Force and Mobility Ellipsoids in the Golf Swing

Dieter Olson

Force and Mobility Ellipsoids in the Golf Swing

theory-core

What Jacobian ellipsoids reveal about motion and force, and why inertia, constraints, compliance and actuator limits must be treated separately.

Author

Dieter Olson

Published

November 28, 2025

Abstract

A small joint rotation can move the clubhead a considerable distance. The same lever arm means that a force at the clubhead can demand a large joint moment. These are two views of a geometric map, but neither determines acceleration, stiffness, tissue strength or the response to impact.

This article derives the velocity and load mappings with explicit units and budgets, examines a two-link example, and connects the result to dynamics, constraints and compliance. The worked cases are illustrative mechanics, not measurements of a golfer or proof of a swing technique.

Critics' Comments

These questions separate what the geometric construction proves from what a swing model must still establish:

Kinematic Limitation:

The Static Fallacy (Kinematic vs. Dynamic)

Does a large velocity radius imply that a small torque will quickly move the club? No. The velocity map answers a joint-rate question. Acceleration also depends on inertia, current motion, applied inputs, retained loads and constraints. The appropriate map depends on the outcome being investigated.

Our Response: The article derives the input-to-acceleration map separately and gives a counterexample where the preferred direction reverses. A force ellipsoid is a generalized-load budget mapped into task space; structural strength, stiffness and impact response require further models. DCR does not supply those missing quantities automatically.

Biological Reality:

The Biological Actuator Problem

Which force and rate budgets represent the golfer? Muscles have state-dependent force capacity, moment arms and activation dynamics. A conveniently chosen unit ball is not a measurement of those capabilities, and independently selected physiological force and velocity limits need not be reciprocal.

Our Response: Jacobian-transpose load mapping follows from the power pairing regardless of actuator physiology. The restriction lies in the chosen feasible load set and in interpreting its image. The derivation makes those budgets explicit; it does not infer muscle forces from net joint loads or treat mechanical advantage as an efficiency multiplier.

Model Fidelity:

Planar Reductionism vs. 3D Reality

What does the planar example omit? It cannot represent general clubface orientation, spatial shaft motion or all coupled upper-limb degrees of freedom. A two-dimensional projection cannot establish the relative importance of those omitted effects in a measured swing.

Our Response: We acknowledge the limits of 2D analogies. This is a pedagogical planar model chosen to make the Jacobian and ellipsoids visible; it omits out-of-plane motion, spatial coupling, face orientation, and three-dimensional hand--club constraints. Spatial Jacobians and ellipsoids can be constructed, but this projection does not establish a quantitative fidelity percentage for a three-dimensional golf swing. Any such percentage requires a governed three-dimensional comparison with a declared dataset, cohort, coordinates, phase window, energy definition, task-relevant outcome metrics, and uncertainty. Spatial interactions must therefore be analyzed and validated explicitly rather than inferred from the planar picture.

Note: These are model checks, not peer-review endorsements or completed human validation.

Introduction

A transmission exchanges force and motion through a geometric ratio. In an ideal transmission, that ratio conserves power; geometry is not an energy loss. Friction and other dissipative mechanisms require additional terms. A kinematic Jacobian maps joint velocity to task velocity, not force to motion. In vibration analysis, “mobility” can instead mean a frequency-dependent velocity/force response. That different quantity requires a dynamic model and a specified frequency.

The golf swing is the same. Your hands are one point, the clubhead is another. The geometry of your arms and the club acts like a transmission. At any instant during the swing, the configuration of these linked bodies (torso, arms, hands, club) imposes constraints on how the clubhead can move and how forces applied to the clubhead propagate back through the joints.

These instantaneous capabilities are captured compactly by two geometric objects:

the mobility ellipsoid, the task velocities produced by a specified joint-rate budget; and
the force ellipsoid, or more generally force preimage, the task loads compatible with a specified generalized-load budget in an idealized balance calculation.

Reciprocal radii require compatible dual budgets, not merely the same Jacobian. Inertia, actuator capacity, contact and compliance must be added before drawing conclusions about a golfer’s motion or load resistance.

Connection to AffineDrift Framework: The “Effective Plant”

Geometry determines how joint motion appears at a selected task and how task loads appear in generalized coordinates. Inertia determines acceleration; compliance and preload determine deformation; actuators and contacts restrict the feasible loads. The AffineDrift investigation needs these pieces together. An ellipsoid based on joint rates is not the torque-input map in a full control-affine state equation, and it does not measure drift dominance or biological intent. Whether a particular alignment helps a golfer is a separate empirical question.

The Jacobian: The Mapping Between Joint Space and Club Motion

Start with independent joint coordinates $q$ and a selected Cartesian point $p(q)$ in a stated inertial frame. Its velocity satisfies

\[ v_p=J_p(q)\dot q, \qquad J_p=\frac{\partial p}{\partial q}. \]

For a spatial task, one consistent ordering is $V=(v_p,\omega)$, paired with the wrench $W=(F,m_p)$ at the same point and in the same frame. Their pairing is power: $W^TV=F^Tv_p+m_p^T\omega$. Moment must be shifted when the reference point changes. Body or spatial twist conventions are equally usable when their velocity, Jacobian and wrench definitions match throughout.

A pose is not globally a vector in $\mathbb R^6$. A local orientation-chart derivative generally differs from angular velocity; its dual generalized moment must be transformed accordingly. Do not pair an analytic pose-coordinate Jacobian directly with a physical wrench without that conversion. Below, $V=J\dot q$ always denotes a consistent task velocity/load pairing.

Each column of $J$ describes the infinitesimal motion of the clubhead resulting from a unit velocity in a single joint.
Thus, the geometry of $J$ encodes the instantaneous mechanical leverage of the entire body at that moment.

The Mobility Ellipsoid

Write a joint-rate budget as $\dot q^T H\dot q\leq1$, where $H$ is positive definite and includes the scales needed to make the expression dimensionless. A diagonal choice can normalize coordinates by selected rate limits. This avoids arbitrarily treating one radian per second as equivalent to one metre per second when revolute and prismatic coordinates coexist.

The allowed velocity set and its shape matrix are

\[ \mathcal E_v=\{J\dot q:\dot q^T H\dot q\leq1\}, \qquad S_v=JH^{-1}J^T. \]

For full row rank, the set is $V^T S_v^{-1}V\leq1$. At deficient rank it lies in $\operatorname{range}(J)$: a pseudoinverse inequality without that range condition would incorrectly admit null-direction velocities. Use a ball including its interior. With redundant joints, the surface of a joint-rate sphere can map into the interior of the task image.

Before taking Euclidean eigenvalues of a six-component twist, choose a task normalization or analyze linear and angular components separately. Scaling metres to millimetres changes raw singular values, not the mechanism. Under $\widehat V=T V$, the dual load transforms as $\widehat W=T^{-T}W$ and the shape as $\widehat S_v=T S_vT^T$. Power is preserved; Euclidean principal axes depend on the metric and are not coordinate-free anatomical properties.

A large semi-axis means a large instantaneous task velocity per declared rate budget. It does not mean low inertia, low effort, a large reachable region by impact or high attained clubhead speed. Actual joint rates can point away from the largest axis. The current rates and momentum depend on earlier inputs and constraints; neither their values nor drift dominance follow from the Jacobian alone.

The Force Ellipsoid

A wrench on the modeled system produces the generalized load $Q_W=J^TW$. The identity does not assume ideal motors or muscle force–velocity behavior. Actuator load equals neither every external load nor the entire inertial load: gravity, inertia, passive loads and other contacts remain in the dynamics. To balance an external wrench $-W$ while preserving an otherwise specified motion, an ideal fully actuated model needs the additional actuator load $+J^TW$. This is the sign convention used in the publisher’s Modern Robotics statics treatment.

If the balancing actuator-load increment is constrained by $\tau^T R\tau\leq1$, its task-wrench preimage is

\[ \mathcal E_F=\{W:W^T J R J^TW\leq1\}. \]

This is a static or prescribed-motion transmission calculation. A contact environment must provide the balancing wrench; the set is not an arbitrary free-space force generator or a measure of structural stiffness.

The familiar reciprocal plot requires dual joint budgets. In consistently normalized coordinates with a common power scale, take $R=H^{-1}$. Then the force quadratic form equals $S_v$: eigenvalues $\lambda_i$ give velocity radii $\sqrt{\lambda_i}$ and force radii $1/\sqrt{\lambda_i}$. This is a polar-set construction, with maximum normalized load–velocity pairing one. The standard manipulability construction must retain its stated joint budgets and task-unit conventions.

Physical force and velocity have different units; their normalizations contain the power scale. Actual torque capacities are not generally reciprocals of the selected rate limits. For independent budgets, $JH^{-1}J^T$ and $JRJ^T$ need not share eigenvectors or reciprocal radii.

For the normalized map $J=\operatorname{diag}(2,1)$, a unit rate ball gives velocity radii $(2,1)$. With $R=\operatorname{diag}(0.01,1)$, the force radii are $(5,1)$. The same direction has the larger radius in both sets because different physical allowances were supplied. This does not claim simultaneous maximal torque and speed; their feasible joint combinations require an actuator model.

Baseline Loads and Other Feasible Sets

If supporting gravity already requires an actuator load $b$, the budget is $(b+J^TW)^TR(b+J^TW)\leq1$. The permitted load set can be offset or empty. For $J=2$, $b=1$ and $|\tau|\leq3$ in consistent scalar units, the admissible wrench interval is $[-2,1]$, rather than the zero-bias interval $[-1.5,1.5]$.

Velocity boxes map to zonotopes; torque boxes give intersections of slabs in wrench space. Unilateral muscle tension, activation history, torque-rate limits, friction and contact mode impose additional constraints. If an ellipsoid approximates these sets, disclose its fitting method and whether it bounds from inside or outside. No joint-load budget alone certifies tissue loads or a golfer’s resistance to impact.

Algorithm: Computing the Ellipsoids

Let $A$ be the normalized rate-to-task map, with Euclidean balls and dual load coordinates after the chosen metric transformations. Its full singular-value decomposition is \[ A=U\Sigma V^T. \] The matrix $\Sigma$ is rectangular when the task and joint dimensions differ. For numerical rank $r$, its retained singular values are positive. The first $r$ columns of $U$ span the retained task range; the remaining columns complete the task basis.

Ellipsoid	Defining set	Principal axes	Semi-axes
Mobility	Image of normalized rate ball	Retained columns of $U$	$\sigma_i$
Force	Preimage of normalized load ball	Same retained directions	$1/\sigma_i$

At deficient rank, the velocity image has zero radius in the remaining task directions and the force preimage is unbounded along $\ker(A^T)$. This is a cylinder, not a bounded force ellipsoid. It says the ideal generalized-load budget imposes no bound there. It does not say that infinite force is required, that infinite load is safe, or that arbitrary force can be commanded.

A reduced SVD of a tall matrix omits some ambient null directions. The example below returns all task directions and pads the radii. The relative tolerance $10^{-10}$ is an illustrative numerical choice, not a material or physiological threshold. Inspect the raw singular values and vary the tolerance according to model/data uncertainty. Truncating a small nonzero singular value changes the approximate model; it does not prove an exact physical singularity.

import numpy as np

def ellipsoid_axes(A, rtol=1e-10):
    """Return full task axes, raw singular values and truncated dual radii."""
    A = np.asarray(A, dtype=float)
    if A.ndim != 2 or min(A.shape) == 0 or not np.isfinite(A).all():
        raise ValueError("A must be a finite, nonempty matrix")
    if not np.isfinite(rtol) or not 0 <= rtol < 1:
        raise ValueError("rtol must lie in [0, 1)")
    U, sigma, _ = np.linalg.svd(A, full_matrices=True)
    mobility = np.zeros(A.shape[0])
    mobility[:sigma.size] = np.where(sigma > rtol * sigma[0], sigma, 0)
    force = np.full(A.shape[0], np.inf)
    np.divide(1, mobility, out=force, where=mobility > 0)
    return U, sigma, mobility, force

For the zero matrix, every velocity radius is zero and the force preimage is unbounded in all task directions. Singular vectors can flip sign or rotate within a repeated-value subspace without changing the set. Report metrics, scales, rank and null subspaces alongside an ellipsoid figure.

A Specified Two-Link Example

Take a fixed-base planar chain with two one-metre links, ideal revolute joints and an endpoint translation task. Angle $q_1$ is counterclockwise from the horizontal; $q_2$ is the second link’s angle relative to the first. Trigonometric arguments and joint rates use radians. Then

\[ p=\begin{bmatrix} \cos q_1+\cos(q_1+q_2)\\ \sin q_1+\sin(q_1+q_2) \end{bmatrix}\ \mathrm m, \]

\[ J_p=\begin{bmatrix} -\sin q_1-\sin(q_1+q_2)&-\sin(q_1+q_2)\\ \cos q_1+\cos(q_1+q_2)&\cos(q_1+q_2) \end{bmatrix}\ \mathrm{m/rad}. \]

At $(q_1,q_2)=(0.3,1.2)$, the singular values are approximately $1.864057$ and $0.500006\ \mathrm{m/rad}$. A Euclidean rate ball of radius $1\ \mathrm{rad/s}$ gives velocity radii $1.864057$ and $0.500006\ \mathrm{m/s}$. A separate torque ball of radius $1\ \mathrm{N\,m}$ gives balancing-force radii $0.536464$ and $1.999978\ \mathrm N$. These chosen scales give reciprocal numerical values; they are not measured human capacities.

At complete extension along the horizontal,

\[ J_p=\begin{bmatrix}0&0\\2&1\end{bmatrix}\ \mathrm{m/rad}. \]

The largest velocity radius under that rate ball is $\sqrt5\ \mathrm{m/s}$, not $2\ \mathrm{m/s}$: the maximizing rate uses both joints. The simpler relation $v_y=2\dot q_1$ fixes the second relative joint rate at zero. The tip-force load is $(2F_y,F_y)$ in consistent SI units. Under the unit torque ball, $|F_y|\leq1/\sqrt5\ \mathrm N$; $F_x$ is absent from that torque constraint.

An axial force lies in $\ker(J_p^T)$ at this ideal posture. It still loads links and bearings. Material stress, deformation, contact slip and buckling limits have not disappeared; the rigid-joint model simply does not bound them.

Lost Instantaneous Motion Is Not Permanent Immobility

Every endpoint velocity is transverse at extension. Nevertheless, if $q_1(t)=t$ and $q_2(t)=0$, then $p(t)=(2\cos t,2\sin t)\ \mathrm m$. At zero time, axial velocity is zero but axial acceleration is $-2\ \mathrm{m/s^2}$. This is the $\dot J_p\dot q$ term: the curved path has second-order axial displacement. It does not restore first-order axial velocity authority at the singular point.

Poor Jacobian conditioning does not prove dynamic instability. Stability requires a specified equilibrium or trajectory, loads, constraints and perturbation model. The same singular geometry can have different stability properties under different constitutive laws or loading conditions.

Connection to the Inclined Plane

The mechanical advantage of an inclined plane provides an even simpler analogy.
Consider pushing a mass up a ramp of angle $\theta$:

\[ F_{\parallel} = mg \sin\theta, \qquad F_{\perp} = mg \cos\theta. \]

These are the gravity components for a frictionless fixed ramp with no other normal loads. Quasistatic ascent needs the opposing tangential force; the ramp supplies the normal support. A shallower ramp reduces the ascent force but increases distance for the same height: $Fs=mgh$ in the ideal limit. Geometry has not dissipated energy.

The normal reaction is not a measured stiffness or a force-ellipsoid radius. Its physical capacity depends on the support and material. The analogy helps explain leverage and ideal constraint work, but it cannot establish passive stability of a golfer’s low-mobility direction.

Why Force and Mobility Ellipsoids Are Duals

The velocity and load mappings preserve the same instantaneous power pairing: \[ W^T V=Q_W^T\dot q. \]

Substituting $V=J\dot q$ gives \[ W^TJ\dot q=(J^TW)^T\dot q. \]

This identity holds for every allowed joint rate. Reciprocal ellipsoids add the dual-budget assumptions above; independently chosen physiological force and velocity limits need not be reciprocal.

Duality is not orthogonality of force and motion. Corresponding ellipsoids share principal directions. In a two-dimensional anisotropic case, their longest reciprocal axes happen to be orthogonal eigenvectors. In an isotropic case, the extreme direction is not unique. Aligned force and velocity can produce nonzero power; the transpose identity does not prohibit them.

Synthesis: Null Space and Parallel Mechanisms

Two hands attached to a club can close a kinematic loop when the arms and torso are included. Closure does not automatically mean redundant constraints or an overconstrained model. Count independent constraints and state which contacts permit slip, rotation or separation.

For fixed ideal bilateral constraints $C(q)\dot q=0$, let $N$ span $\ker C$. Then $\dot q=Nv$ and the reduced task map is $JN$. A reduced rate metric must be obtained consistently; it is not generally the identity. Constraint reactions lie in $\operatorname{range}(C^T)$ and do no work on the allowed rates. They are not the joint-velocity null space $\ker J$.

For contact forces $f_c$ mapped to a net club wrench by a grasp map $W_c=G_c f_c$, internal contact-force distributions lie in $\ker G_c$. Their feasibility still depends on friction, compressive contact and the hands’ ability to supply them. A zero resultant on the club does not imply zero load, muscle activity or deformation in the hands and arms. This map must be distinguished from both $J$ and the constraint matrix $C$.

Co-contraction can change impedance or capacity in an appropriate physiological model. At fixed $J$ and fixed load metric $R$, however, the quadratic form $JRJ^T$ is unchanged. A stiffness change does not literally inflate that force ellipsoid. A capacity change instead requires an updated admissible load set, whose values need evidence.

Dynamics, Compliance and Impact Need Additional Models

Acceleration Authority and Apparent Inertia

For an unconstrained mechanical model with positive-definite inertia $M$, velocity/gravity bias $h$, declared input $u$, input map $B$ and retained loads $Q_{\rm ext}$,

\[ M\ddot q+h=Bu+Q_{\rm ext}, \]

\[ \ddot p=JM^{-1}(Bu+Q_{\rm ext}-h)+\dot J\dot q. \]

The same-state input contribution is $JM^{-1}Bu$, not $J u$. Torque-box or torque-ellipsoid bounds therefore generate an acceleration set through a different map, translated by the retained acceleration. A full control-affine state map also includes the configuration-rate equations and any internal states. The instantaneous acceleration set is not a reachable set at impact.

As an algebraic counterexample in consistently normalized coordinates, take $J=\operatorname{diag}(2,1)$, $B=I$ and $M=\operatorname{diag}(100,0.25)$. Kinematic gain is larger in the first direction, while input-to-acceleration gains are $(0.02,4)$: the second direction now has much larger gain. These specified matrices illustrate why kinematic leverage alone cannot rank dynamic authority; they are not an identified human model.

An additional applied task wrench changes acceleration through $JM^{-1}J^T$. For full row rank, the apparent task inertia is

\[ \Lambda=(JM^{-1}J^T)^{-1}. \]

The square invertible case follows directly from task-space dynamics. For a rectangular full-row-rank map, the same positive quadratic form gives the instantaneous wrench-to-acceleration relation, but does not eliminate internal joint dynamics. Velocity and bias terms remain; an inertia ellipsoid is not a velocity-budget ellipsoid.

Under ideal contact constraints the free inverse inertia must be replaced by the response of the constrained equations. For independent active constraints, one formulation is

\[ \begin{bmatrix}M&-C^T\\C&0\end{bmatrix} \begin{bmatrix}\ddot q\\\lambda\end{bmatrix} =\begin{bmatrix}Bu+Q_{\rm ext}-h\\-\dot C\dot q\end{bmatrix}. \]

This fixed-contact mode still requires feasible reactions, friction and contact-maintenance conditions. Constraint changes require explicit mode transitions. See the publisher’s constrained-dynamics treatment for the separation of motion and ideal reaction loads.

For a selected translational direction $n$, an ideal frozen-configuration impulse model without additional constraints gives $m_{\rm eff}=1/(n^TJM^{-1}J^Tn)$ for a unit vector $n$ and a positive denominator. For independent ideal constraints that remain active through the impulse, replace $M^{-1}$ by \[ D_C=M^{-1}-M^{-1}C^T(CM^{-1}C^T)^{-1}CM^{-1}. \] This projects the velocity response into the permitted motion; an impending contact or a changing contact mode requires its own impulse balance. This scalar can be useful in contact analysis, but a force ellipsoid alone supplies neither it nor restitution, contact duration, dissipated energy or shaft vibration.

Small-Displacement Compliance

At an unloaded equilibrium, suppose joint springs have positive-definite stiffness $K_q$. Neglecting preload terms and using the same small-displacement Jacobian, an applied load increment gives

\[ \delta p=J K_q^{-1}J^T\delta W. \]

This compliance matrix has different units and inputs from the force budget’s quadratic form. With $J=\operatorname{diag}(2,1)$, joint stiffness matrices $\operatorname{diag}(4,1)$ and $\operatorname{diag}(16,1)$ give task compliances $\operatorname{diag}(1,1)$ and $\operatorname{diag}(0.25,1)$ in consistent units. Holding the torque budget fixed leaves its force ellipsoid unchanged while compliance changes by a factor of four.

Preload can add geometric stiffness. For a conservative dead load with potential $-W^T p(q)$, the tangent stiffness includes $-\sum_i W_i\nabla^2 p_i$. For the scalar example $p=q^2$, spring stiffness $10$ and dead load $3$, the local tangent stiffness is $10-2(3)=4$ in the corresponding normalized units. Using the unloaded spring value would miss the load’s contribution. Nonconservative loads, damping, feedback and contact compliance require their own linearization and stability analysis.

The practical connection is therefore conditional: posture changes geometric leverage; activation and material behavior can change load capacity and impedance; motion changes inertial and velocity-dependent effects. Their interaction must be modeled and checked, rather than inferred from one ellipse.

Connection to Drift-Control Decomposition

The force and mobility ellipsoids are more than geometric curiosities: they describe configuration-dependent mappings between generalized and task-space quantities. Mobility alone does not establish that drift dominates a direction. That comparison also requires the current state, applied-input bounds, reference point, and selected norm.

Near impact, a model may combine high task-space mobility with large velocity-dependent drift. Whether the available control is small or large relative to that drift is an empirical model result, not a consequence of the ellipsoid alone. The hand-path attribution study defines the same-state comparison needed to test it.

Under the dual-budget construction, a small velocity radius pairs with a large force radius. That geometric result is not a passive-stability certificate. Testing a golf interpretation requires a declared state, model, reference point, actuator/contact bounds, load or disturbance direction, and outcome horizon. Compare both the kinematic and dynamic sets, then assess sensitivity to measurement errors, coordinate scaling and plausible capacity models.

Conclusion

The Jacobian gives a precise connection between task velocity and generalized load. Ellipsoids add a chosen metric and feasible budget to that map. Their reciprocal construction is useful precisely because its assumptions can be stated and tested; it should not be extended into a universal physiological trade-off.

For the golf investigation, keep the chain of reasoning explicit: specify the task and reference point; derive the geometric map; add inertia, retained loads and contacts; model capacity and compliance; then evaluate the selected outcome over the relevant time interval. The two-link example explains a transmission geometry. Evidence about real release strategies, tissue loading, clubface stabilization or impact performance must come from that fuller model and appropriate measurements.

Scope and Limitations

The example is a rigid linkage. Flexible coordinates and their rates can be included in a more complete kinematic map. Compliance is not a breakdown of Jacobian mathematics; it changes the state, constitutive law and load response that must be modeled. The size of elastic effects requires evidence for the actual shaft, joints, loading and time scale.
Singular limits need a physical interpretation. Exact rank loss, numerical ill-conditioning and dynamic instability are different questions. Small singular values can make inversion sensitive; they do not erase the remaining task directions or prove a particular biological failure. An unbounded force preimage exposes omitted load limits rather than certifying unlimited support.
A capability picture is conditional. State uncertainty, task scaling, load offsets and admissible inputs affect the result. Compare sensitivity to these choices and disclose whether an ellipsoid is exact, fitted, projected or numerically truncated. No numerical human validation or ideal-posture recommendation is supplied here.

In Layman's Terms

A long lever can turn a small angular motion into a large tip motion while a force at its tip demands a substantial joint moment. Ellipsoids picture that geometry under specified budgets; they do not impose a universal choice between speed and strength.

A Velocity Budget

The mobility picture contains the tip velocities allowed by a chosen joint-rate budget at a posture. Its longest axis is not necessarily the direction of the actual swing, and it does not tell us how much torque is needed to reach that motion.

Think of it like: Moving the end of a long ruler with a small rotation. The displacement ratio is geometric; accelerating the ruler also depends on its mass and how it is driven.

A Load Budget

The force picture describes which balanced tip loads fit within a stated joint-effort allowance. Stiffness asks how much the system deforms under load; strength asks which loads it can tolerate. Neither is specified by this force picture alone.

Think of it like: A force through an ideal hinge produces no turning moment about that hinge. The bearing still carries the load, and its material limits still matter.

When Reciprocity Applies

Familiar reciprocal ellipses use matching mathematical budgets. Real rate and torque capacities can differ, so a direction can have large radii in both separately defined sets. Whether the corresponding force and speed can occur simultaneously is another actuator-capability question.

Think of it like: An ideal gear exchanges torque for speed while preserving power. Changing the motor's capacity changes what the complete machine can do; a gear ratio alone does not specify its power, stiffness or stability.

Key Takeaway: Declare the task, posture, units and budgets before interpreting an ellipse. These examples do not establish that elite golfers instantly switch from a speed posture to an impact-stability posture.

Real-World Analogies: Configuration-Dependent Behavior

Posture changes the geometric map. Motion and force capacity also depend on state, inertia, actuator behavior, contact and compliance. Two systems with the same geometry can therefore have different acceleration and load responses.

The same ruler with a different mass: Its rotation-to-tip-motion ratio can remain the same, while the torque needed for a selected acceleration changes. A motion ellipse cannot reveal that mass change by itself.

The same linkage with different springs: Changing spring stiffness changes its deformation under load without necessarily changing its geometric motion map or the chosen torque budget. To understand the swing, evaluate these quantities together while keeping their meanings separate.

Citation

BibTeX citation: Download .bib

@online{olson2025,
  author = {Olson, Dieter},
  title = {Force and {Mobility} {Ellipsoids} in the {Golf} {Swing}},
  date = {2025-11-28},
  url = {https://affinedrift.com/articles/force-mobility-matrices.html},
  langid = {en}
}

For attribution, please cite this work as:

Olson, Dieter. 2025. “Force and Mobility Ellipsoids in the Golf Swing.” November 28. https://affinedrift.com/articles/force-mobility-matrices.html.

--- title: "Force and Mobility Ellipsoids in the Golf Swing" description: "What Jacobian ellipsoids reveal about motion and force, and why inertia, constraints, compliance and actuator limits must be treated separately." author: "Dieter Olson" date: "2025-11-28" date-source: "initial-publication-record" format: html: toc: true toc-depth: 3 number-sections: false code-fold: true css: - /css/technical-explanations.css - /css/force-mobility.css categories: - theory-core --- ::: {.abstract-section} ## Abstract A small joint rotation can move the clubhead a considerable distance. The same lever arm means that a force at the clubhead can demand a large joint moment. These are two views of a geometric map, but neither determines acceleration, stiffness, tissue strength or the response to impact. This article derives the velocity and load mappings with explicit units and budgets, examines a two-link example, and connects the result to dynamics, constraints and compliance. The worked cases are illustrative mechanics, not measurements of a golfer or proof of a swing technique. ::: ```{=html} <details class="technical-explanation"> <summary>Critics' Comments</summary> <p> These questions separate what the geometric construction proves from what a swing model must still establish: </p> <p>Kinematic Limitation:</p> <h3>The Static Fallacy (Kinematic vs. Dynamic)</h3> <p> Does a large velocity radius imply that a small torque will quickly move the club? No. The velocity map answers a joint-rate question. Acceleration also depends on inertia, current motion, applied inputs, retained loads and constraints. The appropriate map depends on the outcome being investigated. </p> <p><strong>Our Response:</strong> The article derives the input-to-acceleration map separately and gives a counterexample where the preferred direction reverses. A force ellipsoid is a generalized-load budget mapped into task space; structural strength, stiffness and impact response require further models. DCR does not supply those missing quantities automatically.</p> <p>Biological Reality:</p> <h3>The Biological Actuator Problem</h3> <p> Which force and rate budgets represent the golfer? Muscles have state-dependent force capacity, moment arms and activation dynamics. A conveniently chosen unit ball is not a measurement of those capabilities, and independently selected physiological force and velocity limits need not be reciprocal. </p> <p><strong>Our Response:</strong> Jacobian-transpose load mapping follows from the power pairing regardless of actuator physiology. The restriction lies in the chosen feasible load set and in interpreting its image. The derivation makes those budgets explicit; it does not infer muscle forces from net joint loads or treat mechanical advantage as an efficiency multiplier.</p> <p>Model Fidelity:</p> <h3>Planar Reductionism vs. 3D Reality</h3> <p> What does the planar example omit? It cannot represent general clubface orientation, spatial shaft motion or all coupled upper-limb degrees of freedom. A two-dimensional projection cannot establish the relative importance of those omitted effects in a measured swing. </p> <p><strong>Our Response:</strong> We acknowledge the limits of 2D analogies. This is a pedagogical planar model chosen to make the Jacobian and ellipsoids visible; it omits out-of-plane motion, spatial coupling, face orientation, and three-dimensional hand--club constraints. Spatial Jacobians and ellipsoids can be constructed, but this projection does not establish a quantitative fidelity percentage for a three-dimensional golf swing. Any such percentage requires a governed three-dimensional comparison with a declared dataset, cohort, coordinates, phase window, energy definition, task-relevant outcome metrics, and uncertainty. Spatial interactions must therefore be analyzed and validated explicitly rather than inferred from the planar picture.</p> <p><strong>Note:</strong> These are model checks, not peer-review endorsements or completed human validation.</p> </details> ``` ## Introduction A transmission exchanges force and motion through a geometric ratio. In an ideal transmission, that ratio conserves power; geometry is not an energy loss. Friction and other dissipative mechanisms require additional terms. A kinematic Jacobian maps **joint velocity to task velocity**, not force to motion. In vibration analysis, "mobility" can instead mean a frequency-dependent velocity/force response. That different quantity requires a dynamic model and a specified frequency. The golf swing is the same. Your hands are one point, the clubhead is another. The geometry of your arms and the club acts like a transmission. At any instant during the swing, the configuration of these linked bodies (torso, arms, hands, club) imposes constraints on how the clubhead can move and how forces applied to the clubhead propagate back through the joints. These instantaneous capabilities are captured compactly by two geometric objects: - the **mobility ellipsoid**, the task velocities produced by a specified joint-rate budget; and - the **force ellipsoid**, or more generally force preimage, the task loads compatible with a specified generalized-load budget in an idealized balance calculation. Reciprocal radii require compatible dual budgets, not merely the same Jacobian. Inertia, actuator capacity, contact and compliance must be added before drawing conclusions about a golfer's motion or load resistance. ::: {.callout-note} ## Connection to AffineDrift Framework: The "Effective Plant" Geometry determines how joint motion appears at a selected task and how task loads appear in generalized coordinates. Inertia determines acceleration; compliance and preload determine deformation; actuators and contacts restrict the feasible loads. The AffineDrift investigation needs these pieces together. An ellipsoid based on joint rates is not the torque-input map in a full control-affine state equation, and it does not measure drift dominance or biological intent. Whether a particular alignment helps a golfer is a separate empirical question. ::: ## The Jacobian: The Mapping Between Joint Space and Club Motion Start with independent joint coordinates $q$ and a selected Cartesian point $p(q)$ in a stated inertial frame. Its velocity satisfies $$ v_p=J_p(q)\dot q, \qquad J_p=\frac{\partial p}{\partial q}. $$ For a spatial task, one consistent ordering is $V=(v_p,\omega)$, paired with the wrench $W=(F,m_p)$ at the **same point** and in the **same frame**. Their pairing is power: $W^TV=F^Tv_p+m_p^T\omega$. Moment must be shifted when the reference point changes. Body or spatial twist conventions are equally usable when their velocity, Jacobian and wrench definitions match throughout. A pose is not globally a vector in $\mathbb R^6$. A local orientation-chart derivative generally differs from angular velocity; its dual generalized moment must be transformed accordingly. Do not pair an analytic pose-coordinate Jacobian directly with a physical wrench without that conversion. Below, $V=J\dot q$ always denotes a consistent task velocity/load pairing. Each column of $J$ describes the infinitesimal motion of the clubhead resulting from a unit velocity in a single joint. Thus, the geometry of $J$ encodes the instantaneous mechanical leverage of the entire body at that moment. ## The Mobility Ellipsoid Write a joint-rate budget as $\dot q^T H\dot q\leq1$, where $H$ is positive definite and includes the scales needed to make the expression dimensionless. A diagonal choice can normalize coordinates by selected rate limits. This avoids arbitrarily treating one radian per second as equivalent to one metre per second when revolute and prismatic coordinates coexist. The allowed velocity set and its shape matrix are $$ \mathcal E_v=\{J\dot q:\dot q^T H\dot q\leq1\}, \qquad S_v=JH^{-1}J^T. $$ For full row rank, the set is $V^T S_v^{-1}V\leq1$. At deficient rank it lies in $\operatorname{range}(J)$: a pseudoinverse inequality without that range condition would incorrectly admit null-direction velocities. Use a ball including its interior. With redundant joints, the surface of a joint-rate sphere can map into the interior of the task image. Before taking Euclidean eigenvalues of a six-component twist, choose a task normalization or analyze linear and angular components separately. Scaling metres to millimetres changes raw singular values, not the mechanism. Under $\widehat V=T V$, the dual load transforms as $\widehat W=T^{-T}W$ and the shape as $\widehat S_v=T S_vT^T$. Power is preserved; Euclidean principal axes depend on the metric and are not coordinate-free anatomical properties. A large semi-axis means a large instantaneous task velocity **per declared rate budget**. It does not mean low inertia, low effort, a large reachable region by impact or high attained clubhead speed. Actual joint rates can point away from the largest axis. The current rates and momentum depend on earlier inputs and constraints; neither their values nor drift dominance follow from the Jacobian alone. ## The Force Ellipsoid A wrench **on the modeled system** produces the generalized load $Q_W=J^TW$. The identity does not assume ideal motors or muscle force–velocity behavior. Actuator load equals neither every external load nor the entire inertial load: gravity, inertia, passive loads and other contacts remain in the dynamics. To balance an external wrench $-W$ while preserving an otherwise specified motion, an ideal fully actuated model needs the additional actuator load $+J^TW$. This is the sign convention used in the publisher's [Modern Robotics statics treatment](https://modernrobotics.northwestern.edu/nu-gm-book-resource/5-2-statics-of-open-chains/). If the balancing actuator-load increment is constrained by $\tau^T R\tau\leq1$, its task-wrench preimage is $$ \mathcal E_F=\{W:W^T J R J^TW\leq1\}. $$ This is a static or prescribed-motion transmission calculation. A contact environment must provide the balancing wrench; the set is not an arbitrary free-space force generator or a measure of structural stiffness. The familiar reciprocal plot requires **dual joint budgets**. In consistently normalized coordinates with a common power scale, take $R=H^{-1}$. Then the force quadratic form equals $S_v$: eigenvalues $\lambda_i$ give velocity radii $\sqrt{\lambda_i}$ and force radii $1/\sqrt{\lambda_i}$. This is a polar-set construction, with maximum normalized load–velocity pairing one. The [standard manipulability construction](https://modernrobotics.northwestern.edu/nu-gm-book-resource/5-4-manipulability/) must retain its stated joint budgets and task-unit conventions. Physical force and velocity have different units; their normalizations contain the power scale. Actual torque capacities are not generally reciprocals of the selected rate limits. For independent budgets, $JH^{-1}J^T$ and $JRJ^T$ need not share eigenvectors or reciprocal radii. For the normalized map $J=\operatorname{diag}(2,1)$, a unit rate ball gives velocity radii $(2,1)$. With $R=\operatorname{diag}(0.01,1)$, the force radii are $(5,1)$. The same direction has the larger radius in both sets because different physical allowances were supplied. This does not claim simultaneous maximal torque and speed; their feasible joint combinations require an actuator model. ### Baseline Loads and Other Feasible Sets If supporting gravity already requires an actuator load $b$, the budget is $(b+J^TW)^TR(b+J^TW)\leq1$. The permitted load set can be offset or empty. For $J=2$, $b=1$ and $|\tau|\leq3$ in consistent scalar units, the admissible wrench interval is $[-2,1]$, rather than the zero-bias interval $[-1.5,1.5]$. Velocity boxes map to zonotopes; torque boxes give intersections of slabs in wrench space. Unilateral muscle tension, activation history, torque-rate limits, friction and contact mode impose additional constraints. If an ellipsoid approximates these sets, disclose its fitting method and whether it bounds from inside or outside. No joint-load budget alone certifies tissue loads or a golfer's resistance to impact. ## Algorithm: Computing the Ellipsoids {#sec-ellipsoid-algorithm} Let $A$ be the normalized rate-to-task map, with Euclidean balls and dual load coordinates after the chosen metric transformations. Its full singular-value decomposition is $$ A=U\Sigma V^T. $$ The matrix $\Sigma$ is rectangular when the task and joint dimensions differ. For numerical rank $r$, its retained singular values are positive. The first $r$ columns of $U$ span the retained task range; the remaining columns complete the task basis. | Ellipsoid | Defining set | Principal axes | Semi-axes | |---|---|---|---| | **Mobility** | Image of normalized rate ball | Retained columns of $U$ | $\sigma_i$ | | **Force** | Preimage of normalized load ball | Same retained directions | $1/\sigma_i$ | At deficient rank, the velocity image has zero radius in the remaining task directions and the force preimage is unbounded along $\ker(A^T)$. This is a cylinder, not a bounded force ellipsoid. It says the ideal generalized-load budget imposes no bound there. It does **not** say that infinite force is required, that infinite load is safe, or that arbitrary force can be commanded. A reduced SVD of a tall matrix omits some ambient null directions. The example below returns all task directions and pads the radii. The relative tolerance $10^{-10}$ is an illustrative numerical choice, not a material or physiological threshold. Inspect the raw singular values and vary the tolerance according to model/data uncertainty. Truncating a small nonzero singular value changes the approximate model; it does not prove an exact physical singularity. ```python import numpy as np def ellipsoid_axes(A, rtol=1e-10): """Return full task axes, raw singular values and truncated dual radii.""" A = np.asarray(A, dtype=float) if A.ndim != 2 or min(A.shape) == 0 or not np.isfinite(A).all(): raise ValueError("A must be a finite, nonempty matrix") if not np.isfinite(rtol) or not 0 <= rtol < 1: raise ValueError("rtol must lie in [0, 1)") U, sigma, _ = np.linalg.svd(A, full_matrices=True) mobility = np.zeros(A.shape[0]) mobility[:sigma.size] = np.where(sigma > rtol * sigma[0], sigma, 0) force = np.full(A.shape[0], np.inf) np.divide(1, mobility, out=force, where=mobility > 0) return U, sigma, mobility, force ``` For the zero matrix, every velocity radius is zero and the force preimage is unbounded in all task directions. Singular vectors can flip sign or rotate within a repeated-value subspace without changing the set. Report metrics, scales, rank and null subspaces alongside an ellipsoid figure. ## A Specified Two-Link Example Take a fixed-base planar chain with two one-metre links, ideal revolute joints and an endpoint translation task. Angle $q_1$ is counterclockwise from the horizontal; $q_2$ is the second link's angle relative to the first. Trigonometric arguments and joint rates use radians. Then $$ p=\begin{bmatrix} \cos q_1+\cos(q_1+q_2)\\ \sin q_1+\sin(q_1+q_2) \end{bmatrix}\ \mathrm m, $$ $$ J_p=\begin{bmatrix} -\sin q_1-\sin(q_1+q_2)&-\sin(q_1+q_2)\\ \cos q_1+\cos(q_1+q_2)&\cos(q_1+q_2) \end{bmatrix}\ \mathrm{m/rad}. $$ At $(q_1,q_2)=(0.3,1.2)$, the singular values are approximately $1.864057$ and $0.500006\ \mathrm{m/rad}$. A Euclidean rate ball of radius $1\ \mathrm{rad/s}$ gives velocity radii $1.864057$ and $0.500006\ \mathrm{m/s}$. A separate torque ball of radius $1\ \mathrm{N\,m}$ gives balancing-force radii $0.536464$ and $1.999978\ \mathrm N$. These chosen scales give reciprocal numerical values; they are not measured human capacities. At complete extension along the horizontal, $$ J_p=\begin{bmatrix}0&0\\2&1\end{bmatrix}\ \mathrm{m/rad}. $$ The largest velocity radius under that rate ball is $\sqrt5\ \mathrm{m/s}$, not $2\ \mathrm{m/s}$: the maximizing rate uses both joints. The simpler relation $v_y=2\dot q_1$ fixes the second relative joint rate at zero. The tip-force load is $(2F_y,F_y)$ in consistent SI units. Under the unit torque ball, $|F_y|\leq1/\sqrt5\ \mathrm N$; $F_x$ is absent from that torque constraint. An axial force lies in $\ker(J_p^T)$ at this ideal posture. It still loads links and bearings. Material stress, deformation, contact slip and buckling limits have not disappeared; the rigid-joint model simply does not bound them. ### Lost Instantaneous Motion Is Not Permanent Immobility Every endpoint velocity is transverse at extension. Nevertheless, if $q_1(t)=t$ and $q_2(t)=0$, then $p(t)=(2\cos t,2\sin t)\ \mathrm m$. At zero time, axial velocity is zero but axial acceleration is $-2\ \mathrm{m/s^2}$. This is the $\dot J_p\dot q$ term: the curved path has second-order axial displacement. It does not restore first-order axial velocity authority at the singular point. Poor Jacobian conditioning does not prove dynamic instability. Stability requires a specified equilibrium or trajectory, loads, constraints and perturbation model. The same singular geometry can have different stability properties under different constitutive laws or loading conditions. ## Connection to the Inclined Plane The mechanical advantage of an inclined plane provides an even simpler analogy. Consider pushing a mass up a ramp of angle $\theta$: $$ F_{\parallel} = mg \sin\theta, \qquad F_{\perp} = mg \cos\theta. $$ These are the gravity components for a frictionless fixed ramp with no other normal loads. Quasistatic ascent needs the opposing tangential force; the ramp supplies the normal support. A shallower ramp reduces the ascent force but increases distance for the same height: $Fs=mgh$ in the ideal limit. Geometry has not dissipated energy. The normal reaction is not a measured stiffness or a force-ellipsoid radius. Its physical capacity depends on the support and material. The analogy helps explain leverage and ideal constraint work, but it cannot establish passive stability of a golfer's low-mobility direction. ## Why Force and Mobility Ellipsoids Are Duals The velocity and load mappings preserve the same instantaneous power pairing: $$ W^T V=Q_W^T\dot q. $$ Substituting $V=J\dot q$ gives $$ W^TJ\dot q=(J^TW)^T\dot q. $$ This identity holds for every allowed joint rate. Reciprocal **ellipsoids** add the dual-budget assumptions above; independently chosen physiological force and velocity limits need not be reciprocal. Duality is not orthogonality of force and motion. Corresponding ellipsoids share principal directions. In a two-dimensional anisotropic case, their longest reciprocal axes happen to be orthogonal eigenvectors. In an isotropic case, the extreme direction is not unique. Aligned force and velocity can produce nonzero power; the transpose identity does not prohibit them. ### Synthesis: Null Space and Parallel Mechanisms Two hands attached to a club can close a kinematic loop when the arms and torso are included. Closure does not automatically mean redundant constraints or an overconstrained model. Count independent constraints and state which contacts permit slip, rotation or separation. For fixed ideal bilateral constraints $C(q)\dot q=0$, let $N$ span $\ker C$. Then $\dot q=Nv$ and the reduced task map is $JN$. A reduced rate metric must be obtained consistently; it is not generally the identity. Constraint reactions lie in $\operatorname{range}(C^T)$ and do no work on the allowed rates. They are not the joint-velocity null space $\ker J$. For contact forces $f_c$ mapped to a net club wrench by a grasp map $W_c=G_c f_c$, internal contact-force distributions lie in $\ker G_c$. Their feasibility still depends on friction, compressive contact and the hands' ability to supply them. A zero resultant on the club does not imply zero load, muscle activity or deformation in the hands and arms. This map must be distinguished from both $J$ and the constraint matrix $C$. Co-contraction can change impedance or capacity in an appropriate physiological model. At fixed $J$ and fixed load metric $R$, however, the quadratic form $JRJ^T$ is unchanged. A stiffness change does not literally inflate that force ellipsoid. A capacity change instead requires an updated admissible load set, whose values need evidence. ## Dynamics, Compliance and Impact Need Additional Models ### Acceleration Authority and Apparent Inertia For an unconstrained mechanical model with positive-definite inertia $M$, velocity/gravity bias $h$, declared input $u$, input map $B$ and retained loads $Q_{\rm ext}$, $$ M\ddot q+h=Bu+Q_{\rm ext}, $$ $$ \ddot p=JM^{-1}(Bu+Q_{\rm ext}-h)+\dot J\dot q. $$ The same-state input contribution is $JM^{-1}Bu$, not $J u$. Torque-box or torque-ellipsoid bounds therefore generate an acceleration set through a different map, translated by the retained acceleration. A full control-affine state map also includes the configuration-rate equations and any internal states. The instantaneous acceleration set is not a reachable set at impact. As an algebraic counterexample in consistently normalized coordinates, take $J=\operatorname{diag}(2,1)$, $B=I$ and $M=\operatorname{diag}(100,0.25)$. Kinematic gain is larger in the first direction, while input-to-acceleration gains are $(0.02,4)$: the second direction now has much larger gain. These specified matrices illustrate why kinematic leverage alone cannot rank dynamic authority; they are not an identified human model. An additional applied task wrench changes acceleration through $JM^{-1}J^T$. For full row rank, the apparent task inertia is $$ \Lambda=(JM^{-1}J^T)^{-1}. $$ The square invertible case follows directly from [task-space dynamics](https://modernrobotics.northwestern.edu/nu-gm-book-resource/8-6-dynamics-in-the-task-space/). For a rectangular full-row-rank map, the same positive quadratic form gives the instantaneous wrench-to-acceleration relation, but does not eliminate internal joint dynamics. Velocity and bias terms remain; an inertia ellipsoid is not a velocity-budget ellipsoid. Under ideal contact constraints the free inverse inertia must be replaced by the response of the constrained equations. For independent active constraints, one formulation is $$ \begin{bmatrix}M&-C^T\\C&0\end{bmatrix} \begin{bmatrix}\ddot q\\\lambda\end{bmatrix} =\begin{bmatrix}Bu+Q_{\rm ext}-h\\-\dot C\dot q\end{bmatrix}. $$ This fixed-contact mode still requires feasible reactions, friction and contact-maintenance conditions. Constraint changes require explicit mode transitions. See the publisher's [constrained-dynamics treatment](https://modernrobotics.northwestern.edu/nu-gm-book-resource/8-7-constrained-dynamics/) for the separation of motion and ideal reaction loads. For a selected translational direction $n$, an ideal frozen-configuration impulse model without additional constraints gives $m_{\rm eff}=1/(n^TJM^{-1}J^Tn)$ for a unit vector $n$ and a positive denominator. For independent ideal constraints that remain active through the impulse, replace $M^{-1}$ by $$ D_C=M^{-1}-M^{-1}C^T(CM^{-1}C^T)^{-1}CM^{-1}. $$ This projects the velocity response into the permitted motion; an impending contact or a changing contact mode requires its own impulse balance. This scalar can be useful in contact analysis, but a force ellipsoid alone supplies neither it nor restitution, contact duration, dissipated energy or shaft vibration. ### Small-Displacement Compliance At an unloaded equilibrium, suppose joint springs have positive-definite stiffness $K_q$. Neglecting preload terms and using the same small-displacement Jacobian, an applied load increment gives $$ \delta p=J K_q^{-1}J^T\delta W. $$ This **compliance** matrix has different units and inputs from the force budget's quadratic form. With $J=\operatorname{diag}(2,1)$, joint stiffness matrices $\operatorname{diag}(4,1)$ and $\operatorname{diag}(16,1)$ give task compliances $\operatorname{diag}(1,1)$ and $\operatorname{diag}(0.25,1)$ in consistent units. Holding the torque budget fixed leaves its force ellipsoid unchanged while compliance changes by a factor of four. Preload can add geometric stiffness. For a conservative dead load with potential $-W^T p(q)$, the tangent stiffness includes $-\sum_i W_i\nabla^2 p_i$. For the scalar example $p=q^2$, spring stiffness $10$ and dead load $3$, the local tangent stiffness is $10-2(3)=4$ in the corresponding normalized units. Using the unloaded spring value would miss the load's contribution. Nonconservative loads, damping, feedback and contact compliance require their own linearization and stability analysis. The practical connection is therefore conditional: posture changes geometric leverage; activation and material behavior can change load capacity and impedance; motion changes inertial and velocity-dependent effects. Their interaction must be modeled and checked, rather than inferred from one ellipse. ## Connection to Drift-Control Decomposition The force and mobility ellipsoids are more than geometric curiosities: they describe configuration-dependent mappings between generalized and task-space quantities. Mobility alone does not establish that drift dominates a direction. That comparison also requires the current state, applied-input bounds, reference point, and selected norm. Near impact, a model may combine high task-space mobility with large velocity-dependent drift. Whether the available control is small or large relative to that drift is an empirical model result, not a consequence of the ellipsoid alone. The [hand-path attribution study](proximal-distal-energy-transfer.html#force-along-the-hand-path) defines the same-state comparison needed to test it. Under the dual-budget construction, a small velocity radius pairs with a large force radius. That geometric result is not a passive-stability certificate. Testing a golf interpretation requires a declared state, model, reference point, actuator/contact bounds, load or disturbance direction, and outcome horizon. Compare both the kinematic and dynamic sets, then assess sensitivity to measurement errors, coordinate scaling and plausible capacity models. ## Conclusion The Jacobian gives a precise connection between task velocity and generalized load. Ellipsoids add a chosen metric and feasible budget to that map. Their reciprocal construction is useful precisely because its assumptions can be stated and tested; it should not be extended into a universal physiological trade-off. For the golf investigation, keep the chain of reasoning explicit: specify the task and reference point; derive the geometric map; add inertia, retained loads and contacts; model capacity and compliance; then evaluate the selected outcome over the relevant time interval. The two-link example explains a transmission geometry. Evidence about real release strategies, tissue loading, clubface stabilization or impact performance must come from that fuller model and appropriate measurements. :::{.callout-warning} ## Scope and Limitations - **The example is a rigid linkage.** Flexible coordinates and their rates can be included in a more complete kinematic map. Compliance is not a breakdown of Jacobian mathematics; it changes the state, constitutive law and load response that must be modeled. The size of elastic effects requires evidence for the actual shaft, joints, loading and time scale. - **Singular limits need a physical interpretation.** Exact rank loss, numerical ill-conditioning and dynamic instability are different questions. Small singular values can make inversion sensitive; they do not erase the remaining task directions or prove a particular biological failure. An unbounded force preimage exposes omitted load limits rather than certifying unlimited support. - **A capability picture is conditional.** State uncertainty, task scaling, load offsets and admissible inputs affect the result. Compare sensitivity to these choices and disclose whether an ellipsoid is exact, fitted, projected or numerically truncated. No numerical human validation or ideal-posture recommendation is supplied here. ::: ```{=html} <details class="technical-explanation"> <summary>In Layman's Terms</summary> <p> A long lever can turn a small angular motion into a large tip motion while a force at its tip demands a substantial joint moment. Ellipsoids picture that geometry under specified budgets; they do not impose a universal choice between speed and strength. </p> <h3>A Velocity Budget</h3> <p> The mobility picture contains the tip velocities allowed by a chosen joint-rate budget at a posture. Its longest axis is not necessarily the direction of the actual swing, and it does not tell us how much torque is needed to reach that motion. </p> <p><strong>Think of it like:</strong> Moving the end of a long ruler with a small rotation. The displacement ratio is geometric; accelerating the ruler also depends on its mass and how it is driven.</p> <h3>A Load Budget</h3> <p> The force picture describes which balanced tip loads fit within a stated joint-effort allowance. Stiffness asks how much the system deforms under load; strength asks which loads it can tolerate. Neither is specified by this force picture alone. </p> <p><strong>Think of it like:</strong> A force through an ideal hinge produces no turning moment about that hinge. The bearing still carries the load, and its material limits still matter.</p> <h3>When Reciprocity Applies</h3> <p> Familiar reciprocal ellipses use matching mathematical budgets. Real rate and torque capacities can differ, so a direction can have large radii in both separately defined sets. Whether the corresponding force and speed can occur simultaneously is another actuator-capability question. </p> <p><strong>Think of it like:</strong> An ideal gear exchanges torque for speed while preserving power. Changing the motor's capacity changes what the complete machine can do; a gear ratio alone does not specify its power, stiffness or stability.</p> <p><strong>Key Takeaway:</strong> Declare the task, posture, units and budgets before interpreting an ellipse. These examples do not establish that elite golfers instantly switch from a speed posture to an impact-stability posture.</p> <h3>Real-World Analogies: Configuration-Dependent Behavior</h3> <p> Posture changes the geometric map. Motion and force capacity also depend on state, inertia, actuator behavior, contact and compliance. Two systems with the same geometry can therefore have different acceleration and load responses. </p> <p><strong>The same ruler with a different mass:</strong> Its rotation-to-tip-motion ratio can remain the same, while the torque needed for a selected acceleration changes. A motion ellipse cannot reveal that mass change by itself.</p> <p><strong>The same linkage with different springs:</strong> Changing spring stiffness changes its deformation under load without necessarily changing its geometric motion map or the chosen torque budget. To understand the swing, evaluate these quantities together while keeping their meanings separate.</p> </details> ``` ## Related Articles ::: {.callout-note title="See Also"} - **[Drift-Control Ratio](drift-control-ratio.html)** — Why drift comparison also needs state, input bounds, direction and a declared norm - **[Screw Theory Reference](screw-theory-reference.html)** — Twists, wrenches, and the spatial force-torque duality underlying mobility mapping - **[Intentional Constraint Collapse](intentional-constraint-collapse.html)** — Separate geometric restriction, impedance and hypotheses about impact - **[The Null Space of the Constraint Jacobian](null-space-constraint-jacobian.html)** — Distinguish allowable velocities, reaction loads and grasp internal forces - **[Superposition in Affine Control Systems](superposition.html)** — Declared fixed-state input contributions and their interpretation limits - **[Tangent Hyperplanes II: Dynamics](tangent-hyperplanes-series/part-2-dynamics.html)** — Connect local dynamics to geometric maps without equating joint rates and torque inputs - **[Tangent-Space Series](../pages/tangent-hyperplanes.html)** — Series hub with the full reading order and provenance notes :::