Stable walking gaits for a three-link planar biped robot with two actuators based on the collocated virtual holonomic constraints and the cyclic unactuated variable ⁎

12th IFAC Symposium on Robot Control 12th IFAC Symposium on Robot Control Budapest, Hungary, August 27-30, 2018 12th on Control 12th IFAC IFAC Symposium Symposium on Robot Robot Control Budapest, Hungary, August 27-30, 2018 Available online at www.sciencedirect.com Budapest, Hungary, August August 27-30, 2018 12th IFAC Symposium on Robot Control Budapest, Hungary, 27-30, 2018 Budapest, Hungary, August 27-30, 2018

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IFAC PapersOnLine 51-22 (2018) 378–385

Stable walking gaits for a three-link planar Stable Stable walking walking gaits gaits for for a a three-link three-link planar planar biped robot with two actuators based on Stable walking gaits for a three-link planar biped robot with two actuators based on biped robot with two actuators based on the collocated virtual holonomic constraints biped robot with two actuators based on the collocated virtual holonomic constraints the collocated virtual holonomic constraints  and the cyclic unactuated variable the and collocated virtual holonomic constraints the unactuated variable  and the cyclic cyclic unactuated variable and the cyclic unactuated variable  ˇ Sergej Celikovsk´ y and Milan Anderle ∗∗ ˇ

ˇ Sergej Celikovsk´ y and Milan Anderle ∗∗ ˇ Sergej y ˇ Sergej Celikovsk´ Celikovsk´ y and and Milan Milan Anderle Anderle ∗ ˇ ∗ Sergej Celikovsk´ y and Milan Anderle and ∗ UTIA - Institute of Information Theory UTIA - Institute of Information Theory and Automation, Automation, Czech Czech ∗ ∗ UTIA of Institute of Information Theory and Automation, Czech Academy Sciences, Pod Vod´ a renskou vˇ e ˇ z ´ ı 4, 182 08 Czech UTIA Institute of Information Theory and Automation, Academy of Sciences, Pod Vod´ a renskou vˇ e ˇ z ´ ı 4, 182 08 Prague, Prague,Czech Czech ∗ Academy of Sciences, Pod Vod´ arenskou renskou vˇeeˇ ˇzz´ ´ııand 4, 182 182 08 Prague, Prague,Czech Czech UTIA of - Institute of Information Theory Automation, Republic, [email protected] Academy Sciences, Pod Vod´ a vˇ 4, 08 Czech Republic, [email protected] Republic, [email protected] Academy of Sciences, Pod Vod´ arenskou vˇeˇz´ı 4, 182 08 Prague, Czech Republic, [email protected] Republic, [email protected] Abstract: Paper presents the current state of the laboratory model of the planar walking Abstract: Paper presents the current state of the laboratory model of the planar walking Abstract: Paper presents the current state of the laboratory model of the planar walking robot built in laboratory in UTIA. Moreover, it presents simulation results of the walkingAbstract: Paper presents the current state of the laboratory model of the planar walking robot built in laboratory in UTIA. Moreover, it presents simulation results of the walkingrobot built in laboratory in UTIA. Moreover, it presents simulation results of the walkingAbstract: Paper presents the current state of the laboratory model of the planar walking like movement of the simplified version the so-called three-link, alternatively also called robotmovement built in laboratory in UTIA. Moreover, presentsthree-link, simulationalternatively results of the walkinglike of the simplified version - the itso-called also called like movement movement of the theMethod simplified version - the the design so-called three-link, alternatively also called robot built in torso. laboratory in UTIA. Moreover, itso-called presents simulation results ofthe the walkingAcrobot with for the walking is based on the use of collocated like of simplified version three-link, alternatively also called Acrobot with torso. Method for the walking design is based on the use of the collocated Acrobot with torso. for the walking is based on the use of the collocated like movement of constraints. theMethod simplified version -constraints the design so-called three-link, alternatively also called virtual holonomic Collocated include the directly actuated variables Acrobot with torso. Method for the walking design is based on the use of the collocated virtual holonomic constraints. Collocated constraints include the directly actuated variables virtual holonomic constraints. Collocated constraints the actuated variables Acrobot with torso. Method for the walking design include isof based on directly the use of theand collocated only. Restricted system then conserves cyclic property the unactuated variable can be virtual holonomic constraints. Collocated constraints include the directly actuated variables only. Restricted system then conserves cyclic property of the unactuated variable and can be only. system then conserves cyclic property of and can virtual holonomic constraints. Collocated constraints include the directlyvariable actuated variables controlled using its three dimensional exact feedback linearization. only. Restricted Restricted system then conserves cyclic property of the the unactuated unactuated variable and can be be controlled using its three dimensional exact feedback linearization. controlled using its three dimensional feedback linearization. only. Restricted system then conservesexact cyclic property of the unactuated variable and can be controlled using its three dimensional exact feedback linearization. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. controlled using its threemethods, dimensional exactrobots, feedback linearization. Keywords: Hamiltonian walking collocated constraints. Keywords: Hamiltonian methods, walking robots, collocated constraints. Keywords: Hamiltonian methods, walking robots, collocated Keywords: Hamiltonian methods, walking robots, collocated constraints. constraints. Keywords: Hamiltonian methods, walking robots, collocated constraints. 1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION 1. continue INTRODUCTION This paper aims to the study study of of the the advantages advantages This paper aims to continue the This paper aims to continue the study of the advantages of handling the so-called collocated virtual holonomic conThis paper the aimsso-called to continue the study of holonomic the advantages of handling collocated virtual conof handling the so-called collocated virtual holonomic conThis paper aims to continue the study of the advantages straints (VHCs) in mechanical systems using its Hamiltoof handling the so-called collocated virtual holonomic constraints (VHCs) in mechanical systems using its Hamiltoˇ virtual straints (VHCs) in mechanical systems using its Hamiltoof handling the so-called collocated holonomic connian representation initiated in Celikovsk´ y and Anderle straints (VHCs) in mechanical systems using its Hamiltoˇ nian representation initiated insystems Celikovsk´ y and Anderle ˇ straints (VHCs) in mechanical using its Hamiltonian representation initiated in Celikovsk´ y and Anderle ˇ (2017b,a) where this this initiated concept was was applied to to the so-called so-called nian representation in Celikovsk´ y and Anderle (2017b,a) where concept applied the ˇunderactuated (2017b,a) where concept was applied the nian representation in Celikovsk´ y and Anderle four-link. Four-link isinitiated the planar planar walking(2017b,a) where this thisis concept was applied to to the so-called so-called four-link. Four-link the underactuated walkingfour-link. Four-link is the planar underactuated walking(2017b,a) where this concept was applied to the so-called like model representing pair of legs with knees. It has four four-link. is the like modelFour-link representing pairplanar of legsunderactuated with knees. It walkinghas four like model representing pair of legs with knees. It has four four-link. Four-link is the planar underactuated walkingdegrees of freedom and three actuators placed at knees like model representing pair of legs with knees. It has four degrees of freedom and three actuators placed at knees degrees of freedom and three actuators placed at knees like model representing pair of legs with knees. It has four and hips. The unactuated angle is at the supporting pivot degrees of freedom and three actuators placed at knees and hips. The unactuated angle is at the supporting pivot and hips. The unactuated angle is at the supporting pivot degrees of freedom and three actuators placed at knees point. Besides, we would like to use this opportunity to and hips. The unactuated at the pivot point. Besides, we would angle like toisuse thissupporting opportunity to point. Besides, we would like to use this opportunity to and hips. The unactuated angle is at the supporting pivot present some further advances in developing planar walkpoint. Besides, we would like to use this opportunity to present some further advances in developing planar walkpresent some advances in planar point. would like to use this opportunity to ing likeBesides, devicefurther inweour our laboratory indeveloping UTIA. present some further advances inin developing planar walkwalking like device in laboratory UTIA. ing device in laboratory UTIA. present ing like like some devicefurther in our our advances laboratoryinin indeveloping UTIA. planar walking like device yy in our laboratory qq2 in UTIA. u 2 u 2 2 yy u u22 qq22 ll3 y u2 q2 llc3 ll333 c3 x m llc3 2 x l2 m c3 l3 2 x x ll22 lc2 m m22 m m33lc3 x l2 llc2 −q u m332 m m33 −q u44 Fig. l2 lc2 c2 −q Fig. 2. 2. The The experimental experimental robot robot in in UTIA UTIA laboratory laboratory -u 3 m −q 3 u44 3 Fig. 2. The experimental robot in UTIA laboratory u version without torso. l 3 c2 Fig. 2. The experimental robot in UTIA laboratory -version without torso. −q3 u 3 u4 u version without torso. Fig. 2. The experimental robot in UTIA laboratory 3 u3 version without torso. llc4 qq44 u3 version without torso. absence of a torso is possible as a theoretical q q44 absence of a torso is possible as a theoretical abstraction abstraction ll1 m llc4 c4 m11 absence of a torso torsoone is possible possible as aa negligible theoreticaltorso abstraction c4 q m4 1 only. In practice need some -- shaft absence of a is as theoretical abstraction l 4 l 4 m only. In practice one need some negligible torso shaft 1 m 1 l l4lc4 m4 m only. In practice one need some negligible torso -- shaft absence of a torso is possible as a theoretical abstraction connecting two drives at hips, see Fig. 2. Indeed, it ll44 4 qq1ml11c1 l11 only. In practice one need some negligible torso shaft m 4 connecting two drives at hips, see Fig. 2. Indeed, it is is 1 lc1 connecting two drives at hips, Fig. 2. Indeed, it is only. In practice one unrealistic need somesee negligible torso - shaft obviously practically to connect two legs m4 qq11 lc1 l4 connecting two drives at hips, see Fig. 2. Indeed, it at is obviously practically unrealistic to connect two legs at lc1 obviously practically unrealistic to connect two legs at connecting two drives at hips, see Fig. 2. Indeed, it is hips trough a single drive. In other words, the four-link q1 lc1 obviously unrealistic to connect twofour-link legs at hips troughpractically a single drive. In other words, the hips trough aa single drive. In other words, the four-link obviously practically unrealistic to connect two legs at realization requires two drives at hips and two at knees. hips trough single drive. In other words, the four-link realization requires two drives at hips and two at knees. realization requires two drives at and at knees. hips a single drive. Intorso other words, the four-link There is negligible providing yet another realization requires two drives at hips hips and two two knees. Theretrough is inertially inertially negligible torso providing yetat another Fig. 1. Four-link mechanical chain. There is inertially negligible torso providing yet another realization requires two drives at hips and two at knees. Fig. 1. Four-link mechanical chain. degree of freedom which requires that torques of two drives There is inertiallywhich negligible torso providing yet another degree of freedom requires that torques of two drives Fig. 1. Four-link mechanical chain. Fig. 1. Four-link mechanical chain. degree of freedom which requires that torques of two There is inertially negligible torso providing yet another at hips should act in mutual dependence preventing fast degree ofshould freedom which requires that torques of two drives drives at hips act in mutual dependence preventing fast Fig. 1. Four-link mechanical chain. The hips should act in mutual dependence preventing fast degree of freedom which requires that torques of two drives rotation of the connecting shaft. In such a way, laboratory The four-link four-link model model does does not not have have a a torso, torso, see see Fig. Fig. 11 at at hips should act in mutual dependence preventing fast rotation of the connecting shaft. In such a way, laboratory The four-link model does have see 1 at for theoretical scheme. It to that The four-link model does not not have a a torso, torso, see Fig. Fig. of the connecting shaft. In such aa way, laboratory hips should act in mutual dependence preventing realization of the four link is actually five degrees of for its its theoretical scheme. It important important to realize realize that1 rotation rotation of the connecting shaft. In such way, laboratory realization of the four link is actually five degreesfast of for its theoretical scheme. It important to realize that The four-link model does not have a torso, see Fig. 1  for its theoretical scheme. It important to realize that realization of the four link is actually five degrees rotation of the connecting shaft. In such a way, laboratory freedom system with four actuators, where one degree of work was supported by the Czech Science Foundation through realization of the four link is actually five degrees  This freedom system with four actuators, where one degree of This work was supported by theIt Czech Science Foundation through for its theoretical scheme. important to realize that  freedom with four actuators, where one degree realization the four is actually five degrees the research grant 17-04682S. is reduced and two actuators act dependently, work supported by  This freedom system system with fourlink actuators, where one degree of of work was was supported by the the Czech Czech Science Science Foundation Foundation through through theThis research grant 17-04682S. freedom is of reduced and two actuators act dependently,  the research grant 17-04682S. freedom is actuators act dependently, withand fourtwo actuators, where degree of work was supported by the Czech Science Foundation through theThis research grant 17-04682S. freedom system is reduced reduced and two actuators act one dependently, the research grant IFAC 17-04682S. freedom is reduced and tworeserved. actuators act dependently, 2405-8963 © IFAC (International Federation of Automatic Control) by Elsevier Ltd. All rights Copyright © 2018, 2018 384Hosting Copyright 2018 responsibility IFAC 384Control. Peer review© under of International Federation of Automatic Copyright © 384 Copyright © 2018 2018 IFAC IFAC 384 10.1016/j.ifacol.2018.11.571 Copyright © 2018 IFAC 384

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resulting in the mathematical model having four degrees of freedom and three actuators. Therefore, further natural step is to consider five-link including real torso as it does not constitute further adding of actuators, see Fig. 3. Adding a torso actually constitutes principally different problem as adding knees. Knees are just needed for bending legs to avoid hitting the ground during the swing phase and therefore it is much easier to tune the corresponding VHCs as their effect onto overall dynamics is not so crucial. Torso, nevertheless, affects the constrained dynamics in a significant way. Moreover, the torso motion is affected strongly by the impact map during switching legs as well. One of interesting and somehow bit paradoxical observations made in experiments later on is that it is easier to tune VHCs for heavier torso, rather than for light torso. Mathematically, this is not so surprising as the transfer from the five-link to the fourlink neglecting the moment of inertia of torso is basically singular perturbation based model reduction presenting all those well-known theoretical and practical peculiarities. Moreover, unlike the straightforward idea that knees should be bended and stretched conveniently, there is no straightforward idea what should do the torso during the step, which further complicates the design of the VHCs for the torso. The above arguments motivated us to start theoretical and simulation study with the so-called three link. One can describe the three link as the device shown on Fig. 3 where the knees are straighten, fixed and the corresponding drives are either idle, or used to keep the knees straight using feedback control. Schematically it is shown in Fig.4 and it is a convenient framework to focus only on the mentioned peculiarities cased by that torso, abstracting from knees control problem for the moment. The three-link has been already studied e.g. in La Herra et al. (2013), but with one actuator only. In our case, there are two actuators, one actuates the relative angle between the stance leg and the torso, while another actuator actuates the angle between the torso and the swing leg. The idea to impose virtual constraints to simplify the complex design has been widely used for quite some time. As a matter of fact, the history of the use of the forced kinematic and dynamic relations goes back to 1920’s B`eghin (1921) and the alternative terminology calling them as the “servoconstraints” was introduced already in 1953 Appel (1953). Detailed survey and history can be found in Kozlov (2015), a rather scarce selection of references from there ˇ is repeated and slightly extended in Celikovsk´ y and Anderle (2016a). During the recent decades, the term virtual holonomic constraints (VHCs) is being used. Some of the ˇ recent advances on this topic can be found in Celikovsk´ y and Anderle (2016a); Maggiori and Consolini (2013); Moˇ ˇ hammadi et al. (2013); Celikovsk´ y (2015); Celikovsk´ y and Anderle (2016b). In particular, the well-known use of the VHCs is the underactuated walking design in Westervelt et al. (2007); Chevallereau et al. (2009). When studying the underactuated mechanical systems, the collocated VHCs are simply those including the actuated coordiˇ ˇ nates only Celikovsk´ y and Anderle (2016a); Celikovsk´ y ˇ (2015); Celikovsk´ y and Anderle (2016b). In particular, the 385

Fig. 3. The experimental robot in UTIA laboratory version with the torso. corresponding constrained dynamics then preserves the cyclic property of the unactuated variable Grizzle et al. (2005); Cambrini et al. (2000); Olfati-Saber (2002), moreover, the constrained dynamics remains to be controlled dynamical system with at least one free actuator for its further control. Note, that prevailing approaches using VHCs, e.g. along line of research in Westervelt et al. (2007); Chevallereau et al. (2009) and its continuation, leads to a restricted system being an uncontrolled system and loosing the cyclic property of unactuated variable. At the same time, there are numerous results being able to use the mentioned property of unactuated variable being cyclic, Grizzle et al. (2005); Cambrini et al. (2000); Anderle ˇ ˇ and Celikovsk´ y (2011); Anderle et al. (2010); Celikovsk´ y ˇ et al. (2013); Anderle and Celikovsk´ y (2010), among many others. The efficient algorithms to impose properly designed collocated VHCs thereby open possibilities to use the cyclic property of unactuated variable in fairly general walking-like configurations. A variable (i.e. a component of generalized coordinates) is called cyclic if the kinetic energy of the mechanical system does not depend on that variable. The cyclic variable does not affect the shape of the mechanical system. Other variables are called as the shape variables. Typical property of the underactuated walking is that unactuated variables are always the cyclic ones. Initial attempts to use the collocated VHCs combined ˇ with that cyclic property were presented in Celikovsk´ y ˇ and Anderle (2016a). Later on, in Celikovsk´ y and Anderle (2016b) conditions were derived characterizing those collocated VHCs that contain trajectories invariant with respect to the so-called impact map during the double support phase and presented sustainable walking of the four ˇ link mechanical chain during 150 steps. Next, Celikovsk´ y and Anderle (2017b) presents alternative and computationally more convenient, and practically robust, technique to impose collocated VHCs using a special canonical form based on Hamiltonian representation of the underactuated

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mechanical systems. It is based on certain backstepping ˇ technique. Finally, Celikovsk´ y and Anderle (2017a) uses this technique to design multi-step sustainable walking. Note, that while the above mentioned results Westervelt et al. (2007); Chevallereau et al. (2009); Grizzle et al. (2014) are mostly related with the so-called path following, the approach presented in this paper provides the design of the periodic trajectory and its tracking as e.g. in Song and Zefran (2006a,b); Majumdar et al. (2013). Purpose of this paper is to apply the approach of ˇ Celikovsk´ y and Anderle (2017a) to the case of three-link. To do so, one has to use the impact invariance conditions ˇ of Celikovsk´ y and Anderle (2016b) to design a single collocated VHC relateing the torso angle to the hip angle. Next, one has to impose this constraint using the technique ˇ of Celikovsk´ y and Anderle (2017a) based on transforming system to the new coordinates where this VHC has the flat form. The flat VHC is then imposed using technique ˇ of Celikovsk´ y and Anderle (2017b). The rest of the paper is organized as follows. Section 2 introduces some preliminary facts and, in a self-contained manner, repeats the previously published results on the transformations of the system and its VHCs in Hamiltonian formalism. Section 3 provides the main ideas of the mechanical three-link chain (alternatively called also as Acrobot with torso) walking design. Section 4 collects the simulations and some conclusions are given in the final section. Notation. Throughout the paper, the gradient of a function is assumed to be a row vector. Furthermore, denote n  d ∂M(q) ˙ M(q) := M(q) := q˙i , (1) dt ∂qi i=1

M(q) being a matrix smoothly dependent on q.



D(q)¨ q + C(q, q) ˙ q˙ + G(q) = [0k , uk+1 , . . . , un ] , (4) where C(q, q) ˙ is the Coriolis and centrifugal forces matrix, while G(q) is the gravity vector. In particular, ∂V (q) ˙ q˙ − Cs (q, q), , C(q, q) ˙ q˙ = D(q) ˙ (5) G (q) = ∂q ∂K(q, q) ˙ Cs (q, q) , (6) ˙ = [Cs1 (q, q), ˙ . . . , Csn (q, q)] ˙ = ∂q   1 ∂D(q) ˙ = q˙ q, ˙ i = 1, . . . , n. (7) Csi (q, q) 2 ∂qi The above representation of the Coriolis terms C(q, q) ˙ q˙ differs from the common one, nevertheless, it will better suit our exposition later on. Hamiltonian representation of (2) uses the generalized coordinates q ∈ Rn and the generalized momenta σ ∈ Rn :  ∂L    σ1  ∂ q˙1    ..    ˙ (8) σ =  ...  =   .  = D(q)q.   ∂L σn ∂ q˙n Next, introduce the so-called Hamiltonian H(q, σ) as:  −1  σ) + V (q), K(q,  σ) = σ D (q)σ . (9) H(q, σ) = K(q, 2  σ) is the kinetic energy as a function of generHere, K(q, alized coordinates and momenta. In particular,  σ) ∂K(q, q) ˙ ∂H(q, σ) ∂L(q, q) ˙ ∂ K(q, =− ⇒ =− , (10) ∂q ∂q ∂q ∂q giving by (2), (3), (8) the following Hamiltonian form       ∂H(q, σ) ∂H(q, σ) 0 , σ˙ = − + . (11) q˙ = u ∂σ ∂q Finally, (11) is straightforwardly equivalent to q˙ = D

2. PRELIMINARY DEFINITIONS AND RESULTS

−1

(q)σ, σ˙ = Cs (q, D

−1

  0 (q)σ) − G(q) + , (12) u

where, G(q), Cs (q, q) ˙ are given by (5)-(7), while u by (4).

2.1 Lagrangian and Hamiltonian representations

2.2 Transformation of generalized coordinate Consider the underactuated controlled Lagrangian system       ∂L d ∂L 0  − = = [0k , uk+1 , . . . , un ] , (2) u dt ∂ q˙ ∂q

1 T q˙ D(q)q˙ − V (q), (3) 2 where q = (q1 , . . . , qn ) , q˙ = (q˙1 , . . . , q˙n ) are the generalized coordinates and velocities, D(q) = D(q) > 0 is the inertia matrix, while K, V are the system kinetic and potential energy. For the general setting replacing in (2) [0, u] by B(q)u, rankB(q) = n − k, and more refined terminology on Lagrangian underactuated systems ˇ see Celikovsk´ y and Anderle (2016a); Mohammadi et al. ˇ (2013); Celikovsk´ y (2015). If k = 0, the system is called fully actuated, otherwise it is called underactuated while k is called as the degree of the underactuation. The coordinates qk+1 , . . . , qn are called actuated while q1 , . . . , qk unactuated. The planar underactuated walking-like mechanical chains have usually k = 1, while the 3-dimensional ones Grizzle et al. (2014) have usually k = 2. L(q, q) ˙ = K(q, q) ˙ − V (q) =

Description (2), (3) gives the second-order dynamics 386

In this subsection, the transformations of the both Lagrangian and Hamiltonian form of the mechanical system into new coordinates are studied. This study is motivated by the possibility, demonstrated later on, to transform system into special coordinates where VHCs take some simplified form, that is more easy to be handled. ˇ Lemma 1. Celikovsk´ y and Anderle (2016a) Consider a region S ⊂ Rn , diffeomorphism F : S → F(S) and let the diffeomorphism S × Rn → F(S) × Rn define new generalized coordinates and velocities q, q˙ as follows:   F1 (q)   ∂F(q) ˙  ..  q. (13) q = F(q) =  .  , q˙ = ∂q Fn (q) Then (13) takes (2), (3) into the following form         d ∂L ∂L ∂F(q) 0 − = , u dt ∂ q˙ ∂q ∂q   ˙ := L F(q), ∂F(q) q˙ . L(q, q) ∂q

(14) (15)

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In particular, (13) takes (4-7) into the following form     0 ˙ q˙ + G(q) = ∂F(q) , (16) D(q)¨q + C(q, q) u ∂q

where

 ∂F(q) G(F(q)), ∂q     ∂F(q) ∂F(q) , D(q) := D(F(q)) ∂q ∂q ˙ ˙ q˙ = D(q) ˙ C(q, q) q˙ − C s (q, q),    ˙ = C s1 (q, q), ˙ . . . , C sn (q, q) ˙ , C s (q, q)   1  ∂D(q) ˙ = q˙ ˙ i = 1, . . . , n. C si (q, q) q, 2 ∂q i G(q) =



(17) (18) (19) (20) (21)

Remark 1. The above lemma actually describes compactly all terms of the second-order description of the transformed system, i.e. inertia matrix, Coriolis terms, gravity vector. Nevertheless, the explicit expression of Coriolis terms in the new coordinates q, q˙ requires due to (19)-(21) some laborious computations giving unhandy expressions omitted here for space reasons. As a matter of fact, the original results presented in this subsection later on show that such a transformation is much more conveniently described in Hamiltonian notation. Denote the generalized momentum corresponding to q, q˙ as σ and the one corresponding to q, q˙ as σ, namely ˙ σ = D(q)q, ˙ σ = D(q)q. (22) Using subsequently (22), (18) and (13) gives     ∂F(q) ∂F(q) ˙ ˙ σ = D(q)q = D(F(q)) q= ∂q ∂q     ∂F(q) ∂F(q) D(q)q˙ = σ ∂q ∂q where the last equality is again by (22). Summarizing   ∂F(q) σ= σ. (23) ∂q Straightforward use of (22), (23) proves the following. Lemma 2. Consider the mechanical system in Lagrangian form (2), (3) and new generalized coordinates and velocities q, q˙ given by (13). Then its Hamiltonian representation (12) related to the generalized coordinates and velocities q, q˙ is −1 q˙ = D (q)σ

   0 ˙σ = C s (q, D−1 (q)σ) − G(q) + ∂F(q) , u ∂q 

(24)

˙ where D(q), G(q) are given by (17), (18), while C s (q, q) by (20), (21) and u by (4). Moreover,     ˙ = ∂F(q) Cs F(q), ∂F(q) q˙ C s (q, q) ∂q ∂q   n   ∂ ∂F(q) + q˙ i , (25) ∂q ∂q i i=1 where Cs (q, q) ˙ is given by (6), (7).

387

381

Remark 2. Note, that (24), (25) provide the explicit form of Hamiltonian dynamics in new coordinates that is more handy than the Lagrangian one provided by Lemma 1. Definition 1. The generalized coordinates and velocities q, q˙ defined by (13) are called as the coordinates collocated with system (2), (3) if the following identities hold ∂Fi (q) ≡ 0 ∀i ∈ {k + 1, . . . , n} ∧ ∀j ∈ {1, . . . , k}. (26) ∂q j The next proposition is straightforward. Proposition 1. Let q, q˙ defined by (13) be collocated with (2), (3) and let u = [uk+1 , . . . , un ]. Then the relation       ∂F(q) 0 0 (27) = u u ∂q defines a smooth one-to-one map between u and u, i.e. the smooth feedback introducing the new input variable u. Remark 3. By Proposition 1 and Lemma 1 state transformation (13) and feedback (27) take (2), (3) into the form       ∂L d ∂L 0 − = , (28) u dt ∂ q˙ ∂q while (13,(23,27) transform (12) into the form −1 q˙ = D (q)σ

  −1 0 σ˙ = C s (q, D (q)σ) − G(q) + . u

(29)

Moreover, if all unactuated variables of the mechanical system (2), (3) are cyclic, then all unactuated variables of the mechanical system (28), (29) are cyclic as well. In other words, collocated coordinate change and feedback (27) preserve all relevant structural properties of (2,3). This important feature will be efficiently used in the sequel. 2.3 Virtual holonomic constraints Definition 2. Virtual holonomic constraints (VHCs) for the system (2-4) are given by l equalities, 1 ≤ l ≤ n, ϕi (q) = 0, i = 1, . . . , l, (30) where ϕ1 , . . . , ϕl are smooth functions of the generalized coordinates having ∀q ∈ Rn linearly independent differentials dϕi (q), i = 1, . . . , l. The VHCs are called global if the functions in (30) can be completed to a diffeomorphism of Rn . Furthermore, the VHCs are called locally regular for (4) at some q 0 , q˙0 , if it holds  i=1,...,l ∂ ϕ¨i rank (q 0 , q˙0 ) = l, (31) ∂uj j=k+1,...,n

where the time derivatives are taken along trajectories of (4). The global VHCs are called globally regular on some subset if they are locally regular at each its point. The VHCs are called flat if there ∃ l mutually distinct integers j1 , . . . , jl ∈ {1, . . . , n} with ϕ1 ≡ qj1 , . . . , ϕl ≡ qjl . Finally, the VHCs (30) are called collocated if they depend only on actuated coordinates. Remark 4. The regular VHCs can be straightforwardly imposed using the well-known input-output exact feedback linearization. In general, even the flat VHCs may not be regular while the global collocated VHCs are always ˇ globally regular Celikovsk´ y (2015). Coordinate free charˇ acterization of the collocated VHCs is given in Celikovsk´ y

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and Anderle (2016a). Further remarks on terminology and ˇ relation to existing results can be found in Celikovsk´ y and ˇ Anderle (2016a); Celikovsk´ y (2015). Lemma 3. Consider VHCs (30) and define the coordinates q = F −1 (q) = [q1 , . . . , qn−l , ϕ1 (q), . . . , ϕl ] . (32) Let l ≤ n−k. Then VHCs (30) are collocated if and only if the coordinates (32) are collocated with the system (2,3). Moreover, in coordinates q given by (32) the VHCs (30) become the flat VHCs of the form q n−l+1 ≡ 0, . . . , q n ≡ 0. In the sequel, only the flat collocated VHCs (30) for the Lagrangian system (2) are to be studied. Indeed, by Lemma 3 and Remark 3 one can assume that the given system has been already transformed using the smooth coordinate change (32) and the related smooth feedback (27) into the Hamiltonian representation with the flat VHCs qn−l+1 ≡ 0, . . . , qn ≡ 0, l ≤ n − k. (33) 2.4 Hamiltonian representation of the collocated VHCs Introduce the following block notation related to (33):     0k un−l+1  uk+1    .. C  uR =  (34) , .  ...  , u =  un un−l     qn−l+1 q1  R q     .. q = C , q R =  ...  , q C =  (35) . . q qn qn−l Further, introduce σ R , σ C via σ in the same way as q R , q C are introduced in (35) via q and denote   DR (q) DRC (q) D(q) = , (36) D (q) DC (q)  R RC   R  Cs (q, q) ˙ G (q) ˙ = ,G = , (37) Cs (q, q) CsC (q, q) GC (q) ˙ where DR , DC are square (n−l)×(n−l) and l×l matrices, correspondingly, while DRC is (n − l) × l matrix. Finally, CsR ∈ R(n−l)×n , GR ∈ Rn−l ,CsC ∈ Rl×n , GC ∈ Rl . Since  D R = DR > 0, D−1 is expressed via Shur complement: −1  DR DRC , Sc (q) = Sc (q) > 0, (38) Sc (q) = DC − DRC D(q)−1 = (39)  −1  −1 −1 −1 −1  DR + DR DRC Sc DRC DR −DR DRC Sc−1 . −1  −Sc−1 DRC DR Sc−1 Definition 3. The variable σ S defined as −1 R C  DR (q , q )σ R , (40) σ S = Sc q˙C = σ C − DRC will be called in the sequel Schur generalized momentum of the VHCs (33). Moreover, the following equalities (41) q C = 0, σ S = 0 will be called the Hamiltonian representation of (33). ˇ Theorem 1. Celikovsk´ y and Anderle (2017b) Consider σ S , C R C q , q and u given by (40), (35) and (34), respectively. Define a new input variable uS = uC − GC (q) + CsC (q, q) ˙ −1  R ˙ − DRC (q)(q˙ + DR (q)DRC (q)q˙C )  (42) −1  − DRC DR (q) uR − GR (q) + CsR (q, q) ˙  − D˙ R (q)(q˙R + D−1 (q)DRC (q)q˙C ) . R

388

Then the mechanical system (2), (3), or (4), is equivalent to −1 −1 q˙R = DR (q)σ R − DR (q)DRC (q)Sc−1 (q)σ S , σ˙ R = −GR (q) + CsR (q, D−1 (q)σ) + uR q˙C = Sc−1 (q)σ S ,

(43)

σ˙ S = uS .

Remark 5. By Theorem 1 uS ≡ 0 is obviously the condition keeping the collocated VHCs (33) invariant. This gives in terms of the original input variable u the condition  uC ≡ GC −CsC + D˙ RC q˙R  (44) −1  R R +DRC DR u − G + CsR − D˙ R q˙R ,

where all omitted arguments are (q R , 0l ) and/or (q˙R , 0l ) . The corresponding invariant restricted dynamics is given by −1 (q)σ R , q˙R = DR (45) σ˙ R = −GR + CsR + uR . Important feature here is that the restricted dynamics can still be controlled by part of the original input components uR and its value is properly attenuated in (44). The purpose of Hamiltonian approach is not obtaining (44) as it is well-known that for any regular VHCs the feedback keeping their invariance should be unique, i.e. (44) is the same as obtained using Lagrangian formalˇ ism in Celikovsk´ y and Anderle (2016b). Nevertheless, the Hamiltonian approach enables to design better feedback imposing the collocated VHCs using (43) combined with the backstepping technique, as shown by the following. ˇ Proposition 2. Celikovsk´ y and Anderle (2017b) Consider the mechanical system (2), (3) and recall the notation of Theorem 1 and (34)-(38). Let C C uC = −Σ1 q˙C − Σ−1 2 Σ3 (Sc q˙ + σ1 q )

C C C −1 ˙ − Σ−1 2 Sc (q)Σ1 q + G (q) − Cs (q, q) −1  R C ˙ + DRC (q)(q˙ + DR (q)DRC (q)q˙ )  −1  + DRC ˙ DR (q) uR − GR (q) + CsR (q, q)  −1 R C − D˙ R (q)(q˙ + D (q)DRC (q)q˙ ) .

(46)

R

Here, Σ1 , Σ2 , Σ3 are l × l positive definite symmetric matrices. Assume that initial conditions q(0), q(0) ˙ and a function uR (t), t ≥ 0, are given such that trajectories of (2), (3) with these initial conditions and input u = (uR (t), uC (t)) with uC given by (46) do not escape to infinity in finite time. Then for any such trajectory it holds that q C (t) → 0 exponentially as t → ∞. 2.5 Impact invariance of the collocated VHCs During the multi-step walking and double support configuration the so-called impact map has to be considered: (47) q + = R(q − ), q˙+ = Φ(q − )q˙− . − − Here, q , q˙ stand for the configurations and velocities “just before” the impact while q + , q˙+ for those “just after” the impact, Φ is the so-called impact matrix, while R(q − ) is the so-called relabeling map. The impact matrix Φ is invertible for the majority of reasonable models and their generic configurations while the relabeling map is usually a simple invertible affine map. The impact matrix can be derived for particular system using the well established

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techniques Westervelt et al. (2007). The relabeling map is just due to switching the legs, otherwise it is clear that configurations angles do not change during the impact. Definition 4. The VHCs (30) are called impact invariant with respect to (47) if ∀i = 1, . . . , l and for all q − corresponding to any double support configuration: ϕi (q − ) = 0 ⇒ ϕi (R(q − )) = 0, (48) − − − − − dϕi (q )q˙ = 0 ⇒ dϕi (R(q ))Φ(q )q˙ = 0. (49) ˇ Proposition 3. Celikovsk´ y and Anderle (2016b) Assume with no loss of generality that the last l columns of the   l × n matrix dϕ := dϕ1 (q) | . . . |dϕl (q) are linearly 1 2 independent and denote dϕ = [Dϕ |Dϕ ], where Dϕ2 is l × l regular matrix. Then (49) are equivalent to    1 2 In−l − − Dϕ |Dϕ (R(q ))Φ(q ) ≡ 0. (50) −[Dϕ2 ]−1 Dϕ1 (q − )

Consider the underactuated mechanical system (2), (3) with k = 1 and the special collocated VHCs with l = n − 2 and (51) ϕi := qi+2 − φi+2 (q2 ), i = 1, . . . , n − 2, where φ3 , . . . , φn are suitable sufficiently smooth functions of a scalar variable q2 . Recall, that q1 is the unactuated variable while q2 , . . . , qn are the actuated ones, i.e. VHCs (51) are, indeed, collocated. Further, for the impact matrix Φ(q) introduce the following notation   11 12   Φ Φ φ11 φ12 − 11 Φ(q ) = , Φ = , (52) φ21 φ22 Φ21 Φ22   φ31 φ32    .. ..  , Φ12 (q − ) = φ13 . . . φ1n 21 − Φ (q ) =  .  . φ23 . . . φ2n φn1 φn2   φ33 (q − ) . . . φ3n (q − )   .. .. .. Φ22 =  . . . . φn3 (q − ) . . . φnn (q − )

ˇ Theorem 2. Celikovsk´ y and Anderle (2016b) Consider VHCs of the form (51) and (52). Assume that the impact occurs at some q − of the form q − = (q1− , q2− , φ3 (q2− ), . . . , φn (q2− )) , such that φ21 (q − ) = 0 and the inverse on the right hand side of (54) given bellow exists and denote q + = R(q − ). Then VHCs (51) are impact invariant if and only if      + φn1 (q − ) φ31 (q − ) +  ,..., , (53) φ3 (q2 ), . . . , φn (q2 ) = φ21 (q − ) φ21 (q − )  −1   −   0 φ31 φ−1 φ3 (q2 ) 21   ..  12  .. .. 22   =  . Φ − Φ   . . φn (q2− )

0 φn1 φ−1 21

 φ32 − φ31 φ22 φ−1 21   −  .. ×  q . . −1 φn2 − φn1 φ22 φ21 

(54)

3. UNDERACTUATED THREE-LINK WALKING ˇ Proposition 4. Celikovsk´ y and Anderle (2016a) Consider the mechanical system having a cyclic unactuated variable qi . Then its restriction with respect to any collocated VHCs also has the cyclic unactuated variable qi . 389

383

Proposition 4 allows the decomposition of a complex design into two simpler steps: (a) enforcing the collocated VHCs; (b) controlling the resulting restriction using a suitable method to control a two-degree of freedom systems having the unactuated cyclic variable, see e.g. Grizzle et al. ˇ (2005); Cambrini et al. (2000); Anderle and Celikovsk´ y ˇ (2011); Anderle et al. (2010); Celikovsk´ y et al. (2013); ˇ Anderle and Celikovsk´ y (2010). l3 q 3 m3

lc3

l1

q2 m1 m2

lc1

l2 lc2

q1

Fig. 4. The Acrobot with torso, or the three-link. Such an approach was successfully applied to generate and to exponentially track the walking-like trajectory for the planar three-link depicted in Fig. 4 in the following way. First, suitable collocated VHCs were designed, namely, the dependence of the angle q3 on the angle q2 , denoted q3 = φ3 (q2 ) such that hybrid invariance conditions of Theorem 2 holds. This simply means that the derivative φ3 at the beginning and at the end of the step are uniquely given by the respective system configurations. That was actually the reason why the torso moment inertia needed to be significantly increased, otherwise those derivatives would be unrealistic. Another factor here is the selection of the three-link configuration at the double support phase. This correspond to initial and final values of the function φ3 (q2 ) which must be the same, after the relabelling, such that the posture of the Acrobot with torso is the same at the beginning and at the end of each step. The course of the function φ3 (q2 ) during the step can be chosen almost arbitrarily according to some other restrictions or requirements. However, the movement of the torso significantly influences the center of mass of the whole Acrobot. Therefore its design was done, in our case, according to a careful and slow movement of the center of mass during the step. The resulting course of the function φ3 is shown in Fig. 5. Secondly, Lemma 2, Proposition 2 and Remark 3 were applied to obtain Hamiltonian representation in new coordinates and with new transformed inputs where this selected VHC is flat by Lemma 3. Then Proposition 2 was applied to generate controllers for torque u3 enforcing this VHC for any input u2 and any states. Finally, the input u2 was designed to control the resulting restricted system ˇ using the method introduced in Anderle and Celikovsk´ y ˇ (2010). This method was used in Anderle and Celikovsk´ y (2010) for the multi-step walking design for the so-called Acrobot (or compass gait walker), but it is applicable to any system having two degrees of freedom with one of

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collocated virtual holonomic constraints with the design based on the cyclic property of the unactuated variable was used. To enforce these constraints, the special Hamiltonian representation based canonical form was used enabling backstepping-like design. The next step is to add knees into legs of Acrobot with torso i.e. the so-called 5-link and use the same approach in order to control the knees in a way that the swing leg of the 5-link do not hit the ground during the step. Moreover, the design of the course of the constraining function for the torso movement is neither straightforward nor unique and therefore it is worth further investigation.

0.1 0.05 0

Φ3 [rad]

-0.05 -0.1 -0.15 -0.2 -0.25 -0.3 2.9

3.5 3

3.1

3.2

3.3

3

3.4

q2 [rad]

2.5

Fig. 5. The constraining function φ3 for the torso. q [rad]

2

them being unactuated cyclic variable. Hence, thank to Proposition 4, this method is applicable to control any restricted system resulting from enforcing the collocated VHCs, including the above described three-link restriction.

1.5 1 0.5

4. SIMULATIONS

0

Simulations demonstrating previously described results for the Acrobot model with a torso depicted in Figure 4 are summarized in this section. Parameters used during the simulations are given in Table 1. The simulations are done using the Simulink. Table 1. Parameters of the Acrobot with torso l1 , l 2 l3 lc1 , lc2 lc3 m1 , m2 m3 I1 , I2 I3

length of each Acrobot leg length of Acrobot torso center of gravity of each Acrobot leg center of gravity of Acrobot torso mass of each Acrobot leg mass of Acrobot torso inertia of each Acrobot leg inertia of Acrobot torso

1 1 0.5 0.75 1 2 0.1 0.3

[m] [m] [m] [m] [Kg] [Kg] [m2 Kg] [m2 Kg]

-0.5 0

0.2

0.4

0.6

0.8

1

Time [s]

Fig. 6. The angular positions of the Acrobot with the torso during two steps. The blue, red and orange lines correspond to q1 , q2 and q3 , respectively. The solid lines represent the Acrobot with the torso whereas the dotted lines represent the reference model to be tracked. 15 10 5

5. CONCLUSIONS The three-link walking like movement was designed and demonstrated in simulations. The combination of the 390

0

dq [rad/s]

In Figures 6, 7 one can see the course of angular positions and velocities of the Acrobot with torso, respectively, and the tracking of the reference multi-step walking trajectory. The dotted lines are the reference and the solid lines are the trajectories of the real Acrobot with the torso during tracking. One can see the impact effect between two steps and cyclic character of the trajectories. The jumps between the steps clearly visible in Figure 6 are due to the relabelling map. ˇ The controller from Anderle and Celikovsk´ y (2009) was used to tracked the Acrobot with the torso along the reference trajectories. At the beginning of the first step are small initial errors in angular velocities. The initial errors are minimalized during the first step via the controller such that the second step starts with smaller errors.

-5 -10 -15 -20 -25 -30 0

0.2

0.4

0.6

0.8

1

Time [s]

Fig. 7. The course of the angular velocities of the Acrobot with the torso during two steps. The blue, red and orange line corresponds to q˙1 , q˙2 and q˙3 , respectively. The solid lines represent the Acrobot with the torso whereas the dotted lines represent the reference model to be tracked.

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