EQUIVARIANT FILTERS ARE EQUIVARIANT PREPRINT Hiya Gada

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EQUIVARIANT FILTERS ARE EQUIVARIANT

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Hiya Gada

Indian Institute of Technology Bombay

Powai, Mumbai, 400 076, India

19d100007@iitb.ac.in

Pieter van Goor

Systems Theory and Robotics Group

Australian National University

ACT, 2601, Australia

Pieter.vanGoor@anu.edu.au

Ravi Banavar

Indian Institute of Technology Bombay

Powai, Mumbai, 400 076, India

banavar@iitb.ac.in

Robert Mahony

Systems Theory and Robotics Group

Australian National University

ACT, 2601, Australia

Robert.Mahony@anu.edu.au

October 26, 2022

ABSTRACT

Observers for systems with Lie group symmetries are an active area of research that is seeing signiﬁcant

impact in a number of practical domains, including aerospace, robotics, and mechatronics. This paper builds

on the theory of the recently proposed Equivariant Filter (EqF), which is a general observer design for systems

on homogeneous spaces that takes advantage of symmetries to yield signiﬁcant performance advantages. It

is shown that the EqF error dynamics are invariant to transformation of the input signal and equivariant as a

parametrised vector ﬁeld. The main theorem shows that two EqF’s with different choices of local coordinates

and origins and with equivalent noise modelling yield identical performance. In other words, the EqF is

intrinsic to the system equations and symmetry. This is veriﬁed in a simulation of a 2D robot localisation

problem, which also shows how the ability to choose an origin for the EqF can yield practical performance

advantages by mitigating ﬂoating point precision errors.

1 Introduction

Systems on Lie groups have been the subject of control research since the 1970s [Brockett, 1972, Jurdjevic and Sussmann,

1972, Brockett, 1973]. In the 2000s, a number of authors in the nonlinear observer community studied systems on Lie groups

for their application to attitude estimation, which was key to enabling the control of small unmanned aerial vehicles [Salcudean,

1991, Thienel and Sanner, 2003, Mahony et al., 2008]. Since then, the practical success of these observer designs has motivated

the development of a general theory of observer design for systems on Lie groups.

Most early work on observer design for general systems on Lie groups sought to extend the results achieved for attitude

estimation, and a number of approaches were taken by different authors. Bonnabel et al. [2006] proposed one of the ﬁrst

frameworks for ‘invariant observers’, and demonstrated their concept with an example application to velocity aided inertial

navigation. Bonnabel [2007] then developed a general theory of left-invariant extended Kalman ﬁlters (IEKFs) for systems on

Lie groups, and demonstrated the effectiveness of the proposed architecture for the attitude estimation problem. Following this,

Bonnabel et al. [2008] developed a theory of symmetry-preserving observers, and provided a constructive method for ﬁnding

an invariant error and a symmetry-preserving correction term for a given system. In parallel work, Lageman et al. [2009]

analysed invariant cost functions on Lie groups - as well as their construction from equivariant measurements - to propose

a design methodology for observers for invariant systems on Lie groups with strong almost-global stability properties. This

was followed by Mahony et al. [2013], which introduced a distinction between a system with a Lie group symmetry and the

corresponding lifted system on the group. They proposed a design method for an observer for the lifted system on the Lie

was supported by the Australian Research Council through the Discovery Project DP210102607.

arXiv:2210.13728v1 [eess.SY] 25 Oct 2022

Equivariant Filters are Equivariant PREPRINT

group, and showed that this respects the system geometry and can provide powerful stability guarantees. Bourmaud et al.

[2013] generalised a discrete extended Kalman ﬁlter (EKF) was to systems on Lie groups using a probabilistic perspective and

a theory of concentrated Gaussian distributions. Extending the work presented in Mahony et al. [2013], Khosravian et al. [2015]

proposed a new constructive observer design methodology for invariant systems on Lie groups with biased inputs. Saccon et al.

[2015] provided explicit formulas for the equations of a second-order-optimal minimum energy ﬁlter for systems on Lie groups

by examining the Hamilton-Jacobi-Bellman equation associated with a loss function constructed from the input and output

measurement disturbances of the system.

Recently, Barrau and Bonnabel [2016] and Barrau and Bonnabel [2018] carefully deﬁned the IEKF and examined its con-

vergence and stability properties for the special class of group-afﬁne systems on Lie groups. They deﬁned an error function

between the observer estimate and the system state which, in this special case of group-afﬁne systems, was shown to evolve

independently of the system and observer states. Moreover, they showed that the evolution of this error function is exactly linear

in the logarithmic coordinates of the Lie group. van Goor and Mahony [2021] also examined the observer design problem for

group-afﬁne systems, and developed a pre-observer that is synchronous with the system trajectories under the chosen deﬁnition

of the error function. Joshi et al. [2021] analysed the observer design problem for systems on general manifolds with a group

action, and determined conditions under which the error dynamics are time-invariant and stable. Finally, van Goor et al. [2020]

and Mahony et al. [2022] developed the Equivariant Filter (EqF), which is a linearisation-based observer design for equivariant

systems deﬁned on manifolds with a transitive group action. They show how to deﬁne a global equivariant error, and how

equivariance of the system output measurements may be exploited to improve the performance.

This paper examines symmetry properties of the EqF developed by van Goor et al. [2020]. Unlike other symmetry-based ﬁlters,

the EqF is parametrised by a choice of origin in the system’s state space manifold and local coordinates about that origin.

This freedom is particularly important for applications to systems on homogeneous spaces, where no natural choice of origin

or local coordinates exists. The EqF error dynamics are shown to be invariant to transformations of the velocity input and

observer state, and equivariant as a parametrised vector ﬁeld; that is, a symmetry applied to the vector ﬁeld may be viewed

as a symmetry on its parameters. This implies that any change to the observer state may be viewed as a change to the input

signal to the error dynamics, and characterises the effect of the observer state on the error dynamics entirely through a single

group action. The main result shows that any two EqF’s for a given system, even with different choices for local coordinates

and origins, yield identical performance when the modelling of system noise is equivalent. Fundamentally, this means that

the EqF design is intrinsic to the system equations and the symmetry; any choice of parametrisation leads to the same ﬁlter.

This result is demonstrated in a simple simulation example involving localisation of a mobile robot in 2D space, where three

different EqF’s provide the same performance up to ﬂoating point precision errors. These results demonstrate not only that the

EqF is theoretically independent of the choice of local coordinates, but also that the freedom to change the local coordinates

and origin of an EqF may be useful to practitioners, such as when developing an observer for localisation (and mapping) over

a large operating environment or for applications requiring high precision.

2 Preliminaries

For a clear introduction to smooth manifolds and Lie groups, the authors recommend Lee [2013].

Let M,Nbe smooth manifolds. The tangent space of Mat a point ξ∈Mis written TξM, the tangent bundle is written TM,

and the space of vector ﬁelds on Mis written X(M). Given a function h:M→N, the differential of hwith respect to ζat

ξ∈Mis denoted

Dζ|ξh(ζ) : TξM→Th(ξ)N,

v7→ Dζ|ξh(ζ)[v].

Alternatively, a shorter form denotes the differential as a map between the tangent bundles; that is,

Dh: TM→TN,

v7→ Dh[v],

where v∈TξMfor some ξ∈Mthat is left implicit.

Given two maps h1:M→N1,h2:N1→N2, denote their composition by

h2◦h1:M→N2,

ξ7→ h2(h1(ξ)).

In the case of linear maps H1, H2, we may instead denote the composition using H2·H1, or simply H2H1to emphasise the

relationship to matrix multiplication. This is particularly useful in applications of the chain rule,

Dζ|ξ(h2◦h1)(ζ) = Dβ|h1(ξ)h2(β)·Dζ|ξh1(ζ),

D(h2◦h1) = Dh2Dh1.

Equivariant Filters are Equivariant PREPRINT

Denote a general Lie group G, and its Lie algebra g. The Lie algebra may be identiﬁed with the tangent space of the group at

the identity; i.e. g'TidG. Denote the Lie exponential map exp : g→G. The left and right translation maps L:G×G→G

and R:G×G→Gare deﬁned by

LZ(X) := ZX, RZ(X) := XZ.

The Adjoint map Ad : G×g→gis deﬁned by

AdXU:= d

ds



s=0

Xexp(sU)X−1.

A right1group action of Gon Mis a map φ:G×M→Mthat satisﬁes

φ(id, ξ) = ξ, φ(Y, φ(X, ξ)) = φ(XY, ξ),

for all ξ∈Mand all X, Y ∈G. For any ξ∈M, the partial map φξ:G→Mis deﬁned to be

φξ(X) = φ(X, ξ).

Likewise, for any X∈G, the partial map φX:M→Mis deﬁned to be

φX(ξ) = φ(X, ξ).

A group action φis said to be transitive if, for all ξ1, ξ2∈M, there exists Z∈Gsuch that φ(Z, ξ1) = ξ2. The map

Φ : G×X(M)→X(M), deﬁned by

ΦX(f)(ξ) := Dζ|φ(X−1,ξ)φ(X, ζ)f(φ(X−1, ξ)),(1)

is a right action of Gon X(M).

2.1 Equivariant System and Lift

A system on a manifold Mis deﬁned by a smooth map f:L→X(M), which associates a vector ﬁeld on Mto every input in

the vector space L. Trajectories of the system are curves ξ: [0,∞)→Msatisfying

ξ(t) = fu(t)(ξ(t)),

for all t≥0, where u: [0,∞)→Lis a given input signal. A measurement or output function is a smooth map

h:M→N⊂Rn.

A symmetry of a system fis a pair of group actions, φ:G×M→Mand ψ:G×L→L, satisfying the equivariance

property

Dζ|ξφX(ζ)·fu(ξ) = fψX(u)(φX(ξ)),(2)

for all X∈G,ξ∈M, and u∈L. In this case, we refer to fas an equivariant system, and the diagram



ψX//L



X(M)ΦX//X(M)

commutes for all X∈G,ξ∈M, and u∈L, where the map ΦXis deﬁned in (1).

Given an equivariant system fwith symmetry (φ, ψ), there exists a Lie-algebraic function Λ : M×L→gsatisfying

DZ|idφξ(Z)Λ(ξ, u) = fu(ξ),(3)

AdX−1Λ(ξ, u) = Λ(φX(ξ), ψX(u)),(4)

for all X∈G,ξ∈M, and u∈L. We refer to Λas an (equivariant) lift of the system f[Mahony et al., 2020], and the

following diagram commutes;

M×L



(φX,ψX)//M×L



gAdX−1//g

1All group actions considered in this paper are right group actions.

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EQUIVARIANTFILTERSAREEQUIVARIANTPREPRINT?HiyaGadaIndianInstituteofTechnologyBombayPowai,Mumbai,400076,India19d100007@iitb.ac.inPietervanGoorSystemsTheoryandRoboticsGroupAustralianNationalUniversityACT,2601,AustraliaPieter.vanGoor@anu.edu.auRaviBanavarIndianInstituteofTechnologyBombayPowai,Mumbai,400...

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