Nonlinear Control Theory Insights, Drift Causality, and Future Research Directions
These entries remain visible until evidence-backed adjudication changes their governed status.
- Open / Medium: Critique: Lie Brackets and "Sequencing" (Misinterpretation of Actuation) (
crit-sequencing-lie-bracket-fallacy)
Introduction: Five Questions About a Coupled Swing
A golfer’s earlier actions establish the posture, velocities, activation and elastic deformation from which later motion develops. Current muscle forces, gravity, contact reactions and equipment dynamics then act on that evolving state. The scientific question is how these contributions interact to produce an impact state, and which changes remain feasible before impact.
Nonlinear control theory helps organize that question, but several different claims must be kept separate:
- Acceleration: How does a specified input change the instantaneous state derivative or clubhead acceleration at a common state?
- Energy: Which mechanisms add, remove, store or redistribute mechanical energy over a declared system boundary?
- Intervention: What trajectory follows from changing one input while retaining the stated plant, internal states, contacts and external loads?
- Finite-Time Control: Which impact states can bounded inputs actually reach within the available time?
- Identification: Which of those quantities can the available measurements distinguish, and with what uncertainty?
These questions are intertwined. An earlier torque changes a later state, and therefore the later drift and torque effectiveness. A contact constraint changes both acceleration and reaction forces. A model that reproduces a trajectory can still misattribute the forces that produced it. A locally effective torque may have little time to change clubhead position, and a large local acceleration contribution need not add energy to the system.
The purpose of the affine formulation is to make such distinctions calculable. It does not establish that the golf swing is universally drift dominated, that muscles supply only small corrections, or that proximal braking is an optimal coaching strategy. The examples below are analytical illustrations, not fitted human-swing results.
From Mechanical Loads to State Rates
For a smooth mechanical model in independent local coordinates, take \(v=\dot q\) and write
\[ M(q)\dot v+h(q,v,t)=B(q,t)u. \]
The coordinates include every retained rigid and flexible mechanical degree of freedom. The inertia matrix \(M\) is positive definite on these independent coordinates. The term \(h\) groups the loads moved to the left of the equation: velocity-dependent terms, gravity, retained elastic and dissipative terms, and any specified external loading with the appropriate sign. The input convention is part of the model. In an effective joint-torque model, \(u\) contains the selected generalized actuator torques and \(B\) maps them into the force balance.
When this mechanical state is complete, stacking \(x=(q,v)\) gives
\[ \begin{aligned} f(x,t)&=\begin{bmatrix}v\\-M^{-1}h\end{bmatrix},\\ G(x,t)u&=\begin{bmatrix}0\\M^{-1}Bu\end{bmatrix},\\ \dot x&=f(x,t)+G(x,t)u. \end{aligned} \]
Thus \(Bu\) is a generalized load, \(M^{-1}Bu\) is an acceleration contribution, and \(G(x,t)u\) is a state-rate contribution. They cannot be substituted for one another. Force and torque are conjugate to displacement and angle; position-rate and acceleration blocks also have different units. Any norm that mixes those blocks needs an explicit scaling convention.
The displayed state is insufficient when retained physics has memory. Muscle activation, tendon history variables, controller states or hysteresis variables must then be included in \(x=(q,v,z)\) with their evolution equations. The enlarged system is control affine only if its full derivative is affine in the declared instantaneous input at fixed complete state. An excitation input can enter an activation equation while the existing activation continues to generate muscle force. Conversely, a model with input-dependent stiffness at fixed state may fail to admit the stated affine split. An additive equation is an assumption to justify, not a physiological validation result.
Generalized velocities need not equal coordinate derivatives globally; other representations require \(\dot q=N(q)v\). Redundant coordinates require constraint handling. The equations in this section use the stated independent-coordinate chart, rather than treating it as a universal human-body parameterization.
Contact Changes the Acceleration Map
For a stationary ideal holonomic constraint with Jacobian \(J(q)\), a redundant coordinate description instead solves
\[ \begin{aligned} M\dot v+h&=Bu+J^T\lambda,\\ J\dot v+\dot Jv&=0. \end{aligned} \]
The reaction \(\lambda\) is solved together with acceleration. After solving this system in a feasible fixed mode, the input-to-acceleration map generally differs from the unconstrained \(M^{-1}B\). Moving supports need the corresponding explicit time terms. Unilateral contact also requires checking the sign of the normal reaction, friction feasibility and the appropriate transition when the assumed mode ceases to apply.
A ground reaction is consequently not automatically an independently commanded input. In a whole-body contact model it can be an output of the constrained dynamics. In a reduced segment model a measured ground or grip wrench may be a prescribed external load. Both are possible conventions, but they answer different intervention questions. The force-measurement article and induced-acceleration chapter develop these boundaries.
Drift Is Instantaneous; the State Reflects Earlier Action
Let \(\Phi_0(t,t_0,x_0)\) be the solution of the declared zero-input system. Along that solution,
\[ \frac{d}{dt}\Phi_0(t,t_0,x_0) =f\bigl(\Phi_0(t,t_0,x_0),t\bigr). \]
The right-hand side must be evaluated at the evolving state. To recover \(f(x_0,t_0)\), evaluate the derivative at the initial time. Drift is an instantaneous vector field; the forward zero-input trajectory is its evolution through time. A complete state summarizes the information needed to predict future evolution under the model. It need not encode a recoverable record of the entire past, and different histories can lead to the same state.
Earlier muscle action can increase velocity, stretch a tendon or bend a shaft. Those changes alter the state at which the later drift is evaluated. Hence independence from the current input at fixed state does not imply independence from previous inputs along a trajectory. This is the precise sense in which preparation and later natural motion are connected.
Writing \(F\) for the full state derivative, input affinity at a common state gives the identity
\[ F(x,u,t)-F(x,0,t)=G(x,t)u. \]
If a reference and a zero-input branch have reached different states, their rate difference additionally contains the difference between their drift fields. Subtracting two separated trajectories is therefore a finite intervention comparison, not a pointwise isolation of the current input term. The canonical ZTCF treatment distinguishes pointwise evaluations, stitched evaluations along a reference, forward branches and velocity-reset diagnostics.
Zero Input Is a Declared Intervention
A forward counterfactual is reproducible only after specifying the plant, parameters, complete initial state, input channels, retained schedules, constraints, external loads and event rules. Within a smooth mode, suitable regularity such as local Lipschitz continuity in the state supplies local uniqueness. It does not guarantee global existence or feasible continuation through every possible contact event.
Zero generalized joint torque, zero muscle excitation and zero activation are three different interventions. Zero net joint torque does not imply absent antagonistic muscle forces. Setting a new excitation command to zero does not instantaneously erase activation or tendon strain. A torque-level model cannot identify that underlying physiological decomposition without further modeling and evidence.
The zero-velocity counterfactual is similarly a model evaluation after a stated reset of selected velocity variables, usually combined with zero selected input. It is not an instrument or an independent measurement of load. Gravity, retained elastic forces, activation and scheduled external loads can remain. A reset must also satisfy constraints; setting velocity to zero may be incompatible with a moving support. Its value is a defined comparison, not a universal passive-force or muscle-effort reading.
Coordinate Changes Preserve an Intervention; Input Offsets Change It
For a smooth invertible state transformation \(y=\Psi(x)\), held within its chart and suppressing explicit time arguments,
\[ \dot y=D\Psi(x)f(x)+D\Psi(x)G(x)u. \]
If the same physical input and all other model choices are preserved, the zero-input solution transforms to the same physical motion. This covariance does not select a unique physiological baseline. It also does not equate models with different constraints, flexible states, parameters or omitted forces.
For a configuration transformation \(y=\psi(q)\), acceleration contains an additional term:
\[ \ddot y=D\psi(q)\ddot q+D^2\psi(q)[v,v]. \]
For example, \(y=q^2\) on the chart \(q>0\) gives \(\ddot y=2q\ddot q+2v^2\). Transforming only the acceleration vector by the Jacobian would miss the velocity-squared term.
Changing the input origin is a different operation. If \(u=\alpha(x)+\beta(x)w\), then
\[ \dot x=\bigl(f+G\alpha\bigr)+G\beta w. \]
Zero \(w\) generally applies a nonzero physical \(u=\alpha(x)\). Feedback linearization or an equilibrium torque offset can therefore change what a “zero-input” trajectory means. A free system with a purely kinetic Lagrangian has geodesic motion under its kinetic metric; adding gravity, dissipation, elastic loads or contacts requires the corresponding forced or constrained dynamics. The general human-club baseline should not be called a geodesic without those qualifications.
Inertial Coupling Without Energy Amplification
A Coupled Acceleration Balance
Partition independent coordinates into proximal and distal groups. The distal row of the unconstrained mechanical equation is
\[ M_{dd}\ddot q_d =(Bu)_d-h_d-M_{dp}\ddot q_p. \]
All gravity, velocity-dependent, elastic and other retained bias terms belong in \(h_d\). The term \(-M_{dp}\ddot q_p\) explains a coupling contribution after rearranging the equation; it is not by itself a separately measured anatomical joint reaction or proof of a particular amount of energy transfer. The complete proximal and distal equations must be solved together.
For a two-coordinate positive-definite inertia matrix, \(M_{pp}>0\), \(M_{dd}>0\) and \(M_{pp}M_{dd}-M_{pd}^2>0\). An arbitrary large coupling coefficient can violate this requirement and permit negative kinetic energy. Moreover, inertias in kg m² multiply angular accelerations expressed in rad/s², not unconverted numerical values in deg/s².
Physical interactions between connected segments are transmitted through their joint forces and moments. The mass matrix encodes that mechanical geometry in the chosen coordinates. Its off-diagonal entries can change under coordinate transformations; an individual entry or ratio is not a coordinate-independent energy-transfer efficiency.
Worked Example: A Braking Torque in the Two-Rod Teaching Model
Use the illustrative two uniform rods from the Chapter 6 worked example: \(m=(2,0.2)\) kg, \(L=(0.4,0.3)\) m, COMs at \(L/2\), COM inertias \(mL^2/12\), and \(g=9.81\) m/s². The fixed base lies in a vertical plane; \(q_1\) is measured counterclockwise from horizontal and \(q_2\) is the second rod’s angle relative to the first. There are no elastic, frictional or moving-support loads in this example. These are teaching parameters, not identified torso or club inertias.
At \(q=(45,70)\) degrees and \(v=(8,12)\) rad/s, the coupled inertia is
\[ M\approx \begin{bmatrix} 0.152875150&0.010104242\\ 0.010104242&0.006000000 \end{bmatrix}\ \mathrm{kg\,m^2}. \]
Its eigenvalues are approximately \(0.005308140\) and \(0.153567010\) kg m², both positive. The complete zero-input acceleration, including velocity and gravity bias, is
\[ \ddot q_0\approx \begin{bmatrix}11.699484\\-119.253631\end{bmatrix} \ \mathrm{rad/s^2}. \]
Apply a proximal braking torque \(u=(-2,0)\) N m at that same state. Solving the full matrix equation gives
\[ \begin{aligned} \Delta\ddot q=M^{-1}u &\approx\begin{bmatrix}-14.721123\\24.790964\end{bmatrix},\\ \ddot q=\ddot q_0+\Delta\ddot q &\approx\begin{bmatrix}-3.021639\\-94.462667\end{bmatrix} \quad\mathrm{rad/s^2}. \end{aligned} \]
The distal acceleration increment is positive, while its total relative acceleration remains negative. A positive contribution is not the same as positive total acceleration. Nor is \(q_2\) the absolute distal orientation: \(\theta_2=q_1+q_2\). Its input acceleration increment is the sum of the two components, \(10.069841\) rad/s². Declaring relative versus absolute angles matters even in this small model.
This calculation establishes a local effect at one state. It does not show that braking increases distal speed at a later impact. That requires a complete trajectory comparison with matched initial conditions, bounded inputs and a declared endpoint or event rule. Calling a torque “braking” also refers to its power at the current velocity; its sign alone is insufficient.
The Energy Ledger
For a time-independent mechanical model with conservative potential \(V\), viscous or other dissipative generalized force \(d\), and external generalized load \(Q_{\rm ext}\), write
\[ M\dot v+C v+\nabla V+d=Bu+Q_{\rm ext}. \]
With a consistent Coriolis representation satisfying the mechanical energy identity, \(E=\tfrac12v^TMv+V\) obeys
\[ \dot E=v^TBu+v^TQ_{\rm ext}-v^Td. \]
Stationary ideal constraint reactions do no work on admissible velocities. Moving supports, contact slip, impacts and internal energy states require their own power or event accounting. The equation is a balance across a stated boundary, not a declaration that all biological work is measured by net joint power.
In the two-rod example, the sole actuator power is \(v^Tu=8(-2)+12(0)=-16\) W. An independent finite difference of the complete kinetic plus gravitational energy along the calculated state derivative also gives \(-16\) W. The braking input removes mechanical energy while changing the distal acceleration contribution. There is no energy amplification by the coupling matrix. Gravity or stored elasticity can release previously stored potential energy, while muscles or external actuators can supply new work.
A segment-level energy balance additionally includes powers of forces and moments crossing that segment’s boundary. Coordinate-wise kinetic-energy terms are not automatically segment energies, because the quadratic form contains cross terms. To identify energy flowing into the club, use its physical body energy and interface wrench powers in consistent frames.
Angular momentum has a separate balance involving external moments about a specified point. Negative actuator power alone does not establish whether total angular momentum is conserved; a support reaction can affect momentum even when it does no work. Consequently, correlated proximal slowing and distal speeding cannot by themselves identify the source, amount or efficiency of energy transfer.
Geometry and Control Before Impact
Accessibility Is Different From Small-Time Local Controllability
Accessibility asks whether a locally reachable set has an interior. Small-time local controllability requires the starting state to lie inside the locally reachable set for arbitrarily short positive horizons while remaining in a sufficiently small neighborhood. The input set and permitted motions matter. An algebraic span of vector fields is not, by itself, a finite-time performance result for a system with drift and bounded actuators.
A scalar counterexample makes the distinction explicit:
\[ \dot x=1+u,\qquad |u|\le\tfrac12. \]
Its vector fields span the one-dimensional state space, and reachable intervals have nonempty interiors. Nevertheless, \(\dot x\ge1/2\), so no admissible input moves the state left from its starting value. It is not small-time locally controllable there. A large enough opposite input would change the conclusion, which is why the actuator set belongs in the claim.
For a driftless system with reversible inputs, noncommuting control flows can generate additional directions. For instance, let \(g_1=(1,0)\) and \(g_2=(0,x)\) on a plane. The sequence \(+g_1,+g_2,-g_1,-g_2\), each applied for time \(\epsilon\), returns the first coordinate to its start and changes the second by \(\epsilon^2\). With the convention
\[ [g_1,g_2]=Dg_2\,g_1-Dg_1\,g_2, \]
the bracket is \((0,1)\). This example shows a direction generated by a sequence, with a second-order displacement and a total time \(4\epsilon\). It does not provide an instantaneous independent actuator in that direction. A system with drift that cannot be reversed does not automatically permit the same four-flow argument.
The publisher-hosted Modern Robotics treatments of local reachable sets and the driftless rank condition make the admissible controls explicit. Applying such tools to a golf model requires its actual fields, constraints and bounds. Lie brackets can be useful in studying controllability, but their existence does not prove that a named proximal-to-distal sequence is necessary or improves impact speed.
Instantaneous Gain Is Only One Ingredient of a Finite-Time Effect
Linearize a smooth control-affine model along a feasible reference \((\bar x(t),\bar u(t))\). For small perturbations within the same mode,
\[ \delta\dot x=A(t)\delta x+G(\bar x(t),t)\delta u, \]
where \(A\) includes both the state derivative of \(f\) and the derivatives of the input fields weighted by the reference inputs. If \(\Phi(T,t)\) is the transition matrix of this variational system, then
\[ \delta x(T)=\Phi(T,t_0)\delta x(t_0) +\int_{t_0}^{T}\Phi(T,s)G(\bar x(s),s)\delta u(s)\,ds. \]
This expression connects present authority to later outcomes through the intervening dynamics. MIT’s trajectory-linearization discussion uses a moving reference, not just a stationary equilibrium. The approximation must be checked against nonlinear trajectories; contact events require event and reset sensitivity, and an impact-defined endpoint can shift in time.
Even the double integrator illustrates the importance of duration. For \(\ddot q=u\), \(|u|\le a\), compare trajectories with equal initial position and velocity against zero input. At a fixed horizon \(T\),
\[ |\Delta q(T)| =\left|\int_0^T(T-s)u(s)\,ds\right| \le\tfrac12aT^2. \]
A saturated constant input attains this bound. Halving the available time quarters the possible position correction, despite unchanged instantaneous acceleration authority. High speed does not make geometry irrelevant; it makes actual horizons, bounds, delays and the evolving state indispensable.
Geometric Phase Is a Specific Mechanism
A geometric-phase claim needs a specified symmetry or group motion, shape variables and a reconstruction rule connecting a shape cycle to net motion. Kelly and Murray’s author-hosted abstract describes that framework for robotic locomotion. It does not demonstrate a golf sequencing effect. A swing is not automatically a closed shape cycle, and a change in clubhead position is not sufficient evidence of geometric phase.
Inertial coupling already explains how joint accelerations interact in a mechanical model. A phase-based explanation may add insight for an appropriately specified model, but it must identify what it predicts beyond that force balance. The linked governed critique remains open: its concern about conflating coupling with nonholonomic steering is relevant, while its own categorical full-actuation and efficiency statements also need model-specific support.
What the Other Control Tools Can Establish
Partial Feedback Linearization
For a selected smooth position output \(y=r(q)\), one can often write \(\ddot y=b_y(x)+A_y(x)u\) within a fixed mode. Prescribing an output acceleration requires the relevant rank condition on \(A_y\), a feasible input solution and attention to the internal dynamics left after controlling the output. Saturation, singular postures and model error can prevent the intended cancellation. The MIT cart-pole and Acrobot treatment illustrates why collocated and noncollocated outputs have different conditions. This tool helps analyze a chosen task; it does not establish that a golfer’s nervous system implements exact nonlinear cancellation.
Optimal Control and Energy Shaping
An optimal-control study must state a terminal objective, a running cost, initial conditions, duration, dynamics and admissible trajectories. For example,
\[ \min_{u(\cdot)}\left[ \phi(x(T))+\int_{t_0}^{T}\ell(x(t),u(t),t)\,dt \right] \]
is incomplete until input, state and contact constraints are supplied. Different weights on impact speed, accuracy, effort or tissue loading can produce different policies. Squared net joint torque is a chosen penalty, not generally a metabolic-energy measurement. A numerical local optimum is not automatically a global optimum or a description of human behavior.
Energy shaping is a particular feedback-design idea: the controller changes closed-loop dynamics to obtain a desired energy structure. It is not synonymous with any redistribution of energy in a swing. Neither underactuation nor an optimal-control formulation guarantees that optimal inputs are small perturbations. Feasibility, actuator limits and the chosen objective can require substantial input throughout a maneuver.
Observability and Unknown-Input Identification
Classical observability asks whether internal states can be distinguished from outputs over time under a specified model and known input. An unknown torque is an input unless a model explicitly augments it as a state with dynamics. Adding that state is an additional assumption, not a free reconstruction of a missing measurement.
For example, if only \(\dot x=d+u\) is observed and both \(d\) and \(u\) are unknown, the same measured derivative can be explained by increasing \(d\) and decreasing \(u\) equally. A control-affine label does not resolve this ambiguity. In a swing, kinematics may constrain effective joint torques only after inertial properties, external loads, contact and derivative estimation are specified. Net torque still does not uniquely determine individual muscle forces or co-contraction.
Sensor selection should therefore be linked to an explicit distinguishability question. Ground forces, grip wrenches, kinematics and shaft measurements observe different boundaries and states. Noise, timing offsets, parameter uncertainty and derivative filtering can further limit an otherwise identifiable model.
Robustness and Data-Driven Identification
A sensitivity calculation asks how a prediction changes with parameters or perturbations. A robust guarantee requires a defined uncertainty set and a specified property, such as feasibility or an impact-error bound, across that set. A stable nominal linearization does not establish robustness throughout a nonlinear swing or across a contact transition.
Learning \(f\) and \(G\) from data likewise requires sufficient input information and variation. If only one unknown-input trajectory is observed, different splits can fit it. Physics constraints, calibrated inputs, held-out interventions and uncertainty analysis can reduce ambiguity; they must be evaluated for the quantity the article interprets. Good trajectory fit alone is insufficient evidence of correct force attribution.
A Research Program That Connects the Questions
A useful study starts with a falsifiable question, such as whether a specified proximal torque change increases a declared clubhead speed component at a specified event. The following sequence connects mechanics to evidence:
- Declare the Plant and Boundary. Specify coordinates, retained flexible and physiological states, input channels, contacts and measurement frames. Document which loads are measured, modeled, prescribed or solved reactions.
- Verify the Mechanics. Check inertia definiteness, units, complete force balance, energy and momentum accounting, and convergence. Preserve the same physical intervention under coordinate changes.
- Compare Defined Interventions. Match initial complete states and retained schedules. Report both pointwise acceleration differences and forward trajectory outcomes, including event timing and feasibility failures.
- Assess Finite-Time Alternatives. Apply input, rate, delay and state bounds. Examine the task directions that remain reachable and the sensitivity to uncertain parameters, rather than inferring authority from one scalar ratio.
- Test Identification and Prediction. Determine whether the sensors can distinguish the proposed mechanisms. Compare against independent measurements or held-out interventions with uncertainty and alternative explanations.
Equipment comparisons must also specify what is held fixed. Replaying identical torque, imposing identical kinematics and allowing an adapted control policy are different experiments when shaft properties change. A conserved quantity under zero input must be derived for the declared loads and constraints; a learned near-constant on one trajectory is only a candidate invariant.
Hybrid motion can be studied, but requires explicit modes, guards, reset maps and a well-posed continuation rule. Counterfactual branches may encounter different events. A fixed-contact calculation should stop or change mode when its contact assumptions fail, rather than extrapolating an impossible reaction. No physical participant test or golf-specific optimal policy is established by this article’s analytical examples.
In Layman’s Terms
Earlier action prepares the moving body and club. Later action then works with that moving state. A mathematical baseline asks what a specified model would do if a specified input were changed. It does not erase the effort that prepared the state, and it does not measure a universal percentage of the swing supplied by muscles.
A braking torque can change how the links accelerate while removing energy from the entire system. That is why “the torso slowed while the club sped up” is an observation to explain, not proof of a beneficial braking strategy. The complete motion, the forces crossing segment boundaries and the time left before impact all matter.
The same reasoning applies to control. A torque that strongly affects acceleration may arrive too late to change position much. A model that predicts motion well may still assign the wrong causes when inputs or contact forces are unknown. Carefully defined comparisons and independent measurements connect these pieces into a testable explanation.
Critics’ Comments
Does a torque-level model establish physiology? No. It can be useful for a declared mechanical question while omitting activation-dependent stiffness, co-contraction and tissue history. Its predictions need validation at the level being interpreted; a net-torque result is not an individual-muscle result.
Does the coupled example establish a sequencing prescription? No. It verifies one local input effect and its energy accounting in a specified two-rod model. A useful golf intervention requires a matched finite-time comparison and appropriate empirical evidence. Neither coupling nor Lie brackets alone supply that result.
Does qualification make the framework unfalsifiable? It should do the opposite. Declare the model and predicted output before comparison, preserve failed and null results, and test alternatives. A model-dependent claim can be precise; an unrestricted assertion about all swings cannot be rescued by moving its assumptions after observing the outcome.
Does this rewrite resolve the governed critique? The existing critique stays visible and open. Correcting the exposition is distinct from adjudicating a scientific claim against its required evidence. This article does not claim a 95% skeletal-baseline accuracy result or a uniquely physiological zero input.
Citation
@online{olson2025,
author = {Olson, Dieter},
title = {Nonlinear {Control} {Theory} {Insights,} {Drift} {Causality,}
and {Future} {Research} {Directions}},
date = {2025-11-28},
url = {https://affinedrift.com/articles/nonlinear-control-insights.html},
langid = {en}
}