Hybrid Closed Loop Diabetes

Leonardo J. Colombo1

Instituto de Ciencias Matem´aticas, Consejo Superior de Investigaciones Cient´ıficas Calle Nicol´as Cabrera 13-15, Cantoblanco, 28049, Madrid, Spain

Mar´Ia Emma Eyrea Iraz´U2

Calle 1 y 115, La Plata 1900, Buenos Aires, Argentina

Abstract

Symmetries are ubiquitous in a wide range of nonlinear systems. Particularly in systems whose dynamics is determined by a Lagrangian or Hamiltonian func- tion. For hybrid systems which possess a continuous-time dynamics determined by a Lagrangian function, with a cyclic variable, the degrees of freedom for the corresponding hybrid Lagrangian system can be reduced by means of a method known as hybrid Routhian reduction. In this paper we study sufficient condi- tions for the existence of periodic orbits in hybrid Routhian systems which also exhibit a time-reversal symmetry. Likewise, we explore some stability aspects of such orbits through the characterization of the eigenvalues for the corresponding linearized Poincar´e map. Finally, we apply the results to find periodic solutions in underactuated hybrid Routhian control systems.

$This work was partially supported by I-Link Project (Ref:

Linka20079) From Csic

and by Ministerio de Econom´ıa, Industria y Competitividad (MINEICO, Spain) under grant MTM2016-76702-P. The project that gave rise to these results received the sup- port of a fellowship from “la Caixa’ Foundation” (ID 100010434).

The Fellowship Code Is

LCF/BQ/PI19/11690016. The graduate student M.E. Eyrea Iraz´u was partially supported by CONICET Argentina. L. Colombo and M.E. Eyrea Iraz´u were partially supported by “Severo Ochoa Programme for Centres of Excellence” in R&D (SEV-2015-0554). The authors are indebted with the reviewers and editor for their recommendations that helped to improve the quality, clarity and exposition of this work.

Preprint submitted to Nonlinear Analysis: Hybrid Systems

Keywords:

Hybrid systems, Symmetries, Routh reduction, Poincar´e map. 2010 MSC: Primary: 37K05, Secondary: 37J15, 37N05, 34A38, 34C14, 34C25

1. Introduction

Hybrid systems are dynamical systems with continuous-time and discrete- time components in its dynamics.

These Dynamical Systems Are Capable Of

modeling several physical systems, such as, multiple UAV systems , , bipedal robots , , , embedded computer systems , , and underactuated mobile vehicles , among others.

Simple hybrid systems are a type of hybrid systems introduced in , called in this manner because of its simple nature. A simple hybrid system is char- acterized by a tuple H = (D, X, S, ∆) where D is a smooth manifold, X is a smooth vector field on D, S is an embedded submanifold of D with co-dimension 1 called the switching surface (or the guard), and ∆: S →D is a smooth em- bedding called the impact map (or the reset map). This type of hybrid system has been mainly employed for the understanding of walking gaits in bipeds and insects , , . In the situation where the vector field X is associated with a mechanical system (Lagrangian or Hamiltonian), alternative approaches for mechanical systems with unilateral constraints have been considered in , , , , and .

A symmetry is a transformation that leave invariant the solutions in a dy- namical system. The type of symmetry that most mechanical systems naturally exhibit is known as reversing symmetry. Reversing symmetries leave the equa- tions of motion invariant if the direction of time is reversed. Dynamical systems possessing this class of symmetry are called reversible if the reversing symmetry is an involution . It is important to mention that the existence of periodic orbits has been a predominant topic of research in dynamical systems since the studies of Poincar´e . The use of reversing symmetries to find periodic orbits has been employed, for instance, in the restricted three-body problem .

When a dynamical system exhibits a symmetry, it produces a conserved

2

quantity for the system. This reduces the degrees of freedom in the dynamics of the system. One of the classical reduction by symmetry procedures in mechanics is the Routh reduction method . During the last few years there has been a growing interest in Routh reduction, mainly motivated by physical applications , , , . Routh reduction for hybrid systems has been introduced by A. Ames and S. Satry and it has been applied in the field of bipedal locomotion , , . The reduced simple hybrid system is called simple hybrid Routhian system. In this work we build in the former approach to that concept by study- ing sufficient conditions for which a simple hybrid Routhian system exhibits a periodic solution.

The search of limit cycles in hybrid systems has been an active research field in the robotics and automatic control community since the works of Mc’Geer due to the study of periodic walking gaits for passive dynamic walkers , . Since these works, the study of orbital stability for hybrid systems has been the more explored analysis in this field. The method of Poincar´e map is frequently used in the legged locomotion community to study orbital stability of walking gaits , , , , , . In most of the studies analyzed in the literature employing such an approach, one assumes the existence of a periodic solution. Then one proceed with the corresponding stability analysis of the orbits by examining the eigenvalues of the linearized Poincar´e Map at its fixed points. In general, since fixed points of the Poincar´e map corresponds with periodic orbits for the underlying dynamical system, to find these orbits one needs to employ computational resources to find the Poincar´e map and its fixed points.

In this work, we show how to ensure the existence of periodic solutions by examining the symmetries for a simple hybrid Routhian system excluding com- putational tools.

Moreover, with the method proposed in this work we also provide a characterization for the eigenvalues associated with the linearized Poincar´e map for these periodic solutions, also from an analytical point of view.

To the best of our knowledge, sufficient conditions for simple hybrid Routhian systems under which one can ensure the existence of periodic solutions and the

3

characterization of its qualitative behavior, excluding the use of computational tools, has not yet been widely discussed in the literature. Similar results for simple hybrid (non-Routhian) systems concerning the existence of periodic so- lutions can be found in and . Sufficient conditions for the existence and uniqueness of Poincar´e maps can be found in and results about stability of periodic solutions for 2D-simple hybrid systems can be found in .

The primary goal of this paper consists on establishing the conditions under which we can ensure the existence of periodic orbits in simple hybrid Routhian systems. By introducing a time-reversal symmetry, trajectories for a simple hy- brid Routhian system will become in a periodic orbit if the trajectory begins at a fixed point of the symmetry map. In this work, we provide characteriza- tions for the eigenvalues associated with the linearized Poincar´e map for these periodic solutions. We also apply the results to the classical example of the 2D one-leg robotic hopper for which, after applying Routh reduction method for simple hybrid system, we reduce the hybrid dynamics to the one for the 2D spring loaded inverted pendulum (SLIP). Then we study the existence of periodic motions and show that periodic orbits are at most marginally stable.

As an application we employ the results given in this work for the search of pe- riodic orbits in underactuated hybrid mechanical control systems, in particular, for the 2D spring loaded inverted pendulum.

The paper is structured as follows: Section 2 presents a review on Lagrangian mechanics and Routh reduction. Section 3 introduces simple hybrid Routhian systems and the 2D-SLIP model is derived as the simple hybrid Routhian system for to the 2D one leg robotic hopper. Section 4 contains the main results of the paper, such as how after introducing time-reversal symmetries we find sufficient conditions to ensure the existence of periodic solutions and we study how to characterize some of the eigenvalues for the linearization of the Poincar´e map corresponding to the periodic orbit. In this section we also explain how to apply the results for the reduced system associated with the 2D one leg robotic hopper.

Finally, Section 5 shows the results obtained in Section 4 to study the existence of periodic solutions in underactuated simple hybrid Routhian control systems.

2. Preliminaries On Routh Reduction

Let Q be the configuration space of a mechanical system, a differentiable manifold of dimension n, with local coordinates q = (q1, . . . , qn). Let TQ be the tangent bundle of Q, locally described by the positions and velocities for the system (q, ˙q) = (q1, . . . , qn, ˙q1, . . . , ˙qn) ∈TQ with dim(TQ) = 2n.

The dynamics of the mechanical system is determined by a Lagrangian func- tion L : TQ →R given by L(q, ˙q) = K(q, ˙q) −V (q) where K : TQ →R is the kinetic energy and V : Q →R the potential energy. Along this work we will assume that the Lagrangian L is regular, that is, det

= 0 For All

i, j = 1, . . . , n. The corresponding equations of motion describing the dynamics of the sys- tem are given by the Euler-Lagrange equations for L, that is, d

∂Qi ,

i = 1, . . . , n; a system of n second-order ordinary differential equations. These equations induce a vector field XL : TQ →T(TQ) describing the dynamics of

!

. There exists a large class of systems for which the Lagrangian does not depend on some of the generalized coordinates.

Such Coordinates Are Called

cyclic or ignorable, and the corresponding generalized momenta is easily checked to be constants of the motion. The Routh reduction procedure is a classical reduction technique which takes advantage of the conservation law to define a reduced Lagrangian function, so- called the Routhian, such that the solutions of the Euler-Lagrange equations for the Routhian are in correspondence with the solutions of Euler-Lagrange equations for the original Lagrangian, when the conservation of momenta is taken into account. The technique is due to Edward Routh, who successfully applied it to the study of the stability of steady motions (today, we call these relative equilibria).

The steps to carry out Routh reduction are described in many classical texts

5

of mechanics (such as ) as follows. Assume qi = (xa, θ) are local coordinates on Q, a = 1, . . . , n −1. 1) Let L(xa, ˙xa, θ, ˙θ) be a G-regular Lagrangian with cyclic coordinate θ, that

Is, ∂L

∂θ = 0, and denote by pθ the generalized momentum corresponding to θ. 2) Fix a value of the momentum µ = pθ, and consider the function

(1)

where the notation means that we have used the relation µ = pθ to replace all the appearances of ˙θ in terms of (xa, ˙xa) and the parameter µ. The function

Rµ

c is the (classical) Routhian.

3) If We Regard Rµ

c as a new Lagrangian in the variables (xa, ˙xa), then the

C Are In Correspondence With

those of L when one takes into account the relation pθ = µ. More precisely,

If L Is G-Regular:

(i) Any solution (xa(t), θ(t)) of the Euler-Lagrange equations for L with momentum pθ = µ projects onto a solutions xa(t) of the Euler-Lagrange

C

∂xa = 0.

(2)

These equations will be referred to as Routh equations. Equations (2)

C : Tq →T(Tq) Describing The Dynamics Of

the reduced system, called Routhian vector field, and given by

−1

. (ii) Conversely, any solution of Routh equations for Rµ

C Can Be Lifted To A

solution of the Euler-Lagrange equations for L with momentum pθ = µ.

6

Example 1. The Lagrangian L : T(R × S1) →R given by

2M( ˙R2 + R2 ˙Θ2) −1

2r2k. describes (in polar coordinates) the motion of a mass m on the plane which is pinned to a fixed point through a spring of elastic constant k (as it is shown in Figure 1).



= (mr)2̸ = 0. The coordinate θ is cyclic, and the associated conservation law reads pθ = mr2 ˙θ. If we set pθ = µ̸ = 0 (a fixed constant µ which represents the fixed value of the momentum), we can work out ˙θ in terms of the (r, ˙r) and find the relation



.

C Happens When One Takes Into

account the conservation law pθ = µ.

More Precisely: A Solution (R(T), Θ(T))

of the Euler Lagrange for the G-regular Lagrangian L with momentum pθ = µ projects onto a solution r(t) of Routh equations for Rµ

C , One Can Use The Conservation Law

pθ = µ to lift r(t) to a solution (r(t), θ(t)) of the Euler Lagrange for L (with momentum µ, obviously).

3.1. Simple Hybrid System

Simple hybrid systems (see also ) are characterized by the 4-tuple H = (D, X, S, ∆) where D is a smooth manifold, the domain, X is a smooth vector field on D, S is an embedded submanifold of D with co-dimension 1 called switching surface, and ∆: S →D is a smooth embedding called the impact map. S and ∆are also called the guard and reset map, respectively, in -.

In this work, the dynamics associated with a simple hybrid system is de- scribed by an autonomous system with impulse effects as in . We denote by ΣH the simple hybrid dynamical system generated by H, that is,

Τ→T+Γ(Τ) Are The Left

and right limits of the state trajectory γ(t), respectively, describing the states immediately before and after the times when integral curves of X intersects S (i.e., pre and post impact of the solution γ(t) with S).

Remark 1. Consider the impact map ∆given by the identity map. When a trajectory crosses S, we will have γ+ = ∆(γ−) = γ−∈S, so that we are again in the regime of discrete dynamics where re-initialization (to γ−) will occur. It is clear that this process will never end. Consequently, there exists an infinite number of resets in a finite amount of time. This situation generates a class of behavior called Zeno behavior. That is, a solution of a hybrid system may experience a Zeno state if infinity many impacts occur in a finite amount of time . This is particularly problematic in applications where numerical work is conducted, as computation time grows infinitely large at these Zeno points.

There are two primary modes through which zeno behavior can occur: (1) A trajectory is reset back onto the guard, prompting additional resets. As seen in the above example where γ+ = ∆(γ−) = γ−∈S, if there is a set of

8

points in the guard which the reset map cycles between, we can get ’stuck’ on the guard. To exclude this type of behavior, we require that S ∩∆(S) = ∅, where ∆(S) denotes the closure as a set of ∆(S). This ensures that the trajectory will always be reset to a point with positive distance from the guard.

(2) The set of times where a solution to our system reaches the guard (and is correspondingly reset) has a limit point. This happens, for example, in the case of the bouncing ball with coefficient of restitution 1/2. If t0 is the time between two impacts, then the time between the next two impacts will be t0/2, then t0/4,

4 + ... = 2T0

we will have infinite resets in a finite time. To exclude these kind of situations, we require the set of impact times to be closed and discrete, as in . Both of the previously established hypothesis will be assumed implicitly through- out the rest of the paper (i.e., ∆(S)∩S = ∅and the set of impact times is closed and discrete). Necessary and sufficient conditions for the existence of Zeno be- havior in the class of hybrid systems studied in this work have been explored in .

3.2. Simple Hybrid Routhian Systems

When the configuration space is Q = P × G (i.e., D = TQ = TP × TG) with G an abelian Lie group and P a smooth manifold, A. Ames and S. Sastry introduced the notion of hybrid Routhian systems in and . This is based on 2 invariance properties allowing the reduction by symmetries: a cyclic regular Lagrangian L : TQ →R and a function h : Q →R describing an unilateral constraint which induces the switching surface, being h cyclic and defined in the same generalized coordinates as L. These two invariance properties allow

C : Tp →R And The Reduced Constraint Function

¯h : P →R. The starting point for symmetry reduction is a Lie group action ψ: G×Q → Q of some Lie group G on the manifold Q. There is a natural lift of the action

Ψt ∗Q : G × T ∗Q →T ∗Q,

(g, (q, p)) 7→T ∗ψg−1(q, p). The cotangent bundle T ∗Q is equipped with the following geometric structure, called canonical symplectic structure (see, for instance, ) Ω= dq ∧dp with (q, p) being local coordinates on T ∗Q. The action ΨT ∗Q enjoys the following

Properties (See ):

• ΨT ∗Q is a symplectic action, meaning that, if we denote ΨT ∗Q

Of

the 2-form Ω. • It admits an Ad∗-equivariant momentum map J : T ∗Q →g∗given by

D

dtψexp(tξ)q is the infinitesimal generator of the element ξ ∈g∗and where g∗denotes the dual of g, the Lie algebra associated with the Lie group G.

Likewise, there is a lift of the action ψ to TQ denoted by ΨT Q

Ψt Q : G × Tq →Tq,

(g, (q, ˙q)) 7→Tψg(q, ˙q). It has been shown in and that to perform a hybrid Routhian reduction one needs to impose some compatibility conditions between the action and the simple hybrid system whose continuous-time dynamics is described by a La- grangian flow associated to L (a.k.a simple hybrid Lagrangian system). By the term hybrid action we mean a Lie group action ψ: G × Q →Q such that (i) L is invariant under ΨT Q, i.e. L ◦ΨT Q = L; (ii) ΨT Q restricts to an action on S; (iii) ∆is equivariant w.r.t. the previous action, namely ∆◦ΨT Q

G

◦∆. In the case of a hybrid action, ΨT Q admits an Ad∗-equivariant (Lagrangian) momentum map JL : TQ →g∗given by JL = J ◦FL, where FL : T →T ∗Q

10

is the Legendre transformation associated with regular Lagrangian L, given by FL(q, ˙q) := (q, p = ∂L/∂˙q). This follows directly from the invariance of L, since it implies that FL is an equivariant diffeomorphism, i.e. FL◦ΨT Q

◦Fl.

The hybrid equivalent of momentum map is the notion of hybrid momentum map introduced in . For example, in the case of TQ, JL is a hybrid momentum

(4)

commutes, where i denotes the canonical inclusion from S to TQ. The situation of interest in this paper is that of an Abelian group action, i.e. G = S1 (the case G = R is analogous; and if G is a product one can iterate the procedure): this corresponds to the classical notion of cyclic coordinates. From now on we will assume Q = P × S1 where P is called the shape space and the

Ψα : S1 × Q →S1 × Q,

(θ, x) 7→(θ + α, x).

(5)

While this is indeed a strong assumption, it is always the case locally, so as long as it applies to the domain of interest of a specific problem the procedure below applies. More general results where the manifold is not a product or the Lie group is arbitrary (not necessarily Abelian) can be handled by using the same tools, but involving more technicalities such as the introduction of a principal connection. These matters will be discussed in a future extension of this work.

A. Ames and S. Sastry also shown that if the trajectory γ for a simple hybrid Lagrangian system starting at γ0 ∈J−1(µ), with J : TQ →g∗the momentum map associated for the conserved quantity µ, being g the Lie algebra associated with G, then the trajectory for a simple hybrid Routhian system starting at π(γ0) with π : TQ →TP the projection over the first factor of TQ = TP ×TG, is determined by π(γ(t)).

11

Definition 3.1. A simple hybrid system H = (D, X, S, ∆) is said to be a simple

Xrµ

c : TP →T(TP) is the Routhian vector field, Sµ is the reduced switching surface given by S |(JL|S)−1(µ) with µ the momentum constraint determined by the conserved quantity which arises from the cyclic coordinate and ∆µ : Sµ → TP is the impact map on Sµ given by ∆|(JL|S)−1(µ).

Definition 3.2. The simple hybrid Routhian dynamical system generated by

(6)

where γ(t) ∈TP. Note that, as before, ∆µ : Sµ →TP is continuous and if we denote ∆µ(Sµ) the closure of ∆µ(Sµ) then we must assume ∆µ(Sµ) ∩Sµ = ∅and therefore, an impact does not lead immediately to another impact. We shall assume that

Hrµ

c satisfies (see Section 4.1 in reference for more details) (A1) Assumption 1: Sµ̸ = ∅and there exists an open subset U ⊂TP and a differentiable function ¯h : U →R such that Sµ = {x ∈U | ¯h(x) = 0} with

∂¯H

∂x(s)̸ = 0 for all s ∈Sµ (that is, Sµ is an embedded submanifold of TP with co-dimension 1) and the Lie derivative of the vector field XRµ

¯H Does Not Vanish On Tp, That Is Lxrµ

c ¯h(w)̸ = 0, ∀w ∈TP. (A2) Assumption 2: A trajectory γ : [0, T] →TP crosses the switching

Surface Sµ At T−

i = inf{t > 0|γ(t) ∈Sµ}. We allow the trajectory γ to be non-

Smooth But Continuous At T−

i . That is, the velocity before the impact ˙x−may be different from the velocity ˙x+ after the impact at Sµ, i.e., ˙x(t−

I )̸ = ˙X(T+

i ). The requirement that the configuration right after the impact does not be- long to Sµ (that is, ∆µ(Sµ) ∩Sµ = ∅), becomes a requirement on the exit velocity which states that the system has to be moving away from the switching surface right after the impact, that is, ∇¯h(γ(ti)) · ˙γ(t+

I ) ≤0, With ¯H : Tp →R

as in A1.

12

A trajectory of a hybrid Routhian system is determined by the dynamics as- sociated with the Routhian until the instant when the state attains the switching surface Sµ. We refer to such instant as the impact time. There is an instan- taneous change in the velocity component of the state at impact times. The impact map gives new initial conditions from which the continuous dynamics evolves until the next impact occurs.

Remark 2. We defined simple hybrid Routhian systems from simple hybrid systems as in -. However our definition is slightly different, but it is not contradictory, with the one given in and . The constraint defining Sµ is smooth satisfying both assumptions A1 and A2, (it is not determined by unilateral constraints as considered in - and , since we use the approach for systems with impulsive effects following and ). Usually, and specially in biped locomotion, the impact map is given by foot placement on the ground and it comes from a Newtonian impact equation , , . In this work we only assume ∆µ is a smooth embedding on Sµ satisfying A1 and A2 .

Solutions for the simple hybrid Routhian dynamical system ΣHRµ

C , Are Con-

sidered right continuous and with finite left and right limits at each impact with

Sµ. More Precisely:

Definition 3.3. A solution for the hybrid Routhian system ΣHRµ

C Is A Curve

γ : [t0, tf) →TP, tf ∈R ∪{∞}, tf > t0, unique from a given initial condition, depending continuously on it, satisfying A2, and such that:

(I) Γ(T) Is Right Continuous On [T0, Tf),

(ii) left and right limits, denoted by γ−(t) and γ+(t), respectively, exists at

Each Point T ∈(T0, Tf),

(iii) there exists a closed discrete subset I ⊂[t0, tf), the impact times, such that, for each t /∈I, γ(t) is differentiable, ˙γ(t) = XRµ

C (Γ(T)), And Γ(T) /∈

Sµ; and for t ∈I, γ−(t) ∈Sµ and γ+(t) = ∆µ(γ−(t)).

13

Note that right continuity of solutions implies γ(t) = γ+(t) at all the points in its domain of definition. If α0 ∈TP denotes the initial state at time t0, the solution at t0 is denoted γ(t0, α0). When α0 /∈Sµ, γ(t0, α0) = α0 and when α0 ∈Sµ, γ(t0, α0) = ∆µ(α0) = γ(t0, ∆µ(α0)) (see Section 4.1 in for details).

Remark 3. As we commented in Section 2, since the Euler-Lagrange equations

For Rµ

c involve less variables, they are easier to solve. If the Lagrangian L is reg- ular, then one usually proceeds to solve these first, and then uses the momentum constraint ∂L/∂˙θ = µ to reconstruct the sought solutions of the Euler-Lagrange equations of L.

Special care should be taken when translating this reduction technique to hybrid systems. The reason is that the collisions with the switching surface will, in general, modify the value of the momentum map. Therefore, if I = {Ii}i∈Λ, where Λ = {0, 1, 2, ...} ⊆N is a finite (or infinite) indexing set, and Ii = [ti, ti+1] if i, i + 1 ∈Λ and IN−1 = [tN−1, tN) or [tN−1, ∞) if |Λ| = N, N finite, with ti, ti+1, tN−1 ∈R and ti ≤ti+1, then the Routhian has to be defined in each Ii taken into account the value of the momentum µi after the collision at time τi. Note that this also has influence in the way the reset map ∆is reduced.

Let us denote: (1) µi the momentum of the system in Ii = [ti, ti+1], (2) ∆µi is given by ∆|(JL|S)−1(µi), and (3) Sµi is given by S |(JL|S)−1(µi), There is a

(. . . )

(. . . )

Coll. Red. Coll.

Coll. Red. Coll.

Red. Similarly as in , the reconstruction procedure from the reduced hybrid flow to the flow for the simple hybrid Lagrangian system involves a recursive inte- gration at each stage in the previous diagram of the cyclic variable using the solution of the reduced hybrid Routhian system. Roughly speaking, this accounts

14

to imposing the momentum constraint on the reconstructed solution. Example 2 (The 2D one leg robotic hopper). The 2D one leg hopper robot consists of a spring loaded inverted pendulum together with a planar rigid body attached at the top of the spring (see Figure 2). This model is a schematic rep- resentation for the stance phase of a running or hopping biped with one foot on the ground at any time (see for details). The common point of attachment is the center of mass of the rigid body (i.e., the sprung leg is attached at a hip joint which is the center of mass).

Figure 2: 2D One Leg Robotic Hopper

The configuration space of the system is Q = (R×S1)×S1, locally parametrized by the coordinates q = (ξ, ϕ, θ) describing the length of the spring, the angle of the spring with respect to the ground (i.e., the angle formed between the line joining the foothold to the center of mass and the vertical gravitational axis) and the attitude for the rigid body, respectively. We denote by m the mass of the rigid body and I its moment of inertia. The spring is considered massless and l0 denotes the non-load length of the spring.

The motion is divided into two phases: The first one is the stance phase, with foothold fixed, the leg under compression, and the body swinging forwards (i.e., θ is increasing). The second one is a flight phase, which occurs when the rigid body moves, describing a ballistic motion under the influence of gravity. The transition between both phases occurs when the spring is uncompressed (i.e., it is unloaded) until the time when the spring touch the ground again. Both phases define a hybrid system.

15

The Lagrangian describing the flight phase is given by the kinetic energy minus the potential energy, which is given by the spring potential V (ξ) and the

2M( ˙Ξ2 + Ξ2 ˙Φ2) + 1

2I ˙θ2 −(mgξ cos ϕ + V (ξ)). To derive the switching surface and the impact map, we note that the flight starts when the spring length reaches its non-load length (i.e., ξ = l0). Therefore the switching surface is given by S = {(ξ, ϕ, θ, ˙ξ, ˙ϕ, ˙θ) ∈TQ| ξ = l0}.

By assuming that at the start of the stance phase the leg is at an angle of −ϕ0, by employing polar coordinates x = ξ sin ϕ, y = ξ cos ϕ and nothing that y+ = l0 cos(−ϕ0) = l0 cos(ϕ0), the impact map is given by ∆(x−, y−, θ−, ˙x−, ˙y−, ˙θ−) = (−l0 sin ϕ0, l0 cos ϕ0, −θ−, ˙x−, −˙y−, −˙θ−).

Note that the Lagrangian and the constraint which define the switching sur- face, i.e., h(q) = ξ −l0, are both cyclic in θ.

2 ( ˙Ξ2 + Ξ2 ˙Φ2) −Μ2

2I −mgξ cos ϕ −V (ξ).

Note That Rµ

c describes the motion for the spring-loaded inverted pendulum (SLIP) which has been used as a model which reasonably provides a template for sagittal plane motions of the center of mass (COM) of diverse legged systems as it was reviewed in and further studied in and .

Figure 3: 2D Spring Loaded Inverted Pendulum

The aerial phase consists of a projectile (or ballistic) motion for the COM

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