Deep Koopman with Control Spectral Analysis of Soft Robot Dynamics Naoto Komeno1 Brendan Michael1 Katharina K uchler12 Edgar Anarossi1and Takamitsu Matsubara1 Abstract Soft robots are challenging to model and control as

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Deep Koopman with Control: Spectral Analysis of Soft Robot Dynamics

Naoto Komeno1,∗, Brendan Michael1,∗, Katharina K¨

uchler1,2, Edgar Anarossi1and Takamitsu Matsubara1

Abstract— Soft robots are challenging to model and control as

inherent non-linearities (e.g., elasticity and deformation), often

requires complex explicit physics-based analytical modelling

(e.g., a priori geometric deﬁnitions). While machine learning

can be used to learn non-linear control models in a data-

driven approach, these models often lack an intuitive internal

physical interpretation and representation, limiting dynamical

analysis. To address this, this paper presents an approach

using Koopman operator theory and deep neural networks to

provide a global linear description of the non-linear control

systems. Speciﬁcally, by globally linearising dynamics, the

Koopman operator is analyzed using spectral decomposition

to characterises important physics-based interpretations, such

as functional growths and oscillations. Experiments in this

paper demonstrate this approach for controlling non-linear

soft robotics, and shows model outputs are interpretable in

the context of spectral analysis.

I. INTRODUCTION

Linear control theory is well suited to developing in-

terpretable control frameworks, through exploration of the

spectral components, i.e., eigenvectors and eigenvalues, of

the associated dynamical system. Spectral analysis can help

determine system stability [1], or provide additional insight

for techniques such as ﬁltering [2]. However, application to

non-linear systems is ill-suited, due to the absence of a linear

evolution of the dynamics, resulting in sub-optimal solutions

and poor control applications.

In particular, soft robots with non-linear properties

(e.g., elasticity) suffer from difﬁculties in both modelling

and control (e.g., unpredictable behaviour due to the high

degree of freedom). As such, predictive control often requires

physics based modelling [3], including analytical descrip-

tions of the geometries [4]. However, dynamical analysis

of non-linear systems remains challenging, and is generally

limited to systems with closed-form derivations (e.g., double

pendulums [5]).

As an alternative to analytical modelling, machine learning

can be employed to learn predictive models of the non-linear

dynamical system, solely through observations of the envi-

ronment. While there exists a large body of work exploring

machine learning methods for soft robots [6], a common

limitation is the lack of interpretability and explainability of

models [7]. Speciﬁcally, in the context of learning predictive

models for control, black-box machine learning systems [8]

may only locally linearise the dynamics [9], thereby not

1Robot Learning Lab, Nara Institue of Science and Technology, Japan

2Institute of Knowledge Based Systems Group, RWTH Aachen Univer-

sity, Germany

∗These authors contributed equally to this work

This work was supported by JSPS KAKENHI Grant Numbers

JP19H01124 and JP22J11687.

Learn globally

linear dynamics

with Koopman theory

Fig. 1: Stabilization of a ﬂexible polyurethane arm. (a)

Complex non-linear robot dynamics are mapped to set of (b)

linearised latent dynamics (polar co-ordinates), via Koopman

operator theory, becoming amenable to spectral analysis.

capturing intrinsic important global physical properties of

the system. This both limits dynamics analysis of the learnt

model, and reduces conﬁdence in model generalisability.

To address this, this paper proposes an approach to con-

trolling and interpreting non-linear soft robotics by learning

globally linearized dynamics models via Koopman operator

theory [10], and applying this to model predictive controllers.

Speciﬁcally, Deep Koopman Networks (DKN) [11] is used

to learn parsimonious dynamics models with control, that

expresses linear dynamics interpretations in the context of

spectral analysis. Prior work applying Koopman theory to

robotics [12]–[15] is limited to improving prediction and

control performance, without considering the interpretability

of the model. This paper presents the ﬁrst evidence of DKN

in soft robotics for non-linear control with dynamics analysis

contextualised within the linear control domain.

Experiments apply the approach to a soft robot system

(soft inverted pendulum) for both modeling the dynamics,

and stabilisation control. Results show an improved control

performance over standard deep learning for a soft ﬂexible

stabilisation task, and models display clear physical interpre-

tations of the dynamic system.

II. BACKGROUND

Koopman operator theory [10] is a dynamical systems

formulation, that provides a global description of non-linear

dynamics, in terms of the linear evolution of a set of observ-

able functions. Speciﬁcally, instead of attempting to model

the non-linear dynamical state space (e.g., positions and

velocities), Koopman operator theory alternatively describes

the dynamical system in terms of the linear evolution of a

arXiv:2210.07563v1 [cs.RO] 14 Oct 2022

set of state-dependent observable functions, forming a latent

linear dynamical system [16].

A. Koopman operator theory

To present the role of Koopman operator theory in dy-

namical systems analysis, a discrete-time dynamical system

is ﬁrst formulated:

xn+1 =F(xn),(1)

where nis the time-index, F:X 7→ X is the ﬂow map,

and Xis a ﬁnite-dimensional metric state-space [16], often

assumed to be a smooth manifold [17] (e.g., Euclidean, X ⊆

RM).

Analysis and modelling of this non-linear Fcan be chal-

lenging. Instead, a reasonable approach is to ﬁnd coordinate

transformations to map from the non-linear dynamics to a

latent linear dynamical system. Koopman operator theory

does this by describing the linear evolution of measurement

functions of the non-linear state [17]:

Kg(xn) = g(F(xn)) = g(xn+1),(2)

where Ktis the Koopman operator, an inﬁnite-dimensional

linear operator, acting on a measurement function gof the

system (also known as an observable.). This observable is

a function of the state, i.e., g:X 7→ C, and an inﬁnite

set of observable functions is deﬁned on a Hilbert space

H[17]. This formulation is often referred to as lifting the

state variables from the ﬁnite non-linear space, to an inﬁnite-

dimensional linear one. Given all observables, Kcontrols

their evolution, i.e., K:H 7→ H. The Koopman operator is

associated with the non-linear state transformation Fthrough

the composition Kg=g◦F,∀g∈ H [16].

B. Koopman theory with control

Koopman operator theory is also applicable to systems

with control inputs, by describing the dynamics of an ex-

tended state space of the product of Xand the space of all

control sequences `(U)[18]. By deﬁning the operator on the

extended space, the Koopman operator remains autonomous

and is equivalent to the operator associated with unforced dy-

namics [19]. Speciﬁcally, considering a non-linear dynamical

system with control u∈ U:

xn+1 =F(xn,un),(3)

a common generalization is to deﬁne the associated Koop-

man operator as evolving an uncontrolled dynamic system

deﬁned by the product F:X ⊗ U 7→ X [18].

As the evolution of Xonly depends on un, the control

is considered as an additional state variable [16], instead

of requiring all control sequences `(U). As such, similar to

Eq. (2), the Koopman operator is described as [16]:

Kg(xn,un) = g(F(xn,un),un+1).(4)

In contrast to linear predictors [18], uis also lifted through

observables galongside the state x.

Applications of Koopman operator theory to domains with

control are are fast appearing in the literature, and generally

focus on strategies for choosing suitable observable functions

[20] or optimal control [18], [21]. For a general review of

Koopman based control frameworks, please see [22]. With

speciﬁc regard to robotics control applications, prior work

has investigated ﬁnding observables [23], LQR [12], [15],

[24] or MPC [25]–[27] control, active learning [13], and

shared human-robot control [28].

C. Spectral analysis and ﬁnite approximations of the Koop-

man operator

Analysis of this inﬁnite-dimensional linear operator is

challenging. However, ﬁnite spectral properties of the opera-

tor are of great importance, as they can outline global prop-

erties of the dynamics [29]. Speciﬁcally, Kcan be spectrally

decomposed into Koopman eigenfunctions φk(.)∈ H\{0}

[16], with corresponding Koopman eigenvalue λk∈C),

which satisﬁes:

Kφk=φk◦F=λkφk.(5)

This Koopman eigenfunction is a linear intrinsic measure-

ment coordinate on which measurements are evolved with a

linear dynamical system [17]. As such, spectral analysis of

this can provide physical intuition of the dynamical system

under investigation.

While there exists a large body of work on ﬁnding analyt-

ical representations of eigenfunctions with knowledge of the

dynamical system [16], [20], data-driven approximations are

increasing prevalent. This is due to the complexity and uncer-

tainty in ﬁnding eigenfunctions, and the modern increase in

computational power for modelling, and availability of large

datasets.

Generally, approximations are often made using the dy-

namic mode decomposition algorithm [17], [30], whereby

spectral components of a linear transition matrix are es-

timated via matrices of state measurements. Speciﬁc im-

plementations that incorporate Koopman operator theory to

handle non-linear transitions include dictionary [31] or deep

learning [32] for ﬁnding observables or eigenfunctions [11],

or utilizing approaches such as time-delay embeddings [33].

III. APPROACH

A. Deep Koopman Network

A promising approach to learning the linearisation via

eigenfunctions and the latent linear dynamics Eq. (5) for

autonomous uncontrolled systems, is the Deep Koopman

Network (DKN) approach [11]. In this, a deep autoen-

coder framework is used to ﬁnd intrinsic latent coordinates

y=φ(x)approximating the Koopman eigenfunctions

φ:RM7→ RP, and associated linear dynamical system

yn+1 =Kyn. This is achieved by using an autoencoder to

learn an encoder φand decoder φ−1, and inner layers to

learn the linear dynamics K. To learn parsimonious models,

continuous spectra dynamics are captured by parameterising

Kby an auxiliary layer, predicting eigenvalues as a function

of y.

An intuitive design constraint for DKN, is to learn la-

tent coordinates ywhich have complex radial symmetry

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DeepKoopmanwithControl:SpectralAnalysisofSoftRobotDynamicsNaotoKomeno1;,BrendanMichael1;,KatharinaK¨uchler1;2,EdgarAnarossi1andTakamitsuMatsubara1AbstractSoftrobotsarechallengingtomodelandcontrolasinherentnon-linearities(e.g.,elasticityanddeformation),oftenrequirescomplexexplicitphysics-basedanal...

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Deep Koopman with Control Spectral Analysis of Soft Robot Dynamics Naoto Komeno1 Brendan Michael1 Katharina K uchler12 Edgar Anarossi1and Takamitsu Matsubara1 Abstract Soft robots are challenging to model and control as

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