions |  Scenario 6: Automatic Do-Not-Disturb Mode
Scenario |  Lisa (general end-user, low AI literacy) has recently been learning how to cook. She wants to try out a new recipe for today’s lunch. She picks “Poached Egg on Avacado Toast” and starts to follow the instructions. After she takes the eggs out of the fridge, the AR glasses prompts to boil the egg for 1 min. Lisa is curious about the time recommendation and wants to understand what the prompt is based on. |  Jeff (general end-user, low AI literacy) is about to drive to work. The AR glasses recommends a new podcast “TEDx Shorts” that Jeff is unfamiliar with. However, the topic is interesting and Jeff wants to give it a try. Meanwhile, Jeff is curious to know the reason for this recommendation.
Platform- Agnostic Key Factors | System Goal |  User Intent Assistance (learn the recipe) |  User Intent Discovery (new podcast)
User Goal |  Reliability |  Informativeness
User Profile |  User Preference: The purpose of learning how to cook; History: Following this instruction for the first time; AI Literacy: General end-user, low |  User Preference: Topic interests; History: Morning driving routine; AI Literacy: General end-user, low
AR-Specific Key Factors | Contextual Info |  Location: Kitchen; Time: Noon; Environment: Cookwares and ingredients |  Location: Outdoor; Time: Morning; Environment: Street conditions
User State |  Activity: Cooking; Cognitive Load: High |  Activity: About to Start Driving to Work; Cognitive Load: High
Explanation Content Type Examples | Input/Output |  The guidance is based on the instruction and your current stage. |  Last week you turned on smart do-not-disturb mode. The mode is based on your location, time, and your ongoing activity.
Why/Why-Not |  Boiling eggs for one minute will result in soft-boiled eggs with slightly firm whites and a runny egg yolk. This is how people prefer soft-boiled eggs with toast. |  This setting automatically blocks notifications when you are at the office during the working hour and working on the laptop.
How |  The algorithm detects your activity and recognizes which stage you are in, then it provides the guidance for the next step. |  The system detects your current context and activity, and checks whether they meet your authored settings. If so, the Do-Not-Disturb mode will be turned on.
Certainty |  The recognition of activity has a high certainty (88%). |  The recognition of the time, location and current activity has a high certainty of 92%.
Example |  N/A |  N/A
What-If |  Other possible ways of cooking eggs, such as scrambled eggs, if you want to explore other recipe instructions. |  When you are not in the office, or it is out of working hours, or you are not working in front of the laptop, the setting will not be turned on.
How-To |  N/A |  You can update any of the three conditions to change the moment the setting it’s activated.
XAI Designs in AR: When |  Availability (G1) |  Always available |  Always available
Delivery (G2) |  Manual-trigger (high cognitive load during cooking). |  Manual-trigger (limited capacity in office).
XAI Designs in AR: What |  Content (G3) |  Input/Output & Why/Why-Not |  Input/Output, Why/Why Not, How to, Confidence, How
Detail - Concise (G4) |  The Why part of the explanation examples. |  A summary of Input, Why, and How-To as the user gets confused and wants to change the output.
Detail - Detailed (G5) |  A list of the two explanation types, plus images of the soft-boiled eggs to support the Why part. |  A list of the five types.
XAI Designs in AR: How |  Modality (G6) |  Visual modality. |  Visual modality
Paradigm - Format (G7) |  Textual format, plus graphical format in the detailed explanations. |  Textual format
Paradigm - Pattern (G8) |  Explicit pattern, presenting texts in the window besides the timer. |  Explicit pattern, presenting texts in front of the user.

Table 5. Additional Application Examples of XAIR on Intelligent Instructions
and Automation.

###  B.3. Explanation Details of the Scenarios in Study 3 & 4

Tab. 6 shows the explanation examples presented to designers in Study 3.

|  |
---|---|---
Scenario Info |  Case 1: Reliable Recipe Recommendation |  Case 2: Wrong Recipe Recommendation
Explanation Content Type Examples | Input/Output |  This recipe comes from the items detected in the fridge: egg and shrimp, and take your diet into account. |  This recipe is based on friends’ food preferences and the detected ingredients in your fridge: salmon and carrot.
Why/Why-Not |  This recipe fits your diet and food preference. It is recommended based on the rich amount of protein: 32g. |  This recipe matches your friends’ preference. It is recommended based on the popularity: 3201 people liked it.
How |  The algorithm recognizes ingredients in the fridge, finds and ranks recipes based on the available ingredients and your diet preference. |  The algorithm first recognizes ingredients in the fridge, finds and ranks recipes based on the available ingredients and food preference.
Certainty |  Match rate between the recipe and the food preference & ingredients : 82%. |  The recognition of salmon is uncertain (confidence 71%). It is not sure whether salmon or steak (recognition confidence 29%).
Example |  N/A |  N/A
What-If |  More recipes if you want to try other cuisines. |  Different recipes if your friends want to try other cuisines.
How-To |  Disable the diet option to see previous recipes before you went on the high-protein diet. |  Select the right ingredients to change the recommendations: salmon or steak [clickable buttons].

Table 6. Details of The Two Cases in Sec. 7. Examples are “N/A” as they are
already multiple examples in the recommendations.