MOTION CONTROL / NUMERICAL STUDY
Servo-axis commissioning study
Executed numerical checks and a declared tracking-model run-through for the selected inspection-shuttle cycle.
The hardware is represented by a second-order tracking ODE, declared motor constraints, friction corners and command quantization. These are synthetic/theoretical results, not physical measurements.
Main calculation report / Acceptance record / Tracking results
Reduced tracking model
ẍ + 2ζωₙ ẋ + ωₙ²x = 2ζωₙ v_command + ωₙ² x_command ωₙ = 2π f_n, ζ = 0.75 e = x_command - x ë + 2ζωₙ ė + ωₙ²e = a_command
For a sufficiently long constant-acceleration segment, the steady tracking error approaches a_command/ωₙ². The integration covers the outward move, its acceleration and deceleration ramps, and a final hold. The model assumes velocity feedforward but no acceleration feedforward. It is a sensitivity model, not an identified servo loop. The 35 Hz and 25 Hz cases must stay within ±0.10 mm; the deliberately degraded 8 Hz case is expected to fail that same allowance.
| Assumed bandwidth (Hz) | Peak error (mm) | Expected | Observed |
|---|---|---|---|
| 35 | 0.049917 | accepted | accepted |
| 25 | 0.098067 | accepted | accepted |
| 8 | 0.970979 | rejected | rejected |
Acceptance record
| Check | Expected | Observed | Matched |
|---|---|---|---|
| Selected setting fits every declared corner | all five corner outcomes accepted | 5 | yes |
| Signed shaft work equals all mechanical losses | absolute residual < 1e-8 J | 2.0095036745715333e-14 | yes |
| High mass and drag exceed the declared drive envelope | rejected | ['peak_torque_Nm', 'rms_torque_Nm', 'peak_current_Arms', 'rms_current_Arms', 'bus_regen_J', 'average_bus_regen_W'] | yes |
| Insufficient peak current is detected | rejected | True | yes |
| Exact torque-square integral agrees with adaptive quadrature | difference < 1e-8 N2 m2 s | 1.433159146912999e-13 | yes |
| Passive losses and separately accumulated energy are nonnegative | minimum >= -1e-9 J | 0.0 | yes |
| Horizontal forward and reverse duties agree | max difference < 1e-9 | 0.0 | yes |
| Position rounding stays within half a 1 micrometer increment | maximum <= 0.5 micrometer | 0.5000000000005 | yes |
| Every jerk ramp spans at least ten 1 ms controller intervals | >= 10 intervals | 49.99999999999999 | yes |
| Reduced tracking model at 35 Hz | within 0.10 mm following-error allowance | 0.04991720150532286 | yes |
| Reduced tracking model at 25 Hz | within 0.10 mm following-error allowance | 0.09806687745672962 | yes |
| Reduced tracking model at 8 Hz | outside 0.10 mm following-error allowance | 0.9709794229880528 | yes |
| Halving the integration step leaves the following-error result stable | maximum pointwise change < 0.001 mm | 8.634648551719692e-09 | yes |
| Lossless deceleration returns the peak kinetic energy | difference < 1e-9 J | 8.881784197001252e-16 | yes |
| Regression record matches the executed source | all regressions pass; hashes current | {'tests': 20, 'hashes_current': True} | yes |
Interpretation
The selected cycle fits the declared load and drive corners. Increasing mass to 160 kg while raising Coulomb drag to 90 N exceeds the envelope; the failed limits are retained in overload.json. A passed acceptance record means the observed outcome matched its stated expectation. It does not turn the overload or degraded-bandwidth case into an acceptable setting.
Drive saturation, structural resonances, flexible couplings, sensor noise, interpolation delay, thermal time constants and safety logic are not modeled. These boundaries are part of the delivered study. No commissioning of physical hardware is claimed.