Cable-Driven Parallel Robot

What is modelled

A platform hangs from four cables that can only pull. Holding a point-mass platform needs two force components but there are four tensions, so a whole family of tension sets works. The controller picks the one with the smallest norm that keeps every cable taut (above T_min) and below the motor limit T_max.

Equations

• Equilibrium: Σ Ti ui = m (a + g), unit vectors u_i from the platform to the anchors.
• All solutions: T = Tp + a n1 + b n2 (null space of the 2×4 structure matrix).
• Choice: minimise |T| subject to Tmin ≤ Ti ≤ Tmax, solved exactly by checking the interior, edges and vertices of the feasible polygon.
• Centre pose: Ttop = Tmin + m g / (2 sin α).

Checks run in the page

• Residual of the equilibrium equation at 40 random poses.
• Analytic tensions at the frame centre.
• Feasible workspace grows monotonically with T_max.
• Along the dynamic figure-eight the tensions reproduce m (a + g).

Honest simplifications

The platform is a point mass (no orientation, no torque balance), cables are massless, straight and inextensible (no sag or stretch), the path is followed exactly (no tracking error) and there are no pulley or friction effects. Real cable robots are 6-DOF with eight cables.

• "Weak motors": the green region shrinks and the figure-eight leaves it.
• Raise the minimum tension: stiffer, but the workspace shrinks and the cables work harder.
• Watch the two bottom cables sit at their minimum while the top cables carry the weight.