Force Measurement Technology: Plates, Pressure, and the Ground Reaction Wrench

Dieter Olson

Force Measurement Technology: Plates, Pressure, and the Ground Reaction Wrench

technology

How force plates work: transducer physics, the six-component ground reaction wrench, centre-of-pressure geometry, and pressure mat limitations.

Author

Dieter Olson

Published

August 2, 2026

A multiaxis force plate measures the load between golfer and ground: upward support, sideways push and turning effects. A pressure-only mat shows how normal pressure is spread out. They answer different questions. Neither measures muscle effort or shows which change will improve a swing.

What Is Under the Floor

Load-sensitive parts turn load into electrical signals. Strain gauges sense resistance changes; piezoelectric sensors generate charge. Calibration turns signals into forces and moments.

What the Centre of Pressure Summarizes

The centre of pressure is a summary point on the contact surface. It is not where your body mass sits. The full load generally includes a force there plus a leftover twisting couple. Different pressure patterns can give the same dot. A small normal load makes the dot error-prone.

Seesaw: A balance point cannot say whether two children sit near the middle or one heavy child sits at the end. Like it, the centre of pressure drops information.

Why Golf Cares About Turning Effects

Opposite sideways forces at separated feet can turn the body even when they cancel as a net force. Each footprint can also carry its own twisting couple. These are separate. Measuring them helps describe how angular momentum changes. Explaining muscle work or better technique needs more evidence.

Key Takeaway: A six-axis plate reports three forces and three moments in a stated frame. A pressure-only mat cannot uniquely find sideways friction or twist. A model may predict them, but that needs validation.

Why This Article Exists

A force plate connects a movement to the loads exchanged with its surroundings. It is especially valuable because a calibrated instrument observes contact loads independently of the kinematic model used to estimate joint moments. It is not the only way to measure an external load: instrumented handles, clubs and wearable force transducers provide other examples. Nor does a plate directly measure muscle force, joint work or the cause of a movement strategy.

The useful chain of reasoning is sensor response → calibrated contact wrench → momentum balance → model-dependent joint kinetics → a testable performance hypothesis. Each arrow introduces assumptions. A precise instrument does not rescue a wrong coordinate convention; an accurate contact wrench does not uniquely identify individual muscles; an association with clubhead speed does not establish an effective coaching intervention.

This article follows that chain through transducer physics, wrench geometry, uncertainty, pressure sensing and the golf literature. The aim is to identify which questions an instrument can answer, how to check the answer, and what additional evidence is needed for the next inference.

Part I: The Measurand Is a Wrench

Six Components in a Declared Frame

A distributed contact load is mechanically equivalent, for the resultant force and moment on a rigid body, to a force and a couple:

\[ \mathcal W_O=(\mathbf F,\mathbf M_O). \]

Here the components are expressed in one right-handed Cartesian frame and the moment is about a declared point $O$. We use the ground-on-golfer sign convention, with $z$ upward. A device may instead export the opposite action or different axis directions; transform its calibrated outputs before using the equations below. A six-component output also says nothing by itself about accuracy, bandwidth or the loads that bypass the sensing structure.

Moving the reference point to $P$, while retaining the same axis orientation, gives

\[ \mathbf M_P=\mathbf M_O-\mathbf r_{OP}\times\mathbf F. \]

The force is unchanged by this translation. Rotating the coordinate frame transforms the components of both vectors. A surface-height correction is therefore a wrench transport, not an optional plotting adjustment.

For a rigid body with angular velocity $\boldsymbol\omega$, the paired twist uses the velocity of the same reference point:

\[ \begin{aligned} P&=\mathbf F\cdot\mathbf v_O+ \mathbf M_O\cdot\boldsymbol\omega,\\ \mathbf v_P&=\mathbf v_O+ \boldsymbol\omega\times\mathbf r_{OP}. \end{aligned} \]

Consistent transport leaves $P$ unchanged. Motion capture observes marker positions or images; fitting segment poses, differentiating and filtering are needed to estimate these velocities. A COP trace is a changing summary of contact loading, so its time derivative is generally not the material-point velocity to use in this power equation. For deforming contact, the more general statement integrates local traction dotted with local material velocity over the contact area.

The Central Axis and Its Exceptional Cases

When $\mathbf F\ne\mathbf0$, a wrench has a central axis on which the residual moment is parallel to its force:

\[ \begin{aligned} h&=\frac{\mathbf F\cdot\mathbf M_O}{\|\mathbf F\|^2},\\ \mathbf r_{OA}(\alpha)&= \frac{\mathbf F\times\mathbf M_O}{\|\mathbf F\|^2} +\alpha\mathbf F. \end{aligned} \]

The first term is the closest point on that axis to $O$; $\alpha$ parametrizes the line. At every point on it the remaining moment is $h\mathbf F$. The pitch $h$ has units of length. A pure couple ($\mathbf F=0$, $\mathbf M_O\ne0$) is already independent of origin and has no finite central axis defined by this formula. The zero wrench has no unique axis either.

The central axis should not be confused with the centre of pressure. The former aligns the residual moment with the force; the latter, for planar contact, aligns it with the surface normal. Those directions differ when shear is present.

Centre of Pressure, Surface Height and the Free Moment

Let the contact surface be the plane $z=z_0$ relative to $O$. For $F_z\ne0$, define $C=(x_C,y_C,z_0)$ so that the transported moment has no $x$ or $y$ component. Expanding $\mathbf M_C=\mathbf M_O-\mathbf r_{OC}\times\mathbf F$ yields

\[ \begin{aligned} x_C&=\frac{z_0F_x-M_y}{F_z},\\ y_C&=\frac{M_x+z_0F_y}{F_z},\\ T_z&=M_z-x_CF_y+y_CF_x. \end{aligned} \]

Thus the equivalent contact description is a force at the COP plus a free normal couple $T_z\mathbf e_z$. A single force at the COP is insufficient unless this residual couple vanishes. The $z_0$ signs follow from the stated axes and moment convention; copying a manufacturer’s formula without its convention can reverse them.

For a flat, unilateral pressure distribution without additional local bending couples, the COP is the centroid of normal pressure and lies within the convex hull of the loaded area. It does not locate the centre of mass or describe the pressure pattern. Two very different heel/forefoot distributions can have the same resultant and COP. Nonplanar contact requires a corresponding surface model rather than blind use of planar formulas.

A Surface-Offset Check

Take $\mathbf F=(100,-50,800)$ N, $C=(0.10,-0.05,0.04)$ m and $T_z=8$ N m. The moment at the sensor origin is

\[ \mathbf M_O=C\times\mathbf F+8\mathbf e_z =(-38,-76,8)\ \mathrm{N\,m}. \]

The equations recover the original COP and free couple. Omitting the height produces $(0.095,-0.0475)$ m: errors of $-5$ mm and $+2.5$ mm. Reversing both height signs doubles these errors. This constructed load is a useful sign check; it is not a measurement of a golfer.

For this same planar wrench, the central axis intersects the surface at

\[ A=C+\frac{T_z}{\|\mathbf F\|^2}(F_y,-F_x,0). \]

It therefore misses the COP when $T_z\ne0$ and shear is nonzero. In the example, the horizontal difference is approximately $(-0.61,-1.23)$ mm. The distinction is geometric, not sensor noise. With a fixed surface and consistent wrench transport, the computed residual free couple is unchanged by moving the reporting origin; the raw component $M_z$ generally is not.

Why COP Uncertainty Grows During Unloading

A denominator near zero makes COP ill-conditioned, but “relative COP error scales as $1/F_z$” is not a general uncertainty law. The numerator, denominator, surface height and their correlations all matter; relative error is also ill-defined when the chosen COP coordinate is zero.

For small perturbations about a nonzero load, first-order propagation gives

\[ \begin{aligned} \delta x_C&\simeq \frac{z_0\delta F_x-\delta M_y+F_x\delta z_0-x_C\delta F_z}{F_z},\\ \delta y_C&\simeq \frac{\delta M_x+z_0\delta F_y+F_y\delta z_0-y_C\delta F_z}{F_z}. \end{aligned} \]

Writing these derivatives as a Jacobian $J_C$ for the measured wrench and height, the covariance estimate is $\Sigma_C\simeq J_C\Sigma J_C^\mathsf T$. Retain cross-channel covariance when calibration or electronics couple the errors. The linear approximation becomes unreliable when the uncertainty in $F_z$ is comparable to $|F_z|$; reporting a confident COP ellipse then is misleading.

A height bias alone gives $\delta x_C=(F_x/F_z)\delta z_0$ and $\delta y_C=(F_y/F_z)\delta z_0$. It depends on the shear-to-normal ratio, not just the size of shear. Moment offsets can dominate during unloading, while gain, alignment, dynamic response or pressure saturation can dominate elsewhere. There is no universal noise source or universal minimum-load threshold.

Report the chosen load gate and how it was established from the instrument’s uncertainty and the intended analysis. Mask or flag unsupported COP samples rather than interpreting an unloading trail-foot excursion as a deliberate control action. Keep force and moment data with the COP: the dot alone hides why the estimate changed.

Per-Foot Torsion Is Not the Whole Turning Moment

For two feet, moments about a common point $O$ add as

\[ \mathbf M_O=\sum_{i=1}^{2} \left(\mathbf r_{OC_i}\times\mathbf F_i+T_{z,i}\mathbf e_z\right). \]

Opposite horizontal forces at separated feet can produce a vertical turning moment with zero net horizontal force even if both per-foot free moments are zero. Conversely, torsional traction within either footprint can produce a free moment. Both effects can change angular momentum; they are distinct terms and should not be given the same label. A reported pivoting moment must be interpreted using the study’s reference point and decomposition.

This distinction is also central to control. Contact loads are reactions compatible with the golfer’s actuation, motion and contact conditions. They are not independent commands that can be added arbitrarily while holding the rest of the swing fixed. The ground-reaction chapter develops the corresponding constrained equations and admissible counterfactuals.

Part II: Transducer Physics

Strain Gauges, Bridges and Calibration

A strain-gauge load cell turns elastic deformation into a resistance change. Gauges placed on selected parts of the structure respond to compression, bending, shear or torsion. A bridge combines these responses; a calibration matrix converts electrical channels into force and moment components. The number of raw channels and the location of this calculation are instrument-specific. Both strain-gauge and piezoelectric systems require calibration and geometric information, even when the software exports six finished channels.

A Wheatstone bridge is a pair of voltage dividers. For one stated labeling of its four resistors,

\[ V_o=V_{\rm ex} \left(\frac{R_2}{R_1+R_2}-\frac{R_4}{R_3+R_4}\right). \]

The bridge balances when the two fractions match. Full-bridge gauge arrangements can increase strain sensitivity and cancel matched common temperature effects, but the output still scales with excitation $V_{\rm ex}$. Four arms alone do not cancel excitation-voltage drift. Ratiometric readout, using the measured excitation or an ADC reference that tracks it, controls that source of gain error. Temperature gradients, imperfect matching, self-heating, creep and electronics remain relevant. NI’s bridge guidance and excitation discussion explain these distinctions.

The ability to observe shear or torsion comes from the mechanical load path and its instrumentation. AMTI’s US 4,493,220 describes strain-gauge arrangements sensitive to shear and vertical-axis moments; Bertec’s US 6,354,155 describes a multicomponent load-transducer architecture. These patents explain designs. They do not independently validate the accuracy of every product using the same transducer family.

A useful measurement model is

\[ \widehat{\mathbf w}=K\,[\mathbf s-\mathbf b], \]

where $\mathbf s$ is the raw channel vector, $\mathbf b$ the zero offset and $K$ the calibrated conversion matrix. This locally linear description may need load-, location- or temperature-dependent correction. Calibration uncertainty, channel noise and uncertainty in the frame transform all propagate into the wrench and derived COP. “Six axes” describes the measurand; it is not an uncertainty specification.

Read Specifications as Qualified Claims

For a concrete example, AMTI’s HPS400600 product page reports typical accuracy within $\pm0.1\%$ of applied load, crosstalk within $\pm0.05\%$ of applied load, and COP accuracy below $0.2$ mm, with a stated minimum applied load of 50 lb. It also lists hysteresis below $0.5\%$ of full-scale output. These are manufacturer specifications with their stated conditions, not an independent comparison against another brand or a guarantee for every swing sample.

Applied load and full scale are different denominators. A percentage of full scale becomes a larger fraction of the measured signal at small loads. Likewise, an impressive vertical-force specification does not establish equally small relative error in a near-zero shear channel. As an illustrative calculation, crosstalk equal to $0.1\%$ of a 2000 N vertical load is 2 N of apparent shear. Whether that matters depends on the actual shear and the downstream inference, rather than the sport’s name.

To compare instruments, require the model, capacity, loading direction and location, temperature range, conditioning settings, uncertainty convention and tested bandwidth. A typical error, a maximum bound, hysteresis and repeatability are not interchangeable. Different calibration procedures can produce different performance from the same transducer principle; attributing a specification difference solely to sensor type or calibration density needs evidence.

Piezoelectric Sensing and the Measurement Time Window

Quartz transducers generate charge under load. Preloaded multiaxis sensors can resolve compression and shear; a platform combines their outputs to recover its wrench. Four-sensor/eight-channel arrangements are familiar examples, not a universal requirement. Quartz’s lack of pyroelectricity does not make a complete mounted system immune to temperature effects, leakage, changing preload or electronic drift.

A simple charge-amplifier model separates two effects that are often conflated. Let the sensor charge be $q=dF$, and let the feedback capacitor and resistor be $C_f$ and $R_f$. With ideal amplifier behavior otherwise and $\tau=R_fC_f$,

\[ \frac{V(s)}{F(s)}=-\frac{d}{C_f} \frac{s\tau}{1+s\tau}. \]

This is a high-pass response from force to voltage, with low-frequency corner $1/(2\pi\tau)$. For a force step applied after zeroing, the magnitude of the response decays as $\exp(-t/\tau)$. That droop is different from an accumulating offset caused by leakage or amplifier input current. Kistler distinguishes these mechanisms in its time-constant explanation and charge-amplifier manual.

For example, a 5 s constant load produces about $4.9\%$ droop with $\tau=100$ s, but about $0.5\%$ with $\tau=1000$ s in this ideal model. Requiring droop no greater than a chosen fraction $0<\epsilon<1$ over duration $T>0$ gives

\[ \tau\ge\frac{-T}{\ln(1-\epsilon)}. \]

This is a design check, not a claim that a particular plate has either time constant. Shorter time constants may be suitable for transient measurements but distort sustained loading and impulse. Longer time constants preserve low-frequency content while making offset control important. An acquisition chain can add further filters, compensation and saturation limits.

Choose the range, time constant, zeroing procedure and permitted trial duration from the actual instrument and error budget. Zeroing while a golfer is already standing on the plate removes that loaded reference unless it is restored explicitly; it does not create an absolute body-weight measurement. Record whether a trace represents absolute load or change from a tare. For impulse, a constant baseline error $b_F$ accumulates an error $b_F T$, even if the waveform looks smooth.

Resonance, Mounting and Sampling Are Separate Limits

A high natural frequency can support a wide useful measurement band, but natural frequency is not usable bandwidth. Amplitude and phase error develop below resonance and depend on damping and the coupled plate–support–load system. For an illustrative single-mode transfer function normalized to unit static gain,

\[ H(j\omega)=\frac{1} {1-(\omega/\omega_n)^2+2j\zeta\omega/\omega_n}. \]

With zero damping, even $\omega/\omega_n=0.2$ gives a gain of about 1.042. A real system has multiple modes and damping; the acceptable band must be established from its measured response and the tolerated amplitude/phase errors. Surface coverings, fixtures, foundations and added mass can change that response.

The HPS400600 page illustrates why the precise model matters: for its 1000 lb version, listed standard natural frequencies are 310 Hz in shear and 370 Hz vertically, with 480/720 Hz for the high-frequency option. The 2000 lb version has different values. These are model-specific natural frequencies, not a validated bandwidth for golf impacts. Neither “portable” nor “piezoelectric” alone determines suitability; geometry, stiffness, mass, mounting and electronics all matter.

Sampling adds another constraint. If $f_s=2000$ Hz, a 1300 Hz sinusoid can alias into a 700 Hz sampled signal. A digital filter applied afterwards cannot reconstruct which original frequency produced it. Effective analog anti-alias filtering, or the equivalent properly designed oversampling/decimation chain, must attenuate unwanted content before it aliases. State the filter response and acquisition architecture, not only the sample rate. A nominal cutoff at Nyquist with a gentle rolloff is not a general guarantee of adequate suppression, and there is no universal 1000 Hz filter fitted to all force plates.

For ordinary swing loading and for impact transients, define the bandwidth of the question separately. Increasing the sample rate does not repair structural resonance, transducer saturation or a low-pass reference that already removed the event of interest. Synchronization matters too: a time shift between force and kinematics can corrupt inverse dynamics even when each signal is individually accurate.

Calibration and In-Situ Verification

Factory calibration and installed-system verification answer different questions. AMTI’s Optima description reports up to 4000 measurements across rated capacity and up to 400 surface positions. Its stated 0.005 mm positioning accuracy describes the calibration machine’s positioning, not the resulting force or COP uncertainty.

List et al. (2017) studied six plates and reported up to approximately 60% reduction in point-of-force-application RMS error with their in-situ correction. Remounting changed correction coefficients substantially. Their recommendation to recalibrate after remounting and at least every five years is evidence from that study, not a universal interval covering every instrument and use. Installation, load range and required uncertainty determine the appropriate local verification plan.

A Force-Platform Standard Exists

ASTM F3109-23, Standard Practice for Verification of Multi-Axis Force Measuring Platforms, addresses performance verification and reporting for multiaxis platforms, including COP and variation with load and surface position. Its public scope excludes dynamic characteristics and platforms integrated into treadmills or stairs. It therefore corrects the claim that no ASTM force-platform standard exists, while leaving important application-specific work. This description uses the published scope; it is not a compliance assessment or a substitute for the complete standard.

A verification plan should expose the failure modes relevant to the proposed inference:

Measure unloaded offset and drift over the intended trial duration, and verify the meaning of any loaded tare.
Apply independently known loads at surveyed locations, including the intended working region and near its boundaries; compare the reconstructed wrench and COP with their uncertainties.
Test cross-axis response using appropriate independent reference loads. A vertical load can reveal spurious shear; inclined loads and known couples probe other components. A pulley is one possible fixture, not the only way to detect crosstalk.
Check coordinate signs, surface height, reference-point transport, synchronization and saturation. Reassess after remounting or changing coverings.
For dynamic claims, verify the installed frequency response and filtering. An impulse comparison is useful only when the reference momentum change and all relevant external impulses are independently known.

The goal is traceability from a stated load to a stated error, rather than a generic certificate used as permission for every downstream calculation.

Part III: Pressure Systems and What Is Observable

Normal Pressure Is Valuable but Incomplete

For a flat horizontal surface with normal pressure $p(x,y)$ and no independent local bending couples,

\[ \begin{aligned} F_z&=\int_A p\,dA,\\ x_C&=\frac{\int_A xp\,dA}{F_z},\qquad y_C=\frac{\int_A yp\,dA}{F_z}. \end{aligned} \]

A sensel array approximates these integrals with calibrated area-weighted sums. It additionally resolves contact area and pressure distribution, information that a resultant force plate discards. Saturation, unsensed regions, spatial resolution, hysteresis and calibration affect the approximation. On a curved or deforming insole, local normals and geometry must also be considered before equating a sensel sum with global vertical ground force. The in-shoe interface is not automatically identical to the shoe–ground interface during motion.

Normal pressure alone does not determine tangential traction. On a planar patch,

\[ \begin{aligned} F_x&=\int_A t_x\,dA,\qquad F_y=\int_A t_y\,dA,\\ T_z&=\int_A[(x-x_C)t_y-(y-y_C)t_x]\,dA. \end{aligned} \]

One can keep $p$ fixed and change $t_x,t_y$ within the admissible friction limits. The pressure data and COP remain identical while shear and torsion change. Friction assumptions can bound possible tractions; they do not generally select a unique one. Higher sensel density improves the measured pressure field but does not add the missing tangential observations.

That is an identifiability limit, rather than a claim that pressure has no statistical relationship to shear. A learned model can exploit relationships within its training population and task. It must be described and validated as an estimator. Additional dedicated shear or torsion sensors can change the measurement capabilities; absence of a displayed horizontal-force channel alone does not logically prove that no independent moment channel exists.

Learned Wrenches Are Predictions

Joo, Oh and Mun (2016) used golf swings from 80 participants to predict force and moment components from plantar-pressure features, with five-fold validation against two force plates. Different feature combinations worked best for different outputs: accumulated pressure with opposite-foot data yielded reported correlations of $0.73$–$0.97$ for selected components; accumulated pressure with COP patterns yielded $0.83$–$0.98$ for others. This is not one universal improvement obtained by sequentially adding every feature.

Correlation measures co-variation, not agreement in magnitude: doubling every predicted force can preserve perfect correlation while producing a large load error. A useful validation therefore includes absolute errors, bias, timing and peak errors, uncertainty across people, and the train/test split. Generalization to unseen golfers and surfaces is a different claim from reproducing trials drawn from the development protocol.

For any commercial output, identify the actual sensing architecture and processing. A pressure-only estimator, a calibrated six-axis platform and a hybrid device should not inherit each other’s validation merely because their screens use the same labels.

COP Error Must Retain Its Units and Study Context

Three examples show why published errors cannot be combined into one device-class verdict:

Study	What Was Evaluated	What the Result Supports
Bincalar et al. (2025)	Simulated low-resolution mat geometry and footprint fitting	Optimization reduced the study’s coordinate-relative error metric; physical validation remained future work.
Davidson et al. (online 2025; 2026 issue)	Three-compartment insoles in 39 participants doing occupational tasks	COP RMSE of 1.7–3.4 cm for that device, estimator and protocol.
Cudejko et al. (2023)	Wireless insoles during functional activities	AP trajectories were comparable; ML trajectories were shorter except during jumps, and vertical forces were lower than plate measurements.

Bincalar’s 17.37%, 5.47% and 3.93% values describe simulations of an 8-by-8, 48 cm square mat. The metric combines errors relative to each true COP coordinate; it is origin-dependent and is neither millimetres nor a percentage of mat width. The optimized geometry and footprint-fitting results are separate alternatives, not consecutive corrections to the same optimized device. These figures should not be presented as measured errors of all inexpensive mats.

Brady and Kiernan (2020) investigated imposed anterior–posterior COP perturbations in children during walking, fast walking and running. Their gait-index threshold was exceeded for 9 and 12 mm perturbations in self-selected walking; the same conclusion did not hold for faster walking and running. Relative joint-moment error decreased with speed in their analysis. This is not a universal 9 mm clinical limit, still less a validated error budget for golf.

The mechanical sensitivity is nevertheless real. Holding force fixed, a position error changes a moment by $\delta\mathbf M=\delta\mathbf r\times\mathbf F$. A 10 mm perpendicular lever-arm error under 1000 N can therefore produce 10 N m of moment error. It could also produce zero error if the displacement is parallel to the force. This calculation identifies a sensitivity to test; it does not predict a whole inverse-dynamics error without the actual directions, time history, force errors and model.

The appropriate golf test propagates plausible measurement and synchronization errors through the intended analysis, reports the resulting uncertainty in the chosen outcome, and compares it with the effect being interpreted. A trace can be useful for pressure-distribution feedback while remaining insufficient for a particular joint-moment claim.

Do Not Confuse the Two Meanings of Mediolateral

Smith et al. (2017) related COP/centre-of-gravity timing and rate-of-change features to clubhead velocity in 22 low-handicap golfers. Those are associations in a declared golf coordinate frame. An insole’s across-foot direction and a golfer’s between-feet direction are not automatically the same global axis, especially with toe-out and changing foot orientation. A reported insole ML bias cannot simply be transferred to the study’s golf ML predictor without transforming frames and validating the relevant motion.

This is one reason the full chain matters: sensor axes, shoe orientation, foot-contact geometry, signal processing and the statistical feature all contribute to what a plotted “weight shift” means.

Part IV: What the Golf Literature Actually Shows

Several Wrench Components Carry Information

Worsfold, Smith and Dyson (2008) studied 24 golfers using two force platforms covered with natural turf. Their driver results showed handicap-related differences in trail-foot torque range across the tested shoe types. For metal spikes, the low- and high-handicap groups had values of $18.2\pm3.1$ and $14.2\pm7.4$ N m. The reported measure was the maximum-to-minimum torque range, not a single positive peak, an angular impulse or work. The lead foot did not show a significant handicap effect in that analysis.

This supports examining torsional loading in its particular shoe–surface and club context. It does not establish that torsion is the sole performance signal, that the trail foot always supplies the dominant moment, or that increasing this torque range will improve an individual golfer’s swing. Surface conditions and footwear are part of the experiment, not incidental details.

Han et al. (2019) examined 63 highly skilled male golfers in three club conditions. Their decomposition distinguished GRF moments, pivoting moments and foot-contact moments. Associations with clubhead speed included lead-foot force components, trail-foot forward/backward force, lead-foot GRF moment and trail-foot pivoting moment. These findings support analyzing the entire wrench and its lever arms. A pivoting moment in that decomposition should not be silently relabeled as the per-foot free moment defined in Part I.

Comparisons need the same reference point and frame. A large moment may result from a large force, a long perpendicular lever arm, a free couple, or their combination. The mechanically relevant quantity for angular-momentum change is the net external moment integrated over the specified interval, with the chosen system boundary and reference-point balance treated consistently. Ranking one channel’s peak is not equivalent to explaining that angular impulse.

Ground Reaction, Joint Work and Muscle Effort

For a system containing the golfer and club, before ball contact and neglecting other environmental loads,

\[ \dot{\mathbf p}=\mathbf F_{\rm ground}+m\mathbf g. \]

For the golfer alone, hand–club interaction is an additional external load; at ball impact the ball’s force is external to the golfer–club system. Thus “100% of the external wrench enters through the feet” is false. Gravity alone already disproves it. Internal muscle and joint actions influence movement and contact reactions, even though their action–reaction pairs cancel from the whole-system momentum balance.

A large GRF also does not, by itself, reveal mechanical power. In ideal stationary no-slip contact, material contact-point velocity is zero and contact does no work on the system. Meanwhile, internal joint actuation can change segment motion and mechanical energy. The product $\mathbf F_{\rm ground}\cdot\mathbf v_{\rm COM}$ enters a centre-of-mass translational-energy balance; it is not generally the actual contact power or the sum of leg-muscle powers.

Nesbit and Serrano (2005) reported modeled work allocations for four golfers. Their leg-joint category accounted for approximately 3.3–3.8%, while their back-and-hip category accounted for approximately 69–72%. The hips were counted with the core, so this is not a general statement that the anatomical lower limbs contribute only a few percent. The numbers depend on the modeled joints, coordinates and work accounting; they are not direct muscle-energy or metabolic measurements.

The justified conclusion is narrower and more useful: force, joint torque, joint power and muscle effort are different quantities. Small net joint work can coexist with substantial support forces, co-contraction, positive and negative work, or energy transfer through multiarticular tissues. To test a claim about leg contribution, specify the joints and tissues, define signed versus positive work, and examine synchronized kinematics and kinetics with model uncertainty. Neither a high force peak nor a low work percentage settles the claim alone.

Variation and the Search for an Ideal Pattern

Ball and Best (2007), Part I, clustered COP trajectories from 62 golfers into front-foot and reverse patterns. Both occurred across skill levels. Part II found different within-group associations with clubhead velocity, including opposite directions of COP movement near contact. This warns against pooling distinct movement patterns without checking the structure of the data.

It does not establish two immutable biological types. Jones, Wallace and Otto (2023) analyzed 104 amateurs and found that the preferred cluster count depended on the criterion. A continuous analysis supported a multidimensional continuum; lower handicap and higher speed tended toward front-foot patterns and faster transfer early in the downswing. That is relevant counterevidence to treating the older two-group classification as settled.

A separate Jones et al. study (2024, online 2021) found lower within-golfer than between-golfer GRF variability, alongside skill-related differences in some variability components. Individual consistency does not prove individual optimality. Conversely, a group association does not prescribe an identical waveform for every person.

A stronger performance argument states the task, outcome and comparison: for example, whether changing a specified force-timing feature improves speed at a controlled accuracy requirement within the same golfer. It then tests that intervention, allowing for learning, fatigue, club choice, surface conditions and uncertainty. Descriptive clustering can help formulate that question; it cannot answer it by itself.

Associations, Mediation and Strength Tests

Rachnavy et al. (2026) report observational regression and mediation analyses in 30 golfers. Adding sequencing and transfer measures improved their model fit. A significant indirect path does not establish causation, and a nonsignificant direct coefficient does not prove the absence of a direct effect. Force impulse, mechanical work and transfer efficiency also remain distinct quantities.

The same discipline applies to strength testing. Leary et al. (2012) studied 12 recreational golfers. Force at 150 ms during an isometric mid-thigh pull had correlations of about $0.46$–$0.47$ with average/maximum clubhead speed; those are force-at-time values. Their reported 0–150 ms rate-of-force-development correlations were about $0.38$–$0.36$, with $p=0.11$–$0.12$. These estimates do not establish either a reliable predictor or the absence of a relationship. The tested RFD is also not the slope of the swing’s ground-force trace.

Johansen et al. (2023) reported positive associations between maximal clubhead speed and several strength/power measures, including leg-press RFD, in 21 national-level male golfers. Different tasks and populations can produce different estimates. Replacing one small observational study with another does not establish the effect of a training intervention; use effect estimates, uncertainty and the actual test definition rather than a binary significant/nonsignificant slogan.

Comparing Body-Weight-Normalized Peaks

A statement such as “players reach twice body weight” needs a cited population, club, trial and definition: one foot or both, instantaneous sum or separate peaks, and total vertical force or force after subtracting body weight.

For the same trial,

\[ \max_t[F_{z,1}(t)+F_{z,2}(t)] \le\max_t F_{z,1}(t)+\max_t F_{z,2}(t). \]

Equality requires compatible peak timing. Published per-foot peaks from one cohort cannot validate or refute a universal combined-force claim about another. Group averages of individual peaks introduce another distinction. Retain the synchronized signals and normalization before comparing numbers.

Screw Theory Connects the Questions, Not Their Answers

Kim (2025) reanalyzed one proficient and one novice female golfer, relating a kinematic instantaneous screw-axis pitch to vertical-force timing. This is a two-person exploratory comparison, not a validated skill classifier or proof that one waveform is necessary for expertise. Kinematic screw estimates require a declared moving body and frame, and become sensitive when angular speed is small.

A motion screw describes velocity geometry; a wrench screw describes force/moment geometry. Their pitches and axes need not coincide. Power pairs the full, consistently referenced twist and wrench; a vertical-force trace and a kinematic pitch alone do not determine that pairing. This makes screw theory useful for asking a precise question without letting an attractive geometric analogy substitute for the missing measurements.

A 2026 systematic review by Watson et al. likewise calls for clearer methods and terminology across the golf GRF/COP literature. Useful synthesis preserves the definitions of forces, moments, timing features and outcomes before comparing studies or drawing coaching conclusions.

Part V: Estimating Force Without a Plate

Whole-Body Dynamics Estimates a Net Load

Given sufficiently accurate segment masses, inertias and kinematics, Newton–Euler balances estimate the net external wrench. To attribute that net load to the ground, subtract gravity and account for other external interactions. Missing club dynamics, uncertain segment parameters, soft-tissue motion, differentiation noise and synchronization all affect the estimate. A force plate observes a contact load through its calibrated sensing chain; a kinematic estimate obtains it through these additional model assumptions.

During double support, the net wrench does not uniquely determine two arbitrary foot wrenches. Ren, Jones and Howard (2008) introduced a smooth-transition assumption for walking to resolve that indeterminacy. It is a closure model based on gait behavior, not an additional measured equation or a universal law of bilateral contact.

Karatsidis et al. (2017) evaluated inertial-motion-based estimates in 11 healthy people walking at different speeds. Their normal-speed results had stronger agreement for vertical force than for some shear and moment components. Importantly, moments were referenced to the ground projection of the ankle, and the distribution curves were derived from the study’s measured data. These results support feasibility under the evaluated gait protocol; they do not establish independent validation in golf, solve vertical-force estimation generally or classify frontal-plane moments as universally unusable.

The appropriate accuracy threshold depends on the question. A useful gross-load estimate may be inadequate for a small change in angular impulse. Conversely, a moment estimate with quantified uncertainty can be useful for a sufficiently large effect. Report absolute as well as normalized errors, their normalization and phase dependence, and test the intended outcome rather than attaching a universal usable/unusable label to a component.

A Golf-Specific Learning Result and Its Boundaries

Li et al. (2026) used lower-limb IMU kinematics to predict three force components in 48 professional golfers, with ten driver swings per participant. Their TCN–BiGRU model reported $R^2\approx0.94$ under subject-level ten-fold validation. Keeping each golfer’s trials in one fold addresses a key source of leakage between training and evaluation.

The measurement target was nevertheless specific: indoor swings, 60 Hz IMUs and force references low-pass filtered at 6 Hz. The result does not establish accuracy for impact transients, moments, other skill groups or outdoor shoe–surface conditions. A bidirectional sequence model’s offline accuracy also does not demonstrate causal real-time prediction, because later samples may contribute to earlier estimates. These are boundaries on the supported claim, not reasons to dismiss the approach.

Any deployment claim should retain the signal-processing pipeline, model version, sensor placement, held-out population and component-wise error distribution. Changes to the target filter or trial segmentation change what is being predicted. A high coefficient of determination is not a substitute for this record.

Why the Per-Foot Partition Remains Underdetermined

At a common reference point,

\[ \mathbf w_{\rm net}=\mathbf w_1+\mathbf w_2. \]

For arbitrary six-component foot wrenches this is six equations for twelve components. Adding $\Delta\mathbf w$ to one foot and subtracting it from the other leaves the net wrench unchanged. Contact geometry and friction restrict the possible choices, but generally leave multiple admissible solutions. When only one foot is in contact, that particular partition ambiguity disappears; golf includes heel lift, rolling and sometimes loss of contact, so the contact mode should be measured or modeled rather than assumed constant throughout every swing.

Even knowing both normal loads and COPs does not generally fix the shear partition. For two coplanar contacts, transfer a small horizontal force $\mathbf a$ from foot 2 to foot 1. The net force is unchanged, while the change in moment is

\[ \Delta\mathbf M=(\mathbf r_1-\mathbf r_2)\times\mathbf a. \]

If $\mathbf a$ is parallel to the line between the COPs, this moment is also unchanged: normal pressure, both COPs and the entire net wrench can all remain identical. Other horizontal transfers can be offset by changes in the per-foot normal couples, when the friction limits permit. This provides a concrete reason that combined net-force and pressure measurements do not automatically recover each foot’s complete wrench.

Two independently calibrated six-axis platforms, with one foot on each and no load paths bypassing them, provide a direct way to resolve the per-foot wrenches. They are not the only possible instrumentation: suitably designed multiaxis wearable sensors or other independently instrumented contact regions can add the necessary observations. Each still needs calibration, frame registration and a check that the measured load belongs to the intended foot.

A pressure mat can separate the normal-pressure contributions of two spatially resolved footprints, subject to its measurement errors. It does not do so “exactly” in a physical instrument, and overlapping or unsensed contacts add difficulties. A hybrid system may combine pressure-derived locations with additional force channels. The Initial Force motion-analysis patent is an example of such architectural ideas; a patent does not prove that a particular commercial implementation uniquely measures every per-foot shear and free-moment component. Identify which additional channels or modeling assumptions resolve each unknown.

Building an Argument From the Available Measurements

Start with the proposed claim and work backward to the observations it needs:

Proposed Claim	Required Reasoning Beyond a Force Trace
The golfer–club system changes momentum	Integrate the complete net external force, with gravity, system mass and other impulses accounted for.
A foot contributes a turning moment	Declare the reference point; include force lever arms and free couples, and identify the per-foot partition.
A joint generates mechanical work	Combine model-consistent joint load with its conjugate velocity; specify sign and integration interval.
Particular muscles supplied that work	Add a muscle/tissue model and supporting observations; net joint kinetics alone is insufficient.
A technique improves performance	Test an intervention with defined performance criteria, uncertainty and appropriate within-person and between-person comparisons.

These questions are intertwined because one measurement feeds the next calculation. Keeping their boundaries explicit strengthens the argument: it identifies which conclusions are already supported, which depend on a model, and which experiment would resolve the remaining uncertainty.

References and Further Reading

Textbooks and Standards

ASTM International (2023). Standard Practice for Verification of Multi-Axis Force Measuring Platforms, F3109-23. Public scope.
Winter, D. A. (2009). Biomechanics and Motor Control of Human Movement, 4th ed. Wiley — body segment parameters, link-segment inverse dynamics, residual analysis for filter selection.
Zatsiorsky, V. M. (2002). Kinetics of Human Motion. Human Kinetics — forces, moments and human-motion kinetics.
Robertson, D. G. E., Caldwell, G. E., Hamill, J., Kamen, G. and Whittlesey, S. (2014). Research Methods in Biomechanics, 2nd ed. Human Kinetics — biomechanical measurement and analysis methods.
de Leva, P. (1996). Adjustments to Zatsiorsky-Seluyanov’s segment inertia parameters. Journal of Biomechanics 29(9), 1223–1230. PubMed.
Wu, G. and Cavanagh, P. R. (1995). ISB recommendations for standardization in the reporting of kinematic data. Journal of Biomechanics 28(10), 1257–1261. PubMed; joint coordinate system definitions Part I (2002) and Part II (2005).

Instrument Physics and Patents

AMTI. Force measuring platform and load cell therefor using strain gages to measure shear forces. US 4,493,220.
Berme, N. (Bertec). Multi-component force and moment measuring platform and load transducer. US 6,354,155; Force measurement system having a plurality of measurement surfaces, US 8,544,347.
Kistler. Force transducers for fitting in force plates. US 4,974,454.
Tekscan. Flexible tactile sensor for measuring foot pressure distributions and for gaskets. US 5,033,291.
Seitz, P. Capacitive measuring assembly for determining forces and pressures. US 4,836,033.
Initial Force AS (Swing Catalyst). Motion Analysis Apparatus. US 2011/0260890.
U.S. Army. Force sensing treadmill. US 6,878,100 — an example of separately instrumented support surfaces.

Measurement Accuracy and Calibration

List, R. et al. (2017). In-situ force plate calibration: 12 years’ experience with an approach for correcting the point of force application. Gait & Posture. PubMed.
Chockalingam, N. et al. (2002). Do strain gauge force platforms need in situ correction? Gait & Posture. PubMed.
Middleton, J., Sinclair, P. and Patton, R. (1999). Accuracy of centre of pressure measurement using a piezoelectric force platform. Clinical Biomechanics. PubMed.
Giacomozzi, C. (2010). Appropriateness of plantar pressure measurement devices: a comparative technical assessment. Gait & Posture. PubMed.
Joo, S. B., Oh, S. E. and Mun, J. H. (2016). Improving the ground reaction force prediction accuracy using one-axis plantar pressure: Expansion of input variable for neural network. Journal of Biomechanics 49(14), 3153–3161. PubMed.

Golf Ground Reaction Force Literature

Watson, A. et al. (2026). Ground reaction force and centre of pressure during the golf swing and associations with clubhead speed and skill level: a systematic review. Sports Medicine 56(5), 1191–1212. PubMed.
Worsfold, P., Smith, N. A. and Dyson, R. J. (2008). Low handicap golfers generate more torque at the shoe–natural grass interface when using a driver. Journal of Sports Science and Medicine 7(3), 408–414. Full text.
Han, K. H. et al. (2019). Effects of the golfer–ground interaction on clubhead speed in skilled male golfers. Sports Biomechanics 18(2), 115–134. PubMed.
Ball, K. A. and Best, R. J. (2007). Different centre of pressure patterns within the golf stroke, Part I and Part II. Journal of Sports Sciences 25(7).
Jones, K. M., Wallace, E. S. and Otto, S. R. (2023). Centre of pressure golf swing movement strategies are better defined using a continuous approach than by segregated styles. Journal of Sports Sciences 41(4), 342–349. PubMed.
McNitt-Gray, J. L. et al. (2013). Regulation of reaction forces during the golf swing. Sports Biomechanics 12(2), 121–131. PubMed.
Peterson, T. J., Wilcox, R. R. and McNitt-Gray, J. L. (2016). Angular impulse and balance regulation during the golf swing. Journal of Applied Biomechanics 32(4), 342–349. PubMed.
Nesbit, S. M. and Serrano, M. (2005). Work and power analysis of the golf swing. Journal of Sports Science and Medicine 4(4), 520–533. PMC.
Wells, J. E. T. et al. (2018). Relationships between highly skilled golfers’ clubhead velocity and force producing capabilities during vertical jumps and an isometric mid-thigh pull. Journal of Sports Sciences 36(16), 1847–1851. PubMed.
Wells, J. E. T. et al. (2019). Relationships between Challenge Tour golfers’ clubhead velocity and force producing capabilities during a countermovement jump and isometric mid-thigh pull. Journal of Sports Sciences 37(12), 1381–1386. PubMed.
Kim, W. (2025). Pitch invariance reveals skill-specific coordination in human movement: a screw-theoretic reanalysis of golf swing dynamics. Journal of Functional Morphology and Kinesiology 10(3), 315. PubMed.

Related Concepts

Twists and screw axes: the kinematic dual of the wrench measured here. (See: Motion Capture Technology)
Inverse dynamics limitations: why joint torques computed from these forces are not muscle forces. (See: Interpretation of Inverse Dynamics)
Ground reaction forces in the swing: (See: The Physics of Golf, Chapter 15)

Citation

BibTeX citation: Download .bib

@online{olson2026,
  author = {Olson, Dieter},
  title = {Force {Measurement} {Technology:} {Plates,} {Pressure,} and
    the {Ground} {Reaction} {Wrench}},
  date = {2026-08-02},
  url = {https://affinedrift.com/articles/technology-force-measurement.html},
  langid = {en}
}

For attribution, please cite this work as:

Olson, Dieter. 2026. “Force Measurement Technology: Plates, Pressure, and the Ground Reaction Wrench.” August 2. https://affinedrift.com/articles/technology-force-measurement.html.

--- title: "Force Measurement Technology: Plates, Pressure, and the Ground Reaction Wrench" description: "How force plates work: transducer physics, the six-component ground reaction wrench, centre-of-pressure geometry, and pressure mat limitations." author: "Dieter Olson" date: "2026-08-02" date-source: "initial-publication-record" css: ../css/force-measurement.css categories: - technology --- ::: {.laymans-terms} ```{=html} <p class="laymans-terms-intro">A multiaxis force plate measures the load between golfer and ground: upward support, sideways push and turning effects. A pressure-only mat shows how normal pressure is spread out. They answer different questions. Neither measures muscle effort or shows which change will improve a swing.</p> <div class="laymans-item"><h3>What Is Under the Floor</h3><p>Load-sensitive parts turn load into electrical signals. Strain gauges sense resistance changes; piezoelectric sensors generate charge. Calibration turns signals into forces and moments.</p></div> <div class="laymans-item"><h3>What the Centre of Pressure Summarizes</h3><p>The <a href="../pages/glossary.html#center-of-pressure">centre of pressure</a> is a summary point on the contact surface. It is not where your body mass sits. The full load generally includes a force there plus a leftover twisting couple. Different pressure patterns can give the same dot. A small normal load makes the dot error-prone.</p><div class="analogy"><strong>Seesaw:</strong> A balance point cannot say whether two children sit near the middle or one heavy child sits at the end. Like it, the centre of pressure drops information.</div></div> <div class="laymans-item"><h3>Why Golf Cares About Turning Effects</h3><p>Opposite sideways forces at separated feet can turn the body even when they cancel as a net force. Each footprint can also carry its own twisting couple. These are separate. Measuring them helps describe how <a href="../pages/glossary.html#angular-momentum">angular momentum</a> changes. Explaining muscle work or better technique needs more evidence.</p></div> <div class="key-takeaway"><strong>Key Takeaway:</strong> A six-axis plate reports three forces and three moments in a stated frame. A pressure-only mat cannot uniquely find sideways friction or twist. A model may predict them, but that needs validation.</div> ``` ::: ## Why This Article Exists A force plate connects a movement to the loads exchanged with its surroundings. It is especially valuable because a calibrated instrument observes contact loads independently of the kinematic model used to estimate joint moments. It is not the only way to measure an external load: instrumented handles, clubs and wearable force transducers provide other examples. Nor does a plate directly measure muscle force, joint work or the cause of a movement strategy. The useful chain of reasoning is **sensor response → calibrated contact wrench → momentum balance → model-dependent joint kinetics → a testable performance hypothesis**. Each arrow introduces assumptions. A precise instrument does not rescue a wrong coordinate convention; an accurate contact wrench does not uniquely identify individual muscles; an association with clubhead speed does not establish an effective coaching intervention. This article follows that chain through transducer physics, wrench geometry, uncertainty, pressure sensing and the golf literature. The aim is to identify which questions an instrument can answer, how to check the answer, and what additional evidence is needed for the next inference. ## Part I: The Measurand Is a Wrench ### Six Components in a Declared Frame A distributed contact load is mechanically equivalent, for the resultant force and moment on a rigid body, to a force and a couple: $$ \mathcal W_O=(\mathbf F,\mathbf M_O). $$ Here the components are expressed in one right-handed Cartesian frame and the moment is about a declared point $O$. We use the **ground-on-golfer** sign convention, with $z$ upward. A device may instead export the opposite action or different axis directions; transform its calibrated outputs before using the equations below. A six-component output also says nothing by itself about accuracy, bandwidth or the loads that bypass the sensing structure. Moving the reference point to $P$, while retaining the same axis orientation, gives $$ \mathbf M_P=\mathbf M_O-\mathbf r_{OP}\times\mathbf F. $$ The force is unchanged by this translation. Rotating the coordinate frame transforms the components of both vectors. A surface-height correction is therefore a wrench transport, not an optional plotting adjustment. For a rigid body with angular velocity $\boldsymbol\omega$, the paired twist uses the velocity of the same reference point: $$ \begin{aligned} P&=\mathbf F\cdot\mathbf v_O+ \mathbf M_O\cdot\boldsymbol\omega,\\ \mathbf v_P&=\mathbf v_O+ \boldsymbol\omega\times\mathbf r_{OP}. \end{aligned} $$ Consistent transport leaves $P$ unchanged. Motion capture observes marker positions or images; fitting segment poses, differentiating and filtering are needed to estimate these velocities. A COP trace is a changing summary of contact loading, so its time derivative is generally **not** the material-point velocity to use in this power equation. For deforming contact, the more general statement integrates local traction dotted with local material velocity over the contact area. ### The Central Axis and Its Exceptional Cases When $\mathbf F\ne\mathbf0$, a wrench has a central axis on which the residual moment is parallel to its force: $$ \begin{aligned} h&=\frac{\mathbf F\cdot\mathbf M_O}{\|\mathbf F\|^2},\\ \mathbf r_{OA}(\alpha)&= \frac{\mathbf F\times\mathbf M_O}{\|\mathbf F\|^2} +\alpha\mathbf F. \end{aligned} $$ The first term is the closest point on that axis to $O$; $\alpha$ parametrizes the line. At every point on it the remaining moment is $h\mathbf F$. The pitch $h$ has units of length. A pure couple ($\mathbf F=0$, $\mathbf M_O\ne0$) is already independent of origin and has no finite central axis defined by this formula. The zero wrench has no unique axis either. The central axis should not be confused with the centre of pressure. The former aligns the residual moment with **the force**; the latter, for planar contact, aligns it with **the surface normal**. Those directions differ when shear is present. ### Centre of Pressure, Surface Height and the Free Moment Let the contact surface be the plane $z=z_0$ relative to $O$. For $F_z\ne0$, define $C=(x_C,y_C,z_0)$ so that the transported moment has no $x$ or $y$ component. Expanding $\mathbf M_C=\mathbf M_O-\mathbf r_{OC}\times\mathbf F$ yields $$ \begin{aligned} x_C&=\frac{z_0F_x-M_y}{F_z},\\ y_C&=\frac{M_x+z_0F_y}{F_z},\\ T_z&=M_z-x_CF_y+y_CF_x. \end{aligned} $$ Thus the equivalent contact description is **a force at the COP plus a free normal couple** $T_z\mathbf e_z$. A single force at the COP is insufficient unless this residual couple vanishes. The $z_0$ signs follow from the stated axes and moment convention; copying a manufacturer's formula without its convention can reverse them. For a flat, unilateral pressure distribution without additional local bending couples, the COP is the centroid of normal pressure and lies within the convex hull of the loaded area. It does not locate the centre of mass or describe the pressure pattern. Two very different heel/forefoot distributions can have the same resultant and COP. Nonplanar contact requires a corresponding surface model rather than blind use of planar formulas. ::: {.callout-note} ## A Surface-Offset Check Take $\mathbf F=(100,-50,800)$ N, $C=(0.10,-0.05,0.04)$ m and $T_z=8$ N m. The moment at the sensor origin is $$ \mathbf M_O=C\times\mathbf F+8\mathbf e_z =(-38,-76,8)\ \mathrm{N\,m}. $$ The equations recover the original COP and free couple. Omitting the height produces $(0.095,-0.0475)$ m: errors of $-5$ mm and $+2.5$ mm. Reversing both height signs doubles these errors. This constructed load is a useful sign check; it is not a measurement of a golfer. ::: For this same planar wrench, the central axis intersects the surface at $$ A=C+\frac{T_z}{\|\mathbf F\|^2}(F_y,-F_x,0). $$ It therefore misses the COP when $T_z\ne0$ and shear is nonzero. In the example, the horizontal difference is approximately $(-0.61,-1.23)$ mm. The distinction is geometric, not sensor noise. With a fixed surface and consistent wrench transport, the computed residual free couple is unchanged by moving the reporting origin; the raw component $M_z$ generally is not. ### Why COP Uncertainty Grows During Unloading A denominator near zero makes COP ill-conditioned, but “relative COP error scales as $1/F_z$” is not a general uncertainty law. The numerator, denominator, surface height and their correlations all matter; relative error is also ill-defined when the chosen COP coordinate is zero. For small perturbations about a nonzero load, first-order propagation gives $$ \begin{aligned} \delta x_C&\simeq \frac{z_0\delta F_x-\delta M_y+F_x\delta z_0-x_C\delta F_z}{F_z},\\ \delta y_C&\simeq \frac{\delta M_x+z_0\delta F_y+F_y\delta z_0-y_C\delta F_z}{F_z}. \end{aligned} $$ Writing these derivatives as a Jacobian $J_C$ for the measured wrench and height, the covariance estimate is $\Sigma_C\simeq J_C\Sigma J_C^\mathsf T$. Retain cross-channel covariance when calibration or electronics couple the errors. The linear approximation becomes unreliable when the uncertainty in $F_z$ is comparable to $|F_z|$; reporting a confident COP ellipse then is misleading. A height bias alone gives $\delta x_C=(F_x/F_z)\delta z_0$ and $\delta y_C=(F_y/F_z)\delta z_0$. It depends on the **shear-to-normal ratio**, not just the size of shear. Moment offsets can dominate during unloading, while gain, alignment, dynamic response or pressure saturation can dominate elsewhere. There is no universal noise source or universal minimum-load threshold. Report the chosen load gate and how it was established from the instrument's uncertainty and the intended analysis. Mask or flag unsupported COP samples rather than interpreting an unloading trail-foot excursion as a deliberate control action. Keep force and moment data with the COP: the dot alone hides why the estimate changed. ### Per-Foot Torsion Is Not the Whole Turning Moment For two feet, moments about a common point $O$ add as $$ \mathbf M_O=\sum_{i=1}^{2} \left(\mathbf r_{OC_i}\times\mathbf F_i+T_{z,i}\mathbf e_z\right). $$ Opposite horizontal forces at separated feet can produce a vertical turning moment with zero net horizontal force **even if both per-foot free moments are zero**. Conversely, torsional traction within either footprint can produce a free moment. Both effects can change angular momentum; they are distinct terms and should not be given the same label. A reported pivoting moment must be interpreted using the study's reference point and decomposition. This distinction is also central to control. Contact loads are reactions compatible with the golfer's actuation, motion and contact conditions. They are not independent commands that can be added arbitrarily while holding the rest of the swing fixed. The [ground-reaction chapter](The_Physics_of_Golf/quarto/ch15_ground_reaction_forces.html) develops the corresponding constrained equations and admissible counterfactuals. ## Part II: Transducer Physics ### Strain Gauges, Bridges and Calibration A strain-gauge load cell turns elastic deformation into a resistance change. Gauges placed on selected parts of the structure respond to compression, bending, shear or torsion. A bridge combines these responses; a calibration matrix converts electrical channels into force and moment components. The number of raw channels and the location of this calculation are instrument-specific. Both strain-gauge and piezoelectric systems require calibration and geometric information, even when the software exports six finished channels. A Wheatstone bridge is a pair of voltage dividers. For one stated labeling of its four resistors, $$ V_o=V_{\rm ex} \left(\frac{R_2}{R_1+R_2}-\frac{R_4}{R_3+R_4}\right). $$ The bridge balances when the two fractions match. Full-bridge gauge arrangements can increase strain sensitivity and cancel matched common temperature effects, but the output still scales with excitation $V_{\rm ex}$. Four arms alone do **not** cancel excitation-voltage drift. Ratiometric readout, using the measured excitation or an ADC reference that tracks it, controls that source of gain error. Temperature gradients, imperfect matching, self-heating, creep and electronics remain relevant. [NI's bridge guidance](https://www.ni.com/getting-started/set-up-hardware/data-acquisition/i/strain-gages) and [excitation discussion](https://www.ni.com/en/support/documentation/supplemental/21/making-accurate-strain-measurements-improving-snr-and-determining-optimal-excitation-levels.html) explain these distinctions. The ability to observe shear or torsion comes from the mechanical load path and its instrumentation. [AMTI's US 4,493,220](https://patents.google.com/patent/US4493220A/en) describes strain-gauge arrangements sensitive to shear and vertical-axis moments; [Bertec's US 6,354,155](https://patents.google.com/patent/US6354155B1/en) describes a multicomponent load-transducer architecture. These patents explain designs. They do not independently validate the accuracy of every product using the same transducer family. A useful measurement model is $$ \widehat{\mathbf w}=K\,[\mathbf s-\mathbf b], $$ where $\mathbf s$ is the raw channel vector, $\mathbf b$ the zero offset and $K$ the calibrated conversion matrix. This locally linear description may need load-, location- or temperature-dependent correction. Calibration uncertainty, channel noise and uncertainty in the frame transform all propagate into the wrench and derived COP. “Six axes” describes the measurand; it is not an uncertainty specification. ### Read Specifications as Qualified Claims For a concrete example, [AMTI's HPS400600 product page](https://www.amti.biz/product/hps400600/) reports typical accuracy within $\pm0.1\%$ of applied load, crosstalk within $\pm0.05\%$ of applied load, and COP accuracy below $0.2$ mm, with a stated minimum applied load of 50 lb. It also lists hysteresis below $0.5\%$ of full-scale output. These are manufacturer specifications with their stated conditions, not an independent comparison against another brand or a guarantee for every swing sample. **Applied load and full scale are different denominators.** A percentage of full scale becomes a larger fraction of the measured signal at small loads. Likewise, an impressive vertical-force specification does not establish equally small relative error in a near-zero shear channel. As an illustrative calculation, crosstalk equal to $0.1\%$ of a 2000 N vertical load is 2 N of apparent shear. Whether that matters depends on the actual shear and the downstream inference, rather than the sport's name. To compare instruments, require the model, capacity, loading direction and location, temperature range, conditioning settings, uncertainty convention and tested bandwidth. A typical error, a maximum bound, hysteresis and repeatability are not interchangeable. Different calibration procedures can produce different performance from the same transducer principle; attributing a specification difference solely to sensor type or calibration density needs evidence. ### Piezoelectric Sensing and the Measurement Time Window Quartz transducers generate charge under load. Preloaded multiaxis sensors can resolve compression and shear; a platform combines their outputs to recover its wrench. Four-sensor/eight-channel arrangements are familiar examples, not a universal requirement. Quartz's lack of pyroelectricity does not make a complete mounted system immune to temperature effects, leakage, changing preload or electronic drift. A simple charge-amplifier model separates two effects that are often conflated. Let the sensor charge be $q=dF$, and let the feedback capacitor and resistor be $C_f$ and $R_f$. With ideal amplifier behavior otherwise and $\tau=R_fC_f$, $$ \frac{V(s)}{F(s)}=-\frac{d}{C_f} \frac{s\tau}{1+s\tau}. $$ This is a **high-pass response** from force to voltage, with low-frequency corner $1/(2\pi\tau)$. For a force step applied after zeroing, the magnitude of the response decays as $\exp(-t/\tau)$. That droop is different from an accumulating offset caused by leakage or amplifier input current. Kistler distinguishes these mechanisms in its [time-constant explanation](https://www.kistler.com/US/en/time-constant-resistor/C00000161) and [charge-amplifier manual](https://assets.kistler.com.cn/SAPCommerce_Download_original/012-067e.pdf). For example, a 5 s constant load produces about $4.9\%$ droop with $\tau=100$ s, but about $0.5\%$ with $\tau=1000$ s in this ideal model. Requiring droop no greater than a chosen fraction $0<\epsilon<1$ over duration $T>0$ gives $$ \tau\ge\frac{-T}{\ln(1-\epsilon)}. $$ This is a design check, not a claim that a particular plate has either time constant. Shorter time constants may be suitable for transient measurements but distort sustained loading and impulse. Longer time constants preserve low-frequency content while making offset control important. An acquisition chain can add further filters, compensation and saturation limits. Choose the range, time constant, zeroing procedure and permitted trial duration from the actual instrument and error budget. Zeroing while a golfer is already standing on the plate removes that loaded reference unless it is restored explicitly; it does not create an absolute body-weight measurement. Record whether a trace represents absolute load or change from a tare. For impulse, a constant baseline error $b_F$ accumulates an error $b_F T$, even if the waveform looks smooth. ### Resonance, Mounting and Sampling Are Separate Limits A high natural frequency can support a wide useful measurement band, but **natural frequency is not usable bandwidth**. Amplitude and phase error develop below resonance and depend on damping and the coupled plate–support–load system. For an illustrative single-mode transfer function normalized to unit static gain, $$ H(j\omega)=\frac{1} {1-(\omega/\omega_n)^2+2j\zeta\omega/\omega_n}. $$ With zero damping, even $\omega/\omega_n=0.2$ gives a gain of about 1.042. A real system has multiple modes and damping; the acceptable band must be established from its measured response and the tolerated amplitude/phase errors. Surface coverings, fixtures, foundations and added mass can change that response. The HPS400600 page illustrates why the precise model matters: for its 1000 lb version, listed standard natural frequencies are 310 Hz in shear and 370 Hz vertically, with 480/720 Hz for the high-frequency option. The 2000 lb version has different values. These are model-specific natural frequencies, not a validated bandwidth for golf impacts. Neither “portable” nor “piezoelectric” alone determines suitability; geometry, stiffness, mass, mounting and electronics all matter. Sampling adds another constraint. If $f_s=2000$ Hz, a 1300 Hz sinusoid can alias into a 700 Hz sampled signal. A digital filter applied afterwards cannot reconstruct which original frequency produced it. Effective analog anti-alias filtering, or the equivalent properly designed oversampling/decimation chain, must attenuate unwanted content **before** it aliases. State the filter response and acquisition architecture, not only the sample rate. A nominal cutoff at Nyquist with a gentle rolloff is not a general guarantee of adequate suppression, and there is no universal 1000 Hz filter fitted to all force plates. For ordinary swing loading and for impact transients, define the bandwidth of the question separately. Increasing the sample rate does not repair structural resonance, transducer saturation or a low-pass reference that already removed the event of interest. Synchronization matters too: a time shift between force and kinematics can corrupt inverse dynamics even when each signal is individually accurate. ### Calibration and In-Situ Verification Factory calibration and installed-system verification answer different questions. [AMTI's Optima description](https://www.amti.biz/optima-technology/) reports up to 4000 measurements across rated capacity and up to 400 surface positions. Its stated 0.005 mm positioning accuracy describes the calibration machine's positioning, not the resulting force or COP uncertainty. [List et al. (2017)](https://pubmed.ncbi.nlm.nih.gov/28763716/) studied six plates and reported up to approximately 60% reduction in point-of-force-application RMS error with their in-situ correction. Remounting changed correction coefficients substantially. Their recommendation to recalibrate after remounting and at least every five years is evidence from that study, not a universal interval covering every instrument and use. Installation, load range and required uncertainty determine the appropriate local verification plan. ::: {.callout-note} ## A Force-Platform Standard Exists [ASTM F3109-23, *Standard Practice for Verification of Multi-Axis Force Measuring Platforms*](https://store.astm.org/f3109-23.html), addresses performance verification and reporting for multiaxis platforms, including COP and variation with load and surface position. Its public scope excludes dynamic characteristics and platforms integrated into treadmills or stairs. It therefore corrects the claim that no ASTM force-platform standard exists, while leaving important application-specific work. This description uses the published scope; it is not a compliance assessment or a substitute for the complete standard. ::: A verification plan should expose the failure modes relevant to the proposed inference: - Measure unloaded offset and drift over the intended trial duration, and verify the meaning of any loaded tare. - Apply independently known loads at surveyed locations, including the intended working region and near its boundaries; compare the reconstructed wrench and COP with their uncertainties. - Test cross-axis response using appropriate independent reference loads. A vertical load can reveal spurious shear; inclined loads and known couples probe other components. A pulley is one possible fixture, not the only way to detect crosstalk. - Check coordinate signs, surface height, reference-point transport, synchronization and saturation. Reassess after remounting or changing coverings. - For dynamic claims, verify the installed frequency response and filtering. An impulse comparison is useful only when the reference momentum change and all relevant external impulses are independently known. The goal is traceability from a stated load to a stated error, rather than a generic certificate used as permission for every downstream calculation. ## Part III: Pressure Systems and What Is Observable ### Normal Pressure Is Valuable but Incomplete For a flat horizontal surface with normal pressure $p(x,y)$ and no independent local bending couples, $$ \begin{aligned} F_z&=\int_A p\,dA,\\ x_C&=\frac{\int_A xp\,dA}{F_z},\qquad y_C=\frac{\int_A yp\,dA}{F_z}. \end{aligned} $$ A sensel array approximates these integrals with calibrated area-weighted sums. It additionally resolves contact area and pressure distribution, information that a resultant force plate discards. Saturation, unsensed regions, spatial resolution, hysteresis and calibration affect the approximation. On a curved or deforming insole, local normals and geometry must also be considered before equating a sensel sum with global vertical ground force. The in-shoe interface is not automatically identical to the shoe–ground interface during motion. Normal pressure alone does not determine tangential traction. On a planar patch, $$ \begin{aligned} F_x&=\int_A t_x\,dA,\qquad F_y=\int_A t_y\,dA,\\ T_z&=\int_A[(x-x_C)t_y-(y-y_C)t_x]\,dA. \end{aligned} $$ One can keep $p$ fixed and change $t_x,t_y$ within the admissible friction limits. The pressure data and COP remain identical while shear and torsion change. Friction assumptions can **bound** possible tractions; they do not generally select a unique one. Higher sensel density improves the measured pressure field but does not add the missing tangential observations. That is an identifiability limit, rather than a claim that pressure has no statistical relationship to shear. A learned model can exploit relationships within its training population and task. It must be described and validated as an estimator. Additional dedicated shear or torsion sensors can change the measurement capabilities; absence of a displayed horizontal-force channel alone does not logically prove that no independent moment channel exists. ### Learned Wrenches Are Predictions [Joo, Oh and Mun (2016)](https://pubmed.ncbi.nlm.nih.gov/27515436/) used golf swings from 80 participants to predict force and moment components from plantar-pressure features, with five-fold validation against two force plates. Different feature combinations worked best for different outputs: accumulated pressure with opposite-foot data yielded reported correlations of $0.73$–$0.97$ for selected components; accumulated pressure with COP patterns yielded $0.83$–$0.98$ for others. This is not one universal improvement obtained by sequentially adding every feature. Correlation measures co-variation, not agreement in magnitude: doubling every predicted force can preserve perfect correlation while producing a large load error. A useful validation therefore includes absolute errors, bias, timing and peak errors, uncertainty across people, and the train/test split. Generalization to unseen golfers and surfaces is a different claim from reproducing trials drawn from the development protocol. For any commercial output, identify the actual sensing architecture and processing. A pressure-only estimator, a calibrated six-axis platform and a hybrid device should not inherit each other's validation merely because their screens use the same labels. ### COP Error Must Retain Its Units and Study Context Three examples show why published errors cannot be combined into one device-class verdict: | Study | What Was Evaluated | What the Result Supports | |---|---|---| | [Bincalar et al. (2025)](https://doi.org/10.3390/s25051283) | Simulated low-resolution mat geometry and footprint fitting | Optimization reduced the study's coordinate-relative error metric; physical validation remained future work. | | [Davidson et al. (online 2025; 2026 issue)](https://pubmed.ncbi.nlm.nih.gov/40704489/) | Three-compartment insoles in 39 participants doing occupational tasks | COP RMSE of 1.7–3.4 cm for that device, estimator and protocol. | | [Cudejko et al. (2023)](https://pubmed.ncbi.nlm.nih.gov/37696840/) | Wireless insoles during functional activities | AP trajectories were comparable; ML trajectories were shorter except during jumps, and vertical forces were lower than plate measurements. | Bincalar's 17.37%, 5.47% and 3.93% values describe simulations of an 8-by-8, 48 cm square mat. The metric combines errors relative to each true COP coordinate; it is origin-dependent and is neither millimetres nor a percentage of mat width. The optimized geometry and footprint-fitting results are separate alternatives, not consecutive corrections to the same optimized device. These figures should not be presented as measured errors of all inexpensive mats. [Brady and Kiernan (2020)](https://pubmed.ncbi.nlm.nih.gov/32911097/) investigated imposed anterior–posterior COP perturbations in children during walking, fast walking and running. Their gait-index threshold was exceeded for 9 and 12 mm perturbations in self-selected walking; the same conclusion did not hold for faster walking and running. Relative joint-moment error decreased with speed in their analysis. This is not a universal 9 mm clinical limit, still less a validated error budget for golf. The mechanical sensitivity is nevertheless real. Holding force fixed, a position error changes a moment by $\delta\mathbf M=\delta\mathbf r\times\mathbf F$. A 10 mm perpendicular lever-arm error under 1000 N can therefore produce 10 N m of moment error. It could also produce zero error if the displacement is parallel to the force. This calculation identifies a sensitivity to test; it does not predict a whole inverse-dynamics error without the actual directions, time history, force errors and model. The appropriate golf test propagates plausible measurement and synchronization errors through the intended analysis, reports the resulting uncertainty in the chosen outcome, and compares it with the effect being interpreted. A trace can be useful for pressure-distribution feedback while remaining insufficient for a particular joint-moment claim. ### Do Not Confuse the Two Meanings of Mediolateral [Smith et al. (2017)](https://pubmed.ncbi.nlm.nih.gov/27737623/) related COP/centre-of-gravity timing and rate-of-change features to clubhead velocity in 22 low-handicap golfers. Those are associations in a declared golf coordinate frame. An insole's across-foot direction and a golfer's between-feet direction are not automatically the same global axis, especially with toe-out and changing foot orientation. A reported insole ML bias cannot simply be transferred to the study's golf ML predictor without transforming frames and validating the relevant motion. This is one reason the full chain matters: sensor axes, shoe orientation, foot-contact geometry, signal processing and the statistical feature all contribute to what a plotted “weight shift” means. ## Part IV: What the Golf Literature Actually Shows ### Several Wrench Components Carry Information [Worsfold, Smith and Dyson (2008)](https://www.jssm.org/volume07/iss3/cap/jssm-07-408.pdf) studied 24 golfers using two force platforms covered with natural turf. Their driver results showed handicap-related differences in trail-foot torque range across the tested shoe types. For metal spikes, the low- and high-handicap groups had values of $18.2\pm3.1$ and $14.2\pm7.4$ N m. The reported measure was the **maximum-to-minimum torque range**, not a single positive peak, an angular impulse or work. The lead foot did not show a significant handicap effect in that analysis. This supports examining torsional loading in its particular shoe–surface and club context. It does not establish that torsion is the sole performance signal, that the trail foot always supplies the dominant moment, or that increasing this torque range will improve an individual golfer's swing. Surface conditions and footwear are part of the experiment, not incidental details. [Han et al. (2019)](https://pubmed.ncbi.nlm.nih.gov/31042142/) examined 63 highly skilled male golfers in three club conditions. Their decomposition distinguished GRF moments, pivoting moments and foot-contact moments. Associations with clubhead speed included lead-foot force components, trail-foot forward/backward force, lead-foot GRF moment and trail-foot pivoting moment. These findings support analyzing the entire wrench and its lever arms. A pivoting moment in that decomposition should not be silently relabeled as the per-foot free moment defined in Part I. Comparisons need the same reference point and frame. A large moment may result from a large force, a long perpendicular lever arm, a free couple, or their combination. The mechanically relevant quantity for angular-momentum change is the **net external moment integrated over the specified interval**, with the chosen system boundary and reference-point balance treated consistently. Ranking one channel's peak is not equivalent to explaining that angular impulse. ### Ground Reaction, Joint Work and Muscle Effort For a system containing the golfer and club, before ball contact and neglecting other environmental loads, $$ \dot{\mathbf p}=\mathbf F_{\rm ground}+m\mathbf g. $$ For the golfer alone, hand–club interaction is an additional external load; at ball impact the ball's force is external to the golfer–club system. Thus “100% of the external wrench enters through the feet” is false. Gravity alone already disproves it. Internal muscle and joint actions influence movement and contact reactions, even though their action–reaction pairs cancel from the whole-system momentum balance. A large GRF also does not, by itself, reveal mechanical power. In ideal stationary no-slip contact, material contact-point velocity is zero and contact does no work on the system. Meanwhile, internal joint actuation can change segment motion and mechanical energy. The product $\mathbf F_{\rm ground}\cdot\mathbf v_{\rm COM}$ enters a centre-of-mass translational-energy balance; it is not generally the actual contact power or the sum of leg-muscle powers. [Nesbit and Serrano (2005)](https://pmc.ncbi.nlm.nih.gov/articles/PMC3899668/) reported modeled work allocations for four golfers. Their leg-joint category accounted for approximately 3.3–3.8%, while their back-and-hip category accounted for approximately 69–72%. **The hips were counted with the core**, so this is not a general statement that the anatomical lower limbs contribute only a few percent. The numbers depend on the modeled joints, coordinates and work accounting; they are not direct muscle-energy or metabolic measurements. The justified conclusion is narrower and more useful: force, joint torque, joint power and muscle effort are different quantities. Small net joint work can coexist with substantial support forces, co-contraction, positive and negative work, or energy transfer through multiarticular tissues. To test a claim about leg contribution, specify the joints and tissues, define signed versus positive work, and examine synchronized kinematics and kinetics with model uncertainty. Neither a high force peak nor a low work percentage settles the claim alone. ### Variation and the Search for an Ideal Pattern [Ball and Best (2007), Part I](https://pubmed.ncbi.nlm.nih.gov/17454544/), clustered COP trajectories from 62 golfers into front-foot and reverse patterns. Both occurred across skill levels. [Part II](https://pubmed.ncbi.nlm.nih.gov/17454545/) found different within-group associations with clubhead velocity, including opposite directions of COP movement near contact. This warns against pooling distinct movement patterns without checking the structure of the data. It does not establish two immutable biological types. [Jones, Wallace and Otto (2023)](https://pubmed.ncbi.nlm.nih.gov/37149899/) analyzed 104 amateurs and found that the preferred cluster count depended on the criterion. A continuous analysis supported a multidimensional continuum; lower handicap and higher speed tended toward front-foot patterns and faster transfer early in the downswing. That is relevant counterevidence to treating the older two-group classification as settled. A [separate Jones et al. study (2024, online 2021)](https://pubmed.ncbi.nlm.nih.gov/34455916/) found lower within-golfer than between-golfer GRF variability, alongside skill-related differences in some variability components. Individual consistency does not prove individual optimality. Conversely, a group association does not prescribe an identical waveform for every person. A stronger performance argument states the task, outcome and comparison: for example, whether changing a specified force-timing feature improves speed at a controlled accuracy requirement within the same golfer. It then tests that intervention, allowing for learning, fatigue, club choice, surface conditions and uncertainty. Descriptive clustering can help formulate that question; it cannot answer it by itself. ### Associations, Mediation and Strength Tests [Rachnavy et al. (2026)](https://www.frontiersin.org/journals/sports-and-active-living/articles/10.3389/fspor.2026.1790645/full) report observational regression and mediation analyses in 30 golfers. Adding sequencing and transfer measures improved their model fit. A significant indirect path does not establish causation, and a nonsignificant direct coefficient does not prove the absence of a direct effect. Force impulse, mechanical work and transfer efficiency also remain distinct quantities. The same discipline applies to strength testing. [Leary et al. (2012)](https://pubmed.ncbi.nlm.nih.gov/22797001/) studied 12 recreational golfers. Force at 150 ms during an isometric mid-thigh pull had correlations of about $0.46$–$0.47$ with average/maximum clubhead speed; those are **force-at-time** values. Their reported 0–150 ms rate-of-force-development correlations were about $0.38$–$0.36$, with $p=0.11$–$0.12$. These estimates do not establish either a reliable predictor or the absence of a relationship. The tested RFD is also not the slope of the swing's ground-force trace. [Johansen et al. (2023)](https://pubmed.ncbi.nlm.nih.gov/37585706/) reported positive associations between maximal clubhead speed and several strength/power measures, including leg-press RFD, in 21 national-level male golfers. Different tasks and populations can produce different estimates. Replacing one small observational study with another does not establish the effect of a training intervention; use effect estimates, uncertainty and the actual test definition rather than a binary significant/nonsignificant slogan. ::: {.callout-note} ## Comparing Body-Weight-Normalized Peaks A statement such as “players reach twice body weight” needs a cited population, club, trial and definition: one foot or both, instantaneous sum or separate peaks, and total vertical force or force after subtracting body weight. For the same trial, $$ \max_t[F_{z,1}(t)+F_{z,2}(t)] \le\max_t F_{z,1}(t)+\max_t F_{z,2}(t). $$ Equality requires compatible peak timing. Published per-foot peaks from one cohort cannot validate or refute a universal combined-force claim about another. Group averages of individual peaks introduce another distinction. Retain the synchronized signals and normalization before comparing numbers. ::: ### Screw Theory Connects the Questions, Not Their Answers [Kim (2025)](https://pubmed.ncbi.nlm.nih.gov/40843846/) reanalyzed one proficient and one novice female golfer, relating a kinematic instantaneous screw-axis pitch to vertical-force timing. This is a two-person exploratory comparison, not a validated skill classifier or proof that one waveform is necessary for expertise. Kinematic screw estimates require a declared moving body and frame, and become sensitive when angular speed is small. A **motion screw** describes velocity geometry; a **wrench screw** describes force/moment geometry. Their pitches and axes need not coincide. Power pairs the full, consistently referenced twist and wrench; a vertical-force trace and a kinematic pitch alone do not determine that pairing. This makes screw theory useful for asking a precise question without letting an attractive geometric analogy substitute for the missing measurements. A [2026 systematic review by Watson et al.](https://pubmed.ncbi.nlm.nih.gov/41653371/) likewise calls for clearer methods and terminology across the golf GRF/COP literature. Useful synthesis preserves the definitions of forces, moments, timing features and outcomes before comparing studies or drawing coaching conclusions. ## Part V: Estimating Force Without a Plate ### Whole-Body Dynamics Estimates a Net Load Given sufficiently accurate segment masses, inertias and kinematics, Newton–Euler balances estimate the **net external wrench**. To attribute that net load to the ground, subtract gravity and account for other external interactions. Missing club dynamics, uncertain segment parameters, soft-tissue motion, differentiation noise and synchronization all affect the estimate. A force plate observes a contact load through its calibrated sensing chain; a kinematic estimate obtains it through these additional model assumptions. During double support, the net wrench does not uniquely determine two arbitrary foot wrenches. [Ren, Jones and Howard (2008)](https://pubmed.ncbi.nlm.nih.gov/18672243/) introduced a smooth-transition assumption for walking to resolve that indeterminacy. It is a closure model based on gait behavior, not an additional measured equation or a universal law of bilateral contact. [Karatsidis et al. (2017)](https://doi.org/10.3390/s17010075) evaluated inertial-motion-based estimates in 11 healthy people walking at different speeds. Their normal-speed results had stronger agreement for vertical force than for some shear and moment components. Importantly, moments were referenced to the ground projection of the ankle, and the distribution curves were derived from the study's measured data. These results support feasibility under the evaluated gait protocol; they do not establish independent validation in golf, solve vertical-force estimation generally or classify frontal-plane moments as universally unusable. The appropriate accuracy threshold depends on the question. A useful gross-load estimate may be inadequate for a small change in angular impulse. Conversely, a moment estimate with quantified uncertainty can be useful for a sufficiently large effect. Report absolute as well as normalized errors, their normalization and phase dependence, and test the intended outcome rather than attaching a universal usable/unusable label to a component. ### A Golf-Specific Learning Result and Its Boundaries [Li et al. (2026)](https://doi.org/10.3390/biomimetics11030159) used lower-limb IMU kinematics to predict three force components in 48 professional golfers, with ten driver swings per participant. Their TCN–BiGRU model reported $R^2\approx0.94$ under subject-level ten-fold validation. Keeping each golfer's trials in one fold addresses a key source of leakage between training and evaluation. The measurement target was nevertheless specific: indoor swings, 60 Hz IMUs and force references low-pass filtered at 6 Hz. The result does not establish accuracy for impact transients, moments, other skill groups or outdoor shoe–surface conditions. A bidirectional sequence model's offline accuracy also does not demonstrate causal real-time prediction, because later samples may contribute to earlier estimates. These are boundaries on the supported claim, not reasons to dismiss the approach. Any deployment claim should retain the signal-processing pipeline, model version, sensor placement, held-out population and component-wise error distribution. Changes to the target filter or trial segmentation change what is being predicted. A high coefficient of determination is not a substitute for this record. ### Why the Per-Foot Partition Remains Underdetermined At a common reference point, $$ \mathbf w_{\rm net}=\mathbf w_1+\mathbf w_2. $$ For arbitrary six-component foot wrenches this is six equations for twelve components. Adding $\Delta\mathbf w$ to one foot and subtracting it from the other leaves the net wrench unchanged. Contact geometry and friction restrict the possible choices, but generally leave multiple admissible solutions. When only one foot is in contact, that particular partition ambiguity disappears; golf includes heel lift, rolling and sometimes loss of contact, so the contact mode should be measured or modeled rather than assumed constant throughout every swing. Even knowing both normal loads and COPs does not generally fix the shear partition. For two coplanar contacts, transfer a small horizontal force $\mathbf a$ from foot 2 to foot 1. The net force is unchanged, while the change in moment is $$ \Delta\mathbf M=(\mathbf r_1-\mathbf r_2)\times\mathbf a. $$ If $\mathbf a$ is parallel to the line between the COPs, this moment is also unchanged: normal pressure, both COPs and the entire net wrench can all remain identical. Other horizontal transfers can be offset by changes in the per-foot normal couples, when the friction limits permit. This provides a concrete reason that combined net-force and pressure measurements do not automatically recover each foot's complete wrench. Two independently calibrated six-axis platforms, with one foot on each and no load paths bypassing them, provide a direct way to resolve the per-foot wrenches. They are not the only possible instrumentation: suitably designed multiaxis wearable sensors or other independently instrumented contact regions can add the necessary observations. Each still needs calibration, frame registration and a check that the measured load belongs to the intended foot. A pressure mat can separate the normal-pressure contributions of two spatially resolved footprints, subject to its measurement errors. It does not do so “exactly” in a physical instrument, and overlapping or unsensed contacts add difficulties. A hybrid system may combine pressure-derived locations with additional force channels. The [Initial Force motion-analysis patent](https://patents.google.com/patent/US20110260890A1/en) is an example of such architectural ideas; a patent does not prove that a particular commercial implementation uniquely measures every per-foot shear and free-moment component. Identify which additional channels or modeling assumptions resolve each unknown. ### Building an Argument From the Available Measurements Start with the proposed claim and work backward to the observations it needs: | Proposed Claim | Required Reasoning Beyond a Force Trace | |---|---| | The golfer–club system changes momentum | Integrate the complete net external force, with gravity, system mass and other impulses accounted for. | | A foot contributes a turning moment | Declare the reference point; include force lever arms and free couples, and identify the per-foot partition. | | A joint generates mechanical work | Combine model-consistent joint load with its conjugate velocity; specify sign and integration interval. | | Particular muscles supplied that work | Add a muscle/tissue model and supporting observations; net joint kinetics alone is insufficient. | | A technique improves performance | Test an intervention with defined performance criteria, uncertainty and appropriate within-person and between-person comparisons. | These questions are intertwined because one measurement feeds the next calculation. Keeping their boundaries explicit strengthens the argument: it identifies which conclusions are already supported, which depend on a model, and which experiment would resolve the remaining uncertainty. ## References and Further Reading **Textbooks and Standards** - ASTM International (2023). *Standard Practice for Verification of Multi-Axis Force Measuring Platforms*, F3109-23. [Public scope](https://store.astm.org/f3109-23.html). - Winter, D. A. (2009). *Biomechanics and Motor Control of Human Movement*, 4th ed. Wiley — body segment parameters, link-segment inverse dynamics, residual analysis for filter selection. - Zatsiorsky, V. M. (2002). *Kinetics of Human Motion*. Human Kinetics — forces, moments and human-motion kinetics. - Robertson, D. G. E., Caldwell, G. E., Hamill, J., Kamen, G. and Whittlesey, S. (2014). *Research Methods in Biomechanics*, 2nd ed. Human Kinetics — biomechanical measurement and analysis methods. - de Leva, P. (1996). Adjustments to Zatsiorsky-Seluyanov's segment inertia parameters. *Journal of Biomechanics* 29(9), 1223–1230. [PubMed](https://pubmed.ncbi.nlm.nih.gov/8872282/). - Wu, G. and Cavanagh, P. R. (1995). ISB recommendations for standardization in the reporting of kinematic data. *Journal of Biomechanics* 28(10), 1257–1261. [PubMed](https://pubmed.ncbi.nlm.nih.gov/8550644/); joint coordinate system definitions [Part I](https://pubmed.ncbi.nlm.nih.gov/11934426/) (2002) and [Part II](https://pubmed.ncbi.nlm.nih.gov/15844264/) (2005). **Instrument Physics and Patents** - AMTI. *Force measuring platform and load cell therefor using strain gages to measure shear forces*. [US 4,493,220](https://patents.google.com/patent/US4493220A/en). - Berme, N. (Bertec). *Multi-component force and moment measuring platform and load transducer*. [US 6,354,155](https://patents.google.com/patent/US6354155B1/en); *Force measurement system having a plurality of measurement surfaces*, [US 8,544,347](https://patents.google.com/patent/US8544347B1/en). - Kistler. *Force transducers for fitting in force plates*. [US 4,974,454](https://patents.google.com/patent/US4974454A/en). - Tekscan. *Flexible tactile sensor for measuring foot pressure distributions and for gaskets*. [US 5,033,291](https://patents.google.com/patent/US5033291A/en). - Seitz, P. *Capacitive measuring assembly for determining forces and pressures*. [US 4,836,033](https://patents.google.com/patent/US4836033A/en). - Initial Force AS (Swing Catalyst). *Motion Analysis Apparatus*. [US 2011/0260890](https://patents.google.com/patent/US20110260890A1/en). - U.S. Army. *Force sensing treadmill*. [US 6,878,100](https://patents.google.com/patent/US6878100B2/en) — an example of separately instrumented support surfaces. **Measurement Accuracy and Calibration** - List, R. et al. (2017). In-situ force plate calibration: 12 years’ experience with an approach for correcting the point of force application. *Gait & Posture*. [PubMed](https://pubmed.ncbi.nlm.nih.gov/28763716/). - Chockalingam, N. et al. (2002). Do strain gauge force platforms need in situ correction? *Gait & Posture*. [PubMed](https://pubmed.ncbi.nlm.nih.gov/12443947/). - Middleton, J., Sinclair, P. and Patton, R. (1999). Accuracy of centre of pressure measurement using a piezoelectric force platform. *Clinical Biomechanics*. [PubMed](https://pubmed.ncbi.nlm.nih.gov/10521614/). - Giacomozzi, C. (2010). Appropriateness of plantar pressure measurement devices: a comparative technical assessment. *Gait & Posture*. [PubMed](https://pubmed.ncbi.nlm.nih.gov/20399101/). - Joo, S. B., Oh, S. E. and Mun, J. H. (2016). Improving the ground reaction force prediction accuracy using one-axis plantar pressure: Expansion of input variable for neural network. *Journal of Biomechanics* 49(14), 3153–3161. [PubMed](https://pubmed.ncbi.nlm.nih.gov/27515436/). **Golf Ground Reaction Force Literature** - Watson, A. et al. (2026). Ground reaction force and centre of pressure during the golf swing and associations with clubhead speed and skill level: a systematic review. *Sports Medicine* 56(5), 1191–1212. [PubMed](https://pubmed.ncbi.nlm.nih.gov/41653371/). - Worsfold, P., Smith, N. A. and Dyson, R. J. (2008). Low handicap golfers generate more torque at the shoe–natural grass interface when using a driver. *Journal of Sports Science and Medicine* 7(3), 408–414. [Full text](https://www.jssm.org/volume07/iss3/cap/jssm-07-408.pdf). - Han, K. H. et al. (2019). Effects of the golfer–ground interaction on clubhead speed in skilled male golfers. *Sports Biomechanics* 18(2), 115–134. [PubMed](https://pubmed.ncbi.nlm.nih.gov/31042142/). - Ball, K. A. and Best, R. J. (2007). Different centre of pressure patterns within the golf stroke, [Part I](https://pubmed.ncbi.nlm.nih.gov/17454544/) and [Part II](https://pubmed.ncbi.nlm.nih.gov/17454545/). *Journal of Sports Sciences* 25(7). - Jones, K. M., Wallace, E. S. and Otto, S. R. (2023). Centre of pressure golf swing movement strategies are better defined using a continuous approach than by segregated styles. *Journal of Sports Sciences* 41(4), 342–349. [PubMed](https://pubmed.ncbi.nlm.nih.gov/37149899/). - McNitt-Gray, J. L. et al. (2013). Regulation of reaction forces during the golf swing. *Sports Biomechanics* 12(2), 121–131. [PubMed](https://pubmed.ncbi.nlm.nih.gov/23898685/). - Peterson, T. J., Wilcox, R. R. and McNitt-Gray, J. L. (2016). Angular impulse and balance regulation during the golf swing. *Journal of Applied Biomechanics* 32(4), 342–349. [PubMed](https://pubmed.ncbi.nlm.nih.gov/26958870/). - Nesbit, S. M. and Serrano, M. (2005). Work and power analysis of the golf swing. *Journal of Sports Science and Medicine* 4(4), 520–533. [PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC3899668/). - Wells, J. E. T. et al. (2018). Relationships between highly skilled golfers' clubhead velocity and force producing capabilities during vertical jumps and an isometric mid-thigh pull. *Journal of Sports Sciences* 36(16), 1847–1851. [PubMed](https://pubmed.ncbi.nlm.nih.gov/29300147/). - Wells, J. E. T. et al. (2019). Relationships between Challenge Tour golfers' clubhead velocity and force producing capabilities during a countermovement jump and isometric mid-thigh pull. *Journal of Sports Sciences* 37(12), 1381–1386. [PubMed](https://pubmed.ncbi.nlm.nih.gov/30572804/). - Kim, W. (2025). Pitch invariance reveals skill-specific coordination in human movement: a screw-theoretic reanalysis of golf swing dynamics. *Journal of Functional Morphology and Kinesiology* 10(3), 315. [PubMed](https://pubmed.ncbi.nlm.nih.gov/40843846/). ::: {.callout-note} ## Related Concepts * **Twists and screw axes**: the kinematic dual of the wrench measured here. (See: [Motion Capture Technology](technology-motion-capture.html)) * **Inverse dynamics limitations**: why joint torques computed from these forces are not muscle forces. (See: [Interpretation of Inverse Dynamics](inverse-dynamics.html)) * **Ground reaction forces in the swing**: (See: [The Physics of Golf, Chapter 15](The_Physics_of_Golf/quarto/ch15_ground_reaction_forces.html)) ::: ## Related Articles - [Interpretation of Inverse Dynamics: Understanding the Fundamental Limitations](inverse-dynamics.html) — why joint torques computed from measured forces are not muscle forces - [Motion Capture: What Golf Data Can Tell Us](technology-motion-capture.html) — the kinematic counterpart to force measurement - [Force and Mobility Ellipsoids in the Golf Swing](force-mobility-matrices.html) — what the measured wrench implies for available force and motion - [Screw Theory Reference for Control-Affine Multibody Dynamics](screw-theory-reference.html) — wrenches and twists as screw quantities - [Intentional Constraint Collapse at Impact: How Golfers Generate High Force With Stable Club Motion](intentional-constraint-collapse.html) — ground-force use through the swing

In Layman's Terms▼