# Safety-Critical Ergodic Exploration in Cluttered Environments via Control
Barrier Functions

Cameron Lerch, Ethan Dong, and Ian Abraham All authors are with the Department
of Mechanical Engineering and Materials Science, Yale University, New Haven
06511, USA. Corresponding author<EMAIL_ADDRESS>
###### Abstract

In this paper, we address the problem of safe trajectory planning for
autonomous search and exploration in constrained, cluttered environments.
Guaranteeing safe navigation is a challenging problem that has garnered
significant attention. This work contributes a method that generates
guaranteed safety-critical search trajectories in a cluttered environment. Our
approach integrates safety-critical constraints using discrete control barrier
functions (DCBFs) with ergodic trajectory optimization to enable safe
exploration. Ergodic trajectory optimization plans continuous exploratory
trajectories that guarantee full coverage of a space. We demonstrate through
simulated and experimental results on a drone that our approach is able to
generate trajectories that enable safe and effective exploration. Furthermore,
we show the efficacy of our approach for safe exploration of real-world
single- and multi- drone platforms.

##  I Introduction

In autonomous search and rescue tasks, robots need to plan effective
exploratory trajectories while avoiding potential hazards to ensure continued
operation. Balancing both the effectiveness of search and the safety of the
robot then becomes a challenge as the environment becomes more complex and
cluttered. As available free space is reduced, the ability of the robot to
reason about where to venture next becomes limited and safety becomes a higher
priority leading to ineffective search behaviors. To address this problem, we
present a safe trajectory planning method for autonomous search in
constrained, cluttered environments through integrated development of ergodic
trajectory optimization methods [1, 2, 3] with safety-critical control
approaches [4, 5, 6].

Ergodic trajectory optimization methods, often referred to as ergodic search
[1, 2], have emerged with the guarantee of complete coverage over a space,
irrespective of the spatial scale [7]. These methods cast the problem of
exploration over a space as a continuous trajectory optimization problem over
time-averaged distributional representations of trajectories. The optimization
leverages spectral methods to synthesize continuous exploration trajectories
over arbitrary scales where the average time spent in a region is proportional
to the measure of importance111Often referred to as a measure of information
or an information measure. assigned to the region. In addition, some recent
adaptations of ergodic search methods have shown the ability to avoid
obstacles [3, 8]; however, they do not provide formal guarantees that the
robot will remain in a safe set of states. This leads to trajectories that can
violate safety conditions and risk collisions with obstacles, themselves, or
other robots. Having formal guarantees prevents an imbalance of task priority
which would ultimately place the robot at risk, but we want to ensure that the
task is successful. Therefore, this work develops an integrated method that
ensures both a complete search over a space and that the generated trajectory
remains safe.

Figure 1: Safe, Multi-Robot Ergodic Exploration:  The proposed Safety-Critical
Ergodic Trajectory Optimization (SC-ETO) applied to a multi-robot system in a
cluttered environment. Our approach generates safe exploratory paths that
provide full coverage over a cluttered region while avoiding safely collision
with other robots.

Our approach poses safe trajectory optimization for exploration as a
constrained ergodic trajectory optimization problem. We leverage discrete-time
control barrier functions (DCBFs) as inequality constraints to ensure the
safety of the robot along a trajectory and jointly optimize an ergodic
trajectory subject to robotic motion constraints to enforce effective ergodic
coverage. We show that our approach is able to compute safety-critical
trajectories in cluttered search environments. We demonstrate in both
experiment and simulation that our approach generates safe, ergodic
exploratory trajectories. Lastly, we test the robustness of our method using
simulations and compare our method with existing approaches. Furthermore, we
show that our method is able to generate efficient search trajectories for
scenarios with multi-robot exploration (see Fig. 1).

Thus, in summary, our contributions are:

  1. 1.

A method for safety-critical ergodic trajectory optimization (SC-ETO) that
integrates discrete-time control barrier functions with ergodic search; and

  2. 2.

Demonstrations of our approach on a single- and multi- robot exploration task
in a cluttered environment.

The paper is organized as follows: Section II provides some background on
related work. Section III-A and section III-B present preliminary information
on safety-critical control via control-barrier functions and ergodic search
methods respectively. Section IV derives our proposed method for safety-
critical ergodic exploration. Results and conclusions are then presented in
Section V and Section VI.

##  II Related Work

Safety Critical Control:  Planning safe trajectories for search and
exploration is a fundamental problem in robotics that involves ensuring the
robot remains in a safe set of states throughout operation, from an initial
configuration to a final (goal) configuration [9, 10, 4, 6, 11]. Generally,
these problems take the form of obstacle avoidance that ensures planned
trajectories are safe for the robot to navigate during operation. Within the
literature, control barrier functions (CBFs) are widely used to enforce these
safety-critical constraints on robotic systems [12, 4]. They have been shown
to be an effective way to generate safe trajectories in tight-fitting
cluttered environments of polytopes [13]; and have proven to be useful in a
wide variety of robotic systems (e.g., locomotion [14, 15, 16], automotive
[17, 18, 19], aerial [20, 21], and multi-robot [22, 23, 21]) for both static
and dynamic obstacles [11]. CBFs maintain forward-set invariance, which
ensures that once a robot enters the safe set, it will stay within the safe-
set, thereby ensuring its safety [24, 25].

Within path planning, several state-of-the-art methods utilize variations of
CBFs (Kinodynamic Barrier Functions and Discrete CBFs) as constraints
incorporated into sampling-based motion planners (such as RRT) to generate
safety-critical trajectories [4, 13, 11]. However, these methods typically
focus on point-to-point planning i.e., getting from some starting
configuration to some final configuration without concern for efficiency or
exploratory coverage.

Ergodic Exploration:  Within the context of exploration, there is a need for
algorithms that generate trajectories that are efficient and guarantee
effective search of an environment. Such methods generate trajectories for
robots that spend time exploring areas of interest, while still guaranteeing
the robot explores unseen areas. Recent methods known as ergodic exploration
have been shown to be an effective way to explore a space. Ergodic exploration
methods balance exploration and exploitation by generating trajectories that
spend time in areas proportional to the utility of being in a region. As a
result, ergodic exploration methods have demonstrated improved information-
gathering behavior compared to prior works [26, 27, 2, 28, 29, 30, 2, 31, 32,
33]. However, few works that consider safety within ergodic trajectories [3].

The difficulty lies in generating continuous exploratory trajectories while
simultaneously respecting safety constraints. Earlier works have used the
metric itself to have robots avoid obstacles in the environment [1, 34]
whereas others have used inequality constraints with stochastic optimization
to avoid objects [3]. However, these methods do not explicitly guarantee the
safety of the robot. Rather, inequality constraints that encode distances to
obstacles are satisfied by generating trajectories that brush up close to the
boundaries of the object. In scenarios where the robot is unable to track
these trajectories or there are modeling inaccuracies, the robot may become
unsafe. As a result, real-world implementations of ergodic exploration methods
on robots are sparse due to the lack of guaranteed safety. This work presents
an integrated approach to jointly plan exploratory ergodic trajectories and
provide explicit safety-critical guarantees through control barrier functions
and demonstrates the effectiveness of the approach on a real robotic system.

Figure 2: The zero level set of the generated functions $h(x)$ for all of the
obstacles in the cluttered environment. Positions of the objects are used for
both simulated and real-world experiments.

##  III Preliminaries

In this section, we present brief preliminary information on control barrier
functions for safety-critical control and ergodic exploration methods for
generating exploratory robot trajectories.

###  III-A Safety-Critical Control via Control Barrier Functions

Let us consider the continuous-time robotic system with states
$x\in\mathcal{X}\subset\mathbb{R}^{n}$ and inputs
$u\in\mathcal{U}\subset\mathbb{R}^{m}$ governed by the differential equation

$\dot{x}=f(x,u),$ (1)

where $f:\mathcal{X}\times\mathcal{U}\to\mathcal{X}$ is the dynamics of the
robot and is continuous and differentiable. Next, let us consider the set

$\mathcal{S}=\\{x\in\mathcal{X}\mid h(x)\geq 0\\}$ (2)

where $h:\mathcal{X}\to\mathbb{R}$ is a continuously differentiable function.
The set $\mathcal{S}$ is considered safe if $\forall$
$x\in\partial\mathcal{S}$ where $\partial\mathcal{S}$ is the interior boundary
of the set $\mathcal{S}$.

$\dot{h}(x)=\nabla h(x)\cdot f(x,u)\geq-\gamma(h(x))\hskip 5.69054pt\forall
u\in\mathcal{U}$ (3)

and that $\frac{\partial h}{\partial x}\neq 0$ for some $\mathcal{K}_{\infty}$
function $\gamma$. The function $h$ is then known as a control barrier
function if (3) holds.

For a discrete-time system,

$x_{t+1}=f(x_{t},u_{t})$ (4)

where $f:\mathcal{X}\times\mathcal{U}\to\mathcal{X}$ now evolves the state
$x_{t}$ in time to $x_{t+1}$. The continuous-time CBF expression in Eq. (3)
has been shown to have a discrete-time analog, i.e., a discrete-time control
barrier function (DCBF) [16, 5]

$\Delta h(x_{t},u_{t})\geq-\gamma h(x_{t})$ (5)

for $0<\gamma\leq 1$, and $\Delta
h(x_{t},u_{t})=h(x_{t+1})-h(x_{t})=h(f(x_{t},u_{t}))$. Ensuring that Eq. (3)
holds, we get that $h(x_{t+1})\geq(1-\gamma)h(x_{t})$ and the lower bound of
the DCBF decreases exponentially with the decay rate $\gamma$ [16].

One can tune the effective strength, i.e. the rate of exponential decay of the
DCBF, by varying $\gamma$ between $(0,1]$. As $\gamma$ approaches close to
$1$, the system converges to $\partial\mathcal{S}$ slowly but can easily
become infeasible [13].

Given a valid DCBF $h(x)$ [17] and imposing it as a constraint in Eq. (5) in
an optimization problem could guarantee system safety, i.e., collision-free
trajectories. If a robotic system described by Eq. (1) is safe with respect to
a set $\mathcal{S}\subset\mathcal{X}$, then any trajectory starting inside
$\mathcal{S}$ will remain inside $\mathcal{S}$.

Figure 3:  Single-Robot Safe Exploration: Time-lapse demonstration of a single
drone robot safely navigating an exploratory plan in a cluttered environment.

###  III-B Ergodic Exploration

Let us consider the state the robot at as
$x\in\mathcal{X}\subset\mathbb{R}^{n}$ and the control input to the robot as
$u\in\mathcal{U}\subset\mathbb{R}^{m}$. In addition, let us define the robot’s
workspace as $\mathcal{W}\in[0,L_{0}]\times\ldots[0,L_{v-1}]$ where $L_{i}$
are the bounds of the exploration space and $v\leq n$ is the dimensionality.
Lastly, we define a map $g:\mathcal{X}\to\mathcal{W}$ that is continuous and
differentiable, and maps the robot’s state $x$ to a point in the work space
(e.g., Euclidean space), that is, $g(x)=w$ and $w\in\mathcal{W}$. A trajectory
of the robot $x(t)$ for some time horizon $t_{f}$ is given by solving (1) for
$t\in[0,t_{f}]$ from some initial condition $x_{0}$.

A trajectory is said to be ergodic if its _time-averaged statistics_ (e.g. its
spatial distribution in time) over a workspace. $\mathcal{W}$, is proportional
to some measure $\phi:\mathcal{W}\to\mathbb{R}$ over the workspace[1]222The
measure $\phi$ can encode any information over the space $\mathcal{W}$, and it
follows that $\int_{\mathcal{W}}\phi(w)dw=1$ and $\phi(w)\neq 0$ $\forall
w\in\mathcal{W}$.. For a continuous deterministic trajectory $x(t)$, we define
the ergodic relationship as

$\lim_{t_{f}\to\infty}\frac{1}{t_{f}}\int_{0}^{t_{f}}F(g(x(t)))dt=\int_{\mathcal{W}}\phi(w)F(w)dw$
(6)

for all Lebesgue integrable functions, $F\in\mathcal{L}^{1}$ [7].

We optimize trajectories $x(t)$ and control signals $u(t)$ to minimize the
errors in Eq. 6 through the use of a Fourier transform, where we define the
_ergodic metric_ :

$\displaystyle\mathcal{E}(x(t),\phi)=\sum_{k\in\mathbb{N}^{v}}\Lambda_{k}\left(c_{k}(x(t))-\phi_{k}\right)^{2}$
(7)
$\displaystyle=\sum_{k\in\mathbb{N}^{v}}\Lambda_{k}\left(\frac{1}{t_{f}}\int_{t=0}^{t_{f}}F_{k}(g(x(t)))dt-\int_{\mathcal{W}}\phi(w)F_{k}(w)dw\right)^{2}$

where $F_{k}(w)=\prod_{i=0}^{v-1}\cos(w_{i}k_{i}\pi/L_{i})/h_{k}$ is the
cosine Fourier transform for the $k^{\text{th}}$ mode, and $h_{k}$ is a
normalization factor [2]. The subsequent trajectory optimization problem is
then defined as:

Method | Success %
---|---
Safety-Critical ETO | 100.0 %
ETO w/ Constr. | 38.0%

Figure 4: Monte-Carlo Analysis: Safety comparison between paths planned by a
Safety Critical Ergodic Trajectory Optimization (SC-ETO) and by Ergodic
Trajectory Optimization with constraints (ETO w/ constraints). We sample
uniformly 50 pairs of random initial and final target positions. SC-ETO
generates collision-free ($100\%$ success) and safe paths while ETO with only
inequality constraints $h(x)$ without CBF condition (5) succeeds only 38% of
the time.

Ergodic Trajectory Optimization:

$\displaystyle\min_{x(t),u(t)}\mathcal{E}(x(t),\phi)+\int_{0}^{t_{f}}u(t)^{\top}Ru(t)dt$
(8) $\displaystyle\text{s.t. }\dot{x}=f(x,u),x\in\mathcal{X},u\in\mathcal{U}$
$\displaystyle x(t)=\bar{x}_{0},x_{t_{f}}=\bar{x}_{f},g(x)\in\mathcal{W}$

where $\bar{x}_{0}$ and $\bar{x}_{f}$ are initial and final conditions, and
$R$ is a positive semi-definite matrix that weights control effort.

Figure 5: Multi-Robot Safe Ergodic Exploration: (a) Time-lapse of a multi-
robot ergodic search of a cluttered exploration space and (b) the
corresponding planned ergodic trajectories for each robot. The four robots’
trajectories are optimized jointly in a single SC-ETO (Eq. 12). Two pairs of
drones start at opposite corners from each other and are tasked to navigate
the cluttered environment, ending at the opposite corner from where they
started. Inter-robot CBFs are used to avoid colliding while exploring. Please
view the attached multimedia to view a demonstration of this example.

##  IV Safety-Critical Ergodic Exploration

Integrating control barrier functions with ergodic exploration requires that
we re-establish the optimization in Eq. (8) in discrete time due to the
intractability of dealing with continuous, infinite trajectories. We begin by
discretizing the state trajectory
$\mathbf{x}=x_{0:T-1}=[x_{0},x_{1},\ldots,x_{T-1}]$, where $x_{t}$ is obtained
through Eq. (4), and the control
$\mathbf{u}=u_{0:T-1}=[u_{0},u_{1},\ldots,u_{T-1}]$. The definition of
ergodicity in Eq. (7) can then be redefined for a discrete-time trajectory as:

$\lim_{T\to\infty}\frac{1}{T}\sum_{t=0}^{T-1}F(g(x_{t}))=\int_{\mathcal{W}}\phi(w)F(w)dw$
(9)

for discrete time horizon $T$. With the time-averaged statistics now defined
over a discrete sum, utilize the same Fourier transform as before and obtain
the following ergodic metric:

$\displaystyle\mathcal{E}(\mathbf{x},\phi)=\sum_{k\in\mathbb{N}^{v}}\Lambda_{k}\left(c_{k}(\mathbf{x})-\phi_{k}\right)^{2}$
(10)
$\displaystyle=\sum_{k\in\mathbb{N}^{v}}\Lambda_{k}\left(\frac{1}{T}\sum_{t=0}^{T-1}F_{k}(g(x_{t}))-\int_{\mathcal{W}}\phi(w)F_{k}(w)dw\right)^{2}$

The discrete-time variation of ergodic trajectory optimization in Eq. (8) is
now given by:

Discrete-time Ergodic Trajectory Optimization (ETO)

$\displaystyle\min_{\mathbf{x},\mathbf{u}}\mathcal{E}(\mathbf{x},\phi)+\sum_{0}^{T-1}u_{t}^{\top}Ru_{t}dt$
(11) $\displaystyle\text{s.t.
}x_{t+1}=f(x_{t},u_{t}),x_{t}\in\mathcal{X},u_{t}\in\mathcal{U}$
$\displaystyle x_{0}=\bar{x}_{0},x_{T-1}=\bar{x}_{f},g(x)\in\mathcal{W}$

Using the discrete-time ETO we integrate the discrete-time CBF into the
optimization problem. We include the discrete-time CBF condition in Eq. (5)
into the ETO in Eq. (11), letting $h(x)$ be a valid barrier function with a
defined $\gamma$ to derive the following safety-critical ETO:

Safety-Critical Ergodic Trajectory Optimization (SC-ETO)

$\displaystyle\min_{\mathbf{x},\mathbf{u}}\mathcal{E}(\mathbf{x},\phi)+\sum_{0}^{T-1}u_{t}^{\top}Ru_{t}dt$
(12) $\displaystyle\text{s.t.
}x_{t+1}=f(x_{t},u_{t}),x_{t}\in\mathcal{X},u_{t}\in\mathcal{U}$
$\displaystyle x_{0}=\bar{x}_{0},x_{T-1}=\bar{x}_{f},g(x)\in\mathcal{W}$
$\displaystyle\Delta h(x_{t},u_{t})\geq-\gamma h(x_{t})$

where the DCBF is introduced into the problem as an inequality constraint.

Solutions that satisfy Eq. (12) result in optimized trajectories that are
guaranteed to be safe and are ergodic with respect to a desired measure $\phi$
over an exploration space $\mathcal{W}$.

Figure 6: Optimized trajectories for various $\gamma$ values: Each trajectory
represented (blue dashed line) was optimized for the same initial and final
conditions in the same exploration space. As $\gamma$ increases, the more the
trajectories become ergodic and follow closely to the objects. The lower the
value of $\gamma$ the more cautious the trajectories become.  Figure 7:
Ergodicity value for various $\gamma$: We evaluated ten different values of
$\gamma$, ranging from 0-1. Depicted above is a trend that shows that
ergodicity decreases as $\gamma$ increases (e.g. there is more coverage of the
exploration space for smaller values of gamma).

##  V Results

We demonstrate the effectiveness of our approach for safety-critical ergodic
exploration using a drone robotic system in a cluttered environment through
two means of validation: 1) simulated results, and 2) empirical evaluations.
For both simulated and experimental results, we assume full knowledge of the
obstacle locations and shapes in the environment (as illustrated in Fig. 2)
and define a uniform measure of information $\phi$ over the exploration space
prior to trajectory optimization. We use a single integrator dynamics model
when carrying out the trajectory optimization outlined in Eq. (11). The drone
dynamics are fully simulated with a proportional, integral, derivative (PID)
controller, converting planned trajectories into low-level motor commands.

For all obstacles and drones, we define $h(x)$ as a distance function:

$h(x)=\Big{\|}\frac{x-\bar{x}}{\ell+b}\Big{\|}_{p}-r$ (13)

where $\ell\in\mathbb{R}^{n}$ is a scaling factor, $\bar{x}$ is the center of
the obstacle, $r$ is a radius term, $b$ is a buffer, and $p\geq 2$ defines the
shape of the norm (e.g., $p=4$ is more square-like). These parameters vary
according to the obstacle’s dimension and shape, and are kept the same for
both simulation and experiment. We use a time horizon of $T=200$ steps with a
$\Delta t=0.1$ which results in a total of $t_{f}=20s$ prediction horizon.

###  V-A Single-Drone Experimental Results

Our experiments are conducted using a Crazyflie 2.0 drone. We gather global
position data using two IR base stations and communicate target positions with
the drones via radio communication.

As shown in Fig. 3, we demonstrate that our approach generates safe, ergodic
exploratory trajectories that can be run on a real-robotic system in a
cluttered environment. SC-ETO is carried out using predefined initial and
final target positions. CBFs are built around each obstacle to ensure the
drone navigates safely around the environment.

The drone tracks the optimized trajectory through the cluttered exploration
space at a rate of 10Hz. A time-lapse of a single-drone flight is shown in
Fig. 3. The drone is able to safely navigate the cluttered environment,
avoiding collisions with obstacles and exploring the area ergodically.

###  V-B Muli-Drone Exploration Experiment:

We additionally demonstrate our method on a multi-drone example (see Fig. 5).
A total of four drones are flown simultaneously that are tasked to safely
explore the space defined in Fig. 2 without colliding with one another. Two
pairs of drones are placed across the opposite sides of the cluttered
environment and tasked to navigate to the opposite side of the environment
while ergodically exploring. Each pair-wise drone defines a CBF as a minimum
distance function equal to the width of the drone to avoid collision. As shown
in Fig. 5, the drones are able to safely navigate and explore the environment
without collision between themselves and obstacles. Note that in this work, we
do not address the computational complexity of the multi-drone exploration
problem (for a connected graph we need $N(N-1)/2$ pair-wise constraints for
$N$ robots), and leave this to future work.

###  V-C Simulated Monte-Carlo Analysis

To investigate the robustness of our SC-ETO method, we sample 50 randomly
generated initial and final target positions in the same environment. We
simulated the drone dynamics using the pybullet gym-pybullet-drones
environment [35] and inspect collisions during the execution of the planned
paths by the drone.

We compare our SC-ETO method against ETO without the CBF constraint inequality
in Eq. (3), which is replaced with the obstacle distance function $h(x)$. Our
results are presented in the table in Fig. 4. We find that our method
generates safe ergodic trajectories with a 100 percent success rate (no
collisions). In contrast, while ETO generates collision-free trajectories
using $h(x)\geq 0$, the resulting trajectories are considered unsafe according
to Eq. 3. As a consequence, a drone tracking this trajectory results in unsafe
exploration that collides with objects in the environment and only succeeds
$38\%$ of the time.

###  V-D Ablation Study

We perform an ablation study on the CBF parameter $\gamma$ to analyze its
effects on the efficacy of ergodic exploration. We evaluate 10 different
values of $\gamma$ between $0$ and $1$. As we increase the value of $\gamma$,
we find that the ergodic trajectories become less cautious and more ergodic as
shown in Fig. 6. This was further reinforced by calculating the ergodicity
values which we found to decrease (more coverage of a space) as $\gamma$
decreased. This effect can be attributed to the effective decay rate of the
CBF as $\gamma$ decreases forcing the optimized trajectories to remain further
away from the barrier. Ultimately, this behavior shows a trade-off between
safely navigating an environment and completely exploring all areas of the
environment. The benefit of our approach is that we obtain this trade-off
through a single value that can be tuned.

##  VI Conclusion and Future Work

In this paper, we demonstrate safe and effective planning for exploration
through the development of safety-critical ergodic trajectory optimization.
Simulated results show the robustness of our approach as a planner for
generating safe ergodic exploratory trajectories. Empirical evaluations
demonstrate the effectiveness of our approach for safe single- and multi-
drone exploration in a cluttered environment. Future work will focus on
implementing these techniques for dynamic obstacles via model-predictive
control (MPC) and integrate a more accurate model of the robot dynamics with
an arbitrary number of robots.

## Acknowledgments

The authors would like to thank Yale’s Center for Collaborative Arts and Media
for their multi-media resources and Samuel Osborne for his assistance with
video editing.

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