Angular Velocity Observer on the Special Orthogonal Group for Velocity-Free Rigid-Body Attitude Tracking Control
AAngular Velocity Observer on the Special Orthogonal Group forVelocity-Free Rigid-Body Attitude Tracking Control
Tse-Huai Wu and Taeyoung Lee ∗ Abstract — This paper studies a rigid body attitude trackingcontrol problem with attitude measurements only, when angularvelocity measurements are not available. An angular velocityobserver is constructed such that the estimated angular velocityis guaranteed to converge to the true angular velocity asymp-totically from almost all initial estimates. As it is developeddirectly on the special orthogonal group, it completely avoidssingularities, complexities, or discontinuities caused by minimalattitude representations or quaternions. Then, the presentedobserver is integrated with a proportional-derivative attitudetracking controller to show a separation type property, whereexponential stability is guaranteed for the combined observerand attitude control system.
I. I
NTRODUCTION
The problem of attitude control of a rigid body is one ofthe most popular research topics in control theory and prac-tice. The corresponding applications include aerial and un-derwater vehicles, robotics, and spacecraft dynamics. Manyapproaches have been studied in the attitude control problemto address various technical challenges [1], [2], [3], [4]. Inmost of the attitude control strategies, full states measure-ments, i.e., both attitude and angular velocity measurements,are required. However, angular velocity measurements arenot available in certain cases, for example, due to limitedsensing, power availability, and costs.Several approaches have been proposed for attitude con-trols without angular velocity measurements, where the valueof the angular velocity is estimated. A nonlinear angularvelocity observer is presented by Salcudean in [5], to con-struct an estimated angular velocity in terms of the attitudemeasurements based on an observer designed for a second-order linear system. However, the observer is designed andanalyzed separately from attitude control systems, assumingthat there is a separation principle-like property, i.e., it isassumed that the convergence of the controller is independentof the observer design. Recently, a switching-type angularvelocity observer is presented to show stability of an attitudecontrol system in terms of quaternions [6].There are other attitude control techniques that do not re-quire an estimate of the angular velocity. An auxiliary systemapproach is proposed based on the passivity property [7],[8], where the auxiliary system generates a damping termsimilar to a derivative term that is directly dependent on theangular velocity [1]. Additionally, a hybrid attitude tracking
Tse-Huai Wu and Taeyoung Lee, Mechanical and Aerospace En-gineering, The George Washington University, Washington DC 20052. { wu52,tylee } @gwu.edu This research has been supported in part by NSF under the grant CMMI-1243000 (transferred from 1029551), CMMI-1335008, and CNS-1337722. controller is proposed in the absence of angular velocityinformation [9], and a velocity-free adaptive controller isdeveloped for rigid-body attitude tracking [10].Most of these prior works on angular velocity observersand velocity-free attitude controls are constructed in termsof local parameterizations of the attitudes, or quaternions.Attitude control systems based on minimal representations,such as Euler angles or modified Rodrigues parameters,suffer from singularities in representing large angle rotationalmaneuvers. Quaternions do not have singularities. However,since the configuration space of quaternions, represented bythree-sphere double-covers the attitude configuration space ofthe special orthogonal group, one physical attitude actuallycorresponds to two antipodal quaternions. This ambiguityshould be carefully resolved for any quaternion-based atti-tude control system to avoid undesirable phenomena such asunwinding, where a rigid body unnecessarily rotates througha large angle, even if the initial attitude error is small, orit may become sensitive to small measurement noise [11],[12].This paper follows the first type of approaches that arebased on an estimated value of the angular velocity. Anangular velocity observer is constructed directly on thespecial orthogonal group, and it is shown that the zeroequilibrium of the estimation errors are almost globallyasymptotically stabile, i.e., it is asymptotically stable andthe region of attraction only excludes a set of zero Lebesguemeasure [2]. The second part of this paper is devoted to aseparation type property by integrating the proposed angularvelocity observer with a separately designed attitude trackingcontrol system. It is shown that the combined system yieldsexponential stability.Compared with the prior work [5], [6], the angular veloc-ity observer presented in this paper completely avoids theaforementioned issues of quaternions. Furthermore, in theswitching-based angular velocity observer [6], the observerperformance depends on the mass distribution of the rigidbody, since the convergence rate becomes slower and thenumber of switching increases as the rigid body becomesmore elongated. Frequent switching may cause undesiredbehaviors or even instability as illustrated by numericalexamples presented later in this paper. The main contributionof this paper can be summarized as (i) developing an angularvelocity observer on the special orthogonal group to avoidthe issues of quaternions and the dependency of convergencerates on the shape of a rigid body, and (ii) showing aseparation property mathematically rigorously and explicitlywithout need for discontinuities caused by switching. To au- a r X i v : . [ m a t h . O C ] M a r hor’s best knowledge, a separation-type property of attitudecontrols and angular velocity estimation has not been studiedbefore without a switching logic.The paper is organized as follows. A rigid-body dynamicmodel is introduced at Section II. An angular velocityobserver is presented at Section III, and a separation-typeproperty is shown at Section IV, followed by numericalexamples at Section V.II. R IGID B ODY A TTITUDE D YNAMICS
Consider the attitude dynamics of a fully-actuated rigidbody. Two coordinate frames are defined: an inertial ref-erence frame and a body-fixed frame. The attitude of therigid body is denoted by R ∈ SO ( ) that represents thetransformation of a representation of a vector from the body-fixed frame to the inertial reference frame. The configurationmanifold of attitude is the special orthogonal group: SO ( ) = { R ∈ R × | R T R = I, det[ R ] = 1 } . Let ω ∈ R and Ω ∈ R denote the angular velocity of therigid body with respect to the inertial reference frame andthe body-fixed frame, respectively. The governing equationsfor the rigid body attitude dynamics are given by ddt ( Jω ) = τ, J = RJ R T , (1) ˙ R = ˆ ωR = R ˆΩ , (2)where J ∈ R × is the fixed inertia matrix expressed inbody-fixed frame and τ is the control moment expressed inthe inertial reference frame. Note that the equation of motionfor the angular velocity, (1) is represented with respect tothe inertial frame. In addition, the hat map ∧ : R → so (3) transforms a vector in R to a × skew-symmetric matrixsuch that ˆ xy = x ∧ y = x × y for any x, y ∈ R . And theinverse of hat map is denoted by the vee map ∨ : so (3) → R . Several properties of hat map are listed as follows:tr [ A ˆ x ] = tr [ˆ xA ] = − x T ( A − A T ) , (3) R ˆ xR T = ( Rx ) ∧ , (4) ˆ xA + A T ˆ x = ( { tr [ A ] I × − A } x ) ∧ , (5)for any x ∈ R , A ∈ R × , R ∈ SO ( ) . The standard innerproduct of two vectors is denoted by x · y = x T y . Throughoutthis paper, I × denotes the × identity matrix and the 2-norm of matrix A is denoted by (cid:107) A (cid:107) . Also, λ M and λ m are defined as the maximum eigenvalue and the minimumeigenvalue of the inertia matrix J , respectively.III. A NGULAR V ELOCITY O BSERVER ON SO ( ) In this section, an observer is constructed such that theangular velocity is estimated when the attitude measurementsand the control input are available.
A. Estimate Frame
Define an orthonormal frame estimated by the observer.The attitude and angular velocity of the estimate frame withrespect to the inertial reference frame are denoted by ¯ R ∈ SO ( ) and ¯ ω ∈ R , respectively. More explicitly, ¯ R denotesthe linear transformation from the inertial reference frame tothe estimate frame. The discrepancy between the true attitude R and the estimated attitude ¯ R is denoted by a rotation matrix Q E ∈ SO ( ) , where Q E = R ¯ R T . (6)Note that Q E = I × when R = ¯ R .To further describe the error dynamics between R and ¯ R ,the estimate error variables are defined as follows: Ψ E = 12 tr [ G E ( I × − Q E )] , (7) e R E = 12 ( Q E G E − G E Q T E ) ∨ , (8) e ω E = Jω − J ¯ ω, (9)where Ψ E ∈ R , e R E ∈ R and e ω E ∈ R denote the estimateerror function, attitude estimate error vector and estimateangular velocity error, respectively. The matrix G E is definedas G E = diag [ (cid:15) , (cid:15) , (cid:15) ] ∈ R × where (cid:15) , (cid:15) , (cid:15) ∈ R aredistinct positive constants. B. Observer Design
The observer dynamics are defined as ddt ( J ¯ ω ) = τ + 12 k E J − e R E , (10) ˙¯ R = (cid:2) Q T E (¯ ω + k v J − e R E ) (cid:3) ∧ ¯ R, (11)where k E , k v ∈ R are positive constants. The observer isdesigned in the inertial reference, and it can be transformedto the body-fixed frame easily since the attitude is assumedto be available.The estimate error variables along the solution of the aboveobserver dynamics satisfy the following properties. Proposition 1:
The estimate error variables Q E , Ψ E , e R E ,and e ω E satisfy:(i) Ψ E is positive definite about R = ¯ R .(ii) Let the positive constants n , . . . , n be n = min { (cid:15) + (cid:15) , (cid:15) + (cid:15) , (cid:15) + (cid:15) } , (12) n = max { ( (cid:15) − (cid:15) ) , ( (cid:15) − (cid:15) ) , ( (cid:15) − (cid:15) ) } ,n = max { ( (cid:15) + (cid:15) ) , ( (cid:15) + (cid:15) ) , ( (cid:15) + (cid:15) ) } ,n = max { (cid:15) + (cid:15) , (cid:15) + (cid:15) , (cid:15) + (cid:15) } ,n = min { ( (cid:15) + (cid:15) ) , ( (cid:15) + (cid:15) ) , ( (cid:15) + (cid:15) ) } , and let ψ E < n . The error function Ψ E is locallyquadratic, i.e., n n + n (cid:107) e R E (cid:107) ≤ Ψ E ≤ n n n ( n − ψ E ) (cid:107) e R E (cid:107) , (13)where the upper bound is satisfied when Ψ E < ψ E .(iii) ˙ Q E = ˆ ω E Q E ,(iv) ˙Ψ E = ω T E e R E ,v) ˙ e R E = E o ( R, ¯ R ) ω E ,(vi) ˙ e ω E = − k E J − e R E ,where ω E ∈ R and E o ( R, ¯ R ) ∈ R × are given by ω E = ω − ¯ ω − k v J − e R E , (14) E o ( R, ¯ R ) = 12 ( tr [ Q E ] I × − e R E − G E Q T E ) . (15) Proof:
It is known that − ≤ tr [ R ] ≤ , for any rotationmatrix R ∈ SO ( ) , then it is clear that Ψ E ≥ and Ψ E = 0 only happens at Q E = I × , which verifies (i). To show (ii),the following properties in [13] are applied: For non-negativeconstants f , f , f , let F = diag [ f , f , f ] ∈ R × , and let P ∈ SO ( ) . Define Φ = 12 tr [ F ( I × − P )] , (16) e P = 12 ( F P − P T F ) ∨ . (17)Then, Φ is bounded by the square of the norm of e P as h h + h (cid:107) e P (cid:107) ≤ Φ ≤ h h h ( h − φ ) (cid:107) e P (cid:107) . (18)If Φ < φ < h for a constant φ , where h i are given by h = min { f + f , f + f , f + f } ,h = max { ( f − f ) , ( f − f ) , ( f − f ) } ,h = max { ( f + f ) , ( f + f ) , ( f + f ) } ,h = max { f + f , f + f , f + f } ,h = min { ( f + f ) , ( f + f ) , ( f + f ) } . Now, we choose F = G E and P = Q E , we then have Φ =Ψ E , e P = e R E , φ = ψ E and h i = n i , for i = { , , . . . , } .This shows (ii).From (2) and (11), the time-derivative of Q E is ˙ Q E = ˆ ωR ¯ R T − R ¯ R T (cid:2) Q T E (¯ ω + k v J − e R E ) (cid:3) ∧ . Using (4) and (14), it is rearranged as ˙ Q E = ˆ ωQ E − Q E Q T E (¯ ω + k v J − e R E ) ∧ Q E = ( ω − ¯ ω − k v J − e R E ) ∧ Q E = ˆ ω E Q E , which shows (iii).Next, the time-derivative of Ψ E is ˙Ψ E = − tr [ ˙ Q E G E ]= − tr (cid:2)(cid:0) ω − ¯ ω − k v J − e R E (cid:1) ∧ Q E G E (cid:3) . From (3), it is rewritten as ˙Ψ E = 12 ( ω − ¯ ω − k v J − e R E ) T ( Q E G E − G E Q T E ) ∨ = ( ω − ¯ ω − k v J − e R E ) T e R E (cid:44) ω T E e R E , which shows (iv). Next, according to (5) and (8), the time-derivative of e R E is ˙ e R E = 12 (cid:0) ˆ ω E Q E G E + G E Q T E ˆ ω E (cid:1) ∨ = 12 ( tr [ Q E G E ] I × − Q E G E ) ω E = 12 ( tr [ Q E ] I × − e R E − G E Q T E ) ω E , which shows (v). Finally, from (1), (9) and (10), we have ˙ e ω E = ddt ( Jω ) − ddt ( J ¯ ω ) = τ − ( τ + 12 k E J − e R E ) , which shows (vi).Next, we show that the zero equilibrium of the estimateerror variables is almost globally asymptotically stable. Proposition 2:
Consider the system given by (1), (2)with the angular velocity observer given by (10), (11). Thefollowing properties holds:(i) There are four equilibrium configurations, given by ( R, ω ) ∈ { ( ¯ R, ¯ ω ) , ( D i ¯ R, ¯ ω ) } , (19)for i = 1 , , , where D = diag [1 , − , − , D = diag [ − , , − and D = diag [ − , − , .(ii) The desired equilibrium ( R, ω ) = ( ¯ R, ¯ ω ) is almostglobally asymptotically stable.(iii) The remaining three undesired equilibrium configura-tions are unstable. Proof:
The equilibrium configurations happen at ( e R E , e ω E ) = (0 , . Clearly, in view of (9), e ω E = 0 yields ω = ¯ ω . From (8), e R E = 0 directly implies that Q E G E − G E Q T E = 0 , which follows that either Q E = I × or tr [ Q E ] = − [14, Theorem 5.1]. Therefore, Q E = R ¯ R T ∈{ I × , D , D , D } , which shows (i).Consider the following Lyapunov function, U = e T ω E e ω E + k E Ψ E , (20)which is positive definite about ( e ω E , e R E ) = (0 , . Fromthe properties (iv) and (vi) of Proposition 1, the time-derivative of U is given by ˙ U = 2 e T ω E ˙ e ω E + k E ˙Ψ E = 2 e T ω E ( − k E J − e R E ) + ( ω − ¯ ω − k v J − e R E ) T e R E = − k E k v e T R E ( J − e R E ) ≤ − k E k v λ M (cid:107) e R E (cid:107) . (21)Hence, one can conclude that e ω E , e R E are globally boundedand lim t →∞ (cid:107) e R E (cid:107) = 0 . Further, one can show that (cid:107) ¨ Q E (cid:107) is bounded and lim t →∞ (cid:82) t (cid:107) ˙ Q E (cid:107) dt (cid:48) = lim t →∞ (cid:107) Q E (cid:107) exits.From Barbalat Lemma, we conclude that lim t →∞ (cid:107) ˙ Q E (cid:107) = lim t →∞ (cid:107) ω − ¯ ω − k v J − e R E (cid:107) = 0 . (22)Since lim t →∞ (cid:107) e R E (cid:107) = 0 , it is clear that lim t →∞ (cid:107) ω − ¯ ω (cid:107) =0 , and this implies lim t →∞ (cid:107) e ω E (cid:107) = 0 . Consequently, theequilibrium ( e R E , e ω E ) = (0 , is asymptotically stable.However, the fact that lim t →∞ e R E = 0 does not necessar-ily imply that the estimated attitude asymptotically convergesto the true attitude. Instead, it asymptotically converges toeither the true attitude or one of the three undesired equilibriagiven by RD i for ≤ i ≤ .ext, we show the undesired equilibria are unstable. At thefirst undesired equilibrium ¯ R = RD , we have Ψ E = (cid:15) + (cid:15) .Define W = k E ( (cid:15) + (cid:15) ) − U , or W = k E ( (cid:15) + (cid:15) − Ψ E ) − (cid:107) e ω E (cid:107) . Then, W = 0 at the undesired equilibrium. Due to thecontinuity of Ψ E , in an arbitrarily small neighborhood of RD in SO ( ) , there exists ¯ R such that ( (cid:15) + (cid:15) ) − Ψ E > .For such attitudes, we can guarantee that W > if (cid:107) e ω E (cid:107) is sufficiently small. In other words, at any arbitrarily smallneighborhood of the undesired equilibrium, there exists adomain, namely U such that W > in U . And we have ˙ W = − ˙ U > from (21). According to Theorem 3.3at [15], the undesired equilibrium is unstable. The instabilityof the other two equilibrium configurations can be shownby the similar way. This shows the almost global asymptoticstability of (ii) as well as (iii).The presented angular velocity observer guarantees thatthe estimation errors asymptotically converge to zero foralmost all initial estimates, i.e., the region of attractionexcludes only a thin set of zero measure. This is thestrongest stability property for any continuous angular ve-locity observer, due to the topological restriction stating thatit is impossible to achieve global attractivity in the specialorthogonal group unless discontinuities are introduced.In contrast to the prior work given by [6] where the the ra-tio of λ M λ m has a crucial impact on the observer performance,the convergence property of the proposed angular velocityobserver is independent of the mass distribution of the rigidbody.IV. A TTITUDE T RACKING WITHOUT A NGULAR V ELOCITY M EASUREMENTS
In this section, we show a separation property of the angu-lar velocity observer developed in this previous section witha proportional-derivative attitude tracking control system on SO ( ) . A. Attitude Tracking Controls
We first review a attitude tracking controller developed on SO ( ) [16, Sec. 11.4.3] and [3]. Suppose the desired attitude R d ( t ) ∈ SO ( ) and the desired angular velocity Ω d ( t ) ∈ R are given as smooth functions of time, and they satisfy thekinematic equation ˙ R d = R d ˆΩ d . Let Q ∈ SO ( ) be therelative attitude of the desired attitude with respect to thecurrent attitude of the rigid body, i.e., Q = R T R d ∈ SO ( ) , The attitude tracking error variables are defined as
Ψ = 12 tr [ G ( I × − Q )] , (23) e R = 12 ( GQ T − QG ) ∨ , (24) e Ω = Ω − Q Ω d , (25)where Ψ ∈ R is the tracking attitude error function; e R , e Ω ∈ R are the tracking attitude error vector and tracking angular velocity error, respectively. The matrix G = diag [ g , g , g ] ∈ R × where g , g , g ∈ R are distinctpositive constants.The corresponding error dynamics are given as ˙Ψ = e T R e Ω , (26) ˙ e R = E c ( Q ) e Ω , (27) J ˙ e Ω = u + ˆ χe Ω − JQ ˙Ω d − (cid:100) Q Ω d J Q Ω d , (28)where E c ( Q ) ∈ R × and χ ∈ R are defined as E c ( Q ) = 12 ( tr [ QG ] I × − QG ) , (29) χ = J e Ω + (2 J − tr [ J ] I × ) Q Ω d , (30)and u ∈ R is the control moment expressed in the body-fixed frame, i.e., u = R T τ . Detailed analysis of the errorvariables has been addressed in [3], [4], [16].A proportional-derivative (PD) type controllers on SO ( ) is introduced as below. Proposition 3: ([3], [4], [16]) Consider the attitude dy-namics given by (1), (2). For positive constants k R , k Ω ∈ R ,let the control input be u = − k R e R − k Ω e Ω + J Q ˙Ω d + (cid:100) Q Ω d ( J Q Ω d ) . (31)Then, the zero equilibrium of the tracking errors ( e R , e Ω ) isalmost globally asymptotically stable. B. Separation-Type Property
The above PD-type attitude tracking control system re-quires that the angular velocity of the rigid body is availablealways. Here we show that the angular velocity observerpresented at Section III satisfies a separation property whencombined with the PD-type controller.Suppose that the true angular velocity Ω is not available tothe control system, and the angular velocity estimated by thepresented observer is applied instead. Define the estimatedangular velocity tracking error as ¯ e Ω = ¯Ω − Q Ω d , (32)where ¯Ω = R T ¯ ω is the estimate angular velocity expressedin the body-fixed frame. The stability properties of the corre-sponding combined observer and controller are summarizedas follows. Proposition 4:
Consider the attitude dynamics given by(1), (2) with the angular velocity observer given by (10),(11). The control input is chosen as u = − k R e R − k Ω ¯ e Ω + J Q ˙Ω d + (cid:100) Q Ω d ( J Q Ω d ) , (33)for positive constants k R , k Ω ∈ R . Assume that the inertiamatrix J of the rigid body and the weighting matrix G E satisfy λ M λ m < tr [ G E ] (cid:107) G E (cid:107) . (34)et ¯ ψ E be a positive constant satisfying ¯ ψ E < min { n , ( tr [ G E ] − λ M λ m (cid:107) G E (cid:107) ) } . Also, assume that the initialconditions satisfy Ψ E (0) < ¯ ψ E < min { n ,
12 ( tr [ G E ] − λ M λ m (cid:107) G E (cid:107) ) } , (35) (cid:107) e ω E (0) (cid:107) < k E ( ψ E − Ψ E (0)) . (36)Then, the desired equilibrium given by ( R, Ω , ¯ R, ¯Ω) =( R d , Ω d , R, Ω) is exponentially stable. Proof:
See Appendix.This proposition implies a separation property that thepresented angular velocity observer guarantees exponentialstability even when combined with an attitude trackingcontrol system. Unlike [6] where a switching logic, thatmay cause frequent switchings, is introduced, the trajectoriesalong the presented velocity-free attitude control schemeis free of discontinuities. This is critical in practice, asillustrated by numerical examples in the next section.While the performance of the angular velocity observerpresented in the previous section is independent of theinertia matrix, the ratio of the maximum eigenvalue to theminimum eigenvalue of the inertia matrix should satisfy(34) for the separation property when combined with theattitude tracking control system. Most of existing spacecraftsatisfy the assumption (34), and several numerical studiesshow that the separation property still holds even for variouselongated rigid bodies that do not satisfy (34). Extension ofthe presented results to eliminate (34) is referred to as futureinvestigation. V. N
UMERICAL E XAMPLES
To illustrate the performance of the presented angularvelocity observer, we consider two cases for attitude sta-bilization and attitude tracking.
A. Attitude Stabilization
We first consider a case of detumbling a rigid body,where the desired attitude and angular velocity are givenby R d = I × and Ω d = 0 . The inertial matrix is givenby J = diag [5 , ,
2] kgm . The initial condition is specifiedas R (0) = exp ( π/ e ) and Ω(0) = [1 , − . , .
5] rad / sec where e = [1 0 0] T . The matrix G and G E are selectedto be G = G E = diag [1 . , , . . In particular, the controlgains are given by k R = 16 J , k Ω = k v = 5 . J , and k E = 10 J . Note that the controller gains k R , k Ω , k v aregiven to be scalars throughout this paper but they can beeasily generalized to symmetric positive definite matrices.Numerical result of the proposed observer is illustrated atFig.1, which exhibits excellent convergence properties.For a comparison, the angular velocity observer andcontroller presented [6] are applied as well, and the cor-responding numerical results are illustrated at Fig. 2. As itis developed in terms of quaternions, the attitude estimationerror and the attitude tracking error are plotted as the scalarpart and the vector part of quaternions. It is shown thatthere are frequent switchings when t ≤ seconds, and thecorresponding control input has high-frequency oscillations. e R E (a) Attitude estimation error e R E ω − ¯ ω (b) Angular velocity estimation error ω − ¯ ω e R (c) Attitude stabilization error e R e Ω (d) Angular velocity stabilization er-ror e Ω u (e) Control moment u Fig. 1. Attitude stabilization with the presented angular velocity observeron SO ( ) B. Attitude Tracking
Next, we consider an attitude tracking problem. Thedesired attitude is given in terms of 3-2-1 Euler angles as R d ( t ) = R d ( α ( t ) , β ( t ) , γ ( t )) where α ( t ) , β ( t ) , and γ ( t ) are chosen as , sin(0 . t ) and cos(0 . t ) + 2 , respectively.The initial conditions and control gains are identical to theattitude stabilization example. The corresponding results areillustrated at Fig. 3, where both the estimation errors and thetracking errors converge to zero nicely.However, when the switching angular velocity [6] isapplied to the same tracking problem, there are persistentswitchings over the entire simulation period as illustratedby Fig. 4, and the estimation errors and tracking errors donot converge for the given simulation period of 40 seconds.Such frequent switchings or high-frequency oscillations inthe control moment may excite unmodelled dynamics orincrease sensitivity to noise. These comparisons illustratethe desirable numerical properties of the proposed angularvelocity observer explicitly.
10 20 30 400.990.99511.005 Time (sec) e q (a) Attitude estimation error (scalarpart of quaternion) e q v (b) Attitude estimation error (vectorpart of quaternion) Ω − ¯ Ω (c) Angular velocity estimation error Ω − ¯Ω u (d) Control moment u q e (e) Attitude stabilization error (scalarpart of quaternion) q e v (f) Attitude stabilization error (vectorpart of quaternion)Fig. 2. Attitude stabilization with the switching angular velocity observerin [6] A PPENDIX P ROOF OF P ROPOSITION G = diag[ g , g , g ] of thecontrol system given at (23), let ψ be a positive constantsatisfying ψ < min { g + g , g + g , g + g } . Considerthe following domain for the configuration of the attitudedynamics and the observer: D = { ( R, Ω , ¯ R, ¯Ω) ∈ ( SO ( ) × R ) | Ψ < ψ, Ψ E < ¯ ψ E } , The subsequent stability proof is developed in this domain.We first show that the estimated attitude and angular velocitytrajectories starting from the initial conditions satisfying (35)and (36) satisfy Ψ E < ¯ ψ E always, i.e., the estimated trag-ictory stay in the domain D . Recall the Lyapunov function U given at (20). For the initial estimates ¯ R (0) and ¯ ω (0) satisfying (35) and (36), we have U (0) < k E ( ¯ ψ E − Ψ E (0)) + Ψ E (0) = k E ¯ ψ E . As U ( t ) is non-increasing from (21), we have k E Ψ E ( t ) ≤ U ( t ) ≤ U (0) < k E ¯ ψ E , e R E (a) Attitude estimation error e R E ω − ¯ ω (b) Angular velocity estimation error ω − ¯ ω e R (c) Attitude tracking error e R e Ω (d) Angular velocity tracking error e Ω u (e) Control moment u Fig. 3. Attitude tracking with the presented angular velocity observer on SO ( ) which yields Ψ E ( t ) < ¯ ψ E < min { n ,
12 ( tr [ G E ] − λ M λ m (cid:107) G E (cid:107) ) } for all t ≥ .Consider the following Lyapunov function V = V c + V o , (37)where V c and V o are related to the controller and theobserver, respectively, and they are defined as V c = 12 e T Ω J e Ω + k R Ψ + c e T R J e Ω , V o = U − c e T ω E e R E = (cid:107) e ω E (cid:107) + k E Ψ E − c e T ω E e R E , for positive constants c and c . It has been shown that V c is positive definite about ( e Ω , e R ) = (0 , when c issufficiently small, and it satisfies α T M α ≤ V c ≤ α T M α, (38)where α = [ (cid:107) e Ω (cid:107) , (cid:107) e R (cid:107) ] T ∈ R , and the matrices M , M ∈ R × are defined as M = 12 (cid:20) λ m − c λ M − c λ M b k R (cid:21) , M = 12 (cid:20) λ M c λ M c λ M b k R (cid:21) ,
10 20 30 400.990.99511.005 Time (sec) e q (a) Attitude estimation error (scalarpart of quaternion) e q v (b) Attitude estimation error (vectorpart of quaternion) Ω − ¯ Ω (c) Angular velocity estimation error Ω − ¯Ω u (d) Control moment u q e (e) Attitude tracking error (scalar partof quaternion) q e v (f) Attitude tracking error (vector partof quaternion)Fig. 4. Attitude tracking with the switching angular velocity observer in [6] for a constant b , b that can be determined by the weightingmatrix G and ψ [3].Similarly, from (13), the second part of the Lyapunovfunction satisfies ξ T M ξ ≤ V o ≤ ξ T M ξ, (39)where ξ = [ (cid:107) e R E (cid:107) , (cid:107) e ω E (cid:107) ] T ∈ R and M = 12 (cid:20) k E n n + n − c − c (cid:21) , M = 12 (cid:20) k E n n n ( n − ψ E ) c c (cid:21) . If the constant c is chosen sufficiently small such that c < min { (cid:114) k E n n + n , (cid:115) k E n n n ( n − ψ E ) } , (40)then the matrices M and M are positive definite. Fromthese, the Lyapunov function V is positive definite about ( e Ω , e R , e ω E , e R E ) = (0 , , , and it is decrescent.From (26)-(28) and (33), the time-derivative of V c is ˙ V c = ( e Ω + c e R ) T J ˙ e Ω + k R ˙Ψ + c ˙ e T R J e Ω + k R e T R e Ω + c ( E c e Ω ) T J e Ω = − k Ω ( e Ω + c e R ) T ¯ e Ω − c k R (cid:107) e R (cid:107) + c e T R ˆ χe Ω + c ( E c e Ω ) T J e Ω . (41) The following properties of the error variables has beenshown in [4]: (cid:107) E c (cid:107) ≤ √ tr [ G ] , (cid:107) χ (cid:107) ≤ λ M (cid:107) e Ω (cid:107) + B ∗ , (cid:107) e R (cid:107) ≤ B ∗ (cid:44) (cid:112) p + 3 p , where the constants p , p are defined as p = max { ( g − g ) , ( g − g ) , ( g − g ) } ,p = max { ( g + g ) , ( g + g ) , ( g + g ) } . Applying these bounds to (41), we obtain ˙ V c ≤ − k Ω ( e Ω + c e R ) T ¯ e Ω − c k R (cid:107) e R (cid:107) + 12 c λ M ( √ tr [ G ] + B ∗ ) (cid:107) e Ω (cid:107) + c ( k Ω + B ∗ ) (cid:107) e R (cid:107)(cid:107) e Ω (cid:107) . (42)From (9) and the second part of (1), we can write e ω E = RJ R T ( ω − ¯ ω ) . Therefore, we have R T e ω E = J R T ( ω − ¯ ω ) = J (Ω − ¯Ω) , which follows that ¯Ω = Ω − J − R T e ω E . From this, the angular velocity error vector given by (32)can be rewritten as ¯ e Ω = e Ω − J − R T e ω E . (43)Substituting (43) into (42), we obtain ˙ V c ≤ − B ∗ (cid:107) e Ω (cid:107) − c k R (cid:107) e R (cid:107) + c ( k Ω + B ∗ ) (cid:107) e R (cid:107)(cid:107) e Ω (cid:107) + k Ω λ m (cid:107) e Ω (cid:107)(cid:107) µ (cid:107) + c k Ω λ m (cid:107) e R (cid:107)(cid:107) µ (cid:107) (44)where B ∗ = [ k Ω − c λ M ( √ tr [ G ] + B ∗ )] .Next, we find the time-derivative of V o . From (21) andproperties (v), (vi) of Proposition 1, we have ˙ V o = ˙ U − c ˙ e T ω E e R E − c e T ω E ˙ e R E = ˙ U + 12 c k E e T R E J − e R E − c e T ω E ( tr [ Q E G E ] I × − Q E G E ) ω E . (45)Equation (14) can be rewritten as ω E = J − [ J ( ω − ¯ ω ) − k v e R E ]= J − ( e ω E − k v e R E ) . (46)Substituting (46) and (21) into (45), we obtain ˙ V o ≤ − ( k v λ M − c λ m ) k E (cid:107) e R E (cid:107) − ( tr [ Q E G E ] λ M − (cid:107) G E (cid:107) λ m ) c (cid:107) e ω E (cid:107) + tr [ Q E G E ] + (cid:107) G E (cid:107) λ m c k v (cid:107) e ω E (cid:107)(cid:107) e R E (cid:107) . (47)ombining (44) and (47), the time-derivative of the Lya-punov function satisfies ˙ V ≤ − B ∗ (cid:107) e Ω (cid:107) − c k R (cid:107) e R (cid:107) + c ( k Ω + B ∗ ) (cid:107) e R (cid:107)(cid:107) e Ω (cid:107)− c k R (cid:107) e R (cid:107) + c k Ω λ m (cid:107) e R (cid:107)(cid:107) e ω E (cid:107) − c A E λ M λ m (cid:107) e ω E (cid:107)− B ∗ | e Ω (cid:107) + k Ω λ m (cid:107) e Ω (cid:107)(cid:107) e ω E (cid:107) − c A E λ M λ m (cid:107) e ω E (cid:107)− k v λ m − c λ M λ M λ m k E (cid:107) e R E (cid:107) − c A E λ M λ m (cid:107) e ω E (cid:107) + c k v λ m B E (cid:107) e ω E (cid:107)(cid:107) e R E (cid:107) , where A E = tr [ Q E G E ] λ m − (cid:107) G E (cid:107) λ M ∈ R and B E = tr [ Q E G E ] + (cid:107) G E (cid:107) ∈ R . Note that (34) ensures that A E > .This is rearranged as the following matrix form: ˙ V ≤ − α T W α − β T W β − ζ T W ζ − ξ T W ξ, where α = [ (cid:107) e Ω (cid:107) , (cid:107) e R (cid:107) ] T , β = [ (cid:107) e R (cid:107) , (cid:107) e ω E (cid:107) ] T , ζ =[ (cid:107) e Ω (cid:107) , (cid:107) e ω E (cid:107) ] T , ξ = [ (cid:107) e R E (cid:107) , (cid:107) e ω E (cid:107) ] T ∈ R and the matricesare defined as W = c (cid:34) B ∗ c k Ω + B ∗ k Ω + B ∗ k R (cid:35) ,W = c (cid:34) k R k Ω λ m k Ω λ m c A E c λ M (cid:35) ,W = 12 (cid:34) B ∗ − k Ω λ m − k Ω λ m c A E λ M λ m (cid:35) ,W = 12 λ m (cid:34) k E k v λ m − c λ M λ M − ck v B E − ck v B E c A E λ M (cid:35) . The constants c , c and the controller gains can be chosensuch that all of the above matrices are positive-definite. 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