Aspects of globally-stable tracking for certain classes of simple ...

Aspects of globally-stable tracking for certain classes of simple mechanical systems

arXiv:1511.00796v3 [cs.SY] 18 Apr 2016

A Nayak1 and R.N. Banavar2 Abstract— Proportional derivative plus feed forward (PD+FF) feedback control laws exist to locally asymptotically track a smooth reference trajectory having bounded velocity for a fully actuated simple mechanical system (SMS) on a Riemannian manifold. Almost-global asymptotically stable (AGAS) tracking for an SMS on a compact Lie group is also addressed in the literature by observing that the closed loop error dynamics is obtained by lifting the gradient vector field of a navigation function and stabilizing the error dynamics about the Lie group identity. Almost-global setpoint stabilization for an SMS on a Riemannian manifold is often tackled by introducing a gradient field of a Morse function as a feedback control force for the dissipative SMS. In this paper we address the AGAS tracking problem for an SMS on a compact Riemannian manifold embedded in Euclidean space by explicitly introducing the error dynamics. A control law, including both feedback and feedforward terms, is then synthesized for the purpose of almost-global stabilization of the error dynamics .

I. I NTRODUCTION The problem of stabilization of a equilibrium point in an SMS on a Lie group and a Riemannian manifold has been well studied in the literature [5], [12], [1], [2], [4] in a geometric framework. Further extensions of these results to the problem of tracking smooth and bounded trajectories in simple mechanical systems can be found in [5], [6], and [8]. A SMS is completely specified by a manifold, the kinetic energy which specifies the Riemannian metric, the potential forces and the external forces or one forms on the manifold. In [5], proportional and derivative plus feed forward (PD+FF) feedback control laws for tracking using error functions for configuration error and transport maps for velocity error are shown to achieve stability with exponential convergence for a fully actuated SMS with bounded energy for certain local initial conditions. However, all these results are local with respect to the choice of initial conditions for a SMS intended to track the desired trajectory. As pointed out in [7] and [6], global stabilization and hence tracking in SMSs on manifolds is not possible unless the configuration manifold is homeomorphic to Rn . This leads us to the question of whether almost-global asymptotic stabilization (AGAS) of an equilibrium point and subsequently AGAS tracking of a suitable class of reference trajectories is possible in a SMS on a Riemannian manifold. Such stabilization problems trace their origin to an early work by Koditschek ( [7]). In [7], the notion of a Morse 1 A. Nayak and 2 R.N. Banavar are with the Systems and Control Enginerring, Indian Institute of Technology Bombay, Mumbai, Maharashtra 400076, India [email protected],

[email protected]

function on the configuration space is introduced and the flow of the negative gradient vector field defined by this Morse function is analysed. It is observed that their is dense set from which the trajectories of the gradient system converge to the minima of the Morse function. By lifting the gradient field to an equivalent dissipative SMS on the tangent bundle, the AGAS property is demonstrated. An immediate application of this is tracking on Lie groups where the notion of error functions and transport maps is naturally defined using the Lie group structure. In [6] the authors propose a control law to almost-globally asymptotically track a reference trajectory for a SMS on compact Lie group or on a group which is a direct product of a compact Lie group and Rn . The controller stabilizes the error dynamics about (e, 0), where e is the Lie group identity. Here the problem of tracking the reference trajectory reduces to stabilizing the error dynamics about the group identity as the error is defined by the group action. The authors use a separation principle so that the error dynamics is a SMS with dissipation and the fact that identity is almost-globally asymptotically stable (AGAS) follows from Koditschek’s theorem. In this paper a configuration error is introduced on a Riemannian manifold embedded in Euclidean space. The error dynamics arising from the configuration error and its derivative is a second order SMS. It follows from Koditschek’s theorem that the error dynamics is AGAS about some equilibrium points. The control law for the actual system is obtained by considering the error between the system trajectory and the reference trajectory. The first section is a brief introduction to notation and definitions on SMS and error maps existing in literature, the second section is a review of existing local PD+FF control law for a SMS on a Riemannian manifold. In the third section we introduce the notion of almost global stabilization and state Koditschek’s theorem for AGAS of certain equilibrium points of a dissipative SMS. In the following section we state our control law for AGAS tracking for a fully actuated SMS on a compact Riemannian manifold in Euclidean space and in the last section we consider the example of tracking problem for a spherical pendulum which is a SMS on S 2 . II. P RELIMINARIES This section introduces known mathematical notions to describe simple mechanical systems which can be found in [5], [9], [3]. A Riemannian manifold is denoted by the 2tuple (Q, G), where Q is a smooth connected manifold and G is the metric on Q. ∇ denotes the Riemannian connection on (Q, G) ( [13], [14]). If Ψ ∶ Q → R is a twice differentiable

function on Q, it is expressed as Ψ ∈ C 2 [Q, R]. The hessian HessΨ of Ψ is the symmetric (0, 2) tensor field on Q defined by HessΨ(q)(vq , wq ) = ⟨⟨vq , ∇wq gradΨ⟩⟩, where vq , wq ∈ Tq Q. If x0 is a critical point of Ψ, and (U, φ) is a local chart on Q about x0 with coordinates {x1 , . . . , xn } then 2 Ψ the hessian at x0 is (HessΨ(x0 ))ij = ∂x∂i ∂x j (x0 ) which is independent of the Riemannian metric. The flat map G♭ ∶ Tq Q → Tq∗ Q is given by G(v1 , v2 ) = ⟨G♭ (v1 ); v2 ⟩ for v1 ,v2 ∈ Tq Q and the sharp map is its dual G♯ ∶ Tq∗ Q → Tq Q, and given by G−1 (w1 , w2 ) = ⟨G♯ (w1 ); w2 ⟩ where w1 , w2 ∈ Tq∗ Q. Therefore if {ei } is a basis for Tq∗ Q, G♭ (v1 ) = Gij v1 j ei and G♯ (w1 ) = Gij w1 j ei . A. SMS on a Riemannian manifold A simple mechanical system (or SMS) on a Riemannian manifold Q is denoted by the 7-tuple (Q, G, V, F, F , U ), where V is a potential function on Q, F is an external uncontrolled force, F = F 1 . . . F m is a collection of covector fields on Q, and U ⊂ Rm is the control set. The system is fully actuated if Tq∗0 Q = span{Fq0 }, ∀q ∈ Q. The governing equations for the above SMS considering no control inputs is given by ∇γ(t) γ(t) ˙ = −gradV (γ(t)) + G♯ (F (γ(t))) ˙ ˙ G

(1)

where gradV (γ(t)) = G♯ dV (γ(t)). The equations for a fully actuated closed loop system with control forces Fu (γ(t)) is given by substituting F + Fu instead of F in (1). B. Error function, tracking error function, and transport map This section briefly goes over the existing notions of error maps and velocity error maps which have been introduced to compare configurations and velocities in a Riemannian manifold. The local tracking control law in [5] utilizes these notions. The configuration error and it’s derivative which define the error dynamics are an extension of ideas in this section. A smooth function Ψ ∶ Q → R is a configuration error function on Q about an element q0 if Ψ is smooth, proper, bounded from below and satisfies (i)dΨ(q0 ) = 0 (the Jacobian of Ψ is zero) and, (ii) HessΨ(q0 ) is positive definite. The definition of error functions may be extended to tracking error functions on Q × Q. A smooth symmetric function Ψ ∶ Q × Q → R is a tracking error function if for r ∈ Q, Ψr ∶ Q → R is a configuration error function about r, where Ψr is defined as Ψr (q) ∶= Ψ(r, q). Therefore for all r ∈ Q, Ψr is smooth, proper and bounded from below and it satisfies (i) d1 Ψ(r, r) ∶= d1 Ψr (r) = 0r and (ii) Hess1 Ψ(r, r) is positive definite. The transport map gives means to compare velocities of two different curves (in our case the reference trajectory and the system trajectory) and define a velocity error which is essential for the purpose of tracking. As the velocities at two configurations belong to different tangent spaces, a map between tangent vector bundles is constructed to transport velocities and hence define velocity error between

two curves. A transport map T ∶ Q×T Q → T Q×Q is smooth such that T (q, vq ) = vq ∀q ∈ Q, vq ∈ Tq Q. Equivalently, T (q, vr ) = (T (q, r).(vr ), q). The pair (Ψ, T ) is said to be compatible if, for all (q, r) ∈ Q × Q, d Ψ(γ(t), η(t)) = ⟨d1 ψ(γ(t), η(t)); ve (t)⟩, (2) dt ¯ + → T Q is the velocity error vector field along where ve ∶ R γ given by ve (t) ∶= γ(t) ˙ − T (γ(t), η(t)).η(t) ˙ ∈ Tγ(t)Q. III. PD+FF CONTROL LAW FOR LOCAL TRACKING AN SMS ON A R IEMANNIAN MANIFOLD

FOR

In this section we state the well known proportional derivative plus feed forward (PD+FF) control for stable tracking for an SMS on a Riemannian manifold from [5] using an error function Ψ and a compatible transport map T . ( [5]) Lemma 1: For the a fully actuated SMS (Q, G, V = 0, F , Rm ), a given bounded smooth reference trajectory ¯ + → Q and the controlled trajectory γ ∶ R ¯+ → Q γref ∶ R , under the following PD+FF control law for the compatible pair (Ψ, T ), γref is stable with respect to Ecl (vq , wr ) ∶= Ψ(q, r) + 21 ∣∣vq − J (q, r).wr ∣∣2G . FP D (t, vq ) = −d1 Ψ(q, γref (t)) + Fdiss (vq − T (q, γref (t)).γ˙ ref (t)),

(3)

FF F (t, vq ) = G♭ (∇vq T (q, γref (t)).γ˙ ref (t) d + (T (q, γref (t)).γ˙ ref (t))) dt where Fdiss is any dissipative force or ⟨Fdiss (vq ), vq ⟩ ≤ 0 for all vq ∈ Tq Q and γref is stable with respect to Ecl if there exists a neighborhood U of γ˙ ref (0) such that for all initial conditions in γ(0) ˙ ∈ U, the error function t → Ecl (t) is nonincreasing. Proof: The Lyapunov function Ecl (t) ∶= Ψ(γ(t), γref (t)) + 12 ∣∣ve (t)∣∣2G is shown to be nonincreasing as follows. The control laws for the SMS are G

FP D (γ)(t)) = −d1 Ψ(γ(t), γref (t)) + Fdiss (ve′ (t))

FF F (γ(t)) = G♭ (∇γ(t) T (γ(t), γref (t)).γ˙ ref (t)); ˙ G

where ve′ (t) = γ(t) ˙ − T (γ(t), γref (t))γ˙ ref (t).

(4)

Therefore, G d d Ecl (t) = Ψ(t) + G(γ(t))(ve′ , ∇γ(t) ve′ ) ˙ dt dt

= ⟨d1 Ψ, ve′ ⟩ + G(ve′ , ∇γ(t) ve′ ) ˙ G

However from (4), ∇γ(t) ve′ =∇γ˙ (γ˙ − T (γ, γref ).γ˙ ref (t)) ˙ G

G

= G (FP D + FF F ) − G FF F = −G♯ (d1 Ψ(γ, γref )) + G♯ (Fdiss (ve′ )) ♯

♯

(5)

Putting together these two equations together along with the compatibility in (2), d Ecl (t) = ⟨d1 Ψ, ve′ ⟩ + G(ve′ , −G♯ (d1 Ψ − Fdiss (ve′ )) dt = ⟨d1 Ψ, ve′ ⟩ − ⟨d1 Ψ, ve′ ⟩ + ⟨Fdiss (ve′ ), ve′ ⟩ ≤ 0.

Remarks: 1) The control law stated above is therefore local with respect to the initial value of the control trajectory or, (γ(0), γ(0)). ˙ The stability can be guaranteed only for all initial conditions in some open subset U ⊂ Tγref (0) Q. 2) From Lemma 1, the only criterion to choose the tracking error function Ψ is that Ψr ∶ q → Ψ(q, r) be a configuration error function about r ∈ Q. 3) An error function is often required to vanish at the configuration about which it is defined. This is not enforced in the definition of error functions here as it does not affect the stability result in Lemma 1. IV. A LMOST- GLOBAL TRACKING The local control law in Lemma 1 relies on the choice of a tracking error function Ψ ∶ Q × Q → R. If a configuration error map E ∶ Q × Q → Q is defined on a manifold, the tracking error function Ψ ∶= ψ ○ E where ψ ∶ Q → R. If ψ is chosen as a navigation function, then the error ˙ are AGAS about (qe , 0) where qe is the dynamics (E, E) minimum of ψ. This means that the error dynamics is stable about (qe , 0) from all initial conditions in an open dense set in T Q. Consequently, the actual dynamics of the SMS are AGAS in closed loop with a corresponding control law. In the first subsection we define navigation functions and the AGAS result for a dissipative SMS on a compact Riemannian manifold from [7]. In the second subsection, we state and prove the control law for the AGAS tracking problem mentioned in the introduction. A. AGAS for a dissipative SMS A dissipative SMS is represented by the 3-tuple (Q, M, fd ) where Q is a Riemannian manifold with metric M and fd is a dissipative force. An elaborate study of stability in dissipative mechanical systems can be found in [11]. A Morse function is real valued and has all non degenerate, hence isolated critical points. A Morse function with a unique minimum is a polar Morse function or a navigation function. Definition 1 ( [7]): A polar Morse function ψ ∈ C 2 [Q, [0, 1]] on a compact manifold Q which takes unique minimum at qd ∈ Q is called a navigation function. Remark: Configuration error functions in stabilization and tracking feedback control require the Hessian to be positive definite and have a local minimum at the desired configuration. However, navigation functions are those which are non degenerate at all critical points and have a unique (or global)

minimum. Lemma 2 ( [7]): Let ∆ be a SMS on (Q, M, fd ) where Q is a smooth, compact, connected Riemannian manifold with metric M and fd is a dissipative force. Suppose ψ ∶ Q → [0, 1] is a navigation function for Q and η is the total energy (kinetic due to M plus potential due to ψ). Then the entire zero section of T Q is included in the closed subset η −1 [0, 1] ⊆ T Q. Also, η −1 [0, 1] positive invariant with respect to the flow of ∆ (called as f∆ ) and includes the entirety of T Q if Q has no boundary. Moreover, there exists an open dense set in η −1 [0, 1] whose limit set under f∆ is the desired configuration qd at zero velocity; where qd is the unique minimum of ψ. Proof: The equations of motion for the dissipative SMS (Q, G, Fdiss ) in local coordinates {q i }, i = 1, . . . , n for a n dimensional manifold without boundary Q are q¨ = G♯ (Fdiss (q)) ˙ − C(q, q) ˙

where C k (q, q) ˙ = Γkij (q)q˙i q˙j are Coriolis force terms quadratic in q˙ and Fdiss ∶ T Q → T ∗ Q is a dissipative force. If a potential function ψ ∶ Q → R is imposed as control force −dψ, the equations of motion for resulting system (which is called ∆) are q¨ = G♯ (−dψ + Fdiss (q)) ˙ − C(q, q) ˙

(6)

Let q = q1 , q˙ = q2 then the equations for ∆ are, q˙ q2 ( 1) = ( ♯ ) q˙2 G (−dψ(q1 ) + Fdiss (q2 )) − C(q1 , q2 )

It is observed that (¯ q, 0) are the equilibrium points of ∆ where q¯ are the critical points of ψ. The local behaviour of ∆ around an equilibrium (q0 , 0) is given by the linearized vector field as follows. 0 In q˙ ( 1 ) = ( ij ∂2ψ ij q˙2 −G (q0 ) ∂q1 i ∂q1 j (q0 ) G (q0 ) ○

q1 ∂Fdiss ) ( ) q2 ∂q2

As the above is a linear system subject to a linear dissipative force Fdiss , therefore the critical point (q0 , 0) is stable if and only if Hessψ(q0 ) is positive definite where Hessψ(q0 ) is the Hessian of ψ at q0 . By La Salle’s invariance theorem, all trajectories in the positively invariant set Ebdd = {v ∈ T Q ∶ η(v) < ∞} = η −1 [0, 1] converge to equilibrium points of ∆, where η(vq ) = Gij vqi vqj + ψ(q). As ψ is a navigation function and has isolated and countable critical points, the stable manifolds of all saddle points in ∆ is a nowhere dense set. The complement of this nowhere dense set, S ⊂ T Q is an open dense set. So all trajectories in S ∩ Ebdd converge to minimum of ψ. Remark: The the negative gradient flow −gradψ on Q determines local behavior of ∆ around equilibrium states in T Q. This fact is referred to as the lifting property of dissipative mechanical systems.

B. Almost-global tracking for an SMS on a compact Riemannian manifold 1) The configuration error map: We introduce the configuration error map E ∶ Q × Q → Q. Let γ ∶ R → Q and γref ∶ R → Q be two parameterized curves on Q and E be a smooth map in both arguements. The derivative of E(γ(t), γref (t)) is ˙ E(γ(t), γref (t)) = d1 E γ˙ + d2 E γ˙ ref

(7)

where the first differential is d1 E(γ, γref ) ∶ Tγ Q → TE(γ,γref ) Q, the second differential is d2 E(γ, γref ) ∶ ˙ Tγref Q → TE(γ,γref ) Q and E(γ, γref ) ∈ TE(γ,γref ) Q. We define Er (q) = E(q, r) for a fixed r ∈ Q. Let fr (q) ∶ Er (q) → q be a local diffeomorphism about all q ∈ Q. Then d1 E(q, r) = dEr (q) is invertible at all q ∈ Q. E is symmetric if ∣∣E(q, r)∣∣Rn = ∣∣E(r, q)∣∣Rn . Definition 2: Consider a manifold Q, a function ψ ∶ Q → R, a smooth and symmetric configuration error E ∶ Q × Q → Q, and the function Ψ ∶ Q × Q → R. The pair (ψ, E) is an appropriate choice for tracking problem if d1 E(q, r) is invertible for all (q, r) ∈ Q × Q. 2) AGAS tracking theorem: Theorem 1: Consider a smooth simple mechanical system on a compact, connected n− dimensional Riemannian manifold without boundary given by (Q, G, Rn ) and an at least twice differentiable trajectory γref ∶ R → Q with bounded velocity. If there exists a navigation function ψ on Q, a configuration error map E so that (ψ, E) is an appropriate choice for the tracking problem (as in Definition 2), then there exists an open dense set S in T Q, such that for all (γ(0), γ(0)) ˙ ∈ S, γref (t) can be tracked with the following control law. u = C(γ) ˙ + (d1 E) [−G♯ dψ(E) + G♯ Fdiss (ve ) −1

− d1 (d1 E)γ˙ 2 − d2 (d1 E)γ˙ ref γ˙ − C(d1 E γ) ˙

(8)

− ∇(d1 E γ) ˙ ref ) − ∇(d2 E γ˙ ref ) (d2 E γ˙ ref )] ˙ (d2 E γ ˙ Fdiss is a dissipative force and C(X) = where ve ∶= E, k i j Γij X X for a vector field X ∶ Q → T Q. Proof: Let the closed loop error dynamics be given by (E, ve ) as follows E˙ = ve

(9)

∇E˙ ve = u1

(10)

where u1 is an intermediate control that is to be designed. ˙ From the expression for E(γ, γref ) in (7), ∇E˙ ve = ∇E˙ (d1 E γ˙ + d2 E γ˙ ref )

= ∇E˙ (d1 E γ) ˙ + ∇E˙ (d2 E γ˙ ref ) d(d1 E γ) ˙ + C(d1 E γ) ˙ + ∇(d1 E γ) ˙ ref ) = ˙ (d2 E γ dt + ∇(d2 E γ˙ ref ) (d2 E γ˙ ref )

= d1 (d1 E)γ˙ 2 + d2 (d1 E)γ˙ ref γ˙ + d1 E¨ γ + C(d1 E γ) ˙ + ∇(d1 E γ) (d E γ ˙ ) + ∇ (d E γ ˙ ) 2 ref 2 ref ˙ (d2 E γ˙ ref )

where C(X) = Γkij X i X j . We have used the following identity, d(d1 E(γ, γref )γ) ˙ = (d1 (d1 E)γ) ˙ γ˙ +d2 (d1 E)γ˙ ref γ˙ +d1 E¨ γ dt

as d1 E(γ, γref ) is dependent on both γ and γref . The quantities d1 d1 E = d21 E and d2 d1 E are tensors taking two vectors as arguments as d1 E(q, r) ∶ Tq Q → TE(q,r) Q takes ones vector as argument. By the assumption in the theorem, d1 E is invertible everywhere. So γ¨ = (d1 E) [u1 − d1 (d1 E)γ˙ 2 − d2 (d1 E)γ˙ ref γ˙ − C(d1 E γ) ˙ − ∇(d1 E γ) ˙ ref ) ˙ (d2 E γ −1

(11)

− ∇(d2 E γ˙ ref ) (d2 E γ˙ ref )]

If the closed loop dynamics for the SMS are ∇γ˙ γ˙ = u then the control u = γ¨ + C(γ). ˙ The control law for the actual system u, depends on the control law u1 for the error dynamics through the equation (11). As these error dynamics are defined for the controlled trajectory γ(t) and the reference trajectory γref (t), stability properties of the actual system will follow from the stability of error dynamics. Let us now define u1 = −G♯ dψ(E)+G♯ Fdiss (ve ). The closed loop error dynamics then are E˙ = ve

∇E˙ ve = −G dψ(E) + G Fdiss (ve ) ♯

♯

(12) (13)

We define a Lyapunov function as Ecl = ψ(E) + 21 ∣∣ve ∣∣ . Then 2

d Ecl (t) = ⟨d1 ψ(E), ve ⟩ + ⟨ve , ∇E˙ ve ⟩ dt = ⟨d1 ψ(E), ve ⟩ + G(ve , −G♯ (d1 Ψ − Fdiss (ve )) = ⟨d1 Ψ, ve ⟩ − ⟨d1 Ψ, ve ⟩ + ⟨Fdiss (ve ), ve ⟩ ≤ 0.

as Fdiss is dissipative. Therefore the error dynamics in (9) are locally stable about (qe , 0) where qe is the minimum of ψ. The sets of bounded energy Ebdd , constructed in Lemma 2 are positively invariant. As Q is compact and without boundary, this positive invariant set is entirely T Q. The equilibria of (9) are (¯ q , 0) where q¯ are the critical points of ψ. So by LaSalle’s invariance theorem, the limit set of all solution trajectories f∆ originating in the positive invariant set T Q is the set (¯ q , 0). Linearizing (9) around an equilibrium point (qe , 0), we have E˙ ( )= v˙ e 0 In ) ( ij (qe , 0) −G (qe )d2 ψ(qe ) Gij (qe ) ○ ∂F∂vdiss e E ( ). ve

As in Lemma 2, the local behaviour of the error dynamics around (¯ q , 0) is the same as that of the negative gradient vector field −G♯ dψ. Lemma 2 can be applied as ψ is a navigation function and Q is a compact Riemannian manifold. So there is an open dense set S ∈ T Q from which all trajectories along (12) converge to (qm , 0) where qm is the unique minimum of ψ. Remarks: 1) The essential fact that the proof uses is the lifting property of dissipative mechanical systems. The controller u1 in (9) is chosen so that the closed loop error dynamics are that of a dissipative SMS. 2) Theorem 1 reduces the problem of almost-global tracking of a given reference trajectory to finding a navigation function ψ on the manifold, a configuration error E so that (ψ, E) is an appropriate choice for the tracking problem. 3) If the Riemannian manifold is a Lie group G, the configuration error E ∶ G × G → G is given by the group action Φ ∶ G × G → G. Considering E(g, h) = Φ(g, h−1 ) for g,h ∈ G, the tracking problem is reduced to stabilizing error dynamics about identity of the Lie group after finding a Morse function with unique minimum at identity. 4) The control law given in [5] uses a single transport map T (γ, γref ) to transport γ˙ ref to Tγ Q. Instead of this, we have two transport maps d1 E and d2 E to transport γ˙ and γ˙ ref respectively to TE(γ,γref ) Q along ˙ The position error is given by the error trajectory E. ψ(E) which is called the tracking error function and denoted by Ψ. However, instead of the velocity error along γ(t), ˙ we consider the velocity error E˙ along E(t). 5) If Ψ is chosen as ψ ○ E for an appropriate choice of (ψ, E) and the PD+FF controller in (3) (as in [5]) is used to track γref , with ve′ ∶= γ˙ − T γ˙ ref then error dynamics are given by E˙ = d1 E.ve′

(14)

v˙ e = G♯ (−dψ(E)d1 E + Fdiss (ve′ ))

(15)

− I(ve′ , T

γ˙ ref ) − C(ve′ )

where I(ve′ , T γ˙ ref ) = Γkij ve′ (T γ˙ ref ) and C(ve′ ) = i j Γkij ve′ ve′ . (14) comes from the equivalent compatibility condition i

j

d2 E(γ, γref ) = −d1 E(γ, γref )T ((γ, γref )).

(16)

Linearizing (14)-(15) about (qe , 0) where qe ∈ q¯, we

get E˙ ( ′) = v˙ e (

0 −Gij d2 ψ(qe )d1 E

E ( ′). ve

d1 E ) Gij ○ Fdiss (qe , 0) − I ♭ (T γ˙ ref )

As I ♭ (T γ˙ ref ) is a time dependent term, the flow of error dynamics around (qe , 0) cannot be approximated by the flow of −G♯ dψ. As a result, the lifting property of dissipative systems cannot be used for the error dynamics and hence AGAS cannot be established. 6) Koditschek (in [7]) considers only proportional derivative controller, whereas the feed forward terms in (8) give additional information about acceleration of the desired trajectory. Moreover, the problem of tracking is considered only on SO(3). Theorem 1 holds for the more general problem of tracking for a SMS on a compact Riemannian manifold. 7) Definition 2 enforces that d1 E(q, r) is invertible at all points on Q × Q. However it sufficient to impose the invertibility condition in S × Q where S is dense in Q and defined in Theorem 1 for a given ψ. 8) Existence of Morse functions on compact manifolds without boundary is well addressed in literature [10]. C. AGAS tracking for a spherical pendulum 1) Coordinate system on S 2 : We use the spherical coordinates f ∶ (−π, π) × (0, π) → R3 for all points in S 2 ∖ {(0, 0, 1), (0, 0, −1)} as S 2 is a manifold in R3 . 2) Morse function on S 2 : The height function on the sphere f (x, y, z) = z, (x, y, z) ∈ S 2 ⊂ R3 is a Morse function. In local spherical coordinates the Morse function is ψ(θ, φ) = 1 − cos φ. Critical values are achieved at φ = 0, π which correspond to points (0, 0, −1) and (0, 0, 1) respectively in R3 . The Hessian at these points is positive definite which can be verified by choosing any local coordinate system. The differential is dψ(θ, φ) = [0 sin φ] 3) The configuration error on S 2 and its differentials: In local coordinates of two configurations (θ1 , φ1 ) and (θ2 , φ2 ) we define the configuration error as E((θ1 , φ1 )(θ2 , φ2 )) = θ − θ2 ) The first differential d1 E = I2 and the second ( 1 φ1 − φ2 differential d2 E = −I2 . E˙ = ve is given as θ˙ − θ˙ ve = ( ˙ 1 ˙ 2 ) φ1 − φ2 . The differential of error function is given as dψ(E(θ1 , φ1 ), (θ2 , φ2 )) = (0 sin(φ1 − φ2 )) . 4) Metric on S 2 : We consider a spherical pendulum of mass m and unit length in gravity free space. The kinetic energy in local coordinates is ˙ φ) ˙ = m (φ˙ 2 + θ˙2 sin2 φ) L(θ, φ, θ, 2

Therefore the kinetic energy metric G in local coordinates (θ, φ) is given by the matrix G=(

2

m sin φ 0

0 ) m

and G♯ is given by G−1 as 1

G−1 = ( m sin 0

2

φ

1) m

0

The Christoffel symbols of second kind in local coordinates (θ, φ) are computed to be Γ111 = Γ122 = Γ212 = Γ221 = Γ222 = 0, φ Γ112 = Γ121 = cos and Γ211 = − sin φ cos φ. sin φ 5) AGAS tracking for the spherical pendulum: As S 2 is a compact Riemannian manifold and (ψ, E) is an appropriate choice for the tracking problem we apply Theorem 1 to write the feedback control. Let (θ˙1 , φ˙ 1 ) be local coordinates of the control trajectory and (θ˙2 , φ˙ 2 ) be local coordinates of the reference trajectory. The first two terms in (8) are A1 = −G♯ dψ(E) + G♯ Fdiss (ve ) and given as A1 =

θ˙1 −θ˙2 ⎛ ⎞ m sin2 φ 1 ˙ ˙ ⎝m (− sin(φ1 − φ2 ) + (φ1 − φ2 ))⎠

A2 = −2d1 (d1 E)γ˙ 2 − 2d2 (d1 E)γ˙ ref γ˙ =0 as

d1 (d1 E) = d2 (d1 E) = 0. A3 = −C(d1 E γ) ˙ + C(γ) ˙ =0 cos φ1 ˙ ˙ 2 sin φ1 θ1 φ1 ). C(γ) ˙ =( − sin φ1 cos φ1 θ˙2 1

j A4 = −Γkij (d1 E γ) ˙ (d2 E γ˙ ref ) = Γkij γ˙ i γ˙ ref cos φ1 ˙ ˙ (θ2 φ1 + φ˙ 2 θ˙1 ) = ( sin φ1 ). − sin φ1 cos φ1 θ˙1 θ˙2 i

j

A5 = −∇(d1 E γ) ˙ ref ) = γ¨ref + A4 . ˙ (d2 E γ A6 = −∇(d2 E γ˙ ref ) (d2 E γ˙ ref ) j i = γ¨ref − Γkij γ˙ ref γ˙ ref φ1 ˙ ˙ −2 cos θ φ sin φ1 2 2 ) , = γ¨ref + ( sin φ1 cos φ1 θ˙2 2

θ¨ where γ¨ref = ( ¨2 ) and u = ∑6i=1 Ai . φ2 V. C ONCLUSIONS

AND FUTURE WORK

In this note we have established almost-global asymptotically stable tracking of a bounded reference trajectory on a compact Riemannian manifold by employing the error dynamics. We are currently working on other examples of SMSs on compact Riemannian manifolds. The problem of AGAS tracking with exponential convergence for SMSs on Riemannian manifolds has not been addressed. This is a possible area of investigation in future.

ACKNOWLEDGMENT The authors thank Sneha Gajbhiye and Amit Sanyal for constructive comments. R EFERENCES [1] N.E. Leonard A.M. Bloch and J.E. Marsden. Controlled lagrangians and the stabilization of mechanical systems. I. The first matching theorem. IEEE Transactions on Automatic Control, 45.12:2253–2270, 2000. [2] N.E. Leonard A.M. Bloch, D.E. Chang and J.E. Marsden. Controlled lagrangians and the stabilization of mechanical systems. II. Potential shaping. IEEE Transactions on Automatic Control, 46.10:1556–1571, 2001. [3] V.I Arnold. Mathematical Methods of Classical Mechanics. SpringerVerlag, New York, 2nd edition, 1989. [4] R. Bayadi and R. N. Banavar. Almost global attitude stabilization of a rigid body for both internal and external actuation schemes. European Journal of Control, 20:45–54, 2014. [5] F. Bullo and A.D. Lewis. Geometric Control of Mechanical Systems. Modelling, Analysis and Design for Simple Mechanical Control Systems. Springer Verlag, Berlin, Germany. [6] J.M. Berg D.H.S. Maithripala and W.P. Dayawansa. Almost-global tracking of simple mechanical systems on general class of Lie groups. IEEE Transactions on Automatic Control, 51:216–225, 2006. [7] D. Koditschek. The application of total energy as a lyapunov function for mechanical control systems. Contemporary Mathematics, 97:131, 1989. [8] T Lee. Geometric tracking control of attitude dynamics of a rigid body on SO(3). American Control Conference, pages 1200–1205, 2011. [9] J.E. Marsden and T.S. Ratiu. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, volume 17. Springer Science and Business, Inc., 2013. [10] M. Morse. The existence of polar non-degenerate functions on differentiable manifolds. Annals of Mathematics, pages 352–383, 1960. [11] M.C. Munoz-Lecanda and F. J. Yaniz-Fernandez. Dissipative control of mechanical systems: a geometric approach. SIAM journal on control and optimization, 40.5:1505–1516, 2002. [12] N. H. McClamroch N. Chaturvedi and D. S. Bernstein. Asymptotic smooth stabilization of the inverted 3-d pendulum. IEEE Transactions on Automatic Control, 54.6:1204–1205, 2009. [13] P. Petersen. Graduate Texts in Mathematics- Riemannian Geometry. Springer, October 1998. [14] K Yano. Integral Formulas in Riemannian Geometry. Marcel Dekker Inc., New York, 1970.