On the Existence and Uniqueness of Poincaré Maps for Systems with Impulse Effects

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1 1 On the Existence and Uniqueness of Poincaré Maps for Systems with Impulse Effects Jacob Goodman and Leonardo Colombo arxiv: v1 [cs.sy] 12 Oct 2018 Abstract The Poincaré map is widely used in the study of the qualitative behavior of dynamical systems. For instance, it can describes the existence and stability behavior of periodic solutions. The Poincaré map for dynamical systems with impulse effects was introduced in the last decade and mainly employed to study the existence of periodic gaits (limit cycles) in locomotion for bipedal robots. We investigate necessary and sufficient conditions for existence and uniqueness of Poincaré maps for dynamical systems with impulse effects evolving on a differentiable manifold. We apply the results to show the existence of Poincaré maps for systems with multiple different domains and for the 2D spring loaded inverted pendulum Index Terms hybrid systems, systems with impulse effects, Poincaré map, hybrid flows. I. INTRODUCTION Hybrid systems are non-smooth dynamical systems which exhibit a combination of smooth and discrete dynamics, where the flow evolves continuously on a state space, and a discrete transition occurs when the flow reaches transversally a codimension one hypersurface of the state space [6], [13], [28], [33]. Due to many advances in control systems, modeling and analysis of switching and robotic systems [32], [34], [35], [41], there has been an increased interest in recent years in studying the existence and stability of limit cycles in hybrid systems for which the Poincaré map became and indispensable tool for the qualitative analysis of hybrid systems [7], [8], [9]. Systems with impulse effects are hybrid systems, generally with a continuous dynamics determined by a mechanical system and where the transition between the continuous and discrete behavior of the system is determined by an impulsive (inelastic) impact giving rise to a change (discontinuity) in the velocities of the system while the trajectory is (left or right) continuous. These class of hybrid systems are also known as simple hybrid systems, by its simple structure, [20], [4], [2], [16], [40]. As opposed to dynamical systems possessing continuous flows, the use of the Poincaré map in systems with impulse effects requires the construction of a hypersurface that is transversal to a candidate periodic trajectory necessary for defining the return map, the re- setting set which determines when the states of the dynamical system are to be reset and providing a natural candidate for the transversal surface on which the Poincaré map of a dynamical system can be defined. Hence, the Poincaré return map is defined by a subset of J. Goodman is with Department of Mathematics, University of Michigan, 530 Church St. Ann Arbor, 48109, Michigan, USA. jdkgbmx@umich.edu L. Colombo is with nstituto de Ciencias Matematicas (CSIC-UAM- UC3M-UCM), Calle Nicolas Cabrera 13-15, 28049, Madrid, Spain. leo.colombo@icmat.es the resetting set that induces a discrete-time mapping from this subset onto the resetting set. This mapping traces the trajectory of the system from a point on the resetting set to its next corresponding intersection with the resetting set. The time when the flow intersects the hypersurface is called impact time. Poincaré map for systems with impulse effects have been introduced in [22] (see also [40]) and mainly employed in the search of periodic gaits (limit cycles) of bipedal robots together with several methods as geometric abelian Routh reduction, Zero hybrid dynamics, virtual constraints, Hamiltonian hybrid systems, systems with symmetries, etc [10], [12], [15], [29], [30], [23], [24], [25], [26], [36], [37], [38]. The goal of this paper is to augment the method of Poincaré maps for the class of nonlinear systems with impulsive effects evolving on a differentiable manifolds, and state conditions that ensure the existence and uniqueness of such a map. Most of the results in the literature which make use of the Poincaré map define the state space as R n, but these constructions does not include important situations in robotics as for instance Lie group configuration spaces as the special orthogonal groups or the special euclidean group for instance. The proposed result of existence and uniqueness of Poincaré maps on manifolds include such a situations which we believe can be further studied based on the results of this work, as for instance, the screw motion of a robotic hand hitting a wall [11], [27]. The proof of the main result is based on the sketch of proof for classical (non-hybrid) dynamical systems given in [1] (Theorem 7.1, Chapter 7, pp. 521). The paper is organized as follows: Section 2 introduces the class of hybrid systems we will study in this work (i.e., nonlinear systems with impulsive effects). In Section 3 we state and prove the main results of this work. Finally in Section 4 we apply the results to hybrid systems with multiple domains and the 2D spring loaded inverted pendulum. II. DYNAMICAL SYSTEMS WITH IMPULSE EFFECTS A dynamical system with impulse effects (SIEs) is a class of hybrid dynamical system (HDS) that exhibits both discrete and continuous behaviors, where the transition from one to the other is determined by the time when the continuous-time flow reaches a co-dimensional one submanifold of the state space, reinitializing the flow for the ODE which specifies the continuous-time dynamics and possibly, giving rise to a discontinuous flow. This class of dynamical systems are characterized by a 4-tuple H = (M, S, X, ), where: 1) M is a differentiable manifold called domain, 2) S M is an embedded co-dimensional one submanifold of M, called the switching surface, or guard,

2 2 3) X is a smooth vector field on M with flow F : D M where D is an open set in M R, 4) : S M is a C 1 function, called the reset or impact map, which re-initializes the trajectories that crosses S. The pair (M, X) describes the continuous -time dynamics of H, whereas (S, ) defines the discrete-time dynamics as x + = (x ). The underlying dynamical system with impulse effects is given by x = X(x) if x S Σ H : x + = (x ) if x S. In general, may be a discontinuity at the point where the flow intersects the guard S. However, as in [40], we are given the choice in deciding whether this point will be leftcontinuous or right-continuous. That is, whether the point of intersection between our trajectory and S is x or x +. Along the paper we will choose the former as in [40]. Note that the results that follow in this work hold regardless of this choice [16]. However, this means that the orbits associated with the flow of this class of HDS will (in general) not be closed. Remark 2.1: (Zeno behavior) Consider the impact map given by the identity map. When a trajectory crosses S, we will have x + = (x ) = x S, so that we are again in the regime of discrete dynamics where re-initialization (to x ) will occur. It is clear that this process will never terminate, so that there exists an infinite number of resets in finite amount of time. This situation generates a class of behaviors called Zeno behavior. It is particularly problematic in applications where numerical work is used, as computation time grows infinitely large at these Zeno points. While there have been proposed models for treating Zeno behavior [2], [3], [13], [18], [19], [31], [42] we exclude it from our systems. While Zeno behavior is typical to systems with rebound after impacts such as the classical bouncing ball problem [6], [14], it is much less relevant in the models for locomotion where completely plastic impacts with no rebound are assumed. Along the paper we will assume that S (S) =, where (S) denotes the closure as a set of (S) (or equivalently, the set of impact times is closed and discrete as in [40]). By Remark 2.1 we may extend the domain of the reset to the entire domain without affecting the dynamics by defining a function M : M M by: (m) if m S, M (m) = m if m S. The map M (m) permits to define the flow for the SIEs Σ H as follows. Definition 2.2: Consider the SIEs Σ H : 1) The flow of Σ H is given by F H = F M and it satisfies F H x (t 0 ) = x + if F H x (t 0 ) S, x M. 2) The integral curve F H m is periodic if there exists some (m, τ) M R such that F H m (t + τ) = F H m (t) for all t R. 3) If the integral curve F H m 0 is periodic, the corresponding periodic orbit is given by γ = m M m = F H m 0 (t) for some t R}. III. EXISTENCE AND UNIQUENESS OF POINCARÉ MAPS FOR DYNAMICAL SYSTEMS WITH IMPULSE EFFECTS In this section we show the main results of this work. Before state the result we introduce the preliminaries notions and results needed. Definition 3.1: A section of M is a codimension one submanifold of M. A section S of M is said to be locally transverse at s S if X(s) T s S. If X(s) T s S for all s S, then S is a local transverse section. We will say that a section S of M is locally transverse with respect to X at s S if X(s) T s S. However, we willoften drop the references to our particular vector field and manifold, as it will be understood by our problem set-up. Proposition 3.2: Let X be a smooth vector field with flow F, and let S be a section of M locally transverse at s S. If s} I D for all s S and for some interval I R, then for all λ I, F λ (S) is locally transverse at F λ (s). Moreover, if S is a local transverse section, then so is F λ (S) Proof: Note that by Lemma 4 (see Appendix) S := F λ (S) is a section of M since S is a section of M and F λ is a diffeomorphism. Moreover, F s : I M is an integral curve at s, and F λ F s is an integral curve at s := F λ (s) on some open interval. Assume that there exists a curve c on S at s that is tangent to F λ F s at s. Since F 1 is a C 1 mapping from a differentiable manifold to itself, we have by Lemma 2 that Fλ 1 F λ F s = F s and Fλ 1 c are tangent at s. Given that Fλ 1 c is a curve on S, it follows that (F 1 λ c) (0) T s S and F s is an integral curve, so that F s(0) = X(s) T s S, as S is locally transverse at s. Hence F s and Fλ 1 c are not tangent at s, and by contradiction, F λ F s and c are not tangent at s. Since c was arbitrary, (F λ F s ) (0) = X(s ) T s S, so S is locally transverse at s. Finally, if S is a local transverse section, the argument above may be applied to each of the points in s, from which it follows that F λ (S) is also a local transverse section. A. Existence of Poincaré maps for dynamical systems with impulse effects Next, we proceed to state and proof the main result of this work. Theorem 3.3: Let H be a SIEs such that S (S) =. Suppose there exists a peridic orbit γ of H such that S is locally transverse at γ S = m 0 } S \ S, (S) is locally transverse at (m 0 ), and the differential of is a linear isomorphism at m 0. Then, there exists a map Θ : W 0 W 1, called Poincaré map, such that 1) W 0 and W 1 are open sections of S containing m 0, and Θ is a diffeomorphism 2) There exists a C 1 function δ : W 0 R such that Θ(w) = (F δ(w) )(w) for all w W 0 Proof: The proof is divided into three stages. First we define the domains needed to construct the Poincaré map, and next, we construct the map. Finally we show that Θ satisfies conditions (1) and (2). λ

3 3 Since S is locally transverse at m 0, we know that X(m 0 ) T m0 S. But 0 T m0 S since it is a vector space, so that X(m 0 ) 0. Hence, we can let (U, φ) be a straightening chart at m 0 with φ : U V I R n 1 R as in Lemma 3 (see Appendix). The flow at (m 0 ) is defined on an open interval containing [0, τ], so for all u U, u} [0, τ] D, by shrinking U. Since the differential of is a linear isomorphism at m 0, by the inverse function theorem, δ is a local diffeomorphism at m 0. Let W 0 be an open subset of S such that : W 0 (W 0 ) is a diffeomorphism, W 0 U S, and W 0 is a local transverse section at m 0 (S is a differentiable manifold and it is locally transverse at m 0, so it must also be locally transverse in some neighborhood of m 0 ). Let V 0 = v V λ I such that (v, λ ) φ(w 0 )}. Then V 0 I V I is open and contains φ(m 0 ) = 0. Further denote U 0 = φ 1 (V 0 I) W 0 = F 1 τ (U 0 ) (W 0 ) W 2 = F τ( W 0 ) where U 0 is an open subset of U, and W 0 is an open section, both containing m 0. Since F τ U and W 0 are diffeomorphisms, W 0 is an open section containing (m 0). Consequently, W 2 is an open section in U 0 containing m 0, and, given that W 2 is locally transverse at (m 0 ), by Proposition 3.2, W 2 is locally transverse at m 0. Next, we proceed to construct the Poincaré map. By construction of V 0 I and the fact that W 0 is a local transverse section, we have that for any (v, λ) V 0 I, there exists a unique point T(v, λ) R such that (v, λ +T(v, λ)) φ(w 0 ). Let T : V 0 I R be this mapping, and L : V 0 I φ(w 0 ) the mapping given by L(v, λ) = (v, λ + T(v, λ)). Now consider the canonical projection π : V 0 I V 0 0}, which sets the n th component of its input to 0. For some (α 1,..., α n ) R n, we can write the tangent plane of φ(w 0 ) at φ(m) = 0 as T 0 (φ(w 0 )) =(x 1,..., x n ) R n α 1 x α n x n = 0} =(x 1,..., x n 1, 1 (α 1 x α n 1 x n 1 )) x j R}, α n where α n 0, since W 0 is locally transverse at m 0. It is clear from this representation that π T0 (φ(w 0 )) is a linear isomorphism, so that π φ(w 0 ) is a local diffeomorphism at 0 by the inverse function theorem. Since W 2 is also a subset of U 0 that is locally transverse at m 0, we have that π φ(w 2 ) is a local diffeomorphism at 0 using the same argument. Hence, we have that (i) π φ(w 0 ) maps an open subset of φ(w 0 ), containing 0, diffeomorphically to an open subset of V 0 0}, say V 1, with 0 V 1. (ii) π φ(w 2 ) maps an open subset of φ(w 2 ), containing 0, diffeomorphically to an open subset of V 0 0}, say V 2, with 0 V 2. Let W 2 = (φ 1 π 1 φ(w 2 ))(V 1 V 2 ). W 2 is an open subset of W 2 containing m 0 and π 1 φ(w 0 ) π φ(w 2 ) maps φ(w 2 ) diffeomorphically onto its image, namely φ(w 1 ), where W 1 is an open subset of W 0. Note that L φ(w2 ) = π 1 φ(w 0 ) π φ(w 2 ), since both maps preserve the element of the base space v V 0, and for all v V 0, there exists a unique fiber λ I such that (v, λ) φ(w 0 ), as W 0 is a local transverse section. Hence L : φ(w 2 ) φ(w 1 ) is a diffeomorphism. This further implies that the mapping T : φ(w 2 ) R is C 1, since L is C 1 and L(v, λ) = (v, λ + T(v, λ)) for all (v, λ) φ(w 2 ). Next, we consider the sets: W 0 = Fτ 1 (W 2 ), open subset of W 0 containing (m 0) W 0 = 1 ( W 0 ), open subset of W 0 containing m 0. Then the composite mapping Θ given by W 0 W 0 F τ W 2 φ φ(w2 ) L φ(w 1 ) φ 1 W 1 is a diffeomorphism between two open neighborhoods of m 0 on S, so that Θ satisfies condition (1) of our theorem. To show condition (2) also holds, consider the C 1 mapping δ : W 0 R defined as δ = T φ F τ. Let π n : R n R be the canonical projection that maps vectors in R n to their n th component. For w W 0, Θ(w) = (φ 1 L φ F τ )(w) = (φ 1 L)((π φ F τ )(w), (π n φ F τ )(w)) = φ 1 ((π φ F τ )(w), (π n φ F τ )(w) + (T φ F τ )(w)) = φ 1 ((π φ F τ )(w), (π n φ F τ )(w) + δ (w)). Furthermore, by the Straightening out Theorem (see Lemma 3 in Appendix), φ 1 ((π φ F τ )(w), ) : I U is an integral curve at φ 1 ((π φ F τ )(w), 0), and by Lemma 4, F (w) : [0, a) M is an integral curve at (w), with a > τ. Noting that F (w) (τ) = (F τ )(w) by Lemma 2, = (φ 1 φ F τ )(w) = φ 1 ((π φ F τ )(w), (π n φ F τ )(w)), φ 1 ((π φ F τ )(w), t + (π n φ F τ )(w)) = F (w) (t + τ) for all t such that t + (π n φ F τ )(w) is in I (this is because the flow F (w) must be defined everywhere that the integral curve given by φ 1 is defined, while the converse is not necessary true). By construction, δ (w) + (π n φ F τ )(w) is on I, so that Θ(w) = φ 1 ((π φ F τ )(w), δ(w) + (π n φ F τ )(w)) = F (w) (τ + δ (w)) = (F τ+δ (w) )(w). Finally, defining δ = δ + τ, we have that Θ(w) = (F δ(w) )(w), and hence, condition (2) holds. For the purposes of demonstrating the applications of Theorem 3.3 we consider the following example.

4 4 Example 3.4: Let M = R 2, parametrized with polar coordinates. Let S be the switching surface given by S = (r, θ) R S 1 θ = π 2 }, that is, a line embedded in the manifold. Consider the smooth vector field on M given by X(r, θ) = (r(1 r 2 ), 1) and the impact map : S R 2 given by the rotation transformation defined by (r, π/2) = (r, 0). Hence we consider the hybrid dynamics system determined by Σ H = ( r, θ) = (r(1 r 2 ), 1) if (r, θ) S (r +, θ + ) = (r, θ ) if (r, θ ) S. Denote by γ the circle arc of radius 1 between 0 and π/2. γ is a periodic orbit of the system with period π/2. Furthermore, S is locally transverse at γ S = (1, π/2), and the differential of is a linear isomorphism at m 0 = (1, π/2). By Theorem 3.3, a Poincaré Map Θ exists and is a diffeomorphism between two open sections of S containing m 0. Next, consider an initial value (r, π/2) on S. After a time greater than 0, we have moved to the point (r, 0), via a rotation specified by. Since θ = 1, it follows that the trajectory will return to S after time t = π/2. Integrating the r-component of our equation, we have that Θ(r) must satisfy: Θ(r) r dr r(1 r 2 ) = π/2 which upon integration and solving for Θ(r) we obtain 0 dt, Θ(r) = [1 + e π (r 2 1)] 1/2. Computing the derivative of Θ, we get: Θ (r) = e π r 3 [e π ( 1 r 2 1) + 1] 3/2, which is non-zero and continuous everywhere except for r = 0. Hence Θ is C 1 on S, and by the inverse function theorem, has an inverse which is C 1 in a neighborhood of (r, θ) = (1, π/2). Thus, Θ is a diffeomorphism between two open neighborhoods of (1, π/2) on S by Theorem 3.3. B. Uniqueness of Poincaré maps for dynamical systems with impulse effects In a continuous-time dynamical system, uniqueness of Poincaré maps is equivalent to local conjugacy (see Theorem 4.3 in the Appendix). This is a desirable property, as local conjugacy preserves the eigenvalues of the Poincaré map s Jacobian, for which stability analysis is concerned (see [1] Chapter 7). If we did not have uniqueness, different Poincaré maps may give rise to conflicting stability results for the analysis of periodic orbits, invalidating one of the primary purpose of these functions. Local conjugacy (and uniqueness) will be defined in a similar fashion than continuous-time systems on differentiable manifolds, however, we must take caution, as a SIEs and its periodic orbits are intimately connected to the chosen reset. To explain this, we consider the following example. Example 3.5: Consider Example 3.4 but now with reset map given by (r, π/2) = (e c(r 1), 0), with c R +. Note that is C 1 and is a diffeomorphism between S and (S). Hence, by Theorem 3.3, a Poincaré map exists. Proceeding as before, integrating the r-component, we find that Θ(r) = [1 + e π (e 2c(r 1) 1)] 1/2, so that Θ (r) = ce 2c(r 1) π [1 + e π (e 2c(r 1) 1 )] 3/2. As in Example 3.4, the circular arc of radius 1 between 0 and π/2 denoted by γ, is a periodic orbit for the system, and it is independent of c, but, Θ (1) = ce π, so that for c > e π, the periodic orbit γ is unstable, and for c < e π, the orbit γ is stable. Therefore, systems with different values of c can not be locally conjugate. Moreover, hybrid systems with the same vector field and/or periodic orbit need not behave similarly, and hence any notion of uniqueness must account appropriately for a reset map. Definition 3.6: Let Σ H1 = (M, S 1, X, 1 ) and Σ H2 = (M, S 2, X, 2 ) are two SIEs and W0 1, W1 1 S1, W0 2, W2 1 S2 open sets of S 1 and S 2, respectively. Given two diffeomorphisms Γ 1 : W0 1 W2 0 and Γ 2 : W0 2 W2 1 are said to be locally conjugate if there exists open sets W2 1 W1 0 W1 1 S1 and W2 2 W2 0 W2 1 S2 and a diffeomorphism h : W2 1 W2 2 such that Γ 2 h = h Γ 1. Theorem 3.7: Suppose that Σ H1 = (M, S 1, X, 1 ) and Σ H2 = (M, S 2, X, 2 ) are two SIEs satisfying the hypothesis of Theorem 3.3. Let Θ 1 : W0 1 W1 1 and Θ2 : W0 2 W2 1 be the corresponding Poincaré Maps, with W j 0, W j 1 S j for j = 1, 2. Assume that 1 (Wk 1) = 2 (Wk 2 ) for k = 0, 1, and that there exists a C 1 -function T : S 2 R such that F T (Wk 2) = W1 k, where F is the flow of the continuous dynamical system (M, X). Then Θ 1 is conjugate to Θ 2, with conjugate function F T. Proof: We wish to show that FT 1 that Θ j = F δ j j for some C 1 function δ j : W j 0 that the expanded mapping of F 1 T F T Θ 1 F T : W 2 0 and for Θ 2, we have Θ 2 : W 2 0 Since F T (W0 2) = W1 0 the map for Θ 2 as Θ 2 : W 2 0 F T W 1 0 Θ1 F T = Θ 2. Knowing R, we see Θ1 F T is: 1 1 (W 1 0 ) F δ 1 W (W 2 0 ) F δ 2 W 2 1. F T W 2 1, and 2 (W0 2) = 1 (W0 1 ), we can expand F T W (W 1 0 ) F δ 2 W 2 1 By uniqueness of the continuous-time flow (see Lemma 1 in Appendix), we must have that F δ 1 T = F δ 2. Hence, F 1 T Θ1 F T =(F T F δ 1) ( 1 F T ) = F δ 1 T 2 = F δ 2 2 = Θ 2.

5 5 Remark 3.8: Note that Σ H1 and Σ H2 need not have the same periodic orbit, but if they are not, one of the orbits is a continuation of the other under the flow. More precisely, let γ 1 and γ 2 be periodic orbits through the points m 1 0 and m2 0 on S 1 and S 2, respectively. Either γ 1 or γ 2 will contain both m 1 0 and m 2 0. Without loss of generality, assume that it is γ 1. Then γ 2 γ 1 and γ 1 \ γ 2 = m M : m = F t (m 2 0 ) for 0 < t < T(m2 0 )}. From this perspective, the result is intuitive. Though Σ 1 and Σ 2 are different systems describing different periodic orbits, they share continuous dynamics and have discrete dynamics that are related though the continuous components. IV. APPLICATIONS In this section we employ Theorems 3.3 and 3.7 for SIEs and with multiple domains, and the 2D spring loaded inverted pendulum. A. Systems with Impulse Effects and with Multiple Domains The notion of a SIEs can be naturally extended to include multiple domains and resets. Definition:: A k-domain SIEs is a tuple H = (Γ, M, S,, X) where (i) Γ = (V, E) is a directed graph such that V = q 1,..., q k } is a set of k vertices, and E Q Q is the set of edges. We further define the maps sor and tar, which return the source and target of the edge. More precisely, if e ij = (q i, q j ), then sor(e ij ) = q i and tar(e ij ) = q j. (ii) M = M q } q V is a collection of differentiable manifolds. (iii) S = S e } e E is a collection of guards, where S e is assumed to be an embedded section of M sor(e). (iv) = e } e E is a collection of reset maps, which are C 1 mappings where e : S e M tar(e). (v) X = X q } q V is a collection of smooth vector fields. We further define (vi) Λ = 0, 1, 2,...} N, an indexing set (vii) ρ : Λ V a map recursively defined by e ρ(i) = (ρ(i), ρ(i + 1)). The underlying dynamical system with impulse effects is then defined by x = X i (x) if x M i and x S i x + = i (x ) if x S i, where it is understood that X i = X ρ(i) and similarly for M i, and S i = S eρ(i) and similarly for i. As before, we will assume the flow to be left continuous and will exclude Zeno behavior from this system by imposing the constraint that S i i (S i ) = for all i Λ Theorem 4.1: Let γ be a periodic orbit of the k-domain SIEs H = (Γ, M, S,, X), and Λ, ρ be defined by conditions (vi) and (vii) respectively. Assume that ρ(n) is N-periodic for some N Λ, that is, ρ(n + N) = ρ(n) for all n Λ, and that S i is locally transverse at γ S i = m i S i \ S i. If the differential of i is a linear isomorphism at m i, then there exists a Poincaré map Θ : W0 0 W1 0 such that: 1) W0 0 and W1 0 are open subsections of S 0 containing m 0, and Θ is a diffeomorphism between them. 2) There exists a collection of C 1 time-to-impact functions δ i : Wi 0 R such that Θ(w) = F δ N 1 0 N 1 F δ N 2 N 1 N 2... F δ 0 1 0(w), where Wi 0 are open subsections of S i containing mi 0, w W0 0 and Ft i is the flow of the vector field X i after time t. Proof: Using the same argument as in the proof of Theorem 3.3, there exists an open subset of S 0 containing m0, say W0 0, an open subset of S 1 containing m1, say W1 1, and a C1 - function δ 1 : W0 0 R such that Fδ : W0 0 W1 1 is a diffeomorphism. By shrinking W0 0 as necessary, we may shrink W1 1 to the point that it is locally transverse and 1 : W1 1 1(W1 1) is a diffeomorphism. We may then again apply the same argument to find an open subset of S 1 containing m1, say W0 1, an open subset of S 2 containing m2, say W1 2, and a C1 -function δ 2 : W1 0 R such that Fδ : W1 0 W1 2 is a diffeomorphism. This in turn implies that F δ F δ is a diffeomorphism onto its image. Continuing this argument until we return to m0 and relabeling as necessary, we obtain the desired result. B. 2D spring loaded pendulum The spring-loaded inverted pendulum (SLIP) has been used as a model which reasonably provide a template for sagittal plane motions of the center of mass (COM) of diverse species as six-legged trotters (cockroaches), two-legged runners (humans and birds), and hoppers (kangaroos) as was reviewed in [17] and further studied in [21]. Fig. 1. 2D spring loaded inverted pendulum Let z denote the length of the spring, and let θ denote the angle of the leg with respect to the center-line. If k is the spring constant, and l 0 is the no-load length of the spring, then the Lagrangian function L : TQ R defined on the tangent bundle of the configuration manifold Q = R S 1 specifying the continuous-time dynamics is given by L(z, θ, z, θ) = m 2 ( z2 + z 2 θ 2 ) mgz cos θ k 2 (z l 0) 2. The Euler-Lagrange equations for L are θ = 2 z z θ g z sin θ, z = z θ 2 + k m (l 0 z) g cos θ.

6 6 The aerial phase starts when the spring length reaches its no-load length (i.e., z = l 0 ); therefore, the switching surface is defined as S = (z, θ, z, θ) TQ z = l 0 }. The aerial phase consists of a projectile (or ballistic) motion for the COM (where the only external force is gravity) at the end of which, when z = l 0, the next stance phase starts. We assume that at the beginning of each step the leg is at an angle θ 0. Therefore, θ + = θ 0 and z + = l 0. Hence, the transition occurs when the height of the mass is l 0 cos(θ 0 ). Writing the equations of motion of a projectile yields x + = x and y + = ( y ) 2 2g(y y 0 ) where, x = z sin(θ), y = z cos(θ) and y 0 = l 0 cos(θ 0 ). Now, using y and y 0, we have that y = y 0 and therefore from x + = x and y + = y. Using the definitions of x and y, z = x sin θ + y cos θ, z θ = x cos θ y sin θ, and therefore, using that x + = x, y + = y and θ + = θ 0, it follows that z + = z. Similarly, with θ = θ 0 and z = l 0, it is easy to see that θ + = θ, and therefore the impact map is given by (l 0, θ 0, z, θ ) = (l 0, θ 0, z, θ ). Next, to ensure the existence of a periodic solution we will use the following result from [5]: Theorem 4.2 (Theorem 4.2 in [5]): Let Σ H be a SIEs with continuous flow specified by Hamilton equations associated to a Hamiltonian function H : T Q R with local coordinates (q, p) T Q. Assume that H invariant under R : T Q T Q given by R(q(t), p(t)) = (F(q(t)), df(q) p(t)) with F : Q Q a smooth involution. If γ is a fixed point of R, γ(t) = (q(t), p(t)) satisfies R(γ(t)) = γ( t). Moreover, if γ(t) crosses the switching surface S at ti = inft > 0 γ(t) S} and the impact map is defined as (γ (t i )) = R(γ (t i )) then γ(t) is a periodic solution for the SIEs Σ H with period 2ti. Given ) that the Lagrangian is hyper-regular, that is, det ( 2 L q q 0, with q = (z, θ), one may define the Hamiltonian function using the Legendre transformation p = L q, ( ) H(z, θ, p z, p θ ) = 1 + mgz cos θ + 1 2m 2 k(z l 0) 2. p 2 z + p2 θ z 2 Hamilton equations are given by z = p z m, θ = p θ z 2 m, p θ = mgz sin θ, p z = mg cos θ k(z l 0 ). Consider the involution F(z, θ) = (z, θ) then R(q, p) = (z, θ, p z, p θ ). The Hamiltonian H : T (R S 1 ) R is invariant under R. Fixed points of R are given by γ = (z, 0, 0, p θ ). Then, the solution γ(t) with γ(0) = γ satisfies R(γ(t)) = γ( t) and there are infinity many solutions γ(t) crossing S (as the quantity of fixed points for R). Consider one solution γ(t), it is easy to verify that (l 0, θ 0, p z, p θ ) = R(l 0, θ 0, p z, p θ ). Therefore, by Theorem 4.2 γ(t) is periodic with period 2ti. Thus, since is a linear isomorphism, we are under the conditions of Theorem 3.3, and therefore there exists a Poincare map for the 2D SLIP system. APPENDIX Because HDS, and in particular SIEs, are a mixture of continuous and discrete dynamics in this appendix we review some properties of vector fields used in the work for both for continuous dynamical systems. Lemmas 1-4 below, and their proofs, can be found in [1] Chapter 2. Let M and N be differentiable manifolds and X a vector field on M. Lemma 1 Suppose c 1 and c 2 are integral curves of X at m M. Then c 1 = c 2 on the intersection of their domains. Lemma 2 Suppose c 1 and c 2 are curves at m M which are tangent at m. Let f : M N be C 1. Then f c 1 and f c 2 are tangent at f (m) N Lemma 3 (Straightening out theorem). Suppose that for m M, X(m) 0. Then there exists a local chart (U, φ) with m U and φ(m) = 0 called straightening chart at m such that 1) φ(u) = V I R n 1 R, V open, and I = ( a, a) with a > 0 2) φ 1 (v, ) : I M is an integral curve of X at φ 1 (v, 0) for all v V 3) Tφ X U = ê n (the image of integral curves in U under φ are straight lines passing orthogonally to the fibers) Lemma 4 Suppose X is smooth and let D denote the set of (m, λ) M R such that there exists an integral curve of X at m whose domain contains λ. Then: 1) M 0} D 2) D is open in M R 3) There exists a unique map F : D M such that the mapping F m : I M defined by F m (t) = F(m, t) is an integral curve for all m M and some open interval I. 4) If U is a subset of M such that u} J D for some open interval J, then for all λ J, the mapping F λ restricted to U is a diffeomorphism onto its image. Theorem 4.3: (Existence and uniqueness for continuoustime Poincaré map) Let X be a smooth vector field on a differentiable manifold M with integral F, γ a closed orbit of X with period τ, and S a local transversal section of X at m γ. Then, 1) there exists a Poincaré Map Θ : W 0 W 1 of γ, where (i) W 0, W 1 S are open neighborhoods of m S, and Θ is a diffeomorphism (ii) There is a C 1 function δ : W 0 R such that for all s W 0, Θ(s) = F(s, δ(s)) 2) If Θ : W 0 W 1 is a Poincaré map of γ (in a locally transverse section S at m γ) and Θ also (in S at m γ), then Θ and Θ are locally conjugate. That is, there are open neighborhoods W 2 of m S, W 2 of m S, and a diffeomorphism H : W 2 W 2 such that W 2 W 0 W 1, W 2 W 0 W 1 and Θ H = H Θ

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