Affine Control Interpretation of the Golf Swing

Drift, Input, and Model-Conditioned Attribution

Foundational derivation showing the golf swing as a control-affine mechanical system, introducing the drift/input decomposition framework for biomechanical analysis.
Author

Dieter Olson

Published

August 30, 2026

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NoteCanonical Version

This is the authoritative treatment of the control-affine framework for the golf swing. An earlier article, Affine Nature of the Golf Swing, covers the same foundational material and is retained for historical reference. Readers are encouraged to use this version.

Abstract

NoteNovelty Status
  • Established (textbook): Control-affine system formulation \(\dot{x} = f(x) + G(x)u\); Lagrangian derivation of rigid-body multibody dynamics; tangent space linearization.
  • Novel application: Applying the drift/input decomposition to golf biomechanics; the “Frozen Strategy” assumption for biological impedance; defining ZTCF family and ZVCF as diagnostic tools for swing analysis.
  • Terminology: The terms “drift” and “input” are standard in nonlinear control. “Zero Torque Counterfactual (ZTCF)” and “Zero Velocity Counterfactual (ZVCF)” are the authors’ naming for these constructions in the golf context.
NoteScope & Exclusions

This model captures the mechanical structure of coupled rigid bodies (golfer’s limbs and torso) and a flexible shaft connected at the grip. It includes inertia, gravity, elastic and damping forces from shaft bending, and joint torques as declared inputs. It explicitly excludes aerodynamic drag, ground reaction forces, marker dynamics, muscle activation delays, and impacts. Terms retained in the chosen equations can be separated into autonomous and declared-input components; what is excluded lies outside the model’s scope and must not be attributed by the decomposition.

ImportantModel-Conditioned Attribution Boundary

The identity \(\dot{x}=f_p(x)+G_p(x)u\) separates the autonomous and declared-input terms for a specified model at a fixed state. Its interpretation is conditional on the model boundary, coordinates, declared input, parameters and frozen variables, and the intervention being evaluated (a pointwise evaluation or a stated simulation horizon). This additivity does not imply orthogonality: under a chosen metric, \(f_p(x)\) and \(G_p(x)u\) may align, oppose, or be oblique. Without a separate identifiability argument and qualifying measurements, this bookkeeping does not identify neural intent, individual-muscle forces, biological effort, or a unique real-world cause.

The golf swing is a high-speed, full-body motion executed by a mechanically complex, constrained multibody system. Conventional inverse dynamics can estimate net generalized loads consistent with motion and external-force measurements, but it does not identify individual-muscle forces, neural intent, or a unique real-world cause. This paper develops a strictly theoretical control-affine model and separates its instantaneous vector field into autonomous (drift) and declared-input terms. The Zero Torque Counterfactual (ZTCF), Zero Velocity Counterfactual (ZVCF), and force taxonomy are model-conditioned diagnostics; empirical interpretation requires parameter identification, qualifying measurements, and validation.

Before writing equations, we ask a bounded question: at a specified model state, how much instantaneous evolution is assigned to the autonomous term, and how much to the declared torque input channel? This is mathematical bookkeeping inside a chosen mechanical model, not a direct reading of what the golfer intended or which muscles produced the motion.

New to control theory? Here's what this foundational article covers in everyday language.

The Core Question

At a specified state in the model, how much instantaneous motion is assigned to the autonomous dynamics, and how much is assigned to the declared input channel?

Imagine a skateboard model with gravity and a declared push force. The equations can report the term contributed by each channel at the same state, but that report does not measure the rider's intent, muscles, or biological effort.

What is a "State"?

The "state" is a complete snapshot of the golfer-club system at any moment—it includes where every joint is positioned and how fast everything is moving. Think of it as pausing a video and noting every detail about the pose and motion.

Given the current state, model parameters, and future declared inputs, the model predicts a trajectory. The state contains the model's represented "memory"; omitted biological states remain outside that prediction.

The Modeling Simplifications

To make the math tractable, this framework makes some simplifying assumptions: body parts are treated as rigid (not squishy), the club shaft can bend but in predictable ways, and we ignore air resistance. These are standard engineering approximations.

It's like how weather models don't track every raindrop—they simplify to make predictions possible while still being accurate enough to be useful.

Every theory faces scrutiny. Here's what skeptics and alternative perspectives say:

Biomechanical Critique:

Muscles vs. Net Torque Abstraction

Biomechanists argue that modeling the golfer's input as a generalized torque vector $u$ is an oversimplification. Real muscles have complex force-length and force-velocity properties, meaning the available torque depends on the state $(q, \dot{q})$. By treating $u$ as an independent input, the affine model ignores the intrinsic impedance of the musculoskeletal system, potentially misattributing passive muscle stiffness (a "drift" effect) to active control.

Our Response: We acknowledge this biological reality. However, the affine decomposition $\dot{x} = f(x) + G(x)u$ remains valid if interpreted correctly: $u$ represents the net non-conservative forcing applied by the neuromotor system, after accounting for passive tissue properties. The "drift" term $f(x)$ captures the dynamics of the "Effective Plant"—the skeletal system plus the baseline impedance of the muscles. The ZTCF (Zero Torque Counterfactual) therefore asks "what if the net driving force vanished?", not "what if the muscles disappeared?".
Energy Dynamics:

The "Free Lunch" Fallacy

Critics suggest that emphasizing the "Drift" component implies that the resulting motion is "free" energy. However, the golfer must expend significant effort early in the swing to create the kinematic state where drift becomes dominant. By separating "Drift" from "Input" at the moment of release, the analysis might undervalue the active work done to set up the passive dynamics.

Our Response: The decomposition does not claim drift is free or passive in a physiological sense. It labels the autonomous term of the declared effective plant at the selected state. Work, metabolic effort, and the history that produced that state require separate calculations and measurements.
Methodological Concern:

Necessity of Affine Control Theory

Why introduce the complexity of affine control theory and Lie brackets? Standard inverse dynamics already calculates the net moments required for a motion. Critics argue that decomposing these moments into $f(x)$ and $G(x)u$ adds mathematical overhead without changing the fundamental physics of $F=ma$.

Our Response: Standard inverse dynamics provides the net generalized load consistent with a motion. A control-affine formulation adds reproducible bookkeeping: it separates the model's state-dependent autonomous term from its declared torque input term. That structure makes a precisely specified zero-input intervention well-posed, but physiological attribution still requires additional models, measurements, and identifiability evidence.
Note: Scientific discourse thrives on debate. These critiques strengthen our understanding by defining the boundaries of the theoretical model.

Introduction

Traditional biomechanical analyses of the golf swing rely heavily on inverse dynamics to estimate generalized joint torques and hand forces from measured kinematics and external forces. Inverse dynamics provides net generalized loads consistent with the declared model and measurements; it does not by itself identify individual-muscle forces, intent, or biological effort.

This distinction is fundamental. Passive forces (drift) can be large—even dominant—during rapid phases of the swing, meaning that inverse-dynamics forces are often misinterpreted as direct indicators of muscular effort. Without a principled decomposition, it is difficult to tell whether a large modeled torque reflects newly applied input or state-dependent dynamics already present in the system.

The aim of this manuscript is to establish a rigorous, strictly theoretical foundation for decomposing the chosen model’s equations into two additive components:

  • Drift terms in the autonomous vector field, and
  • Input terms produced by the declared generalized-torque channel.

The key mathematical observation is that the golfer–club–shaft system can be modeled as a nonlinear control-affine mechanical system. Control-affine systems admit an additive decomposition of their dynamics into passive drift and linear control input channels. Once the affine structure is established, counterfactual tools such as the Zero Torque Counterfactual (ZTCF) and Zero Velocity Counterfactual (ZVCF) follow naturally and enable mechanical attribution of forces within the model. They do not, by themselves, constitute direct physiological identification of muscle activity in a real golfer.

This paper is Part I of a broader project. It develops the theoretical architecture only—no simulation, numerical results, or empirical data are presented here. Subsequent papers (Parts II and III) will implement and validate this framework using high-fidelity multibody simulations, flexible-shaft modeling, and experimental motion-capture data. By isolating the theory in this manuscript, we ensure that the mathematical structure is complete, self-contained, and available for independent scrutiny before numerical or empirical layers are added.

Let us pause and say plainly what we are doing. Imagine you are sitting in a rowboat on a river. Part of your motion downstream comes from the river’s current—that is the “drift.” Part comes from your rowing—that is the “input.” If you stop rowing, the current carries you along at some speed. If the river is still, only your rowing moves you. The total motion is the sum of both. This paper builds the mathematical machinery to decompose the golf swing the same way within a specified mechanical model: what would the club do if the modeled driving torques were removed (drift), and what additional motion do the applied torques create (input)? The mathematical framework is sophisticated, but the physical question is as simple as a boat on a river.

The remainder of the paper is organized as follows:

  • Section 1 defines the golfer–club–shaft system, the modeling assumptions, and the state and input variables.
  • Section 2 derives the unified control-affine form for the coupled rigid–flexible system.
  • Section 3 formalizes the drift–input decomposition.

The subsequent sections (Part II onwards) cover the counterfactual constructions (ZTCF/ZVCF), drift invariance, force taxonomy, and applications.

Detailed derivations, modal approximations, and toy examples are collected in the Appendices (Part IV).

To rigorously distinguish ‘drift’ from ‘input’, we must first define the mechanical universe in which these forces exist. The concepts of active and passive are not absolute; they are defined relative to the system boundary and the level at which the inputs are modeled. A force that is ‘external’ to a sub-component may be ‘internal’ to the whole. Therefore, before deriving equations, we must explicitly fix the boundaries of the golfer–club–shaft system, specifying exactly which degrees of freedom are malleable (flexible modes) and which are directly actuated (joints).

System Definition and Modeling Assumptions

In this section we define the mathematical structure of the golfer–club–shaft system and state the modeling assumptions that bound the theoretical framework. The goal is not to fix a unique anatomical model but to specify a general class of multibody systems for which the subsequent analysis is valid.

Modeling Assumptions

WarningModeling Assumptions for the Theoretical Framework

These assumptions apply strictly to the theoretical decomposition presented in this manuscript (Part I). They define the mathematical boundaries of the model; empirical validation and relaxation of these assumptions will be addressed in later work.

  1. Rigid body segments. All anatomical segments of the golfer (torso, arms, hands) are modeled as rigid bodies with fixed mass and inertia properties.
  2. Flexible shaft via finite-dimensional modal approximation. The shaft is modeled using \(m\) vibration modes obtained from standard beam theory (e.g., Euler–Bernoulli or Timoshenko). Modal truncation approximates the distributed dynamics with a finite-dimensional representation.
  3. No ball–club impact phase. Ball impact is excluded from the domain of analysis due to its hybrid (non-smooth) dynamics. The model applies to the pre-impact and post-impact phases only.
  4. No aerodynamic forces. Drag and lift forces on the clubhead and shaft are omitted in this theoretical treatment. All modeled external forces arise from gravity or internal mechanical interactions.
  5. Torques treated as generalized inputs. Muscular physiology (activation dynamics, force–velocity, force–length, tendon elasticity) is abstracted into a net generalized torque vector \(\tau = B(q) u\). These torques serve as the control inputs in the affine control formulation.
    • Mathematical Justification: At fixed state, the Hill-type expression \(\tau=\tau_{\max}f_{\text{FL}}(q)f_{\text{FV}}(\dot q)a\) is linear in activation \(a\); its state-dependent gain belongs in the input map. We nevertheless declare the mechanical input at the net-torque level and treat physiological feasibility through \(\mathcal{U}(x)\). Nonlinear activation dynamics, input-dependent impedance, saturation, or hysteresis require a richer model.
  6. Ground contact modeled as holonomic constraint. Feet–ground interaction is simplified as a fixed base with no slip or compliance; the golfer is treated as rooted at the ground.
  7. Smooth dynamical evolution (no discontinuities). Between impacts, the system evolves via smooth ordinary differential equations. Shaft deformation and all segmental interactions are continuous in time.
  8. No measurement noise or parameter uncertainty. The theoretical decomposition assumes perfect state knowledge and exact model parameters. Sensitivity to uncertainty and noise is a topic for subsequent empirical work.

To translate these assumptions into a rigorous mathematical structure, we must first define the geometric manifold on which the system evolves. The assumptions above simplify the biomechanical reality into a deterministic state space, allowing us to describe the golfer not as a collection of tissues, but as a point moving along a curved surface defined by the joint constraints.

Generalized Coordinates and State

The golfer’s anatomy is modeled as a multibody kinematic chain with generalized coordinates \(q \in \mathbb{R}^{n}\), each representing a rotational degree of freedom (e.g., shoulder rotations, elbow flexion, wrist deviations, spinal rotations). Any standard representation (Denavit–Hartenberg parameters, exponential coordinates, or spatial vectors) is admissible. The configuration space for the rigid segments is a smooth manifold \(\mathcal{Q}_r\), typically a product of Lie groups (e.g., \(SE(3) \times \mathbb{T}^k\)).

The flexible shaft is modeled via a finite-dimensional modal approximation with modal coordinates

\[\eta = [\eta_{1}, \dots, \eta_{m}]^{T} \in \mathbb{R}^{m},\]

where each \(\eta_{i}\) represents the amplitude of a particular bending or deformation mode. The total configuration manifold is the smooth product manifold \(\mathcal{Q} = \mathcal{Q}_r \times \mathbb{R}^m\) of dimension \(N = n+m\).

We formalize the golfer–club system as a simple mechanical system \((Q, \mathbb{G}, V)\), where the configuration space \(Q\) is a smooth manifold of dimension \(N = n+m\). We endow \(\mathcal{Q}\) with a Riemannian metric \(\mathbb{G}\) defined by the kinetic energy tensor \(M(q, \eta)\). For any tangent vector \(v_q \in T_q \mathcal{Q}\), the kinetic energy is given by the quadratic form: \[T(v_q) = \frac{1}{2} \langle v_q, v_q \rangle_{\mathbb{G}} = \frac{1}{2} v_q^T M(q,\eta) v_q.\]

The dynamics of the system evolve on the tangent bundle \(T\mathcal{Q}\), the smooth manifold formed by the union of all tangent spaces: \[T\mathcal{Q} = \bigcup_{p \in \mathcal{Q}} T_p \mathcal{Q}.\] We define the full state vector \(x\) as a point in this bundle. In local coordinates: \[x = [q^{T}, \dot{q}^{T}, \eta^{T}, \dot{\eta}^{T}]^{T} \in T\mathcal{Q} \cong \mathbb{R}^{2(n+m)}.\]

In plainer terms: the state vector \(x\) captures both where every part of the system is (the joint angles \(q\) and shaft bend \(\eta\)) and how fast each is moving (\(\dot{q}\) and \(\dot{\eta}\)). Just as you need both a car’s position on a highway and its speed to predict where it will be in five seconds, the physics needs both configuration and velocity to predict what comes next. The tangent bundle is simply the mathematical name for the space of all possible “where + how fast” combinations.

The equations of motion are generated by the Levi-Civita connection \(\nabla\) associated with the metric \(\mathbb{G}\). The drift vector field \(f(x)\) is formally the geodesic spray \(S: T\mathcal{Q} \to TT\mathcal{Q}\) of the metric, modified by the vertical lift of the potential and dissipative forces. Explicitly, if \(\nabla_{\dot{x}}\dot{x}\) represents the covariant acceleration, the unforced dynamics satisfy \(\nabla_{\dot{x}}\dot{x} = -M^{-1}(\nabla V + F_{dissip})\). This geometric definition treats \(f(x)\) as a coordinate-independent object—a global section of the tangent bundle—so the drift term is tied to the mechanics rather than to a particular coordinate chart.

NoteNote on Constrained Subsystems (The Wrist)

While the global system evolves on \(T\mathcal{Q}\), specific joints may have internal constraints that reduce the local degrees of freedom. For example, the wrist joint functions as a universal joint with a constrained third axis (forearm rotation). These local constraints generate internal constraint torques that reside in the null space of the actuation map \(B(q)\) but are critical for force transmission. For a detailed derivation of these constraint dynamics, see Constraint Torques at the Wrist.

Torque Inputs and Actuation Mapping

Muscular torques are abstracted as control inputs

\[u \in \mathbb{R}^{m_{u}}, \quad m_{u} \leq n,\]

with

\[\tau = B(q)\, u,\]

where \(\tau \in \mathbb{R}^{n}\) is the generalized joint torque vector and \(B(q) \in \mathbb{R}^{n \times m_{u}}\) encodes actuation pathways and moment arms. Formally, \(B(q)\) is a mapping from the input space to the cotangent bundle \(T^*\mathcal{Q}\) (forces/torques). In the simplest case, \(B\) is constant; more generally, it may depend on configuration. In either case, torques enter linearly in the equations of motion.

Throughout this paper we treat \(u\) as the exogenous input at the mechanical level. The mapping from neural commands to \(u\) is outside the scope of this theoretical work.

While inputs drive the system, they do not act in a vacuum. The golfer must contend with—and exploit—the environmental and inertial field. The effectiveness of any torque input is conditioned by the external forces that define the system’s potential energy landscape and reaction constraints.

External Forces and Constraints

External forces included in the model are:

  • gravitational loading,
  • inertial, Coriolis, and centrifugal effects arising from the multibody kinematics,
  • passive joint contributions (if modeled explicitly),
  • elastic and damping forces from shaft deformation.

Aerodynamic forces, impact forces, and non-holonomic contact effects are excluded under the assumptions above. Ground contact is modeled as a holonomic constraint that effectively fixes the base of the kinematic chain.

The final structural element is the coupling between the actuator (the golfer) and the payload (the club). This interface is not a simple rigid connection; it is the boundary where the active, articulated chain meets the passive, flexible object.

Before we can assemble the global equations of motion, we must rigorously define this boundary. The grip transmits generalized loads between the modeled golfer segments and club; it does not encode intent. To capture that bidirectional mechanical interaction, we cannot simply weld the shaft to the hands. We link rigid-body hand velocity to flexible shaft deformation through explicit Jacobians.

Hand–Club Kinematic Interface and Jacobian Formulation

The structural coupling between the golfer (actuator) and the club (payload) is the critical boundary where active kinematics meet passive dynamics. We model this interface rigorously using differential kinematics.

Let the hand reference frame \(\mathcal{F}_h\) be attached to the distal segment of the kinematic chain. Its spatial configuration \(T_{wh}(q) \in SE(3)\) relative to the world frame is determined by the forward kinematics of the rigid body chain. The spatial velocity of the hands, \(\mathcal{V}_h \in \mathbb{R}^6\) (twist), is related to the joint velocities by the geometric Jacobian \(J_h(q) \in \mathbb{R}^{6 \times n}\): \[\mathcal{V}_h = J_h(q) \dot{q}.\]

The clubshaft is modeled as a flexible beam clamped at the hands. The position of a material point \(s \in [0, L]\) along the shaft in the world frame, denoted \(p(s, q, \eta)\), is the superposition of the rigid hand motion and the local elastic deformation. Using the Assumed Modes Method (AMM), the spatial deformation relative to the hand frame is expressed as \(w(s, t) = \sum_{i=1}^m \phi_i(s) \eta_i(t)\), where \(\phi_i(s) \in \mathbb{R}^3\) are the spatial mode shapes.

The velocity of any differential mass element \(dm = \rho(s)ds\) on the shaft is obtained by differentiating the position vector with respect to time. Applying the chain rule to \(p(s, q, \eta)\): \[ v(s) = \frac{d}{dt} p(s, q, \eta) = \frac{\partial p}{\partial q} \dot{q} + \frac{\partial p}{\partial \eta} \dot{\eta}. \] Identifying the partial derivatives with the geometric Jacobians of the system:

  1. Rigid-Body Jacobian: \(J_{\text{rigid}}(s, q) \equiv \frac{\partial p}{\partial q} \in \mathbb{R}^{3 \times n}\). This Jacobian captures the transport velocity of the point \(s\) induced by the articulation of the golfer’s joints.
  2. Modal Jacobian: \(\Phi(s) \equiv \frac{\partial p}{\partial \eta} \in \mathbb{R}^{3 \times m}\). This is the matrix of mode shapes, mapping generalized modal rates \(\dot{\eta}\) to spatial velocities.

Thus, the velocity field \(v(s)\) is the vector superposition of rigid-body transport and local elastic deformation: \[v(s) = \underbrace{J_{\text{rigid}}(s, q) \dot{q}}_{\text{Rigid Transport}} + \underbrace{\Phi(s) \dot{\eta}}_{\text{Elastic Velocity}}.\]

The total kinetic energy of the club, \(T_{\text{club}}\), is the sum of the distributed shaft kinetic energy and the discrete kinetic energy of the clubhead at the tip (\(s=L\)). Letting \(dm = \rho(s)ds\), and denoting the clubhead mass and inertia as \(m_{head}\) and \(I_{head}\): \[ T_{\text{club}} = \underbrace{\frac{1}{2} \int_0^L \rho(s) \| v(s) \|^2 ds}_{\text{Shaft}} + \underbrace{\frac{1}{2} m_{head} \| v(L) \|^2 + \frac{1}{2} \omega(L)^T I_{head} \omega(L)}_{\text{Clubhead}}. \]

We expand the squared norm of the velocity vector explicitly. Let \(J\) denote the configuration-dependent rigid Jacobian \(J_{\text{rigid}}(s,q)\) and \(\Phi\) denote the spatial mode shape matrix \(\Phi(s)\). The squared norm is the inner product of the velocity vector with itself: \[ \| v(s) \|^2 = v(s)^T v(s) = (J\dot{q} + \Phi\dot{\eta})^T (J\dot{q} + \Phi\dot{\eta}). \] Distributing the transpose and expanding the terms: \[ \| v(s) \|^2 = (\dot{q}^T J^T + \dot{\eta}^T \Phi^T) (J\dot{q} + \Phi\dot{\eta}) \] \[ = \dot{q}^T J^T J \dot{q} + \dot{q}^T J^T \Phi \dot{\eta} + \dot{\eta}^T \Phi^T J \dot{q} + \dot{\eta}^T \Phi^T \Phi \dot{\eta}. \] Since the kinetic energy is a scalar quantity, the cross-terms must be scalars. The transpose of a scalar is itself (\(a = a^T\)), so the two middle terms are identical: \[ (\dot{q}^T J^T \Phi \dot{\eta})^T = \dot{\eta}^T \Phi^T J \dot{q}. \] Thus, we can group them as \(2 \dot{q}^T J^T \Phi \dot{\eta}\). Substituting this back into the integral (and adding the discrete tip terms) and distributing the integration operator over the sum: \[ T_{\text{club}} = \frac{1}{2} \dot{q}^T \left( \int_0^L \rho J^T J \, ds + m_{head} J(L)^T J(L) \right) \dot{q} + \dot{q}^T \left( \int_0^L \rho J^T \Phi \, ds + m_{head} J(L)^T \Phi(L) \right) \dot{\eta} + \frac{1}{2} \dot{\eta}^T \left( \int_0^L \rho \Phi^T \Phi \, ds + m_{head} \Phi(L)^T \Phi(L) \right) \dot{\eta}. \] (Note: For brevity, we omit the rotational inertia terms of the clubhead in the expanded block form, though they follow the same structure using angular velocity Jacobians.)

We define the block mass matrices based on these integrals augmented with the discrete tip mass: \[ M_{qq} \equiv \int_0^L \rho(s) J_{\text{rigid}}(s,q)^T J_{\text{rigid}}(s,q) \, ds + m_{head} J_{\text{rigid}}(L,q)^T J_{\text{rigid}}(L,q), \] \[ M_{q\eta} \equiv \int_0^L \rho(s) J_{\text{rigid}}(s,q)^T \Phi(s) \, ds + m_{head} J_{\text{rigid}}(L,q)^T \Phi(L), \] \[ M_{\eta\eta} \equiv \int_0^L \rho(s) \Phi(s)^T \Phi(s) \, ds + m_{head} \Phi(L)^T \Phi(L). \]

This derivation rigorously identifies the mass matrix blocks as projections of the distributed mass properties onto the generalized coordinate basis:

  1. Rigid Inertia \(M_{qq} = \int \rho J^T J ds\): The standard manipulator inertia, augmented by the “locked-mode” inertia of the shaft.
  2. Modal Inertia \(M_{\eta\eta} = \int \rho \Phi^T \Phi ds\): The mass matrix of the flexible coordinates (typically normalized to identity \(I_m\) via mode orthogonality).
  3. Inertial Coupling \(M_{q\eta} = \int \rho J^T \Phi ds\): The cross-term that couples the rigid and flexible subspaces.

The existence of the non-zero block \(M_{q\eta}\) is the physical mechanism by which hand acceleration (\(\ddot{q}\)) creates an instantaneous inertial load on the shaft (via \(-M_{q\eta}^T \ddot{q}\)), and conversely, how shaft vibration recoil exerts reaction torques on the hands. This coupling exists purely due to the mass distribution of the shaft and is independent of stiffness.

This inertial coupling transmits the effect of the declared generalized-torque channel to the club and transmits club reaction loads through the same model. To represent that bidirectional energy flow, we move from kinematics to dynamics and construct the system Lagrangian.

Notation Table

For reference, the following table summarizes the main symbols used in the manuscript.

Summary of main symbols {#tab:notation}
Symbol Meaning
\(q \in \mathbb{R}^{n}\) Generalized joint coordinates (rigid segments)
\(\dot{q}, \ddot{q}\) Joint angular velocities and accelerations
\(\eta \in \mathbb{R}^{m}\) Modal coordinates of the flexible shaft
\(\dot{\eta}, \ddot{\eta}\) Modal velocities and accelerations
\(w(s, t)\) Local shaft deformation field relative to the hand frame
\(x \in \mathbb{R}^{2(n+m)}\) Full system state: \(x = [q,\dot{q},\eta,\dot{\eta}]^{T}\)
\(u \in \mathbb{R}^{m_{u}}\) Control input vector (abstracted muscular torques)
\(\tau \in \mathbb{R}^{n}\) Generalized torque vector: \(\tau = B(q) u\)
\(B(q)\) Input mapping matrix from control channels to generalized torques
\(M(q,\eta)\) Inertia matrix for coupled rigid–flexible system
\(C(q,\dot{q},\eta,\dot{\eta})\) Coriolis/centrifugal matrix
\(g(q)\) Gravitational torque vector
\(K_{s}\) Modal stiffness matrix of the shaft
\(C_{s}\) Modal damping matrix of the shaft
\(f(x)\) Drift vector field (passive dynamics)
\(G(x)\) Input vector fields defining linear influence of torque inputs
\(F_{\text{drift}}\) Generalized drift force (passive component)
\(F_{\text{input}}\) Generalized input force (torque-driven component)
\(F_{\text{total}}\) Total generalized force
ZTCF Zero Torque Counterfactual
ZVCF Zero Velocity Counterfactual
\(\phi_{i}\) Coefficients associated with shaft mode shapes
\(t\) Time

Unified Control-Affine Derivation (Rigid + Flexible System)

To identify the model-conditioned contribution of the declared generalized-torque channel, we derive the equations of motion in a form that explicitly separates the input term from the autonomous dynamics. We employ the Lagrangian formalism to generate the dynamics, preserve the usual energy bookkeeping of the model, and then transform the result into control-affine state space form to isolate the input channels.

Lagrangian Formulation

We define the composite configuration vector \(q_{\text{sys}} = [q^T, \eta^T]^T \in \mathbb{R}^{n+m}\). The system dynamics are governed by the Euler-Lagrange equations: \[ \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}_{\text{sys}}}\right) - \frac{\partial \mathcal{L}}{\partial q_{\text{sys}}} + \frac{\partial \mathcal{R}}{\partial \dot{q}_{\text{sys}}} = \tau_{\text{ext}}, \] where \(\mathcal{L}(q_{\text{sys}}, \dot{q}_{\text{sys}}) = T - V\) is the Lagrangian and \(\mathcal{R}\) is the Rayleigh dissipation function.

1. Kinetic Energy and the Inertia Tensor

The total kinetic energy \(T\) is the integral of the squared velocity over the system’s mass distribution. As derived in the kinematic interface section, the velocity is linear in \(\dot{q}_{\text{sys}}\), implying \(T\) is a quadratic form: \[T(q_{\text{sys}}, \dot{q}_{\text{sys}}) = \frac{1}{2} \dot{q}_{\text{sys}}^T M(q_{\text{sys}}) \dot{q}_{\text{sys}}.\] The inertia matrix \(M(q_{\text{sys}})\) (the Riemannian metric of the configuration manifold) exhibits a dense block structure due to the kinematic coupling: \[M(q, \eta) = \begin{bmatrix} M_{qq}(q, \eta) & M_{q\eta}(q, \eta) \\ M_{\eta q}(q, \eta) & M_{\eta\eta} \end{bmatrix}.\] Here: * \(M_{qq} \in \mathbb{R}^{n \times n}\) is the effective inertia of the rigid segments, augmented by the instantaneous “locked-mode” inertia of the shaft. * \(M_{\eta\eta} \in \mathbb{R}^{m \times m}\) is the modal mass matrix (typically identity, \(I\), by orthonormal mode normalization). * \(M_{q\eta} = M_{\eta q}^T \in \mathbb{R}^{n \times m}\) is the inertial coupling matrix. This term captures the “recoil” forces: it maps shaft accelerations to torques at the joints and joint accelerations to modal forces. In a Newton-Euler formulation, these terms correspond exactly to the d’Alembert forces propagated back up the chain from the accelerating flexible element.

2. Coriolis and Centrifugal Fields

Expanding the time derivative \(\frac{d}{dt}(M \dot{q}_{\text{sys}})\) yields velocity-dependent inertial forces: \[\frac{d}{dt}(M \dot{q}_{\text{sys}}) = M \ddot{q}_{\text{sys}} + \dot{M} \dot{q}_{\text{sys}}.\] Combined with the partial derivative \(-\frac{\partial T}{\partial q_{\text{sys}}}\), these form the Coriolis and centrifugal matrix \(C(q_{\text{sys}}, \dot{q}_{\text{sys}})\), defined component-wise by Christoffel symbols of the first kind: \[C_{kj} = \sum_{i=1}^{n+m} \frac{1}{2} \left( \frac{\partial M_{kj}}{\partial q_i} + \frac{\partial M_{ki}}{\partial q_j} - \frac{\partial M_{ij}}{\partial q_k} \right) \dot{q}_i.\] A critical property for stability analysis is that \(\dot{M} - 2C\) is skew-symmetric, reflecting the conservation of energy in the absence of work-doing forces.

3. Potentials and Dissipation

The potential energy \(V = V_g(q, \eta) + V_{elastic}(\eta)\) leads to the gravitational vector \(G(q_{\text{sys}}) = \nabla V_g\) and the elastic restoring force \(K_s \eta = \nabla V_{elastic}\). We assume linear modal damping via the Rayleigh function \(\mathcal{R} = \frac{1}{2} \dot{\eta}^T C_s \dot{\eta}\).

4. The Underactuated Equation of Motion

Assembling these terms yields the standard second-order differential equation: \[ \begin{bmatrix} M_{qq} & M_{q\eta} \\ M_{\eta q} & M_{\eta\eta} \end{bmatrix} \begin{bmatrix} \ddot{q} \\ \ddot{\eta} \end{bmatrix} + C(q_{\text{sys}}, \dot{q}_{\text{sys}}) \dot{q}_{\text{sys}} + G(q_{\text{sys}}) + \begin{bmatrix} \tau_{\text{pas}}(q, \dot{q}) \\ K_s \eta + C_s \dot{\eta} \end{bmatrix} = \begin{bmatrix} \tau \\ 0 \end{bmatrix}. \]

We explicitly include the term \(\tau_{\text{pas}}(q, \dot{q})\) to model the passive joint impedance (ligament stiffness, joint friction, and intrinsic muscle viscoelasticity). While biological impedance is technically variable, for the purpose of the affine decomposition we treat it as a property of the ‘Effective Plant’.

NoteNote on Biological Impedance (The “Frozen Strategy” Assumption)

A common critique is that muscle stiffness is input-dependent (\(K(u)\)), which would violate the affine structure. We defend this by adopting the Frozen Strategy assumption: we define the “Drift” \(f(x)\) as the dynamics of the system with the golfer’s impedance fixed at its operational level, but with the net driving torque removed (\(u=0\)). This distinguishes the Effective Plant (which acts like a stiff, damped mechanism) from a “flaccid” ragdoll, ensuring the mathematical drift corresponds to the physically relevant “Zero Torque” baseline used in our simulations.

The zero vector in the lower block of the forcing term is the defining feature of this system: it is underactuated. The golfer has no direct actuator on the shaft modes; control authority over \(\eta\) is achieved solely through dynamic coupling (the \(M_{q\eta}\) term).

Control-Affine State Space Form

To perform the drift–input decomposition, we solve for the accelerations, inverting the inertia matrix to expose the additive terms assigned by the chosen equations.

To isolate the accelerations, we require the inverse inertia matrix \(H(q) = M^{-1}(q)\). We derive the blockwise inverse explicitly using the Schur Complement decomposition. This step is critical because it reveals how the motion of the rigid golfer (\(q\)) is dynamically coupled to the motion of the flexible shaft (\(\eta\)).

We seek a matrix \(H\) partitioned conformably with \(M\): \[ \begin{bmatrix} M_{qq} & M_{q\eta} \\ M_{\eta q} & M_{\eta\eta} \end{bmatrix} \begin{bmatrix} H_{qq} & H_{q\eta} \\ H_{\eta q} & H_{\eta\eta} \end{bmatrix} = \begin{bmatrix} I_n & 0 \\ 0 & I_m \end{bmatrix}. \]

Expanding the matrix multiplication for the first block column yields the system of linear equations: 1. \(M_{qq} H_{qq} + M_{q\eta} H_{\eta q} = I_n\) 2. \(M_{\eta q} H_{qq} + M_{\eta\eta} H_{\eta q} = 0\)

We solve for the blocks \(H_{qq}\) and \(H_{\eta q}\). From equation (2), assuming \(M_{\eta\eta}\) is positive-definite (which holds since \(M\) is a Riemannian metric), we isolate \(H_{\eta q}\): \[ M_{\eta\eta} H_{\eta q} = - M_{\eta q} H_{qq} \quad \implies \quad H_{\eta q} = - M_{\eta\eta}^{-1} M_{\eta q} H_{qq}. \]

This algebraic relationship reveals that the cross-mobility \(H_{\eta q}\) is not an independent property; it is a linear mapping of the rigid-body mobility \(H_{qq}\), determined by the matrix ratio: \[ \Gamma \equiv - M_{\eta\eta}^{-1} M_{\eta q}. \] We define \(\Gamma \in \mathbb{R}^{m \times n}\) as the Inertial Coupling Ratio. It acts as the system’s mechanical ‘gear ratio,’ dictating how much acceleration is transmitted to the flexible modes for every unit of acceleration the golfer produces at the joints. Crucially, this transmission depends solely on the mass distribution (\(M_{q\eta}\) and \(M_{\eta\eta}\)), making it a fixed property of the club’s design that the golfer cannot alter during the swing.

Substituting the expression for \(H_{\eta q}\) into equation (1): \[ M_{qq} H_{qq} + M_{q\eta} \left( - M_{\eta\eta}^{-1} M_{\eta q} H_{qq} \right) = I_n. \] Factoring out \(H_{qq}\) on the right: \[ \left( M_{qq} - M_{q\eta} M_{\eta\eta}^{-1} M_{\eta q} \right) H_{qq} = I_n. \] The term in parentheses is the Schur Complement of \(M_{\eta\eta}\) in \(M\), denoted \(\Delta\): \[ \Delta \equiv M_{qq} - M_{q\eta} M_{\eta\eta}^{-1} M_{\eta q}. \] Physically, \(\Delta\) represents the Articulated Body Inertia projected into the joint space. It is the effective inertia felt by the golfer’s joints when the shaft modes are free to accelerate. This value is strictly less than the “locked-mode” inertia \(M_{qq}\) (in the sense that \(M_{qq} - \Delta\) is positive semi-definite), reflecting the fact that the shaft “gives way” under load, reducing the resistance to hand acceleration.

Thus, the primary diagonal block of the inverse mass matrix is the inverse of the articulated inertia: \[ H_{qq} = \Delta^{-1} = \left( M_{qq} - M_{q\eta} M_{\eta\eta}^{-1} M_{\eta q} \right)^{-1}. \]

This derivation reveals the mechanical structure of the transmission. \(H_{qq}\) represents the Effective Mobility of the joints after accounting for the inertial relief provided by the flexible shaft.

Multiplying the equation of motion by \(H\) isolates the acceleration vector. Let \(h(x)\) denote the vector of all passive forces (Coriolis, gravity, stiffness, damping): \[ h(x) = C(q,\dot{q},\eta,\dot{\eta}) \dot{q}_{\text{sys}} + g(q) + \begin{bmatrix} 0 \\ K_s \eta + C_s \dot{\eta} \end{bmatrix}. \] The acceleration equation is: \[ \begin{bmatrix} \ddot{q} \\ \ddot{\eta} \end{bmatrix} = \underbrace{-M^{-1} h(x)}_{\text{Passive Drift Acceleration } a_{\text{drift}}} + \underbrace{M^{-1} \begin{bmatrix} B(q) \\ 0 \end{bmatrix}}_{\text{Control Efficacy } A_{\text{input}}} u. \]

We examine the Input Acceleration term \(A_{\text{input}} u\) explicitly using the block inverse: \[ \begin{bmatrix} \ddot{q} \\ \ddot{\eta} \end{bmatrix}_{\text{input}} = \begin{bmatrix} H_{qq} & H_{q\eta} \\ H_{\eta q} & H_{\eta\eta} \end{bmatrix} \begin{bmatrix} B(q)u \\ 0 \end{bmatrix} = \begin{bmatrix} H_{qq} B(q) u \\ H_{\eta q} B(q) u \end{bmatrix}. \]

Substituting the derived expression \(H_{\eta q} = \Gamma H_{qq}\): \[ \ddot{\eta}_{\text{input}} = \Gamma H_{qq} B(q) u = \left( - M_{\eta\eta}^{-1} M_{\eta q} \Delta^{-1} \right) B(q) u. \]

This equation provides the rigorous mechanical definition of Input Forces:

  1. Direct Drive: \(\ddot{q}_{in} = H_{qq} B(q) u\). The input torque generates joint acceleration scaled by the effective mobility \(H_{qq}\) (the inverse of the articulated body inertia).
  2. Transmission to Flexible Modes: \(\ddot{\eta}_{in}\) is driven by the joint acceleration \(\ddot{q}_{in}\) transmitted through the Inertial Coupling Ratio \(\Gamma\). Within this model, the golfer does not push the clubhead directly; they accelerate the hands, and the clubhead response is mediated by the inertial coupling \(M_{q\eta}\).

This explicit derivation proves the Mechanical Separability of the system (also called additive decomposition): the golfer’s only “handle” on the system is the torque vector \(u\), and its influence is strictly constrained by the matrices \(H_{qq}\) and \(\Gamma\). The term \(H_{qq} B(q)\) represents the instantaneous Control Efficacy—or Mobility Ellipsoid—which defines the geometric capacity of torque to alter the trajectory.

Substituting the actuation model, we separate the drift and input terms.

Define the drift acceleration \(a_{\text{drift}}(x)\) explicitly in terms of the block inverse \(H\): \[ a_{\text{drift}}(x) = - \begin{bmatrix} H_{qq} & H_{q\eta} \\ H_{\eta q} & H_{\eta\eta} \end{bmatrix} \left( C(x) \dot{q}_{\text{sys}} + G(x) + \begin{bmatrix} \tau_{\text{pas}}(q, \dot{q}) \\ K_s \eta + C_s \dot{\eta} \end{bmatrix} \right). \]

And the input acceleration mapping \(A_{\text{input}}(x)\): \[ A_{\text{input}}(x) = \begin{bmatrix} H_{qq} B(q) \\ H_{\eta q} B(q) \end{bmatrix}. \]

Then the acceleration equation becomes \[ \begin{bmatrix} \ddot{q} \\ \ddot{\eta} \end{bmatrix} = a_{\text{drift}}(x) + A_{\text{input}}(x)\,u. \]

This explicit separation of the acceleration vector is the kinematic foundation for the model-conditioned analysis. The term \(A_{\text{input}}(x)u\) is the instantaneous acceleration contribution of the declared generalized-torque channel in these coordinates. It is not, without further evidence, a measure of power, effort, intent, or individual-muscle action. In Part III, we classify this term relative to the Drift Terms generated by \(a_{\text{drift}}(x)\).

First-Order State-Space Form

We now express the dynamics in first-order form. Recall

\[x = [q^{T}, \dot{q}^{T}, \eta^{T}, \dot{\eta}^{T}]^{T}.\]

The state derivative is

\[\dot{x} = \begin{bmatrix} \dot{q} \\[0.2em] \ddot{q} \\[0.2em] \dot{\eta} \\[0.2em] \ddot{\eta} \end{bmatrix} = f(x) + G(x)\,u,\]

where

\[f(x) = \begin{bmatrix} \dot{q} \\ [a_{\text{drift}}(x)]_{1:n} \\ \dot{\eta} \\ [a_{\text{drift}}(x)]_{n+1:n+m} \end{bmatrix}, \qquad G(x) = \begin{bmatrix} 0 \\ [A_{\text{input}}(x)]_{1:n,:} \\ 0 \\ [A_{\text{input}}(x)]_{n+1:n+m,:} \end{bmatrix}.\]

Here \([\cdot]_{1:n}\) and \([\cdot]_{n+1:n+m}\) denote the rigid and flexible components of the acceleration, and \([\cdot]_{:, :}\) denotes the appropriate block of the input matrix.

This is a nonlinear control-affine system:

\[\dot{x} = f(x) + G(x)\,u.\]

All nonlinearities—including those introduced by shaft flexibility—are contained in the drift vector field \(f(x)\), while the input vector fields \(G(x)\) multiply the control input \(u\) linearly. The purely rigid-body model is recovered by setting \(m = 0\).

Structural Consequences

Three structural observations follow:

  1. Flexibility enriches drift but preserves input linearity. Shaft flexibility introduces additional passive degrees of freedom and modifies \(M, C,\) and the elastic terms, but all shaft forces remain in the drift. The control input still enters linearly through joint torques. This implies a fundamental underactuation: the golfer cannot directly command the shaft modes. Instead, the shaft dynamics act as an environment that must be managed. Mathematically, the shaft’s stiffness and damping are sequestered entirely within the drift vector field \(f(x)\), categorizing them rigorously as passive effects akin to gravity, rather than active outputs of the player.
  2. All torque-driven effects are confined to \(G(x)u\). Any change in the dynamics caused by altering the torque profile must pass through the input term; the drift term depends only on the state and parameters.
  3. The affine structure is independent of parameter values. The decomposition does not rely on particular numerical values of masses, inertias, or shaft stiffness, only on the linearity of torque in the equations of motion. In practice, parameter uncertainty affects the accuracy of the decomposition but not its algebraic form.

This control-affine structure is the backbone for the drift–input decomposition and the counterfactual constructions developed in the next sections.

By isolating the declared torque input \(u\) in \(G(x)u\), we obtain a mathematically precise operation. The term \(f(x)\) evaluates the autonomous dynamics represented at the current state, while \(G(x)u\) evaluates the instantaneous declared-input contribution.

The derivation above provides an additive, model-conditioned map of the instantaneous vector field. It assigns one term to the autonomous dynamics and one to the declared input channel. This algebraic separation does not identify player agency or a unique physical cause; those interpretations require an explicit intervention, identifiable parameters, and qualifying measurements.

Additivity does not imply orthogonality: \(f(x)\) and \(G(x)u\) may align, oppose, or be oblique under a chosen metric. The additive form permits a declared intervention \(u\equiv0\) while holding the stated plant, parameters, and initial state fixed. Because \(\dot{x}=f(x)+G(x)u\) describes only an instantaneous tendency, a trajectory-level counterfactual also requires a stated horizon and numerical integration. Part II defines those interventions without treating them as observations of intent or muscle behavior.

WarningModel Limitations

This Part I analysis is based on the mechanical model only and is subject to the following constraints:

  1. Rigid body segments: All anatomical segments are modeled as rigid bodies connected by ideal joints; soft-tissue compliance and marker dynamics are excluded.

  2. Constant grip impedance: The boundary condition at the grip is treated as a fixed mechanical constraint, not as a state-dependent or input-dependent stiffness parameter.

  3. Aerodynamic forces neglected: Air resistance on clubhead and shaft is not modeled. Aerodynamic loads that depend only on state and fixed parameters belong in \(f(x)\) and do not by themselves break control-affinity. Aerodynamic actuation or input-dependent flow models require a separately declared input map.

  4. Neural control delays not modeled: All torques are applied instantaneously; activation dynamics, sensorimotor delays, and muscle force-velocity relations are outside the model scope.

  5. The declared input level controls the affinity claim: At fixed state, \(\tau = \tau_{\max} f_{\text{FL}}(q) f_{\text{FV}}(\dot{q})a\) is linear in the declared input at fixed state when activation \(a\) is the input; the force–length and force–velocity factors form a state-dependent input map \(G(x)\). Such state-dependent input gains do not by themselves break control-affinity. Nonlinear activation dynamics in excitation, saturation, hysteresis, recruitment, or input-dependent impedance can require a different non-affine or hybrid model. This manuscript nevertheless declares net generalized torque as its mechanical input, and a state-dependent feasibility set \(\mathcal{U}(x)\) does not change that equation-level declaration.

  6. Co-contraction not captured: When antagonist muscles co-contract, they produce zero net torque but modulate joint stiffness in the null space of the torque map. This stiffness modulation is classified as part of the drift term \(f(x)\) under the Frozen Strategy assumption (see §Note on Biological Impedance), and is not separately identifiable from the drift without additional measurements or models.

Within these constraints, the affine decomposition is exact for the declared equations. Outside them, interpretation requires a revised model and new identification evidence. See also sources-of-nonlinearity.qmd for a detailed treatment of biological nonlinearities that fall outside this scope.