Induced Acceleration Analysis: Quantifying Who Moves What
It is not enough to know what forces are present; one must know what each force does.
In the previous chapter, we described the kinetic chain as a sequential cascade of energy from proximal segments (hips, torso) to distal segments (arms, wrists, club). A narrower, model-bounded question is: under one declared coordinate system, contact model, and force partition, how is the instantaneous clubhead acceleration ledger divided among hip torque, shoulder torque, gravity, velocity terms, constraints, and residuals? That ledger does not identify a unique cause.
This chapter introduces a technique from biomechanics research called induced acceleration analysis (IAA). At a frozen state, a declared right-hand-side force partition can be mapped linearly through the constrained forward-dynamics operator. Its terms sum to the model’s acceleration when constraints, contact reactions, and residuals are handled consistently. The terms are calculations under that convention, not observations of forces literally acting alone.
Induced acceleration analysis was pioneered by Zajac and Gordon (1989) for studying muscle function in multi-joint movements and was extended into a comprehensive framework for walking by Zajac et al. (2002), Zajac et al. (2003). The technique has since been applied to throwing Hirashima and Ohtsuki (2008), gait pathology Neptune et al. (2001), sit-to-stand transfers (Caruthers et al. 2016), cycling (Schutte et al. 1993), and postural control (Challis 2011). This chapter brings induced acceleration analysis into the golf swing for the first time, connecting it to the control-affine framework developed in earlier chapters.
An induced-acceleration attribution is a term ledger for a declared model at a declared state. It is not a unique causal history. A publishable result must identify all of the following:
- Model and revision: segment and actuator definitions, parameter set, generalized coordinates, reference frame, output point or task metric, and the state at which the mass matrix is evaluated.
- Computation: engine, solver, and revision; constraint handling; contact model and active contact mode; integration or frozen-state procedure; and numerical tolerance plus closure error.
- Attribution convention: the complete force partition, including sign conventions, passive and applied terms, constraint reactions, and residual treatment. Reassigning a residual between two terms changes their reported contributions while leaving their sum unchanged.
- Identifiability contract: the measurements or assumptions that identify each generalized-force term and the output projection. An algebraic term contribution does not by itself identify anatomical source, neural intent, necessity, sufficiency, or intervention effect.
If any field is absent, label the result unsupported or unqualified. A cross-engine comparison must report an unavailable state as unsupported or unqualified; agreement on one output, or silence about an unsupported state, does not establish solver parity.
Counterexamples to Unique Attribution
- Coordinate representation: under a constant change of generalized coordinates \(q=Tz\), the same dynamics are represented by \(M_z=T^\mathsf{T}M_qT\) and \(Q_z=T^\mathsf{T}Q_q\). The component values of \(\ddot z=M_z^{-1}Q_z\) generally differ from those of \(\ddot q=M_q^{-1}Q_q\), even though \(T\ddot z=\ddot q\). A generalized- acceleration component is therefore not coordinate-invariant causal truth.
- Force partition: if \(Q=Q_A+Q_B\), then for any residual allocation \(r\), \(Q=(Q_A+r)+(Q_B-r)\). The reported terms \(M^{-1}Q_A\) and \(M^{-1}Q_B\) change, while the total acceleration does not. The partition must be declared and justified before a term is interpreted.
An intervention claim requires a separate governed counterfactual that re-solves constraints and contact and states what is held fixed. The frozen- state algebra alone supports neither a unique cause nor an anatomical or behavioral prescription.
The Mathematical Foundation
Starting From the Manipulator Equation
Recall from Chapter 4 that the equations of motion for an \(n\)-degree-of-freedom mechanical system take the form:
\[ \boldsymbol{M}(\boldsymbol{q}) \ddot{\boldsymbol{q}} = \boldsymbol{\tau}_{\text{muscle}} + \boldsymbol{V}(\boldsymbol{q}, \dot{\boldsymbol{q}}) + \boldsymbol{g}(\boldsymbol{q}) \tag{1}\]
where \(\boldsymbol{M}(\boldsymbol{q})\) is the mass (inertia) matrix, \(\boldsymbol{q}\) is the declared vector of independent generalized coordinates, \(\boldsymbol{\tau}_{\text{muscle}}\) is a model-assigned generalized-force term (an anatomical muscle term only when an actuator map and identifiability contract support that label), \(\boldsymbol{V}(\boldsymbol{q}, \dot{\boldsymbol{q}})\) collects the declared velocity-dependent terms, and \(\boldsymbol{g}(\boldsymbol{q})\) is the gravity term under the stated sign convention. Contact, passive, constraint, and residual terms must also appear when the model retains them.
The critical step is to solve for accelerations by premultiplying both sides by \(\boldsymbol{M}^{-1}(\boldsymbol{q})\):
\[ \boxed{\ddot{\boldsymbol{q}} = \boldsymbol{M}^{-1}(\boldsymbol{q}) \left[ \boldsymbol{\tau}_{\text{muscle}} + \boldsymbol{V}(\boldsymbol{q}, \dot{\boldsymbol{q}}) + \boldsymbol{g}(\boldsymbol{q}) \right]} \tag{2}\]
For independent minimal coordinates with a regular kinetic-energy metric, \(\boldsymbol{M}(\boldsymbol{q})\) is symmetric positive definite and invertible. Redundant coordinates or active constraints instead require a declared constrained solve; writing a bare inverse is not sufficient. At a fixed state and fixed active-set convention, the resulting force-to-acceleration map is linear in the declared right-hand-side terms.
At any frozen instant in time (fixed \(\boldsymbol{q}\) and \(\dot{\boldsymbol{q}}\)), the total acceleration \(\ddot{\boldsymbol{q}}\) is the sum of the accelerations induced by each individual force source:
\[ \ddot{\boldsymbol{q}} = \underbrace{\boldsymbol{M}^{-1}(\boldsymbol{q}) \boldsymbol{\tau}_{\text{muscle}}}_{\text{muscle-induced}} + \underbrace{\boldsymbol{M}^{-1}(\boldsymbol{q}) \boldsymbol{V}(\boldsymbol{q}, \dot{\boldsymbol{q}})}_{\text{velocity-induced}} + \underbrace{\boldsymbol{M}^{-1}(\boldsymbol{q}) \boldsymbol{g}(\boldsymbol{q})}_{\text{gravity-induced}} \tag{3}\]
Each term can be computed independently. Furthermore, the muscle torque vector can itself be decomposed into contributions from individual muscles or individual joints:
\[ \boldsymbol{\tau}_{\text{muscle}} = \sum_{k=1}^{m} \boldsymbol{\tau}_k \]
where \(\boldsymbol{\tau}_k\) is the torque vector produced by the \(k\)-th muscle (or the net torque at the \(k\)-th joint). Since \(\boldsymbol{M}^{-1}\) is linear:
\[ \ddot{\boldsymbol{q}}_{\text{muscle}} = \sum_{k=1}^{m} \boldsymbol{M}^{-1}(\boldsymbol{q}) \boldsymbol{\tau}_k = \sum_{k=1}^{m} \ddot{\boldsymbol{q}}_k \]
Each \(\ddot{\boldsymbol{q}}_k = \boldsymbol{M}^{-1}(\boldsymbol{q}) \boldsymbol{\tau}_k\) is the term assigned to the \(k\)-th declared generalized-force channel. The sum recovers the declared applied-force contribution exactly. Calling a channel a muscle requires an explicit muscle-to-generalized-force map and identification evidence.
Imagine you could freeze time at one instant during the downswing. At that instant, multiple forces act on the golfer-club system: hip torque, shoulder torque, wrist torque, gravity, and the centrifugal and Coriolis forces arising from the current velocities.
Induced acceleration analysis asks how the frozen-state operator maps one declared term—say, a hip generalized-torque channel—while the state, active constraints, and partition convention are held fixed. This algebraic term is often called the “acceleration induced by the hip torque.” It is not automatically the result of a realizable intervention that removes other forces; such an intervention must re-solve contact and constraints.
This is not a thought experiment about what would happen over time if we only applied one force (that would be a different and much more complicated question, because the system’s state would diverge). It is a precise statement about the instantaneous contribution of each force to the total acceleration at one frozen moment.
Connection to the Control-Affine Framework
Readers who have followed the development in Chapter 5 will recognize that Equation 3 is closely related to the control-affine decomposition:
\[ \dot{\boldsymbol{x}} = \underbrace{\boldsymbol{f}(\boldsymbol{x})}_{\text{drift}} + \underbrace{\boldsymbol{G}(\boldsymbol{x}) \boldsymbol{u}}_{\text{control}} \]
For a particular unconstrained partition, the acceleration block of the drift field may contain velocity-dependent and gravity terms, while \(\boldsymbol{G}(\boldsymbol{x})\boldsymbol{u}\) maps the declared applied inputs. Passive forces, contact, constraints, internal states, and residuals remain in the complete autonomous field or constrained solve according to the declared plant. A control channel is not synonymous with a measured muscle input.
The connection is direct: the biomechanics community’s “induced acceleration analysis” is the instantaneous superposition that arises naturally from the control-affine structure of mechanical systems. Both frameworks exploit the same mathematical fact—that forces map linearly to accelerations at each frozen state—but they were developed independently by different research communities. The control-affine formulation comes from nonlinear control theory Murray et al. (1994), Bullo and Lewis (2004), while induced acceleration analysis was developed in the biomechanics and rehabilitation literature Zajac and Gordon (1989), Zajac et al. (2002).
In the control-affine framework:
- Drift = the complete autonomous acceleration of the declared effective plant when applied generalized control is zero, including every retained inertial, gravitational, passive, contact, constraint, and internal-state term.
- Control = additional acceleration from declared applied generalized inputs
In induced acceleration analysis:
- Velocity-dependent acceleration = \(\boldsymbol{M}^{-1} \boldsymbol{V}(\boldsymbol{q}, \dot{\boldsymbol{q}})\)
- Gravity-induced acceleration = \(\boldsymbol{M}^{-1} \boldsymbol{g}(\boldsymbol{q})\)
- Muscle-induced acceleration = \(\boldsymbol{M}^{-1} \boldsymbol{\tau}_{\text{muscle}}\)
These expressions coincide only for the same declared state, coordinates, actuator map, constraints, contact mode, and force partition. The complete drift is the autonomous vector field of the declared plant, not necessarily just velocity and gravity, and the control term is an applied-input contribution rather than identified muscular effort.
Dynamic Coupling: A Declared Joint Torque Can Affect Multiple Coordinates
One of the most important and initially counterintuitive consequences of the equations of motion is dynamic coupling: a torque applied at one joint produces accelerations at every joint in the kinematic chain, not just the joint where the torque is applied.
Why This Happens
Consider a two-joint system (shoulder and elbow) in a horizontal plane. If we apply only a shoulder flexion torque \(\tau_1\) (with \(\tau_2 = 0\)), the induced accelerations are:
\[ \begin{bmatrix} \ddot{\theta}_1 \\ \ddot{\theta}_2 \end{bmatrix} = \boldsymbol{M}^{-1}(\boldsymbol{q}) \begin{bmatrix} \tau_1 \\ 0 \end{bmatrix} = \begin{bmatrix} A_{11} \tau_1 \\ A_{21} \tau_1 \end{bmatrix} \tag{4}\]
where \(A_{ij}\) denotes the \((i,j)\) element of \(\boldsymbol{M}^{-1}(\boldsymbol{q})\). The shoulder experiences an acceleration \(A_{11} \tau_1\) (the direct effect), but the elbow also experiences an acceleration \(A_{21} \tau_1\) (the remote or induced effect). The off-diagonal element \(A_{21}\) is generally nonzero because the segments are physically linked: applying a torque at the shoulder creates a reaction force at the elbow joint, which accelerates the forearm.
This phenomenon was articulated by Zajac and Gordon (1989) in a model-bounded review: interpreting a muscle-assigned term in multi-articular movement requires accounting for constraint reactions that propagate through the declared chain. A modeled actuator crossing only the shoulder can receive nonzero terms at elbow, wrist, and club coordinates through the chosen coupling operator. That calculation does not identify an anatomical source without the required actuator and measurement contract.
Posture Dependence: The Same Torque, Different Results
A second important finding, emphasized in the work of Hirashima (2011), is that the induced accelerations depend on the current posture \(\boldsymbol{q}\) of the entire system, because \(\boldsymbol{M}^{-1}(\boldsymbol{q})\) changes with configuration.
Consider the shoulder internal rotation torque applied to a three-segment upper limb model. When the elbow is fully extended (so the entire limb forms a straight line), the principal axes of inertia of the distal chain align with the shoulder joint axes. In this configuration, the shoulder internal rotation torque produces only internal rotation—there is a clean one-to-one mapping from torque to acceleration.
However, when the elbow is flexed at \(90^{\circ}\) in the cited model, the principal axes of inertia no longer align with its joint coordinates. The same declared shoulder-torque channel then receives nonzero internal-rotation, abduction, and horizontal-extension acceleration components Hirashima (2011). These are coordinate- and model-reported terms, not a posture-independent anatomical function.
The term assigned to a modeled actuator depends on:
- The magnitude and direction of the torque vector produced by the muscle (determined by anatomy and moment arms).
- The current posture of the entire kinematic chain (which determines \(\boldsymbol{M}^{-1}(\boldsymbol{q})\)).
Changing posture—even at remote coordinates—can change the model term. This supports a configuration-sensitivity hypothesis; it does not establish equal muscular effort across golfers, proper form, a unique cause, or an intervention effect.
The Induced Acceleration Index
Challis (2011) formalized the configuration-dependent coupling between joints by introducing the induced acceleration index (IAI). For a system where joint \(j\) is “active” (has an applied torque) and joint \(k\) is “inactive” (has no applied torque), the IAI quantifies the potential of the active joint to accelerate the inactive joint:
\[ \text{IAI}_{j \to k} = \left| M_{kk}^{-1} M_{kj} \right| \tag{5}\]
This dimensionless index depends only on the system’s configuration and inertial properties, not on the magnitudes of the torques. It reveals the structural coupling between joints.
For a declared quiet-standing model, Challis (2011) reported an ankle-to-hip index more than 12 times the reverse index. This model-reported structural ratio does not identify the primary neural control strategy or establish necessity or sufficiency.
Comparable asymmetries in a golf model are a hypothesis to calculate, not a result established here. Any such result must use the normative attribution record above and must not be transferred from quiet standing to human swing technique without qualification.
Instantaneous Effects Versus Cumulative Effects
One of the most important distinctions to emerge from the induced acceleration literature—and one with direct relevance to understanding the golf swing—is the difference between instantaneous effects and cumulative effects. This distinction was developed most clearly in the work of Hirashima et al. (2008) and Hirashima (2011) in the context of baseball pitching, and it maps directly onto the drift-control decomposition introduced in Chapter 5.
Instantaneous Effects: What the Current Torques Produce Right Now
The modeled applied-input term at time \(t\) is:
\[ \ddot{\boldsymbol{q}}_{\text{muscle}}(t) = \boldsymbol{M}^{-1}(\boldsymbol{q}(t)) \boldsymbol{\tau}_{\text{muscle}}(t) \]
This is a frozen-state algebraic quantity. It reports the acceleration term assigned to the declared generalized input under the model and partition. It does not simulate muscles vanishing, because that intervention could change activation, passive force, constraint reactions, and contact. Direct and remote labels describe coordinate components of this term, not unique anatomical effects.
Cumulative Effects: The Legacy of Past Forces
The velocity-dependent term \(\boldsymbol{V}(\boldsymbol{q}, \dot{\boldsymbol{q}})\) depends on the current generalized velocities. Those velocities reflect prior dynamics, inputs, constraints, contact transitions, and initial conditions, but the frozen-state term does not uniquely apportion that history. The corresponding ledger entry is:
\[ \ddot{\boldsymbol{q}}_V(t) = \boldsymbol{M}^{-1}(\boldsymbol{q}(t)) \boldsymbol{V}(\boldsymbol{q}(t), \dot{\boldsymbol{q}}(t)) \]
It is often described as a cumulative term because it depends on the current velocity state. Model-bounded language must not relabel it as a uniquely reconstructed force history; no such identifiability contract is provided here.
An applied-input ledger term can match the acceleration block of \(\boldsymbol{G}(\boldsymbol{x})\boldsymbol{u}\) when the actuator map, coordinates, constraints, and force partition match.
Velocity and gravity terms can be components of the drift. The complete drift also retains every autonomous passive, contact, constraint, and internal-state effect in the declared effective plant.
The drift field is a model-defined autonomous vector field at the current state. It is not a uniquely invertible record of earlier muscle torques, gravity, or elastic recoil.
Why the Distinction Matters for the Golf Swing
Fast swings can make velocity-dependent model terms large, but their magnitude relative to applied-input terms is model-, state-, coordinate-, and norm-dependent. This chapter supplies no governed human-golf attribution result satisfying the normative record.
A drift-control ratio is a declared magnitude ratio, not a causal certificate or an induced-acceleration partition. Establishing late-downswing dominance would require a governed computation that reports the complete attribution record, uncertainty, and residual closure.
Hirashima et al. (2008) reported, within its baseball-pitching model and partition, large velocity-dependent terms for elbow extension and wrist flexion and different term balances proximally. These model-reported results are historical context, not cross-engine parity, human-golf evidence, or identification of individual muscle effort.
Lessons From Throwing: Implications for the Golf Swing
The golf swing and the overarm throw share fundamental dynamical features: both are fast, multi-joint movements performed by serial kinematic chains, and both rely on proximal-to-distal energy transfer to achieve high distal segment speeds. The induced acceleration analyses of baseball pitching performed by Hirashima et al. (2007), Hirashima et al. (2008), Hirashima and Ohtsuki (2008) provide some of the clearest demonstrations of how the kinetic chain operates at the level of individual forces and accelerations.
Model-Reported Throwing Terms
Using a 13-degree-of-freedom trunk-and-upper-limb model, Hirashima et al. (2008) reported different balances of generalized-joint-torque and velocity-dependent terms across trunk, shoulder, elbow, and wrist coordinates. The elbow and wrist ledgers contained large velocity-dependent terms near release, while the proximal-coordinate ledgers assigned larger shares to local joint-torque terms. These are model-reported findings under the cited study’s coordinates and partition. This chapter has not reproduced its engine, solver revision, residual closure, or tolerance, so the values remain unqualified here and do not identify muscle source, protective intent, or a golf-swing mechanism.
The Route of the Kinetic Chain
Hirashima et al. (2008) further repartitioned the velocity-dependent term by segment motion. Within that model-bounded ledger, the reported sequence was:
- Trunk and shoulder joint torques accelerate the trunk and upper arm (instantaneous effect, early phase).
- The resulting trunk and upper arm angular velocities produce velocity-dependent torques that accelerate elbow extension (cumulative effect, mid-phase).
- The forearm angular velocity increases, producing additional velocity-dependent torques that accelerate both wrist flexion and (during a brief window near ball release) shoulder internal rotation (cumulative effect, late phase).
The sequence is a source-study interpretation of one force partition, not a coordinate-invariant energy-flow path or unique cause. Transferring it to golf requires a separately qualified golf model, contact treatment, output projection, and human evidence.
Torque Reversal: The Instantaneous Remote Effect
A cited throwing model also contains a torque-reversal interval. Its generalized elbow-torque channel changes sign while the model assigns simultaneous elbow and wrist acceleration terms through off-diagonal coupling. The algebra does not identify which muscles generated that net torque or why.
Hirashima (2011) distinguishes this instantaneous assigned term from the velocity-dependent term. Under the model’s identifiability contract it is not a unique cause, a reconstructed force history, or training guidance.
An analogous wrist term in golf is an open hypothesis. It would require the complete normative record and cannot be asserted from the throwing model or from drift magnitude alone.
Lessons From Walking and Clinical Biomechanics
Foundational walking studies provide methodological examples. They do not validate an AffineDrift golf attribution, and cross-task or cross-engine states remain unsupported or unqualified unless the complete record is supplied.
The Foundational Walking Studies
The comprehensive two-part review by Zajac et al. (2002), Zajac et al. (2003) established the modern framework for induced acceleration analysis in biomechanics. Their key contributions were:
Separating quantities: Joint power, segment energy, and an acceleration term answer different questions. A forward model can compute a declared term ledger; it does not identify a causal relationship between muscle forces and movement outcomes without additional measurement and intervention evidence.
Separating modeled actuator terms: A musculoskeletal simulation can report the term assigned to each modeled actuator for a declared output. The label remains conditional on muscle-force estimation, moment arms, constraints, contact, and residual closure.
Demonstrating counterintuitive muscle functions: Muscles often accelerate segments and joints they do not cross, and their contributions to whole-body function can differ substantially from what anatomy alone would predict.
Soleus Versus Gastrocnemius: A Case Study in Superposition
The study by Neptune et al. (2001) applied induced acceleration analysis to a forward dynamics simulation of normal walking, focusing on the individual contributions of the soleus (a uniarticular ankle plantar flexor) and the gastrocnemius (a biarticular ankle plantar flexor that also crosses the knee).
Despite both muscles being ankle plantar flexors, their contributions to whole-body function were strikingly different:
Soleus: In late stance and pre-swing, the energy generated by soleus was delivered primarily to the trunk, providing forward progression and vertical support.
Gastrocnemius: In the same phase, nearly all the energy generated by gastrocnemius was delivered to the swing leg, initiating swing rather than propelling the trunk.
The source study reported different model-assigned terms for two actuators that share a joint. Net inverse dynamics alone cannot resolve that partition; the forward-model result remains conditional on its actuator, contact, and constraint assumptions rather than uniquely revealing anatomical function.
For golf, this is a reason to test actuator-map and configuration sensitivity, not evidence that similarly located muscles have identified swing roles.
Clinical Applications: Stiff-Legged Gait
Riley and Kerrigan (1999) reported patient-varying hip-, knee-, and ankle-moment terms in a stiff-legged-gait model. This model-bounded result is not a unique cause of reduced knee flexion and does not identify a rehabilitation intervention effect without a separate identifiability contract.
The decomposition can generate patient-specific hypotheses for further clinical testing. Algebraic superposition alone does not identify which impairment is primarily responsible.
Sit-to-Stand Transfers
Caruthers et al. (2016) used a three-dimensional, 46-degree-of-freedom, 194-actuator model and reported positive gluteus-maximus and soleus terms and an opposing quadriceps term for selected center-of-mass outputs. Those are model-reported contributions, not uniquely identified anatomical sources or intervention effects.
Implications for the Golf Swing
Golf Questions, Not Golf Findings
The literature motivates a model-ledger problem for golf. This chapter reports no qualified AffineDrift IAA result and provides no coaching prescription. A future study could test:
How results change across net-joint, muscle-actuator, contact, passive, and residual partitions.
How coordinate, frame, output-point, and posture choices change reported terms while total forward dynamics remain consistent.
Whether a velocity-dependent term is large under preregistered norms and uncertainty, without promoting that ratio into causal, reachability, or effort language.
Whether an intervention that changes an input while re-solving contact and constraints changes a declared performance output.
A Unified Framework
The induced acceleration framework unifies the observations made throughout this book:
| This Book’s Framework | IAA Framework | Physical Meaning |
|---|---|---|
| Complete drift \(\boldsymbol{f}(\boldsymbol{x})\) | Declared autonomous terms | Model-defined zero-input evolution |
| Control \(\boldsymbol{G}(\boldsymbol{x})\boldsymbol{u}\) | Declared applied-input term | Not identified muscle or intent |
| Constrained forward operator | Force-to-acceleration map | Coordinate-, contact-, and model-dependent |
| Term-magnitude comparison | Declared norm and partition | Not a causal or reachability certificate |
| ZTCF family (Ch. Chapter 6) | Zero-control model counterfactual | What the declared effective plant produces under zero applied generalized control |
Both frameworks can use the same frozen-state linear map, but they are not automatically equivalent. The effective plant, input definition, actuator map, constraints, contact, residual allocation, coordinates, and requested output must match before terms can be compared.
Superposition and Its Limits
A declared force partition sums at a frozen state when the constrained solve and residual closure are consistent. Its coordinate components are representation-dependent, and a forward intervention changes the state, constraints, and often contact. Therefore neither a trajectory component nor a unique cause follows from frozen-state superposition.
Hirashima and Ohtsuki (2008) showed why isolated diagonal-inertia calculations do not reproduce the coupled model. A full constrained solve is necessary for closure, but closure alone still does not identify causal contributions; that stronger claim requires the normative identifiability and intervention contract.
For golf biomechanics, any model term must be computed on the declared coupled system. Even then, the result remains a model term rather than an identified statement about what each muscle does.
Summary
A declared ledger can close at a frozen state: Applied, velocity, gravity, passive, contact, constraint, and residual terms must sum under one coordinate and active-set convention.
Coupling is model- and coordinate-dependent: Off-diagonal entries can map one generalized-force channel into several generalized-acceleration components; neither those components nor their labels are invariant causal truth.
Instantaneous is not interventional: Frozen-state terms do not reconstruct force history, intent, necessity, sufficiency, or the outcome of removing a force.
Configuration sensitivity is a hypothesis generator: Changing posture can change the declared term ledger, but does not establish proper mechanics or equal human effort.
Control-affine and IAA terms require a compatibility audit: Compare them only when plant, coordinates, constraints, contact, actuator map, partition, output, and solver qualification match.
Chapter Exercises
For a two-segment planar model (shoulder and elbow in a horizontal plane), write out the \(2 \times 2\) mass matrix \(\boldsymbol{M}(\boldsymbol{q})\) and compute its inverse. Show that the off-diagonal elements of \(\boldsymbol{M}^{-1}\) are generally nonzero, confirming that a shoulder torque induces elbow acceleration.
Using the double-pendulum model from Chapter 3, compute the velocity-dependent torque \(\boldsymbol{V}(\boldsymbol{q}, \dot{\boldsymbol{q}})\) and the gravity torque \(\boldsymbol{g}(\boldsymbol{q})\) at a representative mid-downswing posture (\(\theta_1 = -45^{\circ}, \theta_2 = -90^{\circ}\), with angular velocities \(\dot{\theta}_1 = 500~\text{deg/s}, \dot{\theta}_2 = 200~\text{deg/s}\)). Compute the induced accelerations from each source and verify that they sum to the total acceleration.
The induced acceleration index (IAI) of Challis (2011) quantifies the coupling potential between joints. For a two-segment model of the golfer’s lead arm and club, compute the IAI at three postures: (a) early downswing (large lag angle), (b) mid-downswing (moderate lag angle), and (c) near impact (small lag angle). Discuss how the coupling changes and what this implies for wrist release mechanics.
Re-express the model-reported elbow result from Hirashima et al. (2008) as a term-magnitude comparison. List the additional metadata needed before comparing that result with a control-affine implementation, and explain why neither wording establishes a unique cause.
Consider the model-reported soleus and gastrocnemius terms in Neptune et al. (2001). Identify which parts require an actuator map, contact model, energy calculation, and residual closure. Design a golf study that tests an analogous hypothesis without importing the walking result as anatomical or coaching authority.