Interpretation of Inverse Dynamics: Understanding the Fundamental Limitations
What the Calculation Tells Us
Inverse dynamics answers a precise mechanical question: which net forces and moments are consistent with the measured motion, declared mass properties and known external loads? It does not identify every contact force or muscle force, and it does not directly measure effort, sensation or intent.
For golf, three issues must be kept separate: reconstruction of a net wrench, allocation of that wrench between hands, and attribution within a declared dynamical model. Each needs different information. Their interaction explains why a technically correct force estimate can still support an incorrect coaching interpretation.
Reconstructing the Hand–Club Interaction
Recover the Net Wrench Before Interpreting It
Take the club as the system boundary and use one inertial frame. Let \(C\) be its center of mass, \(P\) a grip reference point, and \(\mathbf g\) gravitational acceleration. For a rigid club with known mass \(m\) and inertia \(I_C\), the net hand-on-club wrench satisfies
\[ \mathbf F_h=m\mathbf a_C-m\mathbf g-\mathbf F_a, \tag{1}\]
\[ \mathbf M_{h,C}=I_C\boldsymbol\alpha+ \boldsymbol\omega\times(I_C\boldsymbol\omega)-\mathbf M_{a,C}. \]
The subscript \(a\) collects declared non-hand external loads such as aerodynamics; gravity has zero moment about \(C\) in a uniform field. Include collision or turf loads when present. Inertia, angular velocity, acceleration and moments must be expressed consistently. For a flexible club, use the sum of segment momentum balances or the distributed model; one rigid-body inertia is an approximation.
Translate the same hand wrench using
\[ \mathbf M_{h,P}=\mathbf M_{h,C} +(\mathbf r_C-\mathbf r_P)\times\mathbf F_h. \tag{2}\]
For any two reporting points, the general rule is
\[ \mathbf M_{h,Q}=\mathbf M_{h,P} +(\mathbf r_P-\mathbf r_Q)\times\mathbf F_h. \tag{3}\]
The sign follows directly from moments of the original forces. A pure couple is independent of reference point; the moment component of a general wrench with nonzero force is not. These are standard wrench transformations (Lynch and Park 2017, 3).
For example, a force \((0,4,0)\) N applied at \((0.5,0,0)\) m produces a moment \((0,0,2)\) N m about the origin and zero moment about its application point. The physical load did not change. It is inappropriate to label the two reported moments different twisting actions by the hands.
What Remains Unidentified
For two contact points with forces \(\mathbf F_1,\mathbf F_2\) and local couples \(\mathbf C_1,\mathbf C_2\), the net wrench gives six balance equations:
\[ \mathbf F_h=\mathbf F_1+\mathbf F_2,\qquad \mathbf M_{h,P}=\sum_{i=1}^2 \left[(\mathbf r_i-\mathbf r_P)\times\mathbf F_i+\mathbf C_i\right]. \]
There are generally more unknowns than observations. Even with zero local couples, an equal-and-opposite force pair along the line joining the two contact points leaves both net force and moment unchanged. Thus a known resultant does not identify bilateral grip squeeze, finger pressures or joint tissue loading. An instrumented grip or independently validated contact model adds information; a pseudoinverse only selects a solution according to a mathematical metric.
The relation \(\tau=J^\mathsf{T}w\) maps a specified external wrench to its generalized load by virtual power. It does not turn an arbitrary net actuator torque into the uniquely measured wrench at one hand. A pseudoinverse requires a consistency check, its null space, a force/moment scaling convention, and a physical allocation model.
Power Is Invariant Under a Consistent Reference Change
For two material points on the same rigid body, \(\mathbf v_Q=\mathbf v_P+\boldsymbol\omega\times(\mathbf r_Q-\mathbf r_P)\). Consequently,
\[ \mathbf F_h\cdot\mathbf v_P+\mathbf M_{h,P}\cdot\boldsymbol\omega =\mathbf F_h\cdot\mathbf v_Q+\mathbf M_{h,Q}\cdot\boldsymbol\omega. \]
The force and moment power components can change separately while their sum does not. A comparison of “force-driven” and “couple-driven” power must declare the reporting point. For a flexible grip/shaft, use velocities at the actual ports and include deformation power; rigid transport is not exact.
Account for Missing Loads Without Double Counting
If aerodynamic force and moment are omitted while measured motion is held fixed, the inferred hand wrench is biased by
\[ \Delta\mathbf F_h=\mathbf F_a,\qquad \Delta\mathbf M_{h,P}=\mathbf M_{a,C} +(\mathbf r_C-\mathbf r_P)\times\mathbf F_a=\mathbf M_{a,P}. \]
The omitted force’s change in moment arm is already included in this wrench translation. Do not add it again to a drag moment already reported about \(P\). For the illustrative force \((0,-8,0)\) N at \((0.6,0,0)\) m relative to the grip, with zero intrinsic aerodynamic couple, the omitted moment is \((0,0,-4.8)\) N m. These are chosen numbers, not measurements of a particular club or golfer.
Drag opposes velocity relative to the air; lift and aerodynamic couples need separate directions. The coefficient, projected area, orientation, relative wind and center of pressure must be supplied to make a quantitative estimate. The fractional error in a small inferred grip moment can be large even when the aerodynamic force is small relative to another load. That is a sensitivity statement, not a general percentage correction to every swing.
Control-Affine Bookkeeping and Its Limits
Use one force convention throughout. In independent generalized coordinates \(z=(q,\eta)\), let \(v=\dot z\) and write
\[ M(z)\dot v+h(z,v)=\bar B(z)u,\qquad h=Cv+\nabla_z V+d,\qquad \bar B=\begin{bmatrix}B\\0\end{bmatrix}. \]
Here \(h\) contains the terms written on the left of the mechanical balance; their signed contribution to acceleration is \(a_d=-M^{-1}h\). Known external loads must either be included with their signs in \(h\) or declared separately. These are the standard forward/inverse dynamics conventions (Lynch and Park 2017; Featherstone 2008).
Inverse dynamics estimates the applied generalized load. Its output is
\[ \tau_{\mathrm{ID}}=\mathrm{ID}(z,v,\dot v)=M\dot v+h=\bar B u. \]
Consequently, \(\mathrm{ID}(z,v,a_d)=0\) for an exactly matched zero-input model, up to numerical error. It does not return a nonzero gravity, elastic, or interaction-force contribution. A constrained model can simultaneously have zero actuator torque and nonzero constraint reactions; the solver’s actuator and reaction outputs must be distinguished.
A resultant-force partition is a different quantity. If we retain \(\tau_{\text{total}}\) for the net generalized right-hand side of \(M\dot v\), the consistent definitions are
\[ \tau_{\text{total}}:=M\dot v, \qquad \tau_{\text{drift}}:=-h=M a_d, \qquad \tau_{\text{input}}:=\bar B u=\tau_{\mathrm{ID}}. \tag{4}\]
Thus
\[ \tau_{\text{input}}=\tau_{\text{total}}-\tau_{\text{drift}} =M(\dot v-a_d). \tag{5}\]
These symbols denote generalized loads, whose units depend on the coordinate: torque for an angle, force for a translation, and the corresponding conjugate load for a normalized modal coordinate. They are not individual muscle forces. The rigid rows require the full acceleration vector: \([M_{qq}\ M_{q\eta}]a_d\) includes recoil; \(M_{qq}(a_d)_q\) alone does not.
For example, an unforced pendulum at an inclined angle has nonzero gravitational acceleration. Applying inverse dynamics to that free fall returns zero pivot torque. Holding it stationary instead requires the gravity-balancing torque. The same distinction persists in a coupled golfer–club model.
A forward ZTCF follows its own state after release. In a shared local chart,
\[ \frac{d}{dt}(x-x_0)=f(x)-f(x_0)+G(x)u, \]
where \(x_0\) is the zero-input trajectory, not the initial-state symbol in this equation. Only at a common state does the drift difference vanish. A later trajectory difference contains the accumulated effect of changed geometry, velocity, deformation, and contact; it is not an instantaneous input-force sample. Specify the output, comparison time or event, horizon, and admissible input set before interpreting closeness to a ZTCF as control authority.
The input term is a net generalized-torque channel. It may reflect voluntary commands, feedback, passive tissues omitted from the bias model, or model error. Its mathematical name does not identify what the golfer consciously controls or perceives. Sensory estimates of load and effort require physiological interpretation beyond the mechanical partition.
A Radial Force Is a Real Contact Load
For an ideal point mass on a massless rod with a fixed pivot, let \(\mathbf e_r\) point outward and let \(T>0\) denote inward rod tension. Radial force balance is
\[ -T+m\mathbf g\cdot\mathbf e_r=-mL\omega^2, \qquad T=mL\omega^2+m\mathbf g\cdot\mathbf e_r. \]
At the bottom of a vertical circle this becomes \(T=mL\omega^2+mg\). The \(mL\omega^2\) term is the required inward resultant, not an additional outward force to subtract from measured grip tension to obtain muscle input. The rod or grip actually transmits the load. Zero pivot torque can coexist with substantial tension.
For a club with translating hands and shaft deformation, the point-mass formula is insufficient: use the center-of-mass acceleration and full moment balance. A grip force normal to the instantaneous path may have small force power at that point, but the total hand wrench can still do work through other force components and its moment.
From Segment Sequencing to Energy Transfer
For \(M\dot v+h=\bar B u\), the following quantities have distinct meanings. Evaluate every row at the same declared state and in the same coordinates.
| Quantity | Definition | Interpretation |
|---|---|---|
| Applied Generalized Load | \(\tau_{\mathrm{ID}}=M\dot v+h\) | Net input required by the declared inverse-dynamics model |
| Resultant Generalized Load | \(\tau_{\text{total}}=M\dot v\) | Right-hand side of the acceleration balance |
| Configuration Drift | \(\tau_{\mathrm{ZVCF}}=-h(z,0)\) | Signed load at the specified velocity-reset state |
| Velocity-Dependent Drift | \(-[h(z,v)-h(z,0)]\) | Difference under that reset, including retained damping |
| Declared Input Contribution | \(\tau_{\text{input}}=\bar B u\) | Applied generalized load, not muscle-force allocation |
The first row must not also be called the sum of drift and input. Doing so counts \(h\) twice. In redundant coordinates, constraint forces require their own balance; uniqueness of net reduced torque does not establish unique actuator, muscle, or bilateral hand-force allocation (Lynch and Park 2017).
For a chosen, identifiable reaction output in a fixed smooth contact mode, substituting acceleration into the reaction equations can yield \(r(x,u)=r_d(x)+R_u(x)u\). A same-state subtraction then gives \(r(x,u)-r(x,0)=R_u(x)u\). Declare the reference point, frame, sign and force/moment components. The result for a wrench is not obtained by relabeling an actuator torque. Changes of mode, friction branches or unidentified internal forces require additional treatment.
There is no universal fourth “mixed force” equal to total minus a forward ZTCF load minus a ZVCF load. That expression can subtract overlapping baseline terms at different states. To investigate interactions, define two interventions and a common outcome \(Y\): the contrast \(Y_{11}-Y_{10}-Y_{01}+Y_{00}\) measures nonadditivity for those interventions and that outcome. It is not a new physical force or a unique anatomical cause.
How Geometry, Energy, and Torque Work Together
Past torque changes the present configuration, speed, and elastic deformation. Those states change both the drift and the input-to-acceleration map. Current torque then acts through that changed geometry. A shaft can store and return energy, but it does not create energy; compare complete endpoint energy and work balances when changing shaft or timing parameters.
For time-independent kinetic and potential energy and ideal stationary constraints, the total mechanical energy satisfies
\[ \dot E=v^\mathsf{T}\bar B u-v^\mathsf{T}d+P_{\mathrm{external}}, \qquad E=\tfrac12v^\mathsf{T}Mv+V. \]
In this energy balance, \(d\) denotes the declared dissipative loads and \(P_{\mathrm{external}}\) the power of the remaining external ports. If an external load was absorbed into \(h\) in the dynamics convention, separate it out before using this balance; do not count it in both \(d\) and \(P_{\mathrm{external}}\). Conservative forces represented by \(V\) are already accounted for through stored energy.
The inertial/Coriolis terms redistribute motion consistently with this balance; their individual magnitudes are not independent energy sources. At an internal interface, equal and opposite wrenches cancel in the whole-system balance but can transfer power between segments. A segment slowing down does not by itself show that its energy went into the club: examine joint force power, moment power, elastic storage and external work (Featherstone 2008).
A torque’s sign alone does not identify braking. For one angular coordinate, \(P=\tau\dot q\) is negative when the torque opposes velocity. Zero power does not imply zero muscle effort, and large normal force can redirect motion while doing little instantaneous work at a specified moving point. This distinction connects hand-path geometry, energy transfer and delivery without conflating force, impulse, work, tissue load and neural strategy.
Measurement and Model Uncertainty
Differentiation amplifies motion noise. Filtering can change acceleration peaks and timing; sample rate alone does not guarantee accurate loads. Report calibration, synchronization, differentiation and filtering choices, uncertainty in mass and inertia, and the force/kinematic bandwidth required by the event of interest.
Closed-chain inverse dynamics can be underdetermined. Regularization selects an allocation consistent with a penalty; it does not measure the selected internal forces. Compare independent grip and ground-force measurements where available, and report sensitivity to alternative plausible allocations.
During impact, contact force and shaft dynamics act on a much shorter time scale than the overall downswing. A smooth pre-impact model cannot supply the collision wrench by extrapolation. Momentum/impulse or resolved compliant-contact equations are needed, with appropriate measurement bandwidth.
How to Build a Stronger Argument
Start with a measured or simulated outcome and define its system boundary. Use force and moment balances to identify the net load, then state what remains unobserved. For a mechanistic hypothesis, compare declared interventions at matched states or events and assess sensitivity to model assumptions. Evaluate energy transfer from power and work balances, not from peak angular-velocity order or an acceleration ratio.
For example, the claim that hand-path curvature increases club speed requires more than a large normal force: compare the complete hand work, initial energy, deformation, final state and input feasibility across paths. If a path changes speed only by requiring more actuator work, that differs from a redistribution effect at matched work. Both may be mechanically interesting; they answer different questions.
Practical Interpretation
A large equivalent grip moment need not mean a separate twisting couple was deliberately applied. A low net joint torque need not mean low muscle force. A large drift acceleration need not imply little control authority. These conclusions follow from the distinctions above and motivate additional measurements rather than a universal swing prescription.
References
Citation
@online{olson2025,
author = {Olson, Dieter},
title = {Interpretation of {Inverse} {Dynamics:} {Understanding} the
{Fundamental} {Limitations}},
date = {2025-11-28},
url = {https://affinedrift.com/articles/inverse-dynamics.html},
langid = {en}
}