Parallel Mechanisms and Loop Constraints

The golf swing combines articulated chains, closed contact paths, muscle-generated loads, and elastic deformation. A useful model must say which connections it represents before it counts freedoms or interprets forces. The two-handed grip closes a path through the arms and club. Two supported feet close another through the legs and ground. The spine connects pelvis and thorax along an articulated path; that connection alone does not form a loop.

The central question is how these connections change the motion produced by a specified set of inputs, and what measurements can distinguish different ways of producing the same motion. Geometry defines feasible motion. Inertia and applied loads determine its acceleration. Contact reactions enforce the declared contact model. Muscle and tissue mechanics determine available forces and energy. None of those roles can be inferred merely by calling the golfer a parallel mechanism.

Draw the Actual Mechanical Graph

Represent each rigid segment as a graph vertex and each joint or maintained contact as an edge. A serial chain has one unbranched path. An open articulated tree can branch without containing a cycle. A closed chain contains a cycle: there must be a distinct return path, not an instruction to go back along the same spine or arm. Serial models can have strongly coupled dynamics even when their joint coordinates are independently selectable.

For a fixed-torso upper-body model, trace the cycle as thorax, left upper arm, left forearm/hand, club, right hand/forearm, right upper arm, thorax. Hand offsets, wrist joints, and shoulder-girdle motion can be resolved further when needed. For the lower body, trace ground, left foot, left leg, pelvis, right leg, right foot, ground. The spine links the pelvis to the thorax between these structures. Removing one hand contact opens the upper loop; removing one foot contact changes the lower contact structure. A raised heel can change the admissible contact wrench without completely detaching the foot.

Two closed paths pass through both arms and the club, and both legs and the ground. The articulated spine connects thorax and pelvis without forming an additional cycle. — Figure 1: Actual Closed Paths and the Connecting Spine Path

A pelvis–spine–thorax sequence is an open path. Defining a relative angle between pelvis and thorax does not add a physical return connection. Soft tissue can introduce additional force paths, but a tendon spanning joints is usually represented by a length-dependent force law, not an inextensible closure constraint. Its finite stiffness must not be silently converted into a rigid kinematic restriction.

The distinction agrees with the humanoid examples in the Modern Robotics constrained-dynamics treatment. The same geometry can be represented by independent joint coordinates plus loop constraints or by a larger set of segment poses plus joint constraints. Mixing those two counts subtracts the same restriction twice.

Mobility and Independent Constraints

Let $q\in\mathbb R^n$ be local coordinates for an open-tree model. Suppose the additional, time-independent holonomic closures are \[ \Phi(q)=0,\qquad J_c(q)=D_q\Phi(q). \tag{1}\] If $J_c$ has constant rank $r$ near a feasible configuration, the local configuration manifold has dimension $n-r$. This is a regular-point statement. At a singular point, the linearized null space can include directions that do not extend to a finite feasible motion; higher-order closure conditions matter. Neither an actuator being passive nor a muscle being inactive adds a kinematic constraint by itself.

For a connected linkage with $N$ bodies including one grounded body, $J$ joints, and $f_i$ freedoms at joint $i$, the generic Gruebler count is \[ d_{\mathrm{generic}}=s(N-1)-\sum_{i=1}^{J}(s-f_i), \qquad s=3\ \text{(planar)},\quad s=6\ \text{(spatial)}. \tag{2}\] It assumes that the counted joint restrictions are independent. In a planar linkage made only of one-freedom revolute or prismatic joints this becomes $d_{\mathrm{generic}}=3(N-1)-2J$. A four-bar with four bodies including ground and four revolute joints therefore has one generic freedom. An unactuated revolute joint still has one freedom. A fixed contact removes freedoms; it is not a positive correction called a passive constraint.

A negative generic count is a reason to inspect the model, independence, and geometry. It is not proof that the actual mechanism is immobile or that tissue elasticity is the only reason a human can move. Omitted arm or leg bodies, counting joint freedoms as joints, and mixing planar and spatial restrictions can all produce meaningless results. Special rigid mechanisms can move because some restrictions are dependent.

A Stewart-Platform Count With All the Parts Included

Consider a six-leg UPS platform: each leg has a universal joint with two freedoms, a prismatic joint with one, and a spherical joint with three. Each leg contains two telescoping rigid members. Including base and moving platform, $N=14$, $J=18$, and $\sum_i f_i=6(2+1+3)=36$. Thus \[ d_{\mathrm{UPS}}=6(14-1-18)+36=6. \tag{3}\] Away from singularities, these six freedoms can describe the moving platform’s pose. Locking the six independent prismatic lengths fixes that pose locally. Actual platforms may use different joint architectures; their counts must follow their own joints rather than the word hexapod.

In particular, two platforms connected by six fixed-length rods with spherical joints at both ends give $N=8$, $J=12$, and a count of six. Those six freedoms are the axial spins of the rods, not six free motions of the platform. The six fixed distance conditions generically fix the platform pose. This example shows why total mechanism mobility, internal passive motion, and output mobility must be distinguished. The joint-counting derivation provides the general starting point; the physical interpretation still requires examining the particular linkage.

Grip Closure With Separated Hand Frames

A rigid two-handed grip fixes each hand frame relative to a different location and orientation on the club. It does not require the hands to coincide. Let $T_L(q)$ and $T_R(q)$ be hand poses in a common world frame, $T_C$ the club pose, and $H_L,H_R$ the constant poses of the hand frames in club coordinates. Then \[ T_L=T_C H_L,\qquad T_R=T_C H_R,\qquad T_L^{-1}T_R=H_L^{-1}H_R. \tag{4}\] The last relation supplies six local independent restrictions at a regular spatial grasp. Pose coordinates require a nonsingular local representation; three position equations alone do not enforce relative orientation. Grip compliance, rolling, or slip replaces some of these ideal equalities with an appropriate contact model.

For two seven-coordinate arms attached to a fixed torso, there are 14 arm coordinates before the grip closes. Seven is a modeling choice; it commonly resolves shoulder rotation, elbow flexion, forearm rotation, and wrist motion. It is not a claim that the wrist alone supplies all three distal rotations. With six independent relative-pose restrictions, the assembly has $14-6=8$ local freedoms. Equivalently, adding a free club contributes six coordinates and fixing each hand to it adds twelve independent restrictions: \[ d=14+6-12=8. \tag{5}\] The fixed torso was already assumed. Subtracting additional pelvis or spine restrictions from this upper-body count is double counting. A whole-body model must instead add its pelvis, trunk, legs, and contact coordinates explicitly. Eight kinematic freedoms also do not mean eight independently available neural commands or eight controllable directions over a finite time interval.

A Planar Example You Can Differentiate

For a deliberately simpler coincident-point grip, fix shoulder bases at $b_L=(-1,0)^T$ and $b_R=(1,0)^T$, in meters. Give each arm two unit-length links and relative elbow coordinates. Its hand position is \[ p_i=b_i+ \begin{bmatrix}\cos q_{i1}+\cos(q_{i1}+q_{i2})\\ \sin q_{i1}+\sin(q_{i1}+q_{i2})\end{bmatrix},\qquad i=L,R. \tag{6}\] The point closure is $\Phi=p_L-p_R=0$, with $J_c=[J_L\;-J_R]$ and $J_i=D_{q_i}p_i$. The difference is essential: fixing the sum of hand positions would allow the hands to move apart while their midpoint stayed fixed.

At $q_L=(0,\pi/2)$ and $q_R=(\pi,-\pi/2)$, both hands are at $(0,1)$ and \[ J_L=\begin{bmatrix}-1&-1\\1&0\end{bmatrix},\quad J_R=\begin{bmatrix}-1&-1\\-1&0\end{bmatrix},\quad J_c=\begin{bmatrix}-1&-1&1&1\\1&0&1&0\end{bmatrix}. \tag{7}\] Here $J_c$ has rank two, leaving two freedoms. The velocity $\dot q=(1,-1,-1,1)^T$ satisfies $J_c\dot q=0$, yet both hands move upward at $1$ meter per second. Preserving closure does not mean holding the club still. A separated rigid club requires its pose and attachment offsets as above; the point example is a lower-fidelity model, not an anatomical measurement.

Three Different Null Spaces

The constraint null space $\ker J_c$ contains compatible velocities. For a stationary holonomic closure, \[ J_c\dot q=0,\qquad J_c\ddot q+\dot J_c\dot q=0. \tag{8}\] The second equation includes the change in tangent directions along the motion. Dropping $\dot J_c\dot q$ can remove required normal acceleration even when the velocity satisfies closure perfectly.

A task has a different Jacobian. Let $y=\psi(q)$ describe the chosen output and $J_y=D_q\psi$. Motions that preserve both closure and this output lie in $\ker J_c\cap\ker J_y$. If the columns of $Z$ form a basis of $\ker J_c$, then $\dot q=Z\nu$ and $\dot y=J_yZ\nu$. The task-preserving dimension is \[ (n-r)-\operatorname{rank}(J_yZ). \tag{9}\] For the regular spatial arm example, an independent six-dimensional club-pose task leaves two posture freedoms out of eight. Holding only club-center position leaves orientation unspecified. A face-angle change can preserve that position task while changing the impact task. It is not invisible to every meaningful output, and kinematic freedom does not establish negligible actuation cost.

For a feasible desired task velocity, a weighted minimum-velocity construction uses $H=J_yZ$ and a positive-definite coordinate weight $W$: \[ \nu_* = W^{-1}H^T(HW^{-1}H^T)^{-1}\dot y_{\mathrm{des}}. \tag{10}\] This formula requires full row rank of $H$. It minimizes $\tfrac12\nu^TW\nu$ subject to $H\nu=\dot y_{\mathrm{des}}$. It does not minimize joint torque, muscle activation, or metabolic cost. Rank-deficient or bounded problems need a suitable pseudoinverse or constrained optimization, with explicit handling of infeasible requests.

Finally, a grasp map $G$ maps contact-load coordinates to a resultant club wrench. Its null space $\ker G$ contains self-equilibrated load changes. Those are forces, not joint velocities. Even the multiplier null space $\ker J_c^T$ is a different object: it describes redundant multiplier representations that produce the same generalized reaction. When $J_c$ has independent rows, $\ker J_c^T=\{0\}$, although a bilateral grip can still have unresolved internal loads when applied inputs are unknown.

Constraint Reactions and the Forward Dynamics

Use minimal open-tree coordinates with $M(q)$ positive definite, input map $B(q)$, and input vector $u$. Put velocity-dependent inertial terms and gravity in $h(q,\dot q)$. Keep passive tissue loads and other applied loads explicitly in $p(q,\dot q,z)$, where $z$ denotes any additional constitutive states. For a fixed, ideal contact mode, \[ M\ddot q+h=Bu+p+J_c^T\lambda. \tag{11}\] Do not put $h$ on the left and then subtract it again inside an additional right-hand-side drift torque. Different bookkeeping conventions are possible, but each physical term must occur exactly once.

Set $a_0=M^{-1}(Bu+p-h)$ and $A=J_cM^{-1}J_c^T$. If $J_c$ has independent rows, $A$ is positive definite. Substitution into the acceleration constraint yields \[ \lambda=-A^{-1}(J_c a_0+\dot J_c\dot q),\qquad \ddot q=a_0-M^{-1}J_c^TA^{-1}(J_c a_0+\dot J_c\dot q). \tag{12}\] The acceleration correction depends on the mass matrix as well as the geometry. The matrix $I-M^{-1}J_c^TA^{-1}J_c$ acts on velocities or acceleration increments; its transpose acts on generalized-force covectors. They cannot be interchanged merely because both are called projections.

This solution is unique for the specified model, state, applied inputs, and regular active equality constraints. Redundant constraints can leave multiplier coordinates nonunique. Unilateral contacts can be infeasible for the computed reaction; frictional contact can require a different mode or contact law. Calling a problem forward dynamics does not remove those conditions.

Reactions Depend on Applied Muscle Loads

For a two-coordinate illustration take $M=\operatorname{diag}(2,1)$, $J_c=[1\;-1]$, $h=p=0$, and $B=I$. The two coordinates must accelerate together. With $u=(1,0)^T$, the solution is \[ \ddot q=(1/3,1/3)^T,\qquad \lambda=-1/3. \tag{13}\] With $u=(2,-1)^T$, the acceleration is unchanged and $\lambda=-4/3$. An additional applied load in $\operatorname{range}(J_c^T)$ is balanced by a changed reaction. Geometry determines the admissible reaction directions, not their magnitudes independently of actuation. A reaction may be zero or large depending on the state and inputs; the existence or number of loops alone gives no force scale.

In inverse dynamics, the observed motion supplies the balance \[ [B\;J_c^T]\begin{bmatrix}u\\\lambda\end{bmatrix} =M\ddot q+h-p. \tag{14}\] Uniqueness depends on the rank of this identification map and on additional measurements and restrictions. Joint-level net torques can be ambiguous as well as individual muscle forces and hand loads. Known segment inertias and motion fix inertial demands; they do not generally separate all unknown sources.

Club Wrenches and Bilateral Load Sharing

For a rigid club, express all vectors in a common inertial frame and take moments about its center of mass $C$. Let $r_i$ point from $C$ to hand contact $i$, $f_i$ be its force on the club, and $m_i$ its contact couple. Then \[ \begin{aligned} m_c a_C &= f_L+f_R+m_c g+f_{\mathrm{other}},\\ I_C\dot\omega+\omega\times(I_C\omega) &=r_L\times f_L+r_R\times f_R+m_L+m_R+m_{\mathrm{other},C}. \end{aligned} \tag{15}\] The current rotational inertia tensor $I_C$ is expressed in that same frame. Other loads include aerodynamic forces or ball contact when applicable. The hand points of a rotating club generally have different linear accelerations: $a_i=a_C+\dot\omega\times r_i+\omega\times(\omega\times r_i)$. Sharing a rigid body does not make those accelerations equal.

Stack each hand load as $w_i=(m_i,f_i)$ and transport its moment to $C$ before summing. The resulting $6\times12$ grasp map $G$ has rank six if both contacts are modeled as unrestricted full wrenches. For a compatible required wrench, \[ w=G^+w_{\mathrm{req}}+N_G\eta,\qquad \operatorname{range}(N_G)=\ker G, \tag{16}\] so there are six unconstrained load-sharing parameters before friction, unilateral pressure, moment capacity, and actuator limits are imposed. Those physical restrictions can reduce the feasible set or make it empty; an arbitrary algebraic split need not be realizable by a hand.

If each contact transmits only a point force, the map is $6\times6$. For two distinct points it generically has rank five, because equal opposite forces along the joining line produce zero resultant wrench. The null dimension is then one, and arbitrary required spatial wrenches cannot be supported. Grip models that allow contact couples and distributed pressure have different capabilities.

A simple scalar illustration is $\tau_L+\tau_R=10$ N m, with both torques expressed about the same axis and reference point after all other terms are accounted for. The pairs $(4,6)$ and $(7,3)$ satisfy the same balance. A minimum-squared-torque choice gives $(5,5)$, but that is an optimization assumption, not an observation of how a golfer shared the load. Relative EMG amplitudes do not by themselves supply the missing torque ratio. Calibrated instrumented grips, muscle models, or additional experimental perturbations can constrain the inference.

Force Transmission, Power, and Internal Loading

A twist is a six-component angular/linear velocity representation, not generally a unit vector. With the matching wrench ordering, $V=(\omega,v)$ and $w=(m,f)$ pair to give power $w^TV=m\cdot\omega+f\cdot v$. Both must refer to compatible points and frames. A change of point changes the moment and linear velocity together, preserving power.

For an ideal rigid internal attachment, the two bodies have the same contact twist and experience opposite wrenches. Their contact powers cancel when both bodies are included in the system. Each body separately can receive or lose power through that attachment. Thus the grip can transfer energy to the club while ideal closure reactions do zero total work on the golfer-plus-club system. In generalized coordinates this cancellation is \[ \dot q^TJ_c^T\lambda=\lambda^TJ_c\dot q=0. \tag{17}\] A constraint can redirect acceleration while doing no total work. An internal load need not produce mechanical work merely because it stresses tissue. Muscle maintenance can have a metabolic cost even when modeled joint power is zero; local tissue deformation and injury outcomes require their own measurements and constitutive models.

Ground Contact: Wrench, Momentum, and Energy

Ground reaction is an external contact wrench on the golfer-plus-club system. Its value depends on the motion, geometry, muscle loading, and contact conditions. It is not independent of muscle action, nor is it an additional muscle force. For a simple unilateral contact with normal gap $\phi_n$, the ideal conditions include $\phi_n\ge0$, $\lambda_n\ge0$, and $\phi_n\lambda_n=0$. A sticking Coulomb contact also requires $\|f_t\|\le\mu\lambda_n$. Distributed foot contact has moment and center-of-pressure restrictions in addition to this force cone. Slip, lift-off, shoe deformation, and moving support change the equations.

For the center of mass $C$ of the selected whole system and angular momentum $H_C$, the external balances are \[ \begin{aligned} m_{\mathrm{tot}}a_C &=m_{\mathrm{tot}}g+\sum_i f_i+f_{\mathrm{other}},\\ \dot H_C &=\sum_i\big((r_i-C)\times f_i+m_i\big) +m_{\mathrm{other},C}. \end{aligned} \tag{18}\] Here $r_i$ is the contact position in world coordinates, whereas the preceding club balance used offsets from the club center. Uniform gravity contributes no net moment about the whole-system center of mass. These equations constrain whole-system momentum, not the pelvis acceleration alone. Hip, spine, and joint reactions exchange angular momentum among segments. Even with zero total external moment, one segment can rotate while others counter-rotate. A free moment measured at a foot is therefore not a complete explanation of pelvis motion or proof of a universally passive shoulder.

A Force Plate’s Reference Point Matters

Declare a right-handed frame with $z$ upward, and let $(F,M_O)$ be the wrench of the ground on the golfer measured about an origin $O$ in the contact plane. For $F_z\ne0$, a center of pressure $r=(x,y,0)$ is defined so that the remaining moment there has only a vertical component. Since $M_O=r\times F+T_z e_z$, \[ x=-\frac{M_y}{F_z},\qquad y=\frac{M_x}{F_z},\qquad T_z=M_z-(xF_y-yF_x). \tag{19}\] Thus the plate-origin yaw moment $M_z$ is not generally the free moment $T_z$. If the contact plane is at height $h$ relative to $O$, use $x=(hF_x-M_y)/F_z$ and $y=(M_x+hF_y)/F_z$. Vendor outputs may use the opposite force-on-plate sign or different axes; transform the full wrench before applying these formulas. Near zero normal load the division is ill-conditioned, so state a justified contact threshold and report unloaded intervals separately.

For a numerical check, a force $F=(20,30,800)$ N applied at $r=(0.2,-0.1,0)$ m with free moment $T_z=5$ N m gives $M_O=(-80,-160,13)$ N m. The formulas recover $x=0.2$ m, $y=-0.1$ m, and $T_z=13-(6+2)=5$ N m. Calling the measured $13$ N m the free moment would conflate the horizontal-force lever arm with the residual couple.

COP is pressure-weighted position, not the center of mass or a direct measure of transferred body mass. A changing foot-load share can move COP while the center of mass barely moves. With two coplanar plates, transform the wrenches to one frame and origin before combining them. Separate plates supply each foot’s boundary wrench; one combined plate generally leaves bilateral sharing unresolved. Six-axis measurements are wrench components, not six motion DOFs. Calibration, cross-talk, sampling, synchronization, and filtering affect accuracy; a universal instrument-accuracy percentage cannot substitute for calibration.

With each foot wrench known, segment poses and inertial parameters allow a Newton–Euler recursion from foot to shank to thigh. Transfer both force and moment, including their lever arms, at each step. This gives model-conditioned net joint loads when other external loads and joint assumptions are specified. It does not uniquely identify individual muscle forces or tissue stress.

Support Can Enable Work Without Supplying Energy

A fully stationary rigid foot has zero velocity at its ground-contact points and zero angular velocity. In this idealization, the ground wrench has zero mechanical power, although its force and moment can change the whole-system momentum. Muscle work can increase body and club energy while ground support prevents an incompatible foot motion. The resultant external force dotted with center-of-mass velocity describes center-of-mass translational-energy change; it is not the contact power of that force on a deformable or articulated system.

For time-independent elastic potentials and ideal stationary contacts, an energy balance has the form \[ \frac{d}{dt}(T+V_g+V_e) =\dot q^TBu+P_{\mathrm{other}}-P_{\mathrm{diss}}, \qquad P_{\mathrm{diss}}\ge0. \tag{20}\] Moving support, slipping friction, changing elastic rest states, and contact impacts require their corresponding boundary or constitutive power terms. Neither a large ground force nor an early force peak establishes the source, amount, or direction of energy transferred to the club. A segment power audit must state the system boundaries, force points, velocities, and sign conventions.

There is no single force-plate curve in this model that defines a good golfer. Lead-foot and combined forces, normalization by body weight, club choice, swing phase, and target all change a comparison. A claim that the best peak must occur exactly at impact needs an identified study and outcome definition, not an illustrative sketch. The chapter’s equations specify what must balance; they do not provide a coaching threshold or an injury-risk cutoff.

X-Factor, Compliance, and Muscle State

In a planar illustration, define $\theta_p$ and $\theta_t$ as pelvis and thorax angles about the same axis. Then $\Delta=\theta_t-\theta_p$ is a relative rotation, not a new loop-closure equation. For $\theta_p=60$ degrees and $\theta_t=20$ degrees, $\Delta=-40$ degrees under this convention. In three dimensions, use the relative orientation $R_p^TR_t$ and declare its reported angle convention. Subtracting two Euler angles or projected shoulder-line angles need not measure the same quantity.

Brown, Selbie, and Wallace’s X-factor methods study examines this measurement issue. A reported intersegment angle is not a direct measurement of lumbar rotation, disc compression, tissue strain, or stored elastic energy. Distribution among spinal levels, rib-cage/shoulder motion, posture, and muscle activation matter. An angle associated with performance is also not by itself evidence that increasing that angle improves an individual swing or is safe for that individual.

For a clearly declared toy torsional spring, \[ V_e=\tfrac12k(\Delta-\Delta_0)^2,\qquad Q_t=-k(\Delta-\Delta_0),\qquad Q_p=+k(\Delta-\Delta_0). \tag{21}\] The two torques are opposite, and their combined power is $-k(\Delta-\Delta_0)\dot\Delta=-\dot V_e$ when $k$ and $\Delta_0$ are fixed. Doubling displacement quadruples this model’s energy only if the stiffness and rest angle stay the same. It does not establish a universal human tissue law or prove that the stored energy reaches the club rather than another segment or dissipation. If stiffness is actively changed, the energy derivative also contains $\tfrac12\dot k(\Delta-\Delta_0)^2$; changing a rest angle adds another term. An impedance schedule is not energetically free by definition.

Joint range of motion, tangent stiffness, damping, geometric constraint rank, and neural feedback are distinct. Finite stiffness resists displacement without reducing the exact kinematic dimension. It can make some motions small over a specified loading range and time scale. A rigid-constraint approximation must be justified against those scales, contact margins, and output errors.

Muscle excitation, activation, fiber length, shortening velocity, and tendon stretch must likewise be distinguished. Eccentric contraction means active lengthening; active shortening is concentric. A tendon can recoil while its muscle fibers behave differently. Treating all release-phase tissue forces as passive because club speed is high is not justified. For a muscle model, $\tau=R(q)F_m(q,\dot q,z)$ maps muscle forces through moment arms; multiplying moment arms by dimensionless activation alone does not produce torque.

Constrained Drift and Forward Counterfactuals

At a fixed admissible state and contact mode, reaction elimination gives \[ \begin{aligned} \ddot q &= f_c(q,\dot q,z)+G_c(q)u,\\ G_c &=\big(I-M^{-1}J_c^TA^{-1}J_c\big)M^{-1}B. \end{aligned} \tag{22}\] The input-independent term includes gravity, velocity-dependent dynamics, retained passive forces, and the geometric acceleration correction. It depends on the declared inputs and state variables. This is the useful local affine structure: closure reshapes both drift and the effect of an input. Removing $u$ does not remove the reactions needed to satisfy the remaining contacts.

Computing these terms requires a model, inertial parameters, calibrated kinematics, and any relevant tissue or contact state. Motion capture alone does not measure masses, inertias, tendon preload, or the chosen decomposition. Although gravitational force can be evaluated before solving for reactions, the resulting constrained gravitational acceleration differs from free fall. Observable motion and model-computable force terms are different evidential categories.

For a particle observed in a rotating frame, the centrifugal and Coriolis fictitious forces are \[ F_{\mathrm{cf}}=-m\omega\times(\omega\times r),\qquad F_{\mathrm{cor}}=-2m\omega\times v_{\mathrm{rel}}. \tag{23}\] An accelerating origin and changing frame angular velocity introduce translational and Euler terms as well. These expressions cannot be added to inertial-frame loads without the matching frame transformation; that would double count the kinematics. Generalized velocity-dependent terms in an articulated model are not automatically identical to one particle’s rotating-frame force.

For a hypothetical $3$ kg point mass rotating at $15$ rad/s on a $0.6$ m radius, $m\omega^2r=405$ N. About that same center, however, the radial force has zero moment because $r\times F_{\mathrm{cf}}=0$. It has zero power on purely tangential relative motion. About another joint it can have a moment, but the perpendicular lever arm and vector directions must be computed. Multiplying $405$ N by an arbitrary segment distance does not establish a shoulder torque.

Similarly, $F_{\mathrm{cor}}\cdot v_{\mathrm{rel}}=0$: the Coriolis term is perpendicular to the relative velocity and is not dissipative resistance to it. For mutually perpendicular $\omega$ and $v_{\mathrm{rel}}$, with magnitudes $10$ rad/s and $8$ m/s and mass $3$ kg, its magnitude is $480$ N. That conditional number is not an observed golf force. A $3$ kg segment weighs $29.43$ N in $9.81$ m/s$^2$ gravity. Neither example establishes a universal ratio of inertial to muscle torque, and ratios of load magnitudes are not fractions of work or clubhead-speed causation.

A forward zero-torque counterfactual (forward ZTCF) branches from a specified initial state and changes a declared torque input or policy. It then evolves the state and recomputes compatible reactions, contact events, and retained passive forces. It must not reset the measured state at every time sample and call the resulting instantaneous accelerations a new trajectory. If the inferred baseline torques or contact model are uncertain, that uncertainty carries into the forward prediction. Forward simulation avoids solving for unspecified inputs during integration; it does not establish that its initial inputs, parameters, or biological interpretation are correct.

Kinematics, Singularities, and Feasible Control

When all compatible open-tree joint coordinates are known, their link poses can be computed recursively and the closure residual checked. If only some actuated coordinates are specified, passive coordinates and platform pose may require solving nonlinear closure equations. Multiple isolated assembly modes are not the same as continuous redundancy or a Jacobian singularity. Inverse kinematics asks for coordinates achieving a specified pose; its difficulty depends on the architecture. For a Stewart-type platform, computing leg lengths from a pose can be much simpler than computing pose from lengths. Constraint forces are not unknowns in a purely kinematic solve.

For local input coordinates $s$ and output coordinates $x$, write differentiated closure as $A_x\dot x+B_s\dot s=0$. In a square nonredundant mechanism, the Gosselin–Angeles classification distinguishes rank loss of the input-side matrix $B_s$ (first kind), output-side matrix $A_x$ (second kind), and simultaneous loss (third kind). When $A_x$ is singular, a nonzero output velocity may satisfy the linearized equations with locked inputs. Architecture singularity concerns geometric design that imposes persistent rank deficiency; it is not a name for any transient unusual posture.

Standing still at address does not demonstrate a singularity, because these ranks depend on geometry rather than whether velocity happens to be zero. A straight two-link planar arm has a singular endpoint-position Jacobian, but its elbow coordinate still exists. A mechanical stop, contact inequality, or a separately imposed lock changes the admissible motion in another way. Neither near extension nor a small singular value proves maximum impact efficiency or safe tissue loading. A first-order rank loss may require nonlinear analysis to determine actual finite self-motion.

Report which Jacobian is tested, its coordinate and unit scaling, the posture, and the numerical threshold. Conditioning of $J_yZ$ concerns task velocity; conditioning of $J_cM^{-1}J_c^T$ concerns reaction elimination; rank of $[B\;J_c^T]$ concerns inverse load identification. Their numerical values answer different questions. Local input rank also does not alone determine finite-time reachability, because drift, input bounds, time horizon, and contact transitions matter.

An Investigation That Connects the Pieces

The defensible research hypothesis is that preparation and changing geometry alter how feasible muscle inputs and stored energy produce the impact state. The two-handed grip and ground contacts are essential parts of that mapping. They can transmit loads, restrict relative motion, and alter the effectiveness of an input without supplying unlimited energy or determining a unique neural strategy. To investigate the hypothesis:

Define the outcome: clubhead velocity and face orientation at contact, strike location, ball launch, or dispersion. A club-center path alone is not a complete impact task.
Declare the graph, joint coordinates, contact modes, hand-frame offsets, inertias, muscle/input representation, and elastic states. Check closure, velocity, and acceleration residuals with measurement uncertainty.
Synchronize segment kinematics, separate foot wrenches, and any hand-load measurements. Identify which aggregate loads are determined and which bilateral or muscle distributions remain compatible with the data.
Audit linear and angular momentum separately from segment power, total energy, and tissue loading. Report sign conventions and reference points. An acceleration contribution, work contribution, and injury hypothesis must not share a percentage without an explicit derivation.
Compare a baseline trajectory with declared forward interventions on torque, timing, grip compliance, or contact. Preserve the same starting state when testing a late intervention; change it deliberately when testing preparation. Include uncertainty in unmeasured load sharing and parameters.
Test predictions on held-out trials and controlled perturbations where feasible. Similar observed kinematics do not prove identical forces, and a model that reproduces a swing does not identify the golfer’s intention.

This approach can reveal when a change in grip loading mainly affects internal stress, when it changes club orientation, or when a different contact condition changes the admissible acceleration. It also allows a passive mechanism to be supported where the evidence supports it, while retaining the muscle work, feedback, and prior preparation needed to explain the entire swing. The energy-transfer chapter develops the next question: how energy moves between the chosen segments and storage elements.

Exercises and Checks

Draw the upper and lower loops and the connecting spine path. Remove one grip edge. Which cycle disappears? Why does defining a pelvis–thorax relative angle not restore it?
Reproduce the UPS platform count and explain the six axial rod spins in the fixed-length SPS example. State which freedoms move the platform.
Differentiate the planar two-arm positions. Verify both closure and common upward hand motion for the stated configuration and velocity. Then compute the stacked closure/task rank when the hand point is fixed.
Solve the two-coordinate reaction example for both applied loads. Verify identical accelerations, different reactions, and zero total ideal constraint power for a common admissible velocity.
Construct the two-point-force grasp map. Show that opposite forces along the joining line lie in its null space. Explain why opposite forces in an arbitrary direction generally leave a couple.
Recover COP and free moment from the numerical wrench. Repeat after moving the reference origin, transporting the moment consistently.
Derive the complete energy derivative of the torsional spring when both stiffness and rest angle vary. Identify which terms require a source or sink beyond the fixed-spring mechanical exchange.
Specify an experiment that distinguishes a grip-load change preserving the resultant club wrench from one that changes club orientation. List the measurements needed; explain why motion alone and raw EMG ratios do not resolve every candidate load distribution.

On This Site

The Null Space of the Constraint Jacobian — Loop constraints, rank deficiency, and reaction forces
Force and Mobility Ellipsoids — Task-space feasibility under closed kinematic chains

Cite this page

Citation

BibTeX citation: Download .bib

@online{untitled,
  author = {},
  url = {https://affinedrift.com/articles/The_Physics_of_Golf/quarto/ch09_parallel_mechanisms.html},
  langid = {en}
}

For attribution, please cite this work as:

n.d. https://affinedrift.com/articles/The_Physics_of_Golf/quarto/ch09_parallel_mechanisms.html.

--- pagetitle: Parallel Mechanisms and Loop Constraints description: "How grip and ground contact connect geometry, feasible motion, load sharing, muscle work, and golf outcomes." --- # Parallel Mechanisms and Loop Constraints {#sec-parallel_mechanisms} The golf swing combines articulated chains, closed contact paths, muscle-generated loads, and elastic deformation. A useful model must say which connections it represents before it counts freedoms or interprets forces. The two-handed grip closes a path through the arms and club. Two supported feet close another through the legs and ground. The spine connects pelvis and thorax along an articulated path; that connection alone does not form a loop. The central question is how these connections change the motion produced by a specified set of inputs, and what measurements can distinguish different ways of producing the same motion. Geometry defines feasible motion. Inertia and applied loads determine its acceleration. Contact reactions enforce the declared contact model. Muscle and tissue mechanics determine available forces and energy. None of those roles can be inferred merely by calling the golfer a parallel mechanism. ## Draw the Actual Mechanical Graph {#sec-serial_vs_parallel} Represent each rigid segment as a graph vertex and each joint or maintained contact as an edge. A serial chain has one unbranched path. An open articulated tree can branch without containing a cycle. A closed chain contains a cycle: there must be a distinct return path, not an instruction to go back along the same spine or arm. Serial models can have strongly coupled dynamics even when their joint coordinates are independently selectable. For a fixed-torso upper-body model, trace the cycle as thorax, left upper arm, left forearm/hand, club, right hand/forearm, right upper arm, thorax. Hand offsets, wrist joints, and shoulder-girdle motion can be resolved further when needed. For the lower body, trace ground, left foot, left leg, pelvis, right leg, right foot, ground. The spine links the pelvis to the thorax between these structures. Removing one hand contact opens the upper loop; removing one foot contact changes the lower contact structure. A raised heel can change the admissible contact wrench without completely detaching the foot. ![Actual Closed Paths and the Connecting Spine Path](../figures/parallel_mechanical_graph.png){#fig-parallel_mechanical_graph fig-alt="Two closed paths pass through both arms and the club, and both legs and the ground. The articulated spine connects thorax and pelvis without forming an additional cycle."} A pelvis–spine–thorax sequence is an open path. Defining a relative angle between pelvis and thorax does not add a physical return connection. Soft tissue can introduce additional force paths, but a tendon spanning joints is usually represented by a length-dependent force law, not an inextensible closure constraint. Its finite stiffness must not be silently converted into a rigid kinematic restriction. The distinction agrees with the humanoid examples in the [Modern Robotics constrained-dynamics treatment](https://modernrobotics.northwestern.edu/nu-gm-book-resource/8-7-constrained-dynamics/). The same geometry can be represented by independent joint coordinates plus loop constraints or by a larger set of segment poses plus joint constraints. Mixing those two counts subtracts the same restriction twice. ## Mobility and Independent Constraints {#sec-gruebler_formula} Let $q\in\mathbb R^n$ be local coordinates for an open-tree model. Suppose the additional, time-independent holonomic closures are $$ \Phi(q)=0,\qquad J_c(q)=D_q\Phi(q). $$ {#eq-loop_jacobian_definition} If $J_c$ has constant rank $r$ near a feasible configuration, the local configuration manifold has dimension $n-r$. This is a regular-point statement. At a singular point, the linearized null space can include directions that do not extend to a finite feasible motion; higher-order closure conditions matter. Neither an actuator being passive nor a muscle being inactive adds a kinematic constraint by itself. For a connected linkage with $N$ bodies including one grounded body, $J$ joints, and $f_i$ freedoms at joint $i$, the generic Gruebler count is $$ d_{\mathrm{generic}}=s(N-1)-\sum_{i=1}^{J}(s-f_i), \qquad s=3\ \text{(planar)},\quad s=6\ \text{(spatial)}. $$ {#eq-gruebler_spatial} It assumes that the counted joint restrictions are independent. In a planar linkage made only of one-freedom revolute or prismatic joints this becomes $d_{\mathrm{generic}}=3(N-1)-2J$. A four-bar with four bodies including ground and four revolute joints therefore has one generic freedom. An unactuated revolute joint still has one freedom. A fixed contact removes freedoms; it is not a positive correction called a passive constraint. A negative generic count is a reason to inspect the model, independence, and geometry. It is not proof that the actual mechanism is immobile or that tissue elasticity is the only reason a human can move. Omitted arm or leg bodies, counting joint freedoms as joints, and mixing planar and spatial restrictions can all produce meaningless results. Special rigid mechanisms can move because some restrictions are dependent. ### A Stewart-Platform Count With All the Parts Included Consider a six-leg UPS platform: each leg has a universal joint with two freedoms, a prismatic joint with one, and a spherical joint with three. Each leg contains two telescoping rigid members. Including base and moving platform, $N=14$, $J=18$, and $\sum_i f_i=6(2+1+3)=36$. Thus $$ d_{\mathrm{UPS}}=6(14-1-18)+36=6. $$ {#eq-stewart_mobility} Away from singularities, these six freedoms can describe the moving platform's pose. Locking the six independent prismatic lengths fixes that pose locally. Actual platforms may use different joint architectures; their counts must follow their own joints rather than the word hexapod. In particular, two platforms connected by six fixed-length rods with spherical joints at both ends give $N=8$, $J=12$, and a count of six. Those six freedoms are the axial spins of the rods, not six free motions of the platform. The six fixed distance conditions generically fix the platform pose. This example shows why total mechanism mobility, internal passive motion, and output mobility must be distinguished. The [joint-counting derivation](https://modernrobotics.northwestern.edu/nu-gm-book-resource/2-2-degrees-of-freedom-of-a-robot/) provides the general starting point; the physical interpretation still requires examining the particular linkage. ## Grip Closure With Separated Hand Frames {#sec-grip_parallel_constraint} A rigid two-handed grip fixes each hand frame relative to a different location and orientation on the club. It does not require the hands to coincide. Let $T_L(q)$ and $T_R(q)$ be hand poses in a common world frame, $T_C$ the club pose, and $H_L,H_R$ the constant poses of the hand frames in club coordinates. Then $$ T_L=T_C H_L,\qquad T_R=T_C H_R,\qquad T_L^{-1}T_R=H_L^{-1}H_R. $$ {#eq-grip_spatial_constraint} The last relation supplies six local independent restrictions at a regular spatial grasp. Pose coordinates require a nonsingular local representation; three position equations alone do not enforce relative orientation. Grip compliance, rolling, or slip replaces some of these ideal equalities with an appropriate contact model. For two seven-coordinate arms attached to a fixed torso, there are 14 arm coordinates before the grip closes. Seven is a modeling choice; it commonly resolves shoulder rotation, elbow flexion, forearm rotation, and wrist motion. It is not a claim that the wrist alone supplies all three distal rotations. With six independent relative-pose restrictions, the assembly has $14-6=8$ local freedoms. Equivalently, adding a free club contributes six coordinates and fixing each hand to it adds twelve independent restrictions: $$ d=14+6-12=8. $$ {#eq-two_arm_dof_with_grip} The fixed torso was already assumed. Subtracting additional pelvis or spine restrictions from this upper-body count is double counting. A whole-body model must instead add its pelvis, trunk, legs, and contact coordinates explicitly. Eight kinematic freedoms also do not mean eight independently available neural commands or eight controllable directions over a finite time interval. ### A Planar Example You Can Differentiate For a deliberately simpler coincident-point grip, fix shoulder bases at $b_L=(-1,0)^T$ and $b_R=(1,0)^T$, in meters. Give each arm two unit-length links and relative elbow coordinates. Its hand position is $$ p_i=b_i+ \begin{bmatrix}\cos q_{i1}+\cos(q_{i1}+q_{i2})\\ \sin q_{i1}+\sin(q_{i1}+q_{i2})\end{bmatrix},\qquad i=L,R. $$ {#eq-two_arm_positions} The point closure is $\Phi=p_L-p_R=0$, with $J_c=[J_L\;-J_R]$ and $J_i=D_{q_i}p_i$. The difference is essential: fixing the sum of hand positions would allow the hands to move apart while their midpoint stayed fixed. At $q_L=(0,\pi/2)$ and $q_R=(\pi,-\pi/2)$, both hands are at $(0,1)$ and $$ J_L=\begin{bmatrix}-1&-1\\1&0\end{bmatrix},\quad J_R=\begin{bmatrix}-1&-1\\-1&0\end{bmatrix},\quad J_c=\begin{bmatrix}-1&-1&1&1\\1&0&1&0\end{bmatrix}. $$ {#eq-two_arm_loop_jacobian} Here $J_c$ has rank two, leaving two freedoms. The velocity $\dot q=(1,-1,-1,1)^T$ satisfies $J_c\dot q=0$, yet both hands move upward at $1$ meter per second. Preserving closure does not mean holding the club still. A separated rigid club requires its pose and attachment offsets as above; the point example is a lower-fidelity model, not an anatomical measurement. ## Three Different Null Spaces {#sec-constraint_jacobian_nullspace} The constraint null space $\ker J_c$ contains compatible velocities. For a stationary holonomic closure, $$ J_c\dot q=0,\qquad J_c\ddot q+\dot J_c\dot q=0. $$ {#eq-velocity_constraint} The second equation includes the change in tangent directions along the motion. Dropping $\dot J_c\dot q$ can remove required normal acceleration even when the velocity satisfies closure perfectly. A task has a different Jacobian. Let $y=\psi(q)$ describe the chosen output and $J_y=D_q\psi$. Motions that preserve both closure and this output lie in $\ker J_c\cap\ker J_y$. If the columns of $Z$ form a basis of $\ker J_c$, then $\dot q=Z\nu$ and $\dot y=J_yZ\nu$. The task-preserving dimension is $$ (n-r)-\operatorname{rank}(J_yZ). $$ {#eq-task_null_dimension} For the regular spatial arm example, an independent six-dimensional club-pose task leaves two posture freedoms out of eight. Holding only club-center position leaves orientation unspecified. A face-angle change can preserve that position task while changing the impact task. It is not invisible to every meaningful output, and kinematic freedom does not establish negligible actuation cost. For a feasible desired task velocity, a weighted minimum-velocity construction uses $H=J_yZ$ and a positive-definite coordinate weight $W$: $$ \nu_* = W^{-1}H^T(HW^{-1}H^T)^{-1}\dot y_{\mathrm{des}}. $$ {#eq-pseudoinverse_solution} This formula requires full row rank of $H$. It minimizes $\tfrac12\nu^TW\nu$ subject to $H\nu=\dot y_{\mathrm{des}}$. It does not minimize joint torque, muscle activation, or metabolic cost. Rank-deficient or bounded problems need a suitable pseudoinverse or constrained optimization, with explicit handling of infeasible requests. Finally, a grasp map $G$ maps contact-load coordinates to a resultant club wrench. Its null space $\ker G$ contains self-equilibrated load changes. Those are forces, not joint velocities. Even the multiplier null space $\ker J_c^T$ is a different object: it describes redundant multiplier representations that produce the same generalized reaction. When $J_c$ has independent rows, $\ker J_c^T=\{0\}$, although a bilateral grip can still have unresolved internal loads when applied inputs are unknown. ## Constraint Reactions and the Forward Dynamics {#sec-drift_control_parallel} Use minimal open-tree coordinates with $M(q)$ positive definite, input map $B(q)$, and input vector $u$. Put velocity-dependent inertial terms and gravity in $h(q,\dot q)$. Keep passive tissue loads and other applied loads explicitly in $p(q,\dot q,z)$, where $z$ denotes any additional constitutive states. For a fixed, ideal contact mode, $$ M\ddot q+h=Bu+p+J_c^T\lambda. $$ {#eq-drift_control_constrained} Do not put $h$ on the left and then subtract it again inside an additional right-hand-side drift torque. Different bookkeeping conventions are possible, but each physical term must occur exactly once. Set $a_0=M^{-1}(Bu+p-h)$ and $A=J_cM^{-1}J_c^T$. If $J_c$ has independent rows, $A$ is positive definite. Substitution into the acceleration constraint yields $$ \lambda=-A^{-1}(J_c a_0+\dot J_c\dot q),\qquad \ddot q=a_0-M^{-1}J_c^TA^{-1}(J_c a_0+\dot J_c\dot q). $$ {#eq-reaction_elimination} The acceleration correction depends on the mass matrix as well as the geometry. The matrix $I-M^{-1}J_c^TA^{-1}J_c$ acts on velocities or acceleration increments; its transpose acts on generalized-force covectors. They cannot be interchanged merely because both are called projections. This solution is unique for the specified model, state, applied inputs, and regular active equality constraints. Redundant constraints can leave multiplier coordinates nonunique. Unilateral contacts can be infeasible for the computed reaction; frictional contact can require a different mode or contact law. Calling a problem forward dynamics does not remove those conditions. ### Reactions Depend on Applied Muscle Loads For a two-coordinate illustration take $M=\operatorname{diag}(2,1)$, $J_c=[1\;-1]$, $h=p=0$, and $B=I$. The two coordinates must accelerate together. With $u=(1,0)^T$, the solution is $$ \ddot q=(1/3,1/3)^T,\qquad \lambda=-1/3. $$ {#eq-reaction_example} With $u=(2,-1)^T$, the acceleration is unchanged and $\lambda=-4/3$. An additional applied load in $\operatorname{range}(J_c^T)$ is balanced by a changed reaction. Geometry determines the admissible reaction directions, not their magnitudes independently of actuation. A reaction may be zero or large depending on the state and inputs; the existence or number of loops alone gives no force scale. In inverse dynamics, the observed motion supplies the balance $$ [B\;J_c^T]\begin{bmatrix}u\\\lambda\end{bmatrix} =M\ddot q+h-p. $$ {#eq-inverse_loop_balance} Uniqueness depends on the rank of this identification map and on additional measurements and restrictions. Joint-level net torques can be ambiguous as well as individual muscle forces and hand loads. Known segment inertias and motion fix inertial demands; they do not generally separate all unknown sources. ## Club Wrenches and Bilateral Load Sharing {#sec-loop_ambiguity_active} For a rigid club, express all vectors in a common inertial frame and take moments about its center of mass $C$. Let $r_i$ point from $C$ to hand contact $i$, $f_i$ be its force on the club, and $m_i$ its contact couple. Then $$ \begin{aligned} m_c a_C &= f_L+f_R+m_c g+f_{\mathrm{other}},\\ I_C\dot\omega+\omega\times(I_C\omega) &=r_L\times f_L+r_R\times f_R+m_L+m_R+m_{\mathrm{other},C}. \end{aligned} $$ {#eq-club_total_force} The current rotational inertia tensor $I_C$ is expressed in that same frame. Other loads include aerodynamic forces or ball contact when applicable. The hand points of a rotating club generally have different linear accelerations: $a_i=a_C+\dot\omega\times r_i+\omega\times(\omega\times r_i)$. Sharing a rigid body does not make those accelerations equal. Stack each hand load as $w_i=(m_i,f_i)$ and transport its moment to $C$ before summing. The resulting $6\times12$ grasp map $G$ has rank six if both contacts are modeled as unrestricted full wrenches. For a compatible required wrench, $$ w=G^+w_{\mathrm{req}}+N_G\eta,\qquad \operatorname{range}(N_G)=\ker G, $$ {#eq-force_decomposition_ambiguous} so there are six unconstrained load-sharing parameters before friction, unilateral pressure, moment capacity, and actuator limits are imposed. Those physical restrictions can reduce the feasible set or make it empty; an arbitrary algebraic split need not be realizable by a hand. If each contact transmits only a point force, the map is $6\times6$. For two distinct points it generically has rank five, because equal opposite forces along the joining line produce zero resultant wrench. The null dimension is then one, and arbitrary required spatial wrenches cannot be supported. Grip models that allow contact couples and distributed pressure have different capabilities. A simple scalar illustration is $\tau_L+\tau_R=10$ N m, with both torques expressed about the same axis and reference point after all other terms are accounted for. The pairs $(4,6)$ and $(7,3)$ satisfy the same balance. A minimum-squared-torque choice gives $(5,5)$, but that is an optimization assumption, not an observation of how a golfer shared the load. Relative EMG amplitudes do not by themselves supply the missing torque ratio. Calibrated instrumented grips, muscle models, or additional experimental perturbations can constrain the inference. ### Force Transmission, Power, and Internal Loading A twist is a six-component angular/linear velocity representation, not generally a unit vector. With the matching wrench ordering, $V=(\omega,v)$ and $w=(m,f)$ pair to give power $w^TV=m\cdot\omega+f\cdot v$. Both must refer to compatible points and frames. A change of point changes the moment and linear velocity together, preserving power. For an ideal rigid internal attachment, the two bodies have the same contact twist and experience opposite wrenches. Their contact powers cancel when both bodies are included in the system. Each body separately can receive or lose power through that attachment. Thus the grip can transfer energy to the club while ideal closure reactions do zero total work on the golfer-plus-club system. In generalized coordinates this cancellation is $$ \dot q^TJ_c^T\lambda=\lambda^TJ_c\dot q=0. $$ {#eq-orthogonality_relation} A constraint can redirect acceleration while doing no total work. An internal load need not produce mechanical work merely because it stresses tissue. Muscle maintenance can have a metabolic cost even when modeled joint power is zero; local tissue deformation and injury outcomes require their own measurements and constitutive models. ## Ground Contact: Wrench, Momentum, and Energy {#sec-grf_lower_body} Ground reaction is an external contact wrench on the golfer-plus-club system. Its value depends on the motion, geometry, muscle loading, and contact conditions. It is not independent of muscle action, nor is it an additional muscle force. For a simple unilateral contact with normal gap $\phi_n$, the ideal conditions include $\phi_n\ge0$, $\lambda_n\ge0$, and $\phi_n\lambda_n=0$. A sticking Coulomb contact also requires $\|f_t\|\le\mu\lambda_n$. Distributed foot contact has moment and center-of-pressure restrictions in addition to this force cone. Slip, lift-off, shoe deformation, and moving support change the equations. For the center of mass $C$ of the selected whole system and angular momentum $H_C$, the external balances are $$ \begin{aligned} m_{\mathrm{tot}}a_C &=m_{\mathrm{tot}}g+\sum_i f_i+f_{\mathrm{other}},\\ \dot H_C &=\sum_i\big((r_i-C)\times f_i+m_i\big) +m_{\mathrm{other},C}. \end{aligned} $$ {#eq-whole_body_momentum} Here $r_i$ is the contact position in world coordinates, whereas the preceding club balance used offsets from the club center. Uniform gravity contributes no net moment about the whole-system center of mass. These equations constrain whole-system momentum, not the pelvis acceleration alone. Hip, spine, and joint reactions exchange angular momentum among segments. Even with zero total external moment, one segment can rotate while others counter-rotate. A free moment measured at a foot is therefore not a complete explanation of pelvis motion or proof of a universally passive shoulder. ### A Force Plate's Reference Point Matters Declare a right-handed frame with $z$ upward, and let $(F,M_O)$ be the wrench of the ground on the golfer measured about an origin $O$ in the contact plane. For $F_z\ne0$, a center of pressure $r=(x,y,0)$ is defined so that the remaining moment there has only a vertical component. Since $M_O=r\times F+T_z e_z$, $$ x=-\frac{M_y}{F_z},\qquad y=\frac{M_x}{F_z},\qquad T_z=M_z-(xF_y-yF_x). $$ {#eq-cop_computation} Thus the plate-origin yaw moment $M_z$ is not generally the free moment $T_z$. If the contact plane is at height $h$ relative to $O$, use $x=(hF_x-M_y)/F_z$ and $y=(M_x+hF_y)/F_z$. Vendor outputs may use the opposite force-on-plate sign or different axes; transform the full wrench before applying these formulas. Near zero normal load the division is ill-conditioned, so state a justified contact threshold and report unloaded intervals separately. For a numerical check, a force $F=(20,30,800)$ N applied at $r=(0.2,-0.1,0)$ m with free moment $T_z=5$ N m gives $M_O=(-80,-160,13)$ N m. The formulas recover $x=0.2$ m, $y=-0.1$ m, and $T_z=13-(6+2)=5$ N m. Calling the measured $13$ N m the free moment would conflate the horizontal-force lever arm with the residual couple. COP is pressure-weighted position, not the center of mass or a direct measure of transferred body mass. A changing foot-load share can move COP while the center of mass barely moves. With two coplanar plates, transform the wrenches to one frame and origin before combining them. Separate plates supply each foot's boundary wrench; one combined plate generally leaves bilateral sharing unresolved. Six-axis measurements are wrench components, not six motion DOFs. Calibration, cross-talk, sampling, synchronization, and filtering affect accuracy; a universal instrument-accuracy percentage cannot substitute for calibration. With each foot wrench known, segment poses and inertial parameters allow a Newton–Euler recursion from foot to shank to thigh. Transfer both force and moment, including their lever arms, at each step. This gives model-conditioned net joint loads when other external loads and joint assumptions are specified. It does not uniquely identify individual muscle forces or tissue stress. ### Support Can Enable Work Without Supplying Energy A fully stationary rigid foot has zero velocity at its ground-contact points and zero angular velocity. In this idealization, the ground wrench has zero mechanical power, although its force and moment can change the whole-system momentum. Muscle work can increase body and club energy while ground support prevents an incompatible foot motion. The resultant external force dotted with center-of-mass velocity describes center-of-mass translational-energy change; it is not the contact power of that force on a deformable or articulated system. For time-independent elastic potentials and ideal stationary contacts, an energy balance has the form $$ \frac{d}{dt}(T+V_g+V_e) =\dot q^TBu+P_{\mathrm{other}}-P_{\mathrm{diss}}, \qquad P_{\mathrm{diss}}\ge0. $$ {#eq-parallel_energy_balance} Moving support, slipping friction, changing elastic rest states, and contact impacts require their corresponding boundary or constitutive power terms. Neither a large ground force nor an early force peak establishes the source, amount, or direction of energy transferred to the club. A segment power audit must state the system boundaries, force points, velocities, and sign conventions. There is no single force-plate curve in this model that defines a good golfer. Lead-foot and combined forces, normalization by body weight, club choice, swing phase, and target all change a comparison. A claim that the best peak must occur exactly at impact needs an identified study and outcome definition, not an illustrative sketch. The chapter's equations specify what must balance; they do not provide a coaching threshold or an injury-risk cutoff. ## X-Factor, Compliance, and Muscle State {#sec-body_as_closed_chain} In a planar illustration, define $\theta_p$ and $\theta_t$ as pelvis and thorax angles about the same axis. Then $\Delta=\theta_t-\theta_p$ is a relative rotation, not a new loop-closure equation. For $\theta_p=60$ degrees and $\theta_t=20$ degrees, $\Delta=-40$ degrees under this convention. In three dimensions, use the relative orientation $R_p^TR_t$ and declare its reported angle convention. Subtracting two Euler angles or projected shoulder-line angles need not measure the same quantity. [Brown, Selbie, and Wallace's X-factor methods study](https://doi.org/10.1080/02640414.2013.775474) examines this measurement issue. A reported intersegment angle is not a direct measurement of lumbar rotation, disc compression, tissue strain, or stored elastic energy. Distribution among spinal levels, rib-cage/shoulder motion, posture, and muscle activation matter. An angle associated with performance is also not by itself evidence that increasing that angle improves an individual swing or is safe for that individual. For a clearly declared toy torsional spring, $$ V_e=\tfrac12k(\Delta-\Delta_0)^2,\qquad Q_t=-k(\Delta-\Delta_0),\qquad Q_p=+k(\Delta-\Delta_0). $$ {#eq-elastic_restoring_torque} The two torques are opposite, and their combined power is $-k(\Delta-\Delta_0)\dot\Delta=-\dot V_e$ when $k$ and $\Delta_0$ are fixed. Doubling displacement quadruples this model's energy only if the stiffness and rest angle stay the same. It does not establish a universal human tissue law or prove that the stored energy reaches the club rather than another segment or dissipation. If stiffness is actively changed, the energy derivative also contains $\tfrac12\dot k(\Delta-\Delta_0)^2$; changing a rest angle adds another term. An impedance schedule is not energetically free by definition. Joint range of motion, tangent stiffness, damping, geometric constraint rank, and neural feedback are distinct. Finite stiffness resists displacement without reducing the exact kinematic dimension. It can make some motions small over a specified loading range and time scale. A rigid-constraint approximation must be justified against those scales, contact margins, and output errors. Muscle excitation, activation, fiber length, shortening velocity, and tendon stretch must likewise be distinguished. Eccentric contraction means active lengthening; active shortening is concentric. A tendon can recoil while its muscle fibers behave differently. Treating all release-phase tissue forces as passive because club speed is high is not justified. For a muscle model, $\tau=R(q)F_m(q,\dot q,z)$ maps muscle forces through moment arms; multiplying moment arms by dimensionless activation alone does not produce torque. ## Constrained Drift and Forward Counterfactuals {#sec-drift_not_constrained} At a fixed admissible state and contact mode, reaction elimination gives $$ \begin{aligned} \ddot q &= f_c(q,\dot q,z)+G_c(q)u,\\ G_c &=\big(I-M^{-1}J_c^TA^{-1}J_c\big)M^{-1}B. \end{aligned} $$ {#eq-constrained_affine_dynamics} The input-independent term includes gravity, velocity-dependent dynamics, retained passive forces, and the geometric acceleration correction. It depends on the declared inputs and state variables. This is the useful local affine structure: closure reshapes both drift and the effect of an input. Removing $u$ does not remove the reactions needed to satisfy the remaining contacts. Computing these terms requires a model, inertial parameters, calibrated kinematics, and any relevant tissue or contact state. Motion capture alone does not measure masses, inertias, tendon preload, or the chosen decomposition. Although gravitational force can be evaluated before solving for reactions, the resulting constrained gravitational acceleration differs from free fall. Observable motion and model-computable force terms are different evidential categories. For a particle observed in a rotating frame, the centrifugal and Coriolis fictitious forces are $$ F_{\mathrm{cf}}=-m\omega\times(\omega\times r),\qquad F_{\mathrm{cor}}=-2m\omega\times v_{\mathrm{rel}}. $$ {#eq-rotating_frame_forces} An accelerating origin and changing frame angular velocity introduce translational and Euler terms as well. These expressions cannot be added to inertial-frame loads without the matching frame transformation; that would double count the kinematics. Generalized velocity-dependent terms in an articulated model are not automatically identical to one particle's rotating-frame force. For a hypothetical $3$ kg point mass rotating at $15$ rad/s on a $0.6$ m radius, $m\omega^2r=405$ N. About that same center, however, the radial force has zero moment because $r\times F_{\mathrm{cf}}=0$. It has zero power on purely tangential relative motion. About another joint it can have a moment, but the perpendicular lever arm and vector directions must be computed. Multiplying $405$ N by an arbitrary segment distance does not establish a shoulder torque. Similarly, $F_{\mathrm{cor}}\cdot v_{\mathrm{rel}}=0$: the Coriolis term is perpendicular to the relative velocity and is not dissipative resistance to it. For mutually perpendicular $\omega$ and $v_{\mathrm{rel}}$, with magnitudes $10$ rad/s and $8$ m/s and mass $3$ kg, its magnitude is $480$ N. That conditional number is not an observed golf force. A $3$ kg segment weighs $29.43$ N in $9.81$ m/s$^2$ gravity. Neither example establishes a universal ratio of inertial to muscle torque, and ratios of load magnitudes are not fractions of work or clubhead-speed causation. A forward zero-torque counterfactual (forward ZTCF) branches from a specified initial state and changes a declared torque input or policy. It then evolves the state and recomputes compatible reactions, contact events, and retained passive forces. It must not reset the measured state at every time sample and call the resulting instantaneous accelerations a new trajectory. If the inferred baseline torques or contact model are uncertain, that uncertainty carries into the forward prediction. Forward simulation avoids solving for unspecified inputs during integration; it does not establish that its initial inputs, parameters, or biological interpretation are correct. ## Kinematics, Singularities, and Feasible Control {#sec-singularities_parallel} When all compatible open-tree joint coordinates are known, their link poses can be computed recursively and the closure residual checked. If only some actuated coordinates are specified, passive coordinates and platform pose may require solving nonlinear closure equations. Multiple isolated assembly modes are not the same as continuous redundancy or a Jacobian singularity. Inverse kinematics asks for coordinates achieving a specified pose; its difficulty depends on the architecture. For a Stewart-type platform, computing leg lengths from a pose can be much simpler than computing pose from lengths. Constraint forces are not unknowns in a purely kinematic solve. For local input coordinates $s$ and output coordinates $x$, write differentiated closure as $A_x\dot x+B_s\dot s=0$. In a square nonredundant mechanism, the [Gosselin–Angeles classification](https://doi.org/10.1109/70.56660) distinguishes rank loss of the input-side matrix $B_s$ (first kind), output-side matrix $A_x$ (second kind), and simultaneous loss (third kind). When $A_x$ is singular, a nonzero output velocity may satisfy the linearized equations with locked inputs. Architecture singularity concerns geometric design that imposes persistent rank deficiency; it is not a name for any transient unusual posture. Standing still at address does not demonstrate a singularity, because these ranks depend on geometry rather than whether velocity happens to be zero. A straight two-link planar arm has a singular endpoint-position Jacobian, but its elbow coordinate still exists. A mechanical stop, contact inequality, or a separately imposed lock changes the admissible motion in another way. Neither near extension nor a small singular value proves maximum impact efficiency or safe tissue loading. A first-order rank loss may require nonlinear analysis to determine actual finite self-motion. Report which Jacobian is tested, its coordinate and unit scaling, the posture, and the numerical threshold. Conditioning of $J_yZ$ concerns task velocity; conditioning of $J_cM^{-1}J_c^T$ concerns reaction elimination; rank of $[B\;J_c^T]$ concerns inverse load identification. Their numerical values answer different questions. Local input rank also does not alone determine finite-time reachability, because drift, input bounds, time horizon, and contact transitions matter. ## An Investigation That Connects the Pieces {#sec-synthesis_golfer} The defensible research hypothesis is that preparation and changing geometry alter how feasible muscle inputs and stored energy produce the impact state. The two-handed grip and ground contacts are essential parts of that mapping. They can transmit loads, restrict relative motion, and alter the effectiveness of an input without supplying unlimited energy or determining a unique neural strategy. To investigate the hypothesis: 1. Define the outcome: clubhead velocity and face orientation at contact, strike location, ball launch, or dispersion. A club-center path alone is not a complete impact task. 1. Declare the graph, joint coordinates, contact modes, hand-frame offsets, inertias, muscle/input representation, and elastic states. Check closure, velocity, and acceleration residuals with measurement uncertainty. 1. Synchronize segment kinematics, separate foot wrenches, and any hand-load measurements. Identify which aggregate loads are determined and which bilateral or muscle distributions remain compatible with the data. 1. Audit linear and angular momentum separately from segment power, total energy, and tissue loading. Report sign conventions and reference points. An acceleration contribution, work contribution, and injury hypothesis must not share a percentage without an explicit derivation. 1. Compare a baseline trajectory with declared forward interventions on torque, timing, grip compliance, or contact. Preserve the same starting state when testing a late intervention; change it deliberately when testing preparation. Include uncertainty in unmeasured load sharing and parameters. 1. Test predictions on held-out trials and controlled perturbations where feasible. Similar observed kinematics do not prove identical forces, and a model that reproduces a swing does not identify the golfer's intention. This approach can reveal when a change in grip loading mainly affects internal stress, when it changes club orientation, or when a different contact condition changes the admissible acceleration. It also allows a passive mechanism to be supported where the evidence supports it, while retaining the muscle work, feedback, and prior preparation needed to explain the entire swing. [The energy-transfer chapter](./ch10_energy_transfer.qmd#sec-energy_transfer) develops the next question: how energy moves between the chosen segments and storage elements. ## Exercises and Checks 1. Draw the upper and lower loops and the connecting spine path. Remove one grip edge. Which cycle disappears? Why does defining a pelvis–thorax relative angle not restore it? 1. Reproduce the UPS platform count and explain the six axial rod spins in the fixed-length SPS example. State which freedoms move the platform. 1. Differentiate the planar two-arm positions. Verify both closure and common upward hand motion for the stated configuration and velocity. Then compute the stacked closure/task rank when the hand point is fixed. 1. Solve the two-coordinate reaction example for both applied loads. Verify identical accelerations, different reactions, and zero total ideal constraint power for a common admissible velocity. 1. Construct the two-point-force grasp map. Show that opposite forces along the joining line lie in its null space. Explain why opposite forces in an arbitrary direction generally leave a couple. 1. Recover COP and free moment from the numerical wrench. Repeat after moving the reference origin, transporting the moment consistently. 1. Derive the complete energy derivative of the torsional spring when both stiffness and rest angle vary. Identify which terms require a source or sink beyond the fixed-spring mechanical exchange. 1. Specify an experiment that distinguishes a grip-load change preserving the resultant club wrench from one that changes club orientation. List the measurements needed; explain why motion alone and raw EMG ratios do not resolve every candidate load distribution. ::: {.callout-note} ## On This Site - **[The Null Space of the Constraint Jacobian](../../null-space-constraint-jacobian.html)** — Loop constraints, rank deficiency, and reaction forces - **[Force and Mobility Ellipsoids](../../force-mobility-matrices.html)** — Task-space feasibility under closed kinematic chains :::