Planar Bipedal Walking with Foot Rotation

Transcription

1 Planar Bipedal Walking with Foot Rotation Jun Ho Choi and J. W. Grizzle Abstract This paper addresses the key problem o walking with both ully actuated and underactuated phases. The studied robot is planar, bipedal, and ully actuated in the sense that it has eet with revolute, actuated ankles. The desired walking motion is assumed to consist o three successive phases: a ully-actuated phase where the stance oot is lat on the ground, an underactuated phase where the stance heel lits rom the ground and the stance oot rotates about the toe, and an instantaneous double support phase where leg exchange takes place. The main contribution o the paper is to provide a provably asymptotically stabilizing controller that integrates the ully-actuated and underactuated phases o walking. By comparison, existing humanoid robots, such as Asimo and Qrio, use only the ully-actuated phase (i.e., they only execute lat-ooted walking), or RABBIT, which uses only the underactuated phase (i.e., it has no eet, and hence walks as i on stilts). The controller proposed here is organized around the hybrid zero dynamics o Westervelt et al. (2003) in order that the stability analysis o the closed-loop system may be reduced to a one-dimensional Poincaré map that can be computed in closed orm. I. INTRODUCTION Over the past seven years, several remarkably anthropomorphic robots have been constructed. Speciically, we have in mind the well known Honda Robot, Asimo 1, 2, Sony s biped, SDR-4X 3, and Jogging Johnnie at the University o Munich 4, 5. The Honda and Sony robots are especially noteworthy or their autonomy, while Jogging Johnnie s construction appears to be particularly light and graceul (1.8 m tall, 40 Kg mass, which includes sensors, actuators and control hardware, though power is supplied through a cable). Each o these robot s control systems is organized around a high-level trajectory generator or the individual joints o the robot, combined with lowlevel servoing to ensure trajectory tracking. There are some dierences in how the low-level servoing is implemented Honda uses PD control 2, Jogging Johnnie uses eedback linearization 5, while Sony has revealed little about their algorithms but these dierences are airly insigniicant. In each case, the overall stability o the robot s motion has been ensured by the zero moment point (ZMP) criterion (see Figure 1), and consequently, these robots literally walk lat ooted. A stability analysis o a lat-ooted walking gait or a ive-link biped with an actuated ankle was carried out numerically in 6, 7, using the Poincaré return map. The control law used eedback linearization to maintain the robot s posture and advance the swing leg; trajectory This work was supported by NSF grant ECS Jun Ho Choi and J. W. Grizzle are with the Department o Electrical Engineering and Computer Science, University o Michigan, Ann Arbor, Michigan , USA {junhoc, grizzle}@umich.edu (a) P Fig. 1. The ZMP (Zero Moment Point) principle in a nutshell. Idealize a robot in single support as a planar inverted pendulum attached to a base consisting o a oot with torque applied at the ankle and all other joints are independently actuated. Assume adequate riction so that the oot is not sliding. In (a), the robot s nominal trajectory has been planned so that the center o pressure o the orces on the oot, P, remains strictly within the interior o the ootprint. In this case, the oot will not rotate (i.e, the oot is acting as a base, as in a normal robotic manipulator), and thus the system is ully actuated. It ollows that small deviations rom the planned trajectory can be attenuated via eedback control, proving stabilizability o the walking motion. In case (b), however, the center o pressure has moved to the toe, allowing the oot to rotate. The system is now underactuated (two degrees o reedom and one actuator), and designing a stabilizing controller is nontrivial, especially when impact events are taken into account. The ZMP principle says to design trajectories so that case (a) holds; i.e., walk lat ooted. Humans, even with prosthetic legs, use oot rotation to decrease energy loss at impact. tracking was only used in the limited sense that the horizontal component o the center o mass was commanded to advance at a constant rate. The unilateral constraints due to oot contact were careully presented. Motivated by energy eiciency, elegant work in 8, 9 has shown how to realize a passive walking gait in a ully actuated biped robot walking on a lat surace. Stability o the resulting walking motion has been rigorously established. The main drawback, however, is that the assumption o ull actuation once again restricts the oot motion to lat-ooted walking. For walking gaits that include oot rotation, various ad hoc control solutions have been proposed in the literature 10, 11, 12, 13, 14, 15, but none o them can guarantee stability in the presence o the underactuation that occurs during heel roll or toe roll. Our previous work on point eet 16, 17, 18, 19, 20 ideally positions us to handle this underactuation; indeed, conceptually, a point oot corresponds to continuous rotation about the toe throughout the entire stance phase (e.g., walking like a ballerina or as i on stilts). Work in 21 shows that plantarlexion o the ankle, which initiates heel rise and toe roll, is the most eicient method to reduce energy loss at the subsequent impact o the swing leg. This motion is also necessary or the esthetics o mechanical walking. In this paper, we extend our analysis o walking with point eet to design a controller that provides asymptotically (b) P

2 p v c p v c q 3, u 3 q 4, u 4 p h c q 4, u 4 p h c q 3, u 3 q 5, u 5 q 2, u 2 q 5, u 5 q 2, u 2 Fig. 2. The three phases o walking modelled in this paper: ully-actuated phase where the stance oot is lat on the ground, underactuated phase where the stance heel rises rom the ground and the stance oot rotates about the toe, and double support phase where the swing oot impacts the ground. stable walking with an anthropomorphic oot motion. In particular, the achieved walking motion consists o three successive phases 1 : a ully-actuated phase where the stance oot is lat on the ground, an underactuated phase where the stance heel lits rom the ground and the stance oot rotates about the toe, and an instantaneous double support phase where leg exchange takes place, see Figure 2. II. ROBOT MODEL The robot considered in this paper is bipedal and planar with N 4 rigid links connected by ideal (rictionless) revolute joints to orm a tree structure (no closed kinematic chains). It is assumed to have two identical open chains called legs that are connected at a point called the hips. The link at the extremity o each leg is called a oot and the joint between the oot and the remainder o the leg is called an ankle. The eet are assumed to be orward acing. The orward end o each oot is called a toe and the back end is called a heel. Each revolute joint is assumed to be independently actuated. It is assumed that walking consists o three successive phases, a ully-actuated phase, an underactuated phase, and a double support phase, see Figure 2 and 3. The detailed assumptions or the robot and the gait are listed in Appendix A. A representative robot is shown in Figure 4 along with a coordinate convention. x Su ẋ = ua (x ) + g (x )ur Fully actuated phase Fig. 3. x + u = u (x ) x + = u (x u ) Diagram o system. ẋu = u(xu) + gu(xu)u Underactuated phase x u S u 1 For simplicity, heel strike with a subsequent heel roll is not addressed. It can be handled in the same manner as toe roll. θ q 6, u 6 q 1, u 1 q 6, u 6 q 1, u 1 Fig. 4. Model o a 7-link robot with coordinate convention. In general, or N-link robot, it is assumed that q N 1 is the angle between the oot and the emur and q N is the angle between the ground and the oot. A. Underactuated phase The underactuated phase is when the stance heel o the robot rises rom the ground and the robot begins to roll over the stance toe; this condition is characterized by the oot rotation indicator (FRI) point o 22 being strictly in ront o the stance oot. The stance toe is assumed to act as a pivot; this condition is characterized by the orces at the toe lying within the allowed riction cone. Both o these conditions (i.e., oot rotation and non-slip) are constraints that must be imposed in the inal controller design phase, as in 17, Sec. VI. Since there is no actuation between the stance toe and the ground, the dynamics o the robot in this phase is equivalent to an N-DOF robot with unactuated point eet and identical legs, as treated in 17. Deine the generalized coordinates as q u = (q 1,, q N ) T Q u, where Q u is a simply connected open subset o IR N. The dynamics are obtained using the method o Lagrange, yielding D u q u + C u q u + G u = B u u, (1) where u = (u 1,, u N 1 ) T is the vector o torques applied at the joints. The dynamic equation in state-variable orm is expressed as ẋ u = u (x u ) + g u (x u )u. B. Fully-actuated phase model During the ully-actuated phase, the stance oot is assumed to remain lat on the ground without slipping. The ankle o the stance leg is assumed to act as an actuated pivot. Since the stance oot is motionless during this phase, the dynamics o the robot during the ully-actuated phase is equivalent to an N 1 DOF robot without the stance oot and with actuation at the stance ankle, as studied in 23. Let q = (q 1,, q N 1 ) T Q be the coniguration variables, where q 1,, q N 2 denote the relative angles o the joints except the stance ankle, q N 1 denotes the angle o the ankle joint, and Q is a simply connected open subset o IR N 1, see Figure 4. Note that because the stance oot remains on the ground, q N 1 is now an absolute angle (i.e., it is reerenced to the world rame). q 7

3 The dynamics or the ully-actuated phase are derived using the method o Lagrange, yielding a model o the orm D (q ) q + C (q, q ) q + G (q ) = B 1 u R + B 2 u A, (2) where q are the velocities, u R = (u 1,, u N 2 ) T are the inputs applied at the joints except the ankle joint, and u A = u N 1 is the input at the ankle joint. The state is taken as x = (q ; q ) T Q and the dynamic equation is given by 2 ẋ = q D 1 ( C q G + B 2 u A ) 0 + D 1 B 1u R =: ua (x ) + g (x )u R. (3) Note that, to satisy the condition that the stance oot is lat on the ground, the FRI point needs to be kept strictly within the support region. This constraint must be imposed on designing the controller as in 17, Sec. V. C. Double support phase During the double support phase, the swing oot impacts the ground. It is assumed that the swing oot is parallel to the ground at impact. It is also assumed that the eet are arc shaped so that the only contact points with the ground are the heel and the toe. Due to the impacts, impulsive orces are applied at the toe and the heel, which cause discontinuous changes in the velocities; however, the position states are assumed to remain continuous 24. Representing the double support phase requires an N+2 DOF model (e.g. N DOF or the joints and 2 DOF or the position o the center o mass). Adding Cartesian coordinates, (p h c, p v c), to the center o mass o the robot gives q d = (q u ; p h c ; p v c) and q d = ( q u ; ṗ h c ; ṗ v c), see Figure 4. Let Υ h (q d ) and Υ t (q d ) denote the Cartesian coordinates o the swing heel and the swing toe, respectively. The method o Lagrange yields the dynamics or the double support phase as ollows: D d (q d ) q d + C d (q d, q d ) q d + G d (q d ) where u = (u T R, u A) T, E h d = ( Υt(q d ) = B d u + E h d δf h + E t dδf t, (4) ( Υh (q d ) q d ) T, E t d = ) T q d, δfh denotes the impulsive ground reaction orce at the swing heel, and δf t denotes the impulsive ground reaction orce at the swing toe. Under the hypothesis IH6 (the actuators not being impulsive) and IH1 (which is the stance oot neither rebounds 2 Note that the ankle torque is included in u A Z (x ); the reason or this will be clear in Section III. nor slips), ollowing the procedure in 16 gives x + = R 0 q d R 0 Π Dd q = u(x d ), (5) where R is a relabeling matrix and Π = E 1T E 2T 0 D d Ed 1 Ed 2 d 0 0 d (6) D. Foot Rotation, or Transition rom Full Actuation to Underactuation The transition rom a lat oot to rotation about the toe can be initiated by causing the angular acceleration about the stance toe to become negative. To characterize the motion o the stance oot or equivalently, when the robot transitions rom ull actuation oot lat on the ground to underactuation oot rotates about the toe, the oot rotation indicator (FRI) point o Goswami is used 22, eq. (6). By enorcing that the FRI point is strictly in ront o the end o the oot, the transition is initiated. I torque discontinuities 3 are allowed as they are assumed to be in this paper when to allow oot rotation becomes a control decision. Here, in view o simpliying the analysis o periodic orbits in Section IV, the transition is assumed to occur at a ixed point in the ully-actuated phase. Hence, H u = θ (q ) θ, where θ (q) is the angle o the hips with respect to the stance ankle (see Figure 4) and θ is a constant to be determined. The positions and the velocities are assumed to remain continuous even with the discontinuous torque. The ensuing initial value o the underactuated phase, x + u, is deined so as to achieve continuity in the position and velocity variables: x + q + u = u q + u q = π q 0 =: u (x ). (7) Continuity o the torques is not imposed, and hence neither is continuity o the accelerations. It is assumed that the control law in the underactuated phase will be designed to achieve the FRI point in ront o the toe. E. Overall Hybrid Model As in 25, the overall model can be expressed as a nonlinear hybrid system containing two state maniolds 3 This is a modeling decision. In practice, the torque is continuous due to actuator dynamics. It is assumed here that the actuator time constant is small enough that it need not be modeled.

4 (called charts in 26): S u X Σ : X = T Q F : ẋ = ua (x ) + g (x )u R S u = {x T Q H u(x ) = 0} T u u = u (x u ) X u = T Q u F u : ẋ u = u (x u ) + g u (x u )u Σ u : Su = {x u T Q u Hu (x u ) = 0} Tu : x + = u(x u ) where, or example, F is the low on state maniold X, is the switching hyper-surace or transitions between and X u, T u : S u X u is the transition unction (8) applied when x S s. The transition rom the underactuated phase to the ullyactuated phase occurs when the swing oot impacts the ground. Hence, H u (x u ) = Υ v h (x u), where Υ v h (x u) denotes the vertical coordinate o the swing heel. III. CONTROL LAW DEVELOPMENT BASED ON THE HYBRID ZERO DYNAMICS In a certain sense, the basic idea o the control law design is quite straightorward. The greatest diiculties in the control design and analysis involve the underactuated phase o the motion. Following the development in 17, 19, we use the method o virtual constraints to create a two-dimensional zero dynamics maniold Z u in the 2Ndimensional state space o the underactuated phase. This requires the use o the ull complement o N 1 actuators on the robot. In the ully-actuated phase, we have one less degree o reedom because the stance oot is motionless and lat on the ground. Consequently, we use N 2 actuators all actuators except the ankle o the stance oot to create a two-dimensional zero dynamics maniold Z that is compatible with Z u in the sense that the ollowing invariance conditions hold: u (Su Z ) Z u and u(s u Z u ) Z. The actuation authority at the ankle is subsequently employed or stability and eiciency augmentation, and or enorcing the non-rotation o the oot. The invariance conditions guarantee the existence o a hybrid zero dynamics or the closed-loop hybrid model. The techniques in 17 are then extended to compute the Poincaré map o the closed-loop system in closed orm. Precise stability conditions then ollow. In the ollowing, the key elements o this program are outlined. Due to space limitations, a more complete analysis, which is actually a simpliication 4 o 25, will be presented elsewhere. 4 The analysis is similar to running because there are multiple phases. The problem treated here is simpler because the light phase o running, which has three degrees o underactuation, is replaced with a ully-actuated phase. A. Control o the underactuated phase Since the stance oot toe acts as a pivot and there is no actuation at the stance oot toe, the eedback design proceeds as in 17. Let y u = h u (x u ) be a (N 1) 1 vector o output unctions satisying the ollowing hypotheses: HHU1) The output unction h u (x u ) depends only the coniguration variables; HHU2) The decoupling matrix L gu L u h u is invertible or an open set Qu Q u ; HHU3) Existence o θ u (q u ) such that h u (q u ); θ u (q u ) is a dieomorphism; HHU4) There exists a point in Q u where h u vanishes. HHU5) There exists a unique point q u Q u such that (h u (q u ), Υ v h (q u )) = (0, 0), Υ h h (q u ) > 0 and the rank o h T u Υ v h T at q u equals to N, where Υ h h (x u) denotes the horizontal coordinate o the swing heel. Then there exists a smooth maniold Z u = {x u T Q u h u (x u ) = 0, L u h u (x u ) = 0}, called the zero dynamics maniold, and S u Z u is smooth. S u Z u is one-dimensional i S u Z u. Dierentiating the output y u twice yields, ÿ u := v u (9) = L 2 u h u (x u ) + L gu L u h u (x u )u. (10) Since the decoupling matrix L gu L u h u (x u ) is invertible, the eedback control u := (L gu L u h u (x u )) 1 (L 2 u h u (x u )) renders the zero dynamics maniold invariant. In addition to the hypotheses HHU1 HHU5, i the hypothesis RH5 is also satisied, then the ully-actuated phase phase zero dynamics in the coordinates o z u := (θ u, σ u ) = (θ u, d u (q u ) q u ) can be written as θ u = κ 1 u(θ u )σ u (11) σ u = κ 2 u(σ u ), (12) where u A is the torque applied at the stance ankle, d u is the last row o D u, and σ u is the angular momentum about the stance toe during the underactuated phase, 27. Equations (11) and (12) are written as ż u = Zu (z u ). The transition map rom the ully-actuated phase to the underactuated phase on the hybrid zero dynamics becomes θ + = θ R 0 q u, (13) σ + = δ uσ u, (14) where δu is a constant that can be calculated using the properties studied in 27. B. Control design or ully-actuated phase Since the stance oot is motionless and acting as a base during this phase, the model only has N 1 DOF. Consequently, the robot is ully actuated, opening up many eedback design possibilities. For example, we could, in principle, design or an empty zero dynamics, eedback linearize the model, etc. though we would run a high

5 risk o requiring so much ankle torque that the oot would rotate, thereby causing underactuation. Instead, we ollow a design where, in principle, the ankle torque could be used exclusively or ensuring that the oot does not rotate, but in most cases, it can also be used to augment stability and eiciency o the overall walking cycle. N 2 virtual constraints are used to create a twodimensional zero dynamics or the ully-actuated phase that is driven by the ankle torque. Let y = h (x ) be a (N 2) 1 vector o output unctions. Let the output unction y satisy the ollowing hypotheses: HHF1) The output unction h (x ) depends only on the coniguration variables o the ully-actuated phase; HHF2) There exists u A IR such that the decoupling matrix L g L u A h is invertible or an open set Q Q ; HHF3) There exists θ (q ) such that h (q ); θ (q ) is dieomorphism; HHF4) There exists a point where h vanishes; HHF5) There exists a unique point q Q such that y = h (q ) = 0, Hu (q ) = 0 and h ; H u has ull rank. Then there exists a smooth maniold Z ua = {x T Q h (x ) = 0, L u A h (x ) = 0}, called the zero dynamics maniold, and S u Z is smooth. S u Z is one-dimensional i S u Z. Dierentiating the output y or the ully-actuated phase twice gives ÿ := v (15) = L 2 u A h (x ) + L g L u A h (x )u R. (16) Since L g L u A h is invertible, the eedback control u R = (L g L u A h (x )) 1 (L 2 u A h (x )) renders the zero dynamics maniold or the ully-actuated phase invariant. In addition to the hypotheses HHF1 HHF5, i the hypothesis RH5 is also satisied, then in the coordinates o z := (θ, σ ) = (θ, d (q ) q ) restricted to the zero dynamics maniold, the ully-actuated phase zero dynamics can be written as θ = κ 1 (θ )σ (17) σ = κ 2 (σ ) + u A, (18) where u A is the torque applied at the stance ankle, d is the last row o D, and σ is the angular momentum about the stance ankle during the ully-actuated phase 27. Equations (17) and (18) are written as ż = ua Z (z ). The transition map rom the ully-actuated phase to the underactuated phase on the zero dynamics becomes q θ u + = θ u, (19) π where δ u σ + u = δ u σ, (20) is a constant that can be calculated using 27. C. Hybrid zero dynamics Let Z be the zero dynamics maniold o the ullyactuated phase and ż = ua (z ) be the associated zero dynamics driven by u A. Let u be the transition map rom the ully-actuated phase to the underactuated phase. Let Z u be the zero dynamics maniold o the underactuated phase and ż u = u (z u ) be the associated zero dynamics. Let u be the transition map rom the underactuated phase to the ully-actuated phase. I z S u Z, u (z ) Z u and z u Su Z u, u(z u ) Z, then ż = ua Z (z ), z Su Z z u + = u (z ), ż u = Zu (z u ), z Su Z zu Su Z u (21) z + = u(z u ), zu Su Z u is an invariant hybrid subsystem o the ull-order hybrid model. The system (21) is called the hybrid zero dynamics and Z and Z u are hybrid zero dynamics maniolds. Remark 1: By deinition, Z and Z u are hybrid zero dynamics maniolds i, and only i, z Su Z, h u u (z ) = 0, (22) L u h u u (z ) = 0, (23) and z u S u Z u and u A = 0, h u(z u ) = 0, (24) L u A h u(z u ) = 0. (25) How to achieve these conditions is not developed here or reasons o space. The required conditions are a straightorward extension o 17, Sec. V. Remark 2: Another very important result not proved here is that asymptotically stable periodic orbits o the hybrid zero dynamics are asymptotically stabilizable periodic orbits in the ull-order model. The proo is based on extending the main result o 16, Thm. 2. For an analogous result in running, see 25. In the next section, the Poincaré map o the hybrid zero dynamics is computed in closed orm. IV. STABILITY ANALYSIS ON THE HYBRID ZERO DYNAMICS In general, due to the input at the stance ankle during the ully-actuated phase, the robot can move backward beore it completes a step. In other words, the angular momentum about the stance ankle can be zero beore entering the unactuated phase. In this paper, the angular momentum is assumed not to be zero during a step; see CH6 in Appendix A. One can think o this hypothesis as a dierence between walking and dancing. Since σ 0 during the ullyactuated phase, ζ = σ2 2 is a valid coordinate transormation. In general, u A can be any unction o the robot s states or the zero dynamics to exist. In this paper, or simplicity,

6 it is assumed to be a uncion o θ only. Then, (17) and (18) become dζ = σ dσ = κ2 (θ ) κ 1 (θ ) + u A(θ ) κ 1 (θ ) dθ. (26) Let z = (θ, σ ) Su Z and θ + (19). For θ + θ θ, deine θ be deined as in κ 2 V ua Z (θ ) = (ξ) θ + κ 1 (ξ) + u A(ξ) κ 1 dξ, (27) (ξ) ua, max VZ = max V ua θ + θ θ Z (θ ). (28) I (δ u)2 ζ ua, max VZ > 0, then (26) can be integrated, which results in 1 2 (σ )2 1 2 (σ+ )2 = ζ ζ+ ua = VZ (θ ). (29) With (14), the Poincaré map or the ully-actuated phase ρ : Su S u on the hybrid zero dynamics is deined as ρ (ζu ) = (δu) 2 ζu V ua Z (θ ). (30) For the underactuated phase, the zero dynamics is equivalent to the robot with unactuated point eet, 17. Let zu = (θu, σu ) Su Z u and let θ u + be deined as in (13). Following the procedure in 17 with (11) and (12), i (δu) 2 ζu VZ max u > 0, then 1 2 (σ u ) (σ+ u ) 2 = ζu ζ u + = V Zu (θu ), (31) where V Zu (θ u ) = θu κ 2 u(ξ) θ u + κ 1 dξ, u(ξ) (32) VZ max u = max θ u + θ u θu V Zu (θ u ) (33) The Poincaré map or the underactuated phase ρ u : S u Su on the hybrid zero dynamics is deined with (14) as ollows. ρ u (ζ ) = (δu ) 2 ζ V Z u (θu ). (34) Hence, the Poincaré map or the overall reduced system in (θ u, ζ u ) coordinates, ρ(ζ u ) : S u Z u S u Z u, is deined as ollows ρ(ζ u ) = ρ u ρ (ζ u ) = (δ u ) 2 (δ u) 2 ζ u (δ u ) 2 V u abs Z (θ ) V Z u (θ u )(35) with domain o deinition D = {ζu > 0 (δu) 2 ζu ua, max VZ > 0, (δ u ) 2 (δu) 2 ζu (δ u ) 2 V ua Z (θ ) V Z max u (θu ) > 0}.(36) Theorem 1: Under the hypotheses RH1 RH5, GH1 GH6, and IH1 IH7 in Appendix A, HHF1 HHF5, and HHU1 HHU5, ζ u = (δu )2 V ua Z (θ ) + V Z u (θ u ) 1 (δ u )2 (δ u) 2 (37) is an exponentially stable ixed point o (35) i, and only i, 0 < (δ u ) 2 (δ u) 2 < 1, (38) (δ u )2 (δ u) 2 V Zu + (δ u )2 V ua Z 1 (δ u )2 (δ u) 2 + V max Z u < 0, (39) (δ u)2 (δu) 2 V ua Z + (δ u)2 V Zu 1 (δ u)2 (δu) + V ua,max 2 Z < 0. (40) Proo: D is non-empty i, and only i, (δ u)2 (δu) 2 > 0. I there exists ζu D satisying ρ(ζu) = (δ u)2 (δu) 2 ζu (δ u)2 V ua Z (θ ) V Z u (θu ), then ζ is an exponentially stable ixed point i, and only i, 0 < (δ u)2 (δu) 2 < 1, and in this case, (37) is the value o ζ. Finally, (39) and (40) are the necessary and suicient conditions or (37) to be in D. Note that the stability o the reduced model is not aected by the choice o u A since δ u does not depend on u A. However, the ixed point ζ is aected by u A. Remark 3: To render these analytical results useul or eedback design, a convenient inite parametrization o the virtual constraints must be introduced as in 17, Sec. V. This will introduce ree parameters into the hybrid zero dynamics, (21). A minimum energy cost criterion can then be posed and parameter optimization applied to the hybrid zero dynamics to design a provably stable, closed-loop system with satisied design constraints, such as walking at a prescribed average speed, the orces on the support leg lying in the allowed riction cone, and the oot rotation indicator point within the hull o the oot during the ullyactuated phase and strictly in ront o the oot in the underactuated phase. V. SPECIAL CASE Since the eedback design or the underactuated phase has been published and illustrated elsewhere 20, 19, we consider the special case o lat-ooted walking, that is, the gait consists only o the ully-actuated phase ollowed by an instantaneous double support phase, in order to illustrate the role o the ankle torque in our eedback design. Specializing the calculations in Section III and IV to this case, the Poincaré map o the hybrid zero dynamics is 5 ρ(ζ ) = (δ u) 2 ζ V ua Z (θ ), (41) where V ua Z, the potential energy (see 19), is given in (27). The stability theorem becomes Corollary 1: Under the hypotheses RH1 RH5, GH1 GH6, and IH1 IH7 in Appendix A, HHF1 HHF5, and HHU1 HHU5, ζ = V ua Z (θ ) 1 (δu) (42) 2 5 Conceptually, we are considering an instantaneous underactuated phase so that V Zu = 0 and δ u = 1

7 is an exponentially stable ixed point o (35) i, and only i, 0 < (δu) 2 < 1, (43) (δu) 2 V ua Z 1 (δu) + V ua,max 2 Z < 0. (44) These conditions are the same as in 17, Th. 3 or pointeet, with the exception that the potential energy term V ua Z can be shaped by choice o the ankle torque, u A ; see second term in (27). Dierent shapes o the potential energy result in dierent ixed points and dierent domains o deinition o the ixed points. Figure 5 shows the potential energy V ua Z (θ ) during the ully-actuated phase with dierent ankle torques. The solid line is when the ankle torque is a aine unction o s and the dashed line is when ankle torque is identically zero. PSrag replacements V ua Z (θ ) s = θ θ+ θ θ+ Fig. 5. Potential energies V u A Z with two dierent ankle inputs or a walking speed o 1.5 m/s. The potential energy can be shaped by the choice o u A. The solid line is when u A = 21.1s 2.9 (Nm) and the dashed line is when u A = 0. VI. CONCLUSION This paper has outlined a solution to the key problem o walking with both ully-actuated and underactuated phases. The studied robot was planar, bipedal, and ully actuated in the sense that it has non-trivial eet with revolute, actuated ankles and all other joints are independently actuated. The desired walking motion was assumed to consist o three successive phases: a ully-actuated phase where the stance oot is lat on the ground, an underactuated phase where the stance heel lits rom the ground and the stance oot rotates about the toe, and an instantaneous double support phase where leg exchange takes place. The main contribution o the paper was to extend the hybrid zero dynamics o 17 to a hybrid model with multiple continuous phases and varying DOF and degrees o actuation. The developed method provides a provably asymptotically stabilizing controller that integrates the ully-actuated and underactuated phases o walking. The role o the ankle torque in the proposed eedback design was emphasized by also considering the special case o a lat-ooted walking gait. The ankle torque was seen as a means to directly adjust the potential energy o the hybrid zero dynamics. In this paper, the ankle-torque eedback was restricted to depend only on position, which is analogous to shaping the potential energy with a nonlinear spring. A larger class o eedbacks will be considered in uture work. A. Hypotheses APPENDIX The hypotheses or the robot are: RH1) The robot consists o N rigid links with revolute joints; RH2) The robot is planar; RH3) The robot is bipedal with identical legs connected at hips; RH4) The joints between adjacent links are actuated; RH5) The coordinate o the robot consists o N 1 relative angles, q 1,, q N 1, and one absolute angle, q N. The hypotheses or gait are: GH1) Walking consists o three successive phases, ullyactuated phase, underactuated phase, and double support phase; GH2) The stance oot remains on the ground during ullyactuated phase; GH3) The stance oot does not slip during ully-actuated phase; GH4) The stance toe acts as a pivot during underactuated phase; GH5) The stance ankle leaves the ground without interaction; GH6) There is no discontinuous change in positions and velocities at transition rom ully-actuated phase to underactuated phase. The hypotheses or impact are: IH1) The swing oot has neither rebound nor slipping during impact; IH2) Ater impact, the stance toe leaves the ground without any interaction with the ground; IH3) The impact is instantaneous; IH4) The reaction orces due to the impact can be modeled as impulses; IH5) The impulsive orces result in discontinous changes in the velocities while the position states remain continuous; IH6) The actuators at joints are not impulsive; IH7) The swing ankle and the swing toe touch the ground at the same time. The hypotheses or the closed-loop chain o double integrators, ÿ = v, are: CH2) Existence o solutions on IR 2N 2 and uniqueness; CH3) Solutions depending continuously on the initial conditions; CH4) The origin being globally asymptotically stable and the convergence being achieved in inite time;

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