Parallel Mechanisms, Constrained Dynamics, and Geometric Control

Parallel Mechanisms, Constrained Dynamics, and Geometric Control

Analyzes closed kinematic loops, holonomic constraints, and geometric control in two-handed grip dynamics and ground-reaction biomechanical systems.

What the Geometry Explains

Holding a club with both hands closes a mechanical loop. Ground contact closes further loops through the legs and pelvis. These connections restrict relative motion and transmit loads, but they do not uniquely prescribe muscle forces, determine a neural control policy, or supply energy merely by exerting a force. The useful geometric question is how configuration, inertia, constraints, and actuation jointly determine feasible motion and load transmission.

We develop that question in a smooth, fixed contact mode. Let $q$ have $n$ local configuration coordinates, $v=\dot q$, and $u\in\mathbb R^m$ denote specified applied inputs. There are $c$ independent, stationary, ideal holonomic constraints $\Phi(q)=0$. Their Jacobian is $J_c=D\Phi\in\mathbb R^{c\times n}$, and $M(q)\succ0$ is the mass matrix. These assumptions permit exact identities. Grip compliance, sliding, lift-off, tissue activation, and ball impact require additional constitutive states or contact modes; they cannot be inferred from the identities alone.

The mechanics below follows the vector/covector and virtual-work treatment of constrained systems (Bullo and Lewis 2004; Murray et al. 1994). Its worked examples are mathematical checks, not measurements of a golfer. The final sections explain what measurements would be needed to turn the framework into an empirical account of a swing.

Configuration, Constraints, and Regularity

Configuration Is Different From State

For $N$ unconstrained spatial rigid bodies, $Q_{\mathrm{free}}=SE(3)^N$ has dimension $6N$; planar bodies contribute three configuration dimensions each. Joint coordinates reduce this description before contact constraints are imposed. Angles have periodic topology: a pair of revolute angles lives on $S^1\times S^1$, with intervals used only as coordinate charts. The mechanical state is in the tangent bundle $TQ$, with local coordinates $(q,v)$ and twice the configuration dimension. Activation and flexible coordinates may enlarge it further.

If $J_c$ has rank $c$ on the level set, the regular level-set theorem gives \[ Q_c=\{q:\Phi(q)=0\},\qquad \dim Q_c=n-c=:d,\qquad T_qQ_c=\ker J_c(q). \tag{1}\] The admissible state satisfies both $\Phi(q)=0$ and $J_cv=0$. A configuration constraint does not imply $J_c\dot v=0$: differentiating again gives \[ J_c\dot v+\dot J_cv=0. \tag{2}\] Thus admissible velocities form a linear tangent space at fixed $q$, whereas admissible coordinate accelerations form an affine set at fixed $(q,v)$.

Rank loss invalidates the regular-level-set argument; it does not by itself prove that the physical set is singular or unstabilizable. The equation $x^2=0$ has zero derivative at its solution, although its solution set is the smooth zero-dimensional set $\{0\}$. Redundant constraint equations can also lose rank without a change in physical mobility. At a true mechanism singularity, inspect independent coordinates, higher-order compatibility, actuator directions, and contact inequalities before drawing a control conclusion.

A Movable Planar Closed Chain

Take a two-link arm based at the origin and a third link based at $(a,b)$. Let all three angles be absolute angles from the same horizontal axis. Loop closure is \[ \begin{aligned} \Phi_1&=\ell_1\cos\theta_1+\ell_2\cos\theta_2-\ell_3\cos\theta_3-a,\\ \Phi_2&=\ell_1\sin\theta_1+\ell_2\sin\theta_2-\ell_3\sin\theta_3-b. \end{aligned} \] The Jacobian has columns $\ell_1(-\sin\theta_1,\cos\theta_1)^\top$, $\ell_2(-\sin\theta_2,\cos\theta_2)^\top$, and $\ell_3(\sin\theta_3,-\cos\theta_3)^\top$. At rank two, three free angles minus two independent constraints leave one local degree of freedom. If instead there were only two grounded links meeting at their tips, two angles minus two regular closure constraints would leave isolated configurations. Equal link lengths alone would not establish a movable chain; coincident bases and compatible geometry would matter too.

For the two-grounded-link Jacobian with columns $\ell_1(-\sin\theta_1,\cos\theta_1)^\top$ and $\ell_2(\sin\theta_2,-\cos\theta_2)^\top$, its determinant is $\ell_1\ell_2\sin(\theta_1-\theta_2)$. This detects first-order rank loss when the links align, but finite mobility still requires examination of the closure equations.

A Revolute Joint and Its Frames

Choose frames at the joint pivot, with their local $z$ axes along the joint axis. The allowed relative transform has $p_{\mathrm{rel}}=0$ and $R_{\mathrm{rel}}=R_z(\theta)$. Locally near aligned axes, five independent constraints are the three components of $p_{\mathrm{rel}}$, $R_{13}=0$, and $R_{23}=0$, on the branch $R_{33}>0$. In contrast, setting $R_{12}=R_{21}=0$ would prohibit ordinary rotation about $z$, because those entries are $-\sin\theta$ and $\sin\theta$. The relative motion has one degree of freedom; two free bodies connected by this joint have seven total configuration degrees of freedom, including the common spatial pose.

Integrability Needs More Than a Derivative Test

For a nonvanishing one-form $\alpha=\sum_i a_i(q)dq_i$, the velocity constraint $\alpha(v)=0$ defines a distribution. Locally, its codimension-one integrability condition is $\alpha\wedge d\alpha=0$. When integrable, it can be written $\alpha=\mu\,d\phi$ with a nonzero integrating factor $\mu$; $\alpha$ itself need not be exact. For example, $e^xdy=0$ means $dy=0$ even though $d(e^xdy)=e^x dx\wedge dy\ne0$. Closed forms are locally exact on a contractible neighborhood, but global exactness requires topological conditions. For several independent forms, use the Frobenius involutivity condition on their common kernel.

A wheel with planar heading $\theta$ has the lateral no-slip constraint $-\sin\theta\,\dot x+\cos\theta\,\dot y=0$. Its distribution is nonintegrable. Rolling on a fixed straight track can instead give $x-r\varphi=\mathrm{constant}$. Rolling is therefore not universally nonholonomic. Sliding friction, unilateral joint limits, and contact opening are different phenomena again; they are not all smooth equality constraints.

Mass Metric, Forces, and Inertial Terms

Vectors and Covectors Have Different Roles

Velocity $v\in T_qQ$ is a vector. A generalized load $F\in T_q^*Q$ is a covector: $F^\top v$ is power. The mass matrix maps velocity to momentum and defines $\langle a,b\rangle_M=a^\top Mb$. Ideal stationary constraint reactions are covectors in $\operatorname{range}J_c^\top$, because \[ (J_c^\top\lambda)^\top v=\lambda^\top J_cv=0. \] Their associated mass-metric normal vectors are in a different space: \[ (\ker J_c)^{\perp_M}=\operatorname{range}(M^{-1}J_c^\top). \tag{3}\] Indeed $v^\top M(M^{-1}J_c^\top\lambda)=0$ for every tangent $v$. With $M=\operatorname{diag}(2,1)$ and $J_c=(1,1)$, the vector $(1,-1)$ is tangent. Its mass inner product with $(1,1)$ is one, but its mass inner product with $(1/2,1)$ is zero. Omitting $M^{-1}$ changes the orthogonality claim.

Constructing the Mass Matrix

For rigid link $k$, let $J_{v,k}$ map generalized rates to center-of-mass velocity in a fixed spatial frame, and let $J_{\omega,k}$ map them to angular velocity in that same frame. If $R_k$ maps body coordinates to that frame and $I_{b,k}$ is inertia about the center of mass in body coordinates, then \[ M(q)=\sum_k\left(m_kJ_{v,k}^\top J_{v,k}+J_{\omega,k}^\top R_kI_{b,k}R_k^\top J_{\omega,k}\right). \] Every summand is an $n\times n$ matrix. Positive definiteness requires a nondegenerate model with independent coordinates and positive inertias in its physical directions. Redundant orientation parameters require their own constraints; one cannot assume an unconstrained positive-definite mass matrix in arbitrary redundant coordinates.

With upward height $z_k$ and downward gravity of magnitude $g>0$, use $V=\sum_km_kgz_k$. Kinetic energy is $T=\tfrac12v^\top Mv$. It measures energy at a velocity, not an unspecified physiological cost of acceleration.

Coriolis Loads and the Energy Identity

Write the forced equations as \[ M\dot v+h(q,v)=Bu+J_c^\top\lambda,\qquad h=c(q,v)+\nabla V, \tag{4}\] with any additional known nonconservative loads added explicitly on the right. The inertial velocity load is \[ \begin{aligned} c_i&=\sum_{j,k}\Gamma_{ijk}v_jv_k,\\ \Gamma_{ijk}&=\tfrac12\left(\partial_kM_{ij}+\partial_jM_{ik}-\partial_iM_{jk}\right). \end{aligned} \tag{5}\] One valid matrix representation is $C_{ij}=\sum_k\Gamma_{ijk}v_k$, so $c=Cv$. It satisfies $\dot M-2C$ skew-symmetric and \[ v^\top c=\tfrac12v^\top\dot Mv. \] In general $C\ne\dot M$. The matrix representation is not unique, while the load $c$ for the specified coordinates is fixed. The second-kind connection coefficients are $\Gamma^i_{jk}=\sum_\ell(M^{-1})_{i\ell}\Gamma_{\ell jk}$, giving $M^{-1}c$ component $\sum_{j,k}\Gamma^i_{jk}v_jv_k$. There is no extra minus sign in this identity; the minus appears when solving the force-free equation for acceleration.

For a simple abstract two-coordinate metric, introduce a length scale $\ell$ so the entries have angular inertia units: \[ M=\ell^2\begin{bmatrix}m_1+m_2&m_2\cos q_2\\m_2\cos q_2&m_2\end{bmatrix},\qquad m_1,m_2>0. \] Its determinant is $\ell^4(m_1m_2+m_2^2\sin^2q_2)>0$. Direct differentiation gives $c=(-\ell^2m_2\sin q_2\,v_2^2,0)^\top$. Thus $\Gamma^1_{22}=-m_2^2\sin q_2/(m_1m_2+m_2^2\sin^2q_2)$ and $\Gamma^1_{12}=0$. Substitution verifies the energy identity. This is a metric example, not a derived anthropometric two-link arm.

Connection coefficients can be nonzero in curvilinear coordinates on a flat space. Their presence does not establish nonzero intrinsic curvature. Coordinate-free equations retain covariant acceleration; they do not eliminate the physical inertial coupling. Curvature concerns how covariant differentiation fails to commute, a separate question from whether a coordinate expression contains centrifugal terms.

Eliminating Reactions Without Losing Their Input Dependence

Define $K_c=J_cM^{-1}J_c^\top\succ0$ and $\gamma=\dot J_cv$. The acceleration constraint and force balance give \[ \lambda=-K_c^{-1}\left[J_cM^{-1}(Bu-h)+\gamma\right]. \tag{6}\] The reaction changes when the input changes, even at the same $(q,v)$. It cannot be held at its measured value and called an input-independent drift.

Define a vector projector and its dual force projector: \[ P_v=I-M^{-1}J_c^\top K_c^{-1}J_c,\qquad P_F=P_v^\top. \] These projectors act on different spaces. They satisfy $P_v^2=P_v$, $J_cP_v=0$, $P_v^\top M=MP_v$, and $P_FJ_c^\top=0$. A load decomposes as $F=P_FF+(I-P_F)F$; the second part belongs to the reaction covector space. This algebraic partition is not a label for useful versus wasted physiological effort.

The reduced acceleration equation is \[ \dot v=f(q,v)+G(q)u, \] \[ f=-P_vM^{-1}h-M^{-1}J_c^\top K_c^{-1}\gamma, \tag{7}\] \[ G=P_vM^{-1}B. \tag{8}\] Here $f$ is an acceleration function in a coordinate chart. The first-order vector fields on $TQ_c$ are $F_0(q,v)=(v,f)$ and $F_i(q,v)=(0,G_i)$. This distinction matters for Lie brackets and state dimension. All columns of $G$ are tangent, while $J_cf=-\gamma$ supplies the normal acceleration required by the changing tangent plane.

For $\Phi(q)=\tfrac12(q^\top q-R^2)$ on a circle with $M=I$, $J_c=q^\top$ and $\gamma=v^\top v$. The admissible acceleration for an unconstrained candidate $a_0$ is \[ a=P_va_0-\frac{q}{R^2}\|v\|^2. \] At $q=(1,0)$ and $v=(0,2)$, its radial component must be $-4$. Projecting $a_0$ onto the tangent line alone gives radial component zero and violates the acceleration constraint. Acceleration-level satisfaction also does not guarantee that an ordinary numerical integrator preserves position and velocity constraints.

A zero-input counterfactual evaluates $u=0$ with the chosen constitutive model and contact mode, then evolves its own state. The drift includes the zero-input reaction computed above. If a computed reaction violates a friction cone or requires tensile ground contact, the assumed mode is infeasible and must change. A prescribed measured base motion is an external boundary intervention, not an isolated passive swing.

Three Different Forms of Redundancy

Constraint Freedom, Task Freedom, and Input Freedom

Constraint-compatible velocity freedom is $\ker J_c$. For a task map $y=\psi(q)$ with Jacobian $J_t$, task-preserving velocity freedom is $\ker J_c\cap\ker J_t$. Neither is automatically an unactuated direction. Input redundancy is the kernel of the effective acceleration map $G$; its elements change input without changing instantaneous acceleration in this ideal model, though reactions can change.

For desired acceleration $a_d$, let $b=a_d-f$. Exact instantaneous tracking requires $b\in\operatorname{range}G$ and some feasible $u\in\mathcal U$ with $Gu=b$. At rank $r$, the unconstrained algebraic solution family is \[ u=G^+b+(I-G^+G)z,\qquad z\in\mathbb R^m, \tag{9}\] provided the range condition holds. Its input redundancy dimension is $m-r$. If $m<d$ and $G$ has full column rank, the system is instantaneously underactuated and each feasible $b$ has a unique input. Fewer inputs do not create an infinite family of solutions. Bounds, force signs, actuator rates, and friction restrictions can eliminate an otherwise algebraically feasible solution.

If reduced coordinates give a full-row-rank acceleration map $\bar G\in\mathbb R^{d\times m}$ and the objective is $\tfrac12u^\top Wu$, $W\succ0$, Lagrange multipliers give \[ u_W=W^{-1}\bar G^\top(\bar G W^{-1}\bar G^\top)^{-1}\bar b. \tag{10}\] For $\bar G=(1,1)$, $\bar b=1$, and $W=\operatorname{diag}(1,4)$, this is $(0.8,0.2)$, not the Euclidean minimum-norm allocation $(0.5,0.5)$. Differentiating $W$ with respect to posture is not a substitute for solving this fixed-state allocation problem. State planning and instantaneous input allocation are distinct optimizations.

Minimum-Energy Velocity and Muscle Allocation

To find a velocity with task rate $\dot y_d$ while preserving the constraint, stack $A=(J_c^\top,J_t^\top)^\top$ and $b_y=(0,\dot y_d)^\top$. With independent rows, the minimizer of $\tfrac12v^\top Mv$ subject to $Av=b_y$ is \[ v_*=M^{-1}A^\top(AM^{-1}A^\top)^{-1}b_y. \] The Euclidean pseudoinverse minimizes Euclidean norm; it minimizes kinetic energy only under the corresponding metric choice. Position-level feasibility and finite-time task execution still require integration and input checks.

Muscle allocation is another inverse problem. If a signed moment-arm matrix $R(q)$ maps muscle forces $s$ to required joint load $\tau$, one may solve $Rs=\tau$ with $0\le s\le s_{\max}$ and a chosen objective such as $\sum_i(s_i/A_i)^p$. For $p=2$ and fixed positive areas this is a convex quadratic objective with linear constraints. A nonzero-load solution is not in $\ker R$; differences between unconstrained solutions lie in $\ker R$. Capacity limits and activation dynamics can remove that freedom. Selecting one solution with an objective supplies a modeling assumption, not a measurement of the nervous system’s objective (Zajac 1989).

Twists, Wrenches, and a Two-Hand Grip

Consistent Frame Transformations

Let $g_{AB}=(R,p)$ map coordinates in frame $B$ into frame $A$. Use twist ordering $\xi=(\omega,v)$, where $v$ is the linear component associated with the chosen frame origin. Then \[ \xi_A=\operatorname{Ad}_{g_{AB}}\xi_B,\qquad \operatorname{Ad}_{g_{AB}}=\begin{bmatrix}R&0\\{}[p]_\times R&R\end{bmatrix}. \] In particular $v_A=Rv_B+p\times R\omega_B$. The equivalent matrix identity is $\widehat\xi_A=g_{AB}\widehat\xi_Bg_{AB}^{-1}$. The translation term must be retained. A six-dimensional twist cannot be transformed by a four-dimensional homogeneous matrix acting directly on its column of components.

For wrench ordering $W=(\tau,F)$, invariance of $W^\top\xi$ requires \[ W_A=\operatorname{Ad}_{g_{AB}}^{-\top}W_B =\begin{bmatrix}R\tau_B+p\times RF_B\\RF_B\end{bmatrix}. \] The reverse transformation is $W_B=\operatorname{Ad}_{g_{AB}}^\top W_A$. Stating the direction of the frame map avoids ambiguous uses of the word coadjoint. Screw reciprocity between a wrench and an admissible twist means $W^\top\xi=0$, not $\xi^\top M\eta=0$ between two velocities.

For integrated twist coordinates $\zeta=(\omega,v)$ and $\theta=\|\omega\|$, the exponential is \[ \begin{aligned} \exp\widehat\zeta&=\begin{bmatrix}e^{[\omega]_\times}&J(\omega)v\\0&1\end{bmatrix},\\ J(\omega)&=I+\frac{1-\cos\theta}{\theta^2}[\omega]_\times +\frac{\theta-\sin\theta}{\theta^3}[\omega]_\times^2. \end{aligned} \] Use the continuous limit $J(0)=I$, with series evaluation for small $\theta$. The last coefficient has denominator $\theta^3$ because $\omega$ already contains the rotation magnitude; mixing unit-axis and integrated-coordinate formulas changes the translation. A time-varying twist generally requires an ordered integration, not the exponential of its simple time integral (Murray et al. 1994).

Different Contact Points Have Different Velocities

At hand contact $i$, let $r_i$ be the vector from the club reference point $O$ to that contact, expressed in one spatial frame. No translational slip imposes \[ J_{h,i}(q)\dot q=v_O+\omega_c\times r_i. \] The left and right hand point velocities are generally unequal when the club rotates. Full rigid attachment also constrains relative angular velocity; point-contact and soft-finger models constrain fewer components. If limb Jacobians are converted to the same club reference point and frame, equality of their full twists can be used. Equating unshifted velocities at different grip points is incorrect.

For two point forces on the club, the resultant wrench at $O$ is \[ F=f_L+f_R,\qquad \tau_O=r_L\times f_L+r_R\times f_R. \] Adding $\delta f_L=a(r_L-r_R)$ and $\delta f_R=-\delta f_L$ changes neither resultant component. An internal squeeze along the hand separation can therefore be invisible to a measured resultant wrench. Distributed contact moments, compliance, and friction alter the admissible internal-load set, but they do not make bilateral force inference unique without further evidence. Internal grip loading may affect stiffness and future contact behavior even when its instantaneous net wrench is zero.

Singular Values Need Physical Scaling

For a task Jacobian, $\sqrt{\det(J_tJ_t^\top)}$ measures the volume of its Euclidean velocity ellipsoid when rows are independent. It is not a complete condition-number measure: $\operatorname{diag}(100,0.01)$ has determinant one but condition number $10^4$. Translational and rotational components need declared scales before such measures can be compared. Report singular values and physical input limits, not just a determinant.

A rank-$k$ screw span defines a point of $\operatorname{Gr}(k,6)$. When the spanning matrix loses rank, it ceases to represent a $k$-plane; the rank-deficient set is not a collection of points inside that same Grassmannian with reduced dimension. Rank-loss codimension depends on the matrix family. Neither arm alignment nor a small manipulability determinant by itself proves high geometric curvature, loss of feedback stabilizability, or beneficial energy transfer.

Ground Contact, Momentum, and Power

Count the Floating Base Before Subtracting Constraints

As an explicitly idealized model, give each leg seven joint coordinates and the pelvis a six-dimensional floating pose. There are then $6+7+7=20$ free configuration dimensions. One rigid stationary foot supplies up to six independent constraints, leaving 14; two such feet supply up to 12, leaving eight at regular configurations. Starting with 14 joint coordinates and subtracting foot constraints while also allowing a floating pelvis omits the base. Anatomical joint models may use different joint counts. Point contacts, rolling contacts, deformable shoes, and dependent constraints also change the count.

A rigid bilateral contact model remains admissible only while its calculated wrench respects nonpenetration, compressive normal force, friction, and the support region. A formal equality solution that pulls a foot into the ground or requires excessive tangential traction is not a feasible physical prediction.

Center of Pressure and Reference-Point Conventions

Express the force exerted by the plate on the body in a right-handed frame with $z$ upward, and shift its measured moment to an origin in the contact plane. For $F_z\ne0$, a resultant applied at $(x_c,y_c,0)$ with a remaining vertical free moment gives \[ x_c=-\frac{M_y}{F_z},\qquad y_c=\frac{M_x}{F_z},\qquad M_{z,\mathrm{free}}=M_z-x_cF_y+y_cF_x. \] These signs follow from $r_c\times F=(y_cF_z,-x_cF_z,x_cF_y-y_cF_x)$. A plate origin below the surface requires a moment shift first; a manufacturer may report the opposite action-reaction convention. Near zero normal load the ratios become ill-conditioned. A center of pressure does not replace the complete wrench, especially its free moment.

Inverse Dynamics With All Loads Declared

For a body-fixed offset $r$ from a moving reference point to the center of mass, expressed spatially, \[ a_C=a_O+\alpha\times r+\omega\times(\omega\times r). \] The last term has a plus sign; its double cross product already points inward for circular motion. Sum all applied forces and all moments about the center of mass: \[ \sum F=m a_C,\qquad \sum\tau_C=I_C\alpha+\omega\times(I_C\omega), \] with $I_C$ represented in the same frame. Gravity, measured contact loads, distal and proximal joint loads, and their moment arms belong in these sums. A backward Newton–Euler recursion is a systematic application of these balances with explicit action-reaction signs and frame shifts. Kinematic accelerations come from the motion model or measurements; measured ground force is a load boundary condition, not an acceleration measurement.

Inverse dynamics gives the required net generalized load after known loads are accounted for. Unknown bilateral load sharing and muscle redundancy remain inverse problems. Recursive inverse dynamics is linear in the number of links for a tree of fixed-size joints; recursive forward-dynamics algorithms can also avoid assembling and inverting a dense mass matrix. Closed loops add constraint solves and cannot be assigned the same computational cost without specifying the solver.

Support Can Redirect Motion Without Supplying Work

With matching reference point and frame, contact power is $P_c=F_c^\top v_c+\tau_c^\top\omega_c$. A stationary rigid foot has zero twist, so its ideal ground wrench delivers zero mechanical power, even if its forces and moments are large. A moment does not supply rotational energy when angular velocity is zero. A foot pivoting or deforming needs the actual contact motion and traction distribution; no-slip at one point does not mean the entire foot is stationary.

Nevertheless, the ground can change total linear and angular momentum. About a fixed inertial origin, $\dot H_O$ equals the total external moment, including ground reactions and gravity where applicable. Zero work does not imply zero impulse or zero angular impulse. This is central to the golf swing: support reactions can redirect motion and enable muscles to do work elsewhere without being an independent source of mechanical energy.

For stationary ideal constraints, stored mechanical energy $E=T+V$ obeys \[ \dot E=u^\top y+P_{\mathrm{other}}-D,\qquad y=B^\top v,\qquad D\ge0, \tag{11}\] when dissipation is modeled by $D$ and all other power sources are declared. Passivity is a storage inequality, $E(t)-E(0)\le\int u^\top y\,dt$ for the chosen port when other sources are absent and storage is bounded below. Equality is the lossless special case. Torque dotted with torque is not power. Muscle activation can maintain force with nonzero metabolic cost even at zero joint power. Mechanical work, squared torque, metabolic expenditure, and fatigue therefore define different optimization problems.

Reachability and Optimal Control

Instantaneous Rank Does Not Decide Finite-Time Motion

For smooth first-order fields, $[X,Y]=DY\,X-DX\,Y$. If $X=(x_2,-x_1)$ and $Y=(0,1)$, then $[X,Y]=(-1,0)$. This is a second state direction relative to the single input $Y$, not a third independent direction in a two-dimensional space.

The Chow–Rashevskii bracket-generating condition concerns driftless systems with reversible admissible controls on an appropriate connected region. Mechanical dynamics have drift, so distinguish accessibility, small-time local controllability, and reachability over a prescribed forward-time horizon. For $\dot x=1$, $\dot y=u$, the drift and input span the plane but no forward trajectory can decrease $x$. For second-order mechanics, input fields $(0,G_i(q))$ have zero mutual brackets when they depend only on $q$; brackets with the drift $(v,f)$ are essential. Checking the acceleration columns alone against the full state dimension is invalid.

As a driftless example, car kinematics have $g_1=(\cos\theta,\sin\theta,\tan\varphi/L,0)$ and $g_2=(0,0,0,1)$. Their bracket $[g_2,g_1]=(0,0,\sec^2\varphi/L,0)$ first gives a heading direction; further brackets produce lateral displacement directions. Steering while stationary does not translate the car. Maneuvers realizing bracket effects use sequences of controls and, for the usual reversible argument, both forward and reverse travel.

Necessary Conditions in Independent State Coordinates

Use a local chart of $TQ_c$ with independent state $z\in\mathbb R^{2d}$ and dynamics $\dot z=\mathcal F(z,u)$. This avoids silently treating dependent ambient position and velocity coordinates as independent. For fixed final time, terminal cost $\phi(z(T))$, and running cost $\ell$, define the normal minimization Hamiltonian \[ H=\ell+p^\top\mathcal F,\qquad \dot p=-H_z,\qquad u_*\in\arg\min_{u\in\mathcal U}H. \] For a free terminal state, $p(T)=\nabla\phi$; endpoint constraints change transversality. These are necessary conditions under the usual regularity assumptions, not a general sufficient certificate. Abnormal extremals need the cost multiplier formulation. Ambient formulations must handle both position and velocity consistency and cannot obtain all costate conditions by adding one unexplained position multiplier.

For a control-linear Hamiltonian $H=H_0+\sigma^\top u$ on a box, minimization selects $u_i=u_{i,\min}$ when $\sigma_i>0$ and $u_i=u_{i,\max}$ when $\sigma_i<0$. A switching function that vanishes on an interval requires higher-order conditions and may admit a singular arc. With $\ell=\ell_0+u^\top Ru$, $R\succ0$, the unconstrained minimizer is $-\tfrac12R^{-1}\mathcal G^\top p$ for $\mathcal F=\mathcal F_0+\mathcal G u$. Under bounds, solve the resulting convex quadratic program. Componentwise clipping is exact for a separable diagonal $R$ and independent box bounds, not for an arbitrary coupled $R$.

Torque input, muscle force input, excitation input, and activation state are different model choices. Smooth activation does not refute a bang-bang excitation solution when activation dynamics filter that excitation. A torque optimum does not establish that muscle activation follows the same profile or that the nervous system optimizes the chosen objective.

Value Functions and Local Numerical Methods

Where differentiable, the finite-horizon value $\mathcal V(z,t)$ satisfies \[ -\mathcal V_t=\min_{u\in\mathcal U}\left(\ell+\nabla\mathcal V^\top\mathcal F\right),\qquad \mathcal V(z,T)=\phi(z). \] Nonsmooth value functions need an appropriate generalized, typically viscosity, solution. Autonomous dynamics do not remove finite-horizon time dependence. A stationary HJB equation belongs to a separately specified infinite-horizon or stationary problem. A grid with ten values in each of 28 independent state dimensions has $10^{28}$ nodes, illustrating why direct grids become impractical; the dimension must come from the chosen constrained model.

For a discrete full-state transition $z_{k+1}=F_k(z_k,u_k)$, iLQR uses $\delta z_{k+1}=A_k\delta z_k+B_k\delta u_k$ and second-order cost terms. Its backward pass produces both a feedforward step and feedback gain, and its forward pass rolls out $u_k^{\mathrm{new}}=u_k+\alpha k_k+K_k(z_k^{\mathrm{new}}-z_k)$. DDP additionally retains value-gradient-weighted second derivatives of the dynamics. Both need regularization, step acceptance, feasibility handling, and convergence checks. Projecting a cost gradient alone does not enforce a nonlinear contact constraint. The optimal-control chapter derives the Riccati and local quadratic calculations; neither algorithm automatically proves a global nonlinear optimum.

What Measurements Can Identify

With known parameters and an explicit zero-input contact model, $f(q,v)$ is computable from an estimated state. That is model evaluation, not a theorem that kinematics alone identify the physical drift. The acceleration constraint supplies $c$ equations for $n$ accelerations and generally leaves $d$ tangent directions undetermined. Actual reactions additionally depend on applied inputs. Unknown inertias, tendon state, compliance, and loads cannot be made known by writing them as functions of $q$.

Observability asks whether different initial states can yield distinguishable output histories under specified inputs. For an autonomous smooth system, full rank of differentials of sufficiently many output Lie derivatives is a sufficient local condition under appropriate hypotheses. Controlled systems require the relevant input fields or a specified known-input trajectory. A local rank condition does not prove global uniqueness, numerical conditioning, or parameter identifiability. Parameters may be augmented as constant states, but adequate excitation and sensitivity remain necessary.

Nonlinear indistinguishable sets need not be affine subspaces. For $\dot x=0$ and output $y=x_1^2+x_2^2$, a nonzero output level is a circle. The kernel of $Dy$ gives its tangent line, not the entire set of indistinguishable states. Linearized null directions describe first-order sensitivity; they must not be promoted to exact finite-state equivalence.

Motion capture and force plates can constrain inverse dynamics when synchronized, calibrated, and accompanied by inertial estimates. Surface EMG adds electrical observations with muscle-specific processing and activation dynamics; integrating its signal does not directly yield force or activation. A universal fixed delay is not an adequate constitutive law. Optimization can select among feasible muscle patterns, but selection by a prior or cost function does not resolve ambiguity empirically (Zajac 1989).

For the two-hand grip, report which sensors constrain the net club wrench and which constrain the internal hand-load family. For the feet, report separate contact wrenches rather than assuming a unique bilateral distribution from a combined resultant. Compare alternative parameter sets and input allocations that fit the same observations; test whether the proposed conclusion survives that ambiguity.

Symmetry, Mechanical Connections, and Geometric Phase

Momentum Conservation Has a Boundary Condition

A cyclic coordinate has conserved conjugate momentum only when the Lagrangian is independent of that coordinate and its generalized applied force vanishes. Constraints and external loads must respect the symmetry. A grounded golfer generally receives external angular impulse, so total angular momentum of the golfer-plus-club subsystem is not automatically conserved. Conservation in a freely floating model cannot be asserted for the grounded experiment.

For a local planar symmetry coordinate $\theta$ and shape $s$, consider \[ T=\tfrac12I(s)\dot\theta^2+\dot\theta\,a(s)^\top\dot s +\tfrac12\dot s^\top H(s)\dot s. \] The conjugate momentum is $p_\theta=I\dot\theta+a^\top\dot s$. If it is conserved, reconstruction gives \[ \dot\theta=I^{-1}p_\theta-A(s)\dot s,\qquad A=I^{-1}a^\top. \tag{12}\] The locked inertia is $I=\partial^2T/\partial\dot\theta^2$; there is no additional factor of one half. The connection needs the group–shape cross block $a$, not the derivative of $I$ alone. At fixed momentum, a conventional Routhian is $L-p_\theta\dot\theta$ with $\dot\theta$ eliminated; forced or noncyclic systems require a modified reduction (Bullo and Lewis 2004).

Separate Dynamic and Geometric Contributions

For a closed shape loop in this planar model, \[ \Delta\theta=\int_0^T I(s(t))^{-1}p_\theta\,dt-\oint A(s)\,ds. \tag{13}\] The first term is a dynamic phase and the second a geometric contribution with the sign fixed by the reconstruction equation. At zero momentum, only the latter remains. On a suitable planar shape patch, Stokes’ theorem converts $\oint A$ to an integral of $dA$. General noncommuting spatial rotations require a group reconstruction and ordered transport; a scalar area formula does not directly apply.

For a checkable example, choose $I=1$, $a=(0,k s_1)^\top$, and $H=I_2+aa^\top$. The full kinetic metric is positive definite because its Schur complement is $I_2$. At zero momentum, $\dot\theta=-k s_1\dot s_2$. A counterclockwise rectangular shape loop of side lengths $a_0,b_0$ gives $\Delta\theta=-k a_0b_0$. This proves a geometric effect in the specified model. It does not identify an analogous contribution in a golf swing until the inertia blocks, external moment, and measured shape trajectory have been established.

Numerical Methods That Preserve the Stated Problem

The position-constrained first-order mechanical DAE is ordinarily differentiation index three under regularity assumptions. Differentiating position once yields velocity consistency and twice yields an algebraic equation for $\lambda$; differentiating that relation yields a differential equation for the multiplier. Position to velocity to acceleration is two differentiations, not three. This terminology does not make a twice-differentiated constraint automatically stable under time stepping.

Baumgarte stabilization replaces the acceleration equation by $J_c\dot v+\gamma+2\alpha J_cv+\beta^2\Phi=0$. In continuous arithmetic the violation obeys $\ddot\Phi+2\alpha\dot\Phi+\beta^2\Phi=0$. Positive gains damp errors, but $\alpha$ and $\beta$ have inverse-time units, and large gains can make integration stiff. Constraint scaling and time-step studies are necessary. Stabilization introduces corrective forces and does not guarantee exact energy conservation.

A local Newton correction for position may use \[ q^{j+1}=q^j-M(q^j)^{-1}J_c(q^j)^\top \left[J_c(q^j)M(q^j)^{-1}J_c(q^j)^\top\right]^{-1}\Phi(q^j). \] It requires a suitable initial guess and stopping criterion. After correcting position, velocity projection is $v=P_v(q)v_{\mathrm{pred}}$. These are projection operations; by themselves they do not specify SHAKE or RATTLE. Those methods combine particular constrained position/momentum updates with an underlying integration scheme. Their symplectic properties require their actual assumptions and solve tolerances; arbitrary after-step projection is not equivalent (Hairer et al. 2006).

For constant mass, a RATTLE-style velocity-Verlet step first computes a half-step momentum with a constraint impulse along $J_c(q_n)^\top$, chooses its multiplier so the updated position satisfies $\Phi(q_{n+1})=0$, then updates momentum with the new applied force and a second multiplier so $J_c(q_{n+1})M^{-1}p_{n+1}=0$. The half-step factors, old/new Jacobians, and both multipliers are part of the method. Configuration-dependent inertia and velocity-dependent forces require an appropriate generalized scheme, not a direct substitution into that separable formula.

SVD can reveal the numerical rank of a scaled $J_c$ and supply a tangent basis $N$. The Euclidean projector $NN^\top$ assumes Euclidean-orthonormal columns; the kinetic-metric projector is $N(N^\top MN)^{-1}N^\top M$. Neither tangent projector alone includes the acceleration offset $-M^{-1}J_c^\top K_c^{-1}\gamma$. Solve linear systems instead of explicitly forming inverses in numerical code.

Validation should report position, velocity, and acceleration residuals separately; input/reaction feasibility; changes under tighter solve tolerances and smaller time steps; and the energy balance including applied work and dissipation. Energy constancy is a test for an isolated lossless model, not for every passive, driven, or damped system. Good energy behavior is not proof of correct contact geometry or scientifically valid parameters.

Connecting the Pieces in a Golf Investigation

Begin with a measurable outcome, such as delivered clubhead speed and face orientation at the model’s own impact event. Specify the body/club model, coordinate frames, grip model, foot contact modes, and inputs. Estimate state and parameter uncertainty before decomposing effects. A useful same-state calculation compares the drift acceleration and individual projected input contributions; a forward intervention branches the state and recomputes reactions and contact events.

Then ask distinct questions. How does posture change the effective input map? Which internal hand loads leave the net wrench unchanged while altering grip compliance? How much mechanical work has already entered the system, and where is it stored? How does ground impulse redirect momentum while doing little or no work in an ideal stationary-contact model? Which feedback corrections remain feasible under torque, activation, and contact limits? Impact depends on the delivered state, so differences earlier in the swing matter through their effects on that state and its uncertainty.

These questions are intertwined without being interchangeable. A favorable configuration may improve load transmission while increasing sensitivity to parameter error. Co-contraction can change impedance without changing the instantaneous net joint torque, yet it can have metabolic cost and change later behavior. A zero-torque continuation can retain substantial speed because of stored momentum and energy, while still being an unrealistic intervention on activation or grip. A force plate can measure a large ground moment without showing that the ground supplied rotational work. An optimizer can expose a feasible strategy without establishing that a golfer uses it.

For a defensible argument, report the model identity, the empirical observation, and the proposed causal mechanism separately. Publish the equations and reference-point conventions, reproducible numerical counterexamples, uncertainty ranges, and interventions that would falsify the proposed explanation. A prediction should survive reasonable alternative contact models and muscle allocations before being presented as a robust swing mechanism. Coaching conclusions require their own outcome evidence; geometric feasibility alone does not establish a beneficial instruction for a person.

Exercises and Reproducible Checks

Verify $P_v^2=P_v$ and $P_v^\top M=MP_v$ for the unequal-mass example. Compare vector and force projections and explain their different domains.
Derive the circle’s normal acceleration and show why a tangent-only acceleration update fails at nonzero speed.
Differentiate the two-coordinate metric to recover $c$ and verify $v^\top c=\tfrac12v^\top\dot Mv$ for arbitrary rates.
Derive the weighted allocation $(0.8,0.2)$ and identify input bounds that make the requested acceleration infeasible.
Transform an arbitrary twist and wrench between translated, rotated frames and verify power invariance. Then construct a nonzero internal hand-force pair with zero resultant wrench.
Derive the force-plate center-of-pressure signs from $r\times F$ and explain why the free moment must be retained.
Integrate the rectangular geometric-phase example and reverse the loop orientation. Add a known external torque and identify the extra momentum-dependent contribution.
Simulate a constrained system at successively smaller time steps. Report constraint residuals and the work-inclusive energy balance rather than certifying accuracy from one plot.

Cite this page

References

Bullo, F., and A. D. Lewis. 2004. Geometric Control of Mechanical Systems. Springer.

Hairer, E., C. Lubich, and G. Wanner. 2006. Geometric Numerical Integration. 2nd ed. Springer.

Murray, R. M., Z. Li, and S. S. Sastry. 1994. A Mathematical Introduction to Robotic Manipulation. CRC Press.

Zajac, F. E. 1989. “Muscle and Tendon: Properties, Models, Scaling, and Application to Biomechanics and Motor Control.” Critical Reviews in Biomedical Engineering 17 (4): 359–411. https://pubmed.ncbi.nlm.nih.gov/2676342/.

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  title = {Parallel {Mechanisms,} {Constrained} {Dynamics,} and
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--- title: "Parallel Mechanisms, Constrained Dynamics, and Geometric Control" description: "Analyzes closed kinematic loops, holonomic constraints, and geometric control in two-handed grip dynamics and ground-reaction biomechanical systems." --- []{#ch-parallel-mechanisms} ## What the Geometry Explains Holding a club with both hands closes a mechanical loop. Ground contact closes further loops through the legs and pelvis. These connections restrict relative motion and transmit loads, but they do not uniquely prescribe muscle forces, determine a neural control policy, or supply energy merely by exerting a force. The useful geometric question is how configuration, inertia, constraints, and actuation jointly determine feasible motion and load transmission. We develop that question in a smooth, fixed contact mode. Let $q$ have $n$ local configuration coordinates, $v=\dot q$, and $u\in\mathbb R^m$ denote specified applied inputs. There are $c$ independent, stationary, ideal holonomic constraints $\Phi(q)=0$. Their Jacobian is $J_c=D\Phi\in\mathbb R^{c\times n}$, and $M(q)\succ0$ is the mass matrix. These assumptions permit exact identities. Grip compliance, sliding, lift-off, tissue activation, and ball impact require additional constitutive states or contact modes; they cannot be inferred from the identities alone. The mechanics below follows the vector/covector and virtual-work treatment of constrained systems [@BulloLewis2004; @Murray1994]. Its worked examples are mathematical checks, not measurements of a golfer. The final sections explain what measurements would be needed to turn the framework into an empirical account of a swing. ## Configuration, Constraints, and Regularity []{#sec-ch9-config-manifolds} ### Configuration Is Different From State For $N$ unconstrained spatial rigid bodies, $Q_{\mathrm{free}}=SE(3)^N$ has dimension $6N$; planar bodies contribute three configuration dimensions each. Joint coordinates reduce this description before contact constraints are imposed. Angles have periodic topology: a pair of revolute angles lives on $S^1\times S^1$, with intervals used only as coordinate charts. The mechanical state is in the tangent bundle $TQ$, with local coordinates $(q,v)$ and twice the configuration dimension. Activation and flexible coordinates may enlarge it further. If $J_c$ has rank $c$ on the level set, the regular level-set theorem gives $$ Q_c=\{q:\Phi(q)=0\},\qquad \dim Q_c=n-c=:d,\qquad T_qQ_c=\ker J_c(q). $$ {#eq-ch9:tangent-constraint} The admissible state satisfies both $\Phi(q)=0$ and $J_cv=0$. A configuration constraint does not imply $J_c\dot v=0$: differentiating again gives $$ J_c\dot v+\dot J_cv=0. $$ {#eq-ch9:accel-constraint} Thus admissible velocities form a linear tangent space at fixed $q$, whereas admissible coordinate accelerations form an affine set at fixed $(q,v)$. Rank loss invalidates the regular-level-set argument; it does not by itself prove that the physical set is singular or unstabilizable. The equation $x^2=0$ has zero derivative at its solution, although its solution set is the smooth zero-dimensional set $\{0\}$. Redundant constraint equations can also lose rank without a change in physical mobility. At a true mechanism singularity, inspect independent coordinates, higher-order compatibility, actuator directions, and contact inequalities before drawing a control conclusion. ### A Movable Planar Closed Chain Take a two-link arm based at the origin and a third link based at $(a,b)$. Let all three angles be absolute angles from the same horizontal axis. Loop closure is $$ \begin{aligned} \Phi_1&=\ell_1\cos\theta_1+\ell_2\cos\theta_2-\ell_3\cos\theta_3-a,\\ \Phi_2&=\ell_1\sin\theta_1+\ell_2\sin\theta_2-\ell_3\sin\theta_3-b. \end{aligned} $$ The Jacobian has columns $\ell_1(-\sin\theta_1,\cos\theta_1)^\top$, $\ell_2(-\sin\theta_2,\cos\theta_2)^\top$, and $\ell_3(\sin\theta_3,-\cos\theta_3)^\top$. At rank two, three free angles minus two independent constraints leave one local degree of freedom. If instead there were only two grounded links meeting at their tips, two angles minus two regular closure constraints would leave isolated configurations. Equal link lengths alone would not establish a movable chain; coincident bases and compatible geometry would matter too. For the two-grounded-link Jacobian with columns $\ell_1(-\sin\theta_1,\cos\theta_1)^\top$ and $\ell_2(\sin\theta_2,-\cos\theta_2)^\top$, its determinant is $\ell_1\ell_2\sin(\theta_1-\theta_2)$. This detects first-order rank loss when the links align, but finite mobility still requires examination of the closure equations. ### A Revolute Joint and Its Frames Choose frames at the joint pivot, with their local $z$ axes along the joint axis. The allowed relative transform has $p_{\mathrm{rel}}=0$ and $R_{\mathrm{rel}}=R_z(\theta)$. Locally near aligned axes, five independent constraints are the three components of $p_{\mathrm{rel}}$, $R_{13}=0$, and $R_{23}=0$, on the branch $R_{33}>0$. In contrast, setting $R_{12}=R_{21}=0$ would prohibit ordinary rotation about $z$, because those entries are $-\sin\theta$ and $\sin\theta$. The relative motion has one degree of freedom; two free bodies connected by this joint have seven total configuration degrees of freedom, including the common spatial pose. ### Integrability Needs More Than a Derivative Test For a nonvanishing one-form $\alpha=\sum_i a_i(q)dq_i$, the velocity constraint $\alpha(v)=0$ defines a distribution. Locally, its codimension-one integrability condition is $\alpha\wedge d\alpha=0$. When integrable, it can be written $\alpha=\mu\,d\phi$ with a nonzero integrating factor $\mu$; $\alpha$ itself need not be exact. For example, $e^xdy=0$ means $dy=0$ even though $d(e^xdy)=e^x dx\wedge dy\ne0$. Closed forms are locally exact on a contractible neighborhood, but global exactness requires topological conditions. For several independent forms, use the Frobenius involutivity condition on their common kernel. A wheel with planar heading $\theta$ has the lateral no-slip constraint $-\sin\theta\,\dot x+\cos\theta\,\dot y=0$. Its distribution is nonintegrable. Rolling on a fixed straight track can instead give $x-r\varphi=\mathrm{constant}$. Rolling is therefore not universally nonholonomic. Sliding friction, unilateral joint limits, and contact opening are different phenomena again; they are not all smooth equality constraints. ## Mass Metric, Forces, and Inertial Terms []{#sec-ch9-lagrangian} ### Vectors and Covectors Have Different Roles Velocity $v\in T_qQ$ is a vector. A generalized load $F\in T_q^*Q$ is a covector: $F^\top v$ is power. The mass matrix maps velocity to momentum and defines $\langle a,b\rangle_M=a^\top Mb$. Ideal stationary constraint reactions are covectors in $\operatorname{range}J_c^\top$, because $$ (J_c^\top\lambda)^\top v=\lambda^\top J_cv=0. $$ Their associated mass-metric normal vectors are in a different space: $$ (\ker J_c)^{\perp_M}=\operatorname{range}(M^{-1}J_c^\top). $$ {#eq-ch9:orthogonal-complement} Indeed $v^\top M(M^{-1}J_c^\top\lambda)=0$ for every tangent $v$. With $M=\operatorname{diag}(2,1)$ and $J_c=(1,1)$, the vector $(1,-1)$ is tangent. Its mass inner product with $(1,1)$ is one, but its mass inner product with $(1/2,1)$ is zero. Omitting $M^{-1}$ changes the orthogonality claim. ### Constructing the Mass Matrix For rigid link $k$, let $J_{v,k}$ map generalized rates to center-of-mass velocity in a fixed spatial frame, and let $J_{\omega,k}$ map them to angular velocity in that same frame. If $R_k$ maps body coordinates to that frame and $I_{b,k}$ is inertia about the center of mass in body coordinates, then $$ M(q)=\sum_k\left(m_kJ_{v,k}^\top J_{v,k}+J_{\omega,k}^\top R_kI_{b,k}R_k^\top J_{\omega,k}\right). $$ Every summand is an $n\times n$ matrix. Positive definiteness requires a nondegenerate model with independent coordinates and positive inertias in its physical directions. Redundant orientation parameters require their own constraints; one cannot assume an unconstrained positive-definite mass matrix in arbitrary redundant coordinates. With upward height $z_k$ and downward gravity of magnitude $g>0$, use $V=\sum_km_kgz_k$. Kinetic energy is $T=\tfrac12v^\top Mv$. It measures energy at a velocity, not an unspecified physiological cost of acceleration. ### Coriolis Loads and the Energy Identity Write the forced equations as $$ M\dot v+h(q,v)=Bu+J_c^\top\lambda,\qquad h=c(q,v)+\nabla V, $$ {#eq-ch9:euler-lagrange-constrained} with any additional known nonconservative loads added explicitly on the right. The inertial velocity load is $$ \begin{aligned} c_i&=\sum_{j,k}\Gamma_{ijk}v_jv_k,\\ \Gamma_{ijk}&=\tfrac12\left(\partial_kM_{ij}+\partial_jM_{ik}-\partial_iM_{jk}\right). \end{aligned} $$ {#eq-ch9:christoffel} One valid matrix representation is $C_{ij}=\sum_k\Gamma_{ijk}v_k$, so $c=Cv$. It satisfies $\dot M-2C$ skew-symmetric and $$ v^\top c=\tfrac12v^\top\dot Mv. $$ In general $C\ne\dot M$. The matrix representation is not unique, while the load $c$ for the specified coordinates is fixed. The second-kind connection coefficients are $\Gamma^i_{jk}=\sum_\ell(M^{-1})_{i\ell}\Gamma_{\ell jk}$, giving $M^{-1}c$ component $\sum_{j,k}\Gamma^i_{jk}v_jv_k$. There is no extra minus sign in this identity; the minus appears when solving the force-free equation for acceleration. For a simple abstract two-coordinate metric, introduce a length scale $\ell$ so the entries have angular inertia units: $$ M=\ell^2\begin{bmatrix}m_1+m_2&m_2\cos q_2\\m_2\cos q_2&m_2\end{bmatrix},\qquad m_1,m_2>0. $$ Its determinant is $\ell^4(m_1m_2+m_2^2\sin^2q_2)>0$. Direct differentiation gives $c=(-\ell^2m_2\sin q_2\,v_2^2,0)^\top$. Thus $\Gamma^1_{22}=-m_2^2\sin q_2/(m_1m_2+m_2^2\sin^2q_2)$ and $\Gamma^1_{12}=0$. Substitution verifies the energy identity. This is a metric example, not a derived anthropometric two-link arm. Connection coefficients can be nonzero in curvilinear coordinates on a flat space. Their presence does not establish nonzero intrinsic curvature. Coordinate-free equations retain covariant acceleration; they do not eliminate the physical inertial coupling. Curvature concerns how covariant differentiation fails to commute, a separate question from whether a coordinate expression contains centrifugal terms. ## Eliminating Reactions Without Losing Their Input Dependence []{#sec-ch9-control-affine} Define $K_c=J_cM^{-1}J_c^\top\succ0$ and $\gamma=\dot J_cv$. The acceleration constraint and force balance give $$ \lambda=-K_c^{-1}\left[J_cM^{-1}(Bu-h)+\gamma\right]. $$ {#eq-ch9:multiplier} The reaction changes when the input changes, even at the same $(q,v)$. It cannot be held at its measured value and called an input-independent drift. Define a vector projector and its dual force projector: $$ P_v=I-M^{-1}J_c^\top K_c^{-1}J_c,\qquad P_F=P_v^\top. $$ These projectors act on different spaces. They satisfy $P_v^2=P_v$, $J_cP_v=0$, $P_v^\top M=MP_v$, and $P_FJ_c^\top=0$. A load decomposes as $F=P_FF+(I-P_F)F$; the second part belongs to the reaction covector space. This algebraic partition is not a label for useful versus wasted physiological effort. The reduced acceleration equation is $$ \dot v=f(q,v)+G(q)u, $$ $$ f=-P_vM^{-1}h-M^{-1}J_c^\top K_c^{-1}\gamma, $$ {#eq-ch9:drift} $$ G=P_vM^{-1}B. $$ {#eq-ch9:input-field} Here $f$ is an acceleration function in a coordinate chart. The first-order vector fields on $TQ_c$ are $F_0(q,v)=(v,f)$ and $F_i(q,v)=(0,G_i)$. This distinction matters for Lie brackets and state dimension. All columns of $G$ are tangent, while $J_cf=-\gamma$ supplies the normal acceleration required by the changing tangent plane. For $\Phi(q)=\tfrac12(q^\top q-R^2)$ on a circle with $M=I$, $J_c=q^\top$ and $\gamma=v^\top v$. The admissible acceleration for an unconstrained candidate $a_0$ is $$ a=P_va_0-\frac{q}{R^2}\|v\|^2. $$ At $q=(1,0)$ and $v=(0,2)$, its radial component must be $-4$. Projecting $a_0$ onto the tangent line alone gives radial component zero and violates the acceleration constraint. Acceleration-level satisfaction also does not guarantee that an ordinary numerical integrator preserves position and velocity constraints. A zero-input counterfactual evaluates $u=0$ with the chosen constitutive model and contact mode, then evolves its own state. The drift includes the zero-input reaction computed above. If a computed reaction violates a friction cone or requires tensile ground contact, the assumed mode is infeasible and must change. A prescribed measured base motion is an external boundary intervention, not an isolated passive swing. ## Three Different Forms of Redundancy []{#sec-ch9-null-space} ### Constraint Freedom, Task Freedom, and Input Freedom Constraint-compatible velocity freedom is $\ker J_c$. For a task map $y=\psi(q)$ with Jacobian $J_t$, task-preserving velocity freedom is $\ker J_c\cap\ker J_t$. Neither is automatically an unactuated direction. Input redundancy is the kernel of the effective acceleration map $G$; its elements change input without changing instantaneous acceleration in this ideal model, though reactions can change. For desired acceleration $a_d$, let $b=a_d-f$. Exact instantaneous tracking requires $b\in\operatorname{range}G$ and some feasible $u\in\mathcal U$ with $Gu=b$. At rank $r$, the unconstrained algebraic solution family is $$ u=G^+b+(I-G^+G)z,\qquad z\in\mathbb R^m, $$ {#eq-ch9:general-control} provided the range condition holds. Its input redundancy dimension is $m-r$. If $m<d$ and $G$ has full column rank, the system is instantaneously underactuated and each feasible $b$ has a unique input. Fewer inputs do not create an infinite family of solutions. Bounds, force signs, actuator rates, and friction restrictions can eliminate an otherwise algebraically feasible solution. If reduced coordinates give a full-row-rank acceleration map $\bar G\in\mathbb R^{d\times m}$ and the objective is $\tfrac12u^\top Wu$, $W\succ0$, Lagrange multipliers give $$ u_W=W^{-1}\bar G^\top(\bar G W^{-1}\bar G^\top)^{-1}\bar b. $$ {#eq-ch9:optimal-null-space} For $\bar G=(1,1)$, $\bar b=1$, and $W=\operatorname{diag}(1,4)$, this is $(0.8,0.2)$, not the Euclidean minimum-norm allocation $(0.5,0.5)$. Differentiating $W$ with respect to posture is not a substitute for solving this fixed-state allocation problem. State planning and instantaneous input allocation are distinct optimizations. ### Minimum-Energy Velocity and Muscle Allocation To find a velocity with task rate $\dot y_d$ while preserving the constraint, stack $A=(J_c^\top,J_t^\top)^\top$ and $b_y=(0,\dot y_d)^\top$. With independent rows, the minimizer of $\tfrac12v^\top Mv$ subject to $Av=b_y$ is $$ v_*=M^{-1}A^\top(AM^{-1}A^\top)^{-1}b_y. $$ The Euclidean pseudoinverse minimizes Euclidean norm; it minimizes kinetic energy only under the corresponding metric choice. Position-level feasibility and finite-time task execution still require integration and input checks. Muscle allocation is another inverse problem. If a signed moment-arm matrix $R(q)$ maps muscle forces $s$ to required joint load $\tau$, one may solve $Rs=\tau$ with $0\le s\le s_{\max}$ and a chosen objective such as $\sum_i(s_i/A_i)^p$. For $p=2$ and fixed positive areas this is a convex quadratic objective with linear constraints. A nonzero-load solution is not in $\ker R$; differences between unconstrained solutions lie in $\ker R$. Capacity limits and activation dynamics can remove that freedom. Selecting one solution with an objective supplies a modeling assumption, not a measurement of the nervous system's objective [@Zajac1989]. ## Twists, Wrenches, and a Two-Hand Grip []{#sec-ch9-screw-theory} ### Consistent Frame Transformations Let $g_{AB}=(R,p)$ map coordinates in frame $B$ into frame $A$. Use twist ordering $\xi=(\omega,v)$, where $v$ is the linear component associated with the chosen frame origin. Then $$ \xi_A=\operatorname{Ad}_{g_{AB}}\xi_B,\qquad \operatorname{Ad}_{g_{AB}}=\begin{bmatrix}R&0\\{}[p]_\times R&R\end{bmatrix}. $$ In particular $v_A=Rv_B+p\times R\omega_B$. The equivalent matrix identity is $\widehat\xi_A=g_{AB}\widehat\xi_Bg_{AB}^{-1}$. The translation term must be retained. A six-dimensional twist cannot be transformed by a four-dimensional homogeneous matrix acting directly on its column of components. For wrench ordering $W=(\tau,F)$, invariance of $W^\top\xi$ requires $$ W_A=\operatorname{Ad}_{g_{AB}}^{-\top}W_B =\begin{bmatrix}R\tau_B+p\times RF_B\\RF_B\end{bmatrix}. $$ The reverse transformation is $W_B=\operatorname{Ad}_{g_{AB}}^\top W_A$. Stating the direction of the frame map avoids ambiguous uses of the word coadjoint. Screw reciprocity between a wrench and an admissible twist means $W^\top\xi=0$, not $\xi^\top M\eta=0$ between two velocities. For integrated twist coordinates $\zeta=(\omega,v)$ and $\theta=\|\omega\|$, the exponential is $$ \begin{aligned} \exp\widehat\zeta&=\begin{bmatrix}e^{[\omega]_\times}&J(\omega)v\\0&1\end{bmatrix},\\ J(\omega)&=I+\frac{1-\cos\theta}{\theta^2}[\omega]_\times +\frac{\theta-\sin\theta}{\theta^3}[\omega]_\times^2. \end{aligned} $$ Use the continuous limit $J(0)=I$, with series evaluation for small $\theta$. The last coefficient has denominator $\theta^3$ because $\omega$ already contains the rotation magnitude; mixing unit-axis and integrated-coordinate formulas changes the translation. A time-varying twist generally requires an ordered integration, not the exponential of its simple time integral [@Murray1994]. ### Different Contact Points Have Different Velocities At hand contact $i$, let $r_i$ be the vector from the club reference point $O$ to that contact, expressed in one spatial frame. No translational slip imposes $$ J_{h,i}(q)\dot q=v_O+\omega_c\times r_i. $$ The left and right hand point velocities are generally unequal when the club rotates. Full rigid attachment also constrains relative angular velocity; point-contact and soft-finger models constrain fewer components. If limb Jacobians are converted to the same club reference point and frame, equality of their full twists can be used. Equating unshifted velocities at different grip points is incorrect. For two point forces on the club, the resultant wrench at $O$ is $$ F=f_L+f_R,\qquad \tau_O=r_L\times f_L+r_R\times f_R. $$ Adding $\delta f_L=a(r_L-r_R)$ and $\delta f_R=-\delta f_L$ changes neither resultant component. An internal squeeze along the hand separation can therefore be invisible to a measured resultant wrench. Distributed contact moments, compliance, and friction alter the admissible internal-load set, but they do not make bilateral force inference unique without further evidence. Internal grip loading may affect stiffness and future contact behavior even when its instantaneous net wrench is zero. ### Singular Values Need Physical Scaling For a task Jacobian, $\sqrt{\det(J_tJ_t^\top)}$ measures the volume of its Euclidean velocity ellipsoid when rows are independent. It is not a complete condition-number measure: $\operatorname{diag}(100,0.01)$ has determinant one but condition number $10^4$. Translational and rotational components need declared scales before such measures can be compared. Report singular values and physical input limits, not just a determinant. A rank-$k$ screw span defines a point of $\operatorname{Gr}(k,6)$. When the spanning matrix loses rank, it ceases to represent a $k$-plane; the rank-deficient set is not a collection of points inside that same Grassmannian with reduced dimension. Rank-loss codimension depends on the matrix family. Neither arm alignment nor a small manipulability determinant by itself proves high geometric curvature, loss of feedback stabilizability, or beneficial energy transfer. ## Ground Contact, Momentum, and Power []{#sec-ch9-grf} ### Count the Floating Base Before Subtracting Constraints As an explicitly idealized model, give each leg seven joint coordinates and the pelvis a six-dimensional floating pose. There are then $6+7+7=20$ free configuration dimensions. One rigid stationary foot supplies up to six independent constraints, leaving 14; two such feet supply up to 12, leaving eight at regular configurations. Starting with 14 joint coordinates and subtracting foot constraints while also allowing a floating pelvis omits the base. Anatomical joint models may use different joint counts. Point contacts, rolling contacts, deformable shoes, and dependent constraints also change the count. A rigid bilateral contact model remains admissible only while its calculated wrench respects nonpenetration, compressive normal force, friction, and the support region. A formal equality solution that pulls a foot into the ground or requires excessive tangential traction is not a feasible physical prediction. ### Center of Pressure and Reference-Point Conventions Express the force exerted by the plate on the body in a right-handed frame with $z$ upward, and shift its measured moment to an origin in the contact plane. For $F_z\ne0$, a resultant applied at $(x_c,y_c,0)$ with a remaining vertical free moment gives $$ x_c=-\frac{M_y}{F_z},\qquad y_c=\frac{M_x}{F_z},\qquad M_{z,\mathrm{free}}=M_z-x_cF_y+y_cF_x. $$ These signs follow from $r_c\times F=(y_cF_z,-x_cF_z,x_cF_y-y_cF_x)$. A plate origin below the surface requires a moment shift first; a manufacturer may report the opposite action-reaction convention. Near zero normal load the ratios become ill-conditioned. A center of pressure does not replace the complete wrench, especially its free moment. ### Inverse Dynamics With All Loads Declared For a body-fixed offset $r$ from a moving reference point to the center of mass, expressed spatially, $$ a_C=a_O+\alpha\times r+\omega\times(\omega\times r). $$ The last term has a plus sign; its double cross product already points inward for circular motion. Sum all applied forces and all moments about the center of mass: $$ \sum F=m a_C,\qquad \sum\tau_C=I_C\alpha+\omega\times(I_C\omega), $$ with $I_C$ represented in the same frame. Gravity, measured contact loads, distal and proximal joint loads, and their moment arms belong in these sums. A backward Newton--Euler recursion is a systematic application of these balances with explicit action-reaction signs and frame shifts. Kinematic accelerations come from the motion model or measurements; measured ground force is a load boundary condition, not an acceleration measurement. Inverse dynamics gives the required net generalized load after known loads are accounted for. Unknown bilateral load sharing and muscle redundancy remain inverse problems. Recursive inverse dynamics is linear in the number of links for a tree of fixed-size joints; recursive forward-dynamics algorithms can also avoid assembling and inverting a dense mass matrix. Closed loops add constraint solves and cannot be assigned the same computational cost without specifying the solver. ### Support Can Redirect Motion Without Supplying Work With matching reference point and frame, contact power is $P_c=F_c^\top v_c+\tau_c^\top\omega_c$. A stationary rigid foot has zero twist, so its ideal ground wrench delivers zero mechanical power, even if its forces and moments are large. A moment does not supply rotational energy when angular velocity is zero. A foot pivoting or deforming needs the actual contact motion and traction distribution; no-slip at one point does not mean the entire foot is stationary. Nevertheless, the ground can change total linear and angular momentum. About a fixed inertial origin, $\dot H_O$ equals the total external moment, including ground reactions and gravity where applicable. Zero work does not imply zero impulse or zero angular impulse. This is central to the golf swing: support reactions can redirect motion and enable muscles to do work elsewhere without being an independent source of mechanical energy. For stationary ideal constraints, stored mechanical energy $E=T+V$ obeys $$ \dot E=u^\top y+P_{\mathrm{other}}-D,\qquad y=B^\top v,\qquad D\ge0, $$ {#eq-ch9:passivity} when dissipation is modeled by $D$ and all other power sources are declared. Passivity is a storage inequality, $E(t)-E(0)\le\int u^\top y\,dt$ for the chosen port when other sources are absent and storage is bounded below. Equality is the lossless special case. Torque dotted with torque is not power. Muscle activation can maintain force with nonzero metabolic cost even at zero joint power. Mechanical work, squared torque, metabolic expenditure, and fatigue therefore define different optimization problems. ## Reachability and Optimal Control []{#sec-ch9:optimal-control} ### Instantaneous Rank Does Not Decide Finite-Time Motion For smooth first-order fields, $[X,Y]=DY\,X-DX\,Y$. If $X=(x_2,-x_1)$ and $Y=(0,1)$, then $[X,Y]=(-1,0)$. This is a second state direction relative to the single input $Y$, not a third independent direction in a two-dimensional space. The Chow--Rashevskii bracket-generating condition concerns driftless systems with reversible admissible controls on an appropriate connected region. Mechanical dynamics have drift, so distinguish accessibility, small-time local controllability, and reachability over a prescribed forward-time horizon. For $\dot x=1$, $\dot y=u$, the drift and input span the plane but no forward trajectory can decrease $x$. For second-order mechanics, input fields $(0,G_i(q))$ have zero mutual brackets when they depend only on $q$; brackets with the drift $(v,f)$ are essential. Checking the acceleration columns alone against the full state dimension is invalid. As a driftless example, car kinematics have $g_1=(\cos\theta,\sin\theta,\tan\varphi/L,0)$ and $g_2=(0,0,0,1)$. Their bracket $[g_2,g_1]=(0,0,\sec^2\varphi/L,0)$ first gives a heading direction; further brackets produce lateral displacement directions. Steering while stationary does not translate the car. Maneuvers realizing bracket effects use sequences of controls and, for the usual reversible argument, both forward and reverse travel. ### Necessary Conditions in Independent State Coordinates Use a local chart of $TQ_c$ with independent state $z\in\mathbb R^{2d}$ and dynamics $\dot z=\mathcal F(z,u)$. This avoids silently treating dependent ambient position and velocity coordinates as independent. For fixed final time, terminal cost $\phi(z(T))$, and running cost $\ell$, define the normal minimization Hamiltonian $$ H=\ell+p^\top\mathcal F,\qquad \dot p=-H_z,\qquad u_*\in\arg\min_{u\in\mathcal U}H. $$ For a free terminal state, $p(T)=\nabla\phi$; endpoint constraints change transversality. These are necessary conditions under the usual regularity assumptions, not a general sufficient certificate. Abnormal extremals need the cost multiplier formulation. Ambient formulations must handle both position and velocity consistency and cannot obtain all costate conditions by adding one unexplained position multiplier. For a control-linear Hamiltonian $H=H_0+\sigma^\top u$ on a box, minimization selects $u_i=u_{i,\min}$ when $\sigma_i>0$ and $u_i=u_{i,\max}$ when $\sigma_i<0$. A switching function that vanishes on an interval requires higher-order conditions and may admit a singular arc. With $\ell=\ell_0+u^\top Ru$, $R\succ0$, the unconstrained minimizer is $-\tfrac12R^{-1}\mathcal G^\top p$ for $\mathcal F=\mathcal F_0+\mathcal G u$. Under bounds, solve the resulting convex quadratic program. Componentwise clipping is exact for a separable diagonal $R$ and independent box bounds, not for an arbitrary coupled $R$. Torque input, muscle force input, excitation input, and activation state are different model choices. Smooth activation does not refute a bang-bang excitation solution when activation dynamics filter that excitation. A torque optimum does not establish that muscle activation follows the same profile or that the nervous system optimizes the chosen objective. ### Value Functions and Local Numerical Methods Where differentiable, the finite-horizon value $\mathcal V(z,t)$ satisfies $$ -\mathcal V_t=\min_{u\in\mathcal U}\left(\ell+\nabla\mathcal V^\top\mathcal F\right),\qquad \mathcal V(z,T)=\phi(z). $$ Nonsmooth value functions need an appropriate generalized, typically viscosity, solution. Autonomous dynamics do not remove finite-horizon time dependence. A stationary HJB equation belongs to a separately specified infinite-horizon or stationary problem. A grid with ten values in each of 28 independent state dimensions has $10^{28}$ nodes, illustrating why direct grids become impractical; the dimension must come from the chosen constrained model. For a discrete full-state transition $z_{k+1}=F_k(z_k,u_k)$, iLQR uses $\delta z_{k+1}=A_k\delta z_k+B_k\delta u_k$ and second-order cost terms. Its backward pass produces both a feedforward step and feedback gain, and its forward pass rolls out $u_k^{\mathrm{new}}=u_k+\alpha k_k+K_k(z_k^{\mathrm{new}}-z_k)$. DDP additionally retains value-gradient-weighted second derivatives of the dynamics. Both need regularization, step acceptance, feasibility handling, and convergence checks. Projecting a cost gradient alone does not enforce a nonlinear contact constraint. The [optimal-control chapter](ch05_optimal_control.html) derives the Riccati and local quadratic calculations; neither algorithm automatically proves a global nonlinear optimum. ## What Measurements Can Identify []{#sec-ch9-observability} With known parameters and an explicit zero-input contact model, $f(q,v)$ is computable from an estimated state. That is model evaluation, not a theorem that kinematics alone identify the physical drift. The acceleration constraint supplies $c$ equations for $n$ accelerations and generally leaves $d$ tangent directions undetermined. Actual reactions additionally depend on applied inputs. Unknown inertias, tendon state, compliance, and loads cannot be made known by writing them as functions of $q$. Observability asks whether different initial states can yield distinguishable output histories under specified inputs. For an autonomous smooth system, full rank of differentials of sufficiently many output Lie derivatives is a sufficient local condition under appropriate hypotheses. Controlled systems require the relevant input fields or a specified known-input trajectory. A local rank condition does not prove global uniqueness, numerical conditioning, or parameter identifiability. Parameters may be augmented as constant states, but adequate excitation and sensitivity remain necessary. Nonlinear indistinguishable sets need not be affine subspaces. For $\dot x=0$ and output $y=x_1^2+x_2^2$, a nonzero output level is a circle. The kernel of $Dy$ gives its tangent line, not the entire set of indistinguishable states. Linearized null directions describe first-order sensitivity; they must not be promoted to exact finite-state equivalence. Motion capture and force plates can constrain inverse dynamics when synchronized, calibrated, and accompanied by inertial estimates. Surface EMG adds electrical observations with muscle-specific processing and activation dynamics; integrating its signal does not directly yield force or activation. A universal fixed delay is not an adequate constitutive law. Optimization can select among feasible muscle patterns, but selection by a prior or cost function does not resolve ambiguity empirically [@Zajac1989]. For the two-hand grip, report which sensors constrain the net club wrench and which constrain the internal hand-load family. For the feet, report separate contact wrenches rather than assuming a unique bilateral distribution from a combined resultant. Compare alternative parameter sets and input allocations that fit the same observations; test whether the proposed conclusion survives that ambiguity. ## Symmetry, Mechanical Connections, and Geometric Phase []{#sec-ch9-connections} ### Momentum Conservation Has a Boundary Condition A cyclic coordinate has conserved conjugate momentum only when the Lagrangian is independent of that coordinate and its generalized applied force vanishes. Constraints and external loads must respect the symmetry. A grounded golfer generally receives external angular impulse, so total angular momentum of the golfer-plus-club subsystem is not automatically conserved. Conservation in a freely floating model cannot be asserted for the grounded experiment. For a local planar symmetry coordinate $\theta$ and shape $s$, consider $$ T=\tfrac12I(s)\dot\theta^2+\dot\theta\,a(s)^\top\dot s +\tfrac12\dot s^\top H(s)\dot s. $$ The conjugate momentum is $p_\theta=I\dot\theta+a^\top\dot s$. If it is conserved, reconstruction gives $$ \dot\theta=I^{-1}p_\theta-A(s)\dot s,\qquad A=I^{-1}a^\top. $$ {#eq-ch9:mechanical-connection} The locked inertia is $I=\partial^2T/\partial\dot\theta^2$; there is no additional factor of one half. The connection needs the group--shape cross block $a$, not the derivative of $I$ alone. At fixed momentum, a conventional Routhian is $L-p_\theta\dot\theta$ with $\dot\theta$ eliminated; forced or noncyclic systems require a modified reduction [@BulloLewis2004]. ### Separate Dynamic and Geometric Contributions For a closed shape loop in this planar model, $$ \Delta\theta=\int_0^T I(s(t))^{-1}p_\theta\,dt-\oint A(s)\,ds. $$ {#eq-ch9:berry-phase-formula} The first term is a dynamic phase and the second a geometric contribution with the sign fixed by the reconstruction equation. At zero momentum, only the latter remains. On a suitable planar shape patch, Stokes' theorem converts $\oint A$ to an integral of $dA$. General noncommuting spatial rotations require a group reconstruction and ordered transport; a scalar area formula does not directly apply. For a checkable example, choose $I=1$, $a=(0,k s_1)^\top$, and $H=I_2+aa^\top$. The full kinetic metric is positive definite because its Schur complement is $I_2$. At zero momentum, $\dot\theta=-k s_1\dot s_2$. A counterclockwise rectangular shape loop of side lengths $a_0,b_0$ gives $\Delta\theta=-k a_0b_0$. This proves a geometric effect in the specified model. It does not identify an analogous contribution in a golf swing until the inertia blocks, external moment, and measured shape trajectory have been established. ## Numerical Methods That Preserve the Stated Problem []{#sec-ch9-numerics} The position-constrained first-order mechanical DAE is ordinarily differentiation index three under regularity assumptions. Differentiating position once yields velocity consistency and twice yields an algebraic equation for $\lambda$; differentiating that relation yields a differential equation for the multiplier. Position to velocity to acceleration is two differentiations, not three. This terminology does not make a twice-differentiated constraint automatically stable under time stepping. Baumgarte stabilization replaces the acceleration equation by $J_c\dot v+\gamma+2\alpha J_cv+\beta^2\Phi=0$. In continuous arithmetic the violation obeys $\ddot\Phi+2\alpha\dot\Phi+\beta^2\Phi=0$. Positive gains damp errors, but $\alpha$ and $\beta$ have inverse-time units, and large gains can make integration stiff. Constraint scaling and time-step studies are necessary. Stabilization introduces corrective forces and does not guarantee exact energy conservation. A local Newton correction for position may use $$ q^{j+1}=q^j-M(q^j)^{-1}J_c(q^j)^\top \left[J_c(q^j)M(q^j)^{-1}J_c(q^j)^\top\right]^{-1}\Phi(q^j). $$ It requires a suitable initial guess and stopping criterion. After correcting position, velocity projection is $v=P_v(q)v_{\mathrm{pred}}$. These are projection operations; by themselves they do not specify SHAKE or RATTLE. Those methods combine particular constrained position/momentum updates with an underlying integration scheme. Their symplectic properties require their actual assumptions and solve tolerances; arbitrary after-step projection is not equivalent [@HairerLubichWanner2006]. For constant mass, a RATTLE-style velocity-Verlet step first computes a half-step momentum with a constraint impulse along $J_c(q_n)^\top$, chooses its multiplier so the updated position satisfies $\Phi(q_{n+1})=0$, then updates momentum with the new applied force and a second multiplier so $J_c(q_{n+1})M^{-1}p_{n+1}=0$. The half-step factors, old/new Jacobians, and both multipliers are part of the method. Configuration-dependent inertia and velocity-dependent forces require an appropriate generalized scheme, not a direct substitution into that separable formula. SVD can reveal the numerical rank of a scaled $J_c$ and supply a tangent basis $N$. The Euclidean projector $NN^\top$ assumes Euclidean-orthonormal columns; the kinetic-metric projector is $N(N^\top MN)^{-1}N^\top M$. Neither tangent projector alone includes the acceleration offset $-M^{-1}J_c^\top K_c^{-1}\gamma$. Solve linear systems instead of explicitly forming inverses in numerical code. Validation should report position, velocity, and acceleration residuals separately; input/reaction feasibility; changes under tighter solve tolerances and smaller time steps; and the energy balance including applied work and dissipation. Energy constancy is a test for an isolated lossless model, not for every passive, driven, or damped system. Good energy behavior is not proof of correct contact geometry or scientifically valid parameters. ## Connecting the Pieces in a Golf Investigation []{#sec-ch9-integration} Begin with a measurable outcome, such as delivered clubhead speed and face orientation at the model's own impact event. Specify the body/club model, coordinate frames, grip model, foot contact modes, and inputs. Estimate state and parameter uncertainty before decomposing effects. A useful same-state calculation compares the drift acceleration and individual projected input contributions; a forward intervention branches the state and recomputes reactions and contact events. Then ask distinct questions. How does posture change the effective input map? Which internal hand loads leave the net wrench unchanged while altering grip compliance? How much mechanical work has already entered the system, and where is it stored? How does ground impulse redirect momentum while doing little or no work in an ideal stationary-contact model? Which feedback corrections remain feasible under torque, activation, and contact limits? Impact depends on the delivered state, so differences earlier in the swing matter through their effects on that state and its uncertainty. These questions are intertwined without being interchangeable. A favorable configuration may improve load transmission while increasing sensitivity to parameter error. Co-contraction can change impedance without changing the instantaneous net joint torque, yet it can have metabolic cost and change later behavior. A zero-torque continuation can retain substantial speed because of stored momentum and energy, while still being an unrealistic intervention on activation or grip. A force plate can measure a large ground moment without showing that the ground supplied rotational work. An optimizer can expose a feasible strategy without establishing that a golfer uses it. For a defensible argument, report the model identity, the empirical observation, and the proposed causal mechanism separately. Publish the equations and reference-point conventions, reproducible numerical counterexamples, uncertainty ranges, and interventions that would falsify the proposed explanation. A prediction should survive reasonable alternative contact models and muscle allocations before being presented as a robust swing mechanism. Coaching conclusions require their own outcome evidence; geometric feasibility alone does not establish a beneficial instruction for a person. ### Exercises and Reproducible Checks 1. Verify $P_v^2=P_v$ and $P_v^\top M=MP_v$ for the unequal-mass example. Compare vector and force projections and explain their different domains. 1. Derive the circle's normal acceleration and show why a tangent-only acceleration update fails at nonzero speed. 1. Differentiate the two-coordinate metric to recover $c$ and verify $v^\top c=\tfrac12v^\top\dot Mv$ for arbitrary rates. 1. Derive the weighted allocation $(0.8,0.2)$ and identify input bounds that make the requested acceleration infeasible. 1. Transform an arbitrary twist and wrench between translated, rotated frames and verify power invariance. Then construct a nonzero internal hand-force pair with zero resultant wrench. 1. Derive the force-plate center-of-pressure signs from $r\times F$ and explain why the free moment must be retained. 1. Integrate the rectangular geometric-phase example and reverse the loop orientation. Add a known external torque and identify the extra momentum-dependent contribution. 1. Simulate a constrained system at successively smaller time steps. Report constraint residuals and the work-inclusive energy balance rather than certifying accuracy from one plot.