Stably Extending Two-Dimensional Bipedal Walking to Three Dimensions

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1 Proceedings of the 2007 Aerican Control Conference Marriott Marquis Hotel at Ties Square New York City USA July ThB0.4 Stably Extending Two-Diensional Bipedal Walking to Three Diensions Aaron D. Aes and Robert D. Gregg Abstract In this paper we develop a feedback control law that results in stable walking gaits on flat ground for a threediensional bipedal robotic walker given stable walking gaits for a two-diensional bipedal robotic walker. This is achieved by cobining disparate techniques that have been eployed in the bipedal robotic counity: controlled syetries geoetric reduction and hybrid ero dynaics. Controlled syetries are utilied to obtain stable walking gaits for a two-diensional bipedal robot walking on flat ground. These are related to walking gaits for a three-diensional hipless bipedal robot through the use of geoetric reduction. Finally these walking gaits in three diensions are ade stable through the use of hybrid ero dynaics. I. INTRODUCTION The central goal of research in bipedal robotic walking is to obtain stable walking gaits i.e. to prove the existence of stable periodic orbits and/or to develop control laws that create such orbits. Due to the inherent coplexity of bipedal robots any approaches have been eployed in order to achieve this goal. While successful in their respective doains of consideration they often do not extend outside of these doains to obtain a ore encopassing solution to the central proble. In this paper we cobine these techniques so as to obtain a ore satisfying overall solution this requires a ethod for gluing together these partial solutions. Before proposing our technique we review preexisting approaches to bipedal walking. Possibly the first studies in obtaining dynaically stable bipedal walking were concerned with passive twodiensional bipeds walking down shallow slopes; such walkers have subsequently been well-studied see [7] and [3] to nae a few. It has been shown that for certain shallow slopes these passive bipeds have stable walking gaits with no control needed. These walking gaits appear very natural but unfortunately they only have been shown to exist for a liited class of bipedal robots. For exaple the techniques eployed by the passive walking counity have since been extended to three-diensional bipedal walkers cf. [5] although the understanding of bipedal walking obtained by this extension sees to be liited since direct control over hip splay is required which ay not be reasonable for certain odels. An interesting variant of the passivity paradig has been applied to a real biped [2]. A. D. Aes is with the Control and Dynaical Systes Departent California Institute of Technology Pasadena CA 925 aes@cds.caltech.edu R. D. Gregg is with the Departent of Electrical and Coputer Engineering University of Illinois at Urbana-Chapaign Urbana IL 680 rgregg@uiuc.edu The liitation of the passive approach to bipedal walking is just that the approach is passive or uncontrolled thus preventing the application of this idea for exaple to bipeds walking on flat ground aong other varying slopes. This issue was addressed in [0] with the introduction of controlled syetries. This fraework allows for the construction of control laws that translate stable periodic orbits for passive bipeds to stable periodic orbits for actuated bipeds walking on arbitrary slopes. The advantage of this technique is that it exploits the natural syetries in bipedal walkers to obtain natural and efficient walking gaits. The disadvantage is that it presupposes the existence of passive gaits these ay not exist for certain odels of bipedal robots. The liitations of controlled syetries highlight the need for a new ore control-oriented ethod for obtaining walking gaits. The theory that successfully eets this challenge is that of hybrid ero dynaics see [4] and [8]. This technique uses virtual constraints in the for of output functions to reduce the proble of finding stable periodic orbits to a very low diensional space where it is tractable. The advantage of this ethod is that it works for very coplex odels of bipedal walkers and in fact has been ipleented on a real robotic syste: Rabbit. The disadvantages are that the resulting walking gaits appear unnatural if one does not choose the correct virtual constraints and there are essentially no results on extending this ethod to bipeds. The abstract goal of this work is to cobine the advantages of the aforeentioned techniques while attepting siultaneously to eliinate the disadvantages in order to obtain natural stable walking gaits in three diensions. This requires a new theoretical technique capable of linking these disparate approaches. The authors ade a first attept at obtaining such a fraework in [] where ethods fro geoetric echanics specifically a variant of Routhian reduction tered functional Routhian reduction were utilied in order to ap walking gaits in two diensions to walking gaits in three diensions. These ethods exploit controlled syetries and by association the passive walking paradig and so the resulting gaits have the advantage of being natural. The proble with this approach is that the resulting gaits are stable only fro a specific subanifold. Hybrid ero dynaics provide the appropriate fraework in which to address this proble thus linking all of the aforeentioned approaches. The ain result of this paper is the construction of a feedback control law that results in stable walking gaits on flat ground for a three-diensional bipedal robotic walker of a specific for given stable walking gaits for /07/$ IEEE. 2848

2 the planar-equivalent two-diensional bipedal robot. To obtain this result we proceed in the following anner: Step : Consider a passive biped able to walk down a shallow slope. Step 2: Use controlled syetries to obtain stable walking gaits on flat ground in two diensions. Step 3: Use functional Routhian reduction to obtain a walking gait in three diensions stable fro a subanifold. Step 4: Use hybrid ero dynaics to stabilie to the subanifold thus achieving stable walking gaits in three diensions. The final control law resulting fro this procedure appears to have the advantage of producing walking gaits that are natural and stable i.e. it sees to deonstrate the positive aspects of walking gaits obtained fro all three of the aforeentioned approaches. II. HYBRID SYSTEMS Hybrid systes are systes that display both continuous and discrete behavior and so bipedal walkers are naturally odeled by systes of this for. This section therefore introduces the basic terinology of hybrid systes. Definition : A siple hybrid syste is a tuple: where H = DG Rf D R n is a subset of R n called the doain G D is subset of D called the guard R : G D is a sooth ap called the reset ap or ipact equations f is a Lipschit vector field on D i.e. ẋ = fx. We also will be interested in studying hybrid control systes. Definition 2: A siple hybrid control syste is a tuple H C = DUG Rfg where D G and R are the doain guard and reset ap as introduced in Definition and U R k is a set of adissible controls fg is a control syste i.e. ẋ = fx + gxu. Hybrid flows. where A hybrid flow is a tuple χ H = ΛIC Λ = {02... N is an indexing set. I = {I i i Λ is a hybrid interval where I i = [τ i τ i+ ] if ii+ Λ and I N = [τ N τ N ] or [τ N τ N or [τ N if Λ = N N finite. Here τ i τ i+ τ N R and τ i τ i+. It is iportant to note that siple hybrid systes correspond to systes with ipulsive effects see [4] and [8] and vice versa. Specifically to a siple hybrid syste there is the associated syste with ipulsive effects of the for: { ẋ = fx x D\G Σ : x + = Rx x G. C = {c i i Λ is a collection of integral curves of f i.e. ċ i t = fc i t for all i Λ. In addition we require that for every ii + Λ i c i τ i+ G ii Rc i τ i+ = c i+ τ i+. The initial condition for the hybrid flow is c 0 τ 0. Hybrid periodic orbits. In the context of bipedal robots we are interested in discussing walking gaits and stable walking gaits these correspond to hybrid periodic orbits and stable hybrid periodic orbits respectively. A hybrid flow χ H = ΛIC of H is periodic if Λ = N li i τ i = c i τ i = c i+ τ i+ for all i Λ. A hybrid periodic orbit O D is a subset of D such that O = i N{c i t : t I i for soe periodic hybrid flow χ H. As is standard denote the distance between a point x and a set Y by dx Y = inf y Y x y. A hybrid periodic orbit O is locally exponentially stable if there exist constants M > 0 α > 0 and δ > 0 such that for all hybrid flows χ H with dc 0 τ 0 O < δ dc i to Me αt τ0 dc 0 τ 0 O for all t I i and i Λ. III. BIPEDAL MODELS We now introduce the hybrid systes that will be of interest throughout this paper those odeling a two-diensional and a three-diensional bipedal walker without a hip. l=a+b Fig.. a b y s M ns Two-diensional bipedal robot. biped odel. We begin by introducing a odel describing a controlled bipedal robot walking in two diensions on flat ground see Figure. That is we explicitly construct the hybrid control syste: H C = D U G R f g describing this syste. The techniques utilied in the construction of this hybrid syste are further explained in []. x 2849

3 The configuration space 2 for the biped is Q = R 2 with coordinates = ns s T where ns is the angle of the non-stance leg fro vertical and s is the angle of the stance leg fro vertical. The doain and guard are constructed by utiliing the constraint that the non-stance swing foot is not allowed to pass through the ground i.e. by utiliing the unilateral constraint function: H = cos s cos ns. In particular: { D = R 4 : H 0 { T G = R 4 : H = 0 H < 0 We put no restrictions on the set of adissible controls and therefore U = R 2. The reset ap R for H C is coputed using the ethods outlined in [] and [4]. In particular it is given by: R = S P where S and P are given in Table I. Finally the dynaics for H C are obtained fro the Euler-Lagrange equations in the standard way. Specifically the Lagrangian describing this syste is: L = 2 T M V where M is the inertial atrix and V is the potential energy these can be found in Table I. The controlled Euler-Lagrange equations yield: M + C + N = B u where C is the coriolis atrix and N = V. In addition we assue that B is invertible. These equations yield the control syste: f = M C N g 0 = 2 2 M B where is a 2 2 atrix of eros. We now discuss how a control law can be constructed which yields stable walking gaits stable periodic orbits for H C utiliing the ethod of controlled syetries. Controlled Syetries. Controlled syetries were introduced in [0] in order to shape the potential of bipedal robotic walkers to allow for stable walking gaits on flat ground based on stable walking gaits down a slope which exist and have been well-studied in [7] and [3]. This is achieved by rotating the world via a group action. 2 Technically the configuration space is given by Q = T 2 the two torus. The otivation for taking the configuration space to be R 2 is that for the subset U of T 2 containing the angular values of interest there is a diffeoorphis sending this subset to a subset of R 2. Therefore we siply view the angles as being eleents of R 2 ; this allows us to consider coordinates which we can view as being globally defined. Consider the group action Ψ : S Q Q denoted by Ψ γ := ns γ s γ T for a slope angle γ S. Define the following feedback control law: u = K γ = B V V Ψ γ. Applying this control law to f g yields the dynaical syste: f γ := f + g K γ. The ain result of [0] is that there exists 3 a γ such that H s := D G R f γ has a stable walking gait i.e. an exponentially stable hybrid periodic orbit. Such a periodic orbit can be seen in Figure 3 in red and blue with γ = π/50 radians M = 0 = 5 l = and a = b = /2. l=a+b x a b Fig. 2. x 0 0 s M ns Three-diensional bipedal robot. biped odel. We now introduce the odel describing a controlled bipedal robot walking in three diensions on flat ground see Figure 2 i.e. we will explicitly construct the hybrid control syste: H C = D U G R f g. Since this construction is very siilar to the construction of H C we will be brief. The configuration space for the biped is taken to be Q = R 3 with coordinates q = ϕ T T where ϕ is the lean or roll fro vertical and = ns s T as in the biped odel. The guard and doain are constructed fro the constraint function: H q = cos s cos ns which again gives the scaled height of the non-stance foot above the ground with the iplicit assuption that ϕ π/2π/2 allowing us to disregard the scaling factor cosϕ that would have been present. In particular { D = R q q 6 : H q 0 { T G = R q q 6 : H q = 0 Hq q q < 0 3 This γ is not unique but we will pick one once and for all. y 2850

4 l 2 l2 cos s ns M = l2 cos s ns l 2 S = + l M 0 V = 0 2 gl3 + 2M coss cosns B = 2cosns P = s 4 + M cos2 ns s 3 4M + 2cos2 s ns 2 + 2M cos ns s = 8 l M + l 2 cos2 ns 8cos ns cos s M cos2 s V ϕ = V cosϕ p = cos2 ns M cos ns cos s 2 + cos2 s 6 + 4M M cos2 ns 8cos ns cos s + cos2 s TABLE I ADDITIONAL EQUATIONS FOR H C AND H C We again put no restrictions on the set of adissible controls and therefore U = R 3. The reset ap R is given by: R q q = S p P ϕ ϕ where p is given in Table I. To obtain the dynaics of H C we first note that the Lagrangian describing this syste is given by: with L q q = 2 qt M q q V q M q = 0 0 M where and V q are given in Table I. Using the controlled Euler-Lagrange equations the dynaics are given by M q q + C q q q + N q = B u with 0 B =. 0 B These equations yield the control syste f g as in the case of the bipedal walker. IV. st CONTROL LAW: RELATING AND BIPEDS The first step in utiliing stable walking gaits for a biped to obtain stable walking gaits for a biped is to relate the dynaics of these systes. The ethod for doing this is the ain result of [] which we suarie here since it fors an integral part of the construction of our second control law that which yields stable walking gaits in three diensions given stable walking gaits in two diensions. The first control law. The ain idea of [] is that ethods fro geoetric echanics and specifically a variant of Routhian reduction [6] can be utilied to relate two- and three-diensional bipedal walkers satisfying certain assuptions. In order to apply these techniques it is necessary to shape the potential energy of L i.e. V so that it is in a for aenable to the application of hybrid functional Routhian reduction see Theore 2 of []. This otivates the introduction of the following control law for H C. Define the feedback control law paraeteried by α R: u = Kq α := B V q V Ψ γ + α 2 ϕ 2 q 2 with γ as in. Applying this control law to the control syste f g yields the dynaical syste: f α q q := f q q + g q qk α q 2 whence we obtain a hybrid syste: H α := D G R f α. The ain result of [] is the following: Theore : χ H α = ΛI {i i ϕ i ϕ i i Λ is a hybrid flow of H α with ϕ 0 τ 0 = αϕ 0τ 0 0 τ 0 3 if and only if χ H s = ΛI {i i i Λ is a hybrid flow of H s and {ϕ i ϕ i i Λ satisfies: ϕ i t = αϕit 4 it ϕ i+ τ i+ = ϕ i τ i+. Iplications of Theore. We now devote soe attention to discussing the iplications of Theore and what it does and does not iply about the behavior of H α. Let ι 0 : R 4 R 6 be the inclusion given by ι 0 = 0 T 0 T T. The first iplication of Theore is the following: Corollary : If O is a hybrid periodic orbit of H s then O := ι 0 O is a hybrid periodic orbit of H α. Therefore if H s has a walking gait in two diensions then H α has a walking gait in three diensions. Herein lies the iportance of Theore although one iediately sees its liitations nothing can be said about the stability of this walking gait in the sense of local exponential stability. Utiliing 3 and 4 we can obtain an understanding of the nature of the periodic orbit O. If χ H α is a hybrid flow of H α satisfying 3 then Theore iplies that 4 holds. Therefore the walker stabilies to the upright 285

5 The ain idea in the construction of the second control law is that we would like to drive hq q to ero i.e. we would like to drive the syste to the surface { Z = R q q 6 : hq q = 0 Fig. 3. Phase portraits for different walking gaits obtained by varying ϕτ 0 and ϕτ 0 and keeping the initial values of and constant. The black region shows values of ϕτ 0 and ϕτ 0 that result in convergence to O. position. This is because the roll ϕ will tend to ero as tie goes to infinity since 4 essentially defines a stable linear syste ϕ = αϕ when α > 0 because i t > 0 by the positive definiteness of M. Thus the lateral and sagittal dynaics effectively have been decoupled. We conclude that the periodic orbit O is stable fro initial conditions satisfying 3; a plot of this set of initial conditions can be seen in Figure 3. Unfortunately the dynaics are unstable for initial conditions not satisfying 3. The goal of the second control law to be introduced in the next section is to render the periodic orbit stable. V. 2 nd CONTROL LAW: STABLE WALKING GAITS IN In this section we introduce our second control law with the goal of stabiliing to the surface fro which trajectories converge to the periodic orbit O that is we render O stable. To achieve this goal we utilie the ain result of [8]. A new hybrid control syste. Before introducing the second control law we define a new hybrid syste that iplicitly utilies the first control law as introduced in the previous section. Specifically let H C α = D RG R f α g α where D G and R are defined as for H C defined in Section III. The control syste f α gα is obtained by applying the control law u = K α q T v to H C where v R. In particular f α q q is given as in 2 and gq α q = T. The second control law. Motivated by our desire to satsify 4 let hq q := ϕ + αϕ. wherein the first control law introduced in the previous section should take over roughly speaking the surface Z is the black region in Figure 3. Fro Theore we know that solutions will then converge to the periodic orbit O. We thus will have rendered this periodic orbit stable. With this in ind and otivated by the standard ethod for driving an output to ero in a nonlinear control syste see [9] we define the following feedback control law: v = K αǫ := q q L g α hq q L f α hq q + ǫ hq q where L g α hq q is the Lie derivative of h with respect to g α and L f α hq q is the Lie derivative of h with respect to f α. Note that we know Kαǫ q q is well-defined since L g α hq q = and > 0 by the positive definiteness of M. Utiliing the feedback control law K αǫ q q we obtain a hybrid syste: H αǫ := D G R f αǫ where f αǫ q q := fα q q+gα q qkαǫ q q. We now have the necessary fraework in which to introduce the ain result of this paper. Theore 2: If H s has a locally exponentially stable hybrid periodic orbit O such that = O G is a singleton L f s H 0 then there exists a δ > 0 such that O = ι 0 O is a locally exponentially stable hybrid periodic orbit of H αǫ for all α > 0 and all ǫ such that δ > ǫ > 0. This theore essentially follows fro the previous results of this paper coupled with the Main Theore of [8]. To see this we first ust introduce a global diffeoorphis that akes the structure of H αǫ ore transparent. We then provide a proof of Theore 2. The coordinate transforation. Consider the following global diffeoorphis: Φq q = hq q ϕ =: η where η R and = T ϕ T T R 5. It is easy to see that this is in fact a global diffeoorphis since it clearly is sooth and has a sooth inverse: Φ η = 2 α η

6 where it follows fro the definition of that = ns 2 = s and 3 = ϕ. Note that in this new coordinate syste the surface Z becoes: { η Z = R 6 : η = 0. Transforing H αǫ αǫ. We now proceed to transfor H to this new coordinate syste; we will denote the transfored syste by: H αǫ = D G R αǫ f. The doain and guard for the transfored syste and the original syste are the sae: D = ΦD = D G = ΦG = G by the definition of D and G i.e. by the fact that H is independent of ϕ. In particular if we let then it follows that: { η D = { η G = R 6 : H = cos 2 cos R 6 : H 0 H = 0 sin 4 sin 2 5 < 0 The reset ap for the transfored syste is given by: since R η = Φ R Φ η p 2 η S = 2 p P 2 p 2 = 2 2 when H = 0. Finally the transfored vector field is given by f αǫ η = DΦΦ η f αǫ Φ η where DΦq q is the Jacobian of Φ evaluated at q q. In particular because of the construction of K αǫ We now prove Theore η = η. 6 ǫ Proof: In order to apply the ain result of [8] we ust verify that assuptions H.-H.6 and H2.-H2.6 hold inasuch as they are needed 4 in the proof of the Main Theore of [8]. It is iediately obvious that ost of the assuptions hold so we only coent on the assuptions 4 That is these assuptions do not hold as they are stated in this paper but they are violated to such a sall degree that it does not affect the ain result of this work. that require soe verification space constraints prevent us fro including a ore detailed proof. H.-H.6 hold with the exception that D is not an H open set and G 0. Both of these violations do not affect the Main Result of [8]. For exaple G H 0 and is a sooth anifold of codiension one. Therefore it can be viewed as the ero-level set of H in the region of interest i.e. in a neighborhood of the periodic orbit O. H2.: In our case the -dynaics are not independent of ǫ as is required by H2. in [8]; despite this the axio is sufficiently satisfied inasuch as it is needed in this paper. Specifically fαǫ 0 is independent of ǫ and Theore iplies that Z is invariant under the continuous part of the odel i.e. the dynaics on Z are given by f α which is independent of ǫ. H2.2: Consider the ap Υ : D R 2 with η Υη :=. H It is easy to verify that rankdυη = 2 when restricted to G and so G Z is a n 2-diensional subanifold of D. In addition it follows fro 5 that R G Z Z since η = 0 iplies that p 2 η = 0. H2.3: Since O = ι 0 O clearly O Z. H2.4: Due to the difference in the notion of solution between [8] and this paper we need not take the closure of O since it is already closed. Now a siple calculation shows that: = O G ι 0 = O G Z and so ΦO G Z = Φι 0 is a singleton. H2.5: Due to the specific for of f αǫ the fact that H is independent of ϕ and ϕ and the fact that Φ leaves and fixed it follows that: L f s H = L f αǫ H ι 0 = L H fαǫ Φι 0. Therefore L f s H 0 L fαǫ H2.6: Follows fro 6. H Φι 0 0. Siulation results. To deonstrate the effectiveness of the proposed control law and the usefulness of Theore 2 we provide soe siulation results. For these siulations we begin with a nuerically-coputed periodic orbit O for H s ; this periodic orbit can be seen in Figure 3 in red and blue. It can be verified nuerically that this periodic orbit is stable and satisfies the assuptions of Theore 2; in particular = O G

7 y x Fig. 4. A walking gait for the three-diensional biped. Fig. 5. The exponential convergence of ϕ for different values of ϕτ 0. Therefore by this theore we know that O = ι 0 O is a stable periodic orbit of H αǫ for α > 0 and for sufficiently sall ǫ > 0. We first note that the stability of O iplies the existence of stable walking gaits. An exaple of a specific walking gait can be seen in Figure 4 for ǫ = /5. For this walking gait the biped exponentially converges to the upright position as the theory would indicate. This exponential convergence can be better visualied in Figure 5 where ϕτ 0 is varied while ϕτ 0 = 0 τ 0 = and τ 0 = are held constant. While we know that the periodic orbit O is locally exponentially stable by Theore 2 this theore does not yield any inforation about the radius of convergence of this periodic orbit. A better understanding can be obtained through siulation. Since the region of stability for O has been well-studied see [3] we are interested in the region of stability of O in the ϕ ϕ-plane. We therefore siulate hybrid flows of H αǫ while fixing τ 0 = and τ 0 = and varying ϕτ 0 and ϕτ 0. The resulting behavior can be seen in Figure 6. In this figure the red and blue regions are a result of perturbations in the - dynaics that occur when initial conditions are chosen off of the surface Z. The lateral and sagittal dynaics are no longer decoupled because Theore does not hold for these values. For these initial conditions off of the surface Z the second control law drives the ϕ ϕ-dynaics to this surface as the figure indicates. Finally we note that this siulation result sees to iply that the region of convergence of O is fairly large indicating that the proposed law can effectively stabilie the biped fro a large set of initial conditions. VI. CONCLUSION In this paper a ethod was presented for stably extending two-diensional walking gaits to three diensions using geoetric reduction. The techniques utilied to obtain this result also can be applied to three-diensional bipeds with a hip; this will be the subject of future papers. Finally in this paper the assuption of full actuation was necessary extensions to the case of underactuated bipedal robots is a very interesting and proising research direction. Fig. 6. and ϕ. The convergent behavior of O for different initial values of ϕ REFERENCES [] A. D. Aes R. D. Gregg E. D. B. Wendel and S. Sastry Towards the geoetric reduction of controlled three-diensional robotic bipedal walkers in 3rd Workshop on Lagrangian and Hailtonian Methods for Nonlinear Control LHMNLC 06 Nagoya Japan [2] S. H. Collins M. Wisse and A. Ruina A 3-D passive dynaic walking robot with two legs and knees International Journal of Robotics Research vol. 20 pp [3] A. Goswai B. Thuilot and B. Espiau Copass-like biped robot part I : Stability and bifurcation of passive gaits 996 rapport de recherche de l INRIA. [4] J. Grile G. Abba and F. Plestan Asyptotically stable walking for biped robots: Analysis via systes with ipulse effects IEEE Transactions on Autoatic Control vol. 46 no. pp [5] A. D. Kuo Stabiliation of lateral otion in passive dynaic walking International Journal of Robotics Research vol. 8 no. 9 pp [6] J. E. Marsden and T. S. Ratiu Introduction to Mechanics and Syetry ser. Texts in Applied Matheatics. Springer 999 vol. 7. [7] T. McGeer Passive dynaic walking International Journal of Robotics Research vol. 9 no. 2 pp [8] B. Morris and J. W. Grile A restricted Poincaré ap for deterining exponentially stable periodic orbits in systes with ipulse effects: Application to bipedal robots in Conference on Decision and Control Seville Spain [9] S. Sastry Nonlinear Systes: Analysis Stability and Control. Springer-Verlag 999. [0] M. W. Spong and F. Bullo Controlled syetries and passive walking IEEE Transactions on Autoatic Control vol. 50 no. 7 pp