Orbital Stabilization of Point-to-Point Maneuvers in Underactuated Mechanical Systems
OOrbital Stabilization of Point-to-Point Maneuvers inUnderactuated Mechanical Systems
Christian Fredrik Sætre ∗ Anton Shiriaev
Department of Engineering Cybernetics, NTNU, Trondheim, Norway.
Abstract
The task of inducing, via continuous static state-feedback, an asymptotically stable heteroclinic orbit ina nonlinear control system is considered in this paper. The main motivation comes from the problem ofensuring convergence to a so-called point-to-point maneuver in an underactuated mechanical system, that is,to a smooth curve in its state–control space that is consistent with the system dynamics and which connectstwo stabilizable equilibrium points. The proposed method uses a particular parameterization, together witha state projection onto the maneuver’s orbit as to combine two linearization techniques for this purpose: theJacobian linearization at the equilibria on the boundaries and a transverse linearization along the orbit. Thisallows for the computation of stabilizing control gains offline by solving a semidefinite programming problem.The resulting nonlinear controller, which simultaneously asymptotically stabilizes both the orbit and thefinal equilibrium, is time-invariant, locally Lipschitz continuous, requires no switching and has a familiarfeedforward plus feedback–like structure. The method is also complemented by synchronization function–based arguments for planning such maneuvers for mechanical systems with one degree of underactuation.Numeric simulations of the non-prehensile manipulation task of a ball rolling between two points upon the“butterfly” robot demonstrates the efficacy of the full synthesis.
Keywords: Orbital stabilization; Underactuated mechanical systems; Nonlinear feedback control; Nonprehensile manipulation. A point-to-point (PtP) motion is perhaps the most fundamental of all motions in robotics: Starting from rest ata certain configuration (point), the task is to steer the system to rest at a different goal configuration. Often itcan be also be beneficial, or even necessary, to know a specific predetermined motion which smoothly connectsthe two configurations, on the form of a curve in the state–control space which is consistent with the systemdynamics, a maneuver [1]. This for instance ensures that the controls stay within the admissible range alongthe nominal motion, as well as that neither any kinematic- nor dynamics constraints are violated along it.Knowledge of a maneuver is also especially important when considering an underactuated mechanical system (UMS), that is, a system having fewer independent controls (actuators) than degrees of freedom [2, 3]. This isdue to the fact that any feasible motion of such a system must comply with the dynamic constraints which arisedue to the underactuation [4].Planning such (open-loop) PtP maneuvers in an UMS, e.g. a swing-up motion of an inverted pendulumwith several passive degrees of freedom, is of course a nontrivial task in general. Supposing, however, thatsuch a maneuver is given, then it naturally has to be complemented by some feedback controller as to handlethe perturbations which inevitably will occur in practice. For non-feedback linearizable systems (i.e. the vastmajority) this is also a nontrivial task. The challenge again lies in the lack of actuation, which may severelylimit the possible actions the controller can take. This can make reference tracking controllers less suited forthis purpose, as they, often unnecessarily so, are tasked with tracking one specific trajectory (among infinitelymany) along the maneuver, meaning that they also have to ensure the timing of the motion, as opposed to onlyits invariance in state space.For the cases in which it is the invariance of the motion itself which is the main concern, one can insteaddesign an orbitally stabilizing feedback : a time-invariant state-feedback controller that (asymptotically) stabilizesthe set of all the states along the maneuver, i.e., its orbit . In regards to PtP maneuvers, such a feedback ∗ Corresponding author. Email addresses: { christian.f.satre, anton.shiriaev } @ntnu.no a r X i v : . [ ee ss . S Y ] F e b ontroller is therefore equivalent to inducing an asymptotically stable heteroclinic orbit in the resulting closed-loop system—an invariant, one-dimensional manifold which (smoothly) connects the initial and final equilibriumpoints. Naturally, there are some clear advantageous to such an approach: First, all solutions starting uponthe orbit asymptotically convergence to the final equilibrium along the maneuver, with the behaviour whenevolving along it known a priori . Secondly, due to the asymptotic stability of the orbit, all solutions startingwithin some neighborhood convergence to it asymptotically, and therefore also to the final equilibrium.In regards to the design of such feedback, the maneuver regulation approach proposed in [1] is of particularinterest. There, the task of stabilizing—by static state-feedback—non-vanishing (i.e. equilibrium-free) maneu-vers of feedback linearizable systems was considered, with the approach later extended to a class of nonminimumphase systems on normal form in [5]. The key idea in these papers is to convert a linear tracking controllerinto a controller stabilizing the maneuver. This is achieved by using a projection of the system states onto themaneuver, a projection operator as we will refer to it here, to recover the corresponding “time” to be used inthe controller, thus eliminating its time dependence. The former tracking error therefore instead becomes a transverse error—a weighted measure of the distance from the current state to the maneuver, or, equivalently,to its orbit. Indeed, it has long been known (see, e.g., [6]) that for non-trivial orbits, such as periodic ones,strict contraction in the directions transverse to it corresponds to the contraction of solutions towards the orbititself; see also [7–10]. Moreover, this contraction can be determined from a specific linearization of the systemdynamics along the nominal orbit [11], a so-called transverse linearization [12–14]. Since this contraction occurson transverse hyperplanes, only the linearization of a set of transverse coordinates of dimension one less thanthe dimension of the state space needs to be stabilized; a fact which has been readily used to stabilize periodicorbits in UMSs (see, e.g., [12] and the references therein).For the purpose we consider in this paper, however, namely the design of a continuous (orbitally) stabilizingfeedback controller for PtP motions with a known maneuver, one must of course also take into consideration theequilibria located at the boundaries of the motion. On the one hand, this directly excludes regular transversecoordinates–based methods such as [4, 12, 13], which would then require some form of control switching and/ororbit jumping `a la [15, 16]. On the other hand, the ideas proposed in regards to maneuver regulations in [1, 5]can, as we will see, to some extent be modified as to also handle the equilibria, but suffers from other short-comings: 1) the choice of projection operator is strictly determined by the tracking controller, thus excludingsimpler operators, e.g., operators only depending on the configuration variables; while most importantly, 2) therestriction of constant feedback gains greatly limits its applicability to stabilize (not necessarily PtP) motionsof both UMSs and nonlinear systems in general. Contributions.
The approach we propose in this paper for stabilizing PtP maneuvers removes the previ-ously mentioned shortcomings by instead utilizing a transverse linearization as in [11] in the feedback designrather than converting an existing tracking controller. This extends the applicability of the ideas in [1] to alarger class of dynamical systems, as well as to different types of behaviours, including PtP maneuvers. Themain novelty in our approach lies in the use of a specific parameterization of the maneuver, together with anoperator to recover the parameterizing variable, which, roughly speaking, allows us to merge the transverse lin-earization with the regular Jacobian linearization at the boundary equilibria. This, in turn, allows us to derivea (locally Lipschitz) Lyapunov function candidate for the nominal orbit as a whole. The main contributionsand results of this paper are listed below.1. We state conditions upon a (Lipschitz continuous) feedback controller for (orbital) stabilization of PtPmaneuvers in nonlinear control-affine systems; see Theorem 9 in Section 4.2. We provide a procedure for constructing such feedback, on the form of a semidefinite programming prob-lem; see Proposition 13 in in Section 4.3. We propose a synchronization function–based method for planning PtP maneuvers for a class of underac-tuated mechanical systems; see Theorem 15 in Section 5.4. We show how to generate a stable PtP motion between any two points upon the “butterfly” robot; seeProposition 16 in Section 6Note also that, with only minor modifications, these statements also provide new approaches to solve the “easier”tasks of planning and orbitally stabilizing periodic motions, or to ensure contraction of non-vanishing motionsdefined over a finite time interval.
Outline.
The rest of the paper is organized as follows. We begin by stating the problem formulation inSection 2. In Section 3 we introduce some preliminary concepts which are used in the main result, namely a2pecific parameterization of the motion (Sec. 3.1), the concept of projection operators to recover the parame-terizing variable (Sec. 3.2) and a special set of variables whose origin correspond to the orbit (Sec. 3.3). Themain result of the paper, i.e. conditions upon an orbitally stabilizing feedback for a PtP maneuver, as well ashow to design one are stated in Section 4. In Section 5 we provide statements for planning PtP maneuvers ofunderactuated systems. A nonprehensile manipulation example is then given in Section 6. Lastly, Section 7contains a brief discussion, while some concluding remarks are given Section 8.
Notation. I n denotes the n × n identity matrix and 0 n × m a n × m matrix of zeros, with 0 n = 0 n × n .For a compact set S , we denote by S o its interior and by ∂ S its boundary. For O ⊂ R n and x ∈ R n weuse dist( O , x ) := inf y ∈O (cid:107) x − y (cid:107) , with (cid:107) x (cid:107) = √ x T x . For some x ∈ R n and (cid:15) > n -ball B (cid:15) ( x ) := { y ∈ R n : dist( x, y ) < (cid:15) } . For column vectors x and y we use the shorthand col( x, y ) := [ x T , y T ] T .For a C mapping h : R n → R m , we denote by J h : R n → R m × n its Jacobian matrix, and if m = 1 then H h : R n → R n × n denotes its Hessian matrix. If s (cid:55)→ h ( s ) is differentiable at s ∈ S ⊆ R , then h (cid:48) ( s ) = dds h ( s ).We use (cid:107) h ( x ) (cid:107) = O ( (cid:107) x (cid:107) k +1 ) to denote that (cid:107) h ( x ) (cid:107) / (cid:107) x (cid:107) k → (cid:107) x (cid:107) → M n (cid:31) ( S ) (resp. M n (cid:23) ( S )) denotes theset of all smooth, symmetric, positive (resp. semi-) definite n × n matrix-valued functions over S ⊆ R , i.e. if M n (cid:31) ( S ) (cid:51) M : S → R n × n then M ( s ) = M T ( s ) (cid:31) s ∈ S . Consider a nonlinear control-affine system ˙ x = f ( x ) + g ( x ) u (1)with state x ∈ R n and with ( m < n ) controls u ∈ R m . It is assumed that both f : R n → R n and the columnsof the full-rank matrix function g : R n → R n × m , denoted g i ( · ), are twice continuously differentiable.Let the pair ( x e , u e ) ∈ R n × R m correspond to an equilibrium of (1), i.e., f ( x e ) + g ( x e ) u e ≡ n × . If wedenote by A := E ( x e , u e ) and B := g ( x e ), with E ( x, u ) := J f ( x ) + m (cid:88) i =1 J g i ( x ) u i , (2)then it is well known that the (forced) equilibrium point x e is linearly stabilizable if there exists some K ∈ R m × n such that A cl := A + BK is Hurwitz (stable). That is, the full-state feedback u = u e + K ( x − x e ) then renders x e a locally asymptotically stable equilibrium of (1).With this in mind, suppose two linearly stabilizable equilibrium states, ( x , u ) and ( x f , u f ), of (1) areknown, for which x (cid:54) = x f . Further suppose that there exists a pair of mappings x (cid:63) : S → R n and u (cid:63) : S → R m of class C and C , respectively, with S := [ s , s f ] ⊂ R , such that M := { ( x, u ) ∈ R n × R m : x = x (cid:63) ( s ) , u = u (cid:63) ( s ) , s ∈ S} is a maneuver of (1) connecting these equilibrium states; that is, x (cid:63) ( · ) and u (cid:63) ( · ) satisfy the following relations: R1. ddt x (cid:63) ( s ) = f ( x (cid:63) ( s )) + g ( x (cid:63) ( s )) u (cid:63) ( s ) for all s ∈ S ; R2. (cid:107) f ( x (cid:63) ( s )) + g ( x (cid:63) ( s )) u (cid:63) ( s ) (cid:107) > s ∈ S o ; R3. ( x (cid:63) ( s i ) , u ∗ ( s i )) = ( x i , u i ) for both i ∈ { , f } .Here time evolution of the parameterizing variable s on M is described by˙ s = ρ ( s ) , (3)where the continuous function ρ : S → R ≥ , which is strictly positive and C over S o , is assumed to be known.Rather than attempting to track a specific time-parameterized trajectory of M , naturally resulting in a time-varying (tracking) control law, we are in this paper instead interested in finding a continuous, time-invariant,static state-feedback controller which asymptotically stabilizes the corresponding heteroclinic orbit . That is tosay, if we denote by O the orbit corresponding to the maneuver M , namely O := { x ∈ R n : x = x (cid:63) ( s ) , s ∈ S} , (4)whose α - and ω -limit points necessarily are x and x f , respectively, then the notion of stability we here considerfalls under the following set-stability notion: 3 efinition 1. A closed invariant set Γ ⊂ R n of the system (1) is said to be asymptotically stable if, for any (cid:15) >
0, there exists a δ > x ( · ) of (1) satisfying dist(Γ , x ( t )) < δ , it is implied that: • ( Stability ) dist(Γ , x ( t )) < (cid:15) for all t ≥ t ; • ( Attractivity ) dist(Γ , x ( t )) → t → ∞ .The main problem we aim solve in this paper can then be stated as follows. Problem 2. (Orbital Stabilization) Find a mapping k : R n → R m , which is locally Lipschitz in a neighborhood N ( O ) of O and satisfies k ( x (cid:63) ( s )) ≡ u (cid:63) ( s ) for all s ∈ S , such that the orbit O is rendered asymptotically stablewith respect to (1) under the control law u = k ( x ).Since we have assumed g ( · ) to be of full rank, the asymptotic stability of O necessarily implies that M isattractive, although the maneuver will generally not be stable. We therefore use the fitting concept of orbitalstabilization, which here is simply taken to mean asymptotic stabilization of the maneuver’s orbit.As the orbit (4) also contains the initial equilibrium x , it follows that, under any feedback which solvesProblem 2, all solution of (1) will (locally) converge either directly to the now unstable equilibrium x or onto O\{ x } . Along O\{ x } the states asymptotically approach the final equilibrium x f , such that the asymptoticstability of O is equivalent to the (local) attractivity of x f . Thus, if in addition x f can be shown to be stable,then (locally) asymptotic stability of x f is necessarily ensured.In order to solve Problem 2, we have to take into consideration that ddt x (cid:63) ( s ) vanishes for s ∈ ∂ S . Wetherefore begin by introducing a parameterization having an additional property next. It will be assumed that a parameterization in accordance with the following is known.
Definition 3. An s -parameterization of M , i.e. one that satisfies R1 - R3 , is said to be regular if (cid:107)F ( s ) (cid:107) > s ∈ S , where F ( s ) := x (cid:48) (cid:63) ( s ).Using F ( · ) and ρ ( · ) (see (3)), we can now rewrite R1 as ρ ( s ) F ( s ) = f ( x (cid:63) ( s )) + g ( x (cid:63) ( s )) u (cid:63) ( s ) . (5)From this and R2 , we can infer that ρ ( · ) must satisfy ρ ( s ) = ρ ( s f ) ≡ ρ ( s ) > ∀ s ∈ S o . (6)and consequently that ρ (cid:48) ( s ) > ρ (cid:48) ( s f ) ≤ s -parameterization as by Def. 3 is just that (cid:107) dds x (cid:63) (cid:107) > ρ ≡ Since we are looking for a time-invariant feedback k ( · ), we require some mapping that allows us to recover theparameterizing variable s from the system’s states within some neighborhood of the orbit, a so-called projectionoperator . Although such a mapping can always be constructed for periodic orbits (see, e.g., [14]), it needs tosatisfy some additional requirements when the orbit begins and terminates at equilibrium points as we considerin this paper. Definition 4.
Let
X ⊂ R n be a compact domain containing the orbit O . A piecewise-defined mapping p : X → S is said to be a projection operator for some x (cid:63) : S → O according to Def. 3 if it satisfies the followingproperties:1) It is a left inverse of x (cid:63) ( · ), i.e., s = p ( x (cid:63) ( s )) ∀ s ∈ S ;4 T Π(ˆ s ) ˆ s H x H f x f Figure 1: Illustration of the moving Poincar´e section Π(ˆ s ) defined by (8) travelling along the orbit O whichbegins and terminates at x and x f , respectively. The gradient of the projection operator is assumed to benonzero and well defined within the blue-shaded tubular neighbourhood T . Within darkly shaded hemispheres H and H f , on the other hand, the gradient vanishes as the projection operator projects the states onto therespective equilibrium therein.2) It is twice (piecewise) continuously differentiable almost everywhere within X ; in particular, there isan open tubular neighborhood T ⊂ X of O within which its gradient J T p ( x ) is nonzero, bounded andcontinuously differentiable, while ω ( s ) := J p ( x (cid:63) ( s )) (7)is such that ω ( s ) F ( s ) ≡ s ∈ S ;3) There exists some (cid:15) , (cid:15) f > p ( B (cid:15) ( x ) \T ) = s and p ( B (cid:15) f ( x f ) \T ) = s f . In order to shed some light on the necessity of these requirements, we define the setΠ(ˆ s ) := { x ∈ X : p ( x ) = ˆ s } . (8)As is illustrated in Figure 1, for some ˆ s ∈ S o this set traces out a hypersurface, a so-called moving Poincar´esection [17], which at s ≡ ˆ s is locally orthogonal to ω ( s ). By condition 2), this in turn implies that it is locallytransverse to the direction of the nominal orbit’s flow given by F . The tubular neighborhood T in the definition(consider the blue-shaded tube in Figure 1) can therefore be taken as any connected subset of (cid:83) s ∈S o Π( s ) suchthat the surfaces Π( s ) ∩ T and Π( s ) ∩ T are locally disjoint for any s , s ∈ S o , s (cid:54) = s . Thus for some x in T , which therefore lies upon a moving Poincar´e section, the gradient J T p ( x ) is nonzero and well defined.This is however not true at the ends of the tube T . This brings us to condition 3), which guarantees theexistence of the two half-ball–like regions H := B (cid:15) ( x ) \T and H f := B (cid:15) f ( x f ) \T of non-zero measure that arecontained within Π( s ) and Π( s f ), respectively (consider the darkly shaded semi-ellipsoids in Figure 1). As aconsequence, (cid:107) J p ( x ) (cid:107) ≡ x ∈ H o ∪ H of . Hence p ( · ) is differentiable almost everywhere within X except atthe hypersurfaces X := lim s → s +0 Π( s ) and X f := lim s → s − f Π( s ), which correspond to the intersections betweenthe tube and half-balls.The following provides a large family of such operators. Lemma 5.
Let Λ ∈ M n (cid:23) ( S ) be such that the following holds for all s ∈ S : F T ( s )Λ( s ) F ( s ) > . Then there isa neighbourhood X ⊂ R n of O within which p ( x ) = arg min s ∈S (cid:2) ( x − x (cid:63) ( s )) T Λ( s )( x − x (cid:63) ( s )) (cid:3) (9) has a unique solution such that p : X → S is projection operator. Moreover, ω ( s ) := J p ( x (cid:63) ( s )) is then given by ω ( s ) = F T ( s )Λ( s ) F T ( s )Λ( s ) F ( s ) . (10) Proof.
See Appendix A. 5 .3 Merging two linearization techniques
Given a regular parameterization x (cid:63) : S → O and a projection operator p : X → S (see definitions 3 and 4),consider now the variables ˜ x := x − x (cid:63) ( s ) with s = p ( x ) . (11)Evidently, the origin of these variables corresponds to the nominal orbit O , while, locally, they are non-zeroaway from it. Moreover, from the discussion in Sec. 3.2, it follows that they are well defined for x ∈ X , locallyLipschitz in a neighborhood of O , and therefore continuously differentiable almost everywhere therein. Due tohow we have defined projection operators, the variables (11) may also be interpreted differently depending onwhere in X the current state is located. To illustrate this fact, we will again consider the half-balls ( H , H f )and the tube T introduced in the Sec. 3.2. We will also use the subscript i to denote some fixed i ∈ { , f } .Suppose the controller in (1) is taken according to u = u (cid:63) ( s ) + K ( s )˜ x with s = p ( x ) (12)where K : S → R m × n is of class C . Let A s ( s ) := E ( x (cid:63) ( s ) , u (cid:63) ( s )) and B s ( s ) := g ( x (cid:63) ( s )) (13)where E ( · ) is given by (2), such that we can define A cl ( s ) := A s ( s ) + B s ( s ) K ( s ) . (14)Since for x ∈ H oi we have p ( x ) = s i , it follows that ˜ x = x − x i therein. It can therefore be shown that x ∈ H oi = ⇒ ddt ˜ x = A cl ( s i )˜ x + O ( (cid:107) ˜ x (cid:107) ) . (15)The first-order approximation of the dynamics of these variables inside either half-ball thus evidently justcorresponds to the Jacobian linearization of (1) about the respective forced equilibria.The variables (11) takes on a somewhat different meaning within the tube T . To see this, consider theJacobian matrix Ω( s ) := J ˜ x ( x (cid:63) ( s )), which can be written asΩ( s ) = I n − F ( s ) ω ( s ) . (16)Let us recall some of its properties. Lemma 6 ([14]) . The matrix function Ω( s ) defined in (16) is a projection matrix, that is Ω ( s ) = Ω( s ) for all s ∈ S ; its rank is always n − ; while ω ( s ) and F ( s ) span its left- and right annihilator spaces, respectively. Simply put, the matrix Ω( s ) projects any vector x ∈ R n upon the hyperplane orthogonal to ω ( s ). Using itsproperties, it can be shown that the Taylor expansion of the coordinates (11) about x (cid:63) ( s ) may be written as˜ x = Ω( s )˜ x + F ( s ) l ( s, ˜ x ) (17)for some scalar function l ( · ) satisfying (cid:107) l ( s, ˜ x ) (cid:107) = O ( (cid:107) ˜ x (cid:107) ). The variables (11) may therefore be interpretedas to form an excessive set of so-called transverse coordinates inside T [14]. This means that the qualitativestability properties of the orbit may be assessed therein using the corresponding transverse linearization —thefirst-order approximation of their dynamics. Lemma 7 (Transverse linearization [11, 14]) . Inside the tube T , the first-order approximation of the dynamicsof the coordinates (11) along the orbit O is described by the differential-algebraic equations: (cid:40) ddt δ ˜ x = [Ω( s ) A cl ( s ) − F ( s ) ω (cid:48) ( s ) ρ ( s )] δ ˜ x, ω ( s ) δ ˜ x, (18) where ω (cid:48) ( s ) := F T ( s ) H p ( x (cid:63) ( s )) . s = p ( x ), it follows from this and (17), that for x ∈ T : ddt ˜ x = [Ω( s ) A cl ( s ) − F ( s ) ω (cid:48) ( s ) ρ ( s )] Ω( s )˜ x + O ( (cid:107) ˜ x (cid:107) ) , (19)or equivalently Ω( s ) ˙˜ x = Ω( s ) A cl ( s )Ω( s )˜ x + O ( (cid:107) ˜ x (cid:107) ). As a consequence, any feedback that ensures contractiontowards the origin of (18) will evidently also ensure local contraction towards the nominal orbit when inside T .In regards to assessing such a contraction, the following statement, which is slight reformulation of Theorem 1in [18] may be used (we omit the s -arguments in order shorten the notation). Note that this statement alsocan be derived from the stronger statements found in [11, 19] (see, respectively, Theorem 5.1 and Theorem 1therein). Lemma 8.
Suppose there exists
R, Q ∈ M n (cid:31) ( S ) such that the projected Lyapunov differential equation (PLDE) Ω T (cid:2) A T cl Ω T R + R Ω A cl + Q (cid:3) Ω + ρ Ω T (cid:2) R (cid:48) − ( ω (cid:48) ) T F T R − R F ω (cid:48) (cid:3) Ω = 0 n (20) is satisfied for all s ∈ S . Then the scalar function V ⊥ = δ ˜ x T R ( s ) δ ˜ x is strictly decreasing over S with respect to the linearized transverse dynamics (18) as ˙ V ⊥ = − δ ˜ x T Q ( s ) δ ˜ x . Before we move on to stating the main result of this paper, we remark that the existence and uniquenessof a (local) solution ˜ x ( · ) is guaranteed under the control law (12). Indeed, this is ensured by the existenceand uniqueness of the solutions to the nonlinear system (1) as the control law (12) is locally Lipschitz in aneighborhood of the nominal orbit (4); a fact which readily follows from ˜ x being locally Lipschitz in X and that K ( · ) is C . Recall the definitions of ρ , A cl and Ω given in (3), (14) and (16), respectively. Theorem 9.
Suppose there exists a symmetric, positive definite (SPD) matrix function R : S → R n × n suchthat: For some SPD matrices Q , Q f ∈ R n × n , it satisfies the algebraic Lyapunov equations: A T cl ( s ) R ( s ) + R ( s ) A cl ( s ) = − Q , (21a) A T cl ( s f ) R ( s f ) + R ( s f ) A cl ( s f ) = − Q f ; (21b) For some smooth SPD matrix function Q : S → R n × n , the matrix function R ⊥ ( s ) := Ω T ( s ) R ( s )Ω( s ) solvesthe following differential matrix equation for all s ∈ S : Ω T ( s ) (cid:2) A T cl ( s ) R ⊥ ( s ) + R ⊥ ( s ) A cl ( s ) + ρ ( s ) R (cid:48)⊥ ( s ) + Q ( s ) (cid:105) Ω( s ) = 0 n . (22) Then taking in (1) the (locally Lipschitz) control law (12) renders both the orbit O , defined by (4) , and the finalequilibrium, x f , asymptotically stable. Remark 10.
Any solution R ⊥ ∈ M n (cid:23) ( S ) to (22) satisfying R ⊥ ( s ) ≡ Ω T ( s ) R ⊥ ( s )Ω( s ) for all s ∈ S is unique. Proof.
The proofs of both Theorem 9 and of Remark 10 are found in Appendix B.
Remark 11.
Again, it is important to note that the control law (12) in Theorem 9 ensures the asymptoticstability of an orbit, i.e. the invariant set O , which also includes x . Thus, if starting sufficiently close tothis orbit, it ensures that the system’s states will eventually converge to either the initial equilibrium x ,which is rendered partially unstable (a “saddle”), or to x f . Any ambiguity in regards to the possibility of thesystem getting “stuck” at x may, however, easily be removed by some ad hoc modification to the controller(12). This may be achieved in a multitude of different ways, with the perhaps simplest just limiting thecodomain of the projection operator used in (12). For example, given an operator of the form (9), one maytake s = arg min s ∈ [ s + (cid:15),s f ] ( · ) for some sufficiently small (cid:15) >
0. As another alternative, one can let (cid:15) ∈ [0 , (cid:15) M ]be a bounded dynamic variable, for example ˙ (cid:15) = λ (cid:15) · sign (cid:0) δ (cid:15) − (cid:107) x − x (cid:107) (cid:1) for small positive numbers (cid:15) M , δ (cid:15) , λ (cid:15) ,although the control law will then no longer be truly static in a neighborhood of x .7efore we move on to show how such a feedback can be constructed, let us also consider a simple exampleas to highlight the effect of the projection operator upon the resulting feedback controller. Example 12.
Consider the double integrator¨ q = u, q ( t ) , u ( t ) ∈ R , with state vector x = col( q, ˙ q ). Starting from rest at q , the task is to drive the system to rest at q f ( > q )along the curve x (cid:63) ( s ) = col( s, ρ ( s )). Here s ∈ S := [ q , q f ] and ρ ( s ) := κ ( s − q )( q f − s ) for some constant κ >
0. As (cid:107)F ( s ) (cid:107) = 1 + ( ρ (cid:48) ( s )) ≥
1, this is a regular parameterization according Def. 3.Suppose p ( · ) is a projection operator in line with Def. 4 (we will provide some candidates for this operatorshortly). Using the shorthand notation s = p ( x ), we define ˜ x := q − s , ˜ x := ˙ q − ρ ( s ) and u (cid:63) ( s ) := ρ (cid:48) ( s ) ρ ( s ),such that u = u (cid:63) ( s ) − k ˜ x − k ˜ x is of the form of (12). Let us therefore check when this feedback, corresponding to a constant K = [ − k , − k ],satisfies the conditions in Theorem 9 for a given p ( · )We begin by considering the algebraic Lyapunov equations (21). Their solutions necessarily correspond tothe constant matrices A cl ( s ) and A cl ( s f ) being Hurwitz, which is the case only if k , k > A cl := (cid:20) − k − k (cid:21) . Suppose, therefore, in the following that k , k > R ∈ R × be the corresponding unique SPDsolution to A T cl R + RA cl = − I . We may then consider the (locally Lipschitz) Lyapunov function candidate V = 2 − ˜ x T R ˜ x , with ˜ x = col(˜ x , ˜ x ), whose origin evidently corresponds to the desired orbit. Within theinteriors of Π( q ) and Π( q f ), with Π( · ) defined in (8), where (cid:107) J p (cid:107) = 0, we therefore have ˙ V = −(cid:107) ˜ x (cid:107) . Thusto determine the stability of the orbit as a whole, we need to check that we also have contraction withinsome tubular neighborhood T for the chosen projection operator. We will therefore consider two different suchoperators next.By taking inspiration from [1], let us first consider the operator p corresponding to taking Λ = R in (9). Using(10), we then observe that Ω T ( s ) R F ( s ) = 0 × for all s ∈ S . Hence (22) is everywhere satisfied for R ⊥ = R and Q = I (see [1] for further details). Moreover, we have ˙ V = −(cid:107) ˜ x (cid:107) for all x such that p ( x ) ∈ ( q , q f ).Consider now instead the operator obtained by taking Λ = diag(1 ,
0) in (9). This is equivalent to p ( x ) =sat q f q ( q ), where sat ba ( · ) is the saturation function with lower- and upper bound a and b , respectively. Clearlythen ˜ x ≡ q ∈ [ q , q f ], while it can be shown that ˙˜ x = − ( k + ρ (cid:48) ( s ))˜ x . Thus, the time derivative of theabove Lyapunov function now satisfies˙ V = − R ( k + ρ (cid:48) ( s ))˜ x = − R ( k + ρ (cid:48) ( s )) (cid:107) ˜ x (cid:107) whenever q ∈ ( q , q f ), with R > R . We may therefore ensure that V will bestrictly decreasing everywhere inside the tube (except, of course, on the nominal orbit) by taking, for instance, k > sup s ∈S | ρ (cid:48) ( s ) | . This is nevertheless in contrast to the previous operator (i.e. Λ = R ) where k > K ( · )upon the choice of projection operator. Let A ⊥ ( s ) := Ω( s ) A s ( s ) − ρ ( s ) F ( s ) F T ( s ) H p ( x (cid:63) ( s ))Ω( s ) and B ⊥ ( s ) := Ω( s ) B s ( s ). By taking inspiration fromthe LMI based method of [20], the following statement can then be used to obtain a feedback matrix K : S → R m × n satisfying the conditions of Theorem 9. Proposition 13.
Suppose that for some strictly positive, smooth function λ : S → R ≥ , there exists a pairof smooth matrix function Y : S → R m × n and W ∈ M n (cid:31) ( S ) such that the following differential linear matrixinequality (LMI) is satisfied for all s ∈ S : ρ ( s ) W (cid:48) ( s ) − W ( s ) A T ⊥ ( s ) − A ⊥ ( s ) W ( s ) − Y ( s ) T B T ⊥ ( s ) − B ⊥ ( s ) Y ( s ) − λ ( s )[Ω( s ) W ( s ) + W ( s )Ω T ( s )] (cid:23) n . (23)8 urther suppose that for some K , K f ∈ R m × n which are such that ( A s ( s )+ B s ( s ) K ) and ( A s ( s f )+ B s ( s f ) K f ) are both Hurwitz, one has K W ( s ) = Y ( s ) and K f W ( s f ) = Y ( s f ) . (24) Then by taking K ( s ) = Y ( s ) W − ( s ) , the matrix function R ( s ) = W − ( s ) satisfies Theorem 9.Proof. See Appendix C.In order to find a solution pair (
W, Y ) to Proposition 13, one can attempt to transcribe (by some means) thedifferential LMI (23) into a finite set of LMIs, which then can be solved as a semi-definite programming problem(SDPP). In regards to handling the constant stabilizing matrices K and K f in such a SDPP formulation, thereare two main options: Add the LMI constraints W ( s ) A T s ( s ) + A s ( s ) W ( s ) + Y T ( s ) B s ( s ) + B s ( s ) Y ( s ) ≺ s ∈ { s , s f } ; or Add the equality constraints (24), in which K and K f have been computedbeforehand, for example by using a LQR: take K = − Γ − B T s ( s ) R and K f = − Γ − f B T s ( s f ) R f , where R , R f ∈ R n × n are the solutions to the algebraic Riccati equations A T s ( s i ) R i + R i A s ( s i ) + Q i − R i B s ( s i )Γ − i B T s ( s i ) R i = 0 n (25)for i ∈ { , f } given some SPD matrices Γ , Γ f ∈ R m × m and Q , Q f ∈ R n × n . We will now consider the task of planning point-to-point maneuvers with a parameterization according toDef. 3 for underactuated mechanical systems with n q degrees of freedom and one degree of underactuation. Theequations of motion of such systems are of the form [3] M ( q )¨ q + C ( q, ˙ q ) ˙ q + G ( q ) = B q u. (26)Here q = col( q , . . . , q n q ) are generalized coordinates and ˙ q ∈ R n q the corresponding generalized velocities, x = col( q, ˙ q ) denotes the n = 2 n q states, while u ∈ R m is a vector of m = n q − M ( · ) ∈ R n q × n q denotes the SPD inertia matrix; the constant matrix B q ∈ R n q × m is of full rank; C ( q, ˙ q ) = C ( q, ˙ q ) + C ( q, ˙ q )is a matrix function corresponding to any Coriolis and centrifugal forces, with C ( q, ˙ q ) := (cid:80) n q i =1 ∂ M ( q ) ∂q i ˙ q i and C ( q, ˙ q ) := − (cid:104) ∂ M ( q ) ∂q ˙ q, . . . , ∂ M ( q ) ∂q nq ˙ q (cid:105) T ; while G ( · ) ∈ R n q is the gradient of the system’s potential energy.Suppose that for a pair of points (configurations) q and q f , q (cid:54) = q f , there exists u , u f ∈ R m such that G ( q ) ≡ B q u and G ( q f ) ≡ B q u f . The task we want solve in this section can then be stated as follows:Starting from rest at q , find a smooth maneuver of (26) with parameterization according to Def. 3, along whichthe system is driven to rest at q f . We propose for this purpose a procedure inspired by the virtual holonomicconstraints approach of [4]. Suppose a maneuver of (26) is known with parameterization according to Def. 3. Since the system (26) ismechanical, we may represent the curve x (cid:63) : S → O in the following form: x (cid:63) ( s ) := col (cid:0) Φ( s ) , Φ (cid:48) ( s ) ρ ( s ) (cid:1) . (27)Here the vector function Φ( s ) = col( φ ( s ) , . . . , φ n q ( s )) traces out a geometric path in the configuration space ofthe system and consists of so-called synchronization functions , that is, smooth functions φ i ( · ) which we assumecan be built up by a finite number of basis functions.The scalar function ρ : S → R ≥ may now, in addition to recovering the nominal velocity of the parame-terizing variable s on O as before (see (3)), also be considered as to set the speed at which the geometric path If s = s ( q ), then the relations φ i ( · ) are sometimes referred to a virtual holonomic constraints [4]. This terminology is somewhatmisleading for the purpose we consider in this paper, however, and we therefore use the more fitting notion of synchronizationfunctions. · ) is traversed. However, as the considered system has one degree of underactuation, we cannotjust take ρ to be any arbitrary function as we did with the fully actuated example in Sec. 4.1. Rather, for (27)to correspond to a feasible solution of (26), then ρ must be a solution to the so-called reduced dynamics —afirst-order differential equation of the form [4]: α ( s ) ρ (cid:48) ( s ) ρ ( s ) + β ( s ) ρ ( s ) + γ ( s ) = 0 . (28)Here α ( s ) := B ⊥ q M (Φ( s ))Φ (cid:48) ( s ), β ( · ) := α (cid:48) ( · )+ ˆ β ( · ) with ˆ β ( s ) := B ⊥ q C (Φ( s ) , Φ (cid:48) ( s ))Φ (cid:48) ( s ) and γ ( s ) := B ⊥ q G (Φ( s )),where B ⊥ q ∈ R × m is any full rank left-annihilator of B q , that is B ⊥ q B q = 0 × m . If a solution ρ ( · ) to the reduceddynamics (28) is found for some Φ( · ), then, using any left inverse B † q of B q , i.e., B † q B q = I m , the nominal controlinput along (27) is given by u (cid:63) ( s ) = B † q (cid:2) A ( s ) ρ (cid:48) ( s ) ρ ( s ) + B ( s ) ρ ( s ) + G ( s ) (cid:3) (29)where A ( s ) := M (Φ( s ))Φ (cid:48) ( s ), B ( s ) := A (cid:48) ( s ) + C (Φ( s ) , Φ (cid:48) ( s ))Φ (cid:48) ( s ) and G ( s ) := G (Φ( s )).The reduced dynamics and its solutions are naturally of particular interest for the task we here consider. Wetherefore recall some of the key properties of this equation next, with the following statement just a reformulationof Theorem 3 in [21]. Lemma 14.
Let s e ∈ S be an equilibrium point of (28) , i.e. γ ( s e ) ≡ , satisfying α ( s e ) (cid:54) = 0 . Let ν ( s ) := γ (cid:48) ( s ) /α ( s ) . (30) Then s e is a center if ν ( s e ) > or a saddle if ν ( s e ) < . Here the conditions for a saddle equilibrium follows directly from the Hartman–Grobman theorem, whereasthe condition for a center equilibrium point, on the other hand, can be attained by noticing that the solutionsof (28) form certain level curves. More specifically, it is not difficult to see that µ ( s r , s ) := 12 α ( s ) e (cid:82) ssr β ( ν ) α ( ν ) dν =: 12 α ( s )Ψ( s r , s ) (31)is an integrating factor of (28) for any s r ∈ S ; that is, dds (cid:0) µ ( s r , s ) α ( s ) ρ ( s ) (cid:1) + µ ( s r , s ) γ ( s ) = 0 . From this (or alternatively from [4, Theorem 1] using our notation), one can obtain the function I ( s , s , S , S ) := α ( s ) S − Ψ( s , s ) (cid:104) α ( s ) S − (cid:90) s s Ψ( s , τ ) α ( τ ) γ ( τ ) dτ (cid:105) (32)whose value is exactly zero for any two points s , s ∈ S if S ≡ ρ ( s ) and S ≡ ρ ( s ), where ρ ( · ) is a solutionto the reduced dynamics (28). Note that for certain systems one has ˆ β ( s ) ≡ s ∈ S , and hence Ψ( · ) ≡ I = α ( s ) S − α ( s ) S + 2 (cid:90) s s α ( τ ) γ ( τ ) dτ. (33)This property holds for all flat systems (i.e., systems whose inertia matrix M ( · ) is constant) and for systemswhere the passive joint is the first in a kinematic chain, such as underactuated systems of Class-I according tothe classification in [22]. Let us now demonstrate how one can use this theory in order to obtain a parameterization of the form as inDef. 3. In this regard, letting ν ( · ) and Ψ( · ) be defined as in (30) and (31), respectively, then the main result ofthis section follows, in which we note that the second part follows from Theorem 1 in [23].10 ~σ ~r f θϕϑ ~τ~n r b Figure 2: The coordinate convention used, with frame having the form of the “butterfly” robot.
Theorem 15.
Suppose that the C -smooth map Φ :
S → R n q satisfying Φ( s ) = q and Φ( s f ) = q f with S := [ s , s f ] , is such that | Φ (cid:48) ( s ) (cid:107) (cid:54) = 0 , (cid:107) Φ (cid:48) ( s f ) (cid:107) (cid:54) = 0 , ν ( s ) < , ν ( s f ) ≤ . (34) Further suppose that one of the following cases hold:
C1.
The open interval S o contains a single isolated equilibrium point s e satisfying ν ( s e ) > , but no singularpoints, that is α ( s ) (cid:54) = 0 for all s ∈ S ; in addition, the following is satisfied: (cid:90) s f s Ψ( s , τ ) α ( τ ) γ ( τ ) dτ ≡ . (35) C2.
The interval S o contains no equilibrium points, with γ ( s ) > for all s ∈ S o , but a single isolated singularpoint s s , that is α ( s s ) ≡ , at which ˆ β ( s s ) < − α (cid:48) ( s s ) < . (36) Then there exists a unique, bounded, C -function ρ : S → R ≥ satisfying ρ ( s ) = ρ ( s f ) ≡ and ρ ( s ) > for all s ∈ S o , such that the curve (27) satisfies Def. 3, with the nominal control input given by (29) .Proof. See Appendix D for the proof of Theorem 15.For systems in which the equilibria of (28) are fixed, one must evidently utilize C2 in order to find a PtPmaneuver between two adjacent equilibria. For system in which ˆ β ( s ) ≡ s ∈ S , on the other hand, e.g.flat systems, then evidently C2 cannot be used. We will now apply both the motion planning method proposed in Section 5 and the feedback design approachoutlined in Section 4 as to solve the following non-prehensile manipulation [24] problem: Generate a stable PtPmaneuver corresponding to a ball rolling between any two points upon a actuated planar frame. We begin bydescribing the system model and provide some necessary assumptions.
Consider a ball of (effective) radius r b rolling-without-slipping upon the boundary of an actuated frame as seenin Figure 2. The edge of the frame is traced out by the polar coordinates ( ϑ, r f ( ϑ )), with ϑ ∈ I ⊆ S and where r f : I → R > is a strictly positive, smooth scalar function. The following relation between the ball’s radius andthe shape of the frame is assumed: A1.
The ball’s center traces out a smooth curve when it traverses the frame. Mathematically, this is equivalent to r b κ f ( ϑ ) < ∀ ϑ ∈ I , where κ f ( ϑ ) is the signed curvature of the planar curve at ϑ . r f ( ϑ ) = const.cos( ϑ ) ), the disk-on-disk [26] ( r f ( ϑ ) = const.), as well as the so-called “butterfly” robot [27],whose frame, as in [28], may be of the form r f ( ϑ ) = a − b cos(2 ϑ ) (37)for some positive constants a and b .In the following, we will further assume that: A2.
The ball is always in contact with the frame;
A3.
The ball always rolls without slipping.The validity of these assumption must of course be checked for any found motion of the system.Now, let θ and ϕ be defined as shown in Figure 2 and take q = col( θ, ϕ ). Then, in light of the aboveassumptions, the system matrices corresponding to (26) are given by M ( q ) = (cid:34) J f + J b + m (cid:107) (cid:126)σ (cid:107) − (cid:0) m(cid:126)σ · (cid:126)n + J b r b (cid:1) ζ (cid:48) − (cid:0) m(cid:126)σ · (cid:126)n + J b r b (cid:1) ζ (cid:48) (cid:0) J b r b + m (cid:1) ζ (cid:48) (cid:35) , C ( q, ˙ q ) = (cid:34) c ˙ ϕ c ˙ θ − c ˙ ϕ − c ˙ θ (cid:16) J b r b + m (cid:17) ζ (cid:48) ζ (cid:48)(cid:48) ˙ ϕ (cid:35) , G ( q ) = (cid:20) m(cid:126)g · (( ddθ Rot( θ ) ) (cid:126)σ ) m(cid:126)g · (Rot( θ ) (cid:126)τ ζ (cid:48) ) (cid:21) , B q = (cid:20) (cid:21) , where c := mζ (cid:48) (cid:126)σ · (cid:126)τ , c := (cid:0) m(cid:126)σ · (cid:126)n + J b r b (cid:1) ζ (cid:48)(cid:48) + c κζ (cid:48) and (cid:126)g = col(0 , g ). Note that a more detailed descriptionof the system parameters and variables can be found in [28], albeit with a slightly different notation. We will now utilize the procedure outlined in Section 5 to plan PtP maneuvers for these systems. For thispurpose, let ψ ( ϕ ) denote the tangential angle of the polar curve at ϕ , namely, the angle such that the unittangent vector (cid:126)τ at ϕ may be written as (cid:126)τ = col(cos( ψ ) , sin( ψ )); or equivalently, the angle such that ∂ψ∂ζ = κ where ζ is the arc length and κ = κ ( ϕ ) is the signed curvature of the curve traced out by the ball. Hence ψ is trivial for systems with constant curvature, e.g., ψ ≡ ψ = − ϕ for thedisk-on-disk.With this in mind, consider Φ( s ) = col (cid:0) Θ( s ) − ψ ( s ) , s (cid:1) , s ∈ S ⊆ S , (38)for some smooth, scalar function Θ( · ). Simply put, if one takes Θ = 0, then the synchronization function (38)aligns (cid:126)τ with the fixed horizontal axis (see Figure 2), such that the ball can be consider as to be rolling on ahorizontal surface. The function Θ( · ) can therefore be used to slow down or speed up the rolling motion byaltering the “slope” upon which the ball rolls.For this choice of Φ( · ), it can be shown that the functions α ( · ) and γ ( · ) in (28) are given by α ( s ) = (cid:18) J b R (cid:18) κ + 1 R (cid:19) + m (1 + (cid:126)σ · (cid:126)κ ) (cid:19) ζ (cid:48) − (cid:18) m(cid:126)σ · (cid:126)n + J b R (cid:19) ζ (cid:48) Θ (cid:48) ( s ) and γ ( s ) = mgζ (cid:48) sin(Θ( s )) . From this and Lemma 14, it is straightforward to deduce the following.
Proposition 16.
A point s e ∈ S , for which α ( s e ) (cid:54) = 0 , is an equilibrium point of (28) if, and only if, Θ( s e ) ≡ .Moreover, it is a center if Θ (cid:48) ( s e ) /α ( s e ) > , or a saddle if Θ (cid:48) ( s e ) /α ( s e ) < . One can therefore choose the equilibrium points of (28) freely through the choice of Θ. Thus
C1. inTheorem 15 may in turn be readily used to construct a PtP maneuver between any two points on the frame.12able 1: Parameter values of the “butterfly” robot used in the numerical simulations. m [kg] r b [m] J b [kg m ] J f [kg m ] g [m s − ] a [m] b [ m]3 . × − . × − . × − . × − . × − . × − e Figure 3: Solutions of the reduced dynamics, with the chosen solution in red.
As an example, we will consider the “butterfly” robot given by (37), with the values of the system parametersgiven in Table 1. The specific task we will consider is to drive the ball from ϕ = 0 rad to ϕ f = 2 rad. Motion planning.
In light of Proposition 16, we consider for this purpose the synchronization function(38) with Θ( s ) = k ( s − s )( s − s e )( s f − s ) (39)where s = 0, s e ≈ . s f = 2, and k = 0 .
01. The solution of the corresponding reduced dynamics (28) (its“phase portrait”) can be seen in Figure 3, with the chosen solution satisfying Theorem 15 marked in red. Thecorresponding nominal control input found from (29) can be seen in Figure 4 measured relative to the rightvertical axis.
Control design.
Since the Jacobian linearization is controllable (and therefore stabilizable) at both x = x (cid:63) ( s ) and x f = x (cid:63) ( s f ), we computed a pair of constant LQR-based feedback matrices K , K f ∈ R m × n bysolving the algebraic Riccati equations (25) using the CARE command in MATLAB with Γ = Γ f = 10 and Q = Q f = I . Note that the magnitude of Γ and Γ f here simply reflects the small parameter values (seeTable 1). Projection operator.
We took Λ = diag(0 , , ,
0) in (9) with S := [ s , s f ], which is equivalent to p ( x ) = sat s f s ( ϕ ), with sat ba : R → [ a, b ] the saturation function. Solving the differential LMI.
By taking λ = 0 .
5, we formulated a semidefinite programming problem tosolve the differential LMI (23) with the equality constraints (24). The elements of the matrix functions W and Y were taken as sixth-order Bezi´er polynomials and (23) was taken as constraints at 200 evenly spaced points.The resulting SDP problem was then solved using the YALMIP toolbox for MATLAB [29] together with theSDPT3 solver [30]. The elements of the obtained K ( s ) = Y ( s ) W − ( s ) ∈ R × are shown in Figure 4. Implementation.
Following the discussion of Remark 11, the projection operator was implemented as p ( x ) = sat s f s + (cid:15) ( ϕ ), where the dynamic variable (cid:15) ∈ [0 , (cid:15) M ] was governed by ˙ (cid:15) = (cid:15) M sign (cid:0) (cid:15) M − (cid:107) x − x (cid:107) (cid:1) with (cid:15) M = 10 − (note that similar results were obtained with a constant (cid:15) = (cid:15) M ). Since exact measurements ofall the states were assumed to be given, the implementation of the controller (12) is straightforward: Step 1.Given x , compute s = p ( x ); Step 2. Compute u (cid:63) ( s ), K ( s ) and x (cid:63) ( s ) (e.g. using splines or lookup tables); Step3. Take u = u (cid:63) ( s ) + K ( s )˜ x with ˜ x := x − x (cid:63) ( s ). Simulation results.
The response of the system when starting with the initial conditions x (0) = x +col(0 . , − . , ,
0) can be seen in Figure 5, with some snapshots of the system’s configuration shown in Figure 6.As the states are initially within the half-ball corresponding to x , it can be seen that the controller thereforefirst brings the states close to x , after which they then follow the nominal orbit to x f . Notice also that thenormal force F n between the ball and the frame is everywhere positive, meaning that the previously made13 -3 -0.03-0.02-0.010 Figure 4: Control gains (left axis) and the nominal control input (right axis). -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.200.20.40.60 5 10 15 20 25 30-202 10 -3 Figure 5: Response of the “butterfly” robot when starting close to x .assumption that the ball does not depart from the frame is not violated.To test the sensitivity of the closed-loop system to noise and uncertainties, we simulated the system withthe same initial conditions, but with a small amount of white noise added to the measurements passed to thecontroller and with the matched disturbances 10 − sin( t ) added to the right-hand side of (26). The resultingsystem response is shown in Figure 7.Figure 8 shows the system response for the initial conditions x (0) = x f + col(0 . , . , , u = u (cid:63) ( s f ) + K ( s f )( x − x f ).Notice also that ϕ becomes less than 2 rad just before t = 1 s, at which the gradient of the projection operatorhas a discontinuity. It can be seen that the smoothness of the control input is violated at this time instant, butit is clear from the highlighted rectangle that Lipschitz continuity is still preserved. Is this Orbital Stabilization?
The main focus of this paper has been upon the stabilization of the set O (see(4)) corresponding to an assumed-to-be-known maneuver M . Even though this set consists of a heteroclinc orbit14 =0 t=4 t=7 t=11 t=15 t=19 t=22 t=26 t=30 Figure 6: Snapshots of the configuration of the “butterfly” robot system corresponding to the response shownin Fig. 5. -3 Figure 7: Response of the “butterfly” robot when starting close to x with white noise added to all the statemeasurements and subject to a small matched disturbance. -3 Figure 8: Response of the “butterfly” robot when starting close to x f .15nd its limit points, it is not immediately clear that this form of set-stabilizing feedback can be referred to as an orbitally stabilizing feedback. We however believe such a classification is not only justified, but that it is in factan important one to make. To illustrate this point, consider the following definition of an orbitally stabilizingfeedback: A (piecewise) continuous (possibly static) state-feedback that induces a unique asymptotically stableorbit (i.e. a bounded, one-dimensional invariant manifold upon which the system has a specific evolution) in theclosed-loop system. The importance of this definition follows from the fact that it simultaneously incorporatesthe problem of stabilizing several important behaviors, including: equilibria (trivial orbits), (hybrid) limit cycles(periodic orbits) and PtP maneuvers (heteroclinic orbits). Indeed, this motivates developing general methods,such as the one presented in this paper, which can be used to control and stabilize these types of maneuvers (andmore). For example, in the case of trivial orbits, Theorem 9 and Proposition 13 condenses down to a standardlinear feedback stabilizing the Jacobian linearization and to the satisfaction of an algebraic Lyapunov equation.For nontrivial periodic orbits, on the other hand, a control law of the form (12) satisfying (20), e.g. found bysolving the then periodic differential LMI (23), will exponentially stabilize the orbit. In a similar manner, themethod can also be extended to maneuvers corresponding to hybrid cycles along which the states undergo anordered series of instantaneous jumps just by adding some additional arguments which ensure that the jumpsdo not destabilize the cycle. Comparison to common approaches.
In the literature, stabilization of point-to-point motions of un-deractuated systems by means of (time-invariant) state-feedback, e.g. swinging up and stabilizing an invertedpendulum, is commonly solved in one of two ways: 1) by (global) asymptotic stabilization of (possibly just someof the states at) the desired equilibrium, or 2) by switching between a separate “swing-up” controller and a local“balance” controller. In contrast to the latter class of approaches, the method we here propose requires noswitching, and therefore always ensures a continuous control signal. While in regards to the former, stabilizationof a maneuver’s orbit provides valuable knowledge of the path along which the system will evolve towards thegoal state. Indeed, this is in contrast to methods which rely on generating a large region of attraction for thegoal state, but at the expense of having limited knowledge of the path which will be taken to reach it for anarbitrary initial state.
Slow convergence.
A major limitation of the proposed scheme is the slow convergence away from theinitial equilibrium. In light of this issue, a possible ad hoc modification was proposed in Remark 11 as to ensurethat the state do not remain too long about x . The suggested modifications were, roughly speaking, basedon removing the initial equilibrium and instead starting part way along the maneuver, either by removing italtogether (static approach) or gradually moving away from it (dynamic approach). In the static approach,it is important not to start too far along the orbit as this might deteriorate the stability close the the initialequilibrium. By similar arguments, one should in the dynamic approach also limit both the bounds and the areaof growth of the dynamic variable. Moreover, the growth rate of this variable should also be kept sufficientlysmall as to not exceed, to a large extent, the nominal speed at which the orbit is traversed. Alternatively, onecan resolve this issue by instead planning a finite-time maneuver . Finite-time maneuvers.
We have limited the scope of this paper to maneuvers with a certain degreeof smoothness. This, however, immediately excludes so-called finite-time maneuvers , that is, maneuvers uponwhich the (negative) time it takes the states to converge to the (initial) final equilibrium is finite. We notethat both the construction and orbital stabilization of such maneuvers are indeed possibly if we loosen thesmoothness requirements at the boundary points. For instance, in the example with the double integrator inSec. 4, one can instead take ρ ( s ) = κ | s − q | n | q f − s | n f for any n , n f ∈ (0 . , n = n f = 3 / υ ( s, ˆ q ) := (cid:112) | s − ˆ q | , then ˙ υ ( s, q f ) = − sgn( q f − s ) | s − q | (cid:112) υ ( s, q f ) demonstrates that q f is reachedin finite time for ˙ s = ρ ( s ) on the interval ( q + (cid:15), q f ], (cid:15) >
0; similarly, one can use υ ( s, q ) to show that thenegative time to reach q on [ q , q f − (cid:15) ) then is finite as well. How to find such a solution for an underactuatedsystem, i.e. as a solution to the reduced dynamics (28), is, however, an interesting research question requiringfurther study. We have introduced a method for inducing, via locally Lipschitz continuous static state-feedback, an asymp-totically stable heteroclinic orbit in a nonlinear control system. Our suggested approach used a particularregular parameterization of a known point-to-point maneuver, together with a projection operator recovering As there is a vast and varied body of literature in regards to both of these directions, we only mention the (by now classical)approaches found in [22] and [3].
Acknowledgement
This research was supported by the Research Council of Norway, grant number: 262363.
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A Proof of Lemma 5
Differentiating the terms inside the brackets in (9) with respect to s , we obtain the function z ( x, s ) := ( x − x (cid:63) ( s )) T [Λ (cid:48) ( s )( x − x (cid:63) ( s )) − s ) F ( s )] . It follows that if there exists some p ( x ) which satisfies z ( x, p ( x )) ≡
0, then s = p ( x ) solves (9). Since also z (cid:48) ( x (cid:63) ( s ) , s ) = 2 F T ( s )Λ( s ) F ( s ) >
0, it is clear that the left-inverse property, i.e. s ≡ p ( x (cid:63) ( s )), is satisfied for all s ∈ S .Suppose, therefore, for the time being that (locally) such a solution exists. Denoting by ˜ x := x − x (cid:63) ( s ) andusing the shorthand notation s = p ( x ), then, by the implicit function theorem, J p ( x ) = F T Λ − ˜ x T Λ (cid:48) F T Λ F + ˜ x T (cid:2) Λ (cid:48)(cid:48) ˜ x − (cid:48) F − Λ F (cid:48) (cid:3) , where we have omitted the s -argument to shorten the notation. Hence J p ( · ) is then non-zero and well definedwithin some tubular neighborhood (with non-zero radius), where ω ( s ) := J p ( x (cid:63) ( s )) is therefore given by (10)and obviously satisfying ω ( s ) F ( s ) ≡ s ∈ S .Now, let X := { x ∈ R n : dist( O , x ) ≤ (cid:15) } , with (cid:15) > T := { x ∈ X : z ( x, p ( x )) = 0 } is simply connected and otherwise well defined (its existence readily followsfrom what we have already shown). Let the ends of this tube, corresponding to p ( x ) = s and p ( x ) = s f , bedenote by X and X f , respectively. Due to the expression for J p ( · ) above, which is valid within T , together with ω F = 1, it follows that sufficiently close to x the states will leave T if they go in the direction −F ( s ) whenon X , but the minimizer of (9) will nevertheless still be s . Using the same arguments for X f , the existence ofthe n-balls in condition 3. follow. Thus all the conditions of Def. 4 are satisfied.18 Proof of Theorem 9 and Remark 10
B.1 Proof of Theorem 9
We will demonstrate that the conditions in Theorem 9 allows to take V = ˜ x T R ( p ( x ))˜ x (40)as a Lyapunov function for the orbit. While this positive definite function is not smooth as it is not differentiableat the hypersurfaces X := lim s → s +0 Π( s ) and X f := lim s → s − f Π( s ) (see (8) for the definition of Π), it is locallyLipschitz in a neighborhood of O . We will therefore show that its time derivative satisfies the inequality ddt V ≤ − µV + O ( (cid:107) ˜ x (cid:107) ) , (41)for some constant µ >
0, almost everywhere. From this and the fact that the nonlinear system (locally) hasa unique solution under the control law (12), the asymptotic stability of the set O can be concluded; see, forexample, Theorems 3.1 and 3.2 in [31]. Indeed, by defining, at some time t , the one-sided (Dini) derivativeof (40) in the direction of ˙ x ( t ) at x ( t ), D + V ( x ; ˙ x ) = lim (cid:15) → + ( V ( x + (cid:15) ˙ x ) − V ( x )) /(cid:15) , the existence of a uniquesolution ensures that the inequality D + V ≤ − µV + O ( (cid:107) ˜ x (cid:107) ) holds everywhere.Let us therefore derive the inequality (41). Under the assumption that V is differentiable at some x in X ,its time derivative then exists and is given by˙ V = ˙˜ x T R ( s )˜ x + ˜ x T [ R (cid:48) ( s ) J p ( x ) ˙ x ] ˜ x + ˜ x T R ( s ) ˙˜ x, (42)where we have used the shorthand notation s = p ( x ). Hence, by (15), the Lyapunov equations (21) imply that(41) holds for x within either of the half-balls H oi , i ∈ { , f } , with µ ≤ min (cid:0) λ Q min /λ R ( s )max , λ Q f min /λ R ( s f )max (cid:1) . Here λ M min (resp., λ M max ) denotes the smallest (resp., largest) eigenvalue of a symmetric matrix M .Let us now also demonstrate that if µ is taken to be less than min s ∈S (cid:0) λ Q ( s )min /λ R ( s )min (cid:1) , then (41) holds withinthe tube T as well. For x in such a neighborhood, one has J p ( x ) ˙ x = ρ ( p ( x )) + O ( (cid:107) ˜ x (cid:107) ) (this follows from thefirst-order Taylor expansion about x (cid:63) ( p ( x )) and by using (5)). Thus by (17), (19) and (42) we obtain, for x ∈ T :˙ V =˜ x T Ω T (cid:2) A T cl Ω T R + R Ω A cl + ρ (cid:2) R (cid:48) − ( ω (cid:48) ) T F T R − R F ω (cid:48) (cid:3)(cid:105) Ω˜ x + O ( (cid:107) ˜ x (cid:107) ) . If we therefore can show that a solution R ⊥ ( s ) to (22) implies a solution to (20), then, by also using (17), wecan derive the inequality ˙ V = − ˜ x T Q ( s )˜ x + O ( (cid:107) ˜ x (cid:107) ) ≤ − µV + O ( (cid:107) ˜ x (cid:107) ) . Let us therefore begin by differentiating R ⊥ ( s ) with respect to s : R (cid:48)⊥ ( s ) = Ω T (cid:48) R ( s )Ω( s ) + Ω T R (cid:48) ( s )Ω( s ) + Ω T R ( s )Ω (cid:48) ( s ) . As Ω (cid:48) ( s )Ω( s ) = −F ( s ) F T ( s ) H p ( x (cid:63) ( s )), inserting the above expression for R (cid:48)⊥ into (22) shows that R ( s ) satisfies(20). The local asymptotic stability of O has therefore been proven.What remains is therefore to show that the final equilibrium x f will also be (locally) asymptotically stable.For this purpose, we denote by ˜ x f := x − x f and utilize the superscript notation A fcl := A cl ( s f ). Then, by thefirst-order approximation of the dynamics of ˜ x f about x f when inside T , we have ddt ˜ x f = A fcl ˜ x f + B fs (cid:0) u (cid:48) s ( s f ) − K f F f (cid:1) ω f ˜ x f + O ( (cid:107) ˜ x f (cid:107) ) . It is easy to verify that F ( s f ) is a constant solution to the linear part of this equation. Hence all solutionsstarting sufficiently close to x f inside T will be funneled into H f (which includes x f ). Inside H f the dynamicsof ˜ x f are given by (15). Evidently, the origin of any system of the form ˙ w = A cl ( s f ) w + O ( (cid:107) w (cid:107) ), correspondingto (15), is asymptotically stable. Now, let V σ ( s ) := { x ∈ Π( s ) : V ≤ σ } for some σ > σ f > V is strictly decreasingwithin V σ f ( s f ) and, moreover, that all solutions of the closed-loop system crossing lim s → s f V σ f ( s ) passes into V σ f ( s f ). Note that the existence of such a σ f follows from the above remarks and the continuous dependenceof solutions on the initial conditions. By the same arguments, we can therefore always find some ˆ s ∈ S o suchthat W (ˆ s ) := { x ∈ X : V υσ f (( s f − ˆ s ) υ + ˆ s ) , υ ∈ [0 , } is invariant with respect to the closed-loop system.In fact, we can choose σ f and ˆ s as to make W (ˆ s ) arbitrarily small, thus ensuring the stability of x f . As x f isalso attractive—a fact which readily follows from it being an ω -limit point of O —its local asymptotic stabilityfollows. This concludes the proof. 19 .2 Proof of Remark 10 Before showing the uniqueness of R ⊥ , we first note that a solution to (20) will not be unique; indeed, givena solution R ⊥ ( s ) to (22), that is R ⊥ ( s ) = Ω T ( s ) R ⊥ ( s )Ω( s ), then it can be shown that ˆ R ( s ) := R ⊥ ( s ) + h R ( s ) ω T ( s ) ω ( s ) will also satisfy (20) for an arbitrary smooth function h ω : S → R > .As to show that R ⊥ will be unique, first note that by Lemma 6 and the relation ω F ≡
1, we can always findsome q T , P : S → R n × n − which are sufficiently smooth and satisfy q ( s ) F ( s ) ≡ n − × , ω ( s ) P ( s ) ≡ × n − and q ( s ) P ( s ) ≡ I n − for all s ∈ S . This allows us to write Ω( s ) = V ( s ) EV − ( s ) in which E := diag(0 , I n − ), V ( s ) := (cid:2) F ( s ) , P ( s ) (cid:3) and ( V − ) T ( s ) = (cid:2) ω T ( s ) , q T ( s ) (cid:3) T . By dropping the s -argument to shorten the notation,we can then rewrite (20) as EV T (cid:2) ˙ R + A T cl ( V − ) T EV T ˆ R ⊥ + ˆ R ⊥ V EV − A cl − ˙ ω T F T R − R F ˙ ω + Q (cid:3) V E = 0 n , with ˙ ω = F T H p ( x (cid:63) ) ρ . It can further be shown that the parts of this equation which are not trivially zerocorrespond to the following matrix differential equation: A T R ⊥ + R ⊥ A + P T (cid:2) ˙ R − ˙ ω T F T R − R F ˙ ω (cid:3) P + Q ⊥ = 0 , where A ( s ) := q ( s ) A cl ( s ) P ( s ), while the matrix functions R ⊥ ( s ) := P T ( s ) R ( s ) P ( s ) and Q ⊥ ( s ) := P T ( s ) Q ( s ) P ( s )evidently are both smooth and SPD. Now using ˙ P = −F ˙ ω P , which is simply a consequence of ω P = 0, onefinds that ˙ R ⊥ = P T (cid:104) ˙ R − ˙ ω T F T R − R F ˙ ω (cid:105) P . We can therefore rewrite the above equation as˙ R ⊥ = −A T R ⊥ − R ⊥ A − Q ⊥ . (43)In order to show uniqueness, we use ˙ R ⊥ ( s ) = 0 as ρ ( s ) = 0. Thus, due to both R ⊥ ( s ) and Q ⊥ ( s ) beingSPD and satisfying the algebraic Lyapunov equation (43) for s ≡ s , it follows that the matrix A ( s ) := q ( s ) A cl ( s ) P ( s ) must necessarily be Hurwitz, which in turn implies that R ⊥ ( s ) is unique [32, Theorem 4.6].Since the right-hand side of (43) is continuous, it then has a unique solution R ⊥ ( s ( t )) satisfying R ⊥ ( s ( t )) = R ⊥ ( s ). Consequently ˆ R ⊥ ( s ) = Ω T ( s ) R ( s )Ω( s ) = q T ( s ) P T ( s ) R ( s ) P ( s ) q ( s ) = q T ( s ) R ⊥ ( s ) q ( s ) is also unique. C Proof of Proposition 13
Let us first demonstrate that the Lyapunov equations (21) are satisfied. To this end, we first note that theconstant matrix A cl ( s ) = A s ( s ) + B s ( s ) Y ( s ) W − ( s ) is Hurwitz. Thus by Theorem 1 in [20] there exists aSPD matrix ˆ Q ∈ R n × n such that A s ( s ) W ( s ) + W ( s ) A T s ( s ) + B s ( s ) Y ( s ) + Y T ( s ) B T s ( s ) = − ˆ Q . For R ( s ) = W − ( s ) we may therefore take Q = W − ( s ) ˆ Q W − ( s ) in (21a). The exact same argumentscan be used for the point s f .Let us now show that a matrix function W ( · ) solving the differential LMI (23) is equivalent to a solution R ( · ) to (20) (and therefore also a solution R ⊥ to (22)). For this purpose, recall that for any smooth nonsingularmatrix function W : S → R n × n one has dds W − ( s ) = − W − ( s )[ dds W ( s )] W − ( s ). Thus taking R ( s ) := W − ( s )and dropping the s -argument to keep the notation short, we obtain the following from (23): ρR (cid:48) (cid:22) − A T ⊥ R − RA ⊥ − K T B T ⊥ R − RB ⊥ K − λ (cid:2) R Ω + Ω T R (cid:3) . Multiplying from the left by Ω T and by Ω from the right, this can be written asΩ T (cid:104) A T cl Ω T R + R Ω A cl + 2 λR + ρ (cid:2) R (cid:48) − H p ( x (cid:63) ) FF T R − R FF T H p ( x (cid:63) ) (cid:3)(cid:105) Ω (cid:22) . Hence, as R = W − is SPD and λ is strictly positive, there must exist a SPD matrix function Q ⊥ : S → R n × n such that R solves the PLDE (20) 20 Proof of Theorem 15
The boundary conditions Φ( s ) = q and Φ( s f ) = q f are self-explanatory. To derive the conditions (cid:107) Φ (cid:48) ( s ) (cid:107) (cid:54) = 0and (cid:107) Φ (cid:48) ( s f ) (cid:107) (cid:54) = 0, we recall that the curve must satisfy the regularity condition (cid:107)F ( s ) (cid:107) > s ∈ S . Hence,as for ˆ s ∈ ∂ S we have ρ (ˆ s ) ≡
0, we therefore obtain (cid:107)F (ˆ s ) (cid:107) = (cid:107) Φ (cid:48) (ˆ s ) (cid:107) (cid:112) ρ (cid:48) (ˆ s ) > s ∈ ∂ S , from whichthese conditions follow.Now, from the conditions (cid:107) G ( q ) − B q u (cid:107) = (cid:107) G ( q f ) − B q u f (cid:107) = 0 it follows that we must have γ ( s ) = γ ( s f ) = 0, such that the remaining conditions in (34), that is ν ( s ) < ν ( s f ) ≤
0, corresponds to s beinga saddle-type equilibrium by Lemma 14, whereas s f is either of type saddle or undefined. This requirementis simply due to the following: Since ρ (ˆ s ) = γ (ˆ s ) ≡ s ∈ { s , s f } , we can infer that (28) is trivially true.Thus, let us instead consider αρ (cid:48)(cid:48) ρ + α ( ρ (cid:48) ) + (cid:0) α (cid:48) + 2 ˆ β (cid:1) ρ (cid:48) ρ + (cid:0) α (cid:48)(cid:48) + ˆ β (cid:48) (cid:1) ρ + γ (cid:48) = 0 , which corresponds to taking the derivative of (28) with respect to s , and which should hold (assuming of course φ i ∈ C ( S ) and ρ ∈ C ( S )) along a solution if (28) holds. Thus setting ρ (ˆ s ) = 0 we get ( ρ (cid:48) (ˆ s )) = − γ (cid:48) (ˆ s ) /α (ˆ s ),such that ν (ˆ s ) = γ (cid:48) (ˆ s ) /α (ˆ s ) ≤ ρ (cid:48) (ˆ s ) to be real. Since the initial equilibrium must necessarily be hyperbolicand as it is required that ρ ( s ) > s ∈ S o , the condition ν ( s ) < ν ( s f ) ≤ ν ( s ) < ν ( s f ) ≤ γ ( s ) = γ ( s f ) ≡ γ ( s ) /α ( s ) must change sign an odd number of times over the open interval ( s , s f ). Considering only one signchange, it follows that either γ changes sign or that α does. The former is equivalent to the existence of theequilibrium point s e in C1 , whereas the latter corresponds to an isolated singular point s s as considered in C2 . Considering C1 , the condition (35) is in itself enough to ensure the existence of a solution using (32)and that we require ρ ( s ) = ρ ( s f ) ≡