Velocity-Free Hybrid Attitude Stabilization Using Inertial Vector Measurements

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1 016 American Control Conference (ACC) Boston Marriott Copley Place July 6-8, 016. Boston, MA, USA Velocity-Free Hybrid Attitude Stabilization Using Inertial Vector Measurements Soulaimane Berkane and Abdelhamid Tayebi Abstract This paper deals with the design of a velocity-free hybrid attitude stabilization scheme relying solely on inertial vector measurements. The proposed control approach, relying on the concept of synergistic potential functions, leads to global asymptotic stability results. Comparative simulation results between the proposed global hybrid control scheme and an existing almost global smooth control scheme have been carried out. I. INTRODUCTION The attitude control problem is a long-lasting challenging problem that generated many interesting research papers in the literature (see, for instance, [1], [], [3] and [4]). The inherent topology of the configuration manifold SO(3) precludes the existence of globally stabilizing continuous timeinvariant feedback laws for attitude systems [5]. Moreover, there is no discontinuous feedback that robustly and globally stabilizes a desired rigid body attitude [6]. To overcome the aforementioned topological obstruction to global asymptotic stability while, in the same time, ensuring robustness to small measurement noise, the authors in [7] proposed a hybrid feedback strategy based on the concept of synergistic potential functions on SO(3). A family of potential functions is said to be synergistic if at each critical point (other than the desired one) of a potential function in the family, there exists another potential function in the family that has a lower value. The coordination of control laws derived from these potential functions leads to global asymptotic stability results (see, for instance, [7], [8], [9]). Output feedback control for rigid body attitude systems has received increasing attention in the last few years. Indeed, in practice, gyroscopes are expensive and prone to failure which rises the challenge of designing attitude control systems that do not necessitate angular velocity measurements. Several approaches exist for the design of velocity-free attitude controllers. Angular velocity observer based control approaches have been proposed in [10], [11], where a separation-like property has been proved. Another velocity-free attitude control approach consists of using an auxiliary system to generate the necessary damping for the system [3]. These solutions are only almost global and/or suffer from pitfalls of using quaternion representation, eg., 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 Western Ontario, London, Ontario, Canada. The second author is also with the Department of Electrical Engineering, Lakehead University, Thunder Bay, Ontario, Canada. sberkane@uwo.ca, tayebi@ieee.org unwinding phenomenon. Globally asymptotically stable hybrid velocity-free attitude controllers have been proposed in [4], [1] assuming that the quaternion attitude is available for feedback. Since there is no sensor that provides directly a measurement of the attitude, an attitude estimation algorithm, which usually relies on angular velocity measurements, is required. In the present paper, we design a hybrid velocity-free attitude stabilization scheme relying solely on inertial vector measurements. The approach [13] for the construction of synergistic potential functions was instrumental in the design of this hybrid controller leading to global asymptotic stability results. This work is an extension of an earlier work proposed in [14] where almost global asymptotic stability results have been obtained. II. ATTITUDE REPRENTATION AND MATHEMATICAL A. Notations PRELIMINARIES The sets of real, non-negative real and natural numbers are denoted as R, R + and N, respectively. R n denotes the n-dimensional Euclidean space and S n denotes the unit n-sphere embedded in R n+1. Given two matrices A, B R m n, their Euclidean inner product is defined as A, B = tr(a B) where ( ) denotes the transpose of ( ). The -norm of a vector x R n is x = x x and the Frobenius norm of a matrix A R n m is A F = A, A. For a given square matrix A R n n, we denote by λ A i, λa min and λ A max the i-th, minimum and maximum eigenvalue of A, respectively. Let f : M R be a differentiable real-valued function and, x : T x M T x M R be a Riemannian metric on M. The gradient of f, denoted f(x) T x M, relative to the Riemannian metric, x is uniquely defined by f(x(t)) = f(x(t)), ẋ(t) x, for all ẋ(t) T x M. The set of critical points of f is defined by Ψ f = {x M : f(x) = 0}. B. Attitude representation Consider the general linear group GL(3). A square matrix R GL(3) is called a rotation matrix if R belongs to the special orthogonal group SO(3) GL(3) where SO(3) := {R R 3 3 det(r) = 1, RR = I}, and I = I 3 3 is the three-dimensional identity matrix. The Lie algebra of SO(3), denoted by so(3), is the vector space of 3-by-3 skew-symmetric matrices so(3) = { Ω R 3 3 Ω = Ω }. The group SO(3) has a compact manifold structure where its tangent spaces are identified /$ AACC 6048

2 by T R SO(3) := {RΩ Ω so(3)}. The Euclidean inner product on R 3 3, when restricted to the Lie-algebra of skew symmetric matrices, defines the following left-invariant Riemannian metric on SO(3) RΩ 1, RΩ R := Ω 1, Ω, (1) for all R SO(3) and Ω 1, Ω so(3). Let denotes the vector cross-product on R 3 and define the map [. ] : R 3 so(3); ω [ω] such that [ω] u = ω u, for all ω, u R 3, where for any vector ω R 3, we have [ω] = 0 ω 3 ω ω 3 0 ω 1 ω ω 1 0 Let vex : so(3) R 3 denotes the inverse isomorphism of the map [. ], such that vex([ω] ) = ω, for all ω R 3 and [vex(ω)] = Ω, for all Ω so(3). By defining P a : R 3 3 so(3) as the projection map on the Lie algebra so(3) such that P a (A) := (A A )/, we can extend the definition of vex to R 3 3 by taking the composition map ψ : R 3 3 R 3 ψ(a) := vex (P a (A)) = 1. A 3 A 3 A 13 A 31 A 1 A 1 Alternatively, an element R SO(3) can be represented as a rotation by an angle θ R about a unit vector axis u S. This is commonly known as the angle-axis parametrization of SO(3) and it is given by the map R a : R S SO(3):. R a (θ, u) = I + sin(θ)[u] + (1 cos(θ))[u]. () A continuously differentiable function U : SO(3) R + is a potential function on SO(3) (with respect to I) if U(R) > 0 for all R SO(3) \ {I} and U(I) = 0. The set of all potential functions on SO(3) is denoted as P. Let U(R) T R SO(3) be the gradient of U relative to the Riemanian metric (1). To U(R), we associate the projected gradient vector of U on R 3 defined as G U (R) = ψ(r U(R)). The time derivative of U(R) along any trajectory Ṙ = R[ω] is given by U(R) = U(R), R[ω] R = G U (R) ω. C. Hybrid Systems Framework In this paper, we make use of the recent framework for dynamical hybrid systems [15], [16]. The closed loop system falls into the following particular form of hybrid systems: ẋ = F(x, q), (x, q) C, q + J (x, q), (x, q) D, where the flow map, F : M Q TM governs the continuous flow of x on the manifold M, the flow set C M Q dictates where the continuous flow could occur. The jump map, J : M Q Q, governs discrete jumps of the state q, and the jump set D M Q defines where the discrete jumps are permitted. A subset E R 0 N is a hybrid time domain, if it is a union of finitely or infinitely many intervals of the form [t j, t j+1 ] {j} where 0 = t 0 t 1 t..., with the last interval being possibly of the form (3) [t j, t j+1 ] {j} or [t j, ) {j}. A hybrid arc is a function h : dom h M Q, where dom h is a hybrid time domain and, for each fixed j, t h(t, j) is a locally absolutely continuous function on the interval I j = {t : (t, j) dom h}. III. HYBRID VELOCITY-FREE ATTITUDE STABILIZATION A. Problem statement We assume that the rigid body is equipped with sensors that provide measurements (in the body-attached frame), denoted by b i R 3 of constant and known inertial vectors r i R 3, i = 1,,..., n, satisfying the following assumption: Assumption 1: At least three vectors, among the n inertial vectors, are not collinear. If we have only two non-collinear vector measurements b 1 and b corresponding to the inertial vectors r 1 and r, then we can always construct the third vector b 3 = b 1 b which corresponds to the body-frame measurement of the inertial vector r 3 = r 1 r. Such measurements can be obtained from an Inertial Measurement Unit (IMU) that typically include an accelerometer and a magnetometer providing, respectively, an estimate of the gravitational field and Earth s magnetic field expressed in the body frame. The rigid body rotational dynamics are governed by Ṙ = R[ω], J ω = [Jω] ω + τ, (4) where R SO(3) represents the attitude, ω R 3 being the angular velocity of the rigid body expressed in the body frame and J R 3 3 is the constant inertia matrix of the rigid body. The control torque, expressed in the body frame, is denoted by τ R 3. Our objective is to design a hybrid control input torque τ, using only vector measurements, guaranteeing global asymptotic stabilization of the attitude R SO(3) to a desired constant reference R d SO(3). B. Synergistic potential functions for hybrid feedback design on SO(3) The concept of synergistic potential functions on SO(3) has been introduced in [17], [7], and further developed in [8], [9], [13], for the purpose of designing attitude feedback systems guaranteeing global stability results. We recall the following definition of synergistic potential functions. Definition 1: [17] For a given finite index set Q N, the family of potential functions F = {U i } i Q P is said to be centrally synergistic if and only if there exists δ > 0 such that 0 < δ < δ, where the synergistic gap δ is given by δ := min i Q R Ψ Ui \ {I} [ U i (R) min j Q U j(r) ], (5) 6049

3 where Ψ Ui = {R SO(3) : U i (R) = 0} such that U i denotes the gradient of U i relative to the Riemanian metric (1). The adjective centrally refers to the fact that all the potential functions U i F share the identity element I as a critical point. We may drop this adjective where not needed. The synergy condition (5) can be interpreted as follows: At any given undesired critical point of a given potential function U i F, there exists another potential function U j F which has a lower value. An existence result for central synergistic potential functions was provided in [17]. Non-central synergistic potential functions were proposed in [8] and [9]. In our recent work [13], we have constructed a family of central synergistic potential functions via angular warping based on the trace form V A (R) = tr(a(i R)), where A is positive definite. We recall the following result. Proposition 1: [13] Let A = A > 0. Assume that A has distinct eigenvalues 0 < λ A 1 < λ A < λ A 3 corresponding to the orthonormal eigenvectors {v 1, v, v 3 }. Let Q = {1, } and let k A (1) = k A () = k A such that 0 < k A < 1 (λ A + λa 3 ) 6 max{1, 4ξ }, (6) where ξ = (λ A 1 + λ A )/(λ A + λ A 3 ). Let u A S be a unit vector chosen as follows. If λ A λ A 1 λ A 3 /(λ A 3 λ A 1 ), then u A v 1 = 0, and (u A v i) = λ A i /(λa + λ A 3 ) for i {1, }. Otherwise, (u A v i) = 1 4 j i λa j / j k λa j λa k. Consider the transformation Γ A q (R) = RR a ( sin 1 (k A (q)v A (R)), u A ), (7) then the following hold: The gradient vector of the potential function V A Γ A q is given by G VA Γ A(R) = ΘA q q (R) ψ(aγ A q (R)), ( where Θ A q (R) = R a sin 1 ) (k A (q)v A (R)), u A + 4k A (q)u A ψ(ar) / 1 ka V A(R). The family F = {V A Γ A q (R)} q Q P is centrally synergistic with a gap exceeding σ(a) where σ(a) = ( ) λ ka + 1 ( 8 k A 8 λ ka + ) 16 λ k A /16kA 4 3, such that λ = λ A 1 + λ A and = λ A 1 if λ A λ A 1 λ A 3 /(λ A 3 λ A 1 ), otherwise = 4 λ A j / j k λa j λa k. Proposition 1 gives a straightforward method to construct a synergistic family that consists of only two potential functions on SO(3). Besides, we will show in the next lemma that, for a suitable choice of the matrix A, these potential functions and their gradients can be explicitly written in terms of the available vector measurements. Lemma 1: Consider a set of vector measurements b i with their corresponding known inertial vectors r i, for i = 1...n under Assumption 1. Let A = n ρ ir i ri for some positive scalars ρ 1, ρ, ρ n and let X = RY for some Y SO(3). Then, the following relations hold: V A(X) = 1 bi ρ i Y r i, (8) ψ (AX) = 1 Y n ρ i(b i Y r i). (9) Furthermore, for q Q let ˆbi (q) = Y R a ( sin 1 (k q V A (R)), u A )r i, then V A Γ A q (X) = 1 bi ρ i ˆb i(q), (10) ( ) ψ AΓ A q (X) = 1 Ra( sin 1 (k qv A(R)), u A) Y ρ i(b i ˆb i(q)). (11) Proof: Let us define the attitude error P := RP, for some P SO(3). Making use of the identity u Au = tr(uu A), one obtains 1 ρ i b i P r i = 1 ρ i ri (I P )(I P )r i = ρ i tr(r i ri (I P )) = V A ( P ). Consequently, equation (8), respectively (10), is obtained by substituting P for Y, respectively P for R a ( sin 1 (k q V A (R)), u A ) Y. Furthermore, one has P a (A P ) = 1 ρ i [r i ri P P ] r i ri = 1 = 1 ρ i P [ P r i ri R R r i ri P ] P [ ρ i P (bi P r i ) ], where the following property has been used (see [18]): R[yx xy ]R = R[x y] R = [R(x y)]. (1) Taking the vex operator on both sides of (1) and substituting P for Y, respectively P for R a ( sin 1 (k q V A (R)), u A ) Y, yields equation (9), respectively equation (11). C. Main result Let us define the positive definite matrix A h = n ρ ihr i ri where ρ ih > 0 for h = 1, and i = 1,,..., n. Assumption 1 ensures that rank(a h ) = 3 for h = 1,. Moreover, the scalars ρ ih can be chosen such that A h has distinct eigenvalues. Let F h = {U hq } q Q be two families of potential functions on SO(3) such that U hq (R) := V Ah Γ A h q (R) for all R SO(3) where the map Γ A q ( ) is as defined in Proposition 1. In view of Lemma 1, the potential functions U hq ( ) and their gradient vectors G Uhqh ( ) can be explicitly measured in terms of the available vector measurements b i and the known inertial vectors r i, for i = 1,, n. In this section, we make use of these measurable quantities to write our velocity-free hybrid 6050

4 attitude stabilization scheme. Let us define the following auxiliary dynamic system ˆR = ˆR[β], (13) with an arbitrary initial condition ˆR(0) SO(3) and a design variable β R 3 to be defined later. Let X h = RYh, h {1, }, with Y 1 = ˆR and Y = R d. The rotation matrix X 1 describes the discrepancy between the actual rigid body orientation and the orientation provided by the auxiliary system (13), and the rotation matrix X describes the discrepancy between the actual rigid body orientation and the desired orientation. Let our state variables be X = (X 1, X ) D X := SO(3) SO(3), and q = (q 1, q ) D q := Q Q. We propose the following hybrid switching law for the control input τ and the input β of the auxiliary system (13) τ = Y h G U hqh (X h ) β = Y1 G U1q1 (X 1 ) q = 0 } {{ } (X,q) C τ + = τ β + = β q + = g(x) }{{} (X,q) D (14) { } where g(x) = (q 1, q ) Q Q : q h = argmin U hp(x h ). p Q and the sets C, D D X D q are given by C := {(X, q) D X D q : µ 1 (X 1, q 1 ) δ 1 and µ (X, q ) δ } D := {(X, q) D X D q : µ 1 (X 1, q 1 ) δ 1 or µ (X, q ) δ } such that µ h (X h, q h ) = U hqh (X h ) min p Q U hp(x h ), for h = 1,. The hybrid controller (14) results in the closed-loop system Ẋ 1 = X 1 [ Y1 ω G U1q1 (X 1 ) ], Ẋ = X [Y ω], J ω = [Jω] ω Y h G U hqh (X h ), q = 0 } {{ } (X,q) C X + = X ω + = ω q + = g(x) }{{} (X,q) D (15) Since Y 1 = X 1 X Y and Y is constant, it is clear that the closed loop dynamics (15) are autonomous. The goal of this hybrid controller is to ensure global asymptotic stability of the set A = {(I, I)} D q. Theorem 1: If the hysteresis gaps δ h satisfy δ h σ(a h ), for h = 1,, then the set A is globally asymptotically stable for the closed-loop system (15). Proof: For h = 1,, let us define the following sets: A h := {(X, q) D X D q : X h = I}, X h := {(X, q) D X D q : G Uhqh (X h ) = 0}, C h := {(X, q) D X D q : µ h (X h, q h ) δ h }. According to Proposition 1, the family F h is synergistic with a gap exceeding σ(a h ), for h = 1,. Then, according to Definition 1, one has 0 < δ h σ(a h ) < µ h (X h, q h ), for all (X, q) X h \ A h. Therefore, in view of the definition of the set C h, one obtains C h X h = A h, (16) where the fact that A h is entirely contained in C h has been used. Consider the Lyapunov function candidate V(X, ω, q) = 1 U hqh (X h ) + 1 ω Jω. The time derivative of V on C, along the trajectories of (15) is given by V(X, ω, q) = 1 U hqh (X h ), Ẋ h Xh + ω J ω = G U1q1 (X 1 ) (Y 1 ω G U1q1 (X 1 ))+ = G U1q1 (X 1 ) 0. G Uq (X ) Y ω + ω τ Thus V is non-increasing along the flows of (15). Moreover, for any (X, q) D and s g(x), one has [ ] V(X, ω, q) V(X, ω, s) = U hqh (X h ) min U hp(x h ) p Q = µ h (X h, q h ) min{δ 1, δ } > 0, which shows that V is strictly decreasing over the jumps of (15). Using [[19], Theorem 7.6], it follows that A is stable. Moreover, applying the invariance principle for hybrid systems given in [[19], Theorem 4.7], one can conclude that any solution must converge to the largest invariant set contained in I = { (X, ω, q) D X R 3 D q : (X, q) C X 1 }. Moreover, from (16), one has C 1 X 1 = A 1 and hence C X 1 = (C 1 C ) X 1 = C A 1, where the fact that C = C 1 C has been used. Since the solutions converge to C A 1, it is clear that X 1 I which leads to Ẋ1 0. Hence, one can conclude from (15) that ω 0. Since ω 0, it follows from (4) that τ must converge to 0. Using this last fact, together with the fact that β 0, one can conclude from (14) that G Uq (X ) = 0. Therefore, the solutions must converge to C A 1 X = A 1 A = A, where the fact that C X = A has been used. Finally, the set A is globally attractive and stable which shows that A is globally asymptotically stable. IV. SIMULATION RESULTS In this section we illustrate the procedure to follow for the implementation of the hybrid scheme derived in Section III. We, then, compare between the smooth feedback law, proposed in [14], and the hybrid feedback of Section III. Let A 1 = diag([1, 3, 5]) and A = diag([0.1, 0.3, 0.5]). The 6051

5 value of k Ah is chosen to verify inequality (6), thus one must have 0 < k A1 < 0.036, 0 < k A < We picked k A1 = 0.03 and k A = 0.3. The rotation unit vectors u A1 and u A are computed as in Proposition 1, u A 1 = [0, 3/8, 5/8], u A = [0, 3/8, 5/8]. Also, one obtains σ(a 1 ) = , σ(a ) = Therefore, it is sufficient to pick δ 1 = 0.5 and δ = 0.05 to implement the switching conditions of the set C and the set D. Once all the parameters have been designed, the hybrid controller (14) can be implemented. We recall the following smooth feedback law proposed in the paper [14] τ = β = ρ ih (b i Yh r i ), ρ i1 (b i Y 1 r i ). (17) Both controllers, hybrid and smooth, were implemented in Simulink. The desired rotation as well as the initial condition ˆR(0) for the auxiliary system were both chosen equal to the identity matrix, i.e., R d = ˆR(0) = I 3 3. The inertia matrix has been taken as J = diag ([1, 1, ]) and the inertial vectors as r i = e i, for i = 1,, 3. The performance of the controllers was evaluated by means of the normilized error 1 e(x h ) := 1 8 I X h F. closed-loop system starts sufficiently close to the undesired equilibria X 1 = X = R(π, e 1 ). Figure 1 shows the evolution of the error e(x ) with respect to time. The convergence is slower for closer initial conditions to the undesired equilibria (smaller choice of ɛ). This phenomenon is the main drawback of the smooth controller. On the other hand, for ɛ = 0, Figure depicts how the hybrid controller reacts immediately to correct its offset rotation, whereas the smooth controller does not react at all, being seemingly unable to correct its rotation. Fig. : Comparison between smooth and hybrid feedback responses with the initial condition X (0) = R a (π, e 1 ) Fig. 3: Comparison between smooth and hybrid feedback responses with the initial condition X (0) = R a (π, e 1 )R a (ϑ 1, u A1 ). Fig. 1: Error between the current state and the desired reference for different initial conditions close to the undesired equilibria R a (π, e 1 )-Smooth controller- To simulate the worst case for the smooth controller, the initial conditions for the rotational dynamics are taken as follows: ω(0) = [0, 0, 0] and R(0) = R a (π + ɛ, cos(ɛ/)e 1 ) for some ɛ 1. Thus, under the smooth feedback (17), the 1 This error can be shown to be equal to e(x h ) = ɛ h where ɛ h is the quaternion vector part corresponding to the orientation described by X h SO(3). For a second comparison, we changed the initial rotation matrix to R(0) = R a (π, e 1 )R a (ϑ 1, u A1 ), where ϑ , so as to start from one of the critical points of the potential function V A1 Γ A1 q 1. Figure 3 depicts the performance of the proposed hybrid feedback law. Starting from an initial configuration q(0) = (1, 1), the system immediately jumps to the configuration q = (, ) since the initial condition (X(0), q(0)) lies inside the jump set D. It is shown in Figure 3 that the hybrid controller still achieves better performance than the continuous controller for this particular initial condition. 605

6 τ1 τ τ Time Fig. 4: Plot of the torque applied by the hybrid feedback with the initial condition X (0) = R a (π, e 1 ). In Figure 4, we give the plot of the torque (control input) applied by the hybrid controller in the first case of X (0) = R a (π, e 1 ). We observe that the torque is quasi-smooth with only three discontinuities which occur during the first few seconds of the control in order to avoid the critical points. The second jump (red) affects only the second component of the torque vector. V. CONCLUSION [10] M. R. Akella, D. Thakur, and F. Mazenc, Partial lyapunov strictification: Smooth angular velocity observers for attitude tracking control, Journal of Guidance, Control, and Dynamics, vol. 38, no. 3, pp , 015. [11] T.Wu and T. Lee, Angular velocity observer on the special orthogonal group for velocity-free rigid-body attitude tracking control, in Proceedings of the European Control Conference, 015. [1] R. Schlanbusch and E. I. Grotli, Hybrid certainty equivalence control of rigid bodies with quaternion measurements, IEEE Transactions on Automatic Control, vol. 60, no. 9, pp , 015. [13] S. Berkane and A. Tayebi, On the design of synergistic potential functions on SO(3), in the 54th IEEE Conference on Decision and Control, Osaka, Japan., vol. 1, 015, pp [14] A. Tayebi, A. Roberts, and A. Benallegue, Inertial vector measurements based velocity-free attitude stabilization, IEEE Transactions on Automatic Control, vol. 58, no. 11, 013. [15] R. Goebel and A. Teel, Solutions to hybrid inclusions via set and graphical convergence with stability theory applications, Automatica, vol. 4, pp , 006. [16] R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid dynamical systems, IEEE Control Systems Magazine, vol. 9, no., pp. 8 93, 009. [17] C. G. Mayhew and A. R. Teel, Synergistic potential functions for hybrid control of rigid-body attitude, in American Control Conference (ACC), 011. IEEE, 011, pp [18] M. Shuster, A survey of attitude representations, The Journal of the Astronautical Sciences, vol. 41, no. 4, pp , [19] R. G. Sanfelice, R. Goebel, and A. Teel, Invariance principles for hybrid systems with connections to detectability and asymptotic stability, IEEE Transactions on Automatic Control, vol. 5, no. 1, 007. A velocity-free hybrid attitude stabilization scheme, relying only on inertial vector measurements, has been proposed. The proposed scheme relies on the concept of synergistic potential functions on SO(3), where the hybrid controller hysteritically switches to the control law derived from the minimal potential function. This control scheme leads to global asymptotic stability results. Some simulation results that illustrate the advantage of the hybrid control scheme over its smooth counterpart have been carried out. REFERENCES [1] G. Meyer, Design and global analysis of spacecraft attitude control systems, NASA, Tech. Rep., [] A. Tayebi and S. McGilvray, Attitude stabilization of a VTOL quadrotor aircraft, IEEE Transactions on Control Systems Technology, vol. 14, no. 3, pp , May 006. [3] A. Tayebi, Unit quaternion-based output feedback for the attitude tracking problem, IEEE Transactions on Automatic Control, vol. 53, no. 6, pp , July 008. [4] C. G. Mayhew, R. G. Sanfelice, and A. R. Teel, Quaternion-based hybrid control for robust global attitude tracking, IEEE Transactions on Automatic Control, vol. 56, pp , 011. [5] S. P. Bhat and D. S. Bernstein, A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon, Systems & Control Letters, vol. 39, no. 1, pp , 000. [6] C. G. Mayhew and A. Teel, On the topological structure of attraction basins for differential inclusions, Systems & Control Letters, vol. 60, no. 1, pp , 011. [7] C. G. Mayhew and A. R. Teel, Hybrid control of rigid-body attitude with synergistic potential functions, in American Control Conference, 011. [8] C. G. Mayhew and A. Teel, Synergistic hybrid feedback for global rigid-body attitude tracking on SO(3), IEEE Transactions on Automatic Control, vol. 58, no. 11, pp , 013. [9] T. Lee, Global exponential attitude tracking controls on SO(3), Automatic Control, IEEE Transactions on, vol. PP, no. 99, pp. 1 1,