A wristed instrument (e.g. Intuitive's EndoWrist) adds pitch and yaw joints behind the jaws, each driven by its own antagonist cable pair wrapped around a capstan pulley of radius r. Commanding a joint angle θ pays out one cable and pulls in the other by the same arc length — the ideal, infinitely-stiff relation:
ideal cable travel Δs = r · θ (capstan relation)
tip orientation (ideal) R = R_x(roll=0) · R_z(θp) · R_y(θy)
effective joint angle θ_eff = sign(θ) · max(0, |θ| − backlash) / (1 + stretch)
cable tension (per side) T = T0 + k · r · max(0, ±θ) · (1 + stretch)
Real cables aren't infinitely stiff. Two non-idealities eat into that ideal relation, both set with the sliders above:
- Cable stretch — under load the cable elongates elastically, so part of the motor's pull is absorbed by stretch instead of rotating the joint; the achieved angle is scaled down by 1/(1 + stretch), and the motor must pull proportionally more cable (and more tension) to compensate.
- Cable backlash — slack in the routing means the first few degrees of motor motion take up play before the joint moves at all; below the backlash threshold the joint angle doesn't move, so it is subtracted from |θ| before the stretch scaling is applied.
- Tip position error — the Euclidean distance between where the tip should be (ideal θ) and where it actually ends up (θ_eff after backlash + stretch). This is the drivetrain-compliance error a real controller has to calibrate out or close with sensor feedback.
- Cable tension — shown per antagonist pair; the driving side tensions up roughly linearly with commanded angle (more so with more stretch, since more pull is needed for the same net travel) while the antagonist stays near preload.
The 3D companion sim models the ideal, zero-compliance forward-kinematics chain; this 2D view adds the stretch/backlash layer that real cable-driven wrists must be manufactured and calibrated to minimise.