Inertial Vector Based Attitude Stabilization of Rigid Body Without Angular Velocity Measurements

0.23 MB·English

by  L. Benziane

1
Inertial Vector Based Attitude Stabilization of Rigid Body Without Angular Velocity
Measurements
L. Benziane, A. Benallegue, Y. Chitour, A. Tayebi
Abstract
We address the problem of attitude stabilization of a rigid body, in which neither the angular velocity nor the instantaneous
5 measurementsoftheattitudeareusedinthefeedback, onlybodyvectormeasurementsareneeded.Thedesignofthecontrolleris
1 basedonanangularvelocityobserver-likesystem,whereafirstorderlinearauxiliarysystembaseddirectlyonvectormeasurements
0 isintroduced. Theintroductionof gainmatricesprovidesmoretuningflexibilityandbetterresultscompared withexistingworks.
2 The proposed controller ensures almost global asymptotic stability. The performance and effectiveness of the proposed solution
are illustrated via simulation results where the gains of the controller are adjusted using non linear optimization.
n
a
J
I. INTRODUCTION
0
Attitude stabilization of rotational motion of rigid body is a classical problem. Despite the considerable existing solutions,
2
it remains until today an active research topic. This is due to the large field of applications such as robotics, unmanned

] aerial vehicles (UAVs), satellites, marine vehicles, etc. The problem of attitude control was treated using several type of
C
parametrizationofthe attitude [1]. Classical solutionsto thisproblemhavebeenproposedusing local-minimalparametrization
O whichliesinR3,suchasEuleranglesormodifiedRodriguezparameters(see,forinstance,[2],[3],[4],[5]).Theglobal-unique
. representation,whichis the naturalparametrizationof the attitudeis the directioncosine rotationmatrixthatlies in the special
h
t orthogonalgroupSO(3).Asaconsequence,manyrecentsolutionsusethisparametrization(see,forinstance,[6],[7],[8],[9]).
a
However, for simplicity of analysis and numericalimplementation reasons, a considerable number of solutions to the problem
m
of attitude stabilization of rigid bodies rather use quaternion parametrization as global representation which lies in the unit
[ sphere S3 (see, for instance, [10], [11], [12], [13], [14]).

1 Since attitude control and stabilization is as an interesting theoretical and technical problem, many scenarii were studied
v in the literature (see for instance [15], [16], [17],[18], [19] and [20]). The interesting and challenging scenario is the attitude
7
stabilization without angular velocity. In fact, the main goal is to stabilize the attitude without the use of gyroscopes, which
6
can beveryexpensiveor vitalto thesystem, like gyroscopesonHubbleusedfor pointingthetelescope.Theymeasureattitude
7
4 when Hubble is changing its pointing from one target (a star or planet) to another, and they help control the telescope’s
0 pointing while scientists are observing targets. There are a total of six gyroscopes on board–three serve as backups. In 2009,
. all six of Hubble’s gyroscopeshad to be replaced and one can imagine the cost generated.At the light of these problems,it is
1
0 conceivable to reduce costs and ensure continuity of the mission of the rigid body despite the failure of the gyroscopes when
5 this type of controllers is used. Many works in the literature dealt with attitude stabilization without angular velocity problem
1 (see, for instance [21], [22], [23], [24], [25], [26], [27], [28], [29]), some of them exploited the passivity of the system such
:
v as [30], [31], [32], [33], [34].
i In almost all results dealing with the case of attitude control without angular velocity, the “instantaneous measurements
X
of the attitude” are used in the control law. As there is no sensor which directly measures the attitude of a rigid body, the
r
a aforementioned velocity-free controllers require some kind of attitude observer relying on the available direction sensors.
However,all static algorithmsbased only on bodyvector measurementsare very sensitive to noise. Also, all the most efficient
dynamic attitude estimation algorithms make use of the body measurements and the angular velocity information to estimate
the attitude of the rigid body. To overcome this problem, a velocity-free attitude control scheme, that incorporates explicitly
vector measurements instead of the attitude itself, has been proposed for the first time in [24]. As claimed in [24], this class
of controllers can be qualified as the class of true velocity-free attitude controllers.
Since it is impossible to achieve a global asymptotic stabilization using continuous time invariant state feedback [35], the
attitude control scheme presented in this work makes use the notion of “Almost Global Asymptotic Stabilization” (AGAS) of
the closed loop system. Therefore, this work and that proposed in [24] present a stronger stability property compared to [21],
where the convergence depends on a non trivial condition on initial conditions.
Theproposedsolutiongiveninthispapercanberegardedasancontinuationof[24].Themaindifferencesarethefollowing:
(a)theuseofanauxiliarysystem intermsofbodyvectormeasurements,definedonR3,ratherthanthatofanauxiliarysystem
definedonS3;(b)theexplicitdesignofanangularvelocityobserverwhichisusedinthedesignofthestabilizingfeedback.As
a consequence,the set of unstable equilibriaofthe closedloop dynamicsof ourauxiliaryerrorsystem is reducedas compared
This research was partially supported by the iCODE institute, research project of the Idex Paris-Saclay. L. Benziane, A. Benallegue are with LISV,
UniversitédeVersaillesSaintQuentin,France.Y.ChitouriswithL2S,UniversitéParis-SudXI,CNRSandSupélec,Gif-sur-Yvette,andTeamGECO,INRIA
Saclay – Ile-de-France, France. A. Tayebi is with the Department of Electrical Engineering, Lakehead University, Thunder Bay, Ontario, Canada, e-mail:
lotfi[email protected], [email protected], [email protected], [email protected].
2
to thatof[24].Itis alsoshownthatourauxiliaryerrorsystemdoesnotmakeusethe inertialfixedreferencevectorsasin [24].
ThequaternionparametrizationisusedinthemainanalysisandthefinalresultsarerewrittenwithrotationsexpressedinSO(3)
by simple projection.We also show thatthe introductionof gainmatricesimprovesdrasticallythe controllerperformancewith
respect to both [24] and [21]. Moreover and contrarily to what is stated in [6], [24] we prove that the set of control gains
leading to a continuumof equilibria of the closed loop system is an algebraic variety of positive co-dimension, independently
on the choice of the observed vectors. Finally, in order to adjust properly the controller gains, we rely a non-linear optimal
tuning method.
The result presented in this paper extends those from [22] where a scalar gain was used in the control law. In addition, a
complete and rigorous mathematical analysis is presented in this version.
II. NOTATIONSAND PROBLEM FORMULATION
A. Notations
Toperforma rotationinEuclideanspace,we usedeitherarotationmatrixRoraunit-quaternionQ=[q ,qT]T.We assume
0
that R∈SO(3)={R∈R3×3 |RTR=RRT =I3×3,det(R)=1} and Q∈S3 ={Q∈R4 |QTQ=1}. The multiplication
p q −pTq
of two quaternions P = (p ,pT)T and Q = (q ,qT)T is denoted by “⊙” and defined as P ⊙Q = 0 0 .
0 0 (cid:20) p0q+q0p+p×q (cid:21)
We use so(3) to denote the Lie algebra of SO(3), i.e., the set of skew symmetric matrices and we set S as the Lie algebra
isomorphism from R3 →so(3) which associates to x=[x ,x ,x ]T the skew-symmetric matrix S(x) given by
1 2 3
0 −x x
3 2
S(x)= x3 0 −x1 .
−x x 0
2 1
 
Note that for every x,y ∈R3, one has S(x)y =x×y where × stands for the vector cross product.
The mapping R:S3 →SO(3) given by Rodrigues’ rotation formula [36]
R(Q)=I3×3+2q0S(q)+2S(q)2, (1)
definesadoublecoveringmapofSO(3)byS3,i.e.,foreveryR∈SO(3)theequationR(Q)=Radmitsexactlytwosolutions
Q and −Q . Asa consequence,a vectorfield f ofS3 projectsontoa vectorfield of SO(3) if andonlyif, foreveryQ∈S3,
R R
f(−Q) = −f(Q) (where we have made the obvious identification between TQS3 the tangent space of S3 at Q and T−QS3
the tangent space of S3 at −Q).
In what follows and for simplicity, the notations below are used.
• Ifm isa positiveinteger,Mm(R)is usedtodenotethesetof mbym matriceswith realentries;03,03n, 0 andI denote
the 3 by 3 zero matrix, the 3n by 1 zero vector, the 3 by 1 zero vector and the 3 by 3 identity matrix respectively;
• {B} and {I} denote an orthonormal body-attached frame with its origin at the center of gravity of the rigid-body and
the inertial reference frame on earth respectively.
For every x,y ∈R3 and a given R∈SO(3) one has the following [37]
S(x)y = −S(y)x, S(x)x=0,
S(x)S(y) = yxT −xTyI, S2(x)=xxT −xTxI,
S(S(x)y) = S(x)S(y)−S(y)S(x), S(Rx)=RS(x)RT.
B. Problem formulation
Let n≥2 be an integer. The attitude kinematics of a rigid body in 3D space is given by
R˙(t)=R(t)S(ω(t)), (2)
where R∈SO(3). The equivalent kinematics evolving in S3 are given by
q˙ (t) −1qT(t)ω(t)
Q˙(t)= 0 = 2 , (3)
(cid:20) q˙(t) (cid:21) (cid:20) 12(q0(t)I +S(q(t)))ω(t) (cid:21)
whereω(t)beingtheangularvelocityoftherigidbodyexpressedin{B}andQ∈S3 istheunitquaternion.Letb (Q(t))∈R3
i
(i=1,··· ,n) be a measured vector expressed in {B}. The relation between b (t) and its corresponding fixed inertial vector
i
r ∈R3 are given by
i
b (Q(t))=RT(t)r (4)
i i
As a consequence we have b (−Q)=b (Q) for 1≤i≤n and Q∈S3.
i i
Using (2) and (4), one can get the reduced attitude kinematics
3
b˙ (Q(t))=−S(ω(t))b (Q(t)), i=1,··· ,n. (5)
i i
The simplified rigid body rotational dynamics are governed by
Jω˙(t)=−S(ω(t))Jω(t)+τ(t), (6)
where
• J ∈R3×3 is a symmetric positive definite constant inertia matrix of the rigid body expressed in {B};
• τ(t) is the external torque applied to the system expressed in {B};
• ω(t) being the angular velocity of the rigid body expressed in {B}.
The problem addressed in this work is the design of an attitude stabilization controlτ(t) based only on inertial measurements
b (t), without using the angular velocity ω(t) in the feedback.
i
C. Assumptions
We make the following assumptions for the rest of the paper.
A1 We assume that only the n vector-valued functions of time b (t) are measured and we do not make any similar
i
assumption on angular velocity vector ω(t). Moreover, note that the b ’s actually depend on the rotation R and one
i
could also write them as b (R(t)) or b (Q(t)) if we choose quaternions instead of rotations. In the sequel, we will
i i
write either b (t) or b (Q(t)).
i i
A2 At least two vectors r , r are non collinear. As a consequence, b (t) and b (t) are linearly independentfor all non
1 2 1 2
negative times.
A3 The desired rigid body attitude is defined by the constant rotation matrix R , relates an inertial vector r to its
d i
corresponding vector in the desired frame, i.e., bd = RTr , with b˙d = 0. An equivalent constant desired unit-
i d i i
quaternion Q is defined as R =R(Q ).
d d d
III. HANDLING THELACK OF ANGULARVELOCITY ANDDESIGN OF THEATTITUDECONTROLLER
A. Angular velocity observer-like system
As well known, the reduced attitude kinematic is defined by (5). Define Γ = diag(Λ ,··· ,Λ ), where Λ is a symmetric
1 n i
positive definite 3×3 matrix, for 1 ≤ i ≤ n. Define the symmetric matrix M(t) = n S(b (t))TΛ S(b (t)), which is
i=1 i i i
positive definite thanks to Assumption A2. P
Multiplying (5) by S(b (t))Λ for 1≤i≤n and doing the sum gives
i i
n
S(b (t))Λ b˙ (t)=−M(t)ω(t). (7)
i i i
Xi=1
From (7) the true angular velocity ω(t) is given by
n
ω(t)=−M−1(t) S(b (t))Λ b˙ (8)
i i i(t)
Xi=1
Since b˙ (t) is not a measured quantity, we propose the following new angular velocity observer-like signal
i
n
ωˆ(t)=−M−1(t) S(b (t))Λ ˆb˙ (t), (9)
i i i
Xi=1
where the vector ˆb˙ (t) can be viewed as an estimate of the vector b˙ (t) using the following linear first-order filter on b
i i i
(i=1,··· ,n).
ˆb˙ (t)=A (b (t)−ˆb (t)), (10)
i i i i
where the constant matrices A ∈R3×3 are chosen as A =RTP (Λ )R , for 1≤i≤n, with P polynomial of degree two
i i d i i d i
which is positive on R∗. As a trivial consequence, one deduces that, for 1≤i≤n, R A RT is symmetric positive definite
+ d i d
and commutes with Λ . Set A=diag(A ,··· ,A ) and A =diag(R A RT,··· ,R A RT). Then Γ and A commute.
i 1 n d d 1 d d n d d
We define an error for the linear first-order filter by ˜b (t) = b (t)−ˆb (t). Using (10), (5) leads to the following error
i i i
dynamics˜b˙ (t)=−A˜b (t)+S(b (t))ω, which can be rewritten using the state vector definedby ξ(t):=[˜bT(t),··· ,˜bT(t)]T,
i i i i 1 n
as
ξ˙(t)=−Aξ(t)+B(t)ω(t), (11)
4
where B(t)= S(b (t))T ··· S(b (t))T T.
1 n
Finally, the a(cid:2)ngular velocity observer-like sig(cid:3)nal can be written as
ωˆ(t)=M−1(t)BT(t)ΓAξ(t). (12)
B. Controller Design
First, the orientation error is defined by
R¯(t)=R(t)RT, (13)
d
where R(t) is a rotation matrix and R is a constant desired rotation matrix. From (2) and (13) one can obtain the attitude
d
dynamics errors in term of matrix rotation as follows
R¯˙(t)=R¯(t)S(R ω(t)), (14)
d
R¯(t) correspond to the quaternion error Q¯(t)=Q(t)⊙Q−1(t)≡[q¯ (t),q¯(t)T]T whose dynamics is governed by
d 0
q¯˙ (t) −1q¯T(t)R ω(t)
0 = 2 d , (15)
(cid:20) q¯˙(t) (cid:21) (cid:20) 12(q¯0(t)I +S(q¯(t)))Rdω(t) (cid:21)
The reduced orientation error is given by ¯b (Q¯(t))=b (Q(t))−bd. Therefore, on can get
i i i
¯b (Q¯(t))=RT(R¯(t)T −I)r , (16)
i d i
where 1≤i≤n which can be rewritten using (1) as
¯b (Q¯(t))=−2RT(q¯ (t)I −S(q¯(t)))S(q¯(t))r . (17)
i d 0 i
We propose the following control law
τ(t)=z (t)−Mωˆ(t), (18)
ρ
where the term z (·) was introduced in [24] and is given by
ρ
n
z (t)= ρ S(bd)b , (19)
ρ i i i
Xi=1
where the coefficients ρ ’s are arbitrary positive constants. Define
i
n
W =− ρ S2(r ), (20)
ρ i i
Xi=1
The matrix W is positive definite, see Lemma 2 of [24]. Then, it has been shown in Lemma 1 of [24] that one can actually
ρ
rewrite z (·) as
ρ
z (t)=−2RT(q¯ (t)I −S(q¯(t)))W q¯(t). (21)
ρ d 0 ρ
One finally gets that the controller τ(·) can be expressed as
τ(t)=−2RT(q¯ (t)I −S(q¯(t)))W q¯(t)−Mωˆ(t). (22)
d 0 ρ
Using (11), (15), (6) and (22), we obtain the following closed loop dynamics
ξ˙ = −Aξ+B(Q¯)ω,
 q¯˙ = −1q¯TR ω,
 q¯˙0 = 12(2q¯0I+dS(q¯))Rdω, , (23)
Jω˙ = −S(ω)Jω−2RT(q¯ I−S(q¯))W q¯−Mωˆ.
One can make a further simplification by changing variables ads f0ollows: ρ
ξ →[R ˜bT(Q(t)),··· ,R ˜bT(Q(t))]T, ω →R ω.
d 1 d n d
By setting
J :=R JRT, B := S(R b )T ··· S(R b )T T ,
d d d d d 1 d n
(cid:2) (cid:3)
5
and by making obvious abuse of notations (i.e., we keep the variables ξ and ω) we end up with an autonomous differential
equation
ξ˙ = −A ξ+B ω,
d d
 q¯˙ = −1q¯Tω,
 q¯˙0 = 12(2q¯0I+S(q¯))ω, , (24)
J ω˙ = −S(ω)J ω−2(q¯ I−S(q¯))W q¯−BTΓA ξ.
 d d 0 ρ d d
q¯
Note that J is a real symmetric positive definite matrix. If one defines the state χ := (ξ, Q¯, ω) where Q¯ ≡ 0 ∈ S3
d (cid:20) q¯ (cid:21)
and the state space Υ:=R3n×S3×R3, one can rewrite (24) as χ˙ =F(χ) where F gathers the right-hand side of (24) and
defines a smooth vector field on Υ. Moreover, note that Q¯ and −Q¯ represents the same physical rotation, implying that (24)
projects on SO(3) as an autonomous differential equation. We will use that fact in Subsection ??.
Lemma 1. With the notations above, one gets that the matrix W defined in (20) has simple eigenvalues generically with
ρ
respect to ρ=(ρ ,··· ,ρ )∈(R∗)n.
1 n +
Proof: For ρ ∈ (R∗)n, let P (·) be the characteristic polynomial of W and ∆(ρ) its discriminant [38]. Recall that
+ ρ ρ
∆(ρ) = 0 if and only if P (·) admits a multiple root. Since W is a 3 by 3 real symmetric positive definite matrix or every
ρ ρ
ρ ∈ (R∗)n, ∆(ρ) is actually a homogeneous polynomial of degree four in ρ. Thus the locus ∆(ρ) = 0 defines an algebraic
+
variety of co-dimension one in (R∗)n and, on its complementary set S in (R∗)n, W has simple eigenvalues.
+ + ρ
This genericity result serves a justification to the following working hypothesis, which will hold for the rest of the paper.
(GEN)W has simple eigenvalues.
ρ
IV. STABILITY ANALYSIS OF THEPROPOSED CONTROLLER
In this section, we present a rigorous analysis using two representations. As often, it turns out that it is simpler for the
stability analysis to use unit quaternionsfor the representation of rotations instead of elements of SO(3), even-thoughwe are
ultimatelyinterestedin a resultformulatedin termsoforthogonalmatrices. Thisiswhywe first completethe stabilityanalysis
and obtain a first theorem (Theorem 1) using unit quaternions and, in a second step, we state our main result in terms of of
elements of SO(3) by simply projecting Theorem 1 using Rodriguez formula (1).
Lemma 2. Under the hypothesis (GEN), the solutions of equation z =0 where z is defined by (21) are the following: (a)
ρ ρ
the two points ±(1,0); the six points ±(0,v ), 1≤i≤3, with (v ,v ,v ) being an orthonormal basis diagonalizing W .
i 1 2 3 ρ
Proof: Let (q ,q)∈S3 such that z =0, i.e.,
0 ρ
(q I−S(q))W q =0.
0 ρ
If q 6= 0, it is immediate to see that q I −S(q) is invertible and thus q = 0, finally implying that q = ±1. If q = 0, we
0 0 0 0
are left with the equation S(q)W q =0. According to the properties of S(q) with q ∈S2, we get that q is an eigenvector of
ρ
W with unit length. We conclude by using (GEN).
ρ
Consider the following non negative differentiable function V : Υ→R+
V =ξTΓA ξ+4q¯TW q¯+ωTJ ω, (25)
d ρ d
which is radially unbounded over Υ since W and J are positive definite. Moreover, since Γ and A commute, the gain
ρ d d
matrix ΓA symmetric bloc diagonal positive definite.
d
Theorem 1. Consider System (3)-(6), under assumptions in sub-section (II-C) and the control law (22) with the auxiliary
system given by (11). Then, if Hypothesis (GEN). holds true, one gets that
(1) there are eight equilibrium points, given by
Ω+ =(0 , 1 , 0), Ω− =(0 , −1 , 0), Ω+ =(0 , 0 , 0), Ω− =(0 , 0 , 0),
1 3n (cid:20) 0 (cid:21) 1 3n (cid:20) 0 (cid:21) 2,3,4 3n (cid:20) vi (cid:21) 2,3,4 3n (cid:20) −vi (cid:21)
with (v ,v ,v ) is an orthonormal basis diagonalizing W .
1 2 3 ρ
(2) All trajectories of (3)-(6) converge to one of the equilibrium points defined in Item (1).
(3) Set c := 4λ (W ), where λ (W ) is the smallest eigenvalue of W . Then the equilibrium point Ω+ is locally
min ρ min ρ ρ 1
asymptotically stable with a domain of attraction containing the set
V+ :={χ∈Υ|V(χ)<c :andq¯ >0} (26)
c 0
6
−
and the equilibrium point Ω is locally asymptotically stable with a domain of attraction containing the set
1
−
V :={χ∈Υ|V(χ)<candq¯ <0}. (27)
c 0
±
(4) The other equilibrium points Ω are hyperbolic and not stable (i.e. the eigenvalues of each of the corresponding
2,3,4
linear systems have non zero real part and at least one of them has positive real part). This implies that System
±
(3)-(6) is almost globallyasymptotically stable with respect to the two equilibrium points Ω in the following sense:
1
there exists an open and dense subset Υ ⊂ Υ such that, for every initial condition χ ∈ Υ , the corresponding
0 0 0
trajectory converges asymptotically to either Ω+ or Ω−.
1 1
Proof: Regarding Item (1), one must solve the equation f(χ) = 0, where f is the nonlinear function describing (24).
Two cases can be considered. Assume first that q¯ 6=0. Both matrices q¯ I+S(q¯) and q¯ I−S(q¯) are non singular. Therefore
0 0 0
from the third equation of (24) ω =0 and thus ξ = 0 from the first equation of (24). The fourth equation of (24) reduces
3n
1
to z = 0 and one concludes that q¯ = 0 and q¯ = ±1 leading to two equilibrium points : Ω+ = (0 , , 0) and
ρ 0 1 3n (cid:20) 0 (cid:21)
Ω− =(0 , −1 , 0).
1 3n (cid:20) 0 (cid:21)
Assume that q¯ = 0. Then kq¯k = 1 and according to the third equation of (24), one gets that ω is parallel to q¯, let say
0
R ω =µq¯andthen µ mustbeequalto zeroaccordingtothe secondequationof(24), implyingthatω =0.As inthe previous
d
case,onededucesthatξ =0 .Thefourthequationof(24)yieldsthatq¯andW q¯areparallel,leadingtothesix pointsΩ± .
3n ρ 2,3,4
We now turn to an argument for Item (2). Using the facts that
ωTS(ω)=0, q¯TW (q¯ I+S(q¯))ω =ωT(q¯ I−S(q¯))W q¯, ωTBTΓξ =ξTΓBω,
ρ 0 0 ρ
the time derivative of (25) in view of (24) yields
V˙ =−ξTΛξ ≤0, (28)
since Λ = ATΓA +ΓA2 = 2ΓA2 is symmetric positive definite. We deduce that all trajectories of (24) are defined for all
d d d d
times and bounded.
Since (24) is autonomousand V is radially unbounded,one can use LaSalle’s invariancetheorem, cf. (28). Therefore every
trajectory converges to a trajectory γ along which V˙ ≡ 0. Then ξ must be identically equal to zero, implying at once that
B ω ≡ 0 as well. The latter assertion yields that ω must be collinear to all the b ’s, which can be true only if ω ≡ 0 since
d i
there are at least two non-collinear vectors b . From the fourth equation of (24) one can conclude that z =0 leading to the
i ρ
conclusion by Lemma 2.
We next address Item (3). We provide a proof only for Ω+ since the other case is entirely similar. Take an initial condition
1
χ¯ in Ω+. Since V is decreasing, the corresponding trajectory stays in V+ for all times and, for every t≥0, q¯(t)TW q¯(t)≤
1 c ρ
λ (W ). This implies that kq¯(t)k<1 for every t ≥0 and thus q¯ (t)6=0 for every t≥0. We deduce that q¯ (t) keeps the
min ρ 0 0
same sign namely that q¯ (0), which is positive. Since the trajectory converges to one of the eight equilibrium points, it must
0
be Ω+ since this is the only one contained in V+.
1 c
±
We finally provide an argument for Item (4). First of all notice the equilibrium points Ω , i = 2,3,4, cannot be locally
i
asymptotically stable. Indeed let Ω be one of these points and U any open neighborhoodof Ω in Υ. Define
−
V :={χ∈Υ|V(χ)<V(Ω},
Ω
and set U− := (V− ∩U). The set U− is obviously non empty since it contains points of the type λΩ with |λ| < 1 close
Ω
enough to 1. Moreover, for every χ∈U−, the trajectory of (24) does not converge to Ω since V is non increasing.
We next prove that the linearization of (24) at Ω is hyperbolic and admits an eigenvalue with positive real part. We first
performachangeofvariables.Ifq¯ =0thenq¯=σv ,whereσ =±1andv isaneigenvectorofW .Letususethefollowing
0 ρ ρ ρ
change of variable (cf. [8], [9], [39])
x 0 q¯ vTq¯
X = 0 = ⊙ 0 =σ ρ . (29)
(cid:20) x (cid:21) (cid:20) −σvρ (cid:21) (cid:20) q¯ (cid:21) (cid:20) −q¯0vρ−S(vρ)q¯ (cid:21)
From (29) we have
q¯ 0 x −vTx
0 = ⊙ 0 =σ ρ . (30)
(cid:20) q¯ (cid:21) (cid:20) σvρ (cid:21) (cid:20) x (cid:21) (cid:20) x0vρ+S(vρ)x (cid:21)
Rewrite (24) using (30) gives
ξ˙ = −A ξ+B (X)ω
d d
 x˙ = −1xTω
 x˙0 = 21(2x0I+S(x))ω (31)
J ω˙ = −BT(X)ΓA ξ−S(ω)J ω+2(x I−S(x))(λ I +S(v )W S(v ))x.
 d d d d 0 ρ ρ ρ ρ
7
SincethetangentspaceofS3 at 1 isgivenbytheequationx =0,thelinearizationofSystem(31)atΩ′ =(ξ, X, ω)=
(cid:20) 0 (cid:21) 0 2
1
(0 , , 0) is given by
3n 0
(cid:20) (cid:21)
−A 0 H
d
Z˙ =AZ, withA= 03 03 I3/2 , (32)
−J−1HTΓA 2J−1G 0
 d d d 3 
where Z = (zT, zT, zT)T, z , z , z are the linearized vectors of ξ, x, ω, respectively. G = λ I +S(v )W S(v ) and
ξ x ω ξ x ω ρ ρ ρ ρ
H = HT ··· HT Twith H =S R I+2S2(v ) bd . Since Ω is not locally asymptotically stable, it is enough to
1 n j d ρ j
show(cid:2)that A does not ad(cid:3)mit any eigenval(cid:0)ue w(cid:0)ith zero real p(cid:1)art.(cid:1)
Reasoning by contradiction, we thus assume that A has an eigenvalue il, l ≥ 0, with Zl = (zT,zT,zT)T ∈ C3n+6 a
1 2 3
corresponding eigenvector. One gets the linear system of equations
−A z + Hz =ilz ,
d 1 3 1
 z /2 =ilz , (33)
 −J−1HTΓA z + 2J−31Gz =ilz2.
d d 1 d 2 3
If l = 0, one gets z = z = 0 (sinceA is positive definite) and J−1Gz = 0. Recalling that W is real symmetric with
3 1 d d 2 ρ
distinct eigenvalues, we have that
W =λ v vT +λ v vT +λ v vT,
ρ ρ ρ ρ 1 1 1 2 2 2
where (v ,v ,v ) is an orthonormalbasis of R3 made of eigenvectorsof W . By using the properties of S(v ), one gets that
ρ 1 2 ρ ρ
G=λ v vT +(λ −λ )v vT +(λ −λ )v vT,
ρ ρ ρ ρ 2 1 1 ρ 1 2 2
implying that det(G) = λ (λ −λ )(λ −λ ) 6= 0 and thus z = 0. Then the eigenvector Z is equal to zero, which is
ρ ρ 1 ρ 2 2
impossible.
We deduce that l >0. One deduces that z =(A +ilI )−1Hz , z =− i z and
1 d 3n 3 2 2l 3
(i(J +G/l)+HTΓA (A +ilI )−1H)z =0. (34)
d d d 3n 3
Note that
n
HTΓA (A +ilI )−1H = HTΛ A (A +ilI )−1H .
d d 3n j j j j 3 j
Xj=1
Recall that, for 1≤j ≤n, A =P (Λ ) where P is a polynomial of degree two which is positive on R∗. One deduces that
j j j j +
3
λ P (λ )
HTΛ A (A +ilI )−1H = jk j jk w wT ,
j j j i 3 j P (λ )+il jk jk
Xk=1 j jk
where ((H−1)Tw ,(H−1)Tw ,(H−1)Tw ) is an orthonormal basis diagonalizing Λ .
j j1 j j2 j j3 j
Multiply Equation (34) on the left by (z∗)T. We get
3
n 3
λ P (λ )(P (λ )−il)
i(z∗)T(lJ +G/l)z + jk j jk j jk ((z∗)Tw )2 =0,
3 d 3 P (λ )2+l2 3 jk
Xj=1kX=1 j jk
where l>0. Since (z∗)T(lJ +G/l)z is a real number, we get
3 d 3
n 3 λ P (λ )2
jk j jk ((z∗)Tw )2 =0
P (λ )2+l2 3 jk
Xj=1Xk=1 j jk
We deduce at once that z =0 and finally Z =0, which is again a contradiction.
3
If A does not have eigenvalues with positive real part, it would have only eigenvalues with negative real part and thus A
would be Hurwitz, implying that (24) would be locally asymptotically stable with respect to Ω. Since this is not true, we get
that A does admit at least one eigenvalue with positive real part. We hence proved that there exists an unstable manifold of
±
dimension at least one in neighborhoodsof the Ω , j =2,3,4, and since all trajectories converge to an equilibrium point we
j
±
deduce that (24) is almost globally asymptotically stable with respect to the two equilibrium points Ω .
1
8
V. CONTROL GAINS TUNING AND SIMULATION RESULTS
This section provides a procedure to have ”optimal” gains (usually local) but approaching as near as possible to the global
solution.Theeffectivenessoftheproposedvelocity-freeattitudestabilizationcontrollerwillbe shownusingsimulationresults.
˜b
We denote a state vector χ=( 1 , Q, ω), where we take two non collinear vectors b , b . For simplicity and without
(cid:20) ˜b (cid:21) 1 2
2
loss of generality we take R = I, which means that q¯= q and bd = r . The matrices Λ , Λ are chosen diagonal such as
d i i 1 2
Λ =diag(γ ,γ ,γ ) where i=1,2, therefore the matrices A , A will be A =a I +a Λ +a Λ2 where i=1,2.
i i1 i2 i3 1 2 i i0 i1 i i2 i
In what follows, the following parameters are the same: the initial angular velocity ω(0)=[0, 0, 0]T, the inertial reference
vectorsr =[0, 0, 1]T and r =[1, 0, 1], the inertiamatrixJ =diag(0.5,0.5, 1), simulationsample time is 0.01swith RK4
1 2
algorithm. The notation “TRB controller” will be used to design the controller proposed in the paper [24].
A. Parameters Tuning
Let us define the problem. Consider the case when we use two non collinear inertial fixed vectors r , r (i.e. n = 2)
1 2
and we use the quaternion formulation of the closed loop dynamics (24). Consider now an objective function g(κ) such
that κ is the vector of all parameters to be tuned. The problem consists of finding min(g(κ)) with the following constraint
κ
l(κ(m))≤κ(m)≤u(κ(m)), where κ=[ ρ1 ρ2 a1(j−1) a2(j−1) γ1j γ2j ]T (j =1,...,3, κ∈(R∗+)14 is the vector
ofparameters),l(κ(m))andu(κ(m))arethelowerandupperboundscorrespondingtoeachparameterandκ(m)isanelement
of κ.
Generally, optimization algorithms find a local optimum. It depends on a basin of attraction of the starting point. Also, the
effectiveness of existing algorithms depends on the lower and upper limits. These last values can be determined based on the
dominant poles of the linearized system around the stable equilibrium point.
1
The linearization of (24) at Ω+ =(0 , , 0) can be written as follows
1 6 (cid:20) 0 (cid:21)
z˙ = −Az +Gz
ξ ξ ω
 z˙ = 1z (35)
 q 2 ω
Jz˙ = −GTΓAz −2W z ,
ω ξ ρ q
where G = GT GT T with G = S(r ), Γ and A are defined in Subsection III-A and W is defined in (20). Setting
1 2 i i ρ
Z =(zξT, zqT(cid:2), zωT)T with(cid:3)zξ ∈R6, zq ∈R3 and zω ∈R3 are the linearized vectors of ξ, q, ω, respectively. Then System (35)
can be rewritten as Z˙ =BZ, where
−A 0 G
3
B = 03 03 I3/2 .
−J−1GTΓA −2J−1W 0
ρ 3
 
Note that we used the fact that z = 0. The linearization of the closed loop dynamics is used to determine the upper and
q0
the lower limits uκ , lκ , respectively, for each parameter. Let’s take an arbitrary initial condition in Euler angle [ϕ, θ, ψ]=
i i
T
[30, 10, 45]° corresponding to Q(0) = 0.8804, 0.2704, −0.02089, 0.3891 . For an arbitrary chosen fixed κ(m), m =
3,...,14 gains values, we start by varyi(cid:2)ng κ(1) and κ(2). After inspecting th(cid:3)e zero-pole map, one can determine an upper
and lower bounds for κ(1) and κ(2) gains based on the placement of the dominant pole, if it exist. Same reasoning gives the
values in Table I.
ObjectiveFunctionsandOptimalControlGainsTuning: Sincethereexistmanypossibilitiestoselecttheobjectivefunction,
we tests different objective functions derived from three well known performance index. The first is Integral of Absolute
Error (IAE), the second is Integral of Time-weighted Absolute Error (ITAE) and the last is Integral of Square Error, with
the possibility to minimize energy and attitude error in the same time or not, by choosing σ ∈ [0 1]. The first con-
clusion after several simulations is that the most appropriate objective function for our application is the ISE function
gise(κ) = ´0∞(cid:0)kq¯k2+σkτk2(cid:1)dt with σ = 0.1. Indeed, it minimizes convergence time of the quaternion error and gives a
comparable energy consumption to the “TRB controller”, as we will see after. Initial gain vector are chosen arbitrary as
κ =[6, 6, 1, 0.4, 0.01, 1, 0.4, 0.01, 12, 11, 1, 10, 10, 10].
0
Togetanideaoftheeffectivenessoftheoptimizationusedmethods,wecomparethreefunctionstocalculategainsoptimally.
The first one uses KNITRO, the second one is based on the use of the Matlab fmincon function and the third method is based
on the use of the same function as the second method with variation of initial conditions of the parameters in a procedure
called global search because the locality of the solution essentially dependson the initial conditions.The best one is the third
one, i.e., the global search method and the final value κ with criterion ISE is presented in Table II. The corresponding
final
gain matrices are presented in Table III.
9
B. Simulation results
Let us now show the impact of the tuned gains on the nonlinear behavior of the new controller and the effectiveness of
the proposed controller compared with “TRB controller”. We therefore choose the same gains presented in [24] for the “TRB
T
controller” and the same initial condition Q(0) = 0.8, 0, 0, 0.6 . The evolution of the unit-quaternion trajectories with
respecttotimeforthenewand“TRBcontroller”are(cid:2)presentedinFig(cid:3)ure1,wherethestatetrajectoriesconvergeasymptotically
to the equilibriumpointΩ+. Figure 2 show the torqueapplied in the two controllers.It is clear that the introductionof matrix
1
gains gives better results with a comparable energy effort for the two controllers.
Figure 3 illustrate that the proposed controller and “TRB controller” can avoid the unwinding phenomenon, where the
−
state trajectories converge asymptotically to the equilibrium point Ω when starting from the initial condition Q(0) =
1
T
−0.8, 0, 0, 0.6 . But, it is clear that the new controller present better performances. Figure 4 show the appearance
o(cid:2)f the real angular(cid:3)velocity for the two controllers.
Remark 1. Note that even if the initial condition is a theoreticalunstable equilibrium point, we verified by simulation that the
numerical errors push the trajectories far from this point.
Remark 2. The controller proposed in [21] was tested. After many simulations, using several initial conditions, the first
conclusion is that the convergenceof quaterniontrajectories correspondingto the proposed controller in the present work and
“TRBcontroller”are,atleast,tentimefaster.Thesecondconclusionisthefactthattheperformanceofthecontrollerproposed
in [21] exhibitpoorperformanceswhenonlytwo inertialvectorsareusedcomparedto whatis presentedin[21],whereresults
use three vectors.
1
New controller
TRB controller
q00.9
0.8
0 1 2 3 4 5 6 7 8 9 10
0.2
New controller
TRB controller
q1 0
−0.2
0 1 2 3 4 5 6 7 8 9 10
0.04
New controller
0.02 TRB controller
q2
0
−0.02
0 1 2 3 4 5 6 7 8 9 10
1
New controller
0.5 TRB controller
q3
0
−0.5
0 1 2 3 4 5 6 7 8 9 10
time (s)
Figure1. Quaternion trajectories forthenewandTRBcontrollers withQ(0)= 0.8, 0, 0, 0.6 T
(cid:2) (cid:3)
VI. CONCLUSIONS
Wehaveproposedanattitudestabilizationcontrollerforrigidbody,inwhichneithertheangularvelocitynortheinstantaneous
measurements of the attitude are used in the feedback. This controller could be of great help (as main or backup controllers)
in applications where prone-to-failure and expensive gyroscopes are used. When almost all existing solutions to this problem
use the instantaneous attitude measurements, while it is well known that efficient attitude observer use the angular velocity
to obtain an accurate results, our approach overcomes totally reconstructing the attitude. It mainly uses an auxiliary system
that can be considered as an observer of the angular velocity using only the inertial measurements. The proposed controller
doesn’tuse the inertial fixed reference vectors, reduces the set of unstable equilibria of the closed loop dynamics with respect
to previous proposed controller, provides an almost global stability of the desirable equilibrium and avoids the "unwinding
phenomenon". In addition, it was shown that the set of control gains leading to a continuum of equilibria of the closed loop
system is an algebraic variety of positive co-dimension, independently on the choice of the observed vectors. A non-linear
optimal tuning method have been used to adjust properly the controller gains. We illustrated that the introduction of matrices
gains gives a better results compared with existing work. The performances and effectiveness of the proposed solution were
illustrated via simulation results.
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New controller
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