A surgical-robot joint needs near-zero play: a fraction of a degree of backlash at the actuator is amplified into millimetres of instrument-tip error inside the patient. Most robotic-surgery joints use a harmonic drive (strain-wave gearing) instead of ordinary spur gears for exactly this reason.
Three parts, one meshing tooth-count difference:
Wave generator (WG) — elliptical cam, driven by the motor
Flexspline (FS) — thin flexible cup, teeth = T_fs = 100
Circular spline (CS) — rigid ring, teeth = T_cs = T_fs + 2 = 102
Gear ratio (circular spline fixed to housing, output = flexspline):
N = -T_fs / (T_cs - T_fs) = -T_fs / 2 = -50 (i.e. -50 : 1)
Output torque model (contact regime):
τ_out = kθ·θ_err − b·ω_out + f_s·sign(ω_out) (elastic + damping + stiction)
Fixed vs. the 3D reference build this simulation is twinned with: that model's own theory text claimed "N ≈ 100" for a 100-tooth flexspline, but its code already computed the ratio correctly as -T_fs/2 = -50 (and even displayed "−50 : 1" once the page finished loading — only its static pre-script placeholder text said "−100 : 1"). A tooth-count difference of 2 always halves T_fs into the ratio; N=100 would require a difference of 1 tooth, which no physical strain-wave gear uses. This 2D build keeps the correct N = -50 and states it plainly everywhere, including in this theory box.
The wave generator's rotation deforms the flexspline into an ellipse, so it meshes fully with the circular spline over ~30% of its circumference at any instant — both flanks of every engaged tooth are loaded simultaneously, which is what drives backlash toward zero. A conventional spur pair only touches one flank per tooth, leaving a dead-zone gap that the output free-wheels through every time the load reverses.
- Wave generator speed — motor-side input speed shared by both transmissions; each divides it by its own ratio.
- Load torque / static friction — how hard the simulated tissue pushes back on the instrument tip; larger values open a wider dead-zone gap in the spur gear and a larger elastic deflection in the harmonic drive.
- Load direction — sign of the load torque relative to motion; flip it to see the backlash gap re-open from the other side.
- Harmonic drive / Spur gear — both models run every frame from the same input trajectory; this button only chooses which one drives the big schematic and the "active" readouts — the oscilloscope always plots both.
- Reverse rotation — flips the input direction so you can watch the backlash dead-zone appear at the reversal instant.
- Drag the schematic to tilt the viewing angle (a 2D stand-in for orbiting a 3D rig); scroll or pinch to zoom.
Real-world relevance: this near-zero-backlash property is why harmonic-drive actuators (or comparable cable/capstan drives) sit inside the wrist and shoulder joints of teleoperated surgical robots such as the da Vinci system — a surgeon's millimetre-scale hand motion must reach the instrument tip with no perceptible lag or looseness.