Chapter III: Gait Synthesis with Hybrid Zero Dynamics
3.2 The Hybrid Zero Dynamics Framework
3.2.4 Hybrid Zero Dynamics
The surface in Equation (3.29) has been designed without taking into account the hybrid transition maps of (2.29), meaning that there is no guarantee that the re- sulting walking cycle will be invariant to impact. To enforce impact invariance, the
Figure 3.1: A depiction of the multi-domain hybrid system geometry and its associ- ated notation.
desired outputs can be shaped through the parametersαv inyvdsuch that the walking satisfies a hybrid invariance condition which necessarily guarantees that the reduced dimensional manifold is invariant to the impact dynamics due to the discrete change of velocities when reaching an edge.
Definition 3.2.0.4. At a given edge e ∈ E, the reduced dimensional hybrid system (3.28) is impact invariant, and thus has hybrid zero dynamics (HZD) if and only if Zvs ∩Se6=∅ and:
∆e(Zvs ∩Se)⊂ Zvt, ∀ e= (vs, vt)∈E, (3.30) where vs and vt are the source and target continuous domains mapped by the edgee.
A submanifoldZS
v∈V Zv is called hybrid invariant if it is invariant over all domains of continuous dynamics and impact invariant through all discrete dynamics.
The main ideas of hybrid zero dynamics are illustrated in Fig. 3.2, which shows how zeroing the outputs renders a reduced dimensional surface that can be shaped to achieve stable periodic motions. In fact, the HZD condition (3.30) implies stability of the overall system dynamics if the zero dynamics are rendered stable through the continuous domains of the hybrid motion. Additionally, the satisfaction of the invariance condition (3.30) guarantees HZD for the full multi-domain hybrid system.
The overarching goal of these constructions is to provide a framework for the synthesis of dynamic walking gaits. With regards to the actual design of walking motions, shaping the zero dynamics to obtain a stable periodic orbit is perhaps the most challenging task. It can thus be beneficial in the development of HZD controllers or in analysis of the hybrid system to directly deal with coordinates on the zero dynamics manifold rather than within full-order state space (3.27). To construct a restricted hybrid system model, let zv ∈ Zv ⊂ Rlv be the local coordinates of the zero dynamics in a given domain, and letηv ∈ Dvα ⊂Rnv−lv be the controlled normal states corresponding to our choice of feedback linearized output dynamics in (3.24) and transversal to Zv. Suppose that there exists local coordinate transformations Φzv : Dv → Zv and Φηv : Dv → Dαv so that (ηv, zv) = (Φηv(x),Φzv(x)) := Φv(x). With these transformations, we can write our controlled system in normal form as:
˙
η= ¯f(η, z) + ¯g(η, z)u (3.31)
˙
z =ω(η, z)
where becauseηis defined in terms ofxdue to our selection of the virtual constraints, y, we can write the dynamics as:
f¯(η(q,q), z(q,˙ q)) =˙
"
˙ y(q,q)˙ Lfy(q,q)˙
#
, ¯g(η(q,q), z(q,˙ q))) =˙
"
0 A(q,q)˙
#
. (3.32)
The closed-loop dynamics of the η coordinates evolve according to the previously shown linear time-invariant system,η˙v =Fclηv, and describe the dynamics transversal toZv. In essence, the feedback linearizing controller and choice of auxiliary feedback control law render the zero dynamics surface attractive and invariant under the con- tinuous dynamics of the system, i.e. η →0andη(0) = 0implies thatη(t)≡0∀t≥0.
Thus, we can write the zero dynamics as the maximal dynamics compatible with the output equal to zero:
˙
z =ω(0, z) :=f|αZv(z). (3.33) Then, according to Definition 3.2.0.4, if the zero dynamics has been shaped to achieve impact invariance, the multi-domain hybrid control system has hybrid zero dynamics, H |αZ. The restricted hybrid dynamical system with impulses can then be written as:
H α|Z ,
˙
z =fα|Zv(z) if z ∈ Zvα\Seα z+ = ∆e|Seα∩Zαv(z−) if z− ∈Seα
(3.34)
The stability of hybrid systems, particularly ones which exhibit periodic behaviors, is often determined by the existence and stability of periodic orbits. If the system (3.34) has HZD, then due to the hybrid invariance of Z, there exits a stable hybrid periodic orbit, O|Z ⊂ Z, for the reduced order zero dynamics evolving on Z, i.e., if we are evolving on the restriction dynamics of fα|Z(z), then O|Z is a stable hybrid periodic orbit for the restricted dynamics in (3.34) [30]. More concretely, using the constructions introduced in Sec. 2.4, let ϕftα|Z(z0) be the (unique) solution to (3.33) at time t ≥ 0 with initial condition z0. For a point z∗ ∈ S, we say that ϕftα|Z is hybrid periodic if there exists a T > 0 such that ϕfTα|Z(∆(z∗)) = z∗. Further, the stability of the resulting hybrid periodic orbit, O|Z = {ϕftα|Z(∆(z∗)) : 0 ≤ t ≤ T}, can be found by analyzing the stability of the Poincaré map, wherein z∗ is a fixed point, as previously presented in (2.33).
On the basis of (3.30), we define the restricted Poincaré map ρv = Pv|Z, where ρv :Svs∩ Zv →Svt∩ Zv. The Poincaré return map for the multi-domain hybrid zero
Figure 3.2: Illustration of the key concepts related to hybrid zero dynamics: contin- uous convergence to a low dimensional zero dynamics surface Z, with an invariance condition: ∆(S ∩ Z)⊂ Z, and the end goal being a stable periodic walking gait.
dynamics can be defined as the composition of all maps for one gait cycle:
ρ:=ρv1 ◦ρv2 ◦...◦ρvnp. (3.35) It is shown in [111], [165] that if there exists a feedback control law rendering the in- variant manifold sufficiently rapidly attractive, then the existence and stability of the periodic orbitOof the full order system can be determined from those of the restricted (reduced-dimensional) zero dynamics periodic orbit, O|Z, through the canonical em- beddingι:Z → D. Thus, if the system has HZD and there exists a periodic orbit for O|Z ofHα|Z, then the full order system also has a hybrid periodic orbitO =ι(O|Z).
Theorem 3.2.1. (Orbital Stability of the Hybrid Zero Dynamics) Suppose that O|Z
is an exponentially stable transverse periodic orbit of the corresponding hybrid zero dynamics system (3.34). Then there exists an ε >¯ 0 such that for all ε > ε >¯ 0, the hybrid periodic orbit O|Z is locally exponentially stable for the full order hybrid periodic orbit O=ι(O|Z) of the hybrid system Hα.