On the Design of Globally Exponentially Stable Hybrid Attitude and Gyro-bias Observers
Soulaimane Berkane, Abdelkader Abdessameud, Abdelhamid Tayebi
aa r X i v : . [ m a t h . O C ] J a n On the Design of Globally Exponentially Stable HybridAttitude and Gyro-bias Observers
Soulaimane Berkane, Abdelkader Abdessameud and Abdelhamid Tayebi ∗† January 26, 2017
Abstract
This paper presents hybrid attitude and gyro-bias observers designed directly onthe Special Orthogonal group SO (3). The proposed hybrid observers, enjoying globalexponential stability, rely on a hysteresis-based switching between different configu-rations derived from a set of potential functions on SO (3). Different sets of potentialfunctions have been designed via an appropriate angular warping transformation ap-plied to some smooth and non-smooth potential functions on SO (3). We show that theproposed hybrid observers can be expressed solely in terms of inertial vector measure-ments and biased angular velocity readings. Simulation results are given to illustratethe effectiveness of the proposed attitude estimation approach. The attitude estimation problem has generated a great deal of research work in the lit-erature. The problem consists in recovering the attitude of a rigid body using availablemeasurements in the body frame. The early attitude estimators were of a static type,designed to reconstruct the attitude from a set of vector measurements (see, for instance,[1, 2]). These static attitude reconstruction techniques are hampered by their inability inhandling measurement noise. To overcome this problem, researchers looked for dynamicestimators (relying on the angular velocity and inertial vector measurements) having theability to recover the attitude while filtering measurement noise. Among these dynamicestimators, Kalman filters played a central role in aerospace applications (see, for instance,[3, 4]).Recently, a new class of dynamic nonlinear attitude estimators (observers) has emerged[5], and proved its ability in handling large rotational motions and measurement noise.This approach, coined nonlinear complementary filtering, was inspired from the linearattitude complementary filters, e.g., [6], used to recover (locally) the attitude using gyroand inertial vector measurements. The smooth nonlinear complementary filters, such asthose proposed in [5], are directly designed on SO (3) and are proved to guarantee almost global asymptotic stability (AGAS), which is as strong as the motion space topology couldpermit [7]. These smooth nonlinear observers ensure the convergence of the estimated at-titude to the actual one from almost all initial conditions except from a set of critical ∗ This work was supported by the National Sciences and Engineering Research Council of Canada(NSERC). † The authors are with the Department of Electrical and Computer Engineering, University of West-ern Ontario, London, Ontario, Canada. A. Tayebi is also with the Department of Electrical Engi-neering, Lakehead University, Thunder Bay, Ontario, Canada. [email protected], [email protected],[email protected] SO (3) has been successfullyaddressed via synergistic hybrid techniques. Following this approach, a non-central hybridattitude observer on SO (3) has been proposed in [14] leading to global asymptotic stabil-ity. Attitude estimators (evolving outside SO (3)) with global asymptotic and exponentialstability properties have also been proposed in [15] and [16], respectively.In the present work, we develop a comprehensive approach for the design of central hybrid attitude and gyro-bias observers on SO (3), using biased angular velocity and iner-tial vector measurements, leading to global exponential stability results. First, we proposea general structure of a hybrid attitude and gyro-bias observer evolving on SO (3), wherethe observer input depends on the gradient of some potential function on SO (3) indexedby a hybrid discrete jump. We show that global exponential stability is guaranteed pro-vided that the family of potential functions under consideration satisfies some properties.Thereafter, we propose four different methods in designing the central family of synergisticpotential functions on SO (3), via angular warping, enjoying the properties required forglobal exponential stability achievement. Two proposed estimation schemes rely on atti-tude information obtained using any reconstruction procedure. The other two proposedhybrid estimation schemes are explicitly expressed in terms of body-frame vector mea-surements. A preliminary and partial version of this work have been published in [17, 18].The present paper generalizes the approach and proposes different other possible designs. Throughout the paper, we use R , R + and N to denote, respectively, the sets of real,nonnegative real and natural numbers. We denote by R n the n -dimensional Euclideanspace and by S n the unit n -sphere embedded in R n +1 . We use k x k to denote the Euclideannorm of a vector x ∈ R n . For matrices A, B ∈ R m × n , the inner product is defined as hh A, B ii = tr( A ⊤ B ), and the Frobenius norm of A is k A k F = p hh A, A ii . We use λ Ai , λ A min and λ A max to denote, respectively, the i -th, minimum and maximum eigenvalues of asymmetric matrix A .Let the map [ · ] × : R → so (3) be defined such that [ x ] × y = x × y , for any x, y ∈ R ,where × is the vector cross-product on R and so (3) := n Ω ∈ R × | Ω ⊤ = − Ω o , is the vector space of 3-by-3 skew-symmetric matrices. Let [ · ] ⊗ : so (3) → R denote theinverse isomorphism of the map [ · ] × , such that [[ ω ] × ] ⊗ = ω, for all ω ∈ R and [[Ω] ⊗ ] × = Ω , for all Ω ∈ so (3). Defining P a : R × → so (3) as the projection map on the vector space so (3) such that P a ( A ) := ( A − A ⊤ ) /
2, one can extend the definition of [ · ] ⊗ to R × by The term central here refers to the use of a central family of potential functions on SO (3), where allthe potential functions in the family share the desired equilibrium point as a critical point. Note that incontrast with the non-central approach, each individual observer configuration derived from each potentialfunction in the central family, guarantees (independently) almost global asymptotic stability results. ψ defined for any matrix A = { a ij } ∈ R × as ψ ( A ) := h P a ( A ) i ⊗ = 12 a − a a − a a − a . (1)The following identities are useful throughout the paper.[ u ] × = −k u k I + uu ⊤ , u ∈ R , (2)[ u × v ] × = vu ⊤ − uv ⊤ , u, v ∈ R , (3)tr( uv ⊤ ) = u ⊤ v, u, v ∈ R , (4) hh A, [ u ] × ii = hh P a ( A ) , [ u ] × ii , A ∈ R × , u ∈ R , (5) hh [ v ] × , [ u ] × ii = 2 u ⊤ v, v, u ∈ R , (6) A [ u ] × + [ u ] × A ⊤ + [ A ⊤ u ] × = tr( A )[ u ] × , A ∈ R × , u ∈ R . (7) The rigid body attitude evolves on the special orthogonal group SO (3) := { X ∈ R × | det( X ) = 1 , XX ⊤ = I } , where I is the three-dimensional identity matrix and X ∈ SO (3) is called a rotationmatrix . The group SO (3) has a compact manifold structure with its tangent spaces beingidentified by T X SO (3) := { X Ω | Ω ∈ so (3) } . The inner product on R × , when restrictedto the Lie algebra of SO (3), defines the following left-invariant Riemannian metric on SO (3) h X Ω , X Ω i X := hh Ω , Ω ii , (8)for all X ∈ SO (3) and Ω , Ω ∈ so (3).A unit quaternion ( η, ǫ ) ∈ Q , consists of a scalar part η and three-dimensional vector ǫ , such that Q := { ( η, ǫ ) ∈ R | η + ǫ ⊤ ǫ = 1 } . A unit quaternion represents a rotationmatrix through the map R Q : Q → SO (3) defined as R Q ( η, ǫ ) = I + 2[ ǫ ] × + 2 η [ ǫ ] × . (9)The set Q forms a group with the quaternion product, denoted by ⊙ , being the groupoperation and quaternion inverse defined by ( η, ǫ ) − = ( η, − ǫ ) as well as the identity-quaternion (1 , × ), where 0 × ∈ R is a column vector of zeros. Given ( η , ǫ ) , ( η , ǫ ) ∈ Q , the quaternion product is defined by ( η , ǫ ) ⊙ ( η , ǫ ) = ( η , ǫ ) such that η = η η − ǫ ⊤ ǫ , ǫ = η ǫ + η ǫ + [ ǫ ] × ǫ , (10)and R Q ( η , ǫ ) R Q ( η , ǫ ) = R Q ( η , ǫ ) . (11)The rotation group SO (3) can be also parametrized by rotations of angle θ ∈ R arounda unit-vector axis u ∈ S . This is commonly known as the angle-axis parametrization of SO (3) and is given by the map R a : R × S → SO (3) such that R a ( θ, u ) = I + sin( θ )[ u ] × + (1 − cos θ )[ u ] × . (12) The reader is referred to [19] for more details on the unit quaternion representation. SO (3) are related through the formulas η = cos (cid:18) θ (cid:19) , ǫ = sin (cid:18) θ (cid:19) u. (13)For any attitude matrix X ∈ SO (3), we define | X | I ∈ [0 ,
1] as the normalized Euclideandistance on SO (3) which is given by | X | I := 18 k I − X k F = 14 tr( I − X ) . (14)The following results (proved in the Appendix section) will be useful in our subsequentanalysis. Lemma 1.
Consider the trajectories ˙ X ( t ) = X ( t )[ u ( t )] × where X ( t ) ∈ SO (3) and u ( t ) ∈ R . Then, ddt tr( A ( I − X )) = 2 ψ ( AX ) ⊤ u, (15) ddt ψ ( AX ) = E ( AX ) u. (16) where E ( AX ) := (tr( AX ) − X ⊤ A ) . Lemma 2.
Let A = A ⊤ and ¯ A := (tr( A ) I − A ) be positive definite. Let X ∈ SO (3) and ( η, ǫ ) ∈ Q be the quaternion representation of X . Then, the following relations hold: tr( A ( I − X )) = 4 ǫ ⊤ ¯ Aǫ, (17) ψ ( AX ) = 2( ηI − [ ǫ ] × ) ¯ Aǫ. (18)
Lemma 3.
Let A = A ⊤ and ¯ A := (tr( A ) I − A ) be positive definite. Then the followingrelations hold: λ ¯ A min | X | I ≤ tr( A ( I − X )) ≤ λ ¯ A max | X | I , (19) k ψ ( AX ) k = α ( A, X )tr( A ( I − X )) , (20)(1 − | X | I ) ≤ α ( A, X ) ≤ (1 − ξ | X | I ) , (21) v ⊤ [ λ ¯ A min − E ( AX )] v ≤
12 tr( A ( I − X )) k v k , ∀ v ∈ R , (22) k E ( AX ) k F ≤ k ¯ A k F , (23) where α ( A, X ) = (1 − | X | I cos ( u, ¯ Au )) , A = tr( ¯ A ) I − A , ξ = λ ¯ A min /λ ¯ A max and u ∈ S isthe axis of rotation X . Lemma 4.
Let A = P ni =1 ρ i v i v ⊤ i with n ≥ , ρ i > and v i ∈ R , i = 1 , . . . , n . Then, thefollowing hold: tr( A ( I − XY ⊤ )) = 12 n X i =1 ρ i k X ⊤ v i − Y ⊤ v i k , (24) ψ ( AXY ⊤ ) = 12 P n X i =1 ρ i ( X ⊤ v i × Y ⊤ v i ) , (25) for any matrices X, Y ∈ SO (3) . .3 Hybrid Systems Framework Let M be a given manifold. A general model of a hybrid system takes the form: (cid:26) ˙ x ∈ F ( x ) , x ∈ C,x + ∈ G ( x ) , x ∈ D, (26)where the flow map , F : M → T M governs the continuous flow of x on the manifold M , the flow set C ⊂ M dictates where the continuous flow could occur. The jump map , G : M → M , governs discrete jumps of the state q , and the jump set D ⊂ M defineswhere the discrete jumps are permitted. In this paper, we consider hybrid systems writtenin the following form ˙ z = f ( z, q ) , (cid:26) ˙ q = 0 , ( z, q ) ∈ C,q + ∈ g ( z, q ) , ( z, q ) ∈ D, (27)which is short-hand notation for a system of the form (26) with x = ( z, q ) , F ( x ) =[ f ( z, q ) ,
0] and G ( x ) = [ z, g ( z, q )].A subset E ⊂ R ≥ × N is a hybrid time domain , if it is a union of finitely or infinitelymany intervals of the form [ t j , t j +1 ] × { j } where 0 = t ≤ t ≤ t ≤ ... , with the lastinterval being possibly of the form [ t j , t j +1 ] × { j } or [ t j , ∞ ) × { j } . The ordering of pointson each hybrid time domain is such that ( t, j ) (cid:22) ( t ′ , j ′ ) if t ≤ t ′ and j ≤ j ′ . A hybridarc is a function h : dom h → M , where dom h is a hybrid time domain and, for eachfixed j , t h ( t, j ) is a locally absolutely continuous function on the interval I j = { t :( t, j ) ∈ dom h } . For more details on the dynamical hybrid systems framework, the readeris referred to [20, 21] and references therein. SO (3) × Q Given a finite index set
Q ⊂ N , let C ( SO (3) × Q , R + ) denote the set of positive-valuedfunctions Φ : SO (3) × Q → R + such that for each q ∈ Q , the map R Φ( R, q ) iscontinuous. If, for each q ∈ Q , the map R Φ( R, q ) is differentiable on the set D q ⊆ SO (3) then the function Φ( R, q ) is continuously differentiable on
D ⊆ SO (3) × Q , where D = ∪ q ∈Q D q × { q } , in which case we denote Φ ∈ C ( D , R + ). Additionally, for all ( R, q ) ∈D , let ∇ Φ( R, q ) ∈ T R SO (3) denote the gradient of Φ, with respect to R , relative to theRiemannian metric (8).A function Φ ∈ C ( SO (3) × Q , R + ) is said to be a potential function on D ⊆ SO (3) ×Q with respect to the set A ⊆ D if: • Φ( R, q ) > R, q ) / ∈ A , • Φ( R, q ) = 0, for all (
R, q ) ∈ A , • Φ ∈ C ( D , R + ).The set of all potential functions on D with respect to A = { I } × Q is denoted as P D ,where a function Φ( R, q ) ∈ P D can be seen as a family of potential functions on SO (3)indexed by the variable q . Let R ∈ SO (3) denote a rotation matrix from the body-fixed frame B to the inertial frame I . The rotation matrix R evolves according to the kinematics equation˙ R = R [ ω ] × , (28)5here ω ∈ R is the angular velocity of the body-fixed frame B with respect to the inertialframe I expressed in the body-fixed frame B . We suppose that a set of n ≥ b i , can be measured in the body-fixed frame and are associated to a set of n known inertial vectors, denoted by a i , such that b i = R ⊤ a i . (29) Assumption 1. n ≥ body-frame vectors b i are available for measurement, and at leastthree of these vectors are non-collinear. The vectors b i can be obtained, for example, from an inertial measurement unit (IMU)that typically includes an accelerometer and a magnetometer measuring, respectively, thegravitational field and Earth’s magnetic field expressed in the body-fixed frame. Also, wesuppose that the measured angular velocity, denoted by ω y , can be subject to a constantor slowly varying bias b ω ∈ R such that ω y = ω + b ω . (30)Our objective consists in designing an attitude and gyro-bias estimation algorithm, usingthe above described available measurements, leading to global exponential stability results. First, we propose a general design of a hybrid attitude and gyro-bias observer depending onan indexed potential function on SO (3) × Q such that Q is a finite index set. In particular,we show that a suitable choice of the potential function (satisfying some conditions) leadsto global exponential stability. Next, we propose different methods for the design of suchpotential functions satisfying our derived conditions. Then, depending on the choice ofthe potential function, and the assumptions on the available measurements described inSection 3, we propose four different hybrid observers achieving our objective. Let
Q ⊂ N be a finite index set and let ˆ R and ˆ b ω denote, respectively, the estimate ofthe rigid body rotation matrix R and the estimate of the constant bias vector b ω . Definethe attitude estimation error ˜ R = R ˆ R ⊤ and the bias estimation error ˜ b ω = b ω − ˆ b ω . Wepropose the following attitude and gyro-bias estimation scheme˙ˆ R = ˆ R h ω y − ˆ b ω + γ P β (cid:0) Φ( ˜
R, q ) (cid:1)i × , (31)˙ˆ b ω = − γ I β (cid:0) Φ( ˜
R, q ) (cid:1) , (32) β (cid:0) Φ( ˜
R, q ) (cid:1) = ˆ R ⊤ h ˜ R ⊤ ∇ Φ( ˜
R, q ) i ⊗ , (33)where ˆ R ( t ) ∈ SO (3), ˆ b ω ( t ) ∈ R , γ P and γ I are strictly positive scalars, and Φ ∈ P D forsome D ⊆ SO (3) × Q . The discrete jump variable q is generated by the following hybridmechanism ( ˙ q = 0 , ( ˜ R, q ) ∈ F ,q + ∈ argmin p ∈Q Φ( ˜
R, p ) , ( ˜ R, q ) ∈ J , (34)6here the flow set F and jump set J are defined by F = { ( ˜ R, q ) : Φ( ˜
R, q ) − min p ∈Q Φ( ˜
R, p ) ≤ δ } , (35) J = { ( ˜ R, q ) : Φ( ˜
R, q ) − min p ∈Q Φ( ˜
R, p ) ≥ δ } , (36)for some δ >
0. A necessary condition to implement this hybrid estimation scheme is that
F ⊆ D . Theorem 1.
Consider the attitude kinematics (28) coupled with the observer (31) - (36) .Assume that the potential function Φ ∈ P D , for some D ⊆ SO (3) × Q , and the hysteresisgap δ > are chosen such that F ⊆ D and α | ˜ R | I ≤ Φ( ˜
R, q ) ≤ α | ˜ R | I , ∀ ( ˜ R, q ) ∈ SO (3) × Q , (37) α | ˜ R | I ≤ k∇ Φ( ˜
R, q ) k F ≤ α | ˜ R | I , ∀ ( ˜ R, q ) ∈ F , (38) where α i > , i = 1 , . . . , are strictly positive scalars. Assume, in addition, that theangular velocity ω ( t ) is uniformly bounded. Then, the number of discrete jumps is finiteand the equilibrium point e = 0 , with e := ( | ˜ R | I , k ˜ b ω k ) ⊤ , is uniformly globally exponentiallystable.Proof. In view of (28), (30) and (31)-(32), one obtains˙˜ R = ˙ R ˆ R ⊤ − R ˆ R ⊤ ˙ˆ R ˆ R ⊤ = R [ ω ] × ˆ R ⊤ − R h ω y − ˆ b ω + γ P β (cid:0) Φ( ˜
R, q ) (cid:1)i × ˆ R ⊤ = ˜ R h ˆ R (cid:0) − ˜ b ω − γ P β (cid:0) Φ( ˜
R, q ) (cid:1)(cid:1)i × , where the fact that [ u ] × P ⊤ = P ⊤ [ P u ] × , for all u ∈ R and P ∈ SO (3), has been used toobtain the last equality. The closed loop dynamics during the flows of F × R are giventhen by ˙˜ R = ˜ R h ˆ R (cid:0) − ˜ b ω − γ P β (cid:0) Φ( ˜
R, q ) (cid:1)(cid:1)i × , (39)˙ q = 0 , (40)˙˜ b ω = γ I β (cid:0) Φ( ˜
R, q ) (cid:1)(cid:1) , (41)First, we show that ˜ b ω is bounded. To this end, we consider the following real-valuedfunction on SO (3) × Q × R L ( ˜ R, q, ˜ b ω ) = Φ( ˜ R, q ) + 1 γ I k ˜ b ω k , (42)which is positive definite with respect to¯ A := { ( ˜ R, q, ˜ b ω ) ∈ SO (3) × Q × R | ˜ R = I, ˜ b ω = 0 } . Now making use of (5)-(6), the time derivative of L , along the trajectories of (39)-(41),7an be shown to satisfy˙ L ( ˜ R, q, ˜ b ω ) = hh∇ Φ( ˜
R, q ) , ˜ R h ˆ R (cid:16) − ˜ b ω − γ P β (cid:0) Φ( ˜
R, q ) (cid:1)(cid:17)i × ii + 2 γ I ˜ b ⊤ ω (cid:16) γ I β (cid:0) Φ( ˜
R, q ) (cid:1)(cid:17) = − γ P hh∇ Φ( ˜
R, q ) , ˜ R h ˆ Rβ (cid:0) Φ( ˜
R, q ) (cid:1)i × ii− h ˜ R ⊤ ∇ Φ( ˜
R, q ) i ⊤⊗ ˆ R ˜ b ω + 2˜ b ⊤ ω β (cid:0) Φ( ˜
R, q ) (cid:1) = − γ P k∇ Φ( ˜
R, q ) k F ≤ , (43)for all ( ˜ R, q, ˜ b ω ) ∈ F × R , where (33) has been used to obtain the last equality. There-fore, L is non-increasing along the flows of F × R . Also, for all ( ˜ R, q ) ∈ J and q + ∈ argmin p ∈Q Φ( ˜
R, p ), one has L ( ˜ R, q + , ˜ b ω ) − L ( ˜ R, q, ˜ b ω ) = Φ( ˜ R, q + ) − Φ( ˜
R, q ) ≤ − δ, (44)which implies that L is strictly decreasing over the jumps. Consequently, the set ¯ A isstable by [22, Theorem 7.6]. Since the positive definite function L is non-increasing, everysolution is bounded preventing escape time. It should be noted that, since jumps map thestate to F \ J , it follows from [20, Proposition 2.4] that every solution is complete .Now, consider the Lyapunov function candidate L ( ˜ R, ˆ R, q, ˜ b ω ) = L ( ˜ R, q, ˜ b ω ) + µ ˜ b ⊤ ω ˆ R ⊤ ψ ( ˜ R ) , (45)where L is given in (42) and µ >
0. Using (19)-(20), it can be verified that k ψ ( ˜ R ) k = 4 | ˜ R | I (cid:0) − | ˜ R | I (cid:1) . (46)Hence, (37) and (46) can be used to show that the function L satisfies the quadraticinequality e ⊤ P e ≤ L ≤ e ⊤ P e, (47)where e = (cid:0) | ˜ R | I , k ˜ b ω k (cid:1) ⊤ and P = (cid:20) α − µ − µ /γ I (cid:21) , P = (cid:20) α µµ /γ I (cid:21) . The matrices P and P are positive definite provided that 0 < µ < p α /γ I and, in thiscase, the function L is positive definite with respect to the equilibrium e = 0.Now, we need to evaluate the time derivative of L along the flows of F × R . Makinguse of (16) with (31) and (39)-(41), the time derivative of X ( ˜ R, ˆ R, ˜ b ω ) := ˜ b ⊤ ω ˆ R ⊤ ψ ( ˜ R ) alongthe flows of F × R is obtained as˙ X ( ˜ R, ˆ R, ˜ b ω ) =˜ b ⊤ ω ˆ R ⊤ ˙ ψ ( ˜ R ) − ˜ b ⊤ ω [ ω − ˜ b ω + γ P β ] × ˆ R ⊤ ψ ( ˜ R ) + ˙˜ b ⊤ ω ˆ R ⊤ ψ ( ˜ R )= ˜ b ⊤ ω ˆ R ⊤ E ( ˜ R ) (cid:16) ˆ R ( − ˜ b ω − γ P β ) (cid:17) − ˜ b ⊤ ω [ ω + γ P β ] × ˆ R ⊤ ψ ( ˜ R ) + γ I β ⊤ ˆ R ⊤ ψ ( ˜ R ) , A solution h ( t, j ) to a hybrid system is complete if dom h is unbounded [23], i.e. , there is an infinitenumber of jumps and/or the continuous time is infinite. β have been omitted for simplicity. Let c ω := sup t ≥ ω ( t ) and c b := sup ( t,j ) (cid:23) ( t , k ˜ b ω ( t, j ) k , which exist, respectively, in view of our assumption on ω ( t )and the fact that all solutions are bounded. One can verify from (33) that k β k = 12 k∇ Φ( ˜
R, q ) k F ≤ α | ˜ R | I . Therefore, in view of inequalities (22)-(23), (46) and the above results, the time derivativeof the cross term X ( ˜ R, ˆ R, ˜ b ω ) during the flows of F × R satisfies˙ X ( ˜ R, ˆ R, ˜ b ω ) ≤ − k ˜ b ω k + 2 | ˜ R | I k ˜ b ω k + γ P √ k ˜ b ω kk β k + γ I k β k (cid:13)(cid:13) ψ ( ˜ R ) (cid:13)(cid:13) + c ω k ˜ b ω k (cid:13)(cid:13) ψ ( ˜ R ) (cid:13)(cid:13) + γ P c b k β k (cid:13)(cid:13) ψ ( ˜ R ) (cid:13)(cid:13) ≤ e ⊤ P X e, (48)with the matrix P X being defined as P X = √ α ( γ I + c b γ P ) + 2 c b c ω + γ P q α c ω + γ P q α − . Now, making use of inequality (48), the expression of the time derivative of L in (43)along with inequality (38), one can show that the time-derivative of L in (45) satisfies˙ L ≤ − e ⊤ (cid:18)(cid:20) γ P α
00 0 (cid:21) − µP X (cid:19)| {z } P e, (49)for all ( ˜ R, q, ˜ b ω ) ∈ F × R . To guarantee that the matrices P , P and P are positivedefinite, it is sufficient to pick µ such that0 < µ < min (cid:26) √ α √ γ I , γ P α P X + P X (cid:27) , where P X = √ α ( γ I + c b γ P ) + 2 c b and P X = c ω + γ P p α / L ≤ − λ F L with λ F := λ P min /λ P max ; L isexponentially decreasing along the flows of F . Equivalently, one has L ( t, j ) ≤ e − λ F ( t − t ′ ) L ( t ′ , j ) , (50)for all ( t, j ) , ( t ′ , j ) ∈ dom ( ˜ R, q, ˜ b ω ) with ( t, j ) (cid:23) ( t ′ , j ). Furthermore, it can verified from(45) and (44) that L ( ˜ R, ˆ R, q + , ˜ b ω ) − L ( ˜ R, ˆ R, q, ˜ b ω ) ≤ − δ, (51)for all ( ˜ R, q ) ∈ J and q + ∈ argmin p ∈Q Φ( ˜
R, p ). Consequently, one can conclude that L isstrictly decreasing over the jumps of J . Equivalently, for all ( t, j ) ∈ dom ( ˜ R, q, ˜ b ω ) suchthat ( t, j + 1) ∈ dom ( ˜ R, q, ˜ b ω ), one has L ( t, j + 1) − L ( t, j ) ≤ − δ. (52) Note that for the sake of presentation simplicity, we used L ( t, j ) to denote L (cid:0) ˜ R ( t, j ) , ˆ R ( t, j ) , q ( t, j ) , ˜ b ω ( t, j ) (cid:1) .
9t is clear from (47), (50) and (52), that the results of the theorem are trivial in the casewhere there is no discrete jumps, i.e., j = 0. Therefore, in the remainder of this proof, wewill consider only the case where j ≥
1. From (50)-(52), one can easily show that0 < L ( t, j ) ≤ L ( t , − δj, (53)which leads to j ≤ L ( t , − L ( t, j ) δ < L ( t , δ . (54)This shows that the number of jumps is finite. Since the solution is complete and thenumber of jumps is bounded, the hybrid time domain of the solution takes the formdom ( ˜ R, q, ˜ b ω ) = ∪ j max − j =0 (cid:0) [ t j , t j +1 ] × { j } (cid:1) ∪ [ t j max , + ∞ ) × { j max } where j max denotes themaximum number of discrete jumps.Now, one can show from (52) that L ( t, j + 1) ≤ (cid:18) − δ L ( t , (cid:19) L ( t, j ) ≤ e − σ L ( t, j ) , (55)for all ( t, j ) ∈ dom ( ˜ R, q, ˜ b ω ) such that ( t, j +1) ∈ dom ( ˜ R, q, ˜ b ω ), where σ = − ln (cid:16) − δ L ( t , (cid:17) .Note that δ < L ( t ,
0) as per (54) (and j ≥ j ≥ L ( t, j ) ≤ e − λ ( t − t + j ) L ( t , , (56)with λ = min { λ F , σ } . Using (50) and (55), one can easily show that (56) is satisfied for j = 1 as follows. Assume that ( t, ∈ dom ( ˜ R, q, ˜ b ω ) then L ( t, ≤ e − λ F ( t − t ) L ( t , ≤ e − λ F ( t − t ) e − σ L ( t , ≤ e − λ ( t − t ) e − λ e − λ ( t − t ) L ( t , ≤ e − λ ( t − t +1) L ( t , . Assuming that (56) holds true for j = k , and using (50) and (55), one has L ( t, k + 1) ≤ e − λ F ( t − t k +1 ) L ( t k +1 , k + 1) ≤ e − λ F ( t − t k +1 ) e − σ L ( t k +1 , k ) ≤ e − λ ( t − t k +1 ) e − λ e − λ ( t k +1 − t + k ) L ( t , ≤ e − λ ( t − t + k +1) L ( t , , for ( t, k + 1) ∈ dom ( ˜ R, q, ˜ b ω ). Therefore, inequality (56) also holds for j = k + 1 and henceit holds true for all j ≥
1. Finally, in view of (56) and (47), one can conclude that | e ( t, j ) | I ≤ k e e − λ ( t − t + j ) | e ( t , | I , (57)for all ( t, j ) ∈ dom ( ˜ R, q, ˜ b ω ), where k e = λ P max /λ P min . The proof is complete.Theorem 1 provides sufficient conditions on the potential function Φ ensuring that thehybrid attitude and gyro-bias estimation scheme (31)-(36) guarantees global exponentialstability of the estimation errors. It can be noticed that the estimator (31)-(36) relieson the rotation matrix ˜ R which is not directly available for feedback. The choice ofthe potential function Φ with the parameters of the hybrid mechanism in (34)-(36), and10he implementation of the proposed observer using the available measurements will bediscussed in detail in the next subsections.Before this, we consider the following two special cases that require some modificationof the estimation algorithm in Theorem 1. In the case where it is required to guarantee apriori bounded bias estimates, which is generally desirable in adaptive control algorithms,the proposed estimation algorithm can be modified using a projection mechanism providedthat the unknown bias b ω satisfies k b ω k ≤ ¯ b ω for some known ¯ b ω . In particular, theadaptation law (32) can be replaced by˙ˆ b ω = Proj (cid:16) − γ I β (Φ( ˜ R, q ) (cid:1) , ˆ b ω (cid:17) , (58)where Proj( µ, ˆ b ω ) = p (ˆ b ω ) µ , with p (ˆ b ω ) = I if k ˆ b ω k ≤ ¯ b ω or ˆ b ⊤ ω µ ≤
0, otherwise p (ˆ b ω ) = I − ˆ b ω ˆ b ⊤ ω / k ˆ b ω k , and satisfies the following properties [24]:(P1) k ˆ b ω ( t, j ) k ≤ ¯ b ω , ∀ ( t, j ) (cid:23) ( t , b ω − b ω ) ⊤ Proj( µ, ˆ b ω ) ≤ (ˆ b ω − b ω ) ⊤ µ, ,(P3) k Proj( µ, ˆ b ω ) k ≤ k µ k . Corollary 1.
Consider system (28) with the observer (31) with (58) and (33) - (36) , where k b ω k ≤ ¯ b ω . Let the potential function Φ and δ be selected as in Theorem 1 and assumethat the angular velocity ω ( t ) is uniformly bounded. Then, the equilibrium point e = 0 ,with e := ( | ˜ R | I , k ˆ b ω − b ω k ) ⊤ , is uniformly globally exponentially stable and the number ofdiscrete jumps is bounded.Proof. Consider the Lyapunov function candidate L , given in (45) with (42), and satisfies(47). Exploiting the properties P1-P3 and following similar steps in (43)-(44) and (48),one can show that the time-derivative of L satisfies (49) and (51) with c b = 2¯ b ω . Then,the result of the corollary follows using the same arguments after (49) in the proof ofTheorem 1.In the case where the angular velocity measurements are given by (30) with b ω ≡ i.e., , unbiased angular velocity measurements, the following corollary of Theorem 1 canbe shown. Corollary 2.
Consider the attitude kinematics (28) coupled with the observer (31) with (33) - (36) and ˆ b ω ≡ . Let the potential function Φ and δ be selected as in Theorem 1.Then, the number of discrete jumps is bounded and the equilibrium point | ˜ R | I = 0 isuniformly globally exponentially stable. More precisely, | ˜ R ( t, j ) | I ≤ α α e − γP α α ( t − t ) | ˜ R ( t , | I − δα j X s =1 e − γP α α ( t − t s ) , (59) for all ( t, j ) ∈ dom( ˜ R, q ) , where α i , i = 1 , ..., are given in (37) - (38) .Proof. Consider the Lyapunov function candidate ¯ L = Φ( ˜ R, q ). Following similar steps asin (42)-(44) in the proof of Theorem 1, the time-derivative of ¯ L is obtained as˙¯ L ≤ − γ P α | ˜ R | I ≤ − λ F Φ( ˜
R, q ) , (60)with λ F = γ P ( α /α ), during the flows of F . Hence, with the definition of ¯ L , one hasΦ( ˜ R ( t, j ) , q ( t, j )) ≤ e − λ F ( t − t ′ ) Φ( ˜ R ( t ′ , j ) , q ( t ′ , j )) , (61)11or all ( t, j ) , ( t ′ , j ) ∈ dom ( ˜ R, q ) such that ( t, j ) (cid:23) ( t ′ , j ). In addition, during the jumps of J , one has Φ( ˜ R ( t, j + 1) , q ( t, j + 1)) − Φ( ˜ R ( t, j ) , q ( t, j )) ≤ − δ, (62)for all ( t, j ) ∈ dom ( ˜ R, q ) such that ( t, j + 1) ∈ dom ( ˜ R, q ). Following similar steps as inthe proof of Theorem 1, it can be verified that the solution is complete, the number ofjumps is bounded and the hybrid time domain of the solution takes the form dom ( ˜
R, q ) = ∪ j max − j =0 (cid:0) [ t j , t j +1 ] × { j } (cid:1) ∪ [ t j max , + ∞ ) × { j max } where j max denotes the maximum numberof discrete jumps. The global exponential stability of | ˜ R | I = 0 also follows using similararguments as in the proof of Theorem 1.In the case where there is no discrete jump i.e., j = 0, the bound (59) follows directlyfrom (61) and (37). Note that the operator P js =1 is understood to be zero when j = 0.Now, let us consider the case when j ≥ holds for all j ≥ t, j ) ≤ e − λ F ( t − t ) Φ( t , − δ j X s =1 e − λ F ( t − t s ) . (63)It is not difficult to show that (63) holds for j = 1. In fact, in view of (62)-(61) and forall ( t, ∈ dom ( ˜ R, q ), one obtainsΦ( t, ≤ e − λ F ( t − t ) Φ( t , ≤ e − λ F ( t − t ) (cid:0) Φ( t , − δ (cid:1) ≤ e − λ F ( t − t ) (cid:0) e − λ F ( t − t ) Φ( t , − δ (cid:1) ≤ e − λ F ( t − t ) Φ( t , − δe − λ F ( t − t ) , which shows that inequality (63) holds for j = 1. Assuming that (63) holds for j = k , andusing (62)-(61), one has for ( t, k + 1) ∈ dom ( ˜ R, q )Φ( t, k + 1) ≤ e − λ F ( t − t k +1 ) Φ( t k +1 , k + 1) ≤ e − λ F ( t − t k +1 ) (cid:0) Φ( t k +1 , k ) − δ (cid:1) ≤ e − λ F ( t − t k +1 ) (cid:0) e − λ F ( t k +1 − t ) Φ( t , − δ P ks =1 e − λ F ( t k +1 − t s ) − δ (cid:1) ≤ e − λ F ( t − t ) Φ( t , − δ P k +1 s =1 e − λ F ( t − t s ) , which shows that (63) also holds for j = k + 1 and, hence, it holds true for all j ≥ Remark 1.
The bound obtained in (59) depends on the properties of the potential function Φ , especially the coefficients α , α and α , the observer gain γ P and the hysteresis gap δ .It should be mentioned here that a similar estimate of the error vector e in Theorem 1 canbe derived, however, it would depend on the unknown eigenvalues of matrix P in (49) . Φ In this subsection, we construct potential functions Φ, along with the hysteresis gap δ > Also for the sake of presentation simplicity, we use Φ( t, j ) to denote Φ (cid:0) ˜ R ( t, j ) , q ( t, j ) (cid:1) . .2.1 Traditional potential functions Consider the following potential function on SO (3) × { } :Φ s ( ˜ R,
1) = U A ( ˜ R ) := tr (cid:0) A ( I − ˜ R ) (cid:1) / λ ¯ A max , (64)where A = A ⊤ such that ¯ A := (tr( A ) I − A ) is positive definite. Note that the smooth function U A has been widely used in attitude control systems design [5, 13, 25, 26]. In viewof (19), it can be seen that Φ s ( ˜ R,
1) satisfies ξ | ˜ R | I ≤ Φ s ( ˜ R, ≤ | ˜ R | I , (65)where ξ := λ ¯ A min /λ ¯ A max and, hence, Φ s ( ˜ R,
1) satisfies the first condition (37) of Theorem 1.However, it can be shown that Φ s ( ˜ R,
1) does not satisfy (38) in Theorem 1. In particular,the relation α | ˜ R | I ≤ k∇ Φ s ( ˜ R, k F , α >
0, requires that ∇ Φ s ( ˜ R,
1) does not vanish onthe flow set F = SO (3) × { } except at ˜ R = I , which is not the case. In fact, the gradientof the potential function Φ s ( ˜ R,
1) is given by (see [13, Lemma 2]) ∇ Φ s ( ˜ R,
1) = ˜ R P a ( A ˜ R ) / λ ¯ A max , (66)which, in view of (5)-(6) and (20), satisfies k∇ Φ s ( ˜ R, k F = 116( λ ¯ A max ) hh P a ( A ˜ R ) , P a ( A ˜ R ) ii = 18( λ ¯ A max ) k ψ ( A ˜ R ) k , = 18( λ ¯ A max ) ǫ ⊤ ¯ A ǫ (cid:0) − k ǫ k cos ( φ ) (cid:1) , where φ := ∠ ( ǫ, ¯ Aǫ ) and ǫ is the vector part of the unit quaternion corresponding to theattitude matrix ˜ R . Consequently, the subset of SO (3) × { } where ∇ Φ s ( ˜ R,
1) = 0, calledalso the set of critical points of Φ s ( ˜ R, S I × { } and S π × { } where S I := { ˜ R ∈ SO (3) | ˜ R = I } , (67) S π := { ˜ R ∈ SO (3) | ˜ R = R Q (0 , v ) , v ∈ E ( A ) } , (68)with E ( A ) being the set of all eigenvectors of A .Therefore, the appearance of the undesired equilibrium points in S π cannot be avoidedwhen using an attitude observer design based on the gradient of U A , as done in [5] withsome matrix A . In addition, performance degradation (slow convergence) is reported inthis case for large attitude errors [9,14]. An alternate approach to design a gradient-basedattitude observer on SO (3) is to consider the following non-differentiable functionΦ ns ( ˜ R,
1) = V A ( ˜ R ) := 2 (cid:18) − q − U A ( ˜ R ) (cid:19) . (69)Since U A ( ˜ R ) ∈ [0 ,
1] and using (65), it is easy to verify that ξ | ˜ R | I ≤ U A ( ˜ R ) ≤ V A ( ˜ R ) ≤ U A ( ˜ R ) ≤ | ˜ R | I , (70)and hence Φ ns ( ˜ R,
1) is also quadratic with respect to | ˜ R | I ; Φ ns ( ˜ R,
1) satisfies (37) inTheorem 1. Note that we consider in (69) a weighted version of the function V I , obtained13rom (69) with A = I , used in [8] where it has been shown that control systems designedbased on V I exhibit faster convergence rates for large attitude manoeuvres as compared tothose designed using the smooth function U A . The potential function V I was also shownto be the solution to the kinematic optimal control problem on SO (3) in [27]. However,gradient-based control systems based on V I are less frequent in the literature, as comparedto those obtained from U A , due to its non-differentiability for rotations of angle 180 o . Infact, it can be verified from (69) that ∇ Φ ns ( ˜ R,
1) = ∇ Φ s ( ˜ R, q − Φ s ( ˜ R, , (71)which is not defined for all ( ˜ R,
1) satisfying Φ s ( ˜ R,
1) = U A ( ˜ R ) = 1. This corresponds tothe set S π, max × { } , with S π, max := { ˜ R ∈ SO (3) | ˜ R = R Q (0 , v ) ,v ∈ E ( A ) , ¯ Av = λ ¯ A max v } , (72)containing the singular points of Φ ns ( ˜ R, ns ( ˜ R, ∈ P D ns ( A ), with D ns = (cid:0) SO (3) \S π, max (cid:1) ×{ } , and, in addition, the critical points of Φ ns ( ˜ R,
1) are containedin (cid:0) S π \ S π, max (cid:1) × { } , with the set S π given in (68). Note that S π, max ⊆ S π ; In particular, S π, max = S π for A = I . It can be verified then that Φ ns ( ˜ R,
1) cannot satisfy (38) inTheorem 1 for all ( ˜ R, ∈ F , where F = SO (3) × { } in this case.Even though the functions Φ s ( ˜ R,
1) and Φ ns ( ˜ R,
1) do not satisfy the conditions of The-orem 1, they can both be used in the design of appropriate potential functions satisfyingour requirements through an adequate transformation described in the following.
To construct potential functions satisfying the conditions of Theorem 1, we introduce thefollowing angular warping transformation [13]Γ A ( ˜ R, q ) = ˜ R R A ( ˜ R, q ) (73) R A ( ˜ R, q ) = R a (cid:16) − ( kU A ( ˜ R )) , ν ( q ) (cid:17) , (74)where U A is defined as in (64) for some matrix A = A ⊤ such that ¯ A = tr( A ) I − A is positive definite, q ∈ Q for some index set Q ⊂ N , the map ν ( q ) : Q → S to bedetermined, and the scalar k satisfies0 < k < ¯ k := 1 p − max { , ξ } , ξ := λ ¯ A min λ ¯ A max . (75)The transformation Γ A ( ˜ R, q ) can be regarded as a perturbation of ˜ R about the unit vector ν ( q ) by an angle 2 sin − ( kU A ( ˜ R )). The above condition on the scalar k guarantees thatthe map ˜ R → Γ A ( ˜ R, q ) is everywhere a local diffeomorphism [13]. We recall the followingLemma which can be derived from [13, Lemma 1 & 3].
Lemma 5.
For any X ∈ SO (3) and ω ∈ R satisfying ˙ X = X [ ω ] × , one has ddt Γ A ( X, q ) = Γ A ( X, q ) [Θ A ( X, q ) ω ] × , (76)14 here the matrix Θ A ( X, q ) is full rank and is given by Θ A ( X, q ) = R A ( X, q ) ⊤ + 4 kν ( q ) ψ ( AX ) ⊤ λ ¯ A max q − k U A ( X ) . (77)Moreover, some useful properties of the transformation Γ A are given in the followinglemma proved in Appendix C.2. Lemma 6.
Consider the transformation Γ A in (73) - (75) . Then, γ | ˜ R | I ≤ | Γ A ( ˜ R, q ) | I ≤ γ | ˜ R | I , (78) for all ( ˜ R, q ) ∈ SO (3) × Q with ¯ γ, γ > given by γ := 1 − k − k p − k , γ := 1 + k + k . (79)It should be noted that γ in (78)-(79) is strictly positive under condition (75). Inequal-ity (78) shows that the transformation Γ A ( ˜ R, q ) acts on the attitude distance on SO (3),namely the norm | ˜ R | I , in a way such that the new attitude Γ A ( ˜ R, q ) has a distance, namely | Γ A ( ˜ R, q ) | I , which is bounded from below and above by a term proportional to the originalattitude distance | ˜ R | I .We show in the following that the potential functions U A (cid:0) Γ A ( ˜ R, q ) (cid:1) and V A (cid:0) Γ A ( ˜ R, q ) (cid:1) satisfy the conditions of Theorem 1 through an appropriate choice of matrix A , the indexset Q , the map ν , and the hysteresis gap δ in (34)-(36). Define Φ U A ( ˜ R, q ) = U A ◦ Γ A ( ˜ R, q ) , Φ V A ( ˜ R, q ) = V A ◦ Γ A ( ˜ R, q ) , (80)where U A and V A are given in (64) and (69), respectively, and Γ A is defined in Section 4.2.2.The main motivation behind introducing the modified functions in (80) is to avoid thecritical/singular points of the functions U A and V A using the transformation Γ A via anappropriate design of the switching mechanism (34)-(36). In fact, the transformationΓ A ( ˜ R, q ) allows to stretch and compress the manifold SO (3) by moving all the points(except the identity rotation I ) to different locations; In particular, for each index q ∈ Q ,the transformation Γ A ( ˜ R, q ) allows to re-locate all the points of the set S π given in (68).Consider the following possible designs of the parameters in (34)-(36) and (73)-(74):D1. Φ = Φ U A , A = I , Q ∈ { , · · · , } , ν ( p ) = e p , p ∈ { , , } , ν ( p + 3) = − ν ( p ) and0 < δ < ∆ I ( k ) := ( − √ k ) / k ; where { e , e , e } is any orthonormalbasis on R and the scalar k satisfies (75).D2. Φ = Φ V A , A = I , Q and ν ( · ) are defined as in D1, and the hysteresis gap δ satisfies0 < δ < ∆ II ( k ) := 2 p ∆ I ( k ), with k satisfying (75).D3. Φ = Φ U A , A is positive definite with the distinct eigenvalues 0 < λ < λ < λ , Q ∈ { , } , ν (1) = u, ν (2) = − u , where the vector u ∈ S satisfies: u ⊤ v = 0 , ( u ⊤ v i ) = λ Ai / ( λ A + λ A ) , (81)15or i ∈ { , } if λ A λ A − λ A λ A − λ A λ A ≥
0, or( u ⊤ v i ) = 1 − Q j = i λ Aj P ℓ P k = ℓ λ Aℓ λ Ak , (82)otherwise, for i ∈ { , , } and v i being the eigenvector of A corresponding to theeigenvalue λ Ai . The hysteresis gap δ satisfies 0 < δ < ∆ III ( k ) := 4 k ¯ V (1 − k ¯ V )Λwith k selected as in (75), ¯ V = [ − p k ξ Λ] / k Λ, andΛ := ( λ A / ( λ A + λ A ) if λ A λ A − λ A λ A − λ A λ A ≥ , Q j λ Aj ( λ A + λ A ) P ℓ P k = ℓ λ Aℓ λ Ak otherwise,D4. Φ = Φ V A , A , Q , and ν ( · ) are given as in D3, and the hysteresis gap satisfies 0 < δ < ∆ IV ( A, k ) := 2[ −√ − ξ + p − ξ + ∆ III ( A, k )], with k and ξ given in (75).Making use of one of the designs above, the following result, proved in Appendix C.3, canbe deduced. Lemma 7.
Consider the functions Φ U A and Φ V A in (80) with the transformation Γ A ( ˜ R, q ) being defined in (73) - (75) and the discrete variable q satisfying (34) - (36) . Suppose that Φ , Q , δ in (34) - (36) as well as matrix A and the map ν in (73) - (74) are selected accordingto one of the designs D1 – D4. Then, for all ( ˜ R, q ) ∈ F , one has Γ A ( ˜ R, q ) / ∈ S π where F is given in (35) and S π is defined in (68) . Moreover, for each of the designs in D1 – D4,we have Φ ∈ P D and F ⊆ D , with D being defined in the proof below. Lemma 7 indicates that Γ A ( ˜ R, q ) / ∈ S π can be guaranteed for all ( ˜ R, q ) ∈ F if oneconsiders one of the potential functions in (80) with an appropriate choice of the designparameters. This property is crucial in the design of hybrid observers ensuring globalstability results.Now, we show that the presented potential functions in Lemma 7 satisfy (37)-(38) inTheorem 1. For this, we need to compute the gradient of each potential function. Forany X ∈ SO (3), and ω ∈ R satisfying ˙ X = X [ ω ] × , one can show using (5)-(6), (66), and(76), that ddt Φ U A ( X, q )= hh∇ U A (Γ A ( X, q )) , ˙Γ A ( X, q ) ii = 14 λ ¯ A max hh P a ( A Γ A ( X, q )) , [Θ A ( X, q ) ω ] × ii = 12 λ ¯ A max ψ ( A Γ A ( X, q )) ⊤ Θ A ( X, q ) ω = 14 λ ¯ A max hh X [Θ A ( X, q ) ⊤ ψ ( A Γ A ( X, q ))] × , X [ ω ] × ii . On the other hand, the gradient of Φ U A verifies ddt Φ U A ( X, q ) = hh∇ Φ U A ( X, q ) , ˙ X ii . Consequently, ∇ Φ U A ( X, q ) = 14 λ ¯ A max X h Θ A ( X, q ) ⊤ ψ ( A Γ A ( X, q )) i × . (83)16n view of (69) and (80), and applying the chain rule, one has ∇ Φ V A ( X, q ) = ∇ Φ U A ( X, q ) p − Φ U A ( X, q ) . (84)It should be noted that the expression of the gradient ∇ Φ V A in (84) is well defined onthe flow set F due to the fact that Γ A ( X, q ) / ∈ S π for all ( X, q ) ∈ F (see Lemma 7). Proposition 1.
Consider the functions Φ U A and Φ V A in (80) with the transformation Γ A ( ˜ R, q ) being defined in (73) - (75) and the discrete variable q satisfying (34) - (36) . Supposethat Φ , Q , δ in (34) - (36) as well as matrix A and the map ν in (73) - (74) are selectedaccording to one of the designs D1 – D4. Then the conditions (37) - (38) in Theorem 1 aresatisfied for the potential function Φ U A with α = ξγ, α = γ,α = λξ γ ( ˜ R,q ) ∈F (cid:16) − | Γ A ( ˜ R, q ) | I cos ( φ q ) (cid:17) , α = λγ , and for the potential function Φ V A with α = ξγ, α = 2 γ,α = λξ γ ( ˜ R,q ) ∈F − | Γ A ( ˜ R, q ) | I cos ( φ q )1 − Φ U A ( ˜ R, q ) ! ,α = λγ ( ˜ R,q ) ∈F − | Γ A ( ˜ R, q ) | I cos ( φ q )1 − Φ U A ( ˜ R, q ) ! , where ( λ, λ ) are given in the proof below, ( γ, γ ) are defined in (79) , φ q = ∠ ( ǫ q , ¯ Aǫ q ) , and ǫ q is the vector part of the unit quaternion corresponding to the rotation Γ A ( ˜ R, q ) .Proof. First, using (65) (respectively (70)) and the results of Lemma 6, it is straightforwardto show that Φ U A (respectively Φ V A ) satisfies (37) with the corresponding α and α givenin the Proposition.Now, for ( ˜ R, q ) ∈ SO (3) × Q , let λ Θmin ( ˜
R, q ) and λ Θmax ( ˜
R, q ) denote, respectively,the smallest and largest eigenvalue of Θ A ( ˜ R, q )Θ A ( ˜ R, q ) ⊤ , and let the constants λ =min SO (3) ×Q ( λ Θmin ( ˜
R, q )) and λ = max SO (3) ×Q ( λ Θmax ( ˜
R, q )). It is clear that λ, λ > A ( ˜ R, q ) is full rank. Then, from (83), one can show that k∇ Φ U A ( ˜ R, q ) k F = 14( λ ¯ A max ) hh ˜ R h Θ A ( ˜ R, q ) ψ ( A Γ A ( ˜ R, q )) i × , ˜ R h Θ A ( ˜ R, q ) ψ ( A Γ A ( ˜ R, q )) i × ii = 14( λ ¯ A max ) (cid:13)(cid:13)(cid:13) Θ A ( ˜ R, q ) ψ ( A Γ A ( ˜ R, q )) (cid:13)(cid:13)(cid:13) ≤ λ λ ¯ A max ) (cid:13)(cid:13)(cid:13) ψ ( A Γ A ( ˜ R, q )) (cid:13)(cid:13)(cid:13) ≤ λ | Γ A ( ˜ R, q ) | I ≤ λγ | ˜ R | I , k∇ Φ U A ( ˜ R, q ) k F = 14( λ ¯ A max ) (cid:13)(cid:13)(cid:13) Θ A ( ˜ R, q ) ψ ( A Γ A ( ˜ R, q )) (cid:13)(cid:13)(cid:13) ≥ λ λ ¯ A max ) (cid:13)(cid:13)(cid:13) ψ ( A Γ A ( ˜ R, q )) (cid:13)(cid:13)(cid:13) ≥ λ ( λ ¯ A min ) λ ¯ A max ) | Γ A ( ˜ R, q ) | I (cid:16) −| Γ A ( ˜ R, q ) | I cos ( φ q ) (cid:17) . with φ q = ∠ ( ǫ q , ¯ Aǫ q ) and ǫ q is the vector part of the unit quaternion corresponding toΓ A ( ˜ R, q ). However, in view of Lemma 7, one has Γ A ( ˜ R, q ) / ∈ S π for all ( ˜ R, q ) ∈ F where v isan eigenvector of ¯ A which implies that ǫ q can not align with ¯ Aǫ q when | Γ A ( ˜ R, q ) | I = 1.Thisimplies that 1 − | Γ A ( ˜ R, q ) | I cos ( φ q ) > , during the flows of F . On the other hand, in view of (69), the gradient of the potentialfunction Φ V A satisfies (84). Therefore, in view of the above obtained results and using theresult of Lemma 6, the result of Proposition 1 follows.Proposition 1, with Lemma 7, provide several methods for the design of potentialfunctions satisfying the conditions in Theorem 1. Each of the proposed potential func-tions can be used for the design of the hybrid attitude and gyro-bias observer presentedin Subsection 4.1 leading to global exponential stability results. This is summarised asfollows: Theorem 2.
Consider the attitude kinematics (28) coupled with the observer (31) - (36) and let the potential function Φ( ˜
R, q ) and the hysteresis gap δ be selected as in Proposition1 and, accordingly, ∇ Φ( ˜
R, q ) in (33) is determined from (83) or (84) . Suppose that theangular velocity ω ( t ) is uniformly bounded. Then, the results of Theorem 1 hold. Remark 2.
The results of Corollary 1 and Corollary 2 can also be shown to hold whenusing the potential function
Φ( ˜
R, q ) and the hysteresis gap δ in Theorem 2. Remark 3.
Despite the fact that a similar stability result is guaranteed in all designcases D1 – D4, it is important to mention some differences between setting
Φ = Φ U A or Φ = Φ V A (for any A in Lemma 7). These differences mainly reside on the parameters α i , i = 1 , . . . , , obtained in Proposition 1 in each case. For example, the scalar α associatedto Φ V A is increased by a factor / (1 − Φ U A ) compared to the one associated to Φ U A . It canbe seen from (59) for instance that a change in α will influence the convergence rate ofthe attitude estimation error. The hybrid schemes described in the previous section depend explicitly on ˜ R = R ˆ R ⊤ ,and hence on R which is not directly measured. Note that it is possible to algebraicallyreconstruct the attitude matrix R from the available inertial measurements, described inSection 3, using static attitude determination algorithms [1, 2]. The resulting estimationscheme can be seen as a filtering algorithm for the reconstructed attitude matrix. It isdesirable in practice, however, to use directly the vector measurements in the estimationalgorithm without reconstructing the rotation matrix. In our preliminary result [17], we18ave shown that the hybrid observer in Theorem 2, with the parameters selected accordingto the second design method D2, (named Observer II) can be written using some newvectors defined as a combination of the measured inertial vectors. While the method in[17] is attractive in the sense that it allows for explicit expressions of the observer in termsof the newly defined vectors, it yet requires an important preconditioning process of themeasured vectors (see [17, Lemma 2]).In this section, we present explicit formulations of the hybrid attitude and gyro-biasobservers obtained using designs D3 and D4 in Theorem 2 using directly the measurementsof inertial vectors satisfying Assumption 1. Let the matrix A be defined as A = n X i =1 ρ i a i a ⊤ i , (85)for some ρ i >
0, and let ¯ A := tr( A ) I − A . Define also the following quantities¯Φ = 18 λ ¯ A max n X i =1 ρ i k b i − ˆ R ⊤ ¯ R a i k , ¯ β = 18 λ ¯ A max ˆ R ⊤ ¯Θ ˆ R n X i =1 ρ i ( b i × ˆ R ⊤ ¯ R a i ) , ¯ R = R a (2 sin − ( kϑ ) , ν ( q )) ,ϑ = n X i =1 ρ i k b i − ˆ R ⊤ a i k / λ ¯ A max , ¯Θ = (cid:16) I + k ˆ R P ni =1 ρ i ( b i × ˆ R ⊤ a i ) ν ( q ) ⊤ λ ¯ A max √ − ϑ (cid:17) . (86)Our result in this subsection is given in the following theorem. Theorem 3.
Consider the four hybrid observers obtained from (31) - (36) with the potentialfunction Φ , the index set Q , the map ν ( · ) , and the hysteresis gap δ being selected accordingto one of designs D3–D4, where Φ U A , Φ V A , Γ A are given in (80) and (73) - (75) . Also, letthe matrix A used in (80) and (73) be defined as in (85) with ρ i selected such that A has distinct eigenvalues. Suppose also that Assumption 1 on the vector measurements issatisfied. Then, the terms Φ( ˜
R, q ) and β (Φ( ˜ R, q )) in (31) - (36) can be written in terms ofthe measured vectors, for each observer, as follows:Observer III : (cid:26) Φ( ˜
R, q ) = ¯Φ ,β (Φ( ˜ R, q )) = ¯ β, (87) Observer IV : Φ( ˜
R, q ) = 2 (cid:0) − p − ¯Φ (cid:1) ,β (Φ( ˜ R, q )) = ¯ β √ − ¯Φ , (88) where ¯Φ , ¯ β are defined in (86) . In addition, both hybrid observers III and IV, when coupledwith system (28) , ensures the result of Theorem 1 provided that the angular velocity of therigid body is uniformly bounded.Proof. Let us show that relation (87) holds for the positive-definite matrix A as defined inthe theorem. Consider the smooth potential function U A given in (64). Using the result19n Lemma 4, one can easily deduce that U A ( ˜ R ) = 18 λ ¯ A max n X i =1 ρ i k b i − ˆ R ⊤ a i k , (89) ψ ( A ˜ R ) = 12 ˆ R n X i =1 ρ i ( b i × ˆ R ⊤ a i ) , (90)where we used ˜ R = R ˆ R ⊤ and b i = R ⊤ a i , i = 1 , . . . , n . It can also be deduced from Lemma4 and (80) with (73) thatΦ U A ( ˜ R, q ) = 18 λ ¯ A max n X i =1 ρ i k b i − ˆ R ⊤ R A ( ˜ R, q ) a i k , (91) ψ (Γ A ( ˜ R, q )) = 12 R A ( ˜ R, q ) ⊤ ˆ R n X i =1 ρ i ( b i × ˆ R ⊤ R A ( ˜ R, q ) a i ) , (92)with R A ( ˜ R, q ) given in (74). In addition, relations (33) with (83) can be used to showthat β (Φ U A ( ˜ R, q )) = 14 λ ¯ A max ˆ R ⊤ Θ A ( ˜ R, q ) ψ ( A Γ( ˜
R, q )) . (93)Then, (87) with (86) can be obtained by taking into account (74) and (77) with (89)-(92).It should be noted from Lemma 7 and (84) that β (cid:0) Φ V A ( ˜ R, q ) (cid:1) = β (cid:0) Φ U A ( ˜ R, q ) (cid:1) q − Φ U A ( ˜ R, q )is well defined on the flows of F . In addition, one can verify from (86) and (89) that ϑ = kU A ( ˜ R ) which, together with (64), imply that ϑ < A = P ni =1 ρ i a i a ⊤ i in this case is positive definite, it is possiblealways to select the weighting scalars ρ i > A are distinct.With such a choice, the parameters Φ, A , Q , ν ( · ) and δ correspond to design methodD3 (respectively D4). Therefore, the result of the theorem can be shown, for each hy-brid observer, using the results of Lemma 7, Proposition 1, and Theorem 1. The proof iscomplete. Remark 4.
Explicit formulation of the hybrid observers satisfying the results of Corollar-ies 1 and 2 can also be derived following the same lines in Theorem 3.
Theorem 3 provides explicit formulations of two hybrid attitude and gyro-bias ob-servers, in terms of the available inertial vector measurements. It can be noticed thatonly two configurations are needed for Observers III and IV. It is important to notice thatthis Assumption 1 is only technical and does not exclude the case where measurementsof only two vectors are available, say b and b corresponding to the non-collinear inertialvectors a and a . In this case, one can always construct a third vector b = b × b whichcorresponds to the measurement of a = a × a . The differences between Observers IIIand IV mentioned in Remark 3 will be further studied through numerical examples.20n [5], an attitude observer of the form (31)-(32) has been proposed with an input β being selected as the sum of the vector-errors between the measured vectors b i and theirestimates ˆ R ⊤ a i , i.e. , β = P ni =1 ρ i ( b i × ˆ R ⊤ a i ). This smooth observer can be obtained from(87), with (86), by setting k = 0 and choosing the parameters corresponding to ObserverIII. As mentioned above, the corresponding hybrid attitude and gyro-bias observer (Ob-server III) employs a switching mechanism between two observer configurations. Eachconfiguration is almost equivalent to the explicit attitude observer in [5] except that afactor proportional to ¯Θ (in (86)) is applied to the input β that is designed, in our case,based on the sum of vector-errors between the inertial measurements b i and their esti-mates perturbed by the rotation matrix ¯ R (in (86)). A key feature here is that, as theestimation error gets small, the values of ¯ R and ¯Θ approach the identity and the proposedhybrid scheme, i.e., Observer III in Theorem 3, becomes identical to the attitude observerproposed in [5]. On the other hand, for extremely large attitude estimation errors theperturbation matrix ¯ R becomes significant to guarantee the necessary gap between thetwo configurations. A similar remark can be made in regards of the attitude observer in [5](with an obvious modification related to the vector measurements) and hybrid Observer Iin Theorem 3, which switches between six observer configurations. In this section, we present numerical examples to validate our theoretical results. Considersystem (28) with ω ( t ) = [0 . . t ) , . . t + π ) , sin(0 . t + π/ ⊤ (rad / sec), and suppose that the measured angular velocity is given by (30) with the slowlyvarying bias b ω ( t ) = (cid:0) . . t ) (cid:1) [0 . , − . , . ⊤ . We also consider measurements of two non-collinear inertial vectors given by a =[1 , − , ⊤ / √ a = [0 , , ⊤ . We implement all the proposed hybrid observers (31)-(36), with different choices of the potential function Φ( ˜ R, q ), with ˆ b ω ( t ) = 0, q ( t ) = 1, γ P = 5 and γ I = 10. The projection operator for the bias estimation law is implementedwith a parameter bound ¯ b ω = 0 .
1. In all simulations, the selected initial attitude es-timates ˆ R ( t ) lead to a large initial attitude estimation error ˜ R ( t ) = R Q (0 , e ), with e = (1 , , ⊤ . We implement the two hybrid observers (Observer I and Observer II corresponding todesigns D1 and D2 and referred to as HO I and HO II , respectively, in the figures below). Theattitude matrix R is reconstructed using any static attitude determination method suchas SVD. The hysteresis gap of the hybrid switching mechanism is chosen as δ = 0 . I ( k ),for Observer I, and δ = 0 . II ( k ) for the Observer II, where the gain k is selected in bothcases as k = 0 . / √ I standing for smooth observer I)˙ˆ R = ˆ R h ω y − ˆ b ω + γ P β i × , ˆ R ( t ) ∈ SO (3) , (94)˙ˆ b ω = Proj (cid:16) − γ I β, ˆ b ω (cid:17) , ˆ b ω ( t ) ∈ R , (95) β = 14 ˆ R ⊤ ψ ( ˜ R ) , (96)which is inspired by the attitude observer proposed in [5]. This smooth attitude observercan be obtained from the hybrid observer (Observer I) by setting k = 0.The obtained results in this example are given in Figs. 1-2 showing, respectively, theattitude estimation error | ˜ R | I and the bias estimation error k ˜ b ω k . It can be seen fromthese figures that both hybrid attitude and gyro-bias observers ensure faster convergenceof the estimation errors as compared to the traditional smooth observer (94)-(33) despitethe large initial attitude estimation error. Also, as mentioned in Remark 3, the hybridobserver (Observer II) shows better performance (in terms of convergence) as compared toObserver I due to the nature of the potential function used in the design of each observer. Time (sec) | ˜ R | I SO I HO I HO II Figure 1:
Example 1. Attitude estimation error.
Time (sec) k ˜ b ω k SO I HO I HO II Figure 2:
Example 1. Bias estimation error. .2 Example 2 In this second example, we implement the two hybrid observers (Observer III and ObserverIV, referred to as HO
III and HO IV in the figures below) given in Theorem 3. Sinceonly two vector measurements are assumed to be available, we construct a third vector b = b × b associated to the inertial vector a = a × a such that Assumption 1 issatisfied. Accordingly, we consider the matrix A = P ni =1 ρ i a i a ⊤ i with ρ = 1 , ρ = 3and ρ = 1. Note that the eigenvalues and eigenvectors of A are used to determine theparameters of the hybrid observers in this example as presented in the design methods D3and D4 in Lemma 7. Similarly to the previous example, we select the hysteresis gap ofthe hybrid switching mechanism as δ = 0 . III ( k ), for Observer III, and δ = 0 . IV ( k )for Observer IV, where the gain k is selected as above so that condition (75) is verified.We also implement the following smooth attitude observer (that we refer to as SO II )for comparison purposes˙ˆ R = ˆ R h ω y − ˆ b ω + γ P β i × , ˆ R ( t ) ∈ SO (3) , (97)˙ˆ b ω = Proj (cid:16) − γ I β, ˆ b ω (cid:17) , ˆ b ω ( t ) ∈ R , (98) β = 18 λ ¯ A max 3 X i =1 ρ i (cid:0) b i × ˆ R ⊤ a i (cid:1) , (99)which is also inspired by [5] and [28], and obtained from Observer III in Theorem 3 bysetting k = 0.The obtained results are given in Figs. 3-4. Similarly to the previous example, one candeduce from these figures that Observer IV exhibits faster convergence as compared toObserver III, and both hybrid observers ensure better performance (in terms of convergencespeed) as compared to the smooth estimation algorithm (97)-(99). Time (sec) | ˜ R | I SO I HO III HO IV Figure 3:
Example 2. Attitude estimation error.
Nonlinear hybrid attitude and gyro-bias observers, leading to global exponential stabilityresults, have been proposed. These observers rely on gyro and inertial vector measurementswithout the need for the reconstruction of the rotation matrix. Different sets of potential23
Time (sec) k ˜ b ω k SO II HO III HO IV Figure 4:
Example 2. Bias estimation error. functions have been designed via an appropriate angular warping transformation appliedto some smooth and non-smooth potential functions on SO (3). These sets of potentialfunctions are the backbones for the switching mechanisms involved in the four proposedhybrid observers. Numerical examples have been given to illustrate the performance of theproposed hybrid observers as compared to smooth (non-hybrid) observers, inspired fromthe literature in the case of large initial attitude estimation errors. A Proof of Lemma 1
Along the trajectories of ˙ X = X [ u ] × , one has ddt tr( A ( I − X )) = − tr( A ˙ X ) = − tr( AX [ u ] × )= hh [ u ] × , AX ii = hh [ u ] × , P a ( AX ) ii = 2 u ⊤ ψ ( AX ) , where we have used identities (4)-(5). Moreover, using (7) one has˙ ψ ( AX ) = ddt [ P a ( AX )] ⊗ = 12 [ AX [ u ] × + [ u ] × X ⊤ A ] ⊗ , = 12 [tr( AX ) I − X ⊤ A ] u := E ( AX ) u. B Proof of Lemma 2
Let ( η, ǫ ) ∈ Q be the quaternion representation of the attitude matrix X . Using theRodrigues formula (9) and (5) one obtainstr( A ( I − R )) = tr( A ( − ǫ ] × − η [ ǫ ] × ))= − A [ ǫ ] × ) + 2 η hh [ ǫ ] × , A ii = − A [ ǫ ] × ) + 2 η hh [ ǫ ] × , P a ( A ) ii = − A [ ǫ ] × )24here we used P a ( A ) = 0 since A is symmetric. Now, using (2)-(4) one getstr( A ( I − R )) = 2tr( ǫ ⊤ ǫA − Aǫǫ ⊤ ) = 2 ǫ ⊤ (tr( A ) I − A ) ǫ := 4 ǫ ⊤ ¯ Aǫ.
Again, using (9) and (2), one has P a ( AR ) = 12 ( AR − R ⊤ A )= Aǫǫ ⊤ − ǫǫ ⊤ A + ηA [ ǫ ] × + η [ ǫ ] × A = [ ǫ × Aǫ ] × + 2 η [ ¯ Aǫ ] × , where (3) and (7) have been used. Consequently, one obtains ψ ( AR ) = ǫ × Aǫ + η ¯ Aǫ = 2( ηI − [ ǫ ] × ) ¯ Aǫ.
C Proof of Lemma 3
Let ( η, ǫ ) ∈ Q be the quaternion representation of the attitude matrix X . In view of (14)and (17), it is clear that | X | I = k ǫ k . Moreover, using again (17), one has4 λ ¯ A min | X | I = 4 λ ¯ A min k ǫ k ≤ tr( A ( I − X )) ≤ λ ¯ A max k ǫ k = 4 λ ¯ A max | X | I . Moreover, making use of (18) and identity (2), one obtains k ψ ( AX ) k = 4 ǫ ⊤ ¯ A ( ηI + [ ǫ ] × )( ηI − [ ǫ ] × ) ¯ Aǫ = 4 ǫ ⊤ ¯ A ( η I − [ ǫ ] × ) ¯ Aǫ = 4 ǫ ⊤ ¯ A ( I − ǫǫ ⊤ ) ¯ Aǫ = 4 ǫ ⊤ ¯ A ǫ (1 − k ǫ k cos ( φ )) , where the facts that η + ǫ ⊤ ǫ = 1 and ǫ ⊤ ¯ Aǫ = k ǫ kk ¯ Aǫ k cos( φ ) with φ = ∠ ( ǫ, ¯ Aǫ ), havebeen used. Finally, in view of (17), one hastr( A ( I − X )) = 4 ǫ ⊤ ¯ A ǫ, where ¯ A = (tr( A ) I − A ) or, equivalently, A = tr( ¯ A ) I − A which proves (20). Fur-thermore, since for any positive definite matrix ¯ A one has λ ¯ A min λ ¯ A max ≤ cos( u, ¯ Au ) ≤ , which implies (21). On the other hand, one has v ⊤ [ λ ¯ A min I − E ( AX )] v = v [ λ ¯ A min −
12 (tr( AX ) I − X ⊤ A )] v =( λ ¯ A min −
12 tr( AX )) k v k + 12 v ⊤ X ⊤ Av ≤ ( λ ¯ A min −
12 tr( AX )) k v k + 12 λ A max k v k = 12 tr( A ( I − X )) k v k , λ ¯ A min = (tr( A ) − λ A max ) has been used. Finally, using the fact thattr( AX ) ≤ tr( A ) for all X ∈ SO (3), it can be verified that k E ( AX ) k F = tr (cid:16) E ( AX ) ⊤ E ( AX ) (cid:17) = 14 tr (cid:16) A + tr ( AX ) I − tr( AX )( AX + X ⊤ A ) (cid:17) = 14 (tr( A ) + tr ( AX )) ≤
14 (tr( A ) + tr ( A ))= 14 tr ((tr( A ) I − A )(tr( A ) I − A ))= k ¯ A k F . C.1 Proof of Lemma 4
Making use of identity (4) one has n X i =1 ρ i k X ⊤ v i − Y ⊤ v i k = n X i =1 ρ i v ⊤ i ( I − Y X ⊤ )( I − XY ⊤ ) v i = n X i =1 ρ i tr( v i v ⊤ i ( I − XY ⊤ ))= tr( A ( I − XY ⊤ )) . Furthermore, making use of (3) one has P a ( AXY ⊤ ) = 12 n X i =1 ρ i (cid:16) v i v ⊤ i XY ⊤ − Y X ⊤ v i v ⊤ i (cid:17) = 12 n X i =1 ρ i Y (cid:16) Y ⊤ v i v ⊤ i X − X ⊤ v i v ⊤ i Y (cid:17) Y ⊤ = 12 n X i =1 ρ i Y h ( X ⊤ v i × Y ⊤ v i ) i × Y ⊤ , = 12 n X i =1 ρ i h Y ( X ⊤ v i × Y ⊤ v i ) i × , (100)where we used the property that Y [ v ] × Y ⊤ = [ Y v ] × for all Y ∈ SO (3) and v ∈ R . Takingthe [ · ] ⊗ operator on both sides of (100) yields (25). C.2 Proof of Lemma 6
Let ( η, ǫ ) and ( η q , ǫ q ) be, respectively, the unit quaternion representation of ˜ R and Γ A ( ˜ R, q ).Using (73)-(74) and (10)-(11), one can deduce that ǫ q = kηU A ( ˜ R ) ν ( q ) + q − k U A ( ˜ R ) ǫ + kU A ( ˜ R )[ ǫ ] × ν ( q ) . (101)26aking the norm square of ǫ q , equality (73) yields k ǫ q k = k η U A ( ˜ R ) + (1 − k U A ( ˜ R )) k ǫ k + k U A ( ˜ R ) k ǫ k sin ( ϕ q )+2 kη k ǫ k U A ( ˜ R ) q − k U A ( ˜ R ) cos( ϕ q ) , where ϕ q is the angle between ǫ and ν ( q ). In view of (64) and (17), one has U A ( ˜ R ) ≤ k ǫ k .Also since k ǫ k ∈ [0 , | η | · k ǫ k = k ǫ k p − k ǫ k ≤ /
2. It follows that k ǫ q k ≤ k ǫ k [1 + k + k / . (102)Moreover, in view of (75) and the fact that U A ( ˜ R ) ≤
1, one has kU A ( ˜ R ) < / √ k ǫ q k ≥ k ǫ k [1 − k U A ( ˜ R ) − kU A ( ˜ R ) q − k U A ( ˜ R )] ≥ k ǫ k [1 − k − k p − k ] , (103)where the fact that the scalar function 1 − x − x √ − x is decreasing on the interval x ∈ [0 , / √
2] has been used to obtain the last inequality. Now, in view of (64),(17),(102),(103) and the fact that | ˜ R | I = tr( I − ˜ R ) / k ǫ k , the result of Lemma 6 follows. C.3 Proof of Lemma 7
We prove the result of Lemma 7 for each design case.
Case of D1
Let Φ = Φ U A with A = I and Q = { , · · · , } . Suppose that Γ A ( ˜ R, q ) ∈ S π for ( ˜ R, q ) ∈ SO (3) × Q . Define Q = ( η, ǫ ), Q q = ( η q , ǫ q ) and Q p = ( η p , ǫ p ) as the unit quaternionrepresentation of the rotation matrices ˜ R , Γ A ( ˜ R, q ) and Γ A ( ˜ R, p ), respectively, for some p ∈ Q .Making use of the quaternion product rule (10) and (11), and the definition of the mapΓ A in (73)-(74), one has η q = η q − k U A ( ˜ R ) − kU A ( ˜ R ) ǫ ⊤ ν ( q ) , (104) η p = η q − k U A ( ˜ R ) − kU A ( ˜ R ) ǫ ⊤ ν ( p ) . (105)On the other hand, since Γ A ( ˜ R, q ) ∈ S π one has, in view of (17), U A (Γ A ( ˜ R, q )) = k ǫ q k = 1and hence η q = 0. Consequently, it follows from (105) that η p = kU A ( ˜ R ) ǫ ⊤ ν ( q ) − kU A ( ˜ R ) ǫ ⊤ ν ( p )= kU A ( ˜ R ) ǫ ⊤ ( ν ( q ) − ν ( p )) . Also, it can be verified, using (17), that U A ( ˜ R ) = k ǫ k , which yields η p = k k ǫ k (cos( ϑ ( q )) − cos( ϑ ( p ))) , (106)where the fact that ν ( q ) ⊤ ǫ = k ǫ k cos( ϑ ( q )), such that ϑ ( q ) = ∠ ( ν ( q ) , ǫ ), has been used.On the other hand, in view of (104) and the fact that η q = 0 and U A ( ˜ R ) = k ǫ k , oneobtains p − k ǫ k p − k k ǫ k = k ǫ k | cos( ϑ ( q )) | , | η | = p − k ǫ k has been used. Squaring both sides of the above equation, it followsthat 1 − k ǫ k = k k ǫ k (1 − sin ( ϑ ( q )) k ǫ k ) which results in the quadratic inequality1 − k ǫ k − k k ǫ k ≤ , where k ǫ k ∈ [0 , k ǫ k ≥ − √ k k . (107)Besides, since ν ( q + 3) = − ν ( q + 3), it follows that ϑ ( q + 3) = π + ϑ ( q ) for all q ∈ Q .Consequently, using relationmax y {| x + y | , | x − y |} = | x | + max y | y | , one can show that max p ∈Q (cid:12)(cid:12) cos( ϑ ( q )) − cos( ϑ ( p )) (cid:12)(cid:12) = max p ∈{ , , } n(cid:12)(cid:12) cos( ϑ ( q )) − cos( ϑ ( p )) (cid:12)(cid:12) , (cid:12)(cid:12) cos( ϑ ( q )) + | cos( ϑ ( p )) (cid:12)(cid:12)o = (cid:12)(cid:12) cos( ϑ ( q )) (cid:12)(cid:12) + max p ∈{ , , } (cid:12)(cid:12) cos( ϑ ( p )) (cid:12)(cid:12) ≥ √ where the fact that X p =1 cos ( ϑ ( p )) = 1 ≤ p ∈Q | cos( ϑ ( p )) | , has been used due to the orthogonality of { ν (1) , ν (2) , ν (3) } . Consequently, in view of(106) and (107) and the above result, one obtainsmax p ∈Q | η p | ≥ [ − √ k ] k , Therefore, it can be shown that, for all Γ A ( ˜ R, q ) ∈ S π , the following holds U A (Γ A ( ˜ R, q )) − min p ∈Q U A (Γ A ( ˜ R, p )) = k ǫ q k − min p ∈Q k ǫ p k = 1 − min p ∈Q (cid:0) − η p (cid:1) = max p ∈Q η p ≥ ∆ I ( k ) > δ. As a result, one can conclude that if Γ A ( ˜ R, q ) ∈ S π then ( ˜ R, q ) / ∈ F . By contraposition,for all ( ˜ R, q ) ∈ F , one has Γ A ( ˜ R, q ) / ∈ S π . Case of D2
Let Φ = Φ V A with A = I and Q = { , · · · , } . Since A = I , the set of all eigenvectors of A is identified by E ( A ) = S . Moreover, following similar steps as in the proof of D1, andfor all Γ A ( ˜ R, q ) ∈ S π , one has V A (Γ A ( ˜ R, q )) − min p ∈Q V A (Γ A ( ˜ R, p ))= 2 − min p ∈Q − q − k ǫ p k ]= 2 max p ∈Q | η | p ≥ p ∆ I ( k ) = ∆ II ( k ) > δ. It follows that if Γ A ( ˜ R, q ) ∈ S π then ( ˜ R, q ) / ∈ F . It follows, by contraposition, that for all( ˜ R, q ) ∈ F , one has Γ A ( ˜ R, q ) / ∈ S π . 28 ase of D3 In [13], we have shown that the function Φ U A with the parameters selected as in D3satisfies: U A (Γ A ( ˜ R, q )) − min p ∈Q U A (Γ A ( ˜ R, p )) ≥ ∆ III ( A, k ) , (108)for all ( ˜ R, q ) satisfying Γ A ( ˜ R, q ) = R Q (0 , v ) where v is an eigenvector of A . Since 0 < δ < ∆ III ( A, k ), it is clear that the set where Γ A ( ˜ R, q ) = R Q (0 , v ) lies entirely in the jump set J . Hence, the attitude Γ A ( ˜ R, q ) can not be equal to R Q (0 , v ) for any v ∈ S (eigenvectorof A ) during the flows of F . Case of D4
Let ( ˜
R, q ) satisfying Γ A ( ˜ R, q ) = R Q (0 , v ) where v is an eigenvector of A , or equivalentlyΓ A ( ˜ R, q ) ∈ S π . In view of (108) and using the fact that ξ ≤ U A ( R Q (0 , v )) ≤
1, one canconclude that U A (Γ A ( ˜ R, q )) − min p ∈Q U A (Γ A ( ˜ R, p )) ≥ ∆ III ( A, k )= ∆ IV ( A, k ) / IV ( A, k ) p − ξ ≥ ∆ IV ( A, k ) / IV ( A, k ) q − U A (Γ A ( ˜ R, q )) , Hence, by completing the squares, one obtainsmax p ∈Q q − U A (Γ A ( ˜ R, p )) − q − U A (Γ A ( ˜ R, q )) ≥ ∆ IV ( A, k ) / , or equivalently V A (Γ A ( ˜ R, q )) − min p ∈Q V A (Γ A ( ˜ R, p )) ≥ ∆ IV ( A, k ) . Hence, if δ < ∆ IV ( A, k ) then it is obvious that the set where Γ A ( ˜ R, q ) = R Q (0 , v ) liesentirely in the jump set J .Now, let us show that F ⊆ D for all the cases. For D1 and D3, the potential functionΦ = Φ U A is differentiable on all SO (3) × Q due to the fact that U A is smooth on SO (3)and the transformation Γ A is differentiable everywhere as shown in Lemma 6. Thus, F ⊆ D = SO (3) × Q holds. The potential function Φ V A , however, is differentiable on theset D = { ( ˜ R, q ) ∈ SO (3) × Q | Φ U A ( ˜ R, q ) = 1 } . Let ( ˜
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