HZD Framework

Give EoM

\[D(q)\ddot q+H(q,\dot q)=Bu\]

where configuration coordinates $q \in \mathcal{Q} \subset \mathbb{R}^n$. Let full system state be given by $x = (q, \dot q) \in \mathcal{X} \subset \mathrm{T} \mathcal{Q}$

In state space the EoM becomes

\[\dot{x}=f(x)+g(x) u\]

By combining the continuous dynamics within a domain and the reset map, the single-domain hybrid control system is obtained

\[\mathcal{H C}= \begin{cases}\dot{x}=f(x)+g(x) u & x \notin \mathcal{S} \\ x^{+}=\Delta\left(x^{-}\right) & x^{-} \in \mathcal{S}\end{cases}\]

The idea behind the HZD framework is to reduce the hybrid dynamical system into a lower dimensional system. Virtual constraints

\[y_\alpha(q):=y^a(q)-y^d(\tau(q), \alpha)\]

When the outputs and their derivatives are equal to zero, we say that the residual dynamics evolve on the zero dynamics surface, defined as

\[\mathcal{Z}_\alpha:=\left\{x \in \mathcal{D} \mid y_\alpha(q)=0, \dot{y}_\alpha(q)=0\right\}\]

to ensure stability of the hybrid system as a whole, we must ensure that the outputs remain zero through impact. This impactinvariance condition is often termed the HZD condition

\[\Delta\left(\mathcal{S} \cap \mathcal{Z}_\alpha\right) \in \mathcal{Z}_\alpha\]


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