Observers for systems with invariant outputs

Transcription

1 Observers for systems with invariant outputs C. Lageman, J. Trumpf an R. Mahony Abstract In this paper we introuce a general esign approach for observers for left-invariant systems on a Lie group with right-invariant outputs on a manifol. This situation is encountere for example in attitue estimation problems where only partial information of the state can be measure. We propose a graient-like observer esign on the output space which can be lifte to a graient-like observer on the state space. The observer on the output space converges for generic initial conitions an has local exponential convergence properties. The lifte observer retains the same convergence properties with respect to the fibers of the output map. Our observer constructions are illustrate on the attitue estimation problem for a rigi boy. I. INTRODUCTION Nonlinear observer esign for invariant systems on Lie groups has receive a growing amount of interest in the last years, riven mainly by applications in state estimation for autonomouos robotic vehicles. The interest in nonlinear observers is motivate by their potential lower computational complexity an their increase robustness compare to stanar filtering approaches. Invariant systems on Lie groups have been stuie in a series of works starting in the seventies, focusing mainly on general, theoretical observability an controllablity questions [1], [2], [3], [4], [5]. Nonlinear observer esign approaches for invariant systems have been generally more closely connecte to specific applications. A large boy of works consiers nonlinear attitue observers for rigi boy ynamics [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] or the full pose estimation problem [18], [19], [20], [21]. However, theoretical aspects of observer esign for invariant systems have receive little attention in the past. Only recent works have begun to tackle such questions an consier the general structure of observers for invariant systems an propose general approaches to observer esign [22], [23], [14], [24], [25], [26], [27]. In this paper, we exten the graient-like observer construction for full state observations propose in [25], [27] to systems with partial state observations given by a rightinvariant output map. This system set up has been consiere in [26], where the authors iscuss a general observer structure which has autonomous error ynamics an yiels a local convergence result for a suitable choice of gain coefficients. C. Lageman is with the Department of Electrical Engineering an Computer Science, Université e Liège, B-4000 Liege Sart-Tilman, Belgium christian.lageman@montefiore.ulg.ac.be J. Trumpf is with the Department of Information Engineering, The Australian National University, Canberra ACT 0200, Australia Jochen.Trumpf@anu.eu.au R. Mahony is with the Department of Engineering, The Australian National University, Canberra ACT 0200, Australia Robert.Mahony@anu.eu.au In this paper, we approach the observer construction problem by first esigning a graient-like observer on the output space. This is justifie by the fact that the system projects to a system with full state observations on the output space. For this projecte system a moification of the graient-like error construction for full state observations on a Lie group [25], [27] can be applie, which yiels an output observer for the system on the group. To obtain an observer for the group state space we use a lift of this output observer. This lift can be interprete as a graientlike observer on the Lie group. This links our approach both to [26] an the graient-like observers for full state observation [25], [27] as the lifte observer fits the observer structures iscusse in these papers. We show that uner mil conitions the errors for the output observer an the group state observer converge to a reference output or the stabilizer of this reference output, respectively. Furthermore, this convergence is locally exponential. To illustrate our concepts, we iscuss an attitue estimation problem for a rigi boy with partial state measurements. In Section II we introuce our notation an the system set up. Section III iscusses the observer construction on the output space an the convergence properties. The lift to the group an the relate convergence results are consiere in Section IV. Section V illustrates the methos on the attitue estimation problem. The paper ens with conclusions in Section VI. II. INVARIANT SYSTEMS WITH OBSERVATIONS ON A A. Notation HOMOGENEOUS SPACE We start with the notation use in the paper. We enote by G a connecte, finite-imensional Lie group with Lie algebra g an ientity e. M will enote a manifol on which G acts by a right action h: G M M, i.e. h(e, y) = y for all y M, e G the ientity an h(x, h( ˆX, y)) = h( ˆXX, y) for all X, ˆX G an y M. The action h is transitive if for all y, ŷ M there exists an X G with h(y, X) = ŷ. The group acts effectively on M if for all X G, X e exists a y M (possibly epening on X) with h X (y) y. To simplify our notation we use h y for the map h y : G M, h y (X) = h(x, y) an h X for the map h X : M M, hx (y) = h(x, y). We enote by stab(y) for y M the stabilizer group stab(y) = {X G h(x, y) = y}. We use the notation T X G an T y M for the tangent spaces of G an M at X an y, respectively. The tangent map for a function φ: G M at X G is enote by T X φ. A Riemannian metric, on G is calle right-invariant if for any X, Y G, v, w g an the

2 right-multiplication map R Y : G G, R Y (X) = XY, one has T e R Y v, T e R Y w G = v, w G. A Riemannian metric on M is calle invariant (uner h) if for all X G, y M, v, w T y M one has v, w = T y hx v, T y hx. A istribution H(X) on G is calle right-invariant if for all X, Y G the equation H(XY ) = T X R Y H(X) hols. We say that a right-invariant Riemannian metric on G inuces an invariant Riemannian metric on M if there is a right-invariant horizontal istribution H(X) such that for all y M, X G, v, w T X G we have T X h y (Xv), T X h y (Xw) = v, w G. We will use the notation (v) H for the horizontal lift of v T y M, y M to T X G, X stab(y) with respect to a horizontal istribution H(X), i.e. the unique w H(X) with T X h y w = v. At last, we recall that the graient of a function f on G or M with respect to a Riemannian metric, is given by the ientity gra f(x), v = f(x)(v) for all v T x G (or all v T x M, respectively). B. System set-up an assumptions We introuce now the systems iscusse in this paper. It is assume that we are given a connecte Lie group G an a smooth manifol M on which G acts by a transitive right action. We consier a left-invariant system on G with rightinvariant outputs on M, i.e. a system of the form Ẋ = Xu, X G y = h(x, ), y M with u: R g a smooth input an a fixe reference output. We will refer to M as the output space. Since our observer construction will be graient-base, we assume that there is a right-invariant metric on G which inuces an invariant metric on M. This assumption is satisfie e.g. for all reuctive homogeneous spaces M (cf. [28] for a efinition) on which G acts effectively, see [29, Proposition 3.16] 1. This inclues in particular G = SO(n) with M = S n 1 an the action h(r, y) = R y for R SO(n) an y S n 1, cf. Section V. The right-invariant horizontal istribution associate with the pair of Riemannian metrics is enote by H(X). In this paper we consier the problem of constructing an observer for systems of the form (1) given only measurements of the input u an the output y. III. GRADIENT-LIKE OBSERVERS ON THE HOMOGENEOUS SPACE Our observer esign approach is base on the fact that the invariant system (1) can be reuce to a system on the output space with full state observations. We will propose an observer for this projecte system similar to the graient-like observers for invariant systems with full state observations on the group propose in [25], [27]. Let us begin with the reuction of the system to the output space. 1 Usually only left actions on M are consiere in the literature but the results exten analogously to right actions by interchanging left- an rightinvariance. (1) Theorem 1: System (1) projects to the system ẏ = T X h y0 (Xu) (2) with X h 1 (y). This system is well efine an a minimal realization of (1). This result is mainly base on the symmetry reuction methos as introuce in [30]. We omit a etaile proof of the theorem. As in the Lie group case with full state observations [25], [27], the observer on the homogeneous space consists of a copy of the system ynamics an a graient-like innovation which will rive a suitably chosen error term into a no error state. For that, we assume that we are given a smooth cost function f : M M R on M. The propose observer has the form: ŷ = T ˆXh y0 ( ˆXu) gra 1 f(ŷ, y). (3) To iscuss the convergence properties of the observer for the projecte system, i.e. the output, we have to introuce a suitable error function which compares the observer state with the state of the original system on the group. Definition 2: We efine the output error E as the function E : M G M, E(ŷ, X) := h( ˆXX 1, ) = h(x 1, ŷ) ˆX h 1 (ŷ). Note that this error is well-efine an a smooth function. The error value correspons to the no error states, i.e. that the state of the observer is the output corresponing to the state X. The use of this error is justifie by the fact that any other smooth error function Ê : M M N which is constant on two copies of the system (2) can be lifte to M G as Ê(ŷ, h (X)) an this lift is the composition g E of a smooth, stab( )-invariant function g : M N with the output error. Uner suitable invariance conitions on the cost, the output error has autonomous graient ynamics. Theorem 3: Assume that f is invariant uner the action of G on M M, i.e. f(h(s, ŷ), h(s, y)) = f(ŷ, y) an that the conitions on the Riemannian metrics of Section II hol. Then we have the following autonomous ynamics for the output error. t E(ŷ, X) = gra 1 f(e(ŷ, X), ). Proof: In this proof we will enote the left an right multiplication in G by Y G as L Y an R Y. Straightforwar calculations using the equivariance of h show that h y L X = h h(x,y) an h X h y = h y R X. Thus we have E(ŷ, X) t = T X 1hŷT e L X 1T X R X 1Xu +Tŷ hx 1T ˆXh y0 ˆXu Tŷ hx 1 gra 1 f(ŷ, y) = T X 1hŷT e R X 1u + T ˆXX 1h y0 T ˆXR X 1 ˆXu Tŷ hx 1 gra 1 f(ŷ, y) = T ˆXX 1h y0 T X 1L ˆXT e R X 1u +T ˆXX 1h y0 T ˆXR X 1 ˆXu Tŷ hx 1 gra 1 f(ŷ, y) = Tŷ hx 1 gra 1 f(ŷ, y).

3 Since the Riemannian metric on M is invariant uner the action of h an f is invariant uner the action of G on M M, we get Tŷ hx 1 gra 1 f(ŷ, y) = gra 1 f( h X 1(ŷ), h X 1(y)) = gra 1 f(e(ŷ, X), ). This yiels the formula for the output error ynamics. The graient error ynamics yiels the following convergence result for the output error on M. Theorem 4: Assume that f is invariant uner the action of G an that the conitions on the Riemannian metrics of Section II hol. Furthermore let y f(y, ) be a Morse-Bott function with one global minimum an no further local minima. Then the output error on M converges to for generic initial conitions an the convergence is locally exponential. Proof: By Theorem 3 the output error has graient ynamics of the function y f(y, ). As the latter function is Morse-Bott, the graient system converges for generic initial conitions to the set of local minima an the convergence at a manifol of local minima is exponential [31]. Since is the only local minimum, we get generic convergence to at a locally exponential rate. Remark 5: Note that for all results in this section, it is sufficient that we are given an invariant Riemannian metric on M. The Riemannian metric on G will only play a role for the observer construction on G. IV. GRADIENT-LIKE OBSERVERS ON THE GROUP We turn now to the construction of an observer on the group. Since the projecte system on the output space is a minimal realization, a natural approach is to lift the observer equation for the projecte system to the Lie group. Here, we use a irect copy of the system ynamics Xu an lift the innovation term on M to the group by a horizontal lift with respect to the horizontal istribution H. This yiels the observer ˆX = ˆXu ( gra 1 f(h y0 ( ˆX), H y)), (4) where the term (v) H T ˆXG enotes the horizontal lift of v T h( ˆX,y0) M, ˆX G. Straightforwar calculations an the invariance properties of H(X) yiel the following projection result for the lifte observer. Proposition 6: Assume that the conitions on the Riemannian metrics of Section II hol. The observer (4) on G projects via the map h y0 to the observer (3) on M, i.e. ( ( T ˆXh y0 ˆXu gra 1 f(h y0 ( ˆX), ) ) H y) = T ˆXh y0 ( ˆXu) gra 1 f(ŷ, y). Furthermore, for any fixe initial values X(0), ˆX(0) G an any smooth input u the error E = ˆXX 1 of the observer on G projects via the map h y0 to the output error of the observer on M for initial values X(0), ŷ(0) = h y0 ( ˆX(0)) an the same input u. Proof: It follows from the efinition of the horizontal lift that (4) projects onto (3). For the statement on the observer error note that E(ŷ(t), X(t)) = h y0 ( ˆX(t)X(t) 1 ) by efinition of the output error, i.e. ˆXX 1 projects onto the output error. The observer (4) can be alternatively obtaine by the esign metho introuce in [25] for graient-like observers with full state observations. Our conitions on the Riemannian metrics on M an G imply that the horizontal lift of the graient of the cost function is the graient of a lifte cost function. Proposition 7: Assume that the conitions on the Riemannian metrics of Section II hol. Let F : G G R be the lift of the cost function f : M M R, i.e. F ( ˆX, X) = f(h( ˆX, ), h(x, )). The graient gra 1 F of the lifte cost function is the horizontal lift of gra 1 f. This property of the lift allows the interpretation of the lifte innovation term as a graient on G. Corollary 8: Uner the conitions of Proposition 7, the observer (4) can be written as ˆX = ˆXu gra 1 F ( ˆX, X). (5) In particular, the observer (5) projects to the observer (3) on the output manifol. Note that (5) is implementable using only observations of y an u since one has F ( ˆX, X) = f(h( ˆX, ), y) an thus the graient term oes only epen on y an ˆX. Since the observer an the error ˆXX 1 project to the observer an the output error on M, it inherits the convergence properties of the observer on M relative to the fibers of the map X h y0 (X). Here, convergence to a fiber h 1 (y) means that inf{ist( ˆX(t)X 1 (t), Z) Z h 1 (y)} converges to 0 with ist the Riemannian istance on G. Theorem 9: Assume that the conitions on f : M M R as in Theorem 4 hol. Then the error ˆXX 1 on G of the observer (4) converges to stab( ) for generic initial conitions an the convergence is locally exponential with respect to the Riemannian istance on G. Proof: Note that our conitions on the Riemannian metrics imply that ist(h y0 (X), ) = inf{ist(x, Z) Z stab( )}. Since the error Ê = ˆXX 1 projects to the output error on M, it converges to the fiber h 1 ( ) = stab( ) if an only if the output error converges to. By efinition h y0 : G M is the projection map of the fiber bunle G over M an hence any open, ense subset of M M whose complement has measure 2 0 lifts by the inverse of ( ˆX, X) (h y0 ( ˆX), h y0 (X)) to an open, ense set in G G whose complement has measure 0. Thus the generic convergence on M for the canonical error yiels the generic convergence on G for the error ˆXX 1. 2 with respect to the Riemannian measure

4 Note, that since the minimal realization of (1) is the projecte system on the output space, the subgroup stab( ) is unobservable an the error ˆXX 1 for any observer of (1) can only converge to the subgroup stab( ). Hence our observer provies the best possible convergence result for generic initial conitions. V. EXAMPLE In this section we will illustrate our observer construction on the attitue estimation problem. The attitue of a rigi boy, with boy-fixe-frame {B}, measure with respect to an inertial frame {A}, can be ientifie with an element R of SO(3). The left invariant ynamics Ṙ = Rω (6) on SO(3) correspon to the natural boy-fixe-frame kinematics of the system. Here ω = (ω 1, ω 2, ω 3 ) {B} is the angular velocity of the rigi-boy expresse in the boyfixe-frame an ω = 0 ω 3 ω 2 ω 3 0 ω 1 ω 2 ω 1 0. Consier the case where only partial information on the rigiboy attitue is measure. The most common situation is when an inertial irection, such as the magnetic fiel, or gravitational fiel, is measure by sensor systems such as magnetometers or accelerometers on boar a vehicle. The inertial irection {A} of the measure irection is known a-priori, for example, the gravitational irection lies in the z-axis of the inertial frame. The measure irection in the boy-fixe-frame, y = R {B}, is obtaine by using the attitue matrix to transform the inertial irection from the inertial frame into the boy fixe frame. The corresponing right action of SO(3) on S 2 is h : SO(3) S 2 S 2, h(r, y) = R y. This yiels the system Ṙ =Rω y =R (7a) (7b) on SO(3) with outputs on S 2. For the observer constructions propose in this paper, we have to choose a Riemannian metric on SO(3) an S 2. We equip the unit sphere with the Riemannian metric inuce by the Eucliean scalar prouct in R 3. To choose a Riemannian metric on SO(3) recall that the tangent spaces T R SO(3) are given by {RΩ Ω = Ω}. We use the Riemannian metric RΩ, RΘ = 1 2 tr(ωθ) on SO(3). The right-invariant horizontal istribution H(R) can be obtaine as the orthogonal complement of ker T e h y0, i.e. H(R) = T e R R (ker T e h y0 ). Straightforwar calculations show that these Riemannian metrics satisfy the conitions from Section II. By Theorem 1, the system (7) projects to a system on S 2. A straighforwar calculation shows that the projecte system can be written as ẏ = ω y, y(0) = h(x(0), ) (8) on S 2. First we esign an observer for the projecte system by the methos introuce in Section III. For constructing a graient-like observer on S 2 we have to choose a cost function. The most natural choice is the Eucliean istance, i.e. f(ŷ, y) = (k/2) ŷ y 2 with k > 0 a positive scaling constant. The constant k is introuce to provie a tunable gain for the observer. The graient of a function restricte to an embee submanifol of the Eucliean space with respect to the inuce metric is given by the Eucliean orthogonal projection of the Eucliean graient to the (embee) tangent space of the submanifol. For the unit sphere this projection is given by (I ŷŷ ). This yiels the graient gra 1 f(ŷ, y) = k(i ŷŷ )(y ŷ) = k ( I ŷŷ ) y. Thus we get from (3) the observer ŷ = ω ŷ + k(i ŷŷ )y (9) on S 2. It s easy to see that the cost function f satisfies the conitions of Theorem 4, i.e. that ŷ f(ŷ, y) is a Morse function with only one global minimum ( ) an no further local minima, an is invariant uner the action (R, (ŷ, y)) (R ŷ, R y) of SO(3) on S 2 S 2. Thus the canonical error on S 2 for this observer converges to for generic initial conitions. Some calculations show that k(i ŷŷ )y = k(ŷ y) ŷ. Hence, the observer (9) can be written as ŷ = ω ŷ + k(ŷ y) ŷ. (10) This observer was originally stuie by Metni et al. [10], [13]. The erivation given in earlier work is base on a Lyapunov analysis an inclues an integral term for compensation of gyro bias that oes not fit the analysis framework presente in this paper, however, the main proportional terms in [10], [13] are exactly (10). To obtain an observer on SO(3) we have to lift (9) to the group. Since the Riemmiann metrics on S 2 an SO(3) satisfy the conitions of Section II, we can apply Corollary 8 an construct the observer from a lift of the cost function to SO(3). The lifte cost function F of f is given by F ( ˆR, R) = k 2 ˆR R 2. To calculate the graient we nee the ifferential 1 of F with respect to the first variable. Some matrix calculations

5 yiel ( 1 F ( ˆR, R)( ˆRΩ) =k tr ˆRΩ( ˆR R) y0 ( k tr ( ˆR R)Ω ˆR ) y0 =k tr ( Ω(ŷ y)ŷ ) k tr (( ˆR R)Ωŷ(ŷ y) ) =2k tr ( Ω(ŷy yŷ ) ). From the ientity ) gra 1 F ( ˆR, R), RΩ = 2 tr ((gra 1 F ( ˆR, R)) ˆR Ω Ω ) = 1 F (R)(RΩ), = Ω, an the fact that ˆR gra1 F ( ˆR, R) is a skew symmetric matrix, one obtains the graient gra 1 F ( ˆR, R) = k ˆR(ŷy yŷ ) = k ˆR(ŷ y). Thus the graient-like observer on the Lie group is t ˆR = ˆRu k ˆR(ŷ y). Some further calculations show that the observer has the error ynamics t ( ˆRR ) = k ˆRR sk(r ˆR y 0 ) with sk(a) = 1 2 (A A ), which is the negative graient of f(e) = k 2 E 2. The observer is the explicit complementary filter propose in [14] excluing the integral introuce in that paper to compensate gyro bias. In particular, our observer esign approach illustrates the connection between the filters in [10], [13] an the complementary filter in [14]. VI. CONCLUSIONS In this paper we introuce a esign metho for constructing an observer for a left-invariant system on a Lie group with right invariant outputs on a manifol. We propose a graient-like observer for the projecte system on the output space which can be lifte to a graient-like observer on the group. The error for the observer for the projecte system converges to the reference output for generic initial states, with a locally exponential convergence rate. This lifts to convergence to the stabilizer of the reference output for the lifte observer on the Lie group, again with a locally exponential convergence rate. We have emonstrate this esign metho on the attitue estimation problem with partial state measurements for a rigi boy. In particular, this gave us new insights into the filters propose in [10], [13], [14]. VII. 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