A non-periodic planning and control framework of dynamic legged locomotion

Abstract

This study proposes an integrated planning and control framework for achieving three-dimensional robust and dynamic legged locomotion over uneven terrain. The proposed framework is composed of three hierarchical layers. The high-level layer is a state-space motion planner designing highly dynamic locomotion behaviors based on a reduced-order robot model. This motion planner incorporates two robust bundles, named as invariant and recoverability bundles, which quantify analytical state-space deviations for robust planning design. The low-level layer is a model-based trajectory tracking controller capable of robustly realizing the planned locomotion behaviors. This controller is synthesized based on full-order hybrid dynamic modeling, model-based state feedback control, and Lyapunov stability analysis. The planning and control layers are concatenated by a middle-level trajectory generator that produces nominal behaviors for a full-order robot model. The proposed framework is validated through flat and uneven terrain walking simulations of a three-dimensional bipedal robot.

Appendix A: Tuple notations in Definition 1

To maintain the notation clarity, we remove the subscript ’r’ which denotes the “reduced-order” model. The tuple components of the robust hybrid automaton in Definition 1 are defined as

• ${\mathcal {Q}}$—set of discrete states. ${\mathcal {Q}} :=\{q_i\}, i = 1, \ldots , n$;

• ${\mathcal {X}}$—set of spaces of continuous states, ${x}_q$, ${\mathcal {X}}=\cup _{q}^{}{\mathcal {X}}_q$. ${\mathcal {X}}_q :=\{{x}_q \in {\mathbb {R}}^{d_q}, q \in {\mathcal {Q}}\}$;

• ${\Sigma }$—system dynamics, $\Sigma =\cup _{q}^{}\Sigma _q$, each described by vector field ${\mathcal {F}}_q$, with ${\mathcal {F}}_q:{\mathcal {X}}_q \rightarrow T_{{\mathcal {X}}_q} \subset {\mathbb {R}}^{d_q}$, where $T_{{\mathcal {X}}_q}$ is the tangent bundle of ${\mathcal {X}}_q$;

• $\mathcal {W}$—disturbance input ${w}$ space, ${w}\in \mathcal {W}\subseteq {\mathbb {R}}^{n_\mathrm{dist}}$;

• ${\mathcal {U}}$—control inputs space ${\mathcal {U}} :=\{{u}_q, q \in {\mathcal {Q}}\}$. ${\mathcal {U}} = \{{u}_c\} \cup \{{u}_d\}$ where ${u}_c, {u}_d$ are continuous and discrete control inputs, respectively;

• ${\mathcal {I}}$—set of allowable initial conditions, ${\mathcal {I}} :=\{\zeta _0\} \times \{q\} \times \{{x}_q\}, \; q \in {\mathcal {Q}}$;

• ${\mathcal {D}}$—domain, ${\mathcal {D}}(q) :=\{{x}_q \}$;

• ${\mathcal {R}}$—recoverability set, ${\mathcal {R}} :=\{{\mathcal {R}}_q, q \in {\mathcal {Q}}\}$;

• ${\mathcal {B}}$—invariant set, ${\mathcal {B}} :=\{{\mathcal {B}}_q, q \in {\mathcal {Q}}\}$;

• $\mathcal {E}$—edges, $\mathcal {E}_{i,j}=\mathcal {E}(q_{i}, q_{j}):{\mathcal {Q}}\times {\mathcal {Q}}$;

• $\mathcal {G}(q_{i}, q_{j})$—guard, $\mathcal {G}(q_{i}, q_{j}) :=\{G_{q_i}, \; q_i \in {\mathcal {Q}}, q_j \in {\mathcal {Q}}\} \subset {\mathcal {X}}_{q_i}$;

• ${\mathcal {T}}(q_i, q_j)$—jump termination set, ${\mathcal {T}}(q_i, q_j) :=\{T_{q_j}, q_i \in {\mathcal {Q}}, q_j \in {\mathcal {Q}}\} \subset {\mathcal {X}}_{q_j}$;

One thing worthy to concern is the well-posedness Frazzoli (2001) of phase-space manifolds. Since our phase-space dynamics are piece-wise continuous, the maneuvers are a finite set of primitives with a finite time durations and the well-posedness of phase-space manifolds is guaranteed for our case.