On the hybrid inverted pendulum dynamics and its relation to the lateral stability of the walking-like mechanical systems⁎

Proceedings, 2nd IFAC Conference on Proceedings, 2nd IFAC Conference on Modelling, Identification and Controlon of Nonlinear Systems Proceedings, 2nd Proceedings, 2nd IFAC IFAC Conference Conference Modelling, Identification and Controlon of Nonlinear Systems Available online at www.sciencedirect.com Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification Identification and Control of Nonlinear Systems Proceedings, 2nd IFAC Conference on Modelling, and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018

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IFAC PapersOnLine 51-13 (2018) 496–501

On On the the hybrid hybrid inverted inverted pendulum pendulum dynamics dynamics On the hybrid inverted pendulum dynamics and its relation to the lateral stability of On the hybrid inverted pendulum dynamics and its relation to the lateral stability of and its relation to the lateral stability of the walking-like mechanical systems and relation to mechanical the lateral systems stabilityof the walking-like theits walking-like mechanical systems the walking-like mechanical systems  ˇ Sergej Celikovsk´ y ∗ Volodymyr Lynnyk ∗

∗ ˇ ∗ Volodymyr Lynnyk ∗ ∗ Sergej Celikovsk´ y ∗ ∗ ˇ Sergej y ˇ Sergej Celikovsk´ Celikovsk´ y ∗∗ Volodymyr Volodymyr Lynnyk Lynnyk ∗∗ ˇ ∗ y Institute Volodymyr Lynnyk Theory and Czech Sergej AcademyCelikovsk´ of Sciences, of Information ∗ ∗ The ∗ The Czech Academy of Sciences, Institute of Information Theory and ∗ The Czech Czech Academy(UTIA), of Sciences, Sciences, Institute of8,Information Information Theory and and Automation 182 08 Prague Czech Republic The Academy of Institute of Theory Automation (UTIA), 182 08 Prague Czech Republic ∗ The(e-mails: Czech Academy of Sciences, Institute of8, Information Theory and Automation (UTIA), 182 08 Prague 8, Czech Republic [email protected], [email protected]) Automation (UTIA), 182 08 Prague 8, Czech Republic (e-mails: [email protected], [email protected]) Automation (UTIA), 182 08 Prague 8, Czech Republic (e-mails: [email protected], [email protected]) (e-mails: [email protected], [email protected]) (e-mails: [email protected], [email protected]) Abstract: A simple hybrid mechanical system, tentatively called as the hybrid inverted Abstract: A simple hybrid mechanical system, tentatively called as the hybrid inverted Abstract: A simple hybrid mechanical system, tentatively called as the hybrid inverted pendulum, is studied. This study is motivated by the problem of the lateral stability of the planar Abstract: A simple hybrid mechanical system, tentatively called as the hybrid inverted pendulum, is studied. This study is motivated by the problem of the lateral stability of the planar Abstract: A simple hybrid mechanical system, tentatively called as the hybrid inverted pendulum, is studied. This study is motivated by the problem of the lateral stability of the planar walking strategies applied to a more realistic settings. Using numerical simulations, the influence pendulum, is studied. Thisto study is motivated by the problem of the lateral stability the of the planar walking strategies applied a more realistic settings. Using numerical simulations, influence pendulum, is studied. This study is motivated by the problem of the lateral stability of the planar walking strategies applied to a a more more realistic realistic settings. Using numerical simulations, the influence influence of the lateral harmonic perturbations is studied as well. Some complex features indicating the walking strategies applied to settings. Using numerical simulations, the of the lateral harmonic perturbations is studied as well. Some complex features indicating the walking strategies applied to a more realistic settings. Using numerical simulations, the influence of the lateral harmonic perturbations is studied as well. Some complex features indicating the possible chaotic behavior of its forced but bounded dynamics is demonstrated as well. of the lateral harmonic perturbations is studied as well. Some complex features indicating the possible chaotic behavior of its forced but bounded dynamics is demonstrated as well. of the lateral harmonic perturbations is studied as well. Some complex features indicating the possible chaotic behavior of its forced but bounded dynamics is demonstrated as well. possible chaotic behavior of its forced but bounded dynamics is demonstrated as well. © 2018, IFAC (International Federation of but Automatic Control) Hostingisby Elsevier Ltd. All rights reserved. possible chaotic behavior of its forced bounded dynamics demonstrated as well. Keywords: Hybrid systems, systems, walking walking robots, robots, hybrid hybrid inverted inverted pendulum, pendulum, chaos. chaos. Keywords: Hybrid Keywords: Keywords: Hybrid Hybrid systems, systems, walking walking robots, robots, hybrid hybrid inverted inverted pendulum, pendulum, chaos. chaos. Keywords: Hybrid systems, walking robots, hybrid inverted pendulum, chaos. 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION 1. INTRODUCTION The aim of this paper is to study the stability of simThe aim aim of of this this paper paper is is to to study study the the stability stability of of aaa simsimThe ple pendulum-like hybrid mechanical system withofimpacts The aim of this paper is to study the stability a simple pendulum-like hybrid mechanical system with impacts The aim of thistopaper is mechanical to study the stability ofimpacts a simple pendulum-like hybrid system with that is related the problem of the lateral stability of ple pendulum-like hybrid mechanical system with impacts that is related to the problem of the lateral stability of ple pendulum-like hybrid mechanical system with impacts that is related to the problem of the lateral stability of the walking robots. For the last decades, the walking that walking is related to theFor problem of the lateralthe stability of the robots. the last last decades, walking that walking is has related tointensively theFor problem of the lateral stability of the robots. the decades, the walking robots been studied with a great great diversity the walking robots. For the last decades, the walking robots has been intensively studied with a diversity the walking robots. Forappealing the lastsubarea decades, the so-called walking robots has been studied with aa is great diversity of approaches. Itsintensively most the robots has been intensively studied with great diversity of approaches. most appealing subarea the so-called robots has beenIts intensively studied withata is great diversity of Its most appealing subarea is the so-called underactuated walking when the angle the pivot supof approaches. approaches. Its most when appealing subarea is thepivot so-called underactuated walking the angle at the supof approaches. Its most appealing subarea is the so-called underactuated walking when the angle at the pivot supporting leg is unactuated and walking-like models have inunderactuated walking when the angle atmodels the pivot porting leg is is unactuated unactuated and walking-like walking-like havesupinunderactuated walkingofwhen thethan anglethe atmodels the pivot supporting leg and have inevitable more degrees freedom actuators. Such porting leg is unactuated and walking-like models have inevitable more degrees of freedom than the actuators. Such leg is unactuated and walking-like models have inevitable more degrees of freedom than the actuators. Such aporting framework represents more adequately the dynamical evitable more degrees of freedom than the actuators. Such a framework represents more adequately the dynamical evitable more degrees of freedom than the actuators. Such anature framework represents more adequately the dynamical of the human gait and it is also theoretically more a framework represents more adequately the dynamical nature of thethan human gait and it is also theoretically more a framework represents the dynamical nature of human gait and it is theoretically more challenging the fully actuated walking studies. nature of the thethan human gaitmore and itadequately is also also theoretically more challenging the fully actuated walking studies. nature of the human gait and it is also theoretically more challenging than the fully actuated walking studies. challenging than the fully actuated walking studies. To simplify the design complexity, the corresponding studchallenging than the fully actuated walking studies. To simplify the design complexity, the corresponding studTo simplify the complexity, the corresponding studies the so-called planar walking models where To started simplifywith the design design complexity, the corresponding studies started with the so-called planar walking models where To simplify the design complexity, the corresponding studies started with the so-called planar walking models where only the sagital plane dynamics is modeled, assuming that ies started with the so-called planar walking models where only the sagital plane dynamics is modeled, assuming that ies started with the so-called planar walking models where only the sagital plane dynamics is modeled, assuming that the lateral stability is dynamics ensured by some ad hoc supporting only the sagital plane is modeled, assuming that the lateral stability is ensured by ad hoc supporting only therod, sagital plane dynamics is some modeled, assuming the lateral stability is ensured by some ad hoc supporting frame, moving platform, etc., as in the case ofthat the the lateral stability is ensured by some ad hoc supporting frame, rod, moving etc., as in case of the the lateral stability isplatform, ensured by some adetthe hoc supporting frame, rod, moving platform, etc., as in the case of the well-known “rabbit” walker Chevallereau al. (2003). For frame, rod, moving platform, etc., as in the case of the Fig. 1. The experimental robot in UTIA laboratory. well-known “rabbit” walker Chevallereau Chevallereau etthe al. case (2003). For frame, rod, “rabbit” moving platform, etc., as picture, inet of For the Fig. 1. The experimental robot in UTIA laboratory. well-known walker al. (2003). some representative, yet notChevallereau complete, see e.g. well-known “rabbit” walker et al. (2003). For Fig. 1. 1. The The experimental experimental robot robot in in UTIA UTIA laboratory. laboratory. some representative, yet not complete, picture, see e.g. the well-known “rabbit” walker Chevallereau et al. (2003). For some representative, yet not complete, picture, see e.g. the monographs Westervelt et al. (2007); Chevallereau et al. Fig. some representative, yet not complete, picture, see e.g. the Fig. 1. The experimental robot in UTIA laboratory. monographs Westervelt et al. (2007); Chevallereau et al. ˇ ˇ some representative, yetintroductory not complete, picture, see e.g. monographs Westervelt et (2007); Chevallereau et al. Celikovsk´ y and Anderle (2016), Celikovsk´ y and (2009), or the general of more recent monographs Westervelt et al. al. (2007);part Chevallereau et the al. (2017), ˇˇ ˇˇ (2017), Celikovsk´ y and Anderle (2016), Celikovsk´ y and (2009), or the general introductory part of more recent (2017), Celikovsk´ y and Anderle (2016), Celikovsk´ y and ˇ(2017). y and Anderle (2016), Celikovsk´ ˇ monographs Westervelt et al.and (2007); Chevallereau et al. (2017), (2009), or the general introductory part of more recent Anderle paper Grizzle et al. (2014) references within there. Celikovsk´ y and (2009), or the general introductory part of more recent Anderle (2017). paper Grizzle et al. (2014) and references within there. ˇ ˇ (2017), Celikovsk´ y and Anderle (2016), Celikovsk´ y and Anderle (2017). (2009), or the general introductory part of more recent paper Grizzle et al. (2014) and references within there. Motivation for et such an approach is the fact that handling Anderle (2017). paper Grizzle al. (2014) and references within there. Nevertheless, for future fully autonomous walking robots Motivation for such an approach is the fact that handling Anderle (2017). paper Grizzle et al. (2014) and references within there. Motivation for such an approach is the fact that handling Nevertheless, for future fully autonomous walking robots the forward instability inside the sagittal plane via the Motivation for such an approach is sagittal the fact plane that handling Nevertheless, for fully autonomous walking three dimensional models be considered. These the forward instability inside the via the Nevertheless, for future future fullyshould autonomous walking robots robots Motivation for such and an approach is sagittal the fact thatswing handling the instability inside the plane via three dimensional models should be considered. These forward movement proper timing of the leg the forward forward instability inside the sagittal plane via the the Nevertheless, for future fully autonomous walking robots three dimensional models should be considered. These forward movement and proper timing of the swing leg models constitute a very sharp increase of complexity with three dimensional models should be considered. These the forward instability inside the sagittal plane via the forward movement and proper timing of the swing leg models constitute a very sharp increase of complexity with hitting the ground seems to be a cornerstone peculiarity forward movement and proper timing of the swing leg three dimensional models should be considered. These models constitute a very sharp increase of complexity with hitting the ground seems to be a cornerstone peculiarity respect to the planar case requiring, in particular, proper models constitute a very sharp increase of complexity with forward movement and proper of swing leg respect hitting the ground seems to be aatiming cornerstone to the planar case requiring, in particular, proper of the dynamic underactuated walking. Onthe thepeculiarity contrary, hitting the ground seems to be cornerstone peculiarity models aangles, very sharp increase complexity with respect constitute to of theEuler planar case requiring, in of particular, proper definition tensors of inertia, etc., for each of the dynamic underactuated walking. On the contrary, respect to the planar case requiring, in particular, proper hitting the ground seems to be a cornerstone peculiarity of the dynamic underactuated walking. On the contrary, definition of Euler angles, tensors of inertia, etc., for each the lateral stability inside the frontal plane seems to be a of the dynamic underactuated walking. On seems the contrary, respect toSong theEuler planar case (2006b) requiring, ina particular, proper definition of angles, tensors of inertia, etc., for each link, see and Zefran for brief introduction the lateral stability inside the frontal plane to be a definition of Euler angles, tensors of inertia, etc., for each of the dynamic underactuated walking. On the contrary, the lateral stability inside the frontal plane seems to be a link, see Song and Zefran (2006b) for a brief introduction somehow simpler phenomenon. Some samples of results on the lateralsimpler stability inside the frontal plane seems to beona or definition of Euler angles, tensors of inertia, etc., for each link, see Song and Zefran (2006b) for a brief introduction somehow phenomenon. Some samples of results Grizzle et al. (2014) for a more comprehensive expolink, see Song and Zefran (2006b) for a brief introduction the lateral stability inside the frontal plane seems to beon somehow simpler phenomenon. Some samples results Grizzle et al. (2014) for aa more comprehensive expoplanar underactuated walking are Shiriaev etof (2014), somehow simpler phenomenon. Some samples ofal. results ona or link, see Song and Zefran (2006b) for a brief introduction or Grizzle et al. (2014) for more comprehensive exposition. Alternative, and somehow simpler, treatment to planar underactuated walking are Shiriaev et al. (2014), Grizzle et al. (2014) for a moresimpler, comprehensive exposomehow simpler phenomenon. samples results on or planar walking are et (2014), sition. Alternative, and somehow treatment to Shiriaev et al. (2006), ShiriaevSome et Shiriaev al. (2005), Song and planar underactuated underactuated walking are Shiriaev etofal. al. (2014), or Grizzle et al. (2014) for a more comprehensive exposition. Alternative, and somehow simpler, treatment to study possibilities of the lateral in stabilization was given Shiriaev et al. (2006), Shiriaev et al. (2005), Song and sition. Alternative, and somehow simpler, treatment to planar underactuated walking are Shiriaev et al. (2014), Shiriaev et al. (2006), Shiriaev et al. (2005), Song and study possibilities of the lateral in stabilization was given Zefran (2006a,b); Majumdar et al. (2013), Pchelkin et al. Shiriaev(2006a,b); et al. (2006), Shiriaev et (2013), al. (2005), Songetand sition. Alternative,of and somehow simpler, treatment to study possibilities the lateral in stabilization was given Kuo (1999). Zefran Majumdar et al. Pchelkin al. study possibilities of the lateral in stabilization was given Shiriaev et al. and (2006), Shiriaev etLa al.Herra (2005), Song Zefran Majumdar et (2013), Pchelkin et al. (1999). (2015), Spong Bullo (2005), et al. (2013) Zefran (2006a,b); (2006a,b); Majumdar et al. al. (2013), Pchelkin etand al. Kuo study possibilities of the lateral in stabilization was given Kuo (1999). (2015), Spong and Bullo (2005), La Herra et al. (2013) Kuo (1999). paper attempts to simplify the latter treatZefran (2006a,b); et al. (2013), Pchelkin et al. ˇand ˇal. (2015), Spong Bullo (2005), La Herra et The current Dolinsk´ y and Celikovsk´ y (2012), Dolinsk´ y and y (2015), Spong andMajumdar Bullo (2005), La Herra et Celikovsk´ al. (2013) (2013) Kuo (1999). paper attempts to simplify the latter treatThe current ˇ ˇ Dolinsk´ y and Celikovsk´ y (2012), Dolinsk´ y and Celikovsk´ y ˇ ˇ (2015), Spong and Bullo (2005), La Herra et al. (2013) The current paper attempts to simplify the treatmentcurrent even further studies possibility to use the Dolinsk´ y y y y ˇ ˇ paper and attempts to the simplify the latter latter treatDolinsk´ y and and Celikovsk´ Celikovsk´ y (2012), (2012), Dolinsk´ Dolinsk´ y and and Celikovsk´ Celikovsk´ y The ment even further and studies the possibility to use the ˇ ˇ  The current paper attempts to simplify the latter treatment even further and studies the possibility to use the Dolinsk´ y and Celikovsk´ y (2012), Dolinsk´ y and Celikovsk´ y planar models and planar walking strategies without any ment even further and studies the possibility to use the  Supported by the Czech Science Foundation through the the planar models and planar walking strategies without any Supported by the Czech Science Foundation through the the  ment even further and studies the possibility to use the planar models and planar walking strategies without any lateral support or actuator action. The idea is that due to research grant No. 17-04682S. Supported by the Czech Science Foundation through the the  planar models and planar walking strategies without any Supported by the Czech Science Foundation through the the lateral support or actuator action. The idea is that due to research grant No. 17-04682S.  planar models and planar walking strategies lateral support support or actuator actuator action. The The idea is iswithout that due dueany to research grantby No.the 17-04682S. Supported Czech Science Foundation through the the lateral or action. idea that to research grant No. 17-04682S. lateral support or actuator action. The idea is that due to research grant No. IFAC 17-04682S. 2405-8963 © 2018, (International Proceedings, 2nd IFAC Conference onFederation of Automatic Control) 496 Hosting by Elsevier Ltd. All rights reserved.

Proceedings, 2nd IFAC Conference on 496 Control. Peer reviewIdentification under responsibility of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd on 496 Proceedings, 2nd IFAC IFAC Conference Conference 496 Modelling, Identification and Controlon of Nonlinear 10.1016/j.ifacol.2018.07.328 Systems Modelling, Identification and Control of Nonlinear Proceedings, 2nd IFAC Conference on 496 Modelling, Systems Identification and Control of Nonlinear Guadalajara, Mexico, June 20-22, 2018 Systems Identification Modelling, and Control of Nonlinear

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Fig. 2. Hybrid inverted pendulum: q1 = 0.

Fig. 3. Hybrid inverted pendulum: q1 < 0.

the distance between hips the pair of legs can act in its lateral direction as a simple pendulum-like system with impact having potential of stable behavior even in the presence of some limited lateral periodic perturbations emulating the influence of the remaining robot dynamics. To illustrate the situation, the picture of the laboratory model of pair of legs with knees is shown in Fig. 1. This model is used to study underactuated planar walking being supported laterally by a special frame, but one may consider idea of its lateral stability even without that frame. Simplified model of its frontal projection is shown in Figs. 2-4. Such a system is tentatively called as the hybrid inverted pendulum and it will be seen that its stability is intrinsically owed to the hybrid and impact effects. It is worth to mention in this respect the passive walkers walking down the moderate slope, where impact effects are also crucial, broadly studied and referred since the seminal paper McGeer (1990), see also Freidovich et al. (2009).

hybrid dynamical model is needed which justifies the term “HIP”. Using the standard Euler-Lagrange formalism 2 2 + lL + d2 )¨ q1 − mg(lR + lL ) sin(q1 ) m(lR (3) q1 = 0, − mg sign(q1 ) cos(q1 ) = F (t), where F (t) stands for possible external generalized force, i.e. the torque. Straightforward computations give  2 2 + lL + d2 )−1 g(lR + lL ) sin(q1 ) q¨1 = (lR  (4) − gd sign(q1 ) cos(q1 ) + m−1 F (t) , q1 = 0. Note, that (4) is not well defined for q1 = 0. For this case, some switching conditions are needed. They can be obtained analyzing the so-called impact, i.e. the effect of one of the vertical links hitting the ground for q1 = 0. The standard assumptions made for the impact are: (i) no slipping; (ii) the impact is instantaneous; (iii) both total momentum and total energy are preserved. As the HIP posses a single degree of freedom only, the switching conditions are simple: both q1 and q˙1 should stay continuous when switching between q1 > 0 and q1 < 0 in (4). All previous considerations are summarized as follows.

The rest of the paper is organized as follows. Next section describes in detail the hybrid inverted pendulum and provides its mathematical model. Some theoretical properties are formulated and proved in Section 3 while Section 4 collects numeric simulations showing, in particular, bounded, yet complex, dynamics of the perturbed hybrid inverted pendulum. Some conclusions and outlooks are given in the final section. Notation. Euclidean norm in Rn is denoted as  · , Bc,r := {x ∈ Rn | x − c < r}, sign(a) = a/|a|, a = 0, g = 9.82 is the gravity constant. 2. HYBRID INVERTED PENDULUM Hybrid inverted pendulum (HIP) is shown in Figs. 2-4. All links are assumed to be massless with two equal masses m placed at different heights lR , lL on the vertical links having equal length l, the horizontal link has the lenght d and the perpendicular angles between links are rigidly fixed. HIP has the only degree of freedom being the angular displacement q1 shown in Figs. 3, 4, while q˙1 stands for the corresponding angular velocity. To obtain the HIP dynamical model, consider Lagrangian 1 2 2 L = K − V, K = mq˙12 (lL + lR + d2 ), (1) 2 V = mg(lR cos q1 + lL cos q1 + d| sin q1 |). (2)

where V is the potential energies with respect to the ground level and K is the kinetic energy. Note, that Lagrangian (1) is not smooth for q1 = 0 and therefore the

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Fig. 4. Hybrid inverted pendulum: q1 > 0. Definition 2.1. The system (4) is called as the hybrid inverted pendulum (HIP). Its trajectory is defined as any everywhere continuously differentiable function q1 (t) satisfying (4) for q1 = 0. If F (t) ≡ 0 (F (t) = 0), (4) is called as the unforced (forced) HIP. Later on, system (4) will be represented in the standard way as the system of the second order differential equations with the discontinuous right hand side and treated as such, cf. Filippov (1988). Before doing so, let us present the following equivalent setting showing that system is actually the hybrid one:

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q¨ =

 2 2 (lR + lL + d2 )−1 g(lR + lL ) sin(q)  − gd cos(q) + m−1 F (t) ,

q > 0;

(5)

q = 0. (6) q + = −q − , q˙+ = −q˙− , In this hybrid setting, q is always positive during the continuous-time part (5), namely, q := q1 in Fig. 3 and q := −q1 in Fig. 4. Discrete-time part (6) then adequately relabels q and reverses for q = 0 the velocity, keeping thereby the set q ≥ 0 invariant with respect to (5-6). E

There are two equilibria q1 1,2 of the unforced HIP (4) in the regions (−π/2, 0) and (0, π/2) indicated by Figs. 3-4:   E q1 1,2 = ± arctan d(lR + lL )−1 . (7) These equilibria corresponds to unstable saddles. By a physics intuition, there is another stable equilibrium of the unforced HIP at q1 = 0 which should be understood in a suitable defined hybrid sense. 3. THEORETICAL ANALYSIS Consider the HIP model introduced in the previous section and introduce its standard state space description using the first order differential equations. To do so, denote x1 = q1 , x2 = x˙ 1 = q˙1 , and assume that the external torque has the form F (t) = −k q˙1 + w(t), where the first term is some friction-like damping while the second one is a bounded external perturbation. Furthermore, denote 2 2 2 2 ω = w/(lR + d2 )m, k1 = k/(lR + lL + d2 )m, + lL then the state space equations can be written as x˙ 1 = x2 ,   g (lR + lL ) sin(x1 ) − d sign(x1 ) cos(x1 ) x˙ 2 = (8) 2 + l 2 + d2 lR L − k1 x2 + ω(t), x(t) := (x2 (t), x2 (t)) .

The equations (8) will be called in the sequel HIP in the state space form (HIPSF). They form the system of ordinary differential equations with the discontinuous dependence of their right hand side with respect to x. Its hybrid reformulation analogous to that of (5-6) is straightforward. Nevertheless, (8) is the well established mathematical concept, cf. Filippov (1988), and it will be used in the sequel. The standard equilibria of (8) are the following unstable hyperbolic points   −1 1 1 , xE xE 1 = arctan d(lR + lL ) 2 = 0, (9)   E2 −1 2 1 = −xE xE 1 = arctan − d(lR + lL ) 1 , x2 = 0. E2 1 The following straightforward relations for xE 1 , x1 given in (9) will be usefull later on: E

sin2 x1 1,2 = d2 ((lR + lL )2 + d2 )−1 , E

cos2 x1 1,2 = (lR + lL )2 ((lR + lL )2 + d2 )−1 .

(10)

Remark 3.1. The mapping x(t) ≡ 0 is the uniquely existing solution in the Fillippov sense Filippov (1988) of (8) for the initial conditions (0, 0) . The origin (0, 0) thereby constitutes yet another equilibrium. Despite its clear meaning (HIP at double support with zero velocity), it can not be directly obtained from equations (8) and one may wish to call it the equilibrium in the Fillipov sense. Definition 3.2. Let T > 0 be given. The mapping [0, T ] → R2 x(t) = (x1 (t), x2 ) (t), t ∈ [0, T ], T > 0, is called the 498

trajectory of (8) on [0, T ] if x1 (t) is differentiable function on [0, T ], x2 (t) is continuous piecewise differentiable function on [0, T ] and (8) for x1 (t) = 0 and t ∈ [0, T ]. HIPSF is called unforced for ω(t) ≡ 0, undamped for k1 = 0, damped for k1 > 0 and forced for ω(t) = 0. The solution of (8) on [0, T ] with the initial condition (x01 , x02 ) is the trajectory (x1 (t), x2 (t)) , t ∈ [0, T ], such that x1 (0) = x01 , x2 (0) = x02 . The solution is called maximal, if it is solution on [0, T ] for all T > 0. Lemma 3.3. Let be given: ε > 0; (x01 )2 + (x02 )2 > ε and a piece-wise continuous and globally bounded ω(t). Then there either exits a unique solution of the HIPSF (8) x1 (t), x2 (t), t ≥ 0, x1 (0) = x01 , x2 (0) = x02 . Moreover, this solution is either maximal, or there is a finite time tf , such that it exists on [0, tf ] and (x1 (tf ), x2 (tf )) ∈ B0,ε . Proof Sketch: First, the right hand side of (8) is obviously Lipschitz separately both on {x ∈ R2 |x1 > 0} and {x ∈ R2 |x1 < 0}, moreover, the appropriate Lipschitz constants are fixed. Secondly, due to the equation x˙ 1 = x2 , for any point (0, xc2 ) ∈ R, xc2 = 0, and any time moment tc > 0 there exists δ > 0 and a unique trajectory on [tc , tc +δ], x ˜(t), such that x ˜(tc ) = (0, xc2 ) and sign(˜ x1 (tc + c δ)) = sign(x2 ). Thirdly, the second equation of (8) gives straightforwardly for any trajectory x(t) that: x2 > 0 : x2 (t) ≥ (x2 (0) + Kk2−1 ) exp(−k2 t) − Kk2−1 , x2 < 0 : x2 (t) ≤ (x2 (0) − Kk2−1 ) exp(−k2 t) + Kk2−1 ,

2 2 + lL + d2 )−1 + ω, K = g(lR + lL + d)(lR where ω stands for upper bound of |ω(t)| existing by the assumption of this proposition. These inequalities simply mean that trajectory x2 (t) starting at some x2 (0) > 0 satisfy x2 (t) ≥ 0 for all t ∈ [0, tg ], where

tg = −k2−1 log(Kk2−1 (x2 (0) + Kk2−1 )−1 ). Analogously, trajectory x2 (t) starting at some x2 (0) < 0 satisfy x2 (t) ≤ 0 for all t ∈ [0, tg ], where

tg = −k2−1 log(−Kk2−1 (x2 (0) − Kk2−1 )−1 ). Now, assume that x(t) stays outside B0,ε , then either x2 > ε or x2 < −ε and therefore the minimal time between crossings the line x1 = 0 should be tmin = −k2−1 log(Kk2−1 (ε + Kk2−1 )−1 ). The rest of the proof then follows easily.  0 0 Proposition 3.4. Let  k1 ≥ 0, ω ≡ 0 and let x1 , x2 , be such that V (x01 , x02 ) < g (lL + lR )2 + d2 , where   V (x1 , x2 ) = g (lL + lR ) cos x1 + d| sin x1 | (11) 2 2 + x22 (lL + lR + d2 )/2. Then there exists a unique solution of (8) x(t), t ∈ [0, ∞), x1 (0) = x01 , x2 (0) = x02 such that ∀ε > 0 ∃δ(ε) > 0 such that (x01 , x02 ) ∈ B0,δ(ε) ⇒ x(t) ∈ B0,ε , ∀t ≥ 0. Finally, if k1 > 0 then x(t) → 0 as t → ∞. Proof. The function V (x) given by (11) is obviously everywhere continuous, positive definite with V (0) = 0 and continuously differentiable ∀x such that x1 = 0. Moreover, its gradient is bounded on R\{x ∈ R : x1 = 0}. The  level sets V (x) = c form the closed curves for all c ≤ (lL + lR )2 + d2 . Indeed, straightforward  computations show that for all x, such that V (x) < (lL + lR )2 + d2 , it holds that Vx1 x1 + Vx2 x2 > 0. The latter implies that

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x1

x1

Fig. 5. Separatrices (14,13) for d = 0.5, lL = lR = 1.  the level curves V (x) = c, c ≤ (lL + lR )2 + d2 , should be closed. Straightforward computations show that dV (x(t)) = −k1 x22 + ω(t)x2 . (12) dt Therefore, for the unperturbed undamped case the function V is constant along trajectories. As a consequence, trajectories belong to level sets and therefore are both bounded and stay away from the origin. In such a way, Lemma 3.3 is applicable and the solution exists for all time moments. The second claim of the proposition follows as well by the fact that trajectories belong to level curves of V , so one can easily show how to choose the required δ(ε) > 0 for a given ε > 0. To prove the last claim, assume its contrary: there exists  > 0 and trajectory x(t) which stays outside the neighborhood of the origin. By Lemma 3.3 then trajectory x(t) exists for all t ≥ 0, by (12) along x(t) it holds V˙ < −k1 x22 and the set {x2 = 0} is not invariant outside that neighborhood of the origin. The function V (x(t)) is strictly decreasing as its time derivative is everywhere continuous and negative except isolate points. This contradicts to x(t) staying outside -neighborhood of the origin and therefore the last claim of the proposition holds as well. . The function V in (11) can be also used to compute separatrices forming the boundary of stable region of unperturbed undamped hybrid inverted pendulum. Indeed, these separatrices clearly form a level set V (x) = csep = E2 1 V (xE 1 ) = V (x1 ). One has straightforwardly by (10)  E E  csep = g d sin x1 1,2 + (lL + lR ) cos x1 1,2 =  (13) d2 + (lL + lR )2 = g (lL + lR )2 + d2 , = g (lL + lR )2 + d2 and the separatrices are given by the following relations:  2csep − 2g(d| sin x1 | + (lL + lR ) cos x1 ) , (14) x2 = ± 2 + l 2 + d2 lL R E1 2 where x1 ∈ [xE 1 , x1 ], cf. Fig. 5.

Fig. 6. State portrait of (8) with k1 = 0, ω(t) = 0. hybrid inverted pendulum is strictly passive for k1 > 0 and it is passive for k1 = 0. In all these cases the input is ω, output is x2 and the storage function is V. First, Fig. 6 shows the phase portrait of the undamped unperturbed hybrid inverted pendulum. As expected, it is periodic-like with frequency changing dependently on the amplitude due to its nonlinear nature, see Tab. 1. One Table 1. Dependence of the frequency f in Hz on the initial condition (x1 , 0) . x1 f

0.24 0.17

0.2 0.32

0.15 0.47

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0.01 2.57

0.005 3.67

10−3 8.25

can observe from Table 1 that frequency goes down with increasing amplitude. This gives interesting clue to study the influence of harmonic disturbance, namely, the disturbance may be in the “bad”, i.e. the destabilizing, phase synchronization for some fixed amplitude only, increasing amplitude by destabilization changes own hybrid pendulum frequency and stabilizing effect may occur. In such a way, even bounded chaotic behavior may be expected. The effect of friction is expected to be negligible for its reasonable values, as shown in Fig. 7. In the sequel, we therefore consider the external perturbation influence only for the case k1 = 0. To start with the above plan, Figs. 8, 9 show that bounded oscillations close to separatrices can be easily destabilized even by small external perturbation. Next, Fig. 10 shows chaotic-like series of behaviors when external perturbation frequency is very slightly changed, other parameters remain unchanged. It is possible to observe both period doubling effect and complex, chaos like behavior. Finally, Fig. 11 illustrates irregular character of bounded behavior showing both full trajectory on a very long time interval and its 3 time subsegments. 5. CONCLUSIONS

4. NUMERICAL SIMULATION In this section, some of the previous results are illustrated by simulations, nevertheless, more importantly, the influence of the external forcing perturbation ω(t) will be systematically studied. As it can be seen by (12), this influence is hard to be estimated theoretically. Obviously, when x2 (t)ω(t) > 0, that influence is destabilizing, when x2 (t)ω(t) < 0, it is stabilizing. As a matter of fact, the 499

The properties of the hybrid inverted pendulum for the unperturbed case has been studied theoretically. Moreover, it has been shown numerically that its behavior remains inside the desired region even under lateral periodic forcing perturbations. This motivates the possible use of the planar walking strategies, as the influence of the forward walking robot motion on its lateral dynamics may have periodic character. Moreover, some external frequencies

2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018 Sergej Čelikovský et al. / IFAC PapersOnLine 51-13 (2018) 496–501 500

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1

Fig. 10. Complex behaviour of (8) for x1 (0) = 0.1, x2 (0) = 0, k1 = 0, t ∈ [0, 100]. From top to bottom: ω(t) = 0.2 sin 1.93t, ω(t) = 0.2 sin 2t, ω(t) = 0.2 sin 2.01t, ω(t) = 0.2 sin 2.03t.

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impose interesting complex, possibly chaotic behavior of the hybrid inverted pendulum model. Study of the adequate chaos indicators of the observed behavior is a subject of the ongoing and future research.

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REFERENCES

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ˇ S. Celikovsk´ y and M. Anderle. Hybrid invariance of the collocated virtual holonomic constraints and its application in underactuated walking. In Preprints of the 10th IFAC Symposium on Nonlinear Control Systems, pages 802–807, Monterey, California, USA, 2016.

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1

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