Hybrid Invariance in Bipedal Robots with Series Compliant Actuators

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1 Hybrid Invariance in Bipedal Robots with Series Compliant Actuators B. Morris and J.W. Grizzle Abstract Stable walking motions in bipedal robots can be modeled as asymptotically stable periodic orbits in nonlinear systems with impulse eects. The method o hybrid zero dynamics, previously used to analyze planar walking in bipeds with one degree o underactuation, is extended to address the increased degrees o underactuation and the additional impact invariance conditions that arise when actuator dynamics are explicitly modeled. The resultant controller is parameterized and includes a discrete eedback in the parameters that is active only in the instantaneous double support phase. The controller design method is illustrated on a ive-link planar walker with series compliant actuation, that is, a robot where a compliant element has been deliberately inserted between each actuated joint and its corresponding motor in order to increase the overall energy eiciency o locomotion. q mot q rel I. INTRODUCTION In legged robots, the physical introduction o tuned springs into an otherwise rigid mechanism can signiicantly improve energy eiciency. The energetic beneits are twoold: within the strides o walking and running, springs can store and release some o the energy that would otherwise be lost as actuators do negative work [2]; and at oot touchdown events, springs isolate relected motor inertias rom the energydissipating eects o rigid collisions. These and other uses o lexible elements have been demonstrated on running robots such as RHex [20], Scout [16], Sprawlita [4], and the notably eicient ARL Monopod II [1]. And while the beneits o energy storage are most evident in running, in practice many robots must quite literally walk beore they can run. In these cases compliance must be taken into account in the design and control o walking gaits, either explicitly by modeling, or implicitly by treating nonrigid eects as disturbances to a ully rigid model. Obtaining the energetic beneits o compliance is not without cost: delivering torque through compliant elements poses several challenges or control design. There is an obvious increase in the degrees o reedom o the robot model, and hence, the degree o underactuation. This is a widely recognized issue in robotics; see [22], [21], [3] and reerences therein. An additional challenge particular to legged robots arises rom the impulsive eects occurring when the swing leg impacts the ground. When torque at a joint is generated by a motor and drivetrain in series with a spring as in this paper) the spring isolates the motor and drivetrain rom the eects o an impact. Post-impact values o rotor position and velocity match their pre-impact This work was supported by NSF grant ECS B. Morris and J.W. Grizzle are with the Control Systems Laboratory, EECS Department, University o Michigan, Ann Arbor, Michigan , USA. morrisbj, grizzle}@umich.edu. Fig. 1. Let: A representative example, intentionally anthropomorphic, o the class o N-link biped robot models considered. Right: A schematic o a rotational joint with series compliant actuation. values, and similar boundary conditions arise or joint torque. Seemingly benign, these additional post-impact boundary conditions alter the structure o the impact map and can signiicantly complicate controller design. The method o hybrid zero dynamics, as presented in [25] or the control o planar walking, assumed that any actuator dynamics were suiciently ast that they could be neglected in the controller design process. The novel element o this paper is the extension o the hybrid zero dynamics ramework to address unique aspects o stabilizing walking motions using actuators with nontrivial series compliance. Speciic attention is given to the post-impact boundary conditions, which no longer satisy the linear structures assumed in [25]. Treating actuator dynamics in this ramework will lead to reduced dimensionality stability tests or closed-loop walking gaits despite the increased degrees o underactuation that accompany compliant actuation. See Fig. 1 or a description o the class o robots considered in this paper, along with a schematic diagram o a lexible actuator. One example o the pictured compliant mechanism is the MIT Series Elastic Actuator, which uses sti springs and an inner-loop eedback controller to achieve reliable orce control [18], [17]. Another, the AMASC Actuator with Mechanically Adjustable Series Compliance) designed by Hurst [11], consists o a drive motor connected in series with a pair o large, variable stiness springs. Unlike the MIT Series Elastic Actuator, the AMASC is designed to mechanically store signiicant amounts o energy that would otherwise be wasted when the actuator does negative work. The rest o the paper is organized as ollows: Section II summarizes background material on systems with impulse eects, periodic orbits in such systems, and the method o Poincaré sections. Section III contains two theorems, the

2 main theoretical results o the paper. The concept o a hybrid zero dynamics HZD) is deined in a more general manner than in [25], ollowed by a theorem that provides suicient conditions and a controller design recipe or stabilizing a periodic orbit in a nonlinear system with impulse eects. One o the conditions o this theorem, that the continuous-phase zero dynamics maniold is invariant under the impact map, can be diicult to meet in practical examples. The second theorem o this section provides conditions or constructing an embedding o the original system with impulse eects into a larger system with impulse eects where the mentioned impact invariance condition is easier to meet. Moving toward applications, Section IV presents a model o a class o planar biped robots with series actuation compliance; properties o the models are summarized that aid in applying the theorems o Section III. Section V presents the application o the results o Section III to a particular instance o the biped models o Section IV. A simulation o the closed-loop walking gait shows properties that are predicted by theory. Conclusions are given in Section VI. II. TECHNICAL BACKGROUND A. Systems with impulse eects Consider a control system ẋ = x) + gx)u, 1) where x X, an open connected subset o IR n, u IR m, and and the columns o g are C vector ields on X. An impact or switching) surace is a co-dimension one C submaniold S := x X Hx) = 0}, where H : X IR is C H and x S, x x) 0. An impact or reset) map is a C unction : S X. A C system with impulse eects is a model o the orm ẋ = x) + gx)u x S x + = x ) x 2) S, where x t) := lim τ րt xτ) and x + t) := lim τ ցt xτ) denote, respectively, the let and right limits o a trajectory, xt). In simple words, a trajectory o the model is speciied by the dierential equation 1) until its state impacts the hypersurace S. At this point, the impact map assigns a new initial condition rom which the dierential equation evolves until the next impact with S. In order to avoid the state having to take on two values at the same instant, the impact event is, roughly speaking, described in terms o the values o the state just prior to impact at time t, and just ater impact at time t +. These values are represented by the let and right limits, x and x +, respectively. A ormal deinition o a solution is easily written down by piecing together appropriately initialized solutions o 1); see [26], [8], [15], [5]. A choice must be made whether to take a solution ϕt) o 2) to be a let- or a right-continuous unction o time at each impact event; here, solutions are assumed to be right continuous as in [8]. B. Periodic orbits A C autonomous system with impulse eects is given by 2) with u = 0, namely, ẋ = x) x S x + = x ) x S. A solution ϕt) o 3) is periodic i there exists a inite T > 0 such that ϕt + T) = ϕt) or all t [t 0, ). A set O X is a periodic orbit i O = ϕt) t t 0 } or some periodic solution ϕt). While a system with impulse eects may certainly have periodic solutions that do not involve impact events, they are not o interest here because they could be studied more simply as solutions o 1). I a periodic solution has an impact event, then the corresponding periodic orbit O is not closed; see [8], [14]. Let Ō denote its set closure. A periodic orbit O is transversal to S i its closure intersects S in exactly one point, and or x := Ō S, L Hx ) := H x x )x ) 0 in words, at the intersection, Ō is not tangent to S). Notions o stability in the sense o Lyapunov, asymptotic stability, and exponential stability o orbits ollow the standard deinitions; see [13, pp. 329], [8], [15]. C. Poincaré return map The method o Poincaré sections is the primary tool used to test the stability o periodic orbits in nonlinear systems. When such orbits occur in systems with impulse eects, it is natural to select the impact surace S as the Poincaré section. To deine the return map, let ϕt, x 0 ) denote the maximal solution o ẋ = x) in 3) with initial condition x 0 at time t 0 = 0. The time-to-impact unction, T I : X IR }, is deined by int 0 ϕt, x 0 ) S} i t such that T I x 0 ) := ϕt, x 0 ) S otherwise. 4) The Poincaré return map, P : S S, is then given as the partial map) 3) Px) := ϕt I x), x)). 5) The method o Poincaré sections can then be stated as ollows: Theorem 0: Method o Poincaré Sections [14], [8], [15]) I the C system with impulse eects 3) has a periodic orbit O with x := Ō S a singleton and L Hx ) 0, then the ollowing are equivalent: i) x is an exponentially stable resp., asymp. stable, or stable i.s.l.) ixed point o P; ii) O is an exponentially stable resp., asymp. stable, or stable i.s.l.) periodic orbit o Σ.

3 III. NEW RESULTS ON HYBRID ZERO DYNAMICS The paper [25] introduced the notion o a hybrid zero dynamics HZD) or systems with impulse eects. The HZD is based on two principles that are ubiquitous in non-hybrid systems, namely invariance and attractivity. Many problems involving the existence and stability o periodic orbits in systems with impulse eects can be simpliied i the system possesses an appropriately deined hybrid invariant maniold. In the context o planar bipedal locomotion, insight can be gained by studying the system restricted to the invariant submaniold, that is, the HZD. The current paper presents two theorems that directly extend the results o [25], [14], and several others. In the irst theorem, using [14], the HZD is extended to outputs with uniorm vector relative degree greater than or equal to one. Previous work o [25] applies to outputs o uniorm vector relative degree two each output component has relative degree two and the associated decoupling matrix is square and invertible [12]). The key contribution o the second theorem is a novel dynamic controller that is active only at the impacts, whose unction is to relax the conditions or achieving impact invariance when constructing an HZD. A. Hybrid Zero Dynamics or Systems with Uniorm Vector Relative Degree k 1 The basic idea o an HZD is that an output should give rise to a zero dynamics or the continuous portion o the model [12] and the resulting zero dynamics maniold should be invariant under the impact map. This is ormalized as ollows: Associate a C output y = hx) 6) to 2) and assume that h has vector relative degree k 1,, k m ) with respect to the continuous portion o the hybrid model that is, h i has relative degree k i with respect to 1) and the decoupling matrix is square and invertible [12]) and that there exists x 0 X such that hx 0 ) = 0. Let Z be the zero dynamics maniold or the continuous portion o the dynamics and let u be the eedback unique on Z) such that or all x Z, x) := x) + gx)u x) T x Z. I in addition, S Z is a C maniold o dimension one less than Z and S Z) Z, 7) then Z is a hybrid zero dynamics maniold or Σ and the restriction dynamics ż = Σ Z : Z z) z S Z z + = S Z z ) z 8) S Z, is called the hybrid zero dynamics o Σ, where Z and S Z are the restrictions o and to Z and S Z, respectively. For notational expediency, we assume in the ollowing that the relative degree is the same or each output component. This level o generality is all that will be needed in the remainder o the paper. The developed results can be extended to systems with general vector relative degree, or to systems or which a vector relative degree is achievable by dynamic eedback; see [12]. Remark 1: When a system with impulse eects 2) has an output hx) with uniorm vector relative degree k, the ollowing are equivalent: a) S Z) Z; b) x S Z) and 0 i k 1 L i hx) = 0. The ollowing theorem gives HZD-based suicient conditions or stabilization o a periodic orbit in an open-loop system with impulse eects. Theorem 1: Consider a C system with impulse eects 2) with a C output 6) and hybrid zero dynamics 8). Suppose that 8) contains a periodic orbit O that is exponentially stable and transversal to S. I in addition 1) the output h has uniorm vector relative degree k; and 2) there exists a vector o unctions φ : X IR n mk such that L g φ 0; and Φx) = φx); hx); L hx); ; L k 1 ) hx) is a dieomorphism on X, then the orbit O is exponentially stabilizable. For any choice o matrices K 0, K 1,, K k 1 satisying that s k + K k 1 s k 1 + K 0 is Hurwitz, or ǫ > 0 suiciently small, the eedback ux) = L g L k 1 ) 1 L k hx) +... hx) k 1 i=0 1 ǫ k i K i L i hx) 10) applied to 2) renders O exponentially stable in the ulldimensional) closed-loop system ẋ = x) + gx)ux) x S x + = x ) x S. ) 9), 11) Proo: As in [12, Prop ], introduce the coordinates z = φx) and η = hx); ; L k 1 hx)). In z; η), the closed-loop dynamics ẋ = x) + gx)ux) satisies all o the hypotheses o [14, Thm. 2]. Hence, or ǫ suiciently small, exponential stability o the orbit in the closed-loop system 11) is equivalent to exponential stability o the orbit in the zero dynamics 8). B. Using a Deadbeat Hybrid Extension to Achieve Impact Invariance One critical aspect o applying Theorem 1 in the context o bipedal locomotion is the selection o an output hx) that leads to a hybrid zero dynamics. Appropriately choosing an output so that Z is impact invariant is a nontrivial task, in general. In previous work, [25, Sec. V, Thm. 4] identiied a class o holonomic 1, uniorm vector relative degree 2 outputs or which it is straightorward to meet the impact invariance condition. The reasoning employed in [25] relied heavily 1 The output unction h depended only on the coniguration variables o the robot, hence the terminology holonomic.

4 on the act that both the impact map and L h were linear in the generalized velocity coordinate. This linearity property breaks down already or L 2 h equivalently, or outputs with uniorm vector relative degree 3), and in the context o compliant actuation, the impact map is aine in the generalized velocity coordinate not strictly linear). Without linearity, giving checkable conditions or impact invariance becomes very hard. We will overcome this obstacle by embedding the system with impulse eects 2) in a larger, parameterized system called a deadbeat hybrid extension. With the introduction o parameterized outputs, a discrete eedback element becomes available the parameter update law. The ollowing theorem illustrates the use o a parameter update law in achieving a hybrid zero dynamics. Under the given hypothesis, the extra dimensionality associated with the parameters does not signiicantly complicate the Poincaré return map. Theorem 2: Consider a C system with impulse eects ẋ = x) + gx)u x S x + = x ) x 12) S, with an n-dimensional state maniold X and m-dimensional inputs u. Let A be an open subset o IR p, or some p 1, and let h : X A IR m be C. Suppose urthermore that 1) α A, y = hx, α), 13) has uniorm vector relative degree k; 2) there exists a non-empty C submaniold Z such that α A, Z α := x X hx, α) = 0,, L k 1 hx, α) = 0} is dieomorphic to Z; 3) S Z α is independent o α; denote it by S Z ; 4) S Z is C and has dimension one less than Z; and 5) there exits a C unction 1 : S A such that, x S Z, the values ξ = x), α = 1 x) result in hξ, α) = 0,, L k 1 hξ, α) = 0. Then the control system, [ ] [ ] ẋ x) + gx)u = x / S α 0 [ ] [ ] x + x ) α + = 1 x x S ) 14) with output y = hx, α) has a hybrid zero dynamics. Moreover, a) the hybrid zero dynamics maniold o 14) is Z A := α A Z α, α), b) Z A S A) = S Z ) A, and c) the Poincaré map o the hybrid zero dynamics, P zero : S Z A S Z A, has the orm [ ] ρz) P zero z, α) =. 15) 1 S Z z) Proo: Hypotheses 1) and 2) imply immediately two things: that α A the continuous part o 12) with output 13) has a well-deined, zero dynamics maniold 2 Z α, and that the continuous portion o 14) with output 13) has a well-deined zero dynamics maniold, denoted temporarily by Z. Again using Hypothesis 2), it ollows that Z = α A Z α, α), and hence the set Z = ZA is a zero dynamics maniold o the continuous portion o 14). Next, note that by Hypothesis 3), Z A S A) = α A Z α, α)) α A S, α)) = α A S Z α, α) = α A S Z, α) = S Z ) A, establishing part b) o the Theorem. This and Hypothesis 4) imply that Z A S A) is a C submaniold o X A, and has dimension one less than Z A. By Hypothesis 5) and Remark 1, Z A S A) is invariant under the impact map o 14). It ollows that 14) has an HZD with zero dynamics maniold Z A, proving part a) o the Theorem. The corresponding restriction dynamics o 14) is [ ] [ ] ż = Zα z, α) z S Z α 0 Σ : [ ] [ ] ZA z + S Z z ) α + = 1 S Z z z S Z, ) 16) rom which the orm o the Poincaré map is immediate, thus proving part c). C. Remarks I a nonlinear system with impulse eects has an HZD, Theorem 1 provides suicient conditions or being able to exponentially stabilize a periodic orbit, such as a walking motion o a bipedal robot. As in [25, Sec. VI], one interesting implication o Theorem 1 is the possibility o creating the periodic orbit on the basis o the HZD, which results in a lower-dimensional design problem. Theorem 2 provides additional lexibility in meeting the hybrid invariance condition 7) by embedding the hybrid system in a special kind o dynamic extension. The dynamic extension is deadbeat in that the additional states are updated only at impacts, and their new values depend only on the original states o the system. This is beneicial because then the existence and stability o a ixed point o the Poincaré map o the restriction dynamics, given by 15), is determined completely by ρ; the term 1 is deadbeat. In other words, the deadbeat hybrid extension has not signiicantly complicated 2 This does not imply that there exists a value o α or which Z α is a hybrid zero dynamics maniold o 12). No such value or α need exist.

5 the determination o the stability properties o a periodic orbit on the basis o the HZD. In 14), the switching surace is S A. It has been written in the given orm to emphasize that the switching condition does not depend the value o α. IV. APPLICATION TO PLANAR BIPEDAL WALKERS This section provides models or a class o planar bipedal walkers, with and without actuator dynamics. Key properties o the models that are useul or checking the hypotheses o Theorems 1 and 2 are then succinctly summarized. These properties take advantage o the act that the hypotheses o Theorems 1 and 2 are invariant under regular static state eedback applied to the continuous portion o the model) and coordinate changes. A. Robot model without actuator dynamics As in [25, Sec. II], consider a bipedal robot consisting o N links connected in a planar tree structure to orm two identical legs with knees, but without eet, with the legs connected at a common point called the hips, and possibly other limbs such as a torso, etc.). All links have mass, are rigid, and are connected in revolute joints see Fig. 2). It is assumed that no actuation is applied between the stance leg and the ground, while all other joints are independently actuated, and hence there are N 1) controls. As shown in [6], [7], addressing the control o robots without actuation between the stance leg and ground is an important step in achieving anthropomorphic walking motions in robots with non-trivial eet and actuated ankles. Further details on the model are given in [25, Sec. II], along with assumptions on the walking gait instantaneous double support phase, no slipping nor rebound at impact, motion rom let to right, symmetric gait). A rigid impact is used to model the contact o the swing leg with the ground. The coniguration coordinates o the robot in single support also called the stance phase) are denoted by q = q 1 ; ; q N ) Q, the state space is denoted by T Q. The method o Lagrange leads to the mechanical model Dq) q + Cq, q) q + Gq) = Bu, 17) where B is an N N 1) constant matrix with rank N 1). Letting x := q; q), and deining and g in the obvious manner, the mechanical model is expressed in state variable orm as ẋ = x) + gx)u. 18) The hybrid model o the robot single support phase Lagrangian dynamics plus impact map) is constructed by speciying the impact or switching surace S = q; q) T Q y 2 q) y 1 q) = 0, x 2 q) x 1 q) > 0}, to be the set o points where the swing leg height is zero and positioned in ront o the stance leg. The impact map : S T Q is computed as in [10], [8]. When the swing leg contacts the ground, a rigid collision gives rise to a jump in the velocity coordinates. So that the same mechanical model x 2,y 2 ) q 2 q 1 q 4 q 3 θ x 1,y 1 ) Fig. 2. A coordinate diagram o the robot model used in this case study. Angles are positive in the counterclockwise direction. All links have length 0.5 m. The thigh and shin links have masses o 0.5 and 0.75 kg respectively. The torso link has a mass o 27.5 kg. Mass distributions are uniorm. The motors relected inertias are kg m 2 at each actuated joint. Each motor acts through a series spring having an eective stiness o 550Nm/rad. can be used independent o which leg is the stance leg, the coordinates must also be relabeled at impact, giving rise to a jump in the coniguration variables as well; see[8], [25]. The corresponding system with impulse eects is written as ẋ = x) + gx)u x / S x + = x ) x 19) S. B. Robot model with compliant actuation Assume now that the vector o torques applied to the robot model 17) is generated through a compliant model o the orm u = kq m q a ) 20) J q m + kq m q a ) = u m, 21) where q m Q m is an N 1) tuple o motor angles, q a is a vector o the relative angles corresponding to the N 1) actuated joints o the robot, u m is the vector o N 1) motor torques, k is a diagonal matrix o positive) spring constants and J is a diagonal matrix o positive) rotor inertias. The stance phase dynamics o the robot is now given by the 2N 1) DOF Lagrangian system Dq) q + Cq, q) q + Gq) = Bkq m q a ) J q m + kq m q a ) = u m. 22) Letting x m := q m ; q m ) and x e := x; x m ), this is easily expressed as ẋ e = e x e ) + g e x e )u m. 23) The corresponding model with impulse eects is written as ẋe = e x e ) + g e x e )u m x e Σ e : / S e x + e = e x e ) x 24) e S e, where the state maniold is X e = T Q T Q m and because the impact condition depends only on the state o the biped

6 and not on the motor angles) S e = S T Q m. Following [10], [8], the impact map e : S e X e has the orm [ ] x e x e ) = ) m x, x m ), 25) where is as in 19) and m imposes continuity 3 in the motor positions and velocities across the impact. Hence, the only jumps in the motor variables x m are those due to leg swapping recall that the robot coordinates are permuted at impact to relect leg swapping). C. Model Properties Some properties o the mechanical models 17) and 22) are now summarized. These properties provide inormation on the zero dynamics o 18) and 23), and hence useul inormation on the HZD o 24) in light o applying Theorem 2 to the biped model with actuator dynamics. In the ollowing, we choose coniguration coordinates or 17) as q = q a ; θ), where θ reerences a position on the robot to the world rame see Fig 2). Proposition 1: Let σ be the angular momentum o the biped about the contact point o the support leg with the ground. In the coordinates, q = q a ; θ), a) the inertia matrix D o 17) is independent o θ; b) 18) is globally eedback equivalent to σ = V θ q) σ θ = d N,N q a ) + Rq a) q a 26) q a = v, where V is the potential energy o the robot model 17), [ dn,1 q a ) Rq a ) = d N,N q a ),, d ] N,N 1q a ) d N,N q a ) and d i,j are the elements o D. The proo and the required eedback are given in [9], and are based on [23], [19]. Because the actuator dynamics 20) is globally eedback equivalent to a vector o double integrators, the next result ollows rom basic results in [12] dealing with dynamic extensions. Proposition 2: Let q = q a ; θ) and let σ be the angular momentum o the biped about the contact point o the support leg with the ground. Then 23) is globally eedback equivalent to σ = V θ q) σ θ = d NN q a ) + Rq a) q a 27) q a = v v = w, 3 In other words, the springs isolate the motors rotor inertias rom the impact dynamics. Since in practice, these would be the rotor inertias relected through a gear ratio on the order o 30 : 1, removing the rotor inertias rom the impact dynamics can result in considerably less energy loss at impacts. where V and R are as in Proposition 1. Proposition 3: Consider an output y = hq, α), where α is ixed. Then the ollowing hold: a) h has uniorm vector relative degree 2 or 18) i, and only i, it has uniorm vector relative degree 4 or 23); b) in both cases, the decoupling matrices depend only on q and they are equal, that is L g L h = L ge L 3 e h; c) or hq, α) = q a h d θ, α), detl g L h)q, α) = 1 Rq a ) h dθ, α) ; θ d) i hq, α) = q a h d θ, α) and has uniorm vector relative degree 2 or 18), then the zero dynamics maniolds and restriction dynamics o 18) and 23) are dieomorphic; moreover, the zero dynamics maniold o 18) is Z α = q; q) T Q q a = h d θ, α), q a = h } dθ, α) θ θ 28) and in the coordinates θ; σ), the restriction dynamics is σ = V θ θ = qa=h d θ,α) σ d N,N θ, α) 1 Rθ, α) h dθ, α) θ 29) ) 1 30), where, dn,n θ, α) = d N,N qa=h d θ,α) and Rθ, α) = R qa=h d θ,α). Proo: a) and b) are immediate rom Propositions 1 and 2. The calculation or c) is a straightorward application o the Sherman-Morrison-Woodbury ormula and is let to the reader. The irst part o d) ollows rom basic results in [12] and equations 28), 29), and 30) ollow rom [25]. V. EXAMPLE The results o the paper are now illustrated on the 5-link planar biped o Fig. 2, with model given in 24). On the basis o Theorem 2 and Proposition 3, an output unction yx e, α) and a parameter update unction 1 will be chosen or 24) so that, with deadbeat parameter updates, it has a valid HZD. Then, using once again Proposition 3, the conditions o Theorem 1 will be veriied, resulting in a eedback u m x e, α). The closed-loop system will then be simulated, showing behaviors that are consistent with the theoretical analysis. A. An Output Function, Feedback, and Update Law Motivated by Proposition 3 and [25], the output is selected as ) θ θi y = hq, α) = q a h d, α 31) θ θ i where h d : IR A IR 4 is a 4 1 vector o Bézier polynomials o degree 4 m 7. The terms θ i and θ are 4 Seventh degree Bézier polynomials have eight independent parameters. This can be shown to be the minimum number o ree parameters needed to design the parameter update law and guarantee that S e Z e,α is independent o α.

7 constants, equal to the values o θ at the beginning and end, respectively, o a steady state gait. For any choice 5 o α = α 0, α 1, α 2, α 3 ) A = IR 4 4, the set o outputs 31) is relative degree 2 with respect to the biped model without actuator dynamics 19), and so by Proposition 3 a) is relative degree 4 with respect to the biped model with compliant actuation 24). In the context o the model with compliant actuation, dierentiating the output our times yields y 4) = L 4 e hx e, α) + L ge L 3 e hx e, α)u m, 32) where the domain o invertibility o the decoupling matrix, L ge L 3 e hx e, α), is computable using Proposition 3 b) and c). The zero dynamics maniold associated with this output is Z e,α = x e X e hx e, α) = 0, L hx e, α) = 0, L 2 hx e, α) = 0, L 3 hx e, α) = 0 and is dieomorphic to the zero dynamics maniold 28). The eedback u m o 10) will render Z e,a := α A Z e,α, α) invariant and exponentially attractive in the continuous phase o the closed-loop system. Note that this eedback is deined using a constant ǫ > 0 that is tuned so that Z e,a can be made exponentially attractive with arbitrarily ast convergence. With h d θ, α) selected as a Bézier polynomial, it may be shown that α A, x e e S e Z e, ), hx e, α) = A 0 x e )α 0 + B 0 x e ) L hx e, α) = A 1 x e )α 1 + B 1 x e, α 0 ) L 2 hx e, α) = A 2 x e )α 2 + B 2 x e, α 0, α 1 ) L 3 hx e, α) = A 3 x e )α 3 + B 3 x e, α 0, α 1, α 2 ) } 33) with A i s invertible. This property guarantees that there exists an impact update law, 1, satisying Hypothesis 5 o Theorem 2. Hypotheses 3 and 4 are satisied by noting that or a ive-link biped without impact updated parameters or actuator dynamics, S Z is smooth and has dimension one less than Z itsel [25], and that the same derivation applies as long as S Z α is independent o α. Such independence has been established earlier by speciying that the last our coeicients o each Bézier polynomial are unaected by the update law). Hypothesis 1 and 2 have already been established by Proposition 3. Thus, the conditions o Theorem 2 are met and the chosen output hq, α) will yield a valid HZD o 24) with deadbeat parameter augmentation. To apply Theorem 1 to the system 14), all that remains is to ind an exponentially stable limit cycle within the HZD o 24) and to veriy that the coordinate transorm o Theorem 1, Hypothesis 2 is valid. This can be done quite eiciently on the basis o the HZD using an optimization technique developed in [24] or inding periodic orbits in the HZD subject to constraints on stability, torque, energy eiciency, ground riction, etc. Using this method, a gait was designed using Matlab s FMINCON unction to achieve, a orward progression rate o 0.8 m/s and to minimize an approximation o motor electrical energy consumed per distance traveled. B. Interpreting Simulation Results Figure 3 illustrates one interpretation o Theorem 1, namely that while the eedback law u m x e, α) o 10) will render Z e,a orward invariant and continuous-phase exponentially attractive or any value o ǫ > 0, only or ǫ suiciently small does it render the maniold exponentially attractive in a hybrid sense. The reason is that or state values outside the zero dynamics maniold, application o the impact map will tend to push the state urther away, an eect that can be overcome by suiciently ast convergence in the continuous phase. This conclusion is reinorced in Fig. 4, where the spectral radius o the Poincaré return map o the closed-loop system is plotted along with the eigenvalue o the return map associated with the HZD. Figure 5 then shows that the trajectories o the HZD converge to a periodic orbit. VI. CONCLUSIONS Motivated by the problem o creating exponentially stable periodic orbits in bipedal robots with underactuation and actuator dynamics, the hybrid zero dynamics HZD) ramework o [25] has been extended to nonlinear systems with impulse eects where the outputs have vector relative degree higher than two. The ormal part o this extension, ormulating the proper conditions o simultaneous invariance under the continuous dynamics and the impact map, was straightorward. Even more, an existing result could be used to show that exponentially stable periodic orbits o the HZD can be rendered exponentially stable in the ull-dimensional) closed-loop system with impulse eects. All o this was ormalized in Theorem 1. The more challenging aspect o the extension was to determine how to meet the impact invariance condition when the relative degree is greater than two. The result in [25] on this aspect o the problem could not be extended in a direct way. A novel embedding o the original system into a system with event-based parameter updates was introduced. The additional dynamic elements in the larger system are tailored to meet the boundary conditions associated with impact invariance, and hence the existence o an HZD. This result was ormalized in Theorem 2. Equipped with Theorems 1 and 2, the motivating problem o creating exponentially stable periodic orbits in a class o planar bipedal robots with underactuation and actuator dynamics was addressed. Key properties o the models were irst summarized, in light o applying the main theorems. A set o detailed simulations on a particular example was then provided in order to illustrate and support the prior theoretical developments. 5 Note that although the Bézier polynomials o hq, α) each have 8 coeicients, only the irst our components are treated by the parameter update unction. The last our must remain constant ater the computation o a periodic orbit in order that S e Z e,α is independent o α.

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The above plot shows that under the eedback 10), or a particular choice o gain matrices, K i, the zero dynamics maniold is attractive or ǫ = but not or ǫ = Plotted on the vertical axis is the euclidian norm o η = hq, α); L e hx e, α); L 2 e hx e, α); L 3 e hx e, α)). The horizontal axis is time. The observed behavior is consistent with Theorem 1, where the zero dynamics maniold y 0 is made exponentially attractive or suiciently small ǫ with the eedback 10). Initial conditions or the two plots are the same and indicated by an asterisk max eig eig t ) ) Px e) x x e e ) Px e) x x e e equal to ρz) z ǫ Fig. 4. The above plot shows how the eigenvalues o the linearization o the return map vary with the parameter ǫ o the eedback 10). As ǫ approaches zero, one eigenvalue remains constant, equal to the eigenvalue o the hybrid zero dynamics ρz)/ z, while all other eigenvalues go to zero. The eigenvalue associated with the 1DOF HZD is σ z Fig. 5. Theorem 1 links the stability o an orbit within the hybrid zero dynamics to the stability o an orbit in the ull system. The above shows the system response to an initial condition noted with an asterisk) within the hybrid zero dynamics but not on the periodic orbit. The state converges exponentially quickly back to the periodic orbit. θ