NONLINEAR ATTITUDE ESTIMATION USING INTERMITTENT AND MULTI -RATEVECTOR MEASUREMENTS PREPRINT

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NONLINEAR ATTITUDE ESTIMATION USING INTERMITTENT AND

MULTI-RATE VECTOR MEASUREMENTS

PREPRINT⋆

Miaomiao Wang

Lakehead University

Thunder Bay, ON P7B 5E1, Canada

mwang8@lakeheadu.ca

Abdelhamid Tayebi

Lakehead University

Thunder Bay, ON P7B 5E1, Canada

atayebi@lakeheadu.ca

December 22, 2023

ABSTRACT

This paper considers the problem of nonlinear attitude estimation for a rigid body system using intermittent

and multi-rate inertial vector measurements as well as continuous (high-rate) angular velocity measurements.

Two types of hybrid attitude observers on Lie group SO(3) are proposed. First, we propose a hybrid attitude

observer where almost global asymptotic stability is guaranteed using the notion of almost global input-to-

state stability on manifolds. Thereafter, this hybrid attitude observer is extended by introducing a switching

mechanism to achieve global asymptotic stability. Both simulation and experimental results are presented to

illustrate the performance of the proposed hybrid observers.

Keywords— Attitude estimation, Lie group SO(3), intermittent measurements, multi-rate measurements, hybrid observers

I. Introduction

The algorithms used for the determination of the attitude

(or orientation) of a rigid body system are instrumental in

many applications related to robotics, aerospace and marine

engineering. Since the attitude is not directly measurable from

any sensor, it can be obtained through the integration of the

angular velocity or determined from body-frame observations

of at least two non-collinear vectors known in the inertial

frame. The latter, known as Wahba’s problem [1], relies on

vector observations obtained from different types of sensors

such as low-cost inertial measurement unit (IMU) sensors (in-

cluding an accelerometer, a gyroscope and a magnetometer),

or sophisticated sensors such as sun sensors and star trackers.

However, although simple, both approaches do not perform

well in the presence of measurement bias and noise. This

motivated the use of Kalman-type ﬁlters leading to dynamic

attitude estimation algorithms (see the survey paper [2], [3]).

Although successfully implemented in many practical appli-

cations, these Kalman-based dynamic estimation techniques

rely on local linearizations/approximations and lack stability

guarantees in the global sense.

Recently, a class of geometric nonlinear attitude observers

have made their appearances in the literature, for instance,

nonlinear complementary ﬁlters on the Special Orthogonal

group SO(3) [4]–[7] and the invariant extended Kalman

⋆This work has been accepted for publication in IEEE TAC.

ﬁlter [8]. Unlike the classical attitude observers/ﬁlters, the

geometric observers take into account the topological prop-

erties of the group SO(3) and can provide almost global

asymptotic stability (AGAS) guarantees, i.e., the estimated

attitude converges asymptotically to the actual one from almost

all initial conditions except from a set of zero Lebesgue

measure. It is important to point out that SO(3) is a compact

odd-dimensional manifold without boundary, which is not

homeomorphic to the vector space Rn, and as such, there

is no smooth vector ﬁeld with a global attractor on SO(3).

Therefore, the strongest result one can achieve via these time-

invariant smooth observers is AGAS (see, for instance [9]).

Motivated by the work in [10]–[12] and the framework of

hybrid dynamical systems in [13], [14], hybrid attitude ob-

servers with global asymptotic stability guarantees have been

proposed in [15], [16]. The idea of these hybrid observers, with

global asymptotic stability guarantees, has been extended to

other more complicated state estimation problems on matrix

Lie groups such as hybrid pose observers on group SE(3)

[17] and hybrid state observers for inertial navigation systems

on group SE2(3) [18].

From the practical point of view, attitude estimation often

involves different types of sensors with different sampling

rates. For instance, in the problem of vision-aided attitude es-

timation, the measurements from a vision sensor are obtained

at rates as low as 20 Hz, which is much lower than the rates

of the IMU measurements (up to 1000 Hz). However, most

of the existing attitude observers in the literature are designed

based on continuous output measurements, for instance, [4],

[6], [16], [19], [20]. One common way to implement these

arXiv:2210.11510v3 [eess.SY] 20 Dec 2023

Nonlinear Attitude Estimation Using Intermittent and Multi-Rate Vector Measurements PREPRINT

continuous observers, using intermittent output measurements,

is to apply the zero-order-hold (ZOH) method. Unfortunately,

the stability and convergence guarantees are not necessarily

preserved under this practical ad-hoc setup. In this context,

some state estimation schemes on Rnrelying on intermittent

output measurements have been proposed, for instance, in

[21]–[25]. Other attitude estimation schemes, using discrete

inertial vector measurements, have been developed on the Lie

group SO(3), such as the discrete-time attitude observers in

[8], [26]–[28], and the continuous-discrete attitude observers

in [29], [30]. The latter category assumes continuous (high-

rate) measurements of the angular velocity and intermittent

measurements of inertial vectors with different sampling rates.

A predictor-observer approach has been proposed in [29]

based on a cascade combination of an output predictor and a

continuous attitude observer. In [30], the authors developed a

(non-smooth) predict-update hybrid estimation scheme, where

the estimated attitude is continuously updated by integrating

the attitude kinematics using the continuous angular velocity

measurements and discretely updated through jumps upon the

arrival of the intermittent vector measurements. Both results

in [29] and [30] only guarantee AGAS due to the topological

obstruction on SO(3) and the nature of the intermittent inertial

vector measurements.

In this paper, we consider the problem of attitude estimation

using continuous (high-rate) angular velocity and intermittent

inertial vector measurements with multiple sampling rates.

We ﬁrst propose a hybrid nonlinear observer on manifold

SO(3)×Rnendowed with AGAS guarantees using the notion

of almost global input-to-state stability (ISS) on manifolds pre-

sented in [31]. In this hybrid observer, the estimated states are

continuously updated through integration using the continuous

angular velocity measurements and discreetly updated upon

the arrival of the intermittent vector measurements. To achieve

global asymptotic stability (GAS), we propose a new hybrid

observer with a switching mechanism motivated from [32].

The contribution of this paper can be summarized as follows:

1) Multi-rate vector measurements: The attitude estimation

observers proposed in this work can handle intermittent

vector measurements with different sampling rates (i.e.,

asynchronously-intermittent measurements) where not all

the measurements are received at the same time. For

instance, in many practical applications, the sampling

rate of the IMU is much higher than that of global

positioning systems (GPS) and vision sensors. This is

a key difference with respect to most of the existing

attitude observers assuming that the vector measurements

are continuous or discrete with the same sampling rate

[4], [6], [16], [26], [27], [33], [34]. Our simulation results

validate that the convergence is not guaranteed when

implementing continuous attitude observers (for instance,

the complementary ﬁlter [4]) with ZOH method.

2) Smooth attitude estimation: The observers proposed in

this paper have a similar continuous-discrete structure as

[30], [33], while the estimated attitude from our hybrid

observers is continuous without any additional smoothing

algorithm as in [30]. The fact that our proposed hybrid

observers generate continuous estimates of the attitude

makes it suitable for practical applications involving

observer-controller implementations.

3) Global asymptotic stability: In contrast to the observers

proposed in this paper, the existing attitude observers can

only guarantee local or almost global asymptotic stability

when dealing with intermittent vector measurements,

for instance, [29], [30], [33], [34]. To the best of our

knowledge, this is the ﬁrst work dealing with nonlinear

smooth attitude estimation with GAS guarantees in terms

of intermittent and multi-rate vector measurements.

4) Experimental validation: In this paper, our proposed hy-

brid attitude observer has been experimentally validated

using the measurements obtained from an IMU and an

RGB-D camera, and compared against some state-of-the-

art attitude estimation/determination algorithms.

The remainder of this paper is organized as follows: Section II

provides the preliminary materials that will be used throughout

this paper. In Section III, we formulate our attitude estimation

problem in terms of intermittent vector measurements. In Sec-

tion IV, a new hybrid attitude observer with AGAS guarantees

is proposed. In Sections V, we propose a new hybrid attitude

observer with GAS guarantees. Simulation and experimental

results are presented in Section VI to illustrate the performance

of the proposed observers.

II. Preliminary Material

A. Notations and Deﬁnitions

The sets of real, non-negative real, and non-zero natural

numbers are denoted by R,R≥0, and N, respectively. We

denote by Rnthe n-dimensional Euclidean space and Sn−1

the set of unit vectors in Rn. The Euclidean norm of a vector

x∈Rnis deﬁned as ∥x∥=√x⊤x. Let Indenote the n-by-n

identity matrix and 0n×mdenote the n-by-mzero matrix. For

a given matrix A∈Rn×n, we deﬁne (λA

i, vA

i)as its i-th pair

of eigenvalue and eigenvector, and E(A) := {v∈Rn:v=

i/∥vA

i∥, AvA

i=λA

ivA

i}as the set of all unit eigenvectors

of A. Given two matrices A, B ∈Rm×n, their Euclidean

inner product is deﬁned as ⟨⟨A, B⟩⟩ = tr(A⊤B)where tr(·)

represents the trace of a square matrix, and the Frobenius norm

of matrix Ais deﬁned as ∥A∥F=p⟨⟨A, A⟩⟩ =ptr(A⊤A).

For each vector x= [x1, x2, x3]⊤∈R3, we deﬁne x×as a

skew-symmetric matrix given by

x×="0−x3x2

x30−x1

−x2x10#.

For a matrix A= [aij ]1≤i,j≤3∈R3×3, we deﬁne ψ(A) :=

2[a32−a23, a13−a31, a21−a12]⊤.For any A∈R3×3, x ∈R3,

one can verify that ⟨⟨A, x×⟩⟩ = 2x⊤ψ(A). We denote

the 3-dimensional Special Orthogonal group by SO(3) :=

R∈R3×3|R⊤R=I3,det(R) = +1and its Lie algebra

by so(3) := Ω∈R3×3|Ω⊤=−Ω.Let the map Ra:

R×S2→SO(3) represent the well-known angle-axis pa-

rameterization of the attitude, which is given by

Ra(θ, u) := I3+ sin(θ)u×+ (1 −cos(θ))(u×)2

with u∈S2indicating the direction of an axis of rotation and

θ∈Rdescribing the angle of the rotation about the axis.

Nonlinear Attitude Estimation Using Intermittent and Multi-Rate Vector Measurements PREPRINT

B. Hybrid Systems Framework

Consider a smooth manifold Membedded in vector space Rn.

Let TxMdenote the tangent space1to Mat x, and TM:=

Sx∈M TxMdenote the tangent bundle of manifold M. A

general model of a hybrid system is given as [14]:

H:˙x=F(x), x ∈ F

x+∈G(x), x ∈ J (1)

where x∈ M denotes the state, x+denotes the state after

an instantaneous jump, the ﬂow map F:M → TM

describes the continuous ﬂow of xon the ﬂow set F ⊆ M,

and the jump map G:M⇒M(a set-valued mapping

from Mto M) describes the discrete jump of xon the

jump set J ⊆ M. A solution xto His parameterized by

(t, j)∈R≥0×N, where tdenotes the amount of time that

has passed and jdenotes the number of discrete jumps that

have occurred. A subset dom x⊂R≥0×Nis a hybrid time

domain if for every (T, J)∈dom x, the set, denoted by

dom xT([0, T ]× {0,1, . . . , J}), is a union of ﬁnite intervals

of the form SJ

j=0([tj, tj+1]× {j})with a time sequence

0 = t0≤t1≤ ··· ≤ tJ+1. A solution xto His said to be

maximal if it cannot be extended by ﬂowing nor jumping, and

complete if its domain dom xis unbounded. Let |x|Adenote

the distance of a point xto a closed set A⊂M, and then

the set Ais said to be: stable for Hif for each ϵ > 0there

exists δ > 0such that each maximal solution xto Hwith

|x(0,0)|A≤δsatisﬁes |x(t, j)|A≤ϵfor all (t, j)∈dom x;

globally attractive for Hif every maximal solution xto H

is complete and satisﬁes limt+j→∞ |x(t, j)|A= 0 for all

(t, j)∈dom x;globally asymptotically stable if it is both

stable and globally attractive for H. Moreover, the Ais said

to be exponentially stable for Hif there exist κ, λ > 0such

that, every maximal solution xto His complete and satisﬁes

|x(t, j)|A≤κe−λ(t+j)|x(0,0)|Afor all (t, j)∈dom x[35].

We refer the reader to [13], [14] and references therein for

more details on hybrid dynamical systems.

III. Problem Statement

Let {I} be an inertial frame and {B} be a body-ﬁxed frame

attached to the center of mass of a rigid body. Consider the

attitude kinematics for a rigid body on SO(3) as

R=Rω×(2)

where the rotation matrix R∈SO(3) denotes the attitude

(orientation) of the body-ﬁxed frame {B} with respect to

the inertial frame {I}, and the vector ω∈R3denotes the

angular velocity of the rigid body expressed in body-ﬁxed

frame {B}. In practice, the angular velocity can be obtained

from the gyroscopes at a very high rate. Therefore, we assume

that the body-ﬁxed frame angular velocity ωis continuously

measurable.

Consider a family of N≥2constant vectors known in the

inertial frame, namely inertial vectors, denoted by ri∈R3for

1For each x∈ M, its tangent space TxMis deﬁned as the set

of all the tangent vectors to Mat x, where the tangent vector at

x∈ M is deﬁned as ˙γ(0) = dγ(t)

dt |t=0, with γbeing a differentiable

curve deﬁned as γ:I→ M satisfying γ(0) = x, with I⊂Rbeing

an open interval containing zero in its interior.

all i∈I:= {1,2,··· , N}. The measurements of the inertial

vectors expressed in the body-ﬁxed frame are modelled as

bi=R⊤ri(3)

for all i∈Iand satisfy the following assumptions:

Assumption 1. There exist at least two non-collinear vectors

among the N≥2inertial vectors.

Assumption 2. For each inertial vector ri, i ∈I, the time

sequence {ti

k}k∈Nof its measurements is strictly increasing,

and there exist two constants 0< T i

m≤Ti

M<∞such that

0≤ti

1≤Ti

Mand Ti

m≤ti

k+1 −ti

k≤Ti

Mfor all 1≤k∈N.

Remark 1. Note that Assumption 1is commonly used, for

observability purposes, for the development of attitude estima-

tion schemes, see for instance [4], [6], [16], [29]. Assumption

2implies that the measurements of the inertial vectors can

be irregular and have different sampling periods. Note also

that the sampling is periodic with a regular sampling period

if Ti

m=Ti

Mfor all i∈I. In practice, the inertial vector

measurements can be obtained from different sensors (such as

magnetometers, accelerometers, vision sensors, star trackers

and sun sensors), asynchronously with different and time-

varying sampling periods. It is important to point out that, as

it is going to be shown later, these lower and upper bounds

are not required for the observer design and stability analysis

of our proposed hybrid observers.

Let ˆ

R∈SO(3) denote the estimate of the attitude Rand

R:= Rˆ

R⊤∈SO(3) denote the attitude estimation error. The

objective of this work is to develop a hybrid attitude estimation

scheme on SO(3) for system (2) guaranteeing that the attitude

estimation error ˜

Rconverges to I3, using continuous angular

velocity measurements and intermittent multi-rate body-frame

vector measurements under Assumptions 1and 2.

IV. Hybrid Observer with AGAS Guarantees

A. Observer Design

In this section, we will design a hybrid estimation scheme to

handle intermittent and multi-rate vector measurements. Let

R∈SO(3) denote the estimate of the attitude R, and ˆri∈R3

denote the estimate of the vector ˆ

Rbicorresponding to the i-th

inertial vector ri. Motivated by [34], we propose the following

hybrid observer on manifold SO(3) ×R3N:

R=ˆ

R(ω+koˆ

R⊤σR)×(4a)

(˙

ˆri=koσ×

Rˆri, t ̸=ti

k, k ∈N

ˆr+

i= ˆri+kr(ˆ

Rbi−ˆri), t =ti

k, k ∈N(4b)

with scalar gains ko>0and 0< kr<1. The innovation term

σRis designed as

σR=

i=1

ρiˆri×ri(5)

where ρi>0for all i∈I. The structure of our proposed

hybrid attitude observer (4) is given in Fig. 1.

The dynamics of the attitude estimate ˆ

Rin (4a) are designed

through a continuous integration using the angular velocity and

the innovation term σR. Note that the nonstandard innovation

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NONLINEARATTITUDEESTIMATIONUSINGINTERMITTENTANDMULTI-RATEVECTORMEASUREMENTSPREPRINT⋆MiaomiaoWangLakeheadUniversityThunderBay,ONP7B5E1,Canadamwang8@lakeheadu.caAbdelhamidTayebiLakeheadUniversityThunderBay,ONP7B5E1,Canadaatayebi@lakeheadu.caDecember22,2023ABSTRACTThispaperconsiderstheproblemofnonlinea...

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