Exact Solutions to a Class of Feedback Systems on SO(n)

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Johan Markdahl, Xiaoming Hu

To cite this version:

Johan Markdahl, Xiaoming Hu. Exact Solutions to a Class of Feedback Systems on SO(n). Rigid-Body

Attitude Control and Related Topics, 2015. �hal-01221414�

Johan Markdahl and Xiaoming Hu

Abstract. This paper explores novel aspects of the attitude tracking problem for a class of almost globally asymptotically stable feedback laws onSO(n). The closed- loop systems are solved exactly for the rotation matrices as explicit functions of time, the initial conditions, and the gain parameters of the control laws. The ex- act solutions provide insight into the transient dynamics of the system and can be used to prove almost global attractiveness of the identity matrix. Applications of these results are found in model predictive control problems where detailed insight into the transient attitude dynamics is utilized to approximately complete a task of secondary importance. Knowledge of the future trajectory of the states can also be used as an alternative to the zero-order hold in systems where the attitude is sampled at discrete time instances.

1. Introduction

The nonlinear control problem of stabilizing the attitude dynamics of a rigid body has a long history of study and is important in a diverse range of engineering applications related toe.g., quadrotors [1], inverted pendulums in three-dimensional space [2], and robotic manipulators [3]. It is interesting from a theoretical point of view due to the nonlinear state equations and the topology of the underlying state spaceSO(3). An often cited result states that global asymptotical stability onSO(3)cannot be achieved by means of a continuous, time-invariant feedback [4]. The literature does however provide results such as almost global asymptotical stability through continuous time-invariant feedback [5, 6], almost semi-global stability [7], or global stability by means of a hybrid control approach [8]. The parameterizations used to representSO(3)have important implications for the limits of control performance [4,5,9]. In particular, the use of local representations yield local results. In most cases, it is preferable to either use global representations such as the unit quaternions or to work with the space of rotation matrices directly [5].

The exact solutions to a closed-loop system give a detailed picture of both its transient and asymptotical behavior and can hence be of use in control applications. The literature on solutions to attitude dynamics can be divided into two categories. Firstly, in a number of works the solutions are obtained during the control design process,e.g., using exact linearization [10] or optimal control design techniques such as the Pontryagin maximum principle [11]. Secondly, there are works whose main focus is solving the equations defin- ing rigid-body dynamics under a set of specific assumptions [12–14]. This paper falls into the second category.

There is a considerable literature on the kinematics and dynamics ofn-dimensional rigid-bodies. This literature includes works on attitude stabilization [15], attitude syn- chronization [16], distributed averaging [17], and generalized Newtonian equations of

motion [18]. It also includes the authors previous work [19,20], which we shall comment on shortly. A key difference between the study ofSO(3)andSO(n)is that parameteriza- tions such as the unit quaternions cannot be used. Another is motivation: work onSO(3) is usually motivated by applications concerning the attitude of rigid bodies. Results re- gardingSO(n)is not only of theoretical concern however; they also finds applications in the visualization of high-dimensional data [21].

The main contribution of this paper is to provide exact solutions to differential equa- tions representing closed feedback loops on SO(n). Recent work on this problem in- clude [19, 20, 22]. Other works such as [12–14] are related in spirit but address somewhat different problems. The work [22] considers the solutions to closed-loop kinematics on SO(3). An application towards model predictive control is proposed but left unexplored.

The more general problem of solving two differential equations onSO(n)is treated in [19].

An application towards the problem of continuous actuation under discrete-time sensing is considered. The work [20] generalizes the results of [19] to a greater class of feed- back laws. This paper in turn generalizes [20] and explores the applications proposed in [19, 22]. Note that many of the results of this paper easily extend to the case of the special euclidean group,SE(n), and may be combined with position control laws in an inner-outer loop feedback scheme to achieve pose stabilization onSE(n)[23].

2. Preliminaries

Let A,B ∈ R^n×n. The spectrum ofA is writtenσ(A). The transpose and conjugate transpose ofAis writtenA^⊤andA^∗respectively. The commutator ofAandBis defined by [A,B] = AB −BA. Their inner product is defined byhA,Bi = tr(A^⊤B) and the Frobenius norm bykAkF =hA,Ai^1/2.

The set of nonsingular matrices over a fieldF is denoted byGL(n,F). The unitary group is denoted byU(n) ={U ∈GL(n,C)|U⁻¹=U^∗}. The orthogonal group isO(n) = {Q ∈ GL(n,R)|Q⁻¹ =Q^⊤}. The special orthogonal group is denoted bySO(n) ={R ∈ O(n)| detR=1}. In this paper we define

N ={R∈SO(n)| −1∈σ(R)}.

It can be shown that {R ∈ SO(n)|R^⊤ = R}/{I} ⊂ N. Equality holds in the cases of n∈ {2,3}. The Lie algebra ofSO(n)is denoted byso(n)={S ∈R^n×n|S^⊤=−S}. In this paper, we useSto denote the matrix LogR ∈so(n)forR ∈SO(n)\N.

The group of symmetric matrices is Sym(n) = {P ∈ R^n×n|P^⊤ = P}. The set of positive-semidefinite matrices is denoted byP ={P ∈Sym(n)|σ(P) ⊂[0,∞)}. The set of positive-definite matrices isP ∩GL(n,R).

The solution to a differential equation ˙X = F(t,X) is denotedX(t;t₀,X₀) wheret is the time, t₀ is the initial time, andX₀ is the initial condition. If the system is time independent we sett₀=0 and omit this dependence.

The principal matrix logarithm is uniquely defined on the set{A ∈GL(n,R)|σ(A)∩ (−∞,0] =∅}[24]. It satisfies Imσ(LogA) ⊂ {z ∈iR| |z| <π}[25]. Since any rotation matrixRis normal, it follows thatR =UΛU^∗and the logarithm ofRmay be calculated as LogR =ULog(Λ)U^∗. Moreover,Λ=exp(iΘ)for a diagonal matrixΘwhich satisfies

Θ_ii ∈(−π,π)for allR ∈SO(n)\N. Hence Log(Λ)=iΘand LogR=iUΘU^∗. The matrix logarithm allows us to calculate the geodesic distance betweenR₁,R₂∈SO(n)using the Riemannian metric

d_R(R₁,R₂)= ^√1₂kLog(R^⊤1R₂)kF.

The matrix valued hyperbolic cosine, sine, and tangent ofA ∈ R^n×n can be de- fined via the matrix exponential as cosh(A) = ¹₂[exp(A)+exp(−A)],sinh(A) = ¹₂[exp (A)−exp(−A)],tanh(A) =sinh(A)cosh⁻¹(A). The hyperbolic arctangent is defined by atanh(A) = ¹₂Log(I+A)− ¹2Log(I−A), for argumentsAsuch that [σ(I+A)∪σ(I− A)]∩(−∞,0]=∅.

By thekth root of a normal matrixA=UΛU^∗we refer to its principal root, the normal matrixA^1/k =UΛ^1/kU^∗. The principal root satisfiesR^1/k =exp(¹/kS) ∈SO(n). Moreover, R^1/k <SO(n)\NifR <SO(n)\N.

3. Problem Statement

From a mathematical perspective, it is appealing to strive for generalization. Consider the evolution of a positively orientedn-dimensional orthogonal frame represented by R ∈ SO(n). The dynamics onSO(n) are given by ˙R = ΩR. This paper concerns the following system.

System 1. Consider the system ˙R =Ω(R)R, whereR ∈SO(n)andΩ :SO(n)→so(n). The input is given byΩ,i.e., the system is actuated on a kinematic level.

The kinematic level stabilization problem onSO(n)concerns the design of anΩthat stabilizes the identity matrix. System 1 states thatRcan be actuated along any direction of so(n), its tangent space at the identity. Note thatSO(n)is invariant under the kinematics of System 1,i.e., any solutionR(t;R₀) for whichR₀ ∈ SO(n) remains inSO(n) for all t ∈[0,∞). This paper concerns a class of almost globally stabilizing feedback lawsΩthat allow System 1 to be solved forRas a function of time, any design parameters, and the initial conditions. It also analyzes the stability of said class of control laws and discusses possible applications of these results.

An equilibrium of System 1 is said to be almost globally asymptotically stable if it is asymptotically stable and the region of attraction is all ofSO(n) except for a set of measure zero. A setS ⊂ SO(n)has measure zero if for every chartϕ:B →R¹2n(n−1)in some atlas ofSO(n), it holds thatϕ(B ∩ S)has Lebesgue measure zero [26].

Problem 2. For a given almost globally stabilizing feedback lawΩ : SO(n) → so(n), solve System 1 forR(t;R₀),i.e., forR as function of the timet ∈ [0,∞) and all initial conditionsR₀ ∈SO(n)belonging to the region of attraction of the identity matrix.

Previous work on global level attitude stabilization apply the stable-unstable mani- fold theorem [5–7] or use Lyapunov function arguments [8] to establish the region of attraction of the identity matrix. The stable-unstable manifold theorem [27] is however ineffective to prove almost global asymptotical stability for systems that are actuated on a

kinematic level when the unstable equilibrium manifold corresponds to the uncountable set{R ∈SO(n)|R^⊤=R}\{I} ⊂ N.

This paper presents a novel approach to establishing almost global asymptotical sta- bility by means of exact solutions to the closed-loop system kinematics. It is possible to establish global existence and uniqueness of the solutions, see Lemma 28 in Appendix A.

Statements regarding control performance can hence be based on the properties of the exact solutions. This paper uses the solutions to show that the region of attraction of the identity matrix for the closed-loop systems generated by Algorithm 4 and 9 isSO(n)\N. The desired result follows sinceN is a set of measure zero inSO(n).

Remark 3. The attitude dynamics of a rigid body is often described by a second order system consisting of a kinematic equation, an equivalent of that in System 1, coupled with Euler’s equation of motion. In that case, the input signal is a torque vector. Kinematic level control design may however be preferable under certain circumstances, for example when an application programming interface restricts actuation to velocity level control commands or as a prerequisite in applying the backstepping control design technique [28]. Models with kinematic level actuation are also common in certain fields such as visual servo control [29, 30]. What is more, there is no compelling reason to impose Newtonian mechanics in the generalSO(n)case.

4. Main Results

This section contains the main results of the paper, the exact solutions to the closed-loop systems resulting from feedback by Algorithm 4 and 9.

4.1 Matrix Riccati Differential Equation. The following algorithm is well-known in the literature.

Algorithm 4. The input signalΩ :SO(n) → so(n) is given byΩ =PR^⊤−RP, where P ∈ Pis either a rankn−1 or a ranknmatrix.

The closed-loop system resulting from plugging Algorithm 4 into System 1 is

R˙ =P−RPR. (1)

Theorem 5. The trajectory of the closed-loop system generated by Algorithm 4 is R(t;R₀)=[sinh(Pt)+cosh(Pt)R₀][cosh(Pt)+sinh(Pt)R₀]⁻¹, where the matrix hyperbolic functions are defined in terms of the exponential matrix.

Proof. Equation (1) is a matrix valued differential Ricatti equation that can be solved using the adjoint equations technique. Introduce two matricesX,Y∈GL(n,R)that satisfy

X˙ =PY, Y˙ =PX (2)

with initial conditionsX(0;R₀) =R₀,Y(0;R₀) =I. Note thatR =XY⁻¹sinceR(0;R₀) = X(0;R₀)Y⁻¹(0;R₀)=R₀and

d

dt(XY⁻¹) =XY˙ ⁻¹−XY⁻¹YY˙ ⁻¹=P−RPR=R.˙ The system (2) is linear and has the transition matrix

exp "

0 P P 0

# t

!

=

"

cosh(Pt) sinh(Pt) sinh(Pt) cosh(Pt)

# .

By reversing the change of variables using the expressions forX(t;R₀)andY(t;R₀) we

findR(t;R₀).

Proposition 6. The identity matrix is an almost globally asymptotically stable equilibrium of System 1 under Algorithm 4. The rate of convergence is locally exponential and the region of attraction isSO(n)\N.

Proof. The proof for the cases of rankP=nand rankP=n−1 are carried out separately.

The proofs rely on the uniqueness property established in Lemma 28 which allows us to draw conclusions regarding control performance based on the exact solutions.

Consider the positive-definite case. The Frobenius norm is submultiplicative whereby kXY⁻¹−IkF =k(X−Y)Y⁻¹kF ≤ kX−YkFkY⁻¹kF.

That lim_t→∞kXY⁻¹−IkF =0 hence follows from

tlim→∞X−Y= lim

t→∞exp(−Pt) (R₀−I)=0,

t→∞limY⁻¹= lim

t→∞[I+tanh(Pt)R₀]⁻¹cosh⁻¹(Pt)=0.

The last limit is given by Lemma 30 in Appendix A. It requires the assumption ofR₀<N. We have shown thatIattracts all system trajectories such thatR₀ ∈SO(n)\N. ThatN does not belong to the region of attraction ofIfollows from Lemma 31 in Appendix A.

Use the first method of Lyapunov to show thatIis a locally exponentially stable equi- librium ofR. TakeZto be the matrix corresponding to the linearization ofR−Iaround 0. Then ˙Z=−PZ−ZP, withZ(0)=Z₀=R₀−I. Hence

Z(t;Z₀)=exp(−Pt)Z₀exp(−Pt), i.e., the linearized system is exponentially stable.

Consider the positive-semidefinite case. The eigenvectorsv₁, . . . ,v_n ofP ∈ Pform an orthogonal basis ofRⁿ by virtue of the spectral theorem. Let 0 be the eigenvalue corresponding tov_n. DenotePexpressed in the basis{v₁, . . . ,v_n}by

Q=



 v^⊤₁

... v_n^⊤



 Pf

v₁ . . . v_ng

=





v^⊤₁Pv₁ . . . v^⊤₁Pv_n

... . .. ...

v^⊤_nPv₁ . . . v^⊤_nPv_n





=







v^⊤₁Pv₁ . . . v^⊤₁Pv_n−1 0

... . .. ... ...

v^⊤_n−1Pv₁ . . . v^⊤_n−1Pv_n−1 0

0 . . . 0 0







=

"

Q₁₁ 0 0 0

# .

DenoteRexpressed in the basis{v₁, . . . ,v_n}byX, and write X=

"

X₁₁ x₁₂ x₂₁ x₂₂

# .

By calculation, one can show that ˙X₁₁=Q₁₁−X₁₁Q₁₁X₁₁, whereby Theorem 5 gives X₁₁=[sinh(Q₁₁t)+cosh(Q₁₁t)][cosh(Q₁₁t)+sinh(Q₁₁t)X_11,0]⁻¹.

Since rankQ = rankP, we find thatQ₁₁ ∈ P(n−1) ∩GL(n−1,R). What is more,

−1 < σ(X_11,0) follows from R < N by Lemma 35. The almost global attractivity and stability ofI as an equilibrium ofX₁₁ follows by reasoning analogously as done in the case of a positive definiteP. The corresponding properties ofx₁₂,x₂₁, andx₂₂follow from

the constraints onX∈SO(n). This carries over toR.

Remark 7. A key step in the above proof makes use of the constraints onX ∈ SO(n)to conclude the attractivity and stability properties ofx₁₂,x₂₁, andx₂₂based on those ofX₁₁. This technique cannot be used if rank P≤n−2 since the lower right block matrix would not be uniquely determined due to the number of constraints being insufficient.

4.2 Commutative Matrix Multiplication.The following closed-loop systems are gen- erated by a class of control laws which all share the property that the state and the input signals commute. This class is of interest since it reduces Problem 2 to Problem 8. Instead of solving a system withn²variables and ¹₂n(n−1)degrees of freedom on the Lie group SO(n), a system that evolves on the Lie algebraso(n)is solved. The Lie algebra is a linear space where the number of variables equals the number of degrees of freedom.

Problem 8. For a given asymptotically stabilizing feedback lawΩ : SO(n) → so(n), solve the autonomous system ˙S =Ω(expS)forS(t;S₀),i.e., forSas function of the time t∈[0,∞), and any initial conditionS₀ ∈so(n).

Algorithm 9. LetF:S →so(n), whereS ⊆so(n), be a mapping that satisfies

[F(S),S]=0. (3)

Moreover, suppose that the zero matrix is an asymptotically stable equilibrium of

S˙ =F(S), (4)

where (4) has a known, unique, continuously differentiable solutionS(t;S₀)for allS₀∈ S and allt ∈ (0,∞), for which{S ∈ so(n)| max_λ∈σ(S)|λ| < π}is a positively invariant set. The input matrix Ω : exp(S)\N → so(n) is given byΩ(R) = F(LogR), where Log :SO(n)\N →so(n)denotes the principal matrix logarithm.

Remark 10. Since the principal logarithm mapsSO(n)\Ninto{S ∈so(n)| max_λ∈σ(S)|λ| <

π}, it suffices to confine the domain ofFto this set or some subset thereof.

The resulting closed-loop system is

R˙ =F(LogR)R. (5)

Theorem 11. The trajectories of (5)are given byR(t;R₀)=exp[S(t; LogR₀)].

Proof. Note thatR(0;R₀) =R₀. Since [Ω,S]=0, it follows that [˙S,S]=0, see Lemma 32 in Appendix A. Hence

ΩR=R˙ = _dt^d exp(S)= d dt

X∞ i=1

1

i!Sⁱ =SR.˙

By multiplying the above identity byR⁻¹from the right, we are left with ˙S=Ω=F. Also note thatR(0;R₀) = R₀. The expression for R(t;R₀) is obtained from the exponential

mapping.

Proposition 12. Algorithm 9 stabilizes System 1. LetRdenote the region of attraction of the zero matrix as an equilibrium of (4)onso(n). The region of attraction of the identity matrix as an equilibrium of System 1 under Algorithm 9 isexp(R)\N.

Proof. The exact solution is unique by Lemma 28. Since 0 is an asymptotically stable equilibrium of (4), we find that

tlim→∞S(t; LogR₀)=0, lim

t→∞R(t;R₀)=I,

for allR₀ such that LogR₀ ∈ R. The region of attraction onSO(3)contains and hence equals exp(R)\N since Algorithm 9 is restricted to this domain.

The identity matrix being a stable equilibrium of (5), follows from the stability of (4) and the continuity of the exponential mapping. More precisely, we require a pair(δ,ε) such thatd_R[I,R(t;R₀)]≤εfor allt∈[0,∞)whend_R(I,R₀) ≤δ. Note that

d_R[I,R(t;R₀)]= ^√1₂kLogR(t;R₀)kF =^√1₂kS(t;S₀)−0kF.

The stability of0as an equilibrium ofSimplies the existence of a pair(δ^′,ε^′)such that kS(t;S₀)−0kF ≤ε^′for allt ∈[0,∞)whenkS₀−0kF ≤δ. Hence we may take^′ (δ,ε) =

1/^√2(δ^′,ε^′).

5. Examples

Algorithm 9 cannot be implemented without choosing a specific functionFwhich satisfies the stated requirements. The class of feedback laws satisfying the commutativity relation (3) includes anyF:so(n)→ so(n)that extends an analytic functionf :R→Rso that Fcan be defined in terms of the Taylor expansion of f [25]. Since it can be a nontrivial task to find such anFthat also stabilizes the zero matrix, we provide the following three control laws, Algorithm 13, 15, and 17, which are special cases of Algorithm 9.

5.1 The Matrix Logarithm. An important special case of Algorithm 9 is the geodesic feedback based on the matrix logarithm [31].

Algorithm 13. The feedbackΩ =−LogRonSO(n)corresponds to the geodesic feed- backF(S)=−Sonso(n).

The closed-loop system resulting from use of the feedback in Algorithm 13 is

R˙ =−Log(R)R. (6)

Proposition 14. The trajectories of (6)are given byR(t;R₀)=exp(e^−tLogR₀).

Proof. Note that [LogR,R]=0 by Lemma 29 in Appendix A. Moreover, the solution to S˙ = −S is given byS(t;S₀) = e^−tS₀. The eigenvalues ofS are decreasing functions of time, whereby any ball containing them is positively invariant. Algorithm 13 is hence a special case of Algorithm 9 wherefore the desired result follows by Theorem 11.

5.2 The Matrix Root.Algorithm 4 withP =Isatisfies [Ω,LogR]=0. This also holds whenRis replaced by itskth rootR^1/kfork ∈N.

Algorithm 15. The input matrix for this control law is given by Ω=k(R^−1/k−R^1/k),

where the proportional gain factorkis used to scale the time dependence ofR,i.e.,F(S)=

−2ksinh(_k¹S).

The closed-loop system resulting from use of the feedback in Algorithm 15 is R˙ =k(R^1−1/k−R^1+1/k). (7) The scalar gaink ∈ Nis introduced so that the limit lim_k→∞Ω =−2 LogR; without it the limit would be zero.

Proposition 16. The trajectories of (7)are given by

R(t;R₀)=[tanh(t)I+R¹₀^/k]^k[I+tanh(t)R¹₀^/k]^−k. Proof. Introduce the variableX=R^1/k ∈SO(n). Then

X˙ =_k¹RR˙ ^1/k−1=_k¹k(R^1−1/k−R^1+1/k)R^1/k−1=I−R^2/k =I−X², (8) which also results from settingP =Iin Algorithm 4. Reversing the change of variables in the solution forXgiven by Theorem 5 yields the desired expression.

ForF(S)=−2ksinh(_k¹S)to satisfy the assumptions of Algorithm 9 requires the invari- ance of an open ball of radiusπ in the space of eigenvalues ofS. The spectral theorem applies since skew-symmetric matrices are normal. Factorize S as done withR in the proof of Lemma 32. The eigenvalues are decreasing functions of time. Further details are omitted, as is the corresponding proof with regard to the Cayley transform.

5.3 The Cayley Transform.Another special case of Algorithm 9 is the Cayley trans- form and the higher order Cayley transforms.

Algorithm 17. The input matrix is given by

Ω=k(I−R^1/k) (I+R^1/k)⁻¹,

i.e., thekth order Cayley transform up to a scalar gain factork ∈N. The corresponding control law onso(n)isF(S) =−ktanh₁

2kS .

The closed-loop system resulting from use of the feedback in Algorithm 17 is R˙ =k(I−R^1/k) (I+R^1/k)⁻¹R. (9) The scalar gaink ∈ Nis introduced so that the limit lim_k→∞Ω =−¹2LogR; without it the limit would be the zero matrix.

Proposition 18. The trajectories of (9)are given by

R(t;R₀)=exp[2katanhY(t,X₀)], where

Y(t;X₀)=sinh(X₀)

sinh²X₀+e^tI₋¹/2

, andX₀= _2k¹ LogR₀.

Proof. ThatY(t;X₀)and atanhY(t;X₀)are well-defined follows from Lemma 34 in Ap- pendix A. Change variables fromRtoX=_2k¹ LogRwhere the scaling is just a matter of notational convenience. Note thatΩ=−ktanhX, whereby [X,Ω]=0and ˙X= _2k¹Ωby Lemma 32.

As an intermediate step, consider the evolution of Y(t;X₀)=sinh(X₀)

sinh²X₀+e^tI₋¹/2

(10) given by

Y˙(t;X₀)= −¹2sinh(X₀) (sinh²X₀+e^tI)^−3/2e^t =−¹2Y(t;X₀) (sinh²X₀+e^tI)⁻¹e^t

= −¹2Y(t;X₀) (sinh²X₀+e^tI)⁻¹(sinh²X₀+e^tI−sinh²X₀)

= −¹2Y(t;X₀)[I−Y(t;X₀)²].

It remains to verify thatX(t;X₀) =atanhY(t;X₀) solves ˙X =−¹2tanhX. Note that X(0;X₀)=atanh(tanhX₀)=X₀. What is more

X(t;˙ X₀)=[I−Y²(t;X₀)]⁻¹Y˙(t;X₀)=−¹2Y(t;X₀)=−¹2tanhX(t;X₀),

where the dynamics ofY(t;X₀)are used.

Proposition 12 reduces the stability analysis for Algorithm 13, 15, and 17 to proving the global asymptotical stability of the zero matrix onso(n). Further details are omitted for the sake of brevity.

5.4 Discussion.Algorithm 4 and 9 differ in several respects. Algorithm 4 has a constant positive-definite gain matrix that can be tuned for desired performance. It provides a continuous feedback but has a low input norm for rotations that are far from the identity, the disadvantage of which is slow convergence in the case of large errors [7]. Algorithm 13 provides a geodesic control law. The feedback laws of Algorithm 13 and 15 have input norms that are increasing functions ofkSkF. This property is useful in attitude control of satellites that are required to make large angle maneuvers [7]. The input norm of Algo- rithm 17 diverges as the error is maximized. Although such behavior is normally unde- sired in practice, it does exemplify the width of the class of algorithms that is Algorithm 9.

The disadvantage of Algorithm 13, 15, 17 as compared to Algorithm 4 is the discontinuity whenR ∈ N. Figure 1 illustrates some of these considerations forR ∈SO(3).

0 0 1

1 2

2 3

3 kΩkF

kLogRkF

Figure 1.Input norms for Algorithm 4 (red), 13 (blue), 15 (pink), and 17 (green). The parameter P =Iin Algorithm 4,k=2 in Algorithm 15, andk=1 in Algorithm 17. The gains are scaled to have equal slope at the origin.

6. Applications

The results of this paper have applications in the field of visual servo control with regards to model predictive control problems and control of sampled systems.

6.1 Model Predictive Control.The exact solutions can be used to pose a model pre- dictive control problem in terms of the feedback gain parameters of the control law. Al- gorithm 4 provides a gain matrix, P ∈ P ∩GL(n,R). The potential benefit of using optimization techniques in lieu with the solutions provided in this paper is hence greater than in [22] where only two parameters are available for tuning. This problem is of in- terest in visual servo control for the case ofn = 3 and in applications that require the visualization of high-dimensional data for the case of generaln[21].

Before turning to the model predictive control problem, consider a switched feed- back control based on the extension of Algorithm 4 to the case ofSO(n), where a time-

dependence is introduced by replacing the gain matrixPwith a piece-wise constant ma- trix valued function of time.

Algorithm 19. Consider a feedback

Ω₃=ΣR^⊤−RΣ,

whereΣ : [0,∞) → P ∩GL(n,R)is a matrix valued switching signal. The matrixΣ ∈ P ∩GL(n,R)is piece-wise constant, right-continuous, has a strictly positive dwell time

∆t, and satisfies

Σ εI (11)

for some constantε∈(0,∞). The closed-loop system is

R˙ =Σ−RΣR. (12)

The next result states the system (12) to be asymptotically stable under all switching signals that respect (11).

Proposition 20. Suppose System 1 withR₀ ∈SO(n)/N is governed by Algorithm 19. The identity matrix is a uniformly asymptotically stable equilibrium ofR. Its region of attraction isSO(n)/N.

Proof. Consider the Lyapunov functionV =tr(I−R)=n−trR. It satisfies V˙ = −tr ˙R=−tr(Σ−RΣR)=−tr(Σ(I−R²))=−hΣ,I−R²i

= −D Σ,I−¹2

R²+R⁻²

−¹2

R²−R⁻² E

−D Σ,I−¹2

R²+R⁻² E

= −

Σ,−¹2

R−R^⊤2

≤ −ε

I,−¹2

R−R^⊤2

=^ε₂D

− R−R^⊤

,R−R^⊤E

=−^ε2R−R^⊤2F, (13) where the inequality follows from utilizing thatΣ=X+εIfor someX0.

Note that ˙V ≤0, and ˙V =0 if and only ifR =R^⊤,i.e., only ifR ∈ {I} ∪ N. The set SO(n)\N is invariant under Algorithm 19 by Lemma 36. The last expression in (13) is therefore negative-definite independently ofΣoverSO(n)\N, makingVa common Lya- punov function for all switching modes. It follows that the identity is uniformly asymp- totically stable [32]. The invariance also implies that all trajectories starting inSO(n)\N

must converge to the identity matrix.

Let the switching times be given by {t_i}^∞i=0. Since Σ is constant on each interval I_i = [t_i,t_i+1), equation (12) has a solution onI_i given by Theorem 5. SetR(t_i;R₀) = lim_t↑t

iR(t;R₀) at isolated switching times. This yields left continuity. Piece together such solutions for

[0,∞)={t_i}^∞i=0∪ [∞

i=0

I_i

to find a solution to (12). The function thus obtained is not continuously differentiable at the switching times but it is a solution in the sense of Carathéodory [33].

Problem 21. Let a set of time instances {t_i}^mi=0 ⊂ [0,∞),m ∈ N∪ {∞}, and an initial conditionR₀ ∈SO(n)be given. Suppose System 1 is governed by Algorithm 19. Denote s =t−t₀. Consider the problem of minimizing a continuous functionf with respect to the inputΣ=Σ_i ∈ P ∩GL(n,R),t∈[ti,t_i+1),i.e., to solve

minimize

Σ f(t,R,Σ),

subject toR(t;t₀,R₀)={sinh[Σ_i(s−t_i)]+ cosh[Σ_i(s−t_i)]R(t_i;t₀,R₀)}

{cosh[Σi(s−t_i)]+ sinh[Σi(s−t_i)]R(t_i;t₀,R₀)}⁻¹, (14) Σ=Σ_i,

εIΣ, (15)

Σ ρI, (16)

for allt∈[t_i,t_i+1)and all i=0, . . . ,m.

The constraint (14) is obtained from solving the closed-loop system generated by Al- gorithm 19. The constraint (15) is imposed in Algorithm 19 to ensure convergence under arbitrary switching. It then follows that lim_t→∞R(t,t₀;Σ,R₀) = I for any feasible so- lution{Σ_i}^mi=0to the model predictive control problem. This frees the specification of f from any concerns regarding the asymptotical stability of the system. The constraint (16) confinesΣto a compact set whenm∈N, thereby guaranteeing the existence of a solution to the model predictive control problem by virtue of Weierstrass’ extreme value theorem.

Note that the assumption ofm ∈ Nposes no restriction in practice but does nullify the attractivity property of Proposition 20.

The model predictive control problem utilizes the transient phase of the system’s evo- lution to carry out a task of secondary importance. The model predictive control problem could also be posed with (14) replaced by (12). The benefit gained from Theorem 5 is to eliminate the computational cost of solving (12) numerically.

Example 22. Consider the problem of stabilizing the orientation of a camera while at some points in time wishing to see a desired view corresponding to the camera orientation R_d ∈SO(3). A possible choice off isf(R,Σ)=min_Σ,td_R(R_d,R)for a constant switching matrixΣ ∈ P(3)∩GL(3,R),i.e., the choice ofΣis made at time zero.

Note that the problem addressed in Example 22 is not solved by tracking a curve in SO(3) that interpolates the pointsR₀,R_d,andI. The key idea is to utilize the transient phase of the system for additional benefit. This can also be done for trajectory tracking.

6.2 Sampled Systems.Consider the problem of continuous time actuation subject to sensing that is either piece-wise unavailable in time or discrete time. The relevance of this problem in the context of attitude stabilization maye.g., be motived by cases where the attitude is calculated from images obtained by a camera for which (i) the reference used

to obtain the attitude from the image is temporarily obscured or outside the image, or (ii) images are shot at a slow frame rate. Problem (i) arises in the field of visual servo control.

The approach of this paper is well-suited for applications in visual servo control since it adopts the same kinematic system model [29, 30]. Problems of type (ii) are commonly addressed using piece-wise constant input signals [34],i.e., by applying a zero-order hold.

This section discusses the zero-order hold approach and an approach based on the flow ofΩR,i.e., the exact solutions to the closed-loop kinematics.

Assume that an outputY = R ∈ SO(n) is available for use in feedback control at timest such thatt ∈I and that it is unavailable whent <I, whereI ⊂[0,∞)is closed and contains 0 (i.e., a sampleY₀ =R₀is taken at timet₀ =0). In the case of continuous sensing, suppose thatI is such that the corresponding switching sequence has a dwell- time. In the case of discrete time sensing the states are sampled at each time instance of a sequence{t_i}^mi=0,m∈N∪ {∞}.

System 23. Consider the system ˙R =ΩR whereR₀ ∈ SO(n) andΩ : SO(n) → so(n) is the input signal. An output given byY =R is available for use in any feedback loop whent∈I.

Problem 24. Design a feedback algorithm for System 23 that stabilizes the identity.

A standard solution to Problem 24 is given by Algorithm 25.

Algorithm 25. The zero-order hold control is a time-varying feedback law given by U(Y,t)=

Ω[Y(t)], t ∈I, Ω[Y(s)], t <I,

whereΩis any control law that stabilizes System 1 ands =maxI∩[0,t].

Proposition 26. Consider System 23 under Algorithm 25 with sample times {t_i}^∞i=0 and U=−k_iLogRfort ∈[t_i,t_i+1). The identity is attractive if and only if

Y∞ i=0

1−k_i∆t_i =0, (17)

where∆t_i =t_i+1−t_i.

Proof. The resulting closed-loop system is a switched linear system which can be inte- grated to yield

R(t_j+1)=exp[−k_j∆t_jLogR(t_j)]R(t_j)=R^1−kj∆tj(t_j)=R

Qj i=01−k_i∆t_i

0 .

Proposition 26 places requirements on{k_i}i^∞=0and{∆t_i}i^∞=0. A deadbeat control,i.e., finite time convergence, is obtained ifk_i∆t_i = 1 for at least onei. In practice however, {∆t_i}^∞i=0 may not be a design parameter. Moreover, there are upper and lower bounds on{k_i}i^∞=0due to requirements on the minimum and maximum angular speed that arise

from time constraints and saturation effects. For large sample times, there may not be any choice of{k_i}i^∞=0that both satisfies (17) and accommodates the additional constraints.

The results of this paper admits an alternative approach to attitude control of sampled systems. LetΦ(R₀,t)denote the flow ofΩR,i.e.,Φ(R₀,t)=RwhereRis the solution at timetto ˙R=ΩRwith initial valueR₀∈SO(n)and the feedbackΩis given by Algorithm 4, 13, 15, or 17.

Algorithm 27. The flow algorithm is a time-varying feedback law given by U(Y,t)=

Ω[Y(t)], t ∈I, Ω{Φ[Y(s),t]}, t <I,

whereΩis a stabilizing feedback law under which the closed-loop system has a known solution ands=maxI∩[0,t].

Algorithm 9 generates the same system trajectory as the feedback would subject to continuous time sensing for allt ∈[0,∞). It is clear that the flow approach has advan- tages over the zero-order hold approach. Algorithm 9 maye.g., be applied as an open loop control based on a single measurement in which case the zero-order hold approach would fail. It is also clear that Algorithm 25 has problems with large hold times which are tolerable for Algorithm 9. Neither algorithm guarantees robustness under such circum- stances, but that is a different matter. Clearly, one could obtain a performance comparable to that of Algorithm 9 without access to exact solutions by means of quadrature but this also introduces discretization errors and a computational cost.

7. Numerical Example

Numerical quadrature transfers System 1 to a discrete-time system that generates a se- quence{R_i}^mi=0. The use of Lie group variational integrators ensures thatR_i ∈SO(n)at all discrete time instances{t_i}^mi=0of the simulation [35]. This is accomplished by setting R_i+1=exp[Ωi(t_i+1−t_i)]Ri, whereΩ_i =Ω(R_i).

System 23 under Algorithm 25 and 27 with the negative matrix logarithm,i.e., Algo- rithm 13, as the underlying attitude control law is simulated onSO(3). The sample time is constant and the gains are set to one,i.e.,k_i =1 for alli ∈ Nin Proposition 26. The initial condition is

R₀=





√1 3

√1 2

√1 1 6

√3 −^√1₂ ^√1₆

√13 0 −

√2

√3



.

The results are displayed in Figure 2. Note that Algorithm 25 behaves as predicted by Proposition 26 with a deadbeat control for ∆t = 1, asymptotical stability for∆t = 1.5, and critical stability for∆t=2. The trajectory of the system generated by Algorithm 27 is invariant of the sample time. Although the deadbeat control yields faster convergence than Algorithm 27 it is not robust to changes in the sample time. Moreover, Algorithm 27 is continuous in time whereas Algorithm 25 is discontinuous in time and gives rise to chattering behavior.

0 0 1

1 2

2 3

3 4 5 6 7 8 9 10

t kR−IkF

Figure 2.The errorkR−IkF for System 23 under Algorithm 25 with∆t ∈ {1,1.5,2}(pink) and Algorithm 27 with∆t=1 (red),∆t=1.5 (blue), and∆t=2 (green).

8. Conclusions

This paper explores the question of whether it is possible to formulate a closed-loop sys- tem onSO(n)that is almost globally asymptotically stable and admits the exact solutions to be determined explicitly. The answer is yes, and it turns out to be possible for a large class of feedback laws. Moreover, the exact solutions can be expressed rather elegantly in terms of the matrix exponential. The main tool used to uncover these solutions is the matrix differential Riccati equation. The technique used to solve the equations can also be generalized to the case ofSE(n). Some results are not straightforwardly generalized since they rely on diagonalizability via the spectral theorem for normal matrices.

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A. Lemmas

Lemma 28. The equations(1),(5), (6),(7), and (9)have unique solutions that belong to SO(n)for allt ∈[0,∞).

Proof. The proof in the case of (1) is similar to that in [19]. The assumptions made in Algorithm 13 ensures uniqueness of the solutionS(t;S₀)to (4) and hence ofR(t;R₀)to (5). In the case of (7), we can use the change of variablesX = R^1/k from the proof of Proposition 16, to obtain (8). The uniqueness of the solution to (1) implies that the solution X(t;R₀) to (8) is also unique. In the case of (9), it can be shown thatSO(n)\N is an invariant set. The change of variablesY=(I+R)⁻¹is hence admissible. Then

Y˙ =Y(I−Y) (I−2Y),

where the right-hand side is polynomial inY. Uniqueness follows by reasoning as in [19].

Reversing the change of variables proves uniqueness ofR(t;R₀)forR₀<N. Lemma 29(N.J. Higham [25]). LetA ∈ C^n×n and f : C→ Cbe given. Suppose fⁱ(λ) exists for alli ∈ {0,1, . . . ,n−1}and allλ ∈σ(A). Thenf(A)is well-defined, see[25], and satisfies(i) f(A^⊤)=f(A)^⊤,(ii)[f(A),A]=0,(iii)if[X,A]=0, then[X,f(A)]=0,(iv) σ[f(A)]={f(λ)|λ∈σ f(A)}, and(v) f(XAX⁻¹) =Xf(A)X⁻¹for anyX∈GL(n,C).

Proof. For a proof, see [25].

Lemma 30. The matricescosh⁻¹PandtanhPare well-defined forP∈ P ∩GL(n,R)and satisfy

tlim→∞cosh⁻¹(Pt)=0, lim

t→∞tanh(Pt)=I.

Proof. This follows from Lemma 29 by using thatPis normal,i.e., unitarily diagonalizable,

and calculating the corresponding scalar limits.

Lemma 31. The setN is invariant under the dynamics(1).

Proof. Consider the time-evolution ofσ(R). Take any eigenpair(λ,v) ofR and impose the constraintkvk2=1. Recall the following relations

Rv=λv, R^⊤v=λ^∗v, v^∗R=λv^∗, v^∗R^⊤=λ^∗v^∗, which hold due toλ⁻¹=λ^∗for any complexλof unit length.

The matrixRbeing normal and analytic implies, as a consequence of Rellich’s Theo- rem, that its eigenpairs are locally analytic functions of the time [36]. Note that

λ˙= _dt^dv^∗Rv=v˙^∗Rv+v^∗Rv˙ +v^∗Rv˙ =v^∗Rv˙ +λ(v^∗v˙ +v˙^∗v)

=v^∗Rv˙ +λ_dt^dkvk2²=v^∗Rv˙ =v^∗Σv−v^∗RPRv=(1−λ²)kP^1/2vk²2

=(1−(Reλ)²+(Imλ)²−2iReλImλ)kP^1/2vk2².

The eigenvalue−1 hence constitutes an unstable node type of equilibrium.

Lemma 32. The statements[˙S,S]=0and[Ω,S]=0are equivalent. Moreover, they imply thatS˙ =Ω.

Proof. From [˙S,S] = 0it follows thatΩR =R˙ =SR. Canceling˙ R yields ˙S = Ωwhich results in [Ω,S]=0.

Conversely, suppose [Ω,S]=0. SinceRis a normal matrix it has a spectral factoriza- tion given byR=UΛU^∗, whereU ∈U(n)andΛis a diagonal matrix. From [Ω,S]=0we get [Ω,R]=0which implies thatΩ=UΞU^∗, whereΞis a diagonal matrix. The matrix Rbeing normal and analytic implies, as a consequence of Rellich’s Theorem, thatΛand Uare locally analytic functions of the time [36]. Then

R˙ =_dt^dUΛU^∗=UΛU˙ ^∗+UΛU˙ ^∗+UΛU˙^∗=U(U^∗UΛ˙ +Λ˙ +ΛU˙^∗U)U^∗

=U([U^∗U,˙ Λ]+Λ)˙ U, ΩR=UΞU^∗UΛU^∗=UΞΛU^∗.

Taken together, we have [U^∗U,˙ Λ]+Λ˙ =ΞΛ, where both ˙ΛandΞΛare diagonal matrices.

The commutator has zero diagonal due toΛbeing diagonal and hence [U^∗U,˙ Λ]=0. It follows that

S˙ =U˙ Log(Λ)U^∗+U_dt^d(LogΛ)U^∗+ULog(Λ)U˙^∗=U[U^∗U,˙ LogΛ]U^∗+U_dt^d(LogΛ)U^∗

=UΛ⁻¹ΛU˙ ^∗=UΛ⁻¹U^∗UΛU˙ ^∗=R⁻¹R˙ =R⁻¹ΩR=Ω,

which yields [˙S,S] =0. The equality [LogΛ,UU˙ ^∗] = 0 follows from [Λ,UU˙ ^∗] = 0by

Lemma 29.

Remark 33. This result is important because it allows us to replace the assumption of [˙S,S] = 0 with [Ω,S] = 0. The latter assumption is preferable sinceΩ is the control input,i.e., we can designΩ. It is not, however, possible to chose ˙Sin general.

Lemma 34. The expression forY(t)given by(10)is well-defined for allt ∈(0,∞), and so isatanhY(t).

Proof. Sinceσ(R) ⊂ {z ∈ C| |z| =1}, we may obtainS =LogRforR < N using the principal logarithm. Thenσ(S)={iλ ∈iR| |λ|<π,e^{i λ} ∈σ(R)}. SinceX= _2k¹Swe find that allλ∈σ(X)satisfy|λ|<^π/2. It follows that

σ(sinh²X+e^tI)={−sin²λ+e^t |iλ∈σ(X)}.

These eigenvalues are strictly positive,i.e., sinh²X+e^tIis nonsingular. It is also normal, whereby its principal square root can be calculated as detailed in Section 2. This shows Y(t)to be well-defined.

Recall the definition of atanh given in Appendix A. Note thatY(t)is skew-symmetric, implying thatσ[Y(t)]⊂iR. Hence atanhY(t)is well-defined.

Lemma 35. LetR∈SO(n)be partitioned as R=

"

R₁₁ r₁₂ r₂₁ r₂₂

# ,

then the spectrum ofRbelongs to the unit disc inC, and in particular it holds that−1<σ(R) implies−1<σ(R₁₁).

Proof. We prove that−1∈σ(R₁₁)implies−1∈σ(R). Take anyv ∈Rⁿ⁻¹. The matrixR being orthogonal gives

"

R₁₁ r₁₂ r₂₁ r₂₂

# "

v 0#

²=kRvk²+kr₂₁vk²=kvk²,

HencekR₁₁vk ≤ kvkfor allv,i.e., the spectrum ofR₁₁is a subset of the unit disc inC. By supposing thatRv=−vwe obtainkr₂₁vk=0 whereby

"

R₁₁ r₁₂ r₂₁ r₂₂

# "

v 0

#

=

"

R₁₁ r₁₂ r₂₁ r₂₂

# "

v 0

#

=−

"

v 0

# ,

i.e.,(−1,[v^⊤0 ]^⊤)is an eigenpair ofR.

Lemma 36. The setSO(n)\N is invariant under the dynamics(12).

Proof. By reasoning as done in the proof of Lemma 31, it can be shown that Re ˙λ=kΣ^1/2vk2²[1−(Reλ)²+(Imλ)²],

Im ˙λ=−2kΣ^1/2vk²2ReλImλ.

Since−2kΣ^1/2vk2²Reλ > ε > 0 for Reλ < −¹/2due toΣ εI, it follows thatλ cannot

converge to−1.