Data-Efﬁcient Characterization of the Global Dynamics of Robot Controllers with Conﬁdence Guarantees Ewerton R. Vieira35 Aravind Sivaramakrishnan1 Yao Song6 Edgar Granados1

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Data-Efﬁcient Characterization of the Global Dynamics

of Robot Controllers with Conﬁdence Guarantees

Ewerton R. Vieira3,5, Aravind Sivaramakrishnan1, Yao Song6, Edgar Granados1,

Marcio Gameiro2, Konstantin Mischaikow2, Ying Hung6, and Kostas E. Bekris1

Abstract— This paper proposes an integration of surrogate

modeling and topology to signiﬁcantly reduce the amount of

data required to describe the underlying global dynamics of

robot controllers, including closed-box ones. A Gaussian Process

(GP), trained with randomized short trajectories over the state-

space, acts as a surrogate model for the underlying dynamical

system. Then, a combinatorial representation is built and used

to describe the dynamics in the form of a directed acyclic

graph, known as Morse graph. The Morse graph is able to

describe the system’s attractors and their corresponding regions

of attraction (RoA). Furthermore, a pointwise conﬁdence level

of the global dynamics estimation over the entire state space is

provided. In contrast to alternatives, the framework does not

require estimation of Lyapunov functions, alleviating the need

for high prediction accuracy of the GP. The framework is suit-

able for data-driven controllers that do not expose an analytical

model as long as Lipschitz-continuity is satisﬁed. The method

is compared against established analytical and recent machine

learning alternatives for estimating RoAs, outperforming them

in data efﬁciency without sacriﬁcing accuracy. Link to code:

https://go.rutgers.edu/49hy35en

I. INTRODUCTION

Multiple tools have been developed to estimate the region

of attraction (RoA) of a dynamical system [1]–[3]. These

tools are useful for understanding the conditions under which

a controller can be safely applied to solve a task. Finding the

true RoA of a controlled system is challenging. Thus, many

efforts try to estimate the largest possible set contained in the

true RoA. For closed-box systems, such as learned controllers

that do not provide an analytical expression, it is impractical

to apply Lyapunov methods directly. Many non-Lyapunov

methods often have signiﬁcant data requirements so as to

estimate RoAs effectively.

This paper addresses the problem of ﬁnding RoAs of

controllers with unknown dynamics by proposing an efﬁcient

way to use data. It explores surrogate modeling together with

topological tools not only to identify the RoA for a speciﬁc

goal region but also to describe the global dynamics. This

also includes data-driven controllers, where a key challenge

in their application is veriﬁcation, i.e. explaining when the

controller works and when it fails. To achieve this objective,

this work uses Gaussian Processes (GPs) as surrogate models

1Dept. of Computer Science, Rutgers, NJ, USA. 2Dept. of Mathematics,

Rutgers, NJ, USA. 3DIMACS, Rutgers, NJ, USA. 4ICMC, Universidade

de S˜

ao Paulo, S˜

ao Carlos, S˜

ao Paulo, Brazil. 5IME, Universidade Federal

de Goi´

as, Goiˆ

ania, GO, Brazil. 6Dept. of Statistics, Rutgers University, NJ,

USA. E-mail: {er691,kb572}@rutgers.edu.

This work is supported in part by NSF HDR TRIPODS award 1934924.

MG and KM were partially supported by NSF under awards DMS-1839294,

DARPA HR0011-16-2-0033, and NIH R01 GM126555. MG was partially

supported by CNPq grant 309073/2019-7.

Fig. 1. Overview of the proposed framework: 1) initial data collection; 2)

a Gaussian Process (GP) is trained as a surrogate model; 3) computation of

Morse Graph and the Region of Attraction (RoA) for veriﬁcation. For an

optional reﬁnement, steps 1-3 can be repeated as necessary.

to compute a Morse graph, which constructs a ﬁnite, com-

binatorial representation of the state space given access to a

discrete-time representation of the dynamics. It achieves data

efﬁciency and improved accuracy relative to alternatives that

are either analytical tools (and can only be used for analytical

systems) or learning-based frameworks. Fig. 1 highlights the

iterative nature of the approach.

In particular, the key contribution of this work is the use

of GPs as a statistical surrogate model of the underlying

controlled system, alongside the Morse Graphs framework

to compactly describe the global dynamics. The integration

results in data efﬁciency: signiﬁcantly fewer samples of

the underlying dynamics are necessary for an informative

representation of the global dynamics. Data efﬁciency in sur-

rogate modeling is achieved by leveraging the effectiveness

of Morse Graphs, alleviating the high prediction accuracy

requirements typically required for this purpose.

Furthermore, this integration allows working with tra-

jectories that may not uniformly cover the state space of

the underlying system. Prior efforts with topological tools

and combinatorial decompositions of the underlying state

space required sampling the dynamics uniformly over a grid-

based discretization of the state space. A GP allows an

incremental approach where the collection of additional data

points is guided to minimize the uncertainty about the global

dynamics. Additionally, GPs provide conﬁdence levels on the

accuracy of the results at each subset of the state space.

arXiv:2210.01292v1 [cs.RO] 4 Oct 2022

II. RELATED WORK

Numerical methods that estimate the RoA given a closed-

form expression of the system dynamics include maximal

Lyapunov functions (LFs) and linear matrix inequalities

(LMIs). Ellipsoidal RoA approximation via LMIs [4], [5] has

been used for mobile robots [6], [7], and LMI relaxations can

also approximate the RoA of polynomial systems [8]. LFs

constructed by restricting them to be sum-of-squares (SoS)

polynomials [9] have been used in building randomized trees

with LQR feedback [10], funnel libraries [11] and stability

certiﬁcates for rigid bodies [12].

Reachability analysis [3], i.e., computing a backward

reachable tube to obtain the RoA without shape imposition,

for computing RoAs of dynamical walkers [13], has been

combined with machine learning to maintain safety over

a given horizon [14]. GPs can learn barrier functions for

ensuring the safety of unknown dynamical systems [15].

Similarly, barrier certiﬁcates (BCs) can identify areas for

exploration to expand the safe set [16].

Machine learning can learn LFs by alternating between

a learner and a veriﬁer [17], [18], or via stable data-driven

Koopman operators [19]. Rectiﬁed Linear Unit (ReLU) acti-

vated neural networks can learn robust LFs for approximated

dynamics [20]. The Lyapunov Neural Network [21] can

incrementally adapt the RoA’s shape given an initial safe set.

As an alternative, GPs can obtain a Lyapunov-like function

[22], or an LF can be synthesized to provide guarantee’s on

a controller’s stability while training [23].

GPs are a popular choice to reduce data requirements

while modeling dynamical systems [24]. Some of their

applications in robotics include model-based policy search

[25], modeling non-smooth dynamics of robots with contacts

[26], and stabilizing controllers for control-afﬁne systems

[27]. For RoA estimation problems, given an initial safe set

computed using a Lyapunov function, a GP can approximate

the model uncertainties on a discrete set of sampling points

from the safe region while expanding it [28].

Topology has multiple applications in robotics, such as

deformable manipulation and others [29]–[34]. Morse theory

can help incrementally build local minima trees for multi-

robot planning [35] and ﬁnds paths to cover 2D or 3D

spaces [36]. In recent work [37], Morse graphs are shown

to be effective in compactly describing the global dynamics

of a control system without an analytical expression of

its dynamics. To the best of the authors’ knowledge, the

current work is the ﬁrst to apply surrogate modeling with

uncertainty quantiﬁcation in conjunction with topological

tools to identify the global dynamics of robot controllers.

III. PROBLEM SETUP

This work aims to provide a data-efﬁcient framework for

the analysis of global dynamics of robot controllers based on

combinatorial dynamics and order theory [37]–[40]. Consider

a non-linear, continuous-time control system:

˙x=f(x, u),(1)

where x(t)∈X⊆RMis the state at time t,Xis a compact

set, u:X7→ U⊆RMis a Lipschitz-continuous control

as deﬁned by a deterministic control policy u(x), and f:

X×U7→ RMis a Lipschitz-continuous function. Neither

f(·)nor u=u(x)are necessarily known analytically. For

a given time τ > 0, let φτ:X→Xdenote the function

obtained by solving Eq. (1) forward in time for duration τ

from everywhere in X. A trajectory (or an orbit) is deﬁned

as a sequence of states obtained by integrating Eq. 1 forward

in time.

The analysis of the global dynamics can reveal the sys-

tem’s attractors, which include ﬁxed points, such as a state

that the control law manages to bring the system to; or limit

cycles, such as a periodic behavior of the system. It will also

reveal a Region of Attraction (RoA) which is a subset of the

basin of attraction of an attractor A. The basin of attraction

is the largest set of points whose forward orbits converge to

A, or more formally, the maximal set Bthat has the property:

A=ω(B) := \

n∈Z+

cl ∞

[

k=n

φk

τ(B)!

where φk

τis the composition φτ◦ · · · ◦ φτ(ktimes) and cl

is topological closure.

Since fand uare Lipschitz-continuous, φτis too; fur-

thermore any RoA of Eq. (1) is an RoA under φτ. Hence, it

is possible to study Eq. (1) by analyzing the behavior of the

dynamics according to φτ, which is not assumed, however,

to be computable and available.

IV. TOPOLOGICAL FRAMEWORK AND UNCERTAINTY

QUANTIFICATION VIA GPS

There are two key components for capturing meaningful

conditions of the dynamics according to φτ. First, identifying

effective combinatorial representations of the attractors and

maximal RoAs. And second, to achieve data efﬁciency by

employing GP-based surrogate modeling with uncertainty

quantiﬁcation.

Morse Graphs for Understanding Global Dynamics:

Fig. 2(left) is used as a running 1-dim. example. The function

φτis ﬁrst approximated by decomposing the state space X

into a collection of regions X, for instance, by deﬁning

agrid. Fig. 2(left) shows a grid on the interval [−3,3]

decomposed into sub-intervals athrough e. Given a region

ξ∈ X (a cell), the system is forward propagated for

multiple initial states within ξfor a time τto identify regions

reachable from ξ. Consider, for example, the sub-interval b

as such a cell ξin Fig. 2(left). The arrows from the boundary

of bdepict the forward propagation of the dynamics where

bmaps to itself given the underlying dynamics

Then, a directed graph representation Fstores each region

in ξ∈ X as a vertex and edges pointing from ξto each

region reachable from ξ. In Fig. 2(left), Fis the graph

containing nodes given by the grid cells ato e. Each edge

represents a pair given by a cell and its image according

to the dynamics. For instance, (b,b)and (b,c)are two

edges added since bboth maps to itself and also maps to c.

Condensing all the nodes belonging to a strongly connected

components (SCCs) of Finto a single node, results in the

condensation graph CG(F). Then, edges on CG(F)reﬂect

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Data-EfcientCharacterizationoftheGlobalDynamicsofRobotControllerswithCondenceGuaranteesEwertonR.Vieira3;5,AravindSivaramakrishnan1,YaoSong6,EdgarGranados1,MarcioGameiro2,KonstantinMischaikow2,YingHung6,andKostasE.Bekris1AbstractThispaperproposesanintegrationofsurrogatemodelingandtopologytosignic...

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Data-Efﬁcient Characterization of the Global Dynamics of Robot Controllers with Conﬁdence Guarantees Ewerton R. Vieira35 Aravind Sivaramakrishnan1 Yao Song6 Edgar Granados1

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