Ground-Reaction Forces and Constrained Dynamics
Ground-reaction force (GRF) is the force exerted by the ground on a body in contact with it. In golf, the practically useful observable is usually a ground-reaction wrench: three force components and three moments expressed about a declared force-plate origin. That wrench changes whole-body linear and angular momentum. It is not a direct neural command, a direct measure of muscular effort, or by itself a measure of energy delivered to the club.
This chapter distinguishes three operations that are often conflated:
- Measurement: a force plate measures an external contact wrench.
- Inverse dynamics: a model reconciles measured kinematics, inertia, and external wrenches to estimate generalized forces (Nesbit 2005; Gatt et al. 1998).
- Counterfactual attribution: the same declared model is reevaluated at a measured state after selected velocity or control terms are set to zero.
The third operation is model dependent. It does not turn a force-plate trace into a unique estimate of muscle activation.
Force, Moment, Center of Pressure, and Free Moment
Let a force plate report force
\[ \bm{F}=(F_x,F_y,F_z) \]
and moment about its origin
\[ \bm{M}_O=(M_x,M_y,M_z). \]
The axes, signs, origin, and units must accompany the data. When the contact surface is the plane \(z=0\) and \(F_z\) is sufficiently large, an equivalent center of pressure (COP) is commonly computed as
\[ x_{\mathrm{COP}}=-\frac{M_y}{F_z}, \qquad y_{\mathrm{COP}}=\frac{M_x}{F_z}. \]
The remaining moment about the surface normal, transported to the COP, is the vertical free moment. It represents a distributed contact couple and cannot be reconstructed from a planar point-contact force alone. COP is also a ratio, so it becomes unstable as \(F_z\) approaches zero. Low-load samples need an explicit mask rather than a silently reported COP.
Pressure insoles measure plantar pressure, primarily the normal component. Shear forces and free moment require additional sensing or an empirically validated estimator. A study that predicts six-axis force-plate quantities from pressure data must label them as predictions, not direct measurements (Joo et al. 2016).
Whole-Body Momentum Balance
For a golfer-plus-club system of total mass \(m\), before ball impact and after neglecting small aerodynamic loads, the linear momentum balance is
\[ \bm{F}_{\mathrm{GRF}}+m\bm{g}=m\ddot{\bm{r}}_{\mathrm{COM}}, \tag{1}\]
where \(\bm{g}\) points downward. With a vertical axis positive upward and \(g=9.81\ \mathrm{m\,s^{-2}}\),
\[ F_z-mg=m\ddot z_{\mathrm{COM}}, \qquad \ddot z_{\mathrm{COM}}=\frac{F_z}{m}-g. \]
For example, \(F_z=1200\) N for a 90 kg system gives \(\ddot z_{\mathrm{COM}}=1200/90-9.81=3.52\ \mathrm{m\,s^{-2}}\) upward. A formula yielding \(23.1\ \mathrm{m\,s^{-2}}\) would have added gravity twice.
Integrating Equation 1 over \([t_1,t_2]\) gives
\[ \int_{t_1}^{t_2}\bm{F}_{\mathrm{GRF}}\,dt =m\Delta\dot{\bm{r}}_{\mathrm{COM}}-m\bm{g}(t_2-t_1). \]
This is a strong data-quality check: the measured resultant impulse should agree with the change in whole-system momentum under the same filtering, events, mass, and coordinate convention.
The angular-momentum balance about the system COM is
\[ \dot{\bm{H}}_{\mathrm{COM}} =\sum_i (\bm{r}_{i}-\bm{r}_{\mathrm{COM}})\times\bm{F}_i +\sum_i\bm{M}_{i,\mathrm{free}}. \]
Consequently, the same resultant force can have different rotational effects when its line of action, bilateral allocation, or free moment differs.
Contact Work Is a Conditional Statement
Contact power for a resultant wrench is
\[ P_c=\bm{F}_c\cdot\bm{v}_c+\bm{M}_c\cdot\bm{\omega}_c. \]
For an ideal rigid, stationary, no-slip contact, the constrained contact velocity is zero, and the ideal constraint wrench does no work. This useful idealization does not imply zero muscular work. It also does not apply without qualification when the shoe deforms, the foot rolls, the contact patch changes, the surface deforms, or slip occurs. A COP is an equivalent-wrench location, not necessarily a material point whose velocity can be inserted into a power calculation.
Muscles and tendons may perform positive, negative, and elastic work while an ideal external contact wrench has zero instantaneous power at its constrained point. GRF can still redirect momentum and alter how internally supplied energy is distributed among segments.
Constrained-Reaction Mechanics
At one state, write the multibody equations as
\[ M(q)\ddot q+h_0(q)+h_v(q,\dot q) =B(q)u+Q_{\mathrm{ext}}+J(q)^\mathsf{T}\lambda, \tag{2}\]
with acceleration constraint
\[ J(q)\ddot q+\gamma(q,\dot q)=0. \]
Here \(h_0\) is velocity independent, \(h_v\) is velocity dependent, \(u\) is the declared controllable generalized input, \(Q_{\mathrm{ext}}\) contains other known applied loads, \(J\) is the active-contact Jacobian, and \(\lambda\) is the reaction in contact coordinates. Define
\[ A=JM^{-1}J^\mathsf{T}. \]
If \(J\) has full row rank and \(A\) is well conditioned, then
\[ \lambda=A^{-1}\left[ JM^{-1}(h_0+h_v-Bu-Q_{\mathrm{ext}})-\gamma \right]. \tag{3}\]
The reaction is affine in the declared generalized inputs, even though it is not itself an independently commanded input. It can be separated as
\[ \lambda=\lambda_0+\lambda_v+\lambda_u+\lambda_e, \]
with
\[ \begin{aligned} \lambda_0&=A^{-1}JM^{-1}h_0,\\ \lambda_v&=A^{-1}(JM^{-1}h_v-\gamma),\\ \lambda_u&=-A^{-1}JM^{-1}Bu,\\ \lambda_e&=-A^{-1}JM^{-1}Q_{\mathrm{ext}}. \end{aligned} \]
These components are exact only for the declared model and state. Errors in kinematics, inertial parameters, contact assignment, external loads, and filtering enter the residual. Dynamic inconsistency is therefore a primary falsification outcome, not a nuisance to hide (Sturdy et al. 2022; Werling et al. 2023).
Pointwise Ground-Reaction ZTCF
The pointwise ground-reaction zero-torque counterfactual (ZTCF) sets the declared controllable input to zero at the achieved \((q,\dot q)\) while retaining declared non-control external loads:
\[ \lambda_{\mathrm{ZTCF}}=\lambda_0+\lambda_v+\lambda_e. \]
It answers what reaction the model requires at this state with zero declared control. It is not a no-muscle experiment and does not describe what the body would do after controls were removed.
Pointwise Ground-Reaction ZVCF
The pointwise ground-reaction zero-velocity counterfactual (ZVCF) evaluates the same configuration and declared inputs after velocity is zeroed:
\[ \lambda_{\mathrm{ZVCF}}=\lambda_0+\lambda_u+\lambda_e. \]
ZTCF and ZVCF are not forward simulations. They overlap in \(\lambda_0+\lambda_e\) and must not be added to reconstruct the total. The additive attribution is configuration plus velocity plus control plus other external load.
Measured GRF minus modeled ZTCF is a residual containing input-induced reaction, model error, measurement error, and omitted external loads. It cannot uniquely identify muscle torques. Even with a perfect resultant wrench, the model cannot uniquely determine the bilateral allocation when the contact system is redundant. A pseudoinverse supplies a numerical allocation, not new physical information.
Interpreting Common Golf Measurements
Vertical Force
A vertical peak may combine configuration support, velocity-dependent whole- body acceleration, and input-induced reaction. Peak magnitude alone does not identify which term dominates. ZTCF and ZVCF provide model-based tests of which features persist after declared terms are removed.
Horizontal Shear
Anterior-posterior and medial-lateral components are particularly sensitive to axis definitions, stance orientation, foot contact, and filtering. Report the laboratory-to-golfer coordinate transform and avoid changing signs implicitly for left-handed participants.
Center of Pressure
COP describes the equivalent line of action of the measured wrench. It is not the center of mass, not “weight transfer,” and not energy transfer. Ball and Best observed more than one recurring COP strategy across skill levels (Ball and Best 2007). The literature therefore does not justify one universal COP or GRF waveform as the correct pattern.
Bilateral Forces and Free Moment
Two force plates provide more information than a combined resultant, but bilateral decomposition remains sensitive to each foot’s frame, crossover, partial contact, and plate assignment. The vertical free moment represents a distributed torsional contact effect. It should not be equated with joint torque or with the two-hand force couple in a club model merely because the signs look similar.
Association With Clubhead Speed
Golf studies have associated selected lead- and trail-foot force/moment features with clubhead speed (Han et al. 2019), and shoe-ground torque differs across some skill and footwear conditions (Worsfold et al. 2008). These associations do not establish that maximizing a peak will increase speed for an individual. Rachnavy and colleagues reported improved clubhead-speed association when foot-ground variables were combined with trunk and transfer variables (Rachnavy et al. 2026), but their cross-sectional mediation analysis is not a controlled causal intervention. The systematic review documents substantial methodological heterogeneity (Watson et al. 2026).
Falsifying a Drift-Based GRF Model
A defensible human study needs synchronized bilateral six-axis force plates, whole-body and club kinematics, participant-specific or uncertainty-bounded segment inertial parameters, and declared filtering and event rules. Model development and threshold selection should be separated from evaluation on held-out participants.
At minimum, report for every force and moment component:
- bias, RMSE, and an explicitly normalized RMSE;
- waveform \(R^2\) with amplitude error reported alongside it;
- vector impulse error over preregistered phases;
- peak magnitude and timing error;
- COP error only above a declared vertical-force threshold;
- free-moment error for three-dimensional contact models;
- residual pelvis forces and moments as dynamic consistency diagnostics; and
- sensitivity to inertial parameters, filtering, coordinate transforms, and contact allocation.
The drift hypothesis is rejected or narrowed when the full model cannot first reproduce measured wrench data, drift-only predictions fail held-out tests, or the inferred drift share changes qualitatively under plausible assumptions. A center-of-mass acceleration baseline should be included; a more complex model must outperform it on held-out observables to justify its additional claims.
Key Takeaways
- GRF is an external contact wrench and a constraint reaction, not a direct motor command.
- Whole-body resultant GRF is strongly constrained by center-of-mass acceleration, but bilateral allocation, COP, and free moment require additional spatial contact information.
- Ideal stationary no-slip contact does zero work at the constrained point; this conditional statement does not eliminate muscular, tendon, shoe, or surface work.
- Pointwise reaction ZTCF and ZVCF are overlapping diagnostics, not additive causes and not forward simulations.
- A measured-minus-ZTCF residual combines control-induced reaction with model and measurement error; it does not reveal unique muscle torques.
- Golfers exhibit multiple viable force and COP patterns. Evaluation should be participant-aware and falsification focused rather than template matching.
Chapter Exercises
- A 95 kg golfer stands still. Compute vertical GRF with upward positive.
- For the same golfer, \(F_z=1350\) N during the downswing. Compute vertical COM acceleration and state the sign convention.
- Explain when an ideal contact wrench does zero work and list three reasons a real shoe-ground interface may violate the idealization.
- Derive Equation 3 from Equation 2 and the acceleration constraint.
- Show algebraically why ZTCF plus ZVCF double-counts configuration reaction.
- Give two different bilateral foot-force allocations with the same combined resultant and explain what additional measurements distinguish them.
- Design a held-out test that compares a COM-only GRF baseline, ZTCF, and the full constrained model without splitting samples from the same participant across training and test sets.