A Landmark Based Nonlinear Observer for Attitude and Position Estimation with Bias Compensation

Transcription

1 Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul Korea July 6-8 A Landmark Based Nonlinear Observer for Attitude and Position Estimation with Bias Compensation J.F. Vasconcelos R. Cunha C. Silvestre P. Oliveira Institute for Systems and Robotics (ISR) Instituto Superior Técnico Lisbon Portugal. Tel: Fax: {jfvasconcelos rita cjs pjcro}@isr.ist.utl.pt Abstract: This paper addresses the problem of estimating position and attitude of a rigid body based on landmark coordinate readings and biased velocity measurements. Using a Lyapunov function conveniently defined by the landmark measurement error a nonlinear observer on SE(3) is derived. The resulting position attitude and biases estimation errors are shown to converge exponentially fast to the desired equilibrium points. The observer terms are explicit functions of the landmark measurements and velocity readings exploiting the sensor information directly. Simulation results for trajectories described by time-varying linear and angular velocities are presented to illustrate the stability and convergence properties of the observer supporting the application of the algorithm to autonomous air vehicles and other robotic platforms.. INTRODUCTION Landmark based navigation is recognized as a promising strategy for providing aerial vehicles with accurate position and attitude information during the critical takeoff landing and hover stages. Among a wide diversity of suitable estimation techniques nonlinear observers stand out as an exciting approach often endowed with stability results Crassidis et al. 7 formulated rigorously in non- Euclidean spaces. Recent advances in this research topic motivate the growing interest in the inclusion of non-ideal sensor effects that have long been accounted for in filtering estimation techniques. The problem of formulating a stabilizing feedback law in non-euclidean spaces such as SO(3) and SE(3) has been addressed in several references namely Malisoff et al. 6 Chaturvedi and McClamroch 6 Fragopoulos and Innocenti 4 Bhat and Bernstein where the analysis of the topological properties of the SE(3) manifold provides important guidelines for the design of observers. Nonlinear attitude observers motivated by aerospace applications are found in Salcudean 99 Thienel and Sanner 3 yielding global exponential convergence of the attitude estimates in the Euler quaternion representation. A nonlinear complementary filter in SE(3) is proposed in Baldwin et al. 7 with almost global exponential convergence and tested in a Vario Benzin-Acrobatic model scale helicopter. The observers proposed in Baldwin et al. 7 Thienel and Sanner 3 Salcudean 99 assume that an explicit quaternion/rotation matrix attitude measurement is available obtained by batch processing sensor measure- This work was partially supported by Fundação para a Ciência e a Tecnologia (ISR/IST plurianual funding) through the POS Conhecimento Program that includes FEDER funds and by the project PTDC/EEA-ACR/7853/6 HELICIM. The work of J.F. Vasconcelos was supported by a PhD Student Scholarship SFRH/BD/8954/4 from the Portuguese FCT POCTI programme. ments such as landmarks image based features and vector readings. In alternative if the landmark measurements are exploited directly in the observer it is possible to analyze the influence of the sensor characteristics and landmark configuration in estimation results. In this work the position and attitude of a rigid body are estimated by exploiting landmark readings and biased velocity measurements directly. The proposed observer yields exponential stability of the position and attitude estimation errors in the presence of biased angular and linear velocity readings. As discussed in Vasconcelos et al. 7 the attitude and position estimation problems can be decoupled assuming perfect angular velocity measurements. However coupling occurs in the presence of angular velocity bias which influences the position feedback law and the attitude and position estimation problems must be addressed together. The stability of the origin is derived based on recent results for parameterized linear time-varying systems presented in Loría and Panteley that also bring about exponential convergence rate bounds. To the best of the authors knowledge no prior work on nonlinear observers has addressed the problem of position estimation in the presence of biased angular velocity readings. The paper is organized as follows. The position and attitude estimation problem is presented in Section. Section 3 introduces the analytical tools adopted in the derivation of the observer. A convenient landmark-based coordinate transformation and Lyapunov function are defined and the necessary and sufficient landmark configuration for attitude determination is discussed. The main contribution of this work is detailed in Section 4 where the exponential convergence of the estimation errors to the desired equilibrium points is presented. It is also shown that the feedback law is an explicit function of the sensor readings and observer estimates. Section 5 illustrates the properties of the observer in simulation. Conclusions and directions of future developments are presented in Section /8/$. 8 IFAC /876-5-KR-.634

2 7th IFAC World Congress (IFAC'8) Seoul Korea July 6-8 n R n. By the definition of frame {L} the landmarks centroid is at the origin of the coordinate frame n L x i = X n =. (3) Fig.. Airborne Landmark Based Navigation NOMENCLATURE The notation adopted is fairly standard. The set of n m matrices with real entries is denoted by M(nm) and M(n) := M(n n). The sets of skew-symmetric orthogonal and special orthogonal matrices are respectively denoted by K(n) := {K M(n) : K = K } O(n) := {U M(n) : U U = I} and SO(n) := {R O(n) : det(r) = }. The special Euclidean group is given by the product space of SO(n) with R n SE(n) := SO(n) R n Murray et al The n-dimensional sphere and unitary ball are described by S(n) := {x R n+ : x x = } and B(n) := {x R n : x x } respectively.. PROBLEM FORMULATION Landmark based navigation illustrated in Fig. can be summarized as the process of determining attitude and position of a rigid body using landmark observations and velocity measurements given by sensors installed onboard the autonomous platform. The rigid body kinematics are described by R = R ω p = v ω p where R is the shorthand notation for the rotation matrix L BR from body frame {B} to local frame {L} coordinates ω and v are the body angular and linear velocities respectively expressed in {B} p is the position of the rigid body with respect to {L} expressed in {B} and a is the skew symmetric matrix defined by the vector a R 3 such that a b = a b b R 3. Without loss of generality the local frame is defined by translating the Earth frame {E} to the landmarks centroid as depicted in Fig.. The body angular and linear velocities are measured by a rate gyro sensor triad and a Doppler sensor respectively ω r = ω v r = v. () The landmark measurements q r i illustrated in Fig. are obtained by on-board sensors that are able to track terrain characteristics such as CCD cameras or ladars so that q r i = q i = R L x i p () where L x i represents the coordinates of landmark i in the local frame {L}. The concatenation of () is expressed in matrix form as Q = R X p n where Q := q... q n X := L x... L x n QX M(3n) and n :=... i= The proposed observer which estimates the position and attitude of the rigid body takes the form ˆR = ˆRˆω ˆp = ˆv ˆω ˆp where ˆω and ˆv are feedback terms that by construction compensate for the estimation errors. The position and attitude errors are defined as p := ˆp p and R := ˆR R respectively. The Euler angleaxis parameterization of the rotation error matrix R is described by the rotation vector λ S() and by the rotation angle θ π yielding the formulation Murray et al. 994 R = rot(θλ) := cos(θ)i + sin(θ)λ + ( cos(θ))λλ. While the observer results are formulated directly in the SE(3) manifold the rotation angle θ is adopted to characterize some of the convergence properties of the observer. The attitude and position error kinematics are given by R = R R(ˆω ω) (4a) p = (ˆv v) ω p + ˆp (ˆω ω) (4b) respectively. The attitude and position observers are obtained by defining ˆω and ˆv as functions of the velocity readings () and landmark observations () so that the closed loop estimation errors converge asymptotically to the origin i.e. R I p as t. 3. OBSERVER CONFIGURATION The attitude feedback law and the analysis of the resulting observer are derived resorting to Lyapunov s stability theory. Based on a judiciously chosen landmark transformation which will be presented shortly a candidate Lyapunov function is defined to synthesize the attitude and position observer. To exploit the landmark readings information vector and position measurements are constructed from a linear combination of () producing respectively Bū n j := a ij ( q i+ q i ) Bū n := n q i (5) n i= where j =...n. To express these transformations n in matrix form define D X := I n In n d p := n n and A X := a ij M(n ) which is assumed to be invertible. Using nd X = the transformation (5) can be written as B Ū X = QD X A X = R U X Bū n = Qd p = p (6) i= where B Ū X := Bū... Bū n M(3n ) and U X := XD X A X M(3n ) is known. The estimates of the transformed landmarks (6) are described by B Û X = ˆR U X Bû n = ˆp. (7) where the columns of B Û X and U X are denoted as B û i and L ū i respectively. 3447

3 7th IFAC World Congress (IFAC'8) Seoul Korea July 6-8 The candidate Lyapunov function is defined by the estimation error of the transformed vectors n V = B û i B ū i. (8) i= It is straightforward to rewrite V as a linear combination of distinct attitude and position components V = V R +V p. The attitude component is given by n V R = B û i B ū i = tr (I R)U X U X i= = 4 I R λ Pλ = ( cos(θ)) λ Pλ (9) where P := tr(u X U X )I U XU X M(3) and the position component described by V p = B û n B ū n = p p. Taking the time derivatives produces V R = U X U R X R U X U X R(ˆω ω) (a) V p = p (ˆp (ˆω ω) + (ˆv v)) (b) where is the unskew operator such that a = aa R Vector Measurement Configuration The Lyapunov function V R measures the error of the transformed landmarks given by B û i B ū i i =...n. The landmark configuration under which zero observation error V R = is equivalent to correct attitude estimation R = I verifies the following assumption: Assumption. (Landmark Configuration). The landmarks are not all collinear that is rank(x). Lemma. The Lyapunov function V R has a unique global minimum (at R = I) if and only if Assumption is verified. Proof. By Vasconcelos et al. 7 Lemma V R > if and only if rank(u X ). The equality rank(u X ) = rank(u X ) = rank(xd X n A X ) = rank(x) given that A X and D X n are nonsingular completes the proof. The necessity of Assumption is illustrated by considering all L x i are collinear which is equivalent to all L ū i collinear and hence any R = rot(θ L ū i / L ū i ) satisfies V R =. 3. Vector Measurement Directionality The derivation of a stabilizing feedback law for attitude estimation relies on analyzing the level sets wherein V R c holds. For c large enough the corresponding level sets contain multiple critical points due to the nonuniform directionality of P see Vasconcelos et al. 7 Lemma for a motivation. Interestingly enough the directionality of matrix P can be made uniform by construction using the transformation A X. Proposition 3. Let H := XD X be full rank there is a nonsingular A X M(n) such that U X U X = I. Proof. Take the SVD decomposition of H = USV where U O(3) V O(n) S = diag(s s s 3 ) 3 (n 3) M(3n) and s > s > s 3 > are the singular values of H. Any A X given by A X = V diag(s s s 3 ) 3 (n 3) V A (n 3) 3 B where B M(n 3) is nonsingular and V A O(n) produces U X U X = HA XA X H = UV A V AU = I. Using the transformation A X defined in Proposition 3 the Lyapunov function (9) is expressed by V R = I R = ( cos(θ)) V R = R R R (ˆω ω) = sin(θ)λ R (ˆω ω). Apparently the case rank(x) = does not satisfy the conditions of Proposition 3 given that rank(h) = rank(x). However by taking two linearly independent columns of H L h i and L h j a full rank matrix is defined as H a = H L h i L h j. The cross product is commutable with coordinate transformations (R L h i ) (R L h j ) = R ( L h i L h j ) hence a modified observer can be rederived without loss of generality by replacing H with H a. 4. OBSERVER SYNTHESIS In this section the feedback law for the observer is derived and the problem of bias in the velocity sensors is addressed. The observer for the case of unbiased velocity measurements which is based on previous work by the authors Vasconcelos et al. 7 is presented first to motivate the derivation of the attitude and position feedback laws and expose topological limitations to global stabilization. 4. Unbiased Velocity Measurements For the case of unbiased velocity readings the Lyapunov function decoupling V = V R + V p allows for the attitude and position estimation problems to be addressed separately. The attitude observer (4a) is derived using the Lyapunov function V R and the position observer (4b) is derived using V p. Attitude Feedback Law Under Assumption and given the Lyapunov function derivatives along the system trajectories (a) consider the following feedback law ˆω = ω K ω s ω () where the feedback term is given by s ω := R R R = sin(θ) R λ () and K ω > is a positive scalar. The attitude feedback yields the autonomous attitude error system R =K ω R( R R) (3) and a negative semi-definite time derivative VR = K ω s ωs ω = 4K ω sin (θ) so it is immediate that the attitude feedback law produces a Lyapunov function that decreases along the system trajectories. Under Assumption the set of points where V R = is given by C R = { R SO(3) : R = I R = rot(πλ)λ S()}. By direct substitution in the closed loop system (3) the subset { R SO(3) : R = rot(πλ)} is invariant hence any initial condition with θ = π does not converge to the desired equilibrium point θ =. This issue is a consequence of the known limitations to global stability on SO(3) discussed in Bhat and Bernstein for details on the present estimation problem see Vasconcelos et al. 7 and references therein. However the set θ = π has 3448

4 7th IFAC World Congress (IFAC'8) Seoul Korea July 6-8 zero measure and as shown in the next theorem every other initial condition converges exponentially fast to the origin yielding almost global convergence. The proof is obtained by adaptation of the Vasconcelos et al. 7 Theorem 3 and omitted due to space constraints. Theorem 4. The closed-loop system (3) has an exponentially stable point at R = I. For any initial condition R(t ) in the region of attraction R A = { R SO(3) : R = rot(θλ) θ < πλ S()} the trajectory satisfies R(t) I k R R(t ) I e γr(t t) (4) where k R = and γ R = K ω ( + cos(θ(t ))). Position Feedback Law The position feedback law for the system (4b) is defined as ˆv = v + ( ω K v I)s v ˆp (ˆω ω) (5) where the feedback term is defined as s v := p and K v is a positive scalar. The position feedback law produces a closed loop linear time-invariant system p = K v p and exponential stability of the origin is immediate. 4. Biased Velocity Measurements This section presents the derivation of an exponentially stabilizing observer for attitude and position estimation in the presence of angular and linear velocity biases ω r = ω + b ω v r = v + b v (6) where the nominal biases are considered constant i.e. b ω = b v =. The proposed Lyapunov function (8) is augmented to account for the effect of the angular and linear velocity bias V b =γ θ ( cos(θ)) + γ p p + ω b ω + v b v V b =γ p s v (ˆp (ˆω ω) + (ˆv v) ω p) + γ θ s ω(ˆω ω) + ω b ω bω + v b v bv (7) where b ω = ˆb ω b ω b v = ˆb v b v are the bias compensation errors ˆb ω ˆb v are the estimated biases and γ θ γ p ω and v are positive scalars. Under Assumption and given Lemma the Lyapunov function V b has an unique global minimum at ( p R b ω b v ) = (I). The feedback laws for the angular and linear velocities are given by ˆω = ( ω + b ω ˆb ω) K ωs ω = ( ω b ω) K ωs ω (8) ˆv = v r ˆb v + ( ω r ˆb ω K vi ) s v ˆp (ˆω (ω r ˆb ω)) = v b v + ( ω b ω K vi ) s v + K ω ˆp s ω. which are obtained by adding bias compensation terms to the feedback laws () and (5) respectively. The augmented Lyapunov time derivative is described by V b = γ p K v s v γ θ K ω s ω +(γ p ˆp p γ θ s ω +ω bω ) b ω + (v bv γ p p) b v. The bias estimates satisfy ˆbω = bω ˆb v = bv and the bias feedback laws are defined as ˆb ω = ω (γ θ s ω γ p ˆp p) The closed loop system can be written as ˆb v = γ p v p. (9) p = p b ω K v p b v R = K ω R( R R ) R R bω () b ω = γ θ R R γ R γ p p p b v = γ p p bω ω v which are nonautonomous and the Lyapunov function time derivative is described by V b = γ p K v s vs v γ θ K ω s ωs ω. Let x b := ( p R b ω b p ) and D b := R 3 SO(3) R 3 R 3 the set of points where V bω = is C b = {x b D b : ( p b ω b p ) = () R C R }. As discussed in Section 4. global asymptotic stability of the origin is precluded by topological limitations associated with the estimation error R = rot(πλ). In the next proposition the boundedness of the estimation errors is shown and used to provide sufficient conditions for excluding convergence to the equilibrium points satisfying R = rot(πλ). Lemma 5. The estimation errors ( p R b ω b p ) are bounded. For any initial condition such that v b v (t ) + γ p p(t ) + ω b ω (t ) < () 4γ θ ( + cos(θ(t ))) the attitude error is bounded by θ(t) θ max < π for all t t. Proof. Define the set Ω ρ = {x b D b : V b ρ}. The Lyapunov function (7) consists of the weighted distance of the state to the origin so α x b αv b and the set Ω ρ is compact. The Lyapunov function decreases along the system trajectories Vb so any trajectory starting in Ω ρ will remain in Ω ρ. Consequently t t x b (t) αv b (x(t )) and the state is bounded. The gain condition () is equivalent to V b (x b (t )) < 4γ θ. The invariance of Ω ρ implies that V b (x b (t)) V b (x b (t )) which implies that γ θ ( cos(θ(t))) V b (x b (t )) < 4γ θ and consequently θmax θ(t) θ max < π for all t t. Adopting the analysis tools for parameterized LTV systems Loría and Panteley the system () in the form ẋ b = f(tx b )x b is rewritten as ẋ = A(λt)x. In this formulation the parameter λ D b R is associated with the initial conditions of the nonlinear system and the solutions of both systems are identical wherever the initial conditions of both systems coincide x (t ) = x(t ) and the parameter satisfies λ = (t x(t )). Sufficient conditions for uniform exponential stability of the parameterized LTV system and thus of the nonlinear system are derived in Loría and Panteley. Using these results exponential convergence of the estimation errors in the presence of biased velocity measurements is shown. Theorem 6. Let v = ω and assume that p v and ω are bounded. For any initial condition that satisfies () the position attitude and bias estimation errors converge exponentially fast to the stable equilibrium point ( p R b ω b v ) = (I). Proof. The stability of () is obtained by a change of coordinates using an attitude representation similar to that proposed in Thienel and Sanner 3. Let the attitude error vector be given by q q = sin( θ )λ the closed loop kinematics are described by p = p b ω K v p b v q q = Q( q)( R b ω 4K ω q q q s) 3449

5 7th IFAC World Congress (IFAC'8) Seoul Korea July 6-8 b ω = 4 γ θ ω R Q ( q) q q γp ω p p b v = γp v p () where Q( q) := q s I + q q q = q q q s qs = cos( θ ) and q s = K ω q q q q q s q q b ω. The vector q is the well known Euler quaternion representation Murray et al Using q q = 8 R I the Lyapunov function in quaternion coordinates is described by V b = 4γ θ q q + γ p p + ω b ω + v b v. Let x q := ( p q q b ω b v ) x q D q and D q := R 3 B(3) R 3 R 3 define the system () in the domain D q = {x D q : V b γ θ (4 ε q )} < ε q < 4. The set D q is given by the interior of the Lyapunov surface so it is positively invariant and well defined. The condition () implies that the initial condition is contained in the set D q for ε q small enough. Let x := ( p q q b ω b v ) D q := R 3 R 3 R 3 R 3 := ω = v and define the parameterized LTV system A(tλ) B ẋ = (tλ) x C(tλ) (3) 3 3 where λ R D q the submatrices are described by A(tλ) = B(tλ) = KvI K ω q s(tλ)q( q(tλ)) p R Q ( q(tλ)) C(tλ) = B(tλ) γpi I 3 3 8γ γ θ I b and the quaternion q(t λ) represents the solution of () with initial condition λ = (t p(t ) q q (t ) b ω (t ) b v (t )). By the boundedness of p the matrices A(tλ) B(tλ) and C(tλ) are bounded and the system is well defined Khalil 996 p. 66. If the parameterized LTV (3) is λ-uges then the nonlinear system () is uniformly exponentially stable in the domain D q Loría and Panteley p.4-5. The parameterized LTV system verifies the assumptions of Loría and Panteley Theorem : ) Given the boundedness of p v and ω p is bounded and the elements of B(tλ) and B(tλ) t as well as the corresponding induced Euclidean norm are bounded for all λ R D q t t. ) The positive definite matrices P(tλ) = γpi 8γ θ I Q(tλ) = Kvγ pi 3 q s (tλ)kωγ satisfy P(tλ)B (tλ) = θi C (tλ) Q(tλ) = A (tλ)p(tλ) + P(tλ)A(tλ) + P(tλ) min(c p )I P(tλ) max(c p )I min(c q )I Q(tλ) max(c q )I with C p = {γ p 8γ θ } C q = {c q c q cos ( θmax )K vγ p } and c q = 3K ω γ θ. The system (3) is λ-uges if and only if B(tλ) is λ- uniformly persistently exciting Loría and Panteley. Algebraic manipulation produces for any y R 3 B(τλ)B (τλ) = R 4 Q ( q)q( q) R p p p 4 y R Q ( q)q( q) Ry = ( y (y R q q ) ) 4 y ( qq ) y ( sin ( 4 4 θ max)) = y c θ where c θ := ( 4 cos θ max). Therefore B(τλ)B (τλ) B(τ) where B(τ) := cθ I p p. Simple but long p I I algebraic manipulations show that the eigenvalues of B(τ) are lower bounded by some α B > independent of τ if p is bounded and θ max < π. It follows that persistency of excitation condition is satisfied B(τλ)B (τλ) α B I the parameterized LTV (3) is λ-uges and the nonlinear system () is exponentially stable in the domain D q. Given γ p γ θ ω and v any initial estimation error x b (t ) satisfying () converges exponentially fast to the origin. The following corollary establishes that the origin is uniform exponential stable i.e. the convergence rate bounds are independent of x b (t ) for a bounded initial estimation error which is a reasonable assumption for most applications. Corollary 7. Assume that the initial estimation errors are bounded p(t ) p max θ(t ) θ max < π b ω (t ) b ω max b v (t ) b (4) v max and let v b v max + γ p p max + ω b ω max < 4γ θ ( + cos(θ max )) ω = v then the equilibrium point x = (I) is exponentially stable uniformly in the set defined by (4). Convergence rate bounds can be obtained by applying Loría 4 Theorem and Remark. However the obtained values were conservative and are omitted due to the necessity of further study. 4.3 Output Feedback Law This section shows that the feedback laws can be expressed in terms of the landmark and velocity readings (6) and () respectively. Theorem 8. The feedback laws are explicit functions of the sensor readings and state estimates ˆω = ω r ˆb ω K ωs ω ˆv = v r ˆb v + ( ω r ˆb ω K vi ) s v + K ω ˆp s ω (5a) (5b) ˆb ω = (γ θ s ω γ p ˆp s v) ω b v = γp s v v (5c) n s ω = ( ˆR XD X A X e i ) (Q rd X A X e i )s v = ˆp Q rd p (5d) i= where Q r := q r q r n is the concatenation of the landmark readings and e i is the unit vector where e i =. Proof. The expressions (5a-5c) are directly obtained from (8-9). Given (6) and Q r = Q produces s v. Using the definition of R the feedback law () can be rewritten as s ω = R ˆR ˆR R. Using B Û B X Ū X = ˆR R B Ū B X Û X = n i= B ū ibû i and B ū ibû i B û B i ū i = (Bû i B ū i ) bears s ω = n i= (B Û X e i ) ( B Ū X e i ). Applying (6) and (7) produces the desired results. 5. SIMULATIONS In this section the simulation results for the proposed observer are presented. The position of the landmarks is described by X = which satisfies the / / / / conditions expressed in Assumption and corresponds to the case discussed in Section 3.. The feedback gains 345

6 7th IFAC World Congress (IFAC'8) Seoul Korea July Position =.6 =5.9 =.6 =5.9 p.5.5 log(v b ) Time (s) I R Attitude =.6 =5.9 Fig. 3. Exponential Convergence of V b. feedback gains. Future work will focus on improving the convergence rate bounds provided by the parameterized systems framework and on the implementation in discrete time and testing of the algorithm on the Vario X-Treme model-scale helicopter platform. bω bv Angular Velocity Bias =.6 = Linear Velocity Bias =.6 = Time (s) Fig.. Estimation Errors. are given by K ω = K v = the values of and γ θ are computed to satisfy the condition of Corollary 7 for p max = 3 m θ max = 9 b ω max = 5 3 /s and b v max = 3 m/s and the initial estimation errors are given by p(t ) = m θ(t ) = 7 b ω (t ) = 5 /s b v = /s. The rigid body trajectory is computed using oscillatory angular and linear velocities of Hz. The estimation errors converge to the origin as depicted in Fig.. The exponential convergence of the Lyapunov function (and of the estimation error) is illustrated in Fig. 3 using a logarithmic scale. The convergence of the estimation error to the origin is faster for larger bias feedback gains. 6. CONCLUSIONS A nonlinear observer for position and attitude estimation on SE(3) was proposed using landmark measurements and biased velocity readings. The estimation errors were shown to converge exponentially fast to the origin by adopting recently derived stability results for parameterized linear time-varying systems. Simulation results illustrated the convergence properties of the observer for different γ p γ θ REFERENCES G. Baldwin R. Mahony J. Trumpf T. Hamel and T. Cheviron. Complementary filter design on the special euclidean group SE(3). In European Control Conference 7 Kos Greece July 7. S. P. Bhat and D. S. Bernstein. A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon. Systems and Control Letters 39():63 7 Jan.. N. Chaturvedi and N. McClamroch. Almost global attitude stabilization of an orbiting satellite including gravity gradient and control saturation effects. In 6 American Control Conference Minnesota USA June 6. J. L. Crassidis F. L. Markley and Y. Cheng. Survey of nonlinear attitude estimation methods. Journal of Guidance Control and Dynamics 3(): 8 Jan.- Feb. 7. D. Fragopoulos and M. Innocenti. Stability considerations in quaternion attitude control using discontinuous Lyapunov functions. IEE Proceedings on Control Theory and Applications 5(3):53 58 May 4. H. K. Khalil. Nonlinear Systems. Prentice Hall nd edition 996. A. Loría. Explicit convergence rates for MRAC-type systems. Automatica 4(8): August 4. A. Loría and E. Panteley. Uniform exponential stability of linear time-varying systems: revisited. Systems and Control Letters 47():3 4 Sept.. M. Malisoff M. Krichman and E. Sontag. Global stabilization for systems evolving on manifolds. Journal of Dynamical and Control Systems ():6 84 April 6. R. M. Murray Z. Li and S. S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC 994. S. Salcudean. A globally convergent angular velocity observer for rigid body motion. IEEE Transactions on Automatic Control 36(): Dec. 99. J. Thienel and R. M. Sanner. A coupled nonlinear spacecraft attitude controller and observer with an unknown constant gyro bias and gyro noise. IEEE Transactions on Automatic Control 48(): 5 Nov. 3. J.F. Vasconcelos R. Cunha C. Silvestre and P. Oliveira. Landmark based nonlinear observer for rigid body attitude and position estimation. In 46th IEEE Conference on Decision and Control Dec