Hand-considered hypotheses, 1501–1600
Control, robotics and mechatronics. Each entry was written on its own and given its own
logical form — a misattribution here, a bound there, an inversion of what the handbook says, an impossibility, a
counting argument. The form is printed above each claim so that structural repetition would be visible if it
occurred. None has been tested. Where a number appears it is an order-of-magnitude estimate and is
labelled as one; no measurement or citation is implied.
Loops, stability and the limits of feedback
Bound
H1501. Suppressing a disturbance at one frequency in a linear loop necessarily amplifies it at another, and the total amplification is fixed by the plant's unstable poles rather than by the controller's cleverness. The bookkeeping is the Poisson integral of the log of the sensitivity, so a loop that buys twenty decibels at ten hertz must give that area back somewhere in the band. Test: measure the sensitivity across two decades, integrate numerically, and compare the area with pi times the sum of the unstable poles' real parts.
Misattribution
H1502. A three-to-six-hertz limit cycle on a positioning axis is routinely booked as stiction, yet the same oscillation appears when a pure integrator faces a quantised measurement: the integrator ramps until the count changes, overshoots by one count, and unwinds. The period is then set by the integrator time and the count size, not by the lubricant. Test: hold the axis still and inject a synthetic one-count square wave into the feedback; if the loop reproduces the same period, friction was never in the story.
Threshold
H1503. The dominant error source in a slow servo changes from friction to encoder quantisation at the velocity where the quantisation's equivalent force equals the friction force, and below that velocity more gain makes the position worse because it amplifies the quantiser's staircase. That crossover is computable from the count size, the loop stiffness and the friction level. Test: sweep velocity over two decades and plot the error spectrum, looking for the count-crossing line to overtake the friction band.
Hidden term
H1504. Noise at a PID output is not white, because the derivative path weights sensor noise by frequency squared; the RMS reaching the amplifier is therefore dominated by the top decade of the sensor's band and is nearly independent of the crossover frequency. The familiar complaint that a loop is noisy describes the sensor's high-frequency floor more than it describes the gains. Test: insert an analog low-pass ahead of the differentiator at two corners and check that the output RMS falls as the inverse square of the corner.
Identity
H1505. Integrator windup and actuator rate limiting produce the same overshoot envelope after a large step, which is why they are confused during commissioning and both blamed on too much integral. They differ in what saturates — the integrator's state or the actuator's derivative — and that difference survives a controlled change. Test: clamp the integrator but keep the rate limit; overshoot that remains belongs to the rate limit, and its height should scale with the step's velocity content.
Count
H1506. Exact rejection of polynomial disturbances extends only to order N minus one in a loop carrying N integrators, so a claim of rejecting a ramp on a proportional-only controller implies an integrator hidden in the drive or the sensor. The mechanism is the internal model principle, which counts poles at the origin rather than tuning effort. Test: apply a constant-velocity disturbance and look for a bounded steady-state error, which a proportional-only loop must show at any gain.
Inversion of standard practice
H1507. On a direct-drive axis with strong cogging, lowering the position gain and adding a disturbance observer gives a smaller position error than raising the gain, because the observer removes the disturbance before the loop amplifies it rather than after. The standing instruction to stiffen the loop assumes the disturbance enters at the plant input, which cogging does not. Test: two runs matched for settling time, one tuned by gain and one by observer bandwidth, compared over a cogging period.
Saturation
H1508. Limit-cycle amplitude under saturation is not free: it settles where the describing function's equivalent gain makes the loop unity at the phase crossover, so the oscillation amplitude is itself a measurement of the saturation level. Reading that amplitude as a friction parameter therefore overestimates friction by the ratio of the two nonlinearities' equivalent gains. Test: change the drive's current limit by a known factor and check that the amplitude scales with it while the frequency does not.
Selection effect
H1509. Published comparisons in which model-based control beats PID are run on plants with strong, identifiable dynamics, because those are the plants on which a model can be built, while plants with weak or nonstationary dynamics are quietly dropped. The reported advantage is thus a selection on identifiability rather than a property of the control law. Test: pre-register a plant sample and report effect size by identifiability quartile; the advantage should vanish in the lowest quartile.
Unit trap
H1510. Gains copied between a drive configured in encoder counts and one configured in radians differ by the encoder resolution times two pi, a factor of a few thousand, and the resulting instability is usually diagnosed as a mechanical resonance. The oscillation appears at the crossover of the mis-scaled loop, which lands in the same band as a real mode. Test: change only the unit convention in the drive's configuration file and see whether the resonance moves with it.
Scale argument
H1511. On a fieldbus the mean latency grows linearly with node count while the jitter grows as its square root, so above some node count it is jitter rather than latency that limits achievable gain. Controllers tuned on a two-node bench therefore fail on a twenty-node machine at the identical nominal update rate. Test: log arrival-time distributions as nodes are added and compare gain margin at fixed mean latency.
Artefact mimics effect
H1512. Improvements credited to adaptive control sometimes come from the estimator having seen the same trajectory repeatedly during tuning, and they disappear on a fresh path with the same spectral content. The parameter estimate has converged to the test's operating point rather than to the plant. Test: hold out a trajectory generated from the same spectrum with different phase and compare the improvement on seen and unseen paths.
Bound
H1513. Static stiffness presented by a position loop is bounded by the sensor resolution times the loop gain, so past a point additional gain buys dither rather than stiffness. Below the bound the measured stiffness rises with gain as predicted; above it the reading is the encoder's noise, dressed as compliance. Test: load the axis at two gains and find where the two stiffness curves part.
Conservation
H1514. Disturbance observers do not create rejection; they relocate it, taking attenuation from the band below the observer's bandwidth and paying for it with amplification above, in an amount fixed by the filter's order. Every claim of disturbance-free operation therefore names a frequency band, whether or not the paper does. Test: integrate the measured noise power across the transition and compare it with the filter order.
Ordering effect
H1515. Swapping a notch filter and an anti-alias filter in the feedback path is not a neutral edit, because the notch's phase is frequency-dependent while the anti-alias filter's is nearly linear, and their group delays add differently in the two orders. A loop tuned with one order can lose several degrees of margin when the blocks are swapped in a firmware update that changes no gain. Test: measure phase at crossover for both orders at identical magnitude responses.
Threshold
H1516. Dither linearises a quantiser only when its RMS exceeds roughly a third of a step; below that the residual is a set of discrete jumps correlated with the signal, which is worse than no dither at all. The threshold belongs to the quantiser's characteristic, not to the actuator. Test: plot the static input-output characteristic at dither amplitudes from a tenth of a step to one step and find the transition from staircase to line.
Aliasing
H1517. Structural modes above the sampling rate fold into the control band and are indistinguishable in a spectrum from genuine low-frequency resonances, except that their apparent damping is wrong and their frequencies move with the sample rate. The fold is exact: the apparent frequency is the mode's distance from the nearest multiple of the sample rate. Test: change the sampling rate by ten percent; a folded mode moves, a real one stays.
Impossibility
H1518. Two sensors with identical transfer functions from the same input cannot separate a load disturbance from a command, however they are filtered, because the pair spans a one-dimensional space. Any observer claiming to do it is using the plant model as a third, silent measurement. Test: inject a disturbance and a command with identical waveforms at different times; identical estimator outputs prove the separation is not being made.
Ratcheting
H1519. Dead band plus integral action ratchets under a periodic disturbance: each cycle the integrator charges through the dead band in one direction and discharges only partly in the other, so the mean position drifts by an amount per cycle proportional to the dead band. The drift is slow and monotone and is therefore mistaken for thermal growth. Test: reverse the disturbance's phase; thermal drift is unchanged while ratcheting reverses.
Common-mode
H1520. Differential sensing rejects the ground it shares but not the supply it shares, because supply ripple moves both channels' references together and the difference survives intact. Differential is a statement about one node, not a claim of isolation. Test: inject ripple on the supply with the ground quiet, then on the ground with the supply quiet, and compare rejection in the two cases.
Actuators and transmissions
Misattribution
H1521. The slow drift after a step in a piezo stack is called creep and modelled as viscoelastic, but most of it is charge redistribution on the electrode capacitance, which a charge amplifier removes almost entirely. The distinction matters because viscoelastic creep is not recoverable while charge redistribution is. Test: drive the same step from a voltage source and from a charge source and compare the drift over the first hundred milliseconds.
Bound
H1522. Blocked force times free stroke, for a piezo stack, equals twice its stored elastic energy, which is fixed by the square of the maximum field and the material's compliance, so no geometry improves the product and only the strain coefficient does. A design quoting both a high blocked force and a long stroke is quoting two different devices. Test: measure both on one stack at one bias and compare the product across geometries.
Hidden term
H1523. Most of the damping measured on a voice coil is electrical: back-EMF drives current through the amplifier's output impedance and that current produces a force opposing motion, so the same actuator is many times more damped on a voltage drive than on a current drive. The mechanical loss is the small remainder. Test: measure the step response on both drive types at identical feedback gain and extract the damping ratio without touching the hardware.
Identity
H1524. Lost motion and backlash in a harmonic drive are one hysteresis measured under two loading histories: backlash is the name used when the output is driven from one side, lost motion when it is cycled. A single quasi-static torque-twist curve contains both, so quoting the numbers separately double-counts the same compliance. Test: superimpose the two protocols on one specimen and check that the curves coincide.
Threshold
H1525. Microstepping improves smoothness but not accuracy once the commanded increment is small compared with the position error set by detent torque, because the rotor's equilibrium is then fixed by the magnet rather than by the current vector. The threshold is the ratio of load torque to detent torque. Test: command one-microstep moves at several loads and measure the actual displacement with an interferometer.
Scale argument
H1526. Flexure stiffness falls as the cube of thickness while its range rises only inversely with thickness, so the force available at full deflection falls as the square and shrinking a stage tenfold cuts that force a hundredfold at constant strain. Miniature stages therefore cannot be compared with their large counterparts through any single similarity number. Test: build one topology at three thicknesses and plot available force against thickness on logarithmic axes; the slope should be two.
Inversion of standard practice
H1527. For a series-elastic actuator, softening the spring raises the achievable force-control bandwidth when the load inertia dominates the rotor inertia, because the spring then resonates with the load instead of with the motor. The rule that a stiffer spring is better holds only when the rotor dominates. Test: measure force bandwidth at two spring stiffnesses with a large and a small load inertia and locate the crossover.
Count
H1528. Tension positivity costs a dimension: a cable-driven mechanism with N cables controls at most N minus one degrees of freedom in directions where every cable can only pull, because the tension-positive solution set loses a dimension to force closure. Designs claiming N degrees from N cables are relying on gravity or a preload spring as the missing actuator. Test: command a wrench that would require one cable to go slack and check whether the pose is still controllable.
Artefact mimics effect
H1529. Ball-screw efficiency is normally measured in the driving direction over a short stroke, so it excludes the return and the preload drag, and the cycle-averaged figure is substantially lower and differs between directions. The asymmetry is largest at low speed where the lubricant film is thinnest. Test: measure input and output torque over complete back-and-forth cycles at several speeds.
Unit trap
H1530. Torque constants in ounce-inches per ampere differ from their SI values by a factor of about 141, and mixing the two in a gain calculation produces a loop that oscillates at a frequency set by the error rather than by the machine. The ambiguity survives because both units appear on the same datasheet. Test: recompute one axis's velocity gain from the SI value alone and see whether the oscillation disappears.
Reversal
H1531. The thermal drift of a preloaded ball screw changes sign as speed rises, because below a threshold the nut friction and the ambient dominate and the screw is held in compression by its fixed bearing, while above it the screw's own heating dominates and it extends. The reversal speed is set by the preload and by the bearing's axial stiffness. Test: run a speed staircase and record the sign of the free-end displacement.
Hidden term
H1532. At low commanded current the inverter's dead time dominates the volt-second balance, so the current loop has a dead band around zero whose width is set by dead time and bus voltage rather than by the controller. The resulting torque error is worst exactly where a direct-drive arm spends most of its time, near zero force. Test: measure current tracking at one, five and twenty percent of rated current and look for a flat region at the zero crossings.
Threshold
H1533. Below a preload threshold a linear guide's rolling elements never unload, so stick-slip disappears outright rather than merely diminishing, the threshold being the load at which the external force variation equals the preload. Above it the same guide shows ordinary Stribeck behaviour. Test: sweep preload at fixed external oscillation amplitude and find where the hysteresis loop collapses.
Bound
H1534. Force resolution in a series-elastic actuator is spring stiffness times encoder resolution, so a softer spring resolves smaller forces but lowers the resonance that sets bandwidth, and the product of resolution and bandwidth is fixed by the encoder alone. Quoting both figures from different prototypes is quoting different machines. Test: measure force noise and force bandwidth on one actuator at two spring stiffnesses.
Conservation
H1535. The energy a regenerative drive recovers over a cycle is bounded by the braking resistor's thermal rating and by the bus capacitor's usable swing, whichever is smaller, so energy-neutral claims usually rest on one cycle at low duty. Over a shift the resistor's thermal time constant, not the mechanics, sets the recovery. Test: repeat a cycle at increasing duty until the recovered fraction falls and compare the fall with the resistor's rating.
Selection effect
H1536. Actuator comparisons are made on new units at room temperature with factory grease, so the ranking of technologies can reverse after a few thousand cycles as grease channels and preload relaxes. The published ranking is a ranking at the initial condition, which is the least representative one. Test: repeat the comparison on the same units after an accelerated life test and look for rank reversal.
Misattribution
H1537. Cogging torque measured with a locked rotor includes the fixture's compliance, and harmonics above the slot order are usually the fixture's resonance rather than the magnet array's. A well-made motor's true cogging spectrum is dominated by the slot and pole counts and falls away quickly. Test: repeat the measurement with a stiffer fixture and a different inertia and see which harmonics move.
Identity
H1538. Detent torque and cogging torque are one physical quantity measured on opposite sides of the gearbox: one is the magnet's preference for a position, the other is the same preference referred through the reduction. Listing them as separate specifications counts the same energy twice. Test: measure at the motor shaft and at the output, refer both to one side, and compare.
Threshold
H1539. PWM acoustic output is dominated by the carrier until the duty approaches zero or one, where the ripple's fundamental enters the audible band and its amplitude grows abruptly at a duty threshold set by the dead time's volt-seconds. The tone is often chased as a mechanical rattle. Test: sweep duty at fixed load and record sound pressure at the switching frequency and its sidebands.
Scale argument
H1540. The oil film in a hydrostatic guide has a Reynolds number scaling with speed times the cube of the clearance, so a small guide at high speed is hydrodynamically different from a large one at low speed even at the same bearing number. Bench tests on small guides therefore do not transfer to large machines by any simple scaling. Test: build two geometrically similar guides at ten-to-one scale and compare stiffness at equal bearing number.
Inversion of standard practice
H1541. Deliberately bleeding a little air from a pneumatic cylinder improves position hold, because the continuous flow keeps the servo valve off its lapped centre where gain collapses. The instinct to seal everything perfectly parks the valve exactly where its dead band is widest. Test: measure holding accuracy with a controlled leak at three valve openings.
Artefact mimics effect
H1542. Force ripple measured on a linear motor with a load cell includes the cell's own mass-spring resonance, and the apparent spatial period changes when the cell is replaced, which is the signature of the instrument rather than of the motor. A genuine magnet-pitch ripple is fixed in space and does not move with the transducer. Test: measure with two cells of different stiffness and compare the spatial spectra.
Sensing and estimation
Hidden term
H1543. A strain-gauge bridge carries the lead resistance's temperature coefficient unless the leads are isothermal, and three-wire compensation removes only the equal part, leaving a residual proportional to the temperature difference between the cable's ends rather than to its absolute temperature. The residual is read as gauge drift and compensated in software at the wrong variable. Test: heat one end of the cable only and compare the indicated strain with a dummy gauge in the same bridge.
Misattribution
H1544. Gradual loss of encoder signal is usually blamed on contamination, but in readheads with automatic gain control the symptom of LED ageing is an interpolation phase shift, which surfaces as position error at the line pitch rather than as a signal loss. The AGC hides precisely the amplitude change that would identify the cause. Test: log the AGC value over life and correlate the error's phase with it.
Threshold
H1545. Below a signal-to-noise ratio the interpolation error of a quadrature encoder changes character from a smooth sinusoid to a staircase, because the arctangent's noise dominates and the output quantises to the readhead's internal resolution. The error amplitude then jumps by a factor of several at a threshold absent from every datasheet. Test: attenuate the readhead signal deliberately and plot error harmonics against attenuation.
Count
H1546. An observer of order N can estimate at most N independent states from one measurement, so a sensorless claim for a plant with N plus one energy-storing elements needs either a second measurement or a structural identity that removes one state. The missing state is usually the one the claim is about. Test: count the plant's storage elements and the observer's independent states and examine the difference.
Identity
H1547. For most position sensors the quoted resolution is the RMS noise in counts at standstill, which makes it the same number as the noise floor, so smallest detectable step and noise are not independent specifications. A resolution claimed below that floor is an average over a time the datasheet does not state. Test: hold the axis still for a minute and compare the Allan deviation at one second with the quoted resolution.
Unit trap
H1548. An accelerometer's sensitivity in millivolts per g differs from the SI value by 9.80665, and the error propagates into every gain that converts acceleration into current. Because g is close to ten the mistake survives a factor-of-ten sanity check and is blamed on the mechanism. Test: calibrate the same sensor on a shaker against laser vibrometry and compare the two sensitivities.
Scale argument
H1549. An accelerometer's bandwidth rises as its seismic mass falls, but at fixed electronics noise its noise floor rises as the inverse square root of that mass, so the product of noise and bandwidth is fixed by the sensing principle. A sensor advertised as both wideband and low-noise is being quoted at two different gains. Test: measure noise floor and resonance on two sensors of one family with different masses.
Artefact mimics effect
H1550. Hysteresis measured on a torque sensor during a slow ramp is often the fixture's stiction rather than the sensor's compliance, and it appears only on the first ramp after a reversal. Genuine sensing hysteresis is rate-independent and survives a dwell at zero torque. Test: reverse at two rates, with and without a hold at zero.
Reversal
H1551. An eddy-current sensor's sensitivity rises with target thickness until the skin depth is reached and then falls, because the target's own eddy currents begin to screen the coil; the peak sits at a thickness comparable to the skin depth. A specification quoting one target thickness hides the peak entirely. Test: sweep thickness at fixed carrier frequency and locate the maximum.
Selection effect
H1552. Tactile-sensing benchmarks are dominated by compliant objects, because compliance makes contact easy to detect, so reported classification accuracy overstates performance on rigid objects, which is where most assembly happens. The selection is on the object set rather than on the sensor. Test: re-evaluate a published sensor on a rigid-only object set matched for shape.
Hidden term
H1553. Camera timestamps refer to the start or middle of the exposure depending on the driver, never to the end of readout, so a vision loop carries half an exposure of latency that no one accounts for. At long exposures that unaccounted half is the difference between a stable loop and an unstable one. Test: image a flashing LED of known phase and measure the offset between the reported stamp and the event.
Bound
H1554. Vision-servo gain is bounded by the product of exposure time and image velocity, since beyond a few pixels of blur the feature centroid's bias grows faster than gain can compensate. Frame rate is irrelevant to this bound; only exposure enters it. Test: hold frame rate fixed, vary exposure, and measure phase margin.
Inversion of standard practice
H1555. In force control against a stiff environment a slower force sensor is often more stable than a fast one, because the sensor's filter attenuates the contact resonance the loop would otherwise chase. The demand for higher sensor bandwidth is right for free motion and wrong for contact. Test: compare contact stability at two filter bandwidths on the same stiff surface.
Identity
H1556. Drift quoted for a force-torque sensor over minutes and thermal zero shift quoted over hours are one process observed at two time scales, since both come from the bridge's temperature gradient with a time constant of minutes. A sensor with good short-term drift and a poor thermal specification is not two different sensors. Test: hold a fixed gradient and record the zero across both windows.
Misattribution
H1557. Position error at once per revolution is assigned to encoder eccentricity, but an angularly misaligned coupling produces the same signature and often a larger one. The two separate because the coupling's error depends on torque direction while the encoder's does not. Test: repeat the error map under two load directions.
Threshold
H1558. Differentiating Hall transitions to estimate velocity produces a dead band at low speed: below one transition per sample the estimate alternates between zero and a large value, and the controller's average is biased low. The threshold is one count per sample period, a few millimetres per second on a typical arm. Test: command a slow ramp and compare the estimate with an interferometer.
Artefact mimics effect
H1559. Lidar range bias with target reflectance mimics a pure time delay when the target moves, because both produce a lag proportional to velocity, so a loop appears to carry latency that is really a ranging error. The two separate when reflectance changes at fixed speed. Test: translate a grey card and a retroreflector at identical speed and compare the apparent delays.
Unit trap
H1560. Body rates in radians per second and quaternion derivatives differ by a factor of one half and a rotation, so integrating the wrong one yields an attitude whose drift is proportional to the angle rather than to the rate error. The mistake is invisible for small motions and grows with the motion itself. Test: integrate a constant-rate rotation through a full turn by both conventions and compare the endpoints.
Conservation
H1561. For a battery-free sensor the product of update rate and energy per update is bounded by the harvester's average power, so a node claiming both a high rate and a long life is claiming two different harvesters. The bound is set by the radio's energy per bit, not by the sensing element. Test: measure average power and update rate across payload sizes.
Ordering effect
H1562. The order of the sample-and-hold and the anti-alias filter changes the effective aperture: with the hold first the filter sees a staircase whose group delay is half a sample, with the filter first it sees the filter's own delay. Swapping them in firmware shifts the loop's phase by a fraction of a sample while changing no gain. Test: measure phase at crossover in both orders.
Calibration and metrology
Symmetry argument
H1563. The reversal technique separates spindle error from artefact error by assuming the artefact's own error is fixed in the artefact's frame, but gravity sag breaks that assumption above a certain artefact mass, because the sag rotates with the artefact and not with the spindle. The separation then folds a real elastic term into the spindle's error map. Test: repeat the reversal with the artefact supported at two different points.
Hidden term
H1564. Thermal uncertainty in a calibration is the artefact's expansion coefficient times the difference between calibration and use temperature, not its absolute value, so a high-coefficient artefact used where it was calibrated beats a stable one used five kelvin away. Datasheets quote the coefficient and conceal the difference. Test: calibrate at two artefact temperatures and compare the residual scale error.
Count
H1565. Averaging many artefacts reduces uncertainty in proportion to the square root of the number of independent calibration lineages, not of artefacts, so a set traced to one master gives no reduction whatever. This is why large inter-comparisons sometimes show no improvement over a single artefact. Test: group artefacts by lineage and compute between-group and within-group variance.
Selection effect
H1566. Inter-laboratory comparisons recruit volunteers, and laboratories expecting to do well volunteer first, so the reported agreement is a selection on expected performance rather than a property of the method. The bias is largest for the laboratories that decline, whose results are never seen. Test: compare a volunteer round with a mandated round on the same artefact set.
Unit trap
H1567. An arc-second is 4.848 microradians, and stage specifications mix the two, so a straightness in microradians and an angular error in arc-seconds can differ by a factor of five inside one document. The mixture survives because both numbers are small. Test: recompute one specification chain entirely in SI and compare it with the vendor's error budget.
Scale argument
H1568. Abbe error grows with the offset between the measurement line and the point of interest times the sine of the tilt, so at a hundred-millimetre offset one arc-second is about half a micrometre — an estimate, but the right order. Machines accurate over short travel and wrong over long travel are usually showing this term, not a scale error. Test: measure one feature at two offsets with the same encoder.
Misattribution
H1569. The hysteresis of a micrometer is assigned to elastic recovery of the frame, but at hand speeds it is mostly flank friction in the thread, which is why it grows when the thimble is turned quickly. Elastic recovery would be rate-independent. Test: approach one reading at two speeds and compare the overshoot on reversal.
Artefact mimics effect
H1570. Dead-path error in a laser interferometer appears as a scale error when the dead path changes between measurements, and it is easily mistaken for the machine's calibration because it is proportional to distance. Measuring two lengths with the same dead path separates the two. Test: compare a short and a long measurement with the reference mirror moved between them.
Count
H1571. Below a certain number of sampled points a least-squares circle fit's radius is biased low by an amount proportional to the form error squared over the radius, and the bias exceeds the random uncertainty at the same point count. The threshold sits at a few dozen points. Test: fit circles to one data set at increasing subsample counts and plot the radius.
Identity
H1572. For a robot that always returns to the same taught pose, quoted accuracy and repeatability are one measurement made with and without a pose-dependent correction, so a repeatability of a few hundredths of a millimetre and an accuracy ten times worse are two views of a single error map. Test: measure both on the same path before and after kinematic identification.
Inversion of standard practice
H1573. Calibrating a robot across its whole workspace is worse than calibrating near the task space when the kinematic model is under-parameterised, because the fit distributes residual over poses that will never be used and leaves the task space worse than before. The instruction to cover the workspace assumes a complete model. Test: compare task-space residuals after whole-workspace and task-space identification.
Hidden term
H1574. Squareness error in the reference artefact folds into the machine's straightness measurement and cannot be separated without a reversal or a second artefact of different size. Most error budgets quote the machine's straightness and silently include the square. Test: measure with two squares of different provenance and compare the difference.
Count
H1575. Identifying N kinematic parameters needs poses whose Jacobian rows span N dimensions, so a planar test identifies at most three per pose and a pose set that varies along one axis cannot identify the rest. Reports of dozens of parameters from a single circular path are rank-deficient. Test: compute the identification Jacobian's condition number for the reported pose set.
Derived-quantity trap
H1576. Volumetric accuracy, as quoted, is a maximum over a sampled grid, so it is a property of the grid as much as of the machine, and a finer grid can only make it worse. Two machines compared on different grids are not comparable at all. Test: evaluate one error map on grids of two densities and compare the reported numbers.
Threshold
H1577. A contacting probe must exceed one force to break the surface film and stay below a second to avoid elastic indentation, and the window narrows as roughness rises because the film's thickness varies over the asperities. Very smooth surfaces are therefore easier to measure than moderately rough ones. Test: sweep contact force on samples of increasing roughness and record scatter.
Inversion of standard practice
H1578. Calibrating at the service temperature rather than at the standard twenty degrees reduces the dominant uncertainty even though it breaks the convention, because the expansion between calibration and use is larger than the coefficient's own uncertainty. The convention optimises comparability, not accuracy. Test: calibrate one artefact both ways and compare in-service residuals.
Artefact mimics effect
H1579. Thermal drift seen on a coordinate machine is often the workpiece's drift rather than the machine's, because the workpiece has less thermal mass and follows the room faster. Warm-up protocols measure the machine and ignore the part. Test: run one program with an unloaded platen and with a large steel part.
Unit trap
H1580. Newton-metres and pound-feet differ by about 1.356, and a wrench calibrated in one unit used against a budget written in the other produces a preload error of that factor, which sits inside the scatter of most bolted joints and is therefore never diagnosed. Test: instrument one joint with a load washer and compute the preload both ways.
Motion systems and mechanisms
Threshold
H1581. The smallest repeatable step of a friction-driven stage is its pre-sliding displacement, which is elastic and fully recoverable, so below it a stage makes steps of consistent size but no consistent absolute position. This is why nanopositioning demonstrations show repeatable increments with a drifting mean. Test: command a hundred one-nanometre steps and plot both the increment distribution and the drift of the mean.
Misattribution
H1582. The long settling tail of a positioning stage is often the machine frame ringing rather than the stage, and it appears at the same frequency on every axis because the frame is shared. Mass added to the stage changes nothing while mass added to the frame moves the tail. Test: instrument the frame with an accelerometer and compare its spectrum with the encoder's tail.
Count
H1583. Folded-beam flexure stages impose 6N constraints against six rigid modes, so they are over-constrained by construction and their surplus constraints appear as parasitic modes rather than as pure stiffness. Adding beams for stiffness adds modes in the same proportion. Test: count the modes in a measured frequency response and compare with the beam count.
Inversion of standard practice
H1584. Raising a scanning stage's closed-loop bandwidth can worsen straightness, because the loop then follows the command's quantisation and the digital-to-analog converter's step edges more faithfully than the mechanics can smooth them. Tighter control helps geometry only below the command's quantisation. Test: scan one profile open-loop and closed-loop and compare the straightness residual's spectrum.
Scale argument
H1585. Angular stiffness in a flexure pivot falls steeply with blade thickness while range rises, so a pivot built for a large stage cannot be scaled down without changing topology, since the ratio of range to stiffness is not invariant. Miniature versions need more blades, not thinner ones. Test: build one topology at three thicknesses and plot range against stiffness.
Hidden term
H1586. Axis cross-coupling includes the digital-to-analog converter's channel-to-channel settling, because a large step on one channel perturbs the shared internal reference seen by the other. The coupling appears only while the other axis is moving, which is exactly when it is not being measured. Test: step one channel with the other actuator disconnected and watch the other channel's output.
Artefact mimics effect
H1587. Resonances in the two-to-four-hundred-hertz band on a positioning machine are frequently the floor rather than the machine, because isolators and slab form a mode excited by the stage's reaction force. Its frequency then depends on where the machine sits, not on how it was designed. Test: move the machine onto a stiff block and repeat the measurement.
Conservation
H1588. Peak power in a minimum-time trajectory is fixed by the drive's voltage and current limits and by the inertia, so a claim of faster moves without more power is a claim about a different inertia or a different limit. Energy is not the binding constraint; the peak is. Test: measure bus power during a minimum-time move and compare it with the drive rating.
Threshold
H1589. Above an acceleration threshold a recirculating guide's rolling elements transition from rolling to sliding, and the threshold is the ratio of inertial force to preload rather than any speed. The transition is sharp, produces a step change in friction, and looks like a controller fault. Test: ramp acceleration slowly at fixed speed and record friction force.
Identity
H1590. The friction parameters identified from a circle test and from a step test are different projections of one bristle model, and neither test excites the state that would distinguish them. A circle at constant speed identifies the velocity branch while a step identifies the pre-sliding stiffness. Test: identify both on one machine and predict a third experiment that neither was fitted to.
Selection effect
H1591. Multi-axis nanopositioning results are usually reported for the best axis, since the axes are built alike but aligned differently and one always has the smallest parasitic error. The published number is a maximum over axes rather than a typical value. Test: ask for per-axis data and compare the spread within one stage.
Ordering effect
H1592. When two axes share a flexure, moving X then Y does not reach the same pose as Y then X, because the first move changes the stiffness the second one works against. The error is path-dependent and does not average away over repeated cycles. Test: command both orders to one endpoint and measure the difference with a capacitance gauge.
Unit trap
H1593. Jerk limits in metres per second cubed and in g per second differ by 9.80665, and trajectory generators often take the number without its unit, so a profile tuned on one machine is violent on another. The symptom is a spectral line at each segment boundary rather than a smooth limit. Test: compute commanded jerk as the third difference of the profile and compare it with the intended limit.
Reversal
H1594. Increasing a linear guide's preload reduces lost motion but raises friction, and the two cross at a preload well below the maximum sold; past that point lost motion is already negligible and only friction grows. Preloading as much as possible therefore buys nothing and costs heat. Test: sweep preload and plot lost motion and friction on one pair of axes.
Bound
H1595. Usable bandwidth of a ball-screw drive is bounded by the screw's first axial resonance, which falls inversely with length, so long-travel machines cannot be fast however strong the motor. The bound is mechanical and no controller moves it. Test: measure the axial resonance on two screws of different length and compare with the predicted inverse relation.
Artefact mimics effect
H1596. Drift measured on a stage in vacuum is often the motor's own heating rather than an ambient change, because there is no air to carry heat away and the coil's time constant stretches to hours. The drift direction follows the coil's location, not gravity. Test: run the profile with the coil energised and again with it disabled but the stage driven through a counterweight.
Hidden term
H1597. The encoder scale's thermal expansion sits outside the loop, so a metre of steel scale at one kelvin moves about eleven micrometres — an estimate — which is larger than most stages' resolution and comparable to their accuracy. Feedback holds the readhead to the scale, not the tool to the part. Test: measure one artefact with the scale at two temperatures.
Threshold
H1598. Dither frees a stuck stage only when its amplitude exceeds the pre-sliding displacement; below that it heats the actuator and leaves the stage where it was, which is why dither appears to work at one setting and not at another. The threshold belongs to the contact, not to the drive. Test: sweep dither amplitude at fixed frequency and locate the onset of motion.
Misattribution
H1599. Lost motion at reversal is often the coupling's wind-up rather than the screw's backlash, and the two separate by locking the output: a coupling shows free input rotation with the output clamped, while backlash does not. Most machines are diagnosed without that clamp in place. Test: clamp the output and measure the input's free travel.
Impossibility
H1600. No motion axis is at once stiff, light and well damped, because stiffness over mass fixes the resonance and damping at that resonance is set by the material's loss factor, which is small for every structural material. Control can add damping electrically, but only by spending bandwidth that the same resonance bounds. Test: measure the three quantities on stages of three materials and plot them against loss factor.
Where the three sharpest ones sit
If three of these were worth an afternoon each, they would be H1518, H1527 and H1563 — an impossibility about twin sensors, an inversion about spring stiffness in series-elastic actuators, and a symmetry failure in the reversal technique. Each has a test that needs one machine and one changed condition, and each would be informative whether or not the claim survives.
100 of 100 written; none tested. Estimates are labelled where they occur; no value here is a measurement.
Incoprea Research · hand-considered hypotheses 1501–1600 · control, robotics and mechatronics · 3 November 2026 · ~/Desktop/Nov3/