Fig. 2.

An overview of the robotically actuated delivery sheath (RADS). The articulating tip of the RADS is actuated in two degrees of freedom by two pairs (red and green) of antagonistic-configured tension wires driven by a two pulleys with radii ($r_p$) and angles (${\psi }_x$ and ${\psi }_y$). Three coordinate systems are assigned to describe the tip pose of the RADS: ${\Psi }_0$ is the reference frame fixed to the shaft, ${\Psi }_i$ is the intermediate frame assigned to the arc section and frame (${\Psi }_t$) is fixed to the articulating tip. Displacement of the tension wires ($t_1,\dots ,t_4$) by ${\delta }_x$ and ${\delta }_y$ (inset centre) results in instrument bending along the x- and y-axes (frame (${\Psi }_0$)), respectively. The arc of the RADS with parameters, bend angle ($\theta$), backbone length ($\ell$), radius (r) and curvature ($\kappa$) lies in a plane described by the arc plane (inset right). The orientation of the arc plane about the z-axis of the reference frame (${\Psi }_0$) is denoted by angle ($\varphi$). Further, the tendon distance to the backbone arc ($\ell$) is denoted $d_b$. A rigid link (not completely shown) of length ($l_t$) is attached to the arc (frame (${\Psi }_i$)) of the RADS.