On the non-robustness of inconsistent quaternion-based attitude control systems using memoryless path-lifting schemes
ABSTRACT The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Quaternion-based feedback control laws that are inconsistent (i.e. do not have a unique value for a given attitude) require an additional mechanism that lifts a continuous attitude trajectory to the unit quaternion space. Lifting mechanisms that are memoryless, for example, selecting the quaternion having positive scalar component, have a limited domain where they remain injective and, when used globally, introduce discontinuities into the closed-loop system. We show that such discontinuities can be exploited by an arbitrarily small measurement disturbance to stabilize attitudes far from the desired attitude and destroy "global" attractivity properties.
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ABSTRACT: This paper provides nonlinear tracking control systems for a quadrotor unmanned aerial vehicle (UAV) that are robust to bounded uncertainties. A mathematical model of a quadrotor UAV is defined on the special Euclidean group, and nonlinear output-tracking controllers are developed to follow (1) an attitude command, and (2) a position command for the vehicle center of mass. The controlled system has the desirable properties that the tracking errors are uniformly ultimately bounded, and the size of the ultimate bound can be arbitrarily reduced by control system parameters. Numerical examples illustrating complex maneuvers are provided.Proceedings of the American Control Conference 09/2011; - SourceAvailable from: Abdelhamid Tayebi[Show abstract] [Hide abstract]
ABSTRACT: The existing attitude controllers (without angular velocity measurements) involve explicitly the orientation (\textit{e.g.,} the unit-quaternion) in the feedback. Unfortunately, there does not exist any sensor that directly measures the orientation of a rigid body, and hence, the attitude must be reconstructed using a set of inertial vector measurements as well as the angular velocity (which is assumed to be unavailable in velocity-free control schemes). To overcome this \textit{circular reasoning}-like problem, we propose a velocity-free attitude stabilization control scheme relying solely on inertial vector measurements. The originality of this control strategy stems from the fact that the reconstruction of the attitude as well as the angular velocity measurements are not required at all. Moreover, as a byproduct of our design approach, the proposed controller does not lead to the unwinding phenomenon encountered in unit-quaternion based attitude controllers.IEEE Transactions on Automatic Control 03/2012; · 2.72 Impact Factor - SourceAvailable from: Hector Gutierrez
Conference Paper: Robust adaptive geometric tracking controls on SO(3) with an application to the attitude dynamics of a quadrotor UAV
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ABSTRACT: This paper provides new results for a robust adaptive tracking control of the attitude dynamics of a rigid body. Both of the attitude dynamics and the proposed control system are globally expressed on the special orthogonal group, to avoid complexities and ambiguities associated with other attitude representations such as Euler angles or quaternions. By designing an adaptive law for the inertia matrix of a rigid body, the proposed control system can asymptotically follow an attitude command without the knowledge of the inertia matrix, and it is extended to guarantee boundedness of tracking errors in the presence of unstructured disturbances. These are illustrated by numerical examples and experiments for the attitude dynamics of a quadrotor UAV.Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on; 01/2011
Page 1
On the Non-Robustness of Inconsistent Quaternion-Based Attitude Control
Systems using Memoryless Path-Lifting Schemes∗
Christopher G. Mayhew♯, Ricardo G. Sanfelice♭, and Andrew R. Teel†
Abstract—The unit quaternion is a pervasive representation
of rigid-body attitude used for the design and analysis of
feedback control laws. Quaternion-based feedback control laws
that are inconsistent (i.e. do not have a unique value for a
given attitude) require an additional mechanism that lifts a
continuous attitude trajectory to the unit quaternion space.
Lifting mechanisms that are memoryless, for example, selecting
the quaternion having positive scalar component, have a limited
domain where they remain injective and, when used globally,
introduce discontinuities into the closed-loop system. We show
that such discontinuities can be exploited by an arbitrarily small
measurement disturbance to stabilize attitudes far from the
desired attitude and destroy “global” attractivity properties.
I. INTRODUCTION
Controlling the attitude of a rigid body is, perhaps, one of
the canonical nonlinear control problems, with applications
in aerospace and publications dating back many decades
[1]–[5]. A fundamental characteristic of attitude control that
imparts a fascinating difficulty is the topological complexity
of the underlying state space, SO(3). In fact, SO(3) is not
a vector space, but a compact manifold without boundary.
As a result of degree theory, this implies that SO(3) does
not have the topological property of contractibility [6, Ex.
2.4.6]. Furthermore, attraction basins of asymptotically sta-
ble equilibrium points of differential equations with locally
Lipschitz right-hand sides are necessarily contractible [7,
Theorem 1], [8, Theorem 21] and in fact, diffeomorphic to
some Euclidean vector space [9, Theorem V.3.4]. These facts
preclude the existence of a continuous, time-invariant, state-
feedback control law that globally asymptotically stabilizes
a particular attitude [8, Corollary 5.9.13].
Often, unit quaternions, which evolve on S3(the set
of unit-magnitude vectors in R4), are used to parametrize
SO(3). This parametrization provides for a minimal globally
nonsingular representation of rigid-body attitude [10] in
terms of a topologically simpler space in several respects;
however, there are exactly two unit quaternions correspond-
ing to the same rigid-body attitude. This creates the need
to stabilize a disconnected set in the covering space [5],
which has its own topological obstructions [11]. As discussed
♯mayhew@ieee.org, Robert Bosch Research and Technology Center, 4005
Miranda Ave., Palo Alto, CA 94304.
♭sricardo@u.arizona.edu, Department of Aerospace and Mechanical En-
gineering, University of Arizona, Tucson, AZ 85721.
†teel@ece.ucsb.edu, Center for Control Engineering and Computation,
Electrical and Computer Engineering Department, University of California,
Santa Barbara, CA 93106-9560.
∗Research partially supported by the National Science Foundation under
grant ECCS-0925637 and grant CNS-0720842, and by the Air Force Office
of Scientific Research under grant FA9550-09-1-0203.
in [7], these topological subtleties can cause confusion and
sometimes lead to dubious claims regarding the globality of
asymptotic stability (see e.g. [1], [12]). Nevertheless, unit
quaternions are still used by many authors (including the
authors of this paper) today to design control algorithms.
A feedback control law designed using a quaternion
representation of attitude may not be consistent with a
control law defined on SO(3). That is, for every rigid-body
attitude, the quaternion-based feedback may take on one
of two possible values. When this is the case, analysis for
quaternion-based feedback is often carried out in S3with
a lifted dynamic equation, but these results are not directly
related to a feedback system that takes measurements from
SO(3). This obviously begs the question, how is a unit
quaternion obtained from a measurement of attitude? While
calculating the set of two quaternions that represent a given
attitude is a fairly simple operation (see e.g. [13]–[18]), the
process of selecting which quaternion to use for feedback
is a less obvious operation. As noted in [4], it is often the
case that the quaternion with positive “scalar” component is
used for feedback. This operation is non-global (the scalar
component could easily be zero) and discontinuous.
In this paper, we show that when a discontinu-
ous quaternion-selection scheme is paired with a widely
used inconsistent quaternion-based feedback, any presumed
“global” attractivity properties are not robust to arbitrarily
small measurement disturbances. In fact, we construct an ex-
plicit malicious measurement disturbance defined on SO(3)
that stabilizes a region about the manifold of 180◦rotations
with zero angular velocity by exploiting the discontinuity
introduced by the quaternion-selection algorithm.
This paper is organized as follows. Section II provides
background material on attitude control and unit quater-
nions. Section III reconstructs the select-the-quaternion-with-
positive-scalar-component mechanism in terms of a metric.
In Section IV, we show by a Lyapunov analysis that,
when composed with a widely used inconsistent feedback,
the aforementioned quaternion-selection scheme makes the
closed-loop system susceptible to arbitrarily small measure-
ment disturbances that can stabilize attitudes far from the
desired attitude. Finally, we make some concluding remarks
in Section VI.
II. ATTITUDE KINEMATICS, DYNAMICS, AND UNIT
QUATERNIONS
The attitude of a rigid body is represented by a 3 × 3
orthogonal matrix with unitary determinant: an element of
2011 American Control Conference
on O'Farrell Street, San Francisco, CA, USA
June 29 - July 01, 2011
978-1-4577-0079-8/11/$26.00 ©2011 AACC1003
Page 2
the special orthogonal group of order three,
SO(3) = {R ∈ R3×3: R⊤R = RR⊤= I,detR = 1},
where I ∈ R3×3denotes the identity matrix. The cross
product between two vectors y,z ∈ R3, is represented here
by a matrix multiplication: y × z = [y]×z, where
The attitude of a rigid body is denoted by R ∈ SO(3),
where R transforms vectors expressed in the local body
frame of the rigid body to an inertial frame. The angular
rate of the rigid body is denoted as ω and J = J⊤> 0 is
the symmetric and positive definite inertia matrix. When τ
is a vector of external torques, the kinematic and dynamic
equations are
[y]×=
0
y3
−y2
−y3
0
y1
y2
−y1
0
.
˙R = R[ω]×
J ˙ ω = [Jω]×ω + τ.
(1a)
(1b)
Let the n-dimensional unit sphere embedded in Rn+1be
denoted as Sn= {x ∈ Rn+1: x⊤x = 1}. Then, members
of SO(3) are often parametrized in terms of a rotation θ ∈
R about a fixed axis u ∈ S2by the so-called Rodrigues
formula–the map U : R × S2→ SO(3) defined as
U(θ,u) = I + sin(θ)[u]×+ (1 − cos(θ))[u]2
In the sense of (2), a unit quaternion, q, is defined as
×.
(2)
q =
?η
ǫ
?
= ±
?cos(θ/2)
sin(θ/2)u
?
∈ S3,
(3)
where η ∈ R and ǫ ∈ R3, represents an element of SO(3)
through the map R : S3→ SO(3) defined as
R(q) = I + 2η [ǫ]×+ 2[ǫ]2
Note that R(q) = R(−q) for each q ∈ S3. We denote the
set-valued inverse map Q : SO(3) ⇉ S3as
Q(R) = {q ∈ S3: R(q) = R}.
For convenience in notation, we will often write a quaternion
as q = (η,ǫ), rather than in the form of a vector.
With the identity element i = (1,0) ∈ S3, each q ∈ S3
has an inverse q−1= (η,−ǫ) under the multiplication rule
?
where qi= (ηi,ǫi) ∈ R4and i ∈ {1,2}. With this definition,
the covering map R is a group homomorphism, satisfying
R(q1)R(q2) = R(q1⊙ q2)
The quaternion state space, S3, is a covering space for
SO(3) and R : S3→ SO(3) is the covering map. Precisely,
for every R ∈ SO(3), there exists an open neighborhood
U ⊂ SO(3) of R such that Q(U) is a disjoint union of open
sets O1,O2, where, for each k ∈ {1,2}, the restriction of R
×.
(4)
(5)
q1⊙ q2=
η1η2− ǫ⊤
η1ǫ2+ η2ǫ1+ [ǫ1]×ǫ2
1ǫ2
?
,
∀q1,q2∈ S3.
(6)
to Okis a diffeomorphism. In particular, this implies that R
is everywhere a local diffeomorphism.
A fundamental property of a covering space is that a
continuous path in the base space can be uniquely “lifted”
to a continuous path in the covering space. In terms of
SO(3) and S3, this means that for every continuous path
R : [0,1] → SO(3) and for every p ∈ Q(R(0)), there exists a
unique continuous path qp: [0,1] → S3satisfying qp(0) = p
and R(qp(t)) = R(t) for every t ∈ [0,1] [19, Theorem 54.1].
We call any such path qpa lifting of R over R.
It is not just paths that can be lifted from SO(3) onto
S3. In fact, flows and vector fields defined on SO(3) can be
lifted onto S3as well [7]. In this direction, given a Lebesgue
measurable function ω : [0,1] → R3and an absolutely
continuous path R : [0,1] → SO(3) satisfying (1a) for almost
all t ∈ [0,1], any q : [0,1] → S3that is a lifting of R over
R satisfies the quaternion kinematic equation
?˙ η
for almost all t ∈ [0,1], where the maps ν : R3→ R4and
Λ : S3→ R4×3are defined as
ν(x) =
x
˙ q =
˙ ǫ
?
=1
2q ⊙ ν(ω) =1
2Λ(q)ω,
(7)
?0
?
,Λ(q) =
?
−ǫ⊤
ηI + [ǫ]×
?
.
(8)
III. INCONSISTENT QUATERNION-BASED CONTROL
LAWS AND PATH LIFTING
It is quite commonplace in the attitude control literature
to design a feedback based upon a quaternion representation
of rigid-body attitude. That is, the control designer creates a
continuous function κ : S3×R3→ R3and closes a feedback
loop around (1) by setting τ(t) = κ(q(t),ω(t)), where q(t)
is selected to satisfy R(q(t)) = R(t), for each t ∈ R≥0.
When the feedback κ satisfies
κ(q,ω) = κ(−q,ω)∀q ∈ S3,
(9)
we say that κ is consistent. When consistent feedbacks are
used, there is little need for a quaternion representation, as
κ might as well be defined in terms of R ∈ SO(3).
When a quaternion-based feedback is inconsistent, that is,
∃q ∈ S3
κ(q,ω) ?= κ(−q,ω),
(10)
the resulting feedback does not define a unique vector field
on SO(3) × R3because for R ∈ SO(3) satisfying Q(R) =
{−q,q}, the feedback κ(Q(R),ω) is a two-element set [7].
At this point, the control designer must, for every t ∈ R≥0,
choose which q(t) ∈ Q(R(t)) to use for feedback. Or, in the
topological terms of lifting, the control designer must choose
how to lift the measured attitude trajectory in SO(3) to S3.
In this direction, we provide a quote from [4]:
“In many quaternion extraction algorithms, the sign
of η is arbitrarily chosen positive. This approach
is not used here, instead, the sign ambiguity is
resolved by choosing the one that satisfies the
associated kinematic differential equation. In im-
plementation, this would probably imply keeping
some immediate past values of the quaternion.”
1004
Page 3
There is much insight to be gained from this quotation,
especially when viewed in the context of lifts over R.
In particular, it suggests that inconsistent quaternion-based
control laws require an extra quaternion memory state to
lift the measured SO(3) trajectory to S3. In what follows,
we reconstruct the memoryless quaternion “extraction” al-
gorithm that discontinuously selects the quaternion with
positive scalar component in terms of a metric and analyze
its properties when used in a feedback control law.
Let P : S3→ [0,2] be defined as
P(q) = P(η,ǫ) = 1 − i⊤q = 1 − η.
Then, the function d : S3× S3→ [0,2] defined as
d(q,p) = P(q−1⊙ p) = 1 − q⊤p
defines a metric on S3. From a geometric viewpoint, d(q,p)
is the height of p on S3“above” the plane perpendicular to
the vector q at q. Given a set Q ⊂ S3, we define the distance
to Q from q (in terms of the metric d) as
(11)
(12)
dist(q,Q) = inf{d(q,p) : p ∈ Q}.
When the set Q in (13) takes the form of Q(R) for some
R ∈ SO(3), the distance function also takes a special form.
In particular, let Q(R) = {p,−p}. Then,
dist(q,Q(R)) = 1 − |q⊤p|.
One candidate method to lift a path from SO(3) to S3
is to simply pick the quaternion representation of R that is
closest to a specific quaternion in terms of the metric d. In
particular, let us define the map Φ : S3× SO(3) ⇉ S3as
Φ(q,R) = argmin
p∈Q(R)
(13)
(14)
d(q,p) = argmax
p∈Q(R)
q⊤p.
(15)
The map Φ has some useful properties, which we summarize
in the following lemmas.
Lemma 1. Let q ∈ S3and R ∈ SO(3). The following are
equivalent:
1) Φ(q,R) is single-valued,
2) 0 ≤ dist(q,Q(R)) < 1,
3) q⊤p ?= 0 for all p ∈ Q(R) so that q⊤Φ(q,R) > 0,
4) R ?= U(π,u)R(q) for some u ∈ S2.
Given a fixed q ∈ S3, Φ can be used to lift curves in
SO(3) to S3, so long as Φ remains single-valued.
Lemma 2. For every ˆ q ∈ S3, every continuous R : [0,1] →
SO(3), and every continuous q : [0,1] → S3satisfying
d(ˆ q,q(0)) < 1, R(q(t)) = R(t), and dist(ˆ q,Q(R(t))) < 1
for all t ∈ [0,1], it follows that Φ(ˆ q,R(t)) = q(t) for all
t ∈ [0,1].
Since a common goal of attitude control is to regulate
R to I, one might choose i as a point of reference (since
R(i) = I) and use the map Φi: SO(3) ⇉ S3defined as
Φi(R) = Φ(i,R).
(16)
Now, following 3) from Lemma 1 we see that i⊤Φi(R) >
0, that is, Φi always chooses the quaternion with positive
scalar component, so long as Φ is single-valued. Further,
Lemma 2 allows one to lift curves with Φi so long as R
does not cross the manifold of 180◦rotations, where Φiis
multi-valued, or else Φiwill produce a quaternion trajectory
that is discontinuous. This discontinuous behavior can have
disastrous effects when Φiis composed with an inconsistent
feedback. We now examine such a feedback.
IV. NON-ROBUSTNESS
Let c > 0 and let L : R3→ R3be a continuous function
satisfying
γ(?ω?2) ≤ ω⊤L(ω),
where γ : R≥0→ R≥0is a continuous and strictly increasing
function satisfying γ(0) = 0. Let
(17)
E(q) = E(η,ǫ) = Λ(q)⊤i = ǫ
(18)
and consider the inconsistent feedback
κ∗(q,ω) = −cǫ − L(ω) = cE(q) − L(ω).
In (19), the cǫ term introduces a rotational spring force and
L(ω) introduces damping. While this control law asymptoti-
cally stabilizes (i,0) for the lifted closed-loop system defined
by (7), (1b), and setting τ = κ∗(q,ω), it renders (−i,0) an
unstable saddle equilibrium. When composed with Φi, one
might expect that the resulting feedback globally asymptot-
ically stabilizes the identity element of SO(3); however, we
show that any such expected global attractivity properties are
not robust to arbitrarily small measurement disturbances. In
particular, we construct a malicious measurement disturbance
that exploits the discontinuity introduced by Φito stabilize
the 180◦manifold.
Define the signum function σ : R → {−1,0,1} as
Then, for 0 ≤ δ < π, consider the (discontinuous) function
∆ : SO(3) × R3→ U(δ,S2) defined as
?
I
(19)
σ(s) =
1
0
−1
s > 0
s = 0
s < 0.
(20)
∆(U(θ,u),ω) =
U(−δσ(ω⊤u),u)cosθ < cos(π + δ)
otherwise.
(21)
For any (R,ω) ∈ SO(3)×R3, the rotation matrix ∆(R,ω)R
constitutes an angular perturbation of R in the amount of
δ and as δ decreases to zero, ∆ converges to the identity
matrix. In particular, the parameter δ controls the size of the
disturbance. We note that (21) is well defined on SO(3).
Lemma 3. For every δ ∈ [0,π) and (R,ω) ∈ SO(3) × R3,
∆(R,ω) is uniquely defined.
Proof. Suppose that R = U(θ,u) for some θ ∈ R and u ∈
S2. Clearly, ∆(R,ω) is uniquely defined when either ω = 0
or cosθ = cos(θ + 2πZ) ≥ cos(π ± δ), since it does not
depend on either R or ω in this case.
Suppose that cosθ < cos(π ± δ) and ω ?= 0. This implies
that R ?= I, since 0 < δ < π. Then, by Euler’s theorem on
1005
Page 4
rotations, for any v ∈ S2and φ such that R = U(φ,v), it
must be the case that u = v or u = −v (only when R ?= I).
Since U(−θ,−u) = U(θ,u), it follows that
∆(U(φ,v),ω) = U(−δσ(ω⊤v),v) = U(−δσ(ω⊤u),u).
So, we have shown that the value of ∆ is independent of
the angle and axis representation used for R, hence, it is
uniquely defined on SO(3) × R3.
Let φi : SO(3) → S3be any single-valued selection
of Φi, that is, φi(R) = Φi(R) for all R ?= U(π,u) and
φi(R) ∈ Φi otherwise. Now, we apply the disturbance ∆
to measurements of attitude before being converted to a
quaternion for use with the inconsistent feedback (19) and
analyze the resulting closed-loop system. That is, we replace
q with φi(∆(R,ω)R) in the control law κ∗defined in (19).
Because φi and ∆ are discontinuous, we use the notion
of Krasovskii solutions for discontinuous systems [20].
Definition 4. Let f : Rn→ Rn. The Krasovskii regulariza-
tion of f is the set-valued mapping
Kf(x) =
?
ǫ>0
convf(x + ǫB)
(22)
where convB denotes the closed convex hull of the set
B ⊂ Rnand B denotes the unit ball in Rn. Then, given
a function f : Rn→ Rn, a Krasovskii solution to ˙ x = f(x)
on an interval I ⊂ R≥0is an absolutely continuous function
satisfying
˙ x(t) ∈ Kf(x(t))
for almost all t ∈ I.
An important property of a Krasovskii regularization is
that Kf(x) = f(x) for every x where the function f is
continuous.
(23)
Theorem 5. Let a > 0, c > 0, and δ > 0 satisfy
0 < δ <1
2
?
−a
c+
??a
c
?2
+ 8
?
<
√2
(24)
and define
B = {(U(θ,u),ω) : cosθ + (1/a)ω⊤Jω ≤ cos(π + δ)}.
Then, the set {U(π,S2)}×{0} is stable and B is invariant
for the Krasovskii regularization of the closed-loop system
˙R = R[ω]×
J ˙ ω = [Jω]×ω − cE(φi(∆(R,ω)R)) − L(ω).
Proof. Since we are studying Krasovskii solutions to (25),
we might normally need to find the Krasovskii regulariza-
tion of (25); however, the analysis in this proof obviates
the need for calculating the Krasovskii regularization for
regions where the calculation is nontrivial. Since the function
(R,ω) ?→ R[ω]×is continuous, its Krasovskii regulariza-
tion is identical to the original map. Also note that, by
definition of ∆ and φi, the map (R,ω) ?→ [Jω]×ω −
κ∗(φi(∆(R,ω)R),ω) is continuous on the set {(U(θ,u),ω) :
(25)
cosθ < cos(π+δ), ω ?= 0}, so its Krasovskii regularization
is also identical to the original map on this set.
Consider the Lyapunov function
V (R,ω) = a(1 − trace(I − R)/4) +1
Expressed in terms of rotation angle, we have equivalently,
2ω⊤Jω.
(26)
V (U(θ,u),ω) =a
since trace(I−U(θ,u)) = 2(1−cosθ), so that V (R,ω) ≥ 0
for all (R,ω) ∈ SO(3)×R3and V (R,ω) = 0 if and only if
R = U(π,v) and ω = 0. Furthermore, the sub-level sets of
V are compact.
Define the function ψ : R3×3→ R3as
ψ(A) =1
2
Then,
ψ
satisfies
traceA[ω]×
ψ(U(θ,u)) = usinθ. In what follows, for a real number
r ∈ R, a vector y ∈ Rnand a set Z ⊂ Rn, we use the
notation r + y⊤Z = {r + y⊤z : z ∈ Z}. Employing the
Krasovskii regularization, we calculate the time derivative
of V as
˙V (R,ω) ∈ −a
+ ω⊤(−KcE(φi(∆(R,ω)R)) − L(ω))
= −ω⊤L(ω)
+ ω⊤?
where we have used the fact that ω⊤[Jω]×ω = 0. Note
that ˙V (R,0) = 0 no matter what values the Krasovskii
regularization may take.
Now, we let R = U(θ,u) and henceforth constrain our
analysis to the case where cosθ < cos(π+δ) and ω ?= 0, so
that ∆(R,ω)R = U(θ − δσ(ω⊤u),u) and φi(∆(R,ω)R) is
single-valued. Also, in this region, the Krasovskii regulariza-
tion of (25) is identical to (25). Recalling that φiselects the
quaternion with positive scalar component and noting that
U(φ,u)U(θ,u) = U(θ + φ,u), we can now write
φi(∆(R,ω)R) =
σ?cos?(θ − δσ(ω⊤u))/2???cos?(θ − δσ(ω⊤u))/2?
and in particular,
2(1 + cosθ) +1
2ω⊤Jω
A32− A23
A13− A31
A21− A12
.
(27)
=−2ω⊤ψ(A)
and
4trace(−R[ω]×)
−a
2ψ(R) − KcE(φi(∆(R,ω)R))
?
(28)
,
sin?(θ − δσ(ω⊤u))/2?u
?
,
E(φi(∆(R,ω)R)) =
σ?cos?(θ − δσ(ω⊤u))/2??sin?(θ − δσ(ω⊤u))/2?u.
Applying (29) and (17) to (28),
(29)
˙V (U(θ,u),ω) ≤ −γ(?ω?2) − ω⊤ua
− ω⊤u?cσ?cos?(θ − δσ(ω⊤u))/2??
2sinθ
∗ sin?(θ − δσ(ω⊤u))/2??.
(30)
1006
Page 5
Note that when ω⊤u = 0, it follows that˙V (U(θ,u),ω) ≤ 0,
so we further constrain our analysis from this point to the
case when ω⊤u ?= 0. Now, without loss of generality, we
assume that π − δ < θ < π + δ, where
σ?cos?(θ − δσ(ω⊤u))/2??= σ?π − (θ − δσ(ω⊤u))?.
Now, since σ(ω⊤u)2= 1 and sσ(s) = |s|, we factor this
term to arrive at
(31)
˙V (U(θ,u),ω) ≤ −γ(?ω?2) − |ω⊤u|a
− |ω⊤u|cσ(ω⊤u)σ?π − (θ − δσ(ω⊤u))?
2σ(ω⊤u)sinθ
∗ sin?(θ − δσ(ω⊤u))/2?.
(32)
Moreover, for any r,s ∈ R, it follows that σ(s)σ(r) =
σ(rσ(s)). Applying this relation to (32), we have
˙V (U(θ,u),ω) ≤ −γ(?ω?2) − |ω⊤u|a
− |ω⊤u|cσ?(π − θ)σ(ω⊤u) + δ)?
It follows that˙V (U(θ,u),ω) < 0 whenever
cσ?(π − θ)σ(ω⊤u) + δ)?sin?(θ − δσ(ω⊤u))/2?
2σ(ω⊤u)sinθ
∗ sin?(θ − δσ(ω⊤u))/2?.
(33)
+a
2σ(ω⊤u)sinθ > 0.
(34)
Now, we can apply trigonometric inequalities to analyze
(34). In particular, we have that |sinθ| ≤ |θ − π| and since
1−cosθ ≤1
deduce that sin(1
(34) holds when
2θ2, we can use the properties of sin and cos to
2(θ − δσ(ω⊤u))) ≥ 1 −1
8(θ − π)2. Hence,
cσ?(π − θ)σ(ω⊤u) + δ)??
1 −?θ − π − δσ(ω⊤u)?2?
/8
>a
2|θ − π|.
(35)
Now, since δ > |π −θ| by a previous assumption, it follows
that σ?(θ − π)σ(ω⊤u) + δ?
(34) holds when
= 1. This assumption also
8(θ − π − δσ(ω⊤u))2≥ 1 −1
implies that 1 −1
2δ2. Hence,
c?1 − δ2/2?> aδ/2
Since δ ≥ 0, we have at least for small δ that 0 > δ2+
aδ/c−2, so we can bound δ by the positive root of λ(x) =
x2+(a/c)x−2 located at x = (−(a/c)±?(a/c)2+ 8)/2.
{(R,ω) : cosθ < cos(π+δ) or ω = 0} ⊃ {U(π,S2)}×{0},
where 0 < δ <
?
that {U(π,S2)} × {0} is stable. We note that this last
inequality implies that 0 < δ <
1
2
⇐⇒0 > δ2+ (a/c)δ − 2. (36)
Hence, we have that˙V (U(θ,u),ω) ≤ 0 on the set W =
−(a/c) +?(a/c)2+ 8
?
/2. This implies
√2, since, if µ(x) =
?−x +√x2+ 8?, it follows that
dµ(x)
dx
=1
2
?
1
√x2+ 8− 1
?
< 0.
That is, as the ratio of a to c increases, the upper bound on
δ given as 0 < δ < µ(a/c) must decrease.
To estimate an invariant set using V , we find a sub-level
set of V contained in the set W. In fact, the set B is a
sub-level set of V corresponding to the set {(U(θ,u),ω) :
V (U(θ,u),ω) ≤
and so it is invariant.
a
2(1 + cos(π + δ))}. Moreover, B ⊂ W
This result shows that the discontinuity created by pairing
an inconsistent quaternion-based feedback with a discontinu-
ous quaternion selection scheme is susceptible to arbitrarily
small measurement disturbances. This is because at the
discontinuity of Φi(on the manifold U(π,S2)), the feedback
term cE(Φi(R)) opposes itself about the discontinuity. Then,
malicious disturbances like (21) can exploit the disconti-
nuity of the vector field to induce a chattering behavior
that stabilizes {U(π,S2)} × {0}. This is quite similar to
the inconsistent feedback κd(q,ω) = −σ(η)ǫ − ω (when
implemented with a lifted trajectory), which was shown
in [5] to exhibit extreme measurement sensitivity that can
destroy “global” attractivity properties of a desired attitude
in a kinematic setting.
The various failures of Φi have led several authors (e.g.
[21]) to derive sufficient conditions on the initial condi-
tions of (1) to ensure that these 180◦attitudes are never
approached, thus obviating the use of a globally nonsingular
representation of attitude like unit quaternions. However,
the issues with using Φias a path-lifting algorithm are not
a problem with the quaternion representation—they arise
because Φi is a memoryless map from SO(3) to S3. In
particular, Φi always chooses the closest quaternion to i.
In general, when one compares Q(R) with ˆ q for some
R ∈ SO(3) and ˆ q ∈ S3, Φ(q,R) is multi-valued on the
2-D manifold {p ∈ S3: p⊤ˆ q = 0}. However, when the
reference point for choosing the closest quaternion is allowed
to change, it is possible to create a dynamic algorithm
for lifting a path from SO(3) to S3. We explore such an
algorithm in the companion paper [22].
V. SIMULATION
In this section, we demonstrate the non-robustness asserted
by Theorem 5 in simulation. For ease of exposition, we let
¯ v = [3 4 5]⊤and define v = ¯ v/|¯ v|. The following simulation
has the parameters J = diag(10v), c = 1, δ = 10π/180, and
L(ω) = ω/10. Initial conditions were selected as R(0) = I,
ω(0) = 2v. Finally, the following simulation was conducted
in MATLAB as follows. The attitude kinematic equation
was implemented using a quaternion representation and the
differential equation ˙ q =
2(q/|q|) ⊙ ν(ω) − 10q(|q| − 1).
This implementation renders S3asymptotically stable and
ensures that q does not drift far from S3during the numerical
integration. The state q was projected to S3before being
subsequently used. The fixed-step solver ode3 was used for
numerical integration with a step size of 1/100.
Fig. 1 shows the effects of the malicious disturbance, ∆,
on the closed-loop system. The top plot shows the signal
θ(R) = cos−1((trace(R) − 1)/2), representing the angle
between R and I, the middle plot shows the components
of ω, and the bottom plot shows the components of τ =
1
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