Existence and stability of a limit cycle in the model of a planar passive biped walking down a slope
EEXISTENCE AND STABILITY OF A LIMIT CYCLE IN THE MODELOF A PLANAR PASSIVE BIPED WALKING DOWN A SLOPE ∗ OLEG MAKARENKOV † Abstract.
We consider the simplest model of a passive biped walking down a slope given by theequations of switched coupled pendula (McGeer, 1990). Following the fundamental work by Garciaet al (1998), we view the slope of the ground as a small parameter γ ≥
0. When γ = 0 the systemcan be solved in closed form and the existence of a family of limit cycles (i.e. potential walkingcycles) can be established explicitly. As observed in Garcia et al (1998), the family of limit cyclesdisappears when γ increases and only isolated asymptotically stable cycles (walking cycles) persist.However, no rigorous proofs of such a bifurcation (often referred to as Melnikov bifurcation) haveever been reported. The present paper fills in this gap in the field and offers the required proof. Key words.
Passive planar biped, limit cycle, perturbation theory, switched system, nonsmoothsystem
AMS subject classifications.
1. Introduction.
In his celebrated paper [13] McGeer proposed to view thepassive bipedal walker of Fig. 1a as a combination of a pendulum with a fixed pivot(Fig. 1b) ¨ α − g sin α = 0 (stance leg)and a pendulum with a moving pivot (Fig. 1c)¨ β + x (cid:48)(cid:48) cos β + ( y (cid:48)(cid:48) + g ) sin β = 0 (swing leg) , which gives ¨ θ − sin( θ − γ ) = 0 , ¨ θ − ¨ φ + ˙ θ sin φ − cos( θ − γ ) sin φ = 0 . (1.1)When the heelstrike occurs (i.e. when φ = 2 θ ), the stance and swing legs swap their ` l l x y (a) (b) (c) Fig. 1 . Building blocks ((a) and (b)) of a planar passive walker (c). roles and the state vector ( θ, ˙ θ, φ, ˙ φ ) T jumps as follows(1.2) θ ( t + )˙ θ ( t + ) φ ( t + )˙ φ ( t + ) = J ( θ ( t )) θ ( t − )˙ θ ( t − ) φ ( t − )˙ φ ( t − ) , if φ ( t ) = 2 θ ( t ) , ∗ Submitted to the editors DATE. † ∼ makarenkov/).1 a r X i v : . [ m a t h . D S ] A p r O. MAKARENKOV where J ( θ ) = − θ − − cos 2 θ ) cos 2 θ . Using Newton’s method, McGeer found that the switched system (1.1)-(1.2) admitsa limit cycle, whose period is close to T = 3 . γ > . Ajustification of the existence of such a limit cycle was offered in Garcia el al [5], wherethe change of the variables(1.3) γ = δ / , θ ( t ) = δ / Θ( t ) , φ ( t ) = δ / Φ( t )is proposed to expand (1.1)-(1.2) in the powers of small parameter δ > δ > θ, ω ) (cid:55)→ P ( θ, ω, δ ) associated to the perturbed switched system (2.1)-(2.2). InSection 4 we show that, when δ = 0 , the Poincare map ( θ, ω ) (cid:55)→ P ( θ, ω, δ ) admitsa family of fixed points ( θ, ω ) = ξ ( s ), where ξ ∈ C ( R , R ) and s is a parameter.In this way the problem of the existence of limit cycles to the perturbed switchedsystem (2.1)-(2.2) reformulates as a problem of bifurcation of asymptotically stablefixed points to the Poincare map ( θ, ω ) (cid:55)→ P ( θ, ω, δ ) from the family ( θ, ω ) = ξ ( s ) as δ crosses 0. The problem obtained is a classical problem of the theory of nonlinearoscillations coming back to Malkin [12] and Melnikov [7, Ch. 4, § θ, ω ) (cid:55)→ P ( θ, ω, δ )of the passive biped in Sections 6 and 7. In Section 8 (Conclusions) we discuss thevalue of this work to the field of perturbation theory. The proof of Theorem 5.1 isgiven in Appendix A and Appendix B contains some technical formulas. All symboliccomputations have been executed in Wolfram Mathematica 11.3.Despite of extensive literature on bifurcation of fixed points from 1-parameter families,the paper by Glover et al [6] on large amplitude oscillations in a suspension bridgemodel seems to be the only example of such a bifurcation accessible to general public.The significant contribution of the present paper is in a rigorous introduction of a onemore example of bifurcation from 1-parameter families that is noticeable to societyon the one hand and is well regarded in engineering community on the other hand.
2. Expanding McGeer’s model of passive biped into the powers ofthe slope of the ground.
Incorporating the change of the variables (1.3) into the
IMIT CYCLES OF THE PLANAR PASSIVE BIPED WALKING DOWN A SLOPE τ = τ − τ
3! + τ − τ
7! + ..., cos τ = 1 − τ
2! + τ − τ
6! + ..., one gets (see Garcia et al [5])¨Θ − (Θ − δ ) + 16 δ Θ + o ( δ ) = 0 , ¨Θ − Φ − ¨Φ + δ ˙Θ Φ + 12 δ Θ Φ + 16 δ Φ + o ( δ ) = 0 , (2.1) Θ( t + )˙Θ( t + )Φ( t + )˙Φ( t + ) = J (Θ( t ) , δ ) Θ( t − )˙Θ( t − )Φ( t − )˙Φ( t − ) , if Φ( t ) = 2Θ( t ) , (2.2)where J (Θ , δ ) = − − δ (2Θ) + o ( δ ) 0 0 − (cid:18) − δ (2Θ) + o ( δ ) (cid:19) (cid:18) δ (2Θ) + o ( δ ) (cid:19) and o i ( δ ) stay for the remainders (perhaps dependent on Θ and Φ) such that o i ( δ ) /δ → δ → , Φ) from any compact set.
3. The Poincare map ( θ, ω ) (cid:55)→ P ( θ, ω, δ ) induced by the heelstrikethreshold. To construct the Poincare map induced by the hyperplane Φ = 2Θ , wewill consider the initial condition (Θ( t + ) , ˙Θ( t + ) , Φ( t + ) , ˙Φ( t + )) T given by (2.2). Be-cause of the properties of the matrix J (Θ , δ ) any vector (Θ( t + ) , ˙Θ( t + ) , Φ( t + ) , ˙Φ( t + )) T coming from (2.2) has the form(3.1) (Θ( t + ) , ˙Θ( t + ) , Φ( t + ) , ˙Φ( t + )) = (cid:0) θ, ω, θ, (cid:0) δθ + o ( δ ) (cid:1) ω (cid:1) . In other words, knowing that (Θ( t + ) , ˙Θ( t + )) = ( θ, ω ), we can use formula (3.1) toobtain the respective values of Φ( t + ) and ˙Φ( t + ). Defining∆( θ, ω, δ ) = (cid:18) − θ (1 − δθ + o ( δ )) ω (cid:19) , we can introduce a 2-dimensional Poincare map as follows(3.2) P ( θ, ω, δ ) = ∆ (cid:104)(cid:16) Θ , ˙Θ (cid:17) ( T ( θ, ω, δ ) , θ, ω, δ ) , δ (cid:105) , where t (cid:55)→ (Θ , ˙Θ , Φ , ˙Φ) T ( t, θ, ω, δ ) is the solution of (2.1) with the initial condition(3.3) (Θ(0) , ˙Θ(0) , Φ(0) , ˙Φ(0)) = (cid:0) θ, ω, θ, δθ ω + o ( δ ) ω (cid:1) and T ( θ, ω, δ ) is the time satisfying(3.4) Φ( T ( θ, ω, δ ) , θ, ω, δ ) = 2Θ( T ( θ, ω, δ ) , θ, ω, δ ) , Φ( t, θ, ω, δ ) (cid:54) = 2Θ( t, θ, ω, δ ) , t ∈ (0 , T ( θ, ω, δ )) . O. MAKARENKOV
4. Families of fixed points of the Poincare map ( θ, ω ) (cid:55)→ P ( θ, ω, δ )when δ = 0. When δ = 0, the system (2.1) and the initial condition (3.1) take theform(4.1) ¨Θ − Θ = 0 , ¨Θ − Φ − ¨Φ = 0 , (Θ(0) , ˙Θ(0) , Φ(0) , ˙Φ(0)) = ( θ, ω, θ, , whose solution is(4.2) Θ( t, θ, ω,
0) = 12 e − t (1 + e t ) θ + 12 e − t ( − e t ) ω, Φ( t, θ, ω,
0) = 2 θ cos t + e − t e t − e t cos t ) θ + e − t e t − − e t sin t ) ω. Observe that ( θ, ω ) = P ( θ, ω,
0) if and only if(4.3) θ = − Θ( T, θ, ω, ,ω = ˙Θ( T, θ, ω, , Φ( T, θ, ω,
0) = 2Θ(
T, θ, ω, . The first two equations of (4.3) give(4.4) ω = α ( T ) θ, where α ( T ) = − e T − e T . Substituting (4.4) into the third equation of (4.3) one obtains the following equationfor T (4.5) − e T + 3( − e T ) cos T + sin T + e T sin T = 0 , whose roots on (0 , π )(4.6) T = π, T = 3 . .... According to Garcia et al [5] only roots within the (0 , π ) correspond to “reasonablyanthropomorphic gaits”. Also, following Garcia et al [5], we will stick to the secondroot T because it corresponds to a symmetric gait in the following sense: plugging ω = α ( T ) θ into the third equation of (4.3) gives approximately − . e − t + 0 . e t + 1 . t + 0 . t = 0 , whose only solution on (0 , T ) is T / T / , θ, ω,
0) = Φ( T / , θ, ω,
0) = 0 . The property (4.7) corresponds to the event where the two legs coincide. Though(4.7) formally implies a heel-strike (the third equation of (4.3) holds at T = T / γ increases, then, formally speaking, an impactoccurs at T = T /
2, but we will still ignore the impact coming from T = T / T = T /
2, see [3]).In other words, for the reasons just explained and following Garcia et al [5], we willconsider the Poincare map (3.2) with T ( θ, ω, δ ) → T as δ → T / . IMIT CYCLES OF THE PLANAR PASSIVE BIPED WALKING DOWN A SLOPE
5. Perturbation theorem for two-dimensional Poincare maps.
Through-out this section we assume that the unperturbed Poincare map ( θ, ω ) (cid:55)→ P ( θ, ω, P ( ξ ( s ) ,
0) = ξ ( s ) for all s ∈ R , where s (cid:55)→ ξ ( s ) isa C curve. Note, the latter property implies that P ( θ,ω ) ( ξ ( s ) , ξ (cid:48) ( s ) = ξ (cid:48) ( s ) , whichmeans that one of the eigenvalues of the matrix P ( θ,ω ) ( ξ ( s ) ,
0) is always 1 for all s ∈ R . To make the notations less bulky we will identify P ( θ, ω, δ ) with P (( θ, ω ) , δ )as it doesn’t seem to cause any confusion.Fix some s ∈ R and put ( θ , ω ) = ξ ( s ) . Denote by y and ˜ y the eigenvectors of P ( θ,ω ) ( θ , ω ,
0) that correspond to the eigen-values 1 and ρ (cid:54) = 1 respectively. We then denote by z and ˜ z the eigenvalues of P ( θ,ω ) ( θ , ω , T that correspond to the eigenvalues 1 and ρ (cid:54) = 1 , and such that(5.1) z T y = ˜ z T ˜ y = 1 . It can be verified that(5.2) z T ˜ y = ˜ z T y = 0 . Properties (5.1) and (5.2) imply that(5.3) ζ = z T ζy + ˜ z T ζ ˜ y, for any ζ ∈ R . We will also assume that z doesn’t depend on the choice of s , in which case we have(5.4) z T ( P ( θ,ω ) ( ξ ( s ) , − I ) = 0 , for all s ∈ R . The following theorem is a corollary of the results of Kamenski et al [8] andMakarenkov-Ortega [11].
Theorem
Let P be a C function. If, for each δ ∈ R , the Poincare map ( θ, ω ) (cid:55)→ P ( θ, ω, δ ) admits a fixed point ( θ δ , ω δ ) such that (5.5) ( θ δ , ω δ ) → ( θ , ω ) as δ → , then (5.6) z T P δ ( θ , ω ,
0) = 0 . Assume that the eigenvector z of P ( θ,ω ) ( θ , ω , T that corresponds to the eigenvalue1 doesn’t depend on s . If, in addition to (5.6), it holds that (5.7) z T ( P δ ) ( θ,ω ) ( θ , ω , y (cid:54) = 0 , then, for all | δ | sufficiently small, the Poincare map ( θ, ω ) (cid:55)→ P ( θ, ω, δ ) does indeedhave a fixed point ( θ δ , ω δ ) that satisfies (5.5). The fixed point ( θ δ , ω δ ) is asymptoticallystable, if the eigenvalue ρ (cid:54) = 1 of P ( θ,ω ) ( θ , ω , satisfies (5.8) | ρ | < , and if (5.7) holds in the stronger sense (5.9) z T ( P δ ) ( θ,ω ) ( θ , ω , y < . O. MAKARENKOV
6. Stability of the family ω = α ( T ) θ of fixed points of the Poincaremap ( θ, ω ) (cid:55)→ P ( θ, ω, δ ) corresponding to δ = 0. As explained in Section 5,one of the eigenvalues of matrix P ( θ,ω ) ( θ, α ( T ) θ,
0) is always 1. In this section wecompute the second eigenvalue (named ρ ) of P ( θ,ω ) ( θ, α ( T ) θ,
0) and verify condition(5.8) of Theorem 5.1. We will see that ρ doesn’t depend on θ , so we write ρ as opposedto ρ ( θ ) from the beginning.Differentiating (3.2) with respect to the vector variable ( θ, ω ) ,P ( θ,ω ) ( θ, α ( T ) θ,
0) = ∆ (cid:18) Θ˙Θ (cid:19) t ( T , θ, α ( T ) θ, T ( θ,ω ) ( θ, α ( T ) θ, (cid:18) Θ˙Θ (cid:19) ( θ,ω ) ( T , θ, α ( T ) θ, , where ∆ = (cid:18) − (cid:19) . Using formulas (4.2) and (4.6) one gets (cid:18)
Θ˙Θ (cid:19) t ( T , θ, ω,
0) = (cid:18) Θ t ( T , θ, ω, tt ( T , θ, ω, (cid:19) = (cid:18) . ω + 22 . θ . ω + 22 . θ (cid:19) and so (cid:18) Θ˙Θ (cid:19) t ( T , θ, α ( T ) θ ) = θ (cid:18) − . − (cid:19) . In the same way, (cid:18)
Θ˙Θ (cid:19) ( θ,ω ) ( τ ) = (cid:18) Θ θ ( τ ) Θ ω ( τ )Θ tθ ( τ ) Θ tω ( τ ) (cid:19) = (cid:18) . . . . (cid:19) , where a shortcut τ = ( T , θ, α ( T ) θ, T ( θ,ω ) ( θ, α ( T ) θ,
0) = − ( F t ( T , θ, α ( T ) θ )) − F ( θ,ω ) ( T , θ, α ( T ) θ ) , where(6.2) F ( t, θ, ω ) = Φ( t, θ, ω, − t, θ, ω, . Plugging formulas (4.2) and (4.6) into (6.1), the function T ( θ,ω ) ( θ, α ( T ) θ,
0) computesas T ( θ,ω ) ( θ, α ( T ) θ,
0) = 1 θ (16 . , . . Combining the above findings together we finally get(6.3) P ( θ,ω ) ( θ, α ( T ) θ,
0) = (cid:18) − . − . . . (cid:19) whose eigenvalues are 1 and ρ = 0 . , so that condition (5.8) holds. IMIT CYCLES OF THE PLANAR PASSIVE BIPED WALKING DOWN A SLOPE
7. Bifurcation of isolated fixed points of the Poincare map ( θ, ω ) (cid:55)→ P ( θ, ω, δ ) from the family ω = α ( T ) θ when δ crosses 0. In this section weverify the remaining conditions (5.6), (5.7) and (5.9) of Theorem 5.1. P δ . Differentiating (3.2) with respect to δ, one gets(7.1) P δ ( θ, ω,
0) = ∆ (cid:18) Θ˙Θ (cid:19) t ( T ( θ, ω, , θ, ω, T δ ( θ, ω, (cid:18) Θ˙Θ (cid:19) δ ( T ( θ, ω, , θ, ω, δ (cid:16)(cid:16) Θ , ˙Θ (cid:17) ( T ( θ, ω, , θ, ω, , (cid:17) . The terms ∆ and (cid:18) Θ˙Θ (cid:19) t ( τ ) were computed in the previous section. For the terms∆ δ ( θ, ω,
0) and (cid:18)
Θ˙Θ (cid:19) ( τ ), the definition of ∆( θ, ω, δ ) and formula (4.3) yield∆ δ ( θ, ω,
0) = (cid:18) − θ ω (cid:19) , (cid:18) Θ˙Θ (cid:19) ( τ ) = (cid:18) − θα ( T ) θ (cid:19) . To compute T δ ( θ, ω,
0) we can use function F of the previous section, which gives(7.2) T δ ( θ, ω,
0) = − ( F t ( T , θ, ω )) − F δ ( T , θ, ω ) . So it remains to compute the function t (cid:55)→ (cid:0) (Θ , Φ) T (cid:1) δ ( t, θ, ω, , which can be foundas the solution t (cid:55)→ ( h ( t ) , f ( t )) T of the δ -derivative of the initial-value problem (2.1)and (3.3):(7.3) ¨ h − h + 1 + 16 Θ( t, σ ) = 0 , ¨ h − f − ¨ f + (cid:16) ˙Θ( t, σ ) (cid:17) Φ( t, σ ) + 12 (cid:16) ˙Θ( t, σ ) (cid:17) Φ( t, σ ) + 16 (Φ( t, σ )) = 0 ,h (0) = 0 , ˙ h (0) = 0 , f (0) = 0 , ˙ f (0) = 2 θ ω, where σ is a shortcut for σ = ( θ, ω, . After plugging (4.2) into (7.3) we get a systemof linear inhomogeneous differential equations, whose solution t (cid:55)→ ( h ( t ) , f ( t )) T isgiven in Appendix B. In particular, plugging t = T , one gets(7.4) (cid:18) ΘΦ (cid:19) δ ( T , θ, ω,
0) = (cid:18) h ( T ) f ( T ) (cid:19) == (cid:18) − . − . ω − . ω θ − . ωθ − . θ − . . ω + 1793 . ω θ + 1582 . ωθ + 458 . θ (cid:19) . and (cid:18) ΘΦ (cid:19) δ ( τ ) = (cid:18) h ( T ) f ( T ) (cid:19) = (cid:18) − . − . θ − . − . θ (cid:19) . Formula (7.2) then provides T δ ( θ, ω,
0) = 0 . . ω + 96 . ω θ + 90 . ωθ + 28 . θ ω + 0 . θ . Plugging all the above findings into formula (7.1), we conclude P δ ( θ, α ( T ) θ,
0) = (cid:18) . . θ − . . θ (cid:19) . O. MAKARENKOV θ , ω ) that satisfies the necessary condition (5.6). Computing an eigenvector z of the transpose of the matrix (6.3) for the eigenvalue 1,we get z = ( − . , − . T . Therefore, taking into account the relation (4.6) between θ and ω , the necessarycondition (5.6) takes the form1 . − . θ ) = 0 . The solution of this equation is θ = 0 . , which coincides with the finding of Garcia et al [5] (see the table at [5, p. 15]). P δ ( θ,ω ) . Differentiating (7.1) with respect to ( θ, ω ) , one gets P δ ( θ,ω ) ( θ, α ( T ) θ,
0) = ∆ (cid:20)(cid:18) Θ˙Θ (cid:19) tt ( τ ) T ( θ,ω ) ( θ, α ( T ) θ, (cid:18) Θ˙Θ (cid:19) t ( θ,ω ) ( τ ) (cid:35) T δ ( θ, α ( T ) θ, (cid:18) Θ˙Θ (cid:19) t ( τ ) T δ ( θ,ω ) ( θ, α ( T ) θ, (cid:34)(cid:18) Θ˙Θ (cid:19) δt ( τ ) T ( θ,ω ) ( θ, α ( T ) θ, (cid:18) Θ˙Θ (cid:19) δ ( θ,ω ) ( τ ) (cid:35) ++∆ δ ( θ,ω ) (cid:16)(cid:16) Θ , ˙Θ (cid:17) ( T ( θ, α ( T ) θ, , θ, ω, , (cid:17) ◦◦ (cid:34)(cid:18) Θ˙Θ (cid:19) t ( τ ) T ( θ,ω ) ( θ, α ( T ) θ, (cid:18) Θ˙Θ (cid:19) ( θ,ω ) ( τ ) (cid:35) . The terms (cid:18)
Θ˙Θ (cid:19) tt ( t, θ, ω,
0) and (cid:18)
Θ˙Θ (cid:19) t ( θ,ω ) ( t, θ, ω,
0) come by taking the derivativesof (cid:18)
Θ˙Θ (cid:19) t ( t, θ, ω,
0) with respect to t and ( θ, ω ) . The formulas for T ( θ,ω ) ( θ, α ( T ) θ, T δ ( θ, ω,
0) were computed in Sections 6 and 7.1. To compute T δ ( θ,ω ) we justdifferentiate the formula for T δ ( θ, ω,
0) of Section 7.1 with respect to ( θ, ω ) obtaining T δ ( θ,ω ) ( θ, ω,
0) = (cid:18) − . . ω + 180 . ω θ + 173 . ωθ + 55 . θ ( ω + 0 . θ ) , − . . ω + 196 . ω θ + 189 . ωθ + 60 . θ ( ω + 0 . θ ) (cid:19) . By analogy with (7.4) we compute (cid:18)
Θ˙Θ (cid:19) δt ( T , θ, ω,
0) = (cid:18) ˙ h ( T )¨ h ( T ) (cid:19) == (cid:18) − . − . ω − . ω θ − . ωθ − . θ − . − . ω − . ω θ − . ωθ − . θ (cid:19) . It remains to find ∆ δ ( θ,ω ) ( θ, ω,
0) which computes as∆ δ ( θ,ω ) ( θ, ω,
0) = (cid:18) − θω − θ (cid:19) . IMIT CYCLES OF THE PLANAR PASSIVE BIPED WALKING DOWN A SLOPE P δ ( θ,ω ) ( θ, α ( T ) θ,
0) finally computesas P δ ( θ,ω ) ( θ, α ( T ) θ,
0) = 1 θ (cid:18) . − . θ . − . θ − . − . θ − . − . θ (cid:19) . To verify condition (5.9), itremains to compute the eigenvector y matrix (6.3) which corresponds to the eigenvalue1 and satisfies the normalization property (5.1) with the vector z of Section 7.2. Sucha computation leads to y = (15 . , − . T . Using the formula for P δ ( θ,ω ) ( θ , α ( T ) θ ,
0) of Section 7.3 and the value θ given bySection 7.2, we get z T P δ ( θ,ω ) ( θ , α ( T ) θ , y = − . , so that both the conditions (5.7) and (5.9) hold.
8. Conclusions.
In this paper we built upon the fundamental paper by Garciaet al [5] and then used the results by Kamenskii et al [8] and Makarenkov-Ortega [11]in order to offer a step-by-step guide as for how the classical perturbation theory needsto be applied in order to establish the existence and stability of a walking limit cyclein a model of passive biped by McGeer [13]. Since the dynamics of a passive walkerconstitutes an important building block of more complex robotics models (engineersuse the passive walker dynamics to diminish the energy required for locomotion), welike to think that the present work will stimulate the use of perturbation theory inthe field of robotics.
Appendix A. Derivation of the perturbation theorem of Section 5 fromthe results of Kamenskii et al [8] and Makarenkov-Ortega [11].
The following two results have been established in Kamenskii et al [8] and they willplay the central role in the perturbation theorem (Theorem 5.1) that this sectiondevelops. We now reformulate the required results of [8] in the notations of thepresent paper to avoid confusion.
Theorem
A.1. (two-dimensional version of a combination of [8, Theorem 1] and[8, Remark 2])
Consider a C -function ( θ, ω, δ ) (cid:55)→ F ( θ, ω, δ ) . Let Π : R → R be a linear projector invariant with respect to F ( θ,ω ) ( θ , ω , with F ( θ,ω ) ( θ , ω , invertible on ( I − Π) R . Assume that Π F δ ( θ , ω ,
0) = 0 , Π F ( θ,ω ) ( θ , ω , h Π h =0 for any h , h ∈ R , and that (A.1) − Π F ( θ,ω ) ( θ , ω , h + Π( F δ ) ( θ,ω ) ( θ , ω , , where h = ( I − Π) (cid:16) F ( θ,ω ) ( θ , ω , (cid:12)(cid:12) ( I − Π) R (cid:17) − F δ ( θ , ω , , is invertible on Π R . Then, there exists a unique ( θ , ω ) ∈ R such that, for all | δ | (cid:54) = 0 sufficiently small, one can find ( θ ,δ , ω ,δ ) ∈ R that satisfies both F ( θ + δθ ,δ , ω + δω ,δ , δ ) = 0 , and ( θ ,δ , ω ,δ ) → ( θ , ω ) as δ → . O. MAKARENKOV
Theorem
A.2. (two-dimensional version of [8, Theorem 2])
Assume all the con-ditions of Theorem A.1. Let ( θ ,δ , ω ,δ ) be as given by Theorem A.1. Denote by λ ∗ ∈ R the eigenvalue of the linear map (A.2) Π F ( θ ,ω ) ( θ , ω ) T (cid:12)(cid:12) Π R + Π( F δ ) ( θ,ω ) ( θ , ω , (cid:12)(cid:12) Π R . Then λ δ = δλ ∗ + o ( δ ) . In order to apply Theorem A.1 to the Poincare map ( θ, ω ) (cid:55)→ P ( θ, ω, δ ), we consider(A.3) F ( θ, ω, δ ) = P ( θ, ω, δ ) − ( θ, ω ) T , Π ζ = z T ζy, and notice that (5.4) implies(A.4) z T P ( θ,ω ) ( ξ ( s ) , y = 0 , for all s ∈ R , which allows (as we show in the proof of Theorem 5.1), to ignore all the expressionsof Theorem A.1 that involve the second derivative. Proof of Theorem 5.1.
The necessity part.
Here we follow the idea of Makarenkov-Ortega [11, Lemma 2]. Assume that P ( θ δ , ω δ , δ ) = ( θ δ , ω δ ) T , δ ∈ R , for some family { ( θ δ , ω δ ) } δ ∈ R satisfying (5.5). We claim that (5.6) holds.The derivative F (cid:48) ( θ, ω, δ ) of the C function (A.3) is a 2 × F (cid:48) ( ξ ( s ) ,
0) = 1 . Otherwise the equation F ( θ, ω, δ ) = 0 should describe a curvein a small neighborhood of ( ξ ( s ) , { ( θ, ω, δ ) : F ( θ, ω, δ ) = 0 } contains both the curve { ( ξ ( s ) , } s ∈ R and also the set { ( θ δ , ω δ , δ ) } δ ∈ R . Now we knowthat rank F (cid:48) ( ξ ( s ) ,
0) = 1 and it remains to prove that(A.5) rank F (cid:48) ( ξ ( s ) ,
0) = 2 , if z T F δ ( ξ ( s ) , (cid:54) = 0 . By Fredholm alternative for matrices (see e.g. [10, Theorem 4.5.3]),Im F ( θ,ω ) ( ξ ( s ) ,
0) = (cid:0)
Ker F ( θ,ω ) ( ξ ( s ) , T (cid:1) ⊥ Since Ker Φ ( θ,ω ) ( ξ ( s ) , T = span( z ), we conclude that (cid:0) Ker F ( θ,ω ) ( ξ ( s ) , T (cid:1) ⊥ =span(˜ y ), where ˜ y is an eigenvector of F ( θ,ω ) ( ξ ( s ) ,
0) that corresponds to the non-zero eigenvalue of F ( θ,ω ) ( ξ ( s ) , F ( θ,ω ) ( ξ ( s ) ,
0) = span(˜ y ). But z T F δ ( ξ ( s ) , (cid:54) = 0 implies, see formula (5.3), that the vectors ˜ y and F δ ( ξ ( s ) , The sufficiency part.
Here we use Theorem A.1. The projector Π defined by (A.3)is invariant with respect to F ( θ,ω ) ( θ , ω ,
0) and the projector I − Π is given by, seeformula (5.3), ( I − Π) ζ = ˜ z T ζ ˜ y, so that F ( θ,ω ) ( θ , ω ,
0) is invertible on ( I − Π) R . The requirement Π F δ ( θ , ω ,
0) = 0of Theorem A.1 holds by (5.6), and the requirement Π F ( θ,ω ) ( θ , ω , h Π h = 0holds by (A.4). The properties (A.3) and (A.4) imply that the expression (A.1) isinvertible on span( y ) if and only if (5.7) holds. Therefore, the conclusion of thetheorem follows by applying Theorem A.1. IMIT CYCLES OF THE PLANAR PASSIVE BIPED WALKING DOWN A SLOPE The stability part.
Assume that conditions (5.8) and (5.9) hold. Let ρ δ be the eigen-value of P ( θ,ω ) ( θ δ , ω δ , δ ) such that ρ δ → δ → . We have to show that | ρ δ | < | δ | > λ δ = ρ δ − F ( θ,ω ) ( θ δ , ω δ , δ ). As it was established in the sufficiency partof the proof, the expression (A.2) coincides with z T ( P δ ) ( θ,ω ) ( θ , ω , y . Therefore,condition (5.9) ensures that λ ∗ of Theorem A.2 verifies λ ∗ < λ δ < δ > Appendix B. The solution of equation (7.3).
The solution ( h ( t ) , f ( t )) ofequation (7.3) is given by h ( t ) = 1384 e − t [384 e t + ( ω − θ ) − e t ( ω + θ ) + e t {−
192 + 3 ω (3 + 4 t ) ++3 ω (1 − t ) θ − ω (7 + 4 t ) θ + (1 + 12 t ) θ } + e t ( −
192 + 3 ω ( − t ) −− ω ( − t ) θ + (1 − t ) θ + 3 ω ( θ + 4 tθ ))] ,f ( t ) = 17680 e − t [ − e t − e t − ω + 60 e t ω − e t ω + 56 e t ω ++120 e t ω t + 120 e t ω t + 168 ω θ + 60 e tω θ + 60 e t ω θ + 168 e t ω θ −− e t ω tθ + 120 e t ω tθ − ωθ − e t ωθ + 780 e t ωθ + 168 e t ωθ −− e t ωtθ − e t ωtθ + 56 θ + 580 e t θ + 580 e t θ + 56 e t θ ++120 e t tθ − e t tθ + 3 e t {− ω − θ )( ω − θ ) + 65 e t ( ω + θ ) ( ω + 3 θ ) ++ e t (1280 + 140 ω t − ω θ + 60 ωtθ − θ ) } cos t + 12 e t { ( − e t ) ω ++13(1 + e t ) ω θ + 3( − e t ) ωθ − e t ) θ } cos(2 t ) + 45 e t ω θ cos(3 t ) −− e t θ cos(3 t ) − e t ω sin t − e t ω sin t − e t ω sin t −− e t ω θ sin t + 195 e t ω θ sin t + 1260 e t ω tθ sin t + 975 e t ωθ sin t ++3813 e t ωθ sin t + 975 e t ωθ sin t − e t θ sin t + 585 e t θ sin t ++540 e t tθ sin t − e t ω sin(2 t ) − e t ω sin(2 t ) − e t ω θ sin(2 t ) ++48 e t ω θ sin(2 t ) + 288 e t ωθ sin(2 t ) + 288 e t ωθ sin(2 t ) − e t θ sin(2 t ) ++216 e t θ sin(2 t ) − e t ω sin(3 t ) + 135 e t ωθ sin(3 t )] . Compliance with Ethical Standards. Conflict of Interest:
The authorshave no conflict of interest.
REFERENCES[1] A. Buica, J.-P. Francoise, J. Llibre, Periodic solutions of nonlinear periodic differential sys-tems, Comm. Pure Appl. Anal. 6 (2007) 103–111.[2] A. Buica, J. Llibre, O. Makarenkov, Asymptotic stability of periodic solutions for nonsmoothdifferential equations with application to the nonsmooth van der Pol oscillator, SIAM J.Math. Anal. 40 (2009) 2478–2495.2