Cable-Driven Coaxial Spherical Parallel Mechanism

• CDC-SPM is a cable-driven spherical parallel mechanism that provides three pure rotational degrees of freedom about a remote center, ideal for precise teleoperation in medical settings.
• The design minimizes end‐effector mass through Bowden-cable remote actuation, enhancing stiffness and isotropic force/torque transmission for improved dynamic performance.
• Parametric design and kinematic mapping ensure an optimized workspace and manipulability, supporting accurate haptic feedback and control in ultrasound probe applications.

A Cable-Driven Coaxial Spherical Parallel Mechanism (CDC-SPM) is a parallel manipulator architecture characterized by cable-driven actuation and uniquely coaxial placement of all actuated rotational axes. This mechanism yields three pure rotational degrees of freedom about a remote center of rotation (CoR), typically coincident with the tip of an ultrasound probe. The CDC-SPM achieves high fidelity in force and motion transmission—a requisite for haptic teleoperation in medical applications—by minimizing moving mass via Bowden-cable remote actuation, maximizing isotropy in force/torque transmission, and maintaining a workspace geometrically tailored for clinical utility (Seraj et al., 7 Dec 2025).

1. Geometric Architecture and Cable Actuation

The CDC-SPM consists of three identical legs, each forming a 3-RRR serial chain. Each chain comprises:

• An active revolute joint, axis $\mathbf{u}_i$ (motorized, coaxial to the base Z-axis),
• Two passive revolute joints, axes $\mathbf{v}_i$ and $\mathbf{w}_i$,
• Curved links that geometrically guide all axes to intersect at the remote CoR.

Heavy motors are off-board, transmitting torque via polymer rope in PTFE Bowden tubes routed around mini pulleys at each active joint. This arrangement reduces the end-effector mass to $\approx 0.55$ kg in the aluminium prototype. The coaxial configuration ($\gamma = 0$) ensures all actuated axes are aligned with the base frame Z-direction, while the passive axes converge at the CoR above the moving platform.

2. Parametric Design Variables and Performance Trade-offs

CDC-SPM geometry is defined by variables:

• $\alpha_1$, $\alpha_2$: Curvature angles for proximal and distal links
• $\beta$: Half-angle of moving-platform pyramid
• $R_1$, $R_2$: Radii for joint loci
• $z_{CoR}$: Vertical offset (CoR height)
• $L_{tool}$: Probe length
• $d_{1i}$: Base offsets per leg

Performance is directly influenced by these choices:

• Increasing $\alpha_1$/$\alpha_2$ enlarges the roll/pitch workspace but decreases structural stiffness and can induce near-singular configurations.
• Larger $R_1$, $R_2$ expand workspace but increase moving inertia.
• $z_{CoR}$ trades probe-tip dexterity and structural deflection.
• Pulley diameter and Bowden tube layout affect torque bandwidth (larger pulley increases cable travel/rad but raises inertia). The inclusion of appropriately chosen $d_{1i}$ offsets avoids inter-leg collisions, critical for maximizing joint-space feasibility.

3. Kinematic Analysis: Forward, Inverse, and Jacobian Mapping

Forward Kinematics

The closed-loop leg vector is:

$$\vec r_{BT} = \vec r_{B j_{1i}} + \vec r_{j_{1i} j_{2i}} + \vec r_{j_{2i} j_{3i}} + \vec r_{j_{3i} C} + \vec r_{CT}, \quad i = 1,2,3$$

Denavit–Hartenberg (D–H) parameterization converts geometric primitives into analytic chain parameters tied to $(R_1, R_2, \alpha_1, \alpha_2, z_{CoR})$ and joint positions $(\phi_{1i}, \phi_{2i}, \phi_{3i})$.

Orientation is modeled in unit quaternion form $q = [e_0, e_1, e_2, e_3]$, constrained by:

$$v_i|_B \cdot w_i|_T = \cos\alpha_2 \quad (i = 1,2,3)$$

$w_i|_T$ depends on the quaternion, and $v_i|_B$ on actuated angles. The closure yields three scalar constraints and the normalization condition $e_0^2 + e_1^2 + e_2^2 + e_3^2 = 1$.

Inverse Kinematics

Given desired $q$, scalar equations in $\phi_{1i}$ can be solved directly:

$$v_i|_{B}(\phi_{1i}) \cdot w_i|_{T}(q) = \cos\alpha_2$$

Passive joint angles are then extracted via axis alignment constraints.

Force and Velocity Mapping

The implicit kinematic constraint $F(\mathbf{x}, \mathbf{q}) = \mathbf{0}$ relates configuration and orientation. The effective Jacobian is:

$$\dot{\mathbf{x}} = J\,\dot{\boldsymbol{\phi}}, \qquad J = -J_x^{-1} J_q^{(a)}$$

Torque-tension relationships are:

$w = (J^T)^{-1} F t$

where $t$ are cable tensions, $F$ the pulley-radius matrix, and $J^T$ the transpose Jacobian. The velocity–tension map $w = J_c t$ with $J_c = (J^T)^{-1} F$ describes wrench generation at the CoR.

4. Stiffness, Inertia, and Dynamic Bandwidth

FEA and analytical modeling confirm that under a 50 N load, the mechanism's deformation is $<0.075$ mm (aluminium, safety factor $>5.5$), with stiffness exceeding $0.7$ MN/m along maximally loaded axes. Cartesian stiffness is given by:

$K = J_c \mathrm{diag}(k_1, k_2, k_3) J_c^T$

where $k_i$ denotes individual cable axial stiffness. Dynamic performance benefits from the minimal moving mass (links and pulleys only), with inertia tensor $I_e$ mapped to the base as $M(\mathbf{x}) = R(\mathbf{x}) I_e R(\mathbf{x})^T$. This configuration supports high control bandwidth, with force transients up to $\sim 50$ Hz rendered without noticeable lag in pilot tests using a 200 Hz controller.

5. Workspace, Manipulability, and Isotropy

Simulation demonstrates CDC-SPM workspace predominantly encompasses the clinical “useful cone”: $\pm 35^\circ$ roll/pitch and $\pm 180^\circ$ yaw. Physical constraints—such as Bowden-cable interference—can limit yaw coverage ($\sim \pm 135^\circ$ in the PLA prototype), but design modifications (e.g., cable-tensioning idlers) can restore full range.

The manipulability condition number ${\rm cn}(J)$ exceeds $0.2$ across feasible joint configurations, and remains near unity over the central $\sim50^\circ$ roll/pitch, indicating isotropic transmission and haptic transparency. The normalized workspace and manipulability metrics ensure safe and responsive operation in critical teleoperation tasks.

6. Implementation Guidelines and Clinical Optimization

For ultrasound scanning, parameter tuning recommendations are:

• $\alpha_2 \approx 43.5^\circ$ for full $\pm 35^\circ$ roll/pitch coverage with sub-0.1 mm tip deflection under 50 N load.
• Platform angle $\beta \approx 50^\circ$ for maximal yaw range without Bowden tube collision.
• Base offsets $d_{1i}$ should differ by $8$–$10$ mm to avoid leg–leg collision.
• Cable pre-tensioning to $\approx 20$ N yields compliance $<0.1^\circ$ under 5 Nm torque.
• Condition number ${\rm cn}(J)>0.3$ maintained by avoiding joint limits within $10^\circ$.
• Employ IMU instrumentation (accurate to $\pm 0.2^\circ$) and sensor fusion for residual compliance compensation.
• FEA stress validation is required when substituting aluminium for composite links.

These implementation practices yield mechanisms capable of pure rotational manipulation about a remote pivot, high force feedback fidelity, dynamic responsiveness, and workspace congruent with clinical requirements for ultrasound imaging.

7. Comparative Advantages of the CDC-SPM Architecture

The CDC-SPM's cable-driven, coaxial configuration offers:

• Mass minimization at the end-effector by remote actuation, direct inertia reduction from $>2$ kg (motorized) to $\sim0.55$ kg.
• True RCM mechanics—intersecting rotational axes at the probe tip—obviating the need for software compensation of complex movement.
• Elimination of conventional lower-pyramid singularities in parallel mechanisms by the coaxial actuator layout, yielding enlarged usable workspace and simpler mechanical integration.
• High stiffness and isotropy over the clinical workspace, supporting accurate and intuitive force/motion transmission for haptic teleoperation (Seraj et al., 7 Dec 2025).

A plausible implication is that adoption of the CDC-SPM design in medical robotics can improve operator sensory fidelity and reduce control latency in teleoperated procedures requiring precise, pivoted manipulations.