Core Actuation Promotes Self-manipulability on a Direct-Drive Quadrupedal Robot

Abstract

For direct-drive legged robots operating in unstructured environments, workspace volume and force generation are competing, scarce resources. In this paper we demonstrate that introducing geared core actuation (i.e., proximal to rather than distal from the mass center) increases workspace volume and can provide a disproportionate amount of work-producing-force to the mass center without affecting leg linkage transparency. These effects are analytically quantifiable up to modest assumptions, and are demonstrated empirically on a spined quadruped performing a leap both on level ground and from an isolated foothold (an archetypal feature of unstructured terrain).

Appendix 1: Analytic Leg Force Generation Versus Workspace Volume Trade-off via Linkage Scaling

We explicitly show the trade-off between leg force generation and workspace volume confronting the designer by considering a simple scaling of a nominal leg linkage design by a scaling factor $\lambda$, assuming a fully actuated leg interacting with the ground through a point contact. Let the forward kinematic map of the nominal leg linkage with a point toe and origin at the hip be given by $f: Q\rightarrow \mathbb {R}^n$, where $q\in Q$ denotes the actuated joint positions. Consider a uniform scaling transformation applied to this linkage, scaling the length of all vectors by a factor of $\lambda \in \mathbb {R}^{+}$, and let $f_{\lambda }(q) :=\lambda f(q)$ denote the forward kinematic map of the scaled linkage. The nominal leg linkage has a workspace volume given by $V:=\int _{f(Q)} \varOmega$, where $\varOmega$ indicates the standard volume form on $\mathbb {R}^n$ [20]. The forces $F$ generated at the toe from motor torques $\tau$ is then given by $F(q) :=Df^{-T}(q) \tau$ assuming the leg linkage is not at singularity, where $Df:=\frac{\partial f}{\partial q}$. Denoting the workspace volume of the scaled linkage by $V_{\lambda }:=\int _{f_{\lambda }(Q)} \varOmega$ and the forces generated at the toe by $F_{\lambda }(q) :=Df_{\lambda }^{-T}(q) \tau$, we have that

$$\begin{aligned} V_{\lambda }&= \int _{\lambda f(Q)} \varOmega = \int ... \int _{\lambda f(Q)} {dx_1} ... {dx}_n = \int ... \int _{f(Q)} {\lambda dy_1} ... {\lambda dy}_n \\&= \lambda ^n \int ... \int _{f(Q)} {dy_1} ... {dy}_n = \lambda ^n V\end{aligned}$$

and

$$ F_{\lambda }(q) = (\lambda Df(q))^{-T} \tau = \frac{1}{\lambda } Df^{-T}(q) \tau = \frac{1}{\lambda } F(q), $$

so that increasing scale has the dual effect of decreasing end effector force magnitude for a given motor torque vector while increasing workspace volume.⁹