The Null Space of the Constraint Jacobian: Unconstrained Motion in Constrained Systems With Applications to Golf Swing Biomechanics

Dieter Olson

The Null Space of the Constraint Jacobian: Unconstrained Motion in Constrained Systems With Applications to Golf Swing Biomechanics

theory-core

Rigorous account of admissible motion, constrained acceleration, reduced speeds, and control authority, with checked examples and model boundaries.

Author

Dieter Olson

Published

November 28, 2025

Abstract

A constraint null space describes admissible instantaneous velocities. Turning that geometry into a prediction of a golf swing requires additional information: inertia, applied forces, constraint curvature, contact mode and an input model. This article derives the connections explicitly. It distinguishes reduced speeds from coordinates, force projections from velocity projections, instantaneous acceleration authority from controllability over time, and kinematic redundancy from measured physiological coordination. Independently checked examples expose errors that these distinctions prevent. A declared planar two-arm and club model provides a reproducible closed-chain example; it is a teaching model, not a fitted golfer.

Conceptual Overview

Holding a club with two hands couples the possible motions of the arms and club. A small movement of one grip must agree with the movement of the same point on the club. The constraint Jacobian expresses that agreement locally. Its null space collects combinations of joint and club velocities that preserve it.

This answers which motions are compatible with a chosen model. It does not identify which muscles should activate, which motion maximizes clubhead speed, or which forces cost metabolic energy. Those questions involve dynamics, actuation and an objective. The connection is useful precisely when we keep these layers separate and then join them through explicit equations.

In Layman’s Terms

Imagine two articulated drawing arms attached to a ruler. Their joint motions must agree about where the ruler’s attachment points are going. There may be several compatible ways to move. Some move a selected point on the ruler; some rotate the ruler while leaving that point instantaneously stationary. None of this tells us how hard the motors must work. To answer that, we need the masses, velocities, applied loads and available motor forces.

Introduction

We begin with independent generalized coordinates $q\in\mathbb R^n$ before loop constraints are imposed, a symmetric positive-definite mass matrix $M(q)$, and smooth bilateral constraints of locally constant rank. Unless a section says otherwise, the constraints are stationary and ideal, and the contact mode is fixed. Ideal means that the reaction does no work on admissible virtual displacements. It does not mean that a biological joint or grip is perfectly rigid.

A generalized force $f$ is defined by its work pairing $f^\top\delta q$. Some components may be forces and others torques. Coordinate scaling therefore matters when interpreting Euclidean norms, singular values or “orthogonality.” The mass metric supplies the dynamically relevant pairing of velocities.

The derivations below are independently checkable from these assumptions. For complementary formulations, see the generalized-speed and bilateral-constraint sections of Tedrake’s multibody notes (Tedrake, n.d.-a). The worked examples are reproduced by the numerical builder and checked against direct saddle solves, analytic trajectories, finite differences and forward integration in the verification suite.

Holonomic Constraints and the Constraint Jacobian

Constraint Formulation

A holonomic constraint is a position relation \[ \Phi(q,t)=0,\qquad \Phi:\mathbb R^n\times\mathbb R\longrightarrow\mathbb R^m. \] For stationary constraints, suppress $t$. If $J_c=\Phi_q$ has full row rank $m$ in a neighborhood of the configuration, the level set is locally a smooth manifold of dimension $n-m$. Redundant equations should be reduced to an independent description before using the inverses below.

Bilateral constraints enforce equality. Ground contact is generally unilateral: a foot may separate from the ground, and sticking requires friction conditions. Treating a foot as fixed is a declared contact-mode approximation, not a consequence of merely touching the ground.

The Constraint Jacobian

Differentiating along a trajectory gives \[ J_c\dot q+\Phi_t=0,\qquad J_c\ddot q+\gamma=0, \] where, component by component, \[ \gamma_i=\dot q^\top\Phi_{i,qq}\dot q +2\Phi_{i,qt}\dot q+\Phi_{i,tt}. \] For stationary constraints, $\gamma=\dot J_c\dot q$. For prescribed moving constraints, the feasible velocities form an affine set, not generally $\ker J_c$. Differences between two feasible velocities still belong to that kernel.

The acceleration equation is also affine. Even when velocity lies in the tangent space, acceleration need not: a point moving around a circle needs normal acceleration.

Example: Closed Kinematic Chain in Golf

Let $p_L(q_b)$ and $p_R(q_b)$ locate two hand attachment points from body coordinates $q_b$. A rigid club has position $c$, orientation $R$, and fixed local grip offsets $r_L,r_R$. Point attachment requires \[ \Phi_L=p_L(q_b)-c-Rr_L=0,\qquad \Phi_R=p_R(q_b)-c-Rr_R=0. \] Differentiating $Rr_i$ produces a club-rotation contribution. Omitting it changes the model and gives the wrong admissible motions.

These equations constrain points, not hand orientation relative to the handle. A welded grasp needs additional orientation relations; a compliant grasp needs constitutive force laws and possibly extra states. Nesbit’s golf model, for example, omitted explicit hands and introduced translational wrist flexibility to avoid an indeterminate closed loop. Its methods therefore do not validate a perfectly rigid two-hand grasp (Nesbit 2005, 501). The planar construction below states its own assumptions.

The Null Space: Mathematical Foundations

Definition and Basic Properties

At a fixed configuration, \[ \ker J_c=\{w:J_cw=0\}. \] Rank-nullity gives $\dim\ker J_c=n-\operatorname{rank}J_c$ as a linear-algebra statement. Identifying this kernel with the actual tangent space also requires a regular description of the constraint set.

For example, $\Phi(x,y)=x^2+y^2=0$ defines only one point. At that point $J_c=0$, so its kernel is all of $\mathbb R^2$, although no nonconstant feasible path exists. A singular Jacobian does not automatically demonstrate two physical degrees of freedom.

Null Space Basis and Projection

Let $N(q)\in\mathbb R^{n\times k}$ have independent columns spanning $\ker J_c$, with $k=n-m$. For stationary constraints, \[ \dot q=N(q)\nu . \] The components of $\nu$ are reduced speeds. Calling them $\dot z$ presupposes a coordinate chart $q=\psi(z)$ with $N=\psi_z$, or a separately declared transformation between chart velocities and speeds.

The Euclidean projector onto the column space is \[ P_E=N(N^\top N)^{-1}N^\top. \] If the columns are orthonormal this reduces to $NN^\top$. Neither expression is automatically the force-to-acceleration map of a system with nonidentity inertia.

Geometric Interpretation

A tangent distribution can be integrable while a particular basis is not a coordinate basis. On the dimensionless coordinate plane $q_3=0$, take \[ n_1=(\cos q_1,\sin q_1,0)^\top,\qquad n_2=(-\sin q_1,\cos q_1,0)^\top. \] These are an orthonormal tangent frame. With the convention $[n_1,n_2]=Dn_2\,n_1-Dn_1\,n_2$, their bracket is $(-1,0,0)^\top$. Coordinate basis fields commute, so these columns cannot both be the derivatives of one local chart. They still span the tangent plane everywhere.

This distinction matters computationally: integrating $\nu$ as if it were a vector of coordinate derivatives generally reconstructs the wrong configuration. Integrate $\dot q=N\nu$ or supply an actual chart and its velocity transformation.

Constrained Equations of Motion

Lagrangian Formulation With Constraints

Collect applied generalized forces, gravity and velocity-dependent inertial terms into $f(q,\dot q,u,t)$ so that \[ M\ddot q=f+J_c^\top\lambda,\qquad J_c\ddot q=-\gamma . \] The sign convention defines $\lambda$; changing the sign of a constraint changes its multiplier, not the physical reaction $J_c^\top\lambda$. The instantaneous saddle system is \[ \begin{bmatrix}M&-J_c^\top\\J_c&0\end{bmatrix} \begin{bmatrix}\ddot q\\\lambda\end{bmatrix} =\begin{bmatrix}f\\-\gamma\end{bmatrix}. \] Define $W=J_cM^{-1}J_c^\top$. Positive-definite $M$ and full row rank $J_c$ make $W$ invertible. Elimination yields \[ \lambda=-W^{-1}(J_cM^{-1}f+\gamma), \] \[ \ddot q=A_cf-M^{-1}J_c^\top W^{-1}\gamma,\qquad A_c=M^{-1}-M^{-1}J_c^\top W^{-1}J_cM^{-1}. \] Both terms are essential. Substitution verifies $J_c\ddot q=-\gamma$; the first term alone gives zero constraint acceleration and loses curvature.

The original first-order system with the position equation $\Phi(q)=0$ has differentiation index 3 under these regularity assumptions. Replacing it with the acceleration relation gives an index-1 formulation with algebraically solvable multipliers; eliminating them yields an ODE on the consistent state manifold. These are different formulations of the same exact motion only with consistent initialization. Numerical integration can violate both position and velocity constraints, so both residuals need monitoring or correction (Hairer 2011, sec. IV.4).

Null Space Method for Constraint Elimination

For stationary constraints, differentiate $\dot q=N\nu$: \[ \ddot q=N\dot\nu+\dot N\nu,\qquad \dot N=DN(q)[\dot q]. \] Premultiplication by $N^\top$ eliminates reactions: \[ M_r\dot\nu=N^\top f-N^\top M\dot N\nu,\qquad M_r=N^\top MN. \] The reduced mass matrix is positive definite. This is an equation for speed derivatives, together with the reconstruction equation for $q$. Dropping $\dot N\nu$ loses the acceleration caused by the changing tangent frame. Differentiating $J_cN=0$ confirms that this term supplies the required normal acceleration.

Physical Interpretation: Force Projection

Define the velocity and force projectors \[ P_v=I-M^{-1}J_c^\top W^{-1}J_c,\qquad P_\tau=P_v^\top=I-J_c^\top W^{-1}J_cM^{-1}. \] They satisfy $P_v^2=P_v$, $J_cP_v=0$ and $P_v^\top M=MP_v$. Thus $P_v$ is orthogonal in the mass metric. Its transpose acts on force covectors, preserving the work pairing. In general they are different matrices. Moreover, \[ A_c=P_vM^{-1}=M^{-1}P_\tau =N(N^\top MN)^{-1}N^\top. \]

A numerical counterexample prevents an easy mistake. Choose two translational coordinates with \[ M=\begin{bmatrix}2&1\\1&3\end{bmatrix}\ {\rm kg},\quad J_c=\begin{bmatrix}1&1\end{bmatrix},\quad f=\begin{bmatrix}1\\0\end{bmatrix}\ {\rm N},\quad\gamma=0 . \] Then \[ P_\tau=\frac13\begin{bmatrix}1&-1\\-2&2\end{bmatrix},\qquad A_c=\frac13\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\ {\rm kg}^{-1}. \] The acceleration is $(1/3,-1/3)^\top\ {\rm m\,s^{-2}}$ and $\lambda=-2/3\ {\rm N}$. Using $M^{-1}P_Ef$ instead gives $(0.4,-0.3)^\top\ {\rm m\,s^{-2}}$, which violates $J_c\ddot q=0$. An ordinary Euclidean projection followed by an inertia inverse is not generally valid.

Connection to Control-Affine Systems and Drift

Control-Affine Structure

Suppose $f=f_0(q,\dot q)+B(q,\dot q)u$, with stationary constraints and inputs that enter affinely. On \[ \mathcal X=\{(q,v):\Phi(q)=0,\ J_c(q)v=0\}, \] the ambient representation of the state vector field is \[ \begin{bmatrix}\dot q\\\dot v\end{bmatrix} = \underbrace{\begin{bmatrix} v\\A_cf_0-M^{-1}J_c^\top W^{-1}\gamma \end{bmatrix}}_{\text{drift}} + \underbrace{\begin{bmatrix}0\\A_cB\end{bmatrix}}_{\text{input map}}u . \] This vector field is tangent to the state manifold, although its acceleration component need not lie in $\ker J_c$. A velocity-dependent $B$ remains control-affine if it is independent of $u$. Muscle activation dynamics or nonlinear actuator laws may require augmenting the state or changing the input definition.

Drift Dynamics and the Null Space

A mass moving on a circle demonstrates the missing term. Let $M=2I\ {\rm kg}$, $R=1\ {\rm m}$ and \[ \Phi(q)=\frac{x^2+y^2-R^2}{2R}. \] At $q=(1,0)\ {\rm m}$ and $v=(0,3)\ {\rm m\,s^{-1}}$, the dimensionless Jacobian is $(1,0)$ and $\gamma=9\ {\rm m\,s^{-2}}$. Under gravity $f_0=(0,-19.62)\ {\rm N}$, \[ \ddot q=(-9,-9.81)\ {\rm m\,s^{-2}},\qquad \lambda=-18\ {\rm N}. \] The horizontal acceleration keeps the mass on the circle. It exists even though the ideal reaction power is zero. The tangent representation gives the same result: at this point $N=(0,1)^\top$, $\nu=3\ {\rm m\,s^{-1}}$, $\dot N=(-3,0)^\top\ {\rm s^{-1}}$ and $\dot\nu=-9.81\ {\rm m\,s^{-2}}$.

“Drift” means zero chosen input in the chosen model. It need not mean zero neural command, relaxed muscles or a feasible free response after a foot lifts. If the computed reaction violates a unilateral contact condition, the fixed-mode drift no longer describes the actual motion.

The Golf Swing as a Control Problem

A useful golf question specifies an initial state, an impact target, a time horizon, permissible inputs and a cost. Clubhead speed alone is not the entire target: club orientation, path, strike location and contact timing can constrain the solution.

The constrained equations connect these choices. Geometry determines admissible motion; inertia and velocity determine acceleration and reaction; actuators bound achievable changes; the objective selects among feasible trajectories. “Exploit drift” becomes a testable hypothesis only after those ingredients are declared and competing trajectories are compared under the same conditions.

Controllability and the Null Space

Low instantaneous input rank does not preclude control of the full state over time. Consider two unit masses with unit springs, stiffness \[ K=\begin{bmatrix}2&-1\\-1&2\end{bmatrix}, \] and a force $u$ on the first mass. With $x=(q_1,q_2,\dot q_1,\dot q_2)^\top$, \[ \dot x=Ax+bu,\qquad A=\begin{bmatrix}0&I_2\\-K&0\end{bmatrix},\quad b=(0,0,1,0)^\top . \] In SI units the masses are $1\ {\rm kg}$ and the spring constants $1\ {\rm N\,m^{-1}}$. The acceleration input rank is one, yet \[ [b,\ Ab,\ A^2b,\ A^3b] =\begin{bmatrix} 0&1&0&-2\\ 0&0&0&1\\ 1&0&-2&0\\ 0&0&1&0 \end{bmatrix} \] has rank four and determinant of magnitude one. Coupling distributes the effect of the input over time. This is the linear state-controllability test, not an instantaneous force-rank test (Tedrake, n.d.-b).

A stronger check constructs an actual control. From rest, target $x_f=(0,1,0,0)^\top$ at $T=2\ {\rm s}$. For unconstrained real-valued inputs, set \[ W_T=\int_0^T e^{As}bb^\top e^{A^\top s}\,ds,\qquad u_*(t)=b^\top e^{A^\top(T-t)}W_T^{-1}x_f . \] The minimum squared-force integral is $x_f^\top W_T^{-1}x_f =59.6394\ {\rm N^2\,s}$. Independent forward integration reaches the specified terminal state. This cost is not mechanical work or metabolic energy.

Input limits change reachability. If $|u|\le5\ {\rm N}$ over two seconds, then $\int u^2dt\le50\ {\rm N^2\,s}$, below the unconstrained minimum. Therefore that bounded actuator cannot reach this target in two seconds, despite full state controllability with unrestricted inputs. A rank result alone cannot settle a late-swing timing question.

Computational Methods

SVD-Based Null Space Computation

For $J_c=U\Sigma V^\top$, retain the right singular vectors associated with zero singular values, including the structural null columns when $m<n$. A compact SVD can omit those columns; use a complete right basis or a routine that explicitly returns the null space.

Floating-point “zero” requires a declared tolerance. SciPy’s documented default uses a threshold relative to the largest singular value, with relative factor equal to machine precision times the larger matrix dimension (The SciPy Community, n.d.). That is an arithmetic default, not an uncertainty model for marker data or anatomical calibration. Normalize coordinate and constraint scales meaningfully, inspect the spectrum and test sensitivity to plausible perturbations.

Check $J_cN$, independence of the columns, and any intended normalization. Near a singularity, report both the residual and the rank decision. A pseudoinverse may handle redundant equations, but it does not establish regularity or repair inconsistent constraint accelerations.

QR Decomposition Approach

A complete QR factorization of $J_c^\top$ can produce an orthonormal complement to its column space. When the rank is uncertain, pivoting and a rank criterion are needed. Do not assume that the last columns of an incomplete factorization contain the entire kernel.

Different valid bases may flip signs or rotate between evaluations. Physical predictions should be invariant to the basis, but numerically differentiating an unaligned basis can create a spurious $\dot N$. Use a smooth local construction, align neighboring frames where appropriate, or solve the ambient saddle system instead. The latter still requires accurate constraint derivatives.

Recursive Null Space Formulation

Suppose $N_0$ spans $\ker J_0$ and new constraints add rows $J_1$. Compute a full-column-rank basis $Z$ of $\ker(J_1N_0)$ and set \[ N=N_0Z . \] Then $J_0N=0$ and $J_1N=0$. Conversely, any vector satisfying both constraints is $N_0a$ with $a\in\ker(J_1N_0)$, so this construction spans the complete combined kernel. Its dimension depends on the additional independent rank, not merely the number of appended equations.

If the bases vary with configuration, derivatives of both factors enter $\dot N=\dot N_0Z+N_0\dot Z$. Recursion reduces algebraic size; it does not remove the moving-frame terms.

Application: Golf Swing Multi-Body Model

Model Description

Use a planar teaching model with two fixed shoulder points, two two-link arms, and a freely translating and rotating rigid club. The seven coordinates are \[ q=(\theta_L,\beta_L,\theta_R,\beta_R,c_x,c_y,\varphi)^\top . \] The elbow angles $\beta_i$ are relative to the corresponding upper-arm angles. Both arms have lengths $a=0.6\ {\rm m}$ and $b=0.5\ {\rm m}$. Shoulder positions are $s_L=(-0.5,0)$ and $s_R=(0.5,0)\ {\rm m}$; club-local grip offsets are $r_L=(-0.1,0)$ and $r_R=(0.1,0)\ {\rm m}$.

Only point coincidence is imposed at the grips. The shoulders are fixed supports, the club is rigid, and neither wrist orientation nor a pelvis coordinate is silently added. These assumptions make the model reproducible; they deliberately omit the three-dimensional torso, legs, grip compliance, muscle physiology and shaft deformation needed for a golfer model.

Constraint Equations

Write $e(\theta)=(\cos\theta,\sin\theta)^\top$. The arm endpoints are \[ p_i=s_i+a e(\theta_i)+b e(\theta_i+\beta_i). \] The four scalar closure equations are the two components of each \[ p_i-c-R(\varphi)r_i=0,\qquad i\in\{L,R\}. \] One consistent pose is \[ q\approx(0.06860223,\ 1.91063324,\ 1.41808412,\ 1.91063324,\ 0,\ 0.5,\ 0)^\top . \] Angles are in radians and translations in meters. The stored example uses full precision; rounded coordinates here incur small closure residuals.

Constraint Jacobian Structure

Let $S=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ and \[ D_i=\begin{bmatrix} aSe(\theta_i)+bSe(\theta_i+\beta_i)&bSe(\theta_i+\beta_i) \end{bmatrix}. \] Then \[ J_c= \begin{bmatrix} D_L&0_{2\times2}&-I_2&-R(\varphi)Sr_L\\ 0_{2\times2}&D_R&-I_2&-R(\varphi)Sr_R \end{bmatrix}. \] It has rank four at the stated pose, leaving three independent instantaneous speeds. Finite differences of all seven coordinates verify every block, including club rotation.

For a selected club point $p_H=c+R(\varphi)r_H$, with $r_H=(0.8,0)\ {\rm m}$, \[ J_H=\begin{bmatrix}0_{2\times4}&I_2&R(\varphi)Sr_H\end{bmatrix}. \] Admissibility is governed by $\ker J_c$; keeping that point instantaneously stationary requires the smaller space \[ \ker\begin{bmatrix}J_c\\J_H\end{bmatrix}. \] At this pose the smaller kernel is one-dimensional. Its nonzero direction includes club rotation: zero velocity of a selected point does not imply zero angular velocity of the club. Nor does instantaneous stationarity ensure that the point remains fixed over a finite interval.

Kinematic Synergies via Null Space Analysis

The columns of $N$ are not uniquely determined. For an invertible matrix $Q$, $NQ$ spans the same admissible motions. With an orthonormal basis and orthogonal $Q$, even normalization is unchanged. Naming a column an “arm synergy” or “wrist synergy” therefore depends on a basis convention unless additional structure identifies it.

PCA of $NN^\top$ cannot discover preferred coordination in this setting: its eigenvalues are one on the tangent space and zero on the normal space, with no preferred directions within either degenerate eigenspace. PCA of measured, consistently scaled trajectories can describe observed covariance; it still does not establish a neural command mechanism or an optimal recruitment strategy. Data, preprocessing, task conditions and a physiological hypothesis are separate evidence requirements.

Drift-Control Separation in Golf Swing Dynamics

The Natural Swing: Exploiting Drift

A model can investigate whether initial configuration and velocity make a desired impact state reachable with less actuator effort. Define that effort first: squared torque, activation, mechanical work, peak force and metabolic cost are different objectives and can favor different trajectories.

Compare candidate motions with matched boundary conditions, duration, contact assumptions and actuator limits. Recompute reactions and task accelerations throughout. A low-input trajectory is useful evidence for that model and objective; it does not establish that an elite golfer is passive, that large reaction forces are wasteful, or that one wrist posture is universally preferable.

The muscle-to-joint torque chapter supplies the next layer: signed moment arms, feasible force sharing and the distinction between net torque and individual muscle force. The constraint chapter develops compatible motion and contact assumptions. Together these layers connect geometry to a bounded actuator model without treating either as a complete account of motor control.

Control Authority Analysis

For a position task $p_H=h(q)$, \[ \dot p_H=J_H\dot q,\qquad \ddot p_H=J_H\ddot q+\dot J_H\dot q. \] Substituting constrained dynamics gives \[ \ddot p_H= \underbrace{J_H(A_cf_0-M^{-1}J_c^\top W^{-1}\gamma) +\dot J_H\dot q}_{\text{task drift acceleration}} +J_HA_cBu . \] At a fixed state, $J_HA_cB$ maps an input change to a task-acceleration change. Its singular values require input and output scaling before they can be compared meaningfully. With bounded inputs, examine the image of the actual admissible input set, not only its linear span.

The changing-Jacobian term is physical. A point at radius $0.5\ {\rm m}$ rotating at constant $2\ {\rm rad\,s^{-1}}$ has acceleration magnitude $2\ {\rm m\,s^{-2}}$, although angular acceleration is zero. In three dimensions, \[ \ddot p_H=\ddot c+\alpha\times(Rr_H) +\omega\times\bigl(\omega\times(Rr_H)\bigr). \] This includes tangential and centripetal acceleration. It also makes clear why a zero clubhead-velocity direction need not be a zero clubhead-acceleration direction.

Finally, impact control concerns the endpoint of a trajectory. Available time, state evolution, actuator bandwidth and limits determine whether an instantaneous acceleration direction can produce a useful change before contact.

Energy Flow Analysis

For stationary ideal constraints, \[ \mathcal P_c=\dot q^\top J_c^\top\lambda =\lambda^\top J_c\dot q=0. \] This is total reaction power on the modeled constrained system. A joint or grip can transfer power between connected bodies while contributing zero net power to their combination. Large internal loads can also matter for tissue stress and muscle recruitment, neither of which follows from the zero-net-power identity.

For time-independent kinetic and potential energy, write \[ M\ddot q+C(q,\dot q)\dot q+\nabla V+d=Bu+J_c^\top\lambda, \] with $C$ consistent with the kinetic-energy identity and $d$ the declared dissipative generalized load. Then \[ \frac{d}{dt}(T+V)=\dot q^\top Bu-\dot q^\top d \] under stationary ideal constraints. This accounting distinguishes actuator power, storage and dissipation; it does not infer metabolic expenditure.

For moving constraints, $J_c\dot q=-\Phi_t$, so \[ \mathcal P_c=-\lambda^\top\Phi_t \] need not vanish. A $2\ {\rm kg}$ mass prescribed to follow $x=t^2$ in SI units has $v=2\ {\rm m\,s^{-1}}$ and $a=2\ {\rm m\,s^{-2}}$ at $t=1\ {\rm s}$. With an applied force of $3\ {\rm N}$, the guide supplies another $1\ {\rm N}$. Applied and guide powers are $6$ and $2\ {\rm W}$, adding to the $8\ {\rm W}$ kinetic-energy rate. Ideal virtual-work behavior is compatible with actual power supplied by a moving support.

Advanced Topics

Configuration-Dependent Null Space Dimension

Within a regular fixed constraint description, rank is locally constant and so is nullity. At a rank change, a larger linearized kernel may reflect a singular description, a genuine mechanism singularity, or both. Examine feasible paths and higher-order constraints before assigning extra physical mobility.

A contact change is different again. A foot lifting, a grip slipping or impact beginning changes the active equations. New contact can require an impulse and a velocity reset. A smooth fixed-mode acceleration solve does not describe that reset. Model each mode and its transition conditions explicitly.

Velocity-Level vs. Acceleration-Level Null Spaces

For prescribed moving holonomic constraints, choose a particular velocity $b(q,t)$ satisfying $J_cb=-\Phi_t$ and write \[ \dot q=b+N\nu,\qquad \ddot q=\dot b+\dot N\nu+N\dot\nu. \] The dots here are total derivatives along the trajectory, including explicit time dependence. Projecting the dynamics gives \[ (N^\top MN)\dot\nu =N^\top\!\left[f-M(\dot b+\dot N\nu)\right]. \] The choice of $b$ is not unique; adding a tangent component changes the speed representation, not the set of physical velocities.

For a genuinely nonholonomic velocity constraint $H(q)\dot q=0$, the kernel still describes admissible velocities, but it need not be tangent to any lower-dimensional configuration manifold. Do not use a holonomic coordinate reduction without checking integrability.

Also distinguish a null-space basis from a generalized-speed map. If a representation uses $\dot q=E(q)v$, the relevant constraint matrix is $H=J_cE$, and the velocity-coordinate mass matrix and force covectors must be consistent with $v$. Reactions then enter that equation through $H^\top\lambda$. Quaternion coordinate rates, for example, are not interchangeable with a three-component angular velocity. Tedrake’s multibody treatment explicitly separates configuration rates and generalized velocities (Tedrake, n.d.-a).

Lie Bracket Relationships

For smooth control-affine vector fields, brackets describe directions produced through state-dependent combinations of flows. They help analyze accessibility when instantaneous input directions are insufficient. They must be computed on the consistent state manifold, using the complete drift, including curvature and reconstruction.

Accessibility, local controllability, small-time local controllability and reachability under bounded unilateral inputs are distinct properties. With drift, a bracket-rank calculation alone does not generally establish all of them. The two-mass example already demonstrates the more basic point: deficient instantaneous acceleration rank is not a necessary obstruction to state controllability. Conversely, full linear controllability does not eliminate force and deadline constraints.

Conclusions

The constraint Jacobian organizes a sequence of questions. Its kernel describes locally admissible velocity under regular stationary constraints. The mass metric and differentiated constraint determine acceleration and reaction. An actuator model limits the accelerations that can be changed. Integration over time determines reachable task states, and a declared objective defines what counts as improvement.

For golf and humanoid modeling, the value of this framework is the ability to test those connections. A claim about a grip, posture or coordination pattern should survive its own geometry, dynamics, force limits and task conditions before it becomes an instruction to a player.

Limitations

The analytic results assume smooth local models with independent constraint rows and positive-definite inertia unless explicitly extended. The examples are constructed. The planar grasp has no fitted human inertias, muscle data, ground mechanics or measured swing trajectory. Its rank and task redundancy establish properties of that construction only.

Critics’ Comments

Does this prove an effortless swing? No. Zero ideal reaction power is not zero muscle force or metabolic cost. The relevant optimization and physiological measurements remain separate.

Does a large null space mean better control? No. It means more first-order compatible velocity directions in a regular description. Input limits, inertia, task geometry and time determine how those directions can be used.

Can the examples validate a golfer model? They can catch mathematical and implementation errors. Empirical validation requires measurements, uncertainty estimates and out-of-sample comparisons for the intended physical model.

Can a pseudoinverse solve a singularity? It can select an algebraic solution under specified conditions. It cannot decide whether a rank loss represents physical mobility, inconsistent constraints or a change in contact mode.

Appendix

Proof: Orthogonality of Constraint Forces

An admissible virtual displacement at fixed time obeys $J_c\delta q=0$. Thus \[ (J_c^\top\lambda)^\top\delta q =\lambda^\top J_c\delta q=0. \] This is a force–displacement duality statement. With mixed coordinates, interpreting it as an ordinary geometric angle requires a coordinate metric. Actual velocities for moving constraints need not satisfy the homogeneous virtual-displacement equation, which is why their reaction power can be nonzero.

Derivation of the Null Space Projector

For a full-column-rank $N$, minimize $\|w-Na\|_2^2$ over $a$. The normal equations give $a=(N^\top N)^{-1}N^\top w$, hence $P_E=N(N^\top N)^{-1}N^\top$.

For the mass metric, minimize $(w-Na)^\top M(w-Na)$. This gives \[ P_v=N(N^\top MN)^{-1}N^\top M . \] An equivalent constrained minimization over $J_cz=0$ gives $z=w-M^{-1}J_c^\top W^{-1}J_cw$. Equating the two forms proves the stated mass projector and, after right multiplication by $M^{-1}$, the identity for $A_c$.

Consequently a force of the form $J_c^\top\mu$ has zero reduced force $N^\top J_c^\top\mu$ and zero $A_c$ acceleration contribution. This does not say the actual reaction is absent or that total acceleration is tangent: the curvature term remains.

Summary Table of Key Equations

Question	Relation	Conditions and Meaning
Compatible velocity	$J_c\dot q=0$	Stationary holonomic constraint; regularity identifies the tangent space.
Moving constraint	$J_c\dot q=-\Phi_t$	Affine velocity set; its direction space is $\ker J_c$.
Compatible acceleration	$J_c\ddot q=-\gamma$	Includes curvature and prescribed time dependence.
Reduced motion	$\dot q=N\nu$	Stationary case; $\nu$ denotes speeds unless a chart is supplied.
Reduced dynamics	$M_r\dot\nu=N^\top f-N^\top M\dot N\nu$	Full-column-rank tangent frame and positive-definite inertia.
Force-to-acceleration map	$A_c=N(N^\top MN)^{-1}N^\top$	Fixed-state force contribution; total acceleration also includes constraint curvature.
Task acceleration	$\ddot p_H=J_H\ddot q+\dot J_H\dot q$	Differentiable position task; changing-Jacobian term retained.
Reaction power	$\mathcal P_c=-\lambda^\top\Phi_t$	Ideal holonomic constraint; zero for stationary constraints.

References

The annotated bibliography maps the cited passages to the claims they support and distinguishes source reading from the independent examples. It does not treat conceptual relationships as verified citation links between publications.

Hairer, Ernst. 2011. Solving Differential Equations on Manifolds. Université de Genève. https://www.unige.ch/~hairer/poly-sde-mani.pdf.

Nesbit, Steven M. 2005. “A Three Dimensional Kinematic and Kinetic Study of the Golf Swing.” Journal of Sports Science and Medicine 4: 499–519. https://www.jssm.org/volume04/iss4/cap/jssm-04-499.pdf.

Tedrake, Russ. n.d.-a. Underactuated Robotics: Multi-Body Dynamics. Accessed September 12, 2026. https://underactuated.mit.edu/multibody.html.

Tedrake, Russ. n.d.-b. Underactuated Robotics: The Acrobot and Cart-Pole. Accessed September 12, 2026. https://underactuated.mit.edu/acrobot.html.

The SciPy Community. n.d. Scipy.linalg.null_space. Accessed September 12, 2026. https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.null_space.html.

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@online{olson2025,
  author = {Olson, Dieter},
  title = {The {Null} {Space} of the {Constraint} {Jacobian:}
    {Unconstrained} {Motion} in {Constrained} {Systems} {With}
    {Applications} to {Golf} {Swing} {Biomechanics}},
  date = {2025-11-28},
  url = {https://affinedrift.com/articles/null-space-constraint-jacobian.html},
  langid = {en}
}

For attribution, please cite this work as:

Olson, Dieter. 2025. “The Null Space of the Constraint Jacobian: Unconstrained Motion in Constrained Systems With Applications to Golf Swing Biomechanics.” November 28. https://affinedrift.com/articles/null-space-constraint-jacobian.html.

--- title: "The Null Space of the Constraint Jacobian: Unconstrained Motion in Constrained Systems With Applications to Golf Swing Biomechanics" description: "Rigorous account of admissible motion, constrained acceleration, reduced speeds, and control authority, with checked examples and model boundaries." author: "Dieter Olson" date: "2025-11-28" date-source: "initial-publication-record" bibliography: ../references/nullspace-rigor.bib css: ../css/duality-chapter.css format: html: toc: true toc-depth: 3 number-sections: false code-fold: true categories: - theory-core --- ::: {.abstract-section} ## Abstract A constraint null space describes admissible instantaneous velocities. Turning that geometry into a prediction of a golf swing requires additional information: inertia, applied forces, constraint curvature, contact mode and an input model. This article derives the connections explicitly. It distinguishes reduced speeds from coordinates, force projections from velocity projections, instantaneous acceleration authority from controllability over time, and kinematic redundancy from measured physiological coordination. Independently checked examples expose errors that these distinctions prevent. A declared planar two-arm and club model provides a reproducible closed-chain example; it is a teaching model, not a fitted golfer. ::: ## Conceptual Overview Holding a club with two hands couples the possible motions of the arms and club. A small movement of one grip must agree with the movement of the same point on the club. The constraint Jacobian expresses that agreement locally. Its null space collects combinations of joint and club velocities that preserve it. This answers **which motions are compatible with a chosen model**. It does not identify which muscles should activate, which motion maximizes clubhead speed, or which forces cost metabolic energy. Those questions involve dynamics, actuation and an objective. The connection is useful precisely when we keep these layers separate and then join them through explicit equations. ::: {.callout-note} ## In Layman's Terms Imagine two articulated drawing arms attached to a ruler. Their joint motions must agree about where the ruler's attachment points are going. There may be several compatible ways to move. Some move a selected point on the ruler; some rotate the ruler while leaving that point instantaneously stationary. None of this tells us how hard the motors must work. To answer that, we need the masses, velocities, applied loads and available motor forces. ::: ## Introduction We begin with independent generalized coordinates $q\in\mathbb R^n$ before loop constraints are imposed, a symmetric positive-definite mass matrix $M(q)$, and smooth bilateral constraints of locally constant rank. Unless a section says otherwise, the constraints are stationary and ideal, and the contact mode is fixed. Ideal means that the reaction does no work on admissible virtual displacements. It does not mean that a biological joint or grip is perfectly rigid. A generalized force $f$ is defined by its work pairing $f^\top\delta q$. Some components may be forces and others torques. Coordinate scaling therefore matters when interpreting Euclidean norms, singular values or “orthogonality.” The mass metric supplies the dynamically relevant pairing of velocities. The derivations below are independently checkable from these assumptions. For complementary formulations, see the generalized-speed and bilateral-constraint sections of Tedrake's multibody notes [@TedrakeMultibody]. The worked examples are reproduced by the [numerical builder](https://github.com/D-sorganization/AffineDrift/blob/main/scripts/build_nullspace_examples.py) and checked against direct saddle solves, analytic trajectories, finite differences and forward integration in the [verification suite](https://github.com/D-sorganization/AffineDrift/blob/main/tests/test_nullspace_article_rigor.py). ## Holonomic Constraints and the Constraint Jacobian ### Constraint Formulation A holonomic constraint is a position relation $$ \Phi(q,t)=0,\qquad \Phi:\mathbb R^n\times\mathbb R\longrightarrow\mathbb R^m. $$ For stationary constraints, suppress $t$. If $J_c=\Phi_q$ has full row rank $m$ in a neighborhood of the configuration, the level set is locally a smooth manifold of dimension $n-m$. Redundant equations should be reduced to an independent description before using the inverses below. Bilateral constraints enforce equality. Ground contact is generally unilateral: a foot may separate from the ground, and sticking requires friction conditions. Treating a foot as fixed is a declared contact-mode approximation, not a consequence of merely touching the ground. ### The Constraint Jacobian Differentiating along a trajectory gives $$ J_c\dot q+\Phi_t=0,\qquad J_c\ddot q+\gamma=0, $$ where, component by component, $$ \gamma_i=\dot q^\top\Phi_{i,qq}\dot q +2\Phi_{i,qt}\dot q+\Phi_{i,tt}. $$ For stationary constraints, $\gamma=\dot J_c\dot q$. For prescribed moving constraints, the feasible velocities form an **affine set**, not generally $\ker J_c$. Differences between two feasible velocities still belong to that kernel. The acceleration equation is also affine. Even when velocity lies in the tangent space, acceleration need not: a point moving around a circle needs normal acceleration. ### Example: Closed Kinematic Chain in Golf Let $p_L(q_b)$ and $p_R(q_b)$ locate two hand attachment points from body coordinates $q_b$. A rigid club has position $c$, orientation $R$, and fixed local grip offsets $r_L,r_R$. Point attachment requires $$ \Phi_L=p_L(q_b)-c-Rr_L=0,\qquad \Phi_R=p_R(q_b)-c-Rr_R=0. $$ Differentiating $Rr_i$ produces a club-rotation contribution. Omitting it changes the model and gives the wrong admissible motions. These equations constrain points, not hand orientation relative to the handle. A welded grasp needs additional orientation relations; a compliant grasp needs constitutive force laws and possibly extra states. Nesbit's golf model, for example, omitted explicit hands and introduced translational wrist flexibility to avoid an indeterminate closed loop. Its methods therefore do not validate a perfectly rigid two-hand grasp [@Nesbit2005Kinematics, pp. 501]. The planar construction below states its own assumptions. ## The Null Space: Mathematical Foundations ### Definition and Basic Properties At a fixed configuration, $$ \ker J_c=\{w:J_cw=0\}. $$ Rank-nullity gives $\dim\ker J_c=n-\operatorname{rank}J_c$ as a linear-algebra statement. Identifying this kernel with the actual tangent space also requires a regular description of the constraint set. For example, $\Phi(x,y)=x^2+y^2=0$ defines only one point. At that point $J_c=0$, so its kernel is all of $\mathbb R^2$, although no nonconstant feasible path exists. A singular Jacobian does not automatically demonstrate two physical degrees of freedom. ### Null Space Basis and Projection Let $N(q)\in\mathbb R^{n\times k}$ have independent columns spanning $\ker J_c$, with $k=n-m$. For stationary constraints, $$ \dot q=N(q)\nu . $$ The components of $\nu$ are **reduced speeds**. Calling them $\dot z$ presupposes a coordinate chart $q=\psi(z)$ with $N=\psi_z$, or a separately declared transformation between chart velocities and speeds. The Euclidean projector onto the column space is $$ P_E=N(N^\top N)^{-1}N^\top. $$ If the columns are orthonormal this reduces to $NN^\top$. Neither expression is automatically the force-to-acceleration map of a system with nonidentity inertia. ### Geometric Interpretation A tangent distribution can be integrable while a particular basis is not a coordinate basis. On the dimensionless coordinate plane $q_3=0$, take $$ n_1=(\cos q_1,\sin q_1,0)^\top,\qquad n_2=(-\sin q_1,\cos q_1,0)^\top. $$ These are an orthonormal tangent frame. With the convention $[n_1,n_2]=Dn_2\,n_1-Dn_1\,n_2$, their bracket is $(-1,0,0)^\top$. Coordinate basis fields commute, so these columns cannot both be the derivatives of one local chart. They still span the tangent plane everywhere. This distinction matters computationally: integrating $\nu$ as if it were a vector of coordinate derivatives generally reconstructs the wrong configuration. Integrate $\dot q=N\nu$ or supply an actual chart and its velocity transformation. ## Constrained Equations of Motion ### Lagrangian Formulation With Constraints Collect applied generalized forces, gravity and velocity-dependent inertial terms into $f(q,\dot q,u,t)$ so that $$ M\ddot q=f+J_c^\top\lambda,\qquad J_c\ddot q=-\gamma . $$ The sign convention defines $\lambda$; changing the sign of a constraint changes its multiplier, not the physical reaction $J_c^\top\lambda$. The instantaneous saddle system is $$ \begin{bmatrix}M&-J_c^\top\\J_c&0\end{bmatrix} \begin{bmatrix}\ddot q\\\lambda\end{bmatrix} =\begin{bmatrix}f\\-\gamma\end{bmatrix}. $$ Define $W=J_cM^{-1}J_c^\top$. Positive-definite $M$ and full row rank $J_c$ make $W$ invertible. Elimination yields $$ \lambda=-W^{-1}(J_cM^{-1}f+\gamma), $$ $$ \ddot q=A_cf-M^{-1}J_c^\top W^{-1}\gamma,\qquad A_c=M^{-1}-M^{-1}J_c^\top W^{-1}J_cM^{-1}. $$ Both terms are essential. Substitution verifies $J_c\ddot q=-\gamma$; the first term alone gives zero constraint acceleration and loses curvature. The original first-order system with the position equation $\Phi(q)=0$ has differentiation index 3 under these regularity assumptions. Replacing it with the acceleration relation gives an index-1 formulation with algebraically solvable multipliers; eliminating them yields an ODE on the consistent state manifold. These are different formulations of the same exact motion only with consistent initialization. Numerical integration can violate both position and velocity constraints, so both residuals need monitoring or correction [@HairerManifoldNotes, sec. IV.4]. ### Null Space Method for Constraint Elimination For stationary constraints, differentiate $\dot q=N\nu$: $$ \ddot q=N\dot\nu+\dot N\nu,\qquad \dot N=DN(q)[\dot q]. $$ Premultiplication by $N^\top$ eliminates reactions: $$ M_r\dot\nu=N^\top f-N^\top M\dot N\nu,\qquad M_r=N^\top MN. $$ The reduced mass matrix is positive definite. This is an equation for **speed derivatives**, together with the reconstruction equation for $q$. Dropping $\dot N\nu$ loses the acceleration caused by the changing tangent frame. Differentiating $J_cN=0$ confirms that this term supplies the required normal acceleration. ### Physical Interpretation: Force Projection Define the velocity and force projectors $$ P_v=I-M^{-1}J_c^\top W^{-1}J_c,\qquad P_\tau=P_v^\top=I-J_c^\top W^{-1}J_cM^{-1}. $$ They satisfy $P_v^2=P_v$, $J_cP_v=0$ and $P_v^\top M=MP_v$. Thus $P_v$ is orthogonal in the mass metric. Its transpose acts on force covectors, preserving the work pairing. In general they are different matrices. Moreover, $$ A_c=P_vM^{-1}=M^{-1}P_\tau =N(N^\top MN)^{-1}N^\top. $$ A numerical counterexample prevents an easy mistake. Choose two translational coordinates with $$ M=\begin{bmatrix}2&1\\1&3\end{bmatrix}\ {\rm kg},\quad J_c=\begin{bmatrix}1&1\end{bmatrix},\quad f=\begin{bmatrix}1\\0\end{bmatrix}\ {\rm N},\quad\gamma=0 . $$ Then $$ P_\tau=\frac13\begin{bmatrix}1&-1\\-2&2\end{bmatrix},\qquad A_c=\frac13\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\ {\rm kg}^{-1}. $$ The acceleration is $(1/3,-1/3)^\top\ {\rm m\,s^{-2}}$ and $\lambda=-2/3\ {\rm N}$. Using $M^{-1}P_Ef$ instead gives $(0.4,-0.3)^\top\ {\rm m\,s^{-2}}$, which violates $J_c\ddot q=0$. An ordinary Euclidean projection followed by an inertia inverse is not generally valid. ## Connection to Control-Affine Systems and Drift ### Control-Affine Structure Suppose $f=f_0(q,\dot q)+B(q,\dot q)u$, with stationary constraints and inputs that enter affinely. On $$ \mathcal X=\{(q,v):\Phi(q)=0,\ J_c(q)v=0\}, $$ the ambient representation of the state vector field is $$ \begin{bmatrix}\dot q\\\dot v\end{bmatrix} = \underbrace{\begin{bmatrix} v\\A_cf_0-M^{-1}J_c^\top W^{-1}\gamma \end{bmatrix}}_{\text{drift}} + \underbrace{\begin{bmatrix}0\\A_cB\end{bmatrix}}_{\text{input map}}u . $$ This vector field is tangent to the **state manifold**, although its acceleration component need not lie in $\ker J_c$. A velocity-dependent $B$ remains control-affine if it is independent of $u$. Muscle activation dynamics or nonlinear actuator laws may require augmenting the state or changing the input definition. ### Drift Dynamics and the Null Space A mass moving on a circle demonstrates the missing term. Let $M=2I\ {\rm kg}$, $R=1\ {\rm m}$ and $$ \Phi(q)=\frac{x^2+y^2-R^2}{2R}. $$ At $q=(1,0)\ {\rm m}$ and $v=(0,3)\ {\rm m\,s^{-1}}$, the dimensionless Jacobian is $(1,0)$ and $\gamma=9\ {\rm m\,s^{-2}}$. Under gravity $f_0=(0,-19.62)\ {\rm N}$, $$ \ddot q=(-9,-9.81)\ {\rm m\,s^{-2}},\qquad \lambda=-18\ {\rm N}. $$ The horizontal acceleration keeps the mass on the circle. It exists even though the ideal reaction power is zero. The tangent representation gives the same result: at this point $N=(0,1)^\top$, $\nu=3\ {\rm m\,s^{-1}}$, $\dot N=(-3,0)^\top\ {\rm s^{-1}}$ and $\dot\nu=-9.81\ {\rm m\,s^{-2}}$. “Drift” means zero chosen input in the chosen model. It need not mean zero neural command, relaxed muscles or a feasible free response after a foot lifts. If the computed reaction violates a unilateral contact condition, the fixed-mode drift no longer describes the actual motion. ### The Golf Swing as a Control Problem A useful golf question specifies an initial state, an impact target, a time horizon, permissible inputs and a cost. Clubhead speed alone is not the entire target: club orientation, path, strike location and contact timing can constrain the solution. The constrained equations connect these choices. Geometry determines admissible motion; inertia and velocity determine acceleration and reaction; actuators bound achievable changes; the objective selects among feasible trajectories. “Exploit drift” becomes a testable hypothesis only after those ingredients are declared and competing trajectories are compared under the same conditions. ### Controllability and the Null Space Low instantaneous input rank does not preclude control of the full state over time. Consider two unit masses with unit springs, stiffness $$ K=\begin{bmatrix}2&-1\\-1&2\end{bmatrix}, $$ and a force $u$ on the first mass. With $x=(q_1,q_2,\dot q_1,\dot q_2)^\top$, $$ \dot x=Ax+bu,\qquad A=\begin{bmatrix}0&I_2\\-K&0\end{bmatrix},\quad b=(0,0,1,0)^\top . $$ In SI units the masses are $1\ {\rm kg}$ and the spring constants $1\ {\rm N\,m^{-1}}$. The acceleration input rank is one, yet $$ [b,\ Ab,\ A^2b,\ A^3b] =\begin{bmatrix} 0&1&0&-2\\ 0&0&0&1\\ 1&0&-2&0\\ 0&0&1&0 \end{bmatrix} $$ has rank four and determinant of magnitude one. Coupling distributes the effect of the input over time. This is the linear state-controllability test, not an instantaneous force-rank test [@TedrakeAcrobot]. A stronger check constructs an actual control. From rest, target $x_f=(0,1,0,0)^\top$ at $T=2\ {\rm s}$. For unconstrained real-valued inputs, set $$ W_T=\int_0^T e^{As}bb^\top e^{A^\top s}\,ds,\qquad u_*(t)=b^\top e^{A^\top(T-t)}W_T^{-1}x_f . $$ The minimum squared-force integral is $x_f^\top W_T^{-1}x_f =59.6394\ {\rm N^2\,s}$. Independent forward integration reaches the specified terminal state. This cost is not mechanical work or metabolic energy. Input limits change reachability. If $|u|\le5\ {\rm N}$ over two seconds, then $\int u^2dt\le50\ {\rm N^2\,s}$, below the unconstrained minimum. Therefore that bounded actuator cannot reach this target in two seconds, despite full state controllability with unrestricted inputs. A rank result alone cannot settle a late-swing timing question. ## Computational Methods ### SVD-Based Null Space Computation For $J_c=U\Sigma V^\top$, retain the right singular vectors associated with zero singular values, including the structural null columns when $m<n$. A compact SVD can omit those columns; use a complete right basis or a routine that explicitly returns the null space. Floating-point “zero” requires a declared tolerance. SciPy's documented default uses a threshold relative to the largest singular value, with relative factor equal to machine precision times the larger matrix dimension [@SciPyNullSpace]. That is an arithmetic default, not an uncertainty model for marker data or anatomical calibration. Normalize coordinate and constraint scales meaningfully, inspect the spectrum and test sensitivity to plausible perturbations. Check $J_cN$, independence of the columns, and any intended normalization. Near a singularity, report both the residual and the rank decision. A pseudoinverse may handle redundant equations, but it does not establish regularity or repair inconsistent constraint accelerations. ### QR Decomposition Approach A complete QR factorization of $J_c^\top$ can produce an orthonormal complement to its column space. When the rank is uncertain, pivoting and a rank criterion are needed. Do not assume that the last columns of an incomplete factorization contain the entire kernel. Different valid bases may flip signs or rotate between evaluations. Physical predictions should be invariant to the basis, but numerically differentiating an unaligned basis can create a spurious $\dot N$. Use a smooth local construction, align neighboring frames where appropriate, or solve the ambient saddle system instead. The latter still requires accurate constraint derivatives. ### Recursive Null Space Formulation Suppose $N_0$ spans $\ker J_0$ and new constraints add rows $J_1$. Compute a full-column-rank basis $Z$ of $\ker(J_1N_0)$ and set $$ N=N_0Z . $$ Then $J_0N=0$ and $J_1N=0$. Conversely, any vector satisfying both constraints is $N_0a$ with $a\in\ker(J_1N_0)$, so this construction spans the complete combined kernel. Its dimension depends on the additional independent rank, not merely the number of appended equations. If the bases vary with configuration, derivatives of both factors enter $\dot N=\dot N_0Z+N_0\dot Z$. Recursion reduces algebraic size; it does not remove the moving-frame terms. ## Application: Golf Swing Multi-Body Model ### Model Description Use a planar teaching model with two fixed shoulder points, two two-link arms, and a freely translating and rotating rigid club. The seven coordinates are $$ q=(\theta_L,\beta_L,\theta_R,\beta_R,c_x,c_y,\varphi)^\top . $$ The elbow angles $\beta_i$ are relative to the corresponding upper-arm angles. Both arms have lengths $a=0.6\ {\rm m}$ and $b=0.5\ {\rm m}$. Shoulder positions are $s_L=(-0.5,0)$ and $s_R=(0.5,0)\ {\rm m}$; club-local grip offsets are $r_L=(-0.1,0)$ and $r_R=(0.1,0)\ {\rm m}$. Only point coincidence is imposed at the grips. The shoulders are fixed supports, the club is rigid, and neither wrist orientation nor a pelvis coordinate is silently added. These assumptions make the model reproducible; they deliberately omit the three-dimensional torso, legs, grip compliance, muscle physiology and shaft deformation needed for a golfer model. ### Constraint Equations Write $e(\theta)=(\cos\theta,\sin\theta)^\top$. The arm endpoints are $$ p_i=s_i+a e(\theta_i)+b e(\theta_i+\beta_i). $$ The four scalar closure equations are the two components of each $$ p_i-c-R(\varphi)r_i=0,\qquad i\in\{L,R\}. $$ One consistent pose is $$ q\approx(0.06860223,\ 1.91063324,\ 1.41808412,\ 1.91063324,\ 0,\ 0.5,\ 0)^\top . $$ Angles are in radians and translations in meters. The stored example uses full precision; rounded coordinates here incur small closure residuals. ### Constraint Jacobian Structure Let $S=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ and $$ D_i=\begin{bmatrix} aSe(\theta_i)+bSe(\theta_i+\beta_i)&bSe(\theta_i+\beta_i) \end{bmatrix}. $$ Then $$ J_c= \begin{bmatrix} D_L&0_{2\times2}&-I_2&-R(\varphi)Sr_L\\ 0_{2\times2}&D_R&-I_2&-R(\varphi)Sr_R \end{bmatrix}. $$ It has rank four at the stated pose, leaving three independent instantaneous speeds. Finite differences of all seven coordinates verify every block, including club rotation. For a selected club point $p_H=c+R(\varphi)r_H$, with $r_H=(0.8,0)\ {\rm m}$, $$ J_H=\begin{bmatrix}0_{2\times4}&I_2&R(\varphi)Sr_H\end{bmatrix}. $$ Admissibility is governed by $\ker J_c$; keeping that point instantaneously stationary requires the smaller space $$ \ker\begin{bmatrix}J_c\\J_H\end{bmatrix}. $$ At this pose the smaller kernel is one-dimensional. Its nonzero direction includes club rotation: zero velocity of a selected point does not imply zero angular velocity of the club. Nor does instantaneous stationarity ensure that the point remains fixed over a finite interval. ### Kinematic Synergies via Null Space Analysis The columns of $N$ are not uniquely determined. For an invertible matrix $Q$, $NQ$ spans the same admissible motions. With an orthonormal basis and orthogonal $Q$, even normalization is unchanged. Naming a column an “arm synergy” or “wrist synergy” therefore depends on a basis convention unless additional structure identifies it. PCA of $NN^\top$ cannot discover preferred coordination in this setting: its eigenvalues are one on the tangent space and zero on the normal space, with no preferred directions within either degenerate eigenspace. PCA of measured, consistently scaled trajectories can describe observed covariance; it still does not establish a neural command mechanism or an optimal recruitment strategy. Data, preprocessing, task conditions and a physiological hypothesis are separate evidence requirements. ## Drift-Control Separation in Golf Swing Dynamics ### The Natural Swing: Exploiting Drift A model can investigate whether initial configuration and velocity make a desired impact state reachable with less actuator effort. Define that effort first: squared torque, activation, mechanical work, peak force and metabolic cost are different objectives and can favor different trajectories. Compare candidate motions with matched boundary conditions, duration, contact assumptions and actuator limits. Recompute reactions and task accelerations throughout. A low-input trajectory is useful evidence for that model and objective; it does not establish that an elite golfer is passive, that large reaction forces are wasteful, or that one wrist posture is universally preferable. The [muscle-to-joint torque chapter](The_Physics_of_Golf/quarto/ch16_muscle_to_joint_torques.html) supplies the next layer: signed moment arms, feasible force sharing and the distinction between net torque and individual muscle force. The [constraint chapter](The_Physics_of_Golf/quarto/ch07_constraint_forces.html) develops compatible motion and contact assumptions. Together these layers connect geometry to a bounded actuator model without treating either as a complete account of motor control. ### Control Authority Analysis For a position task $p_H=h(q)$, $$ \dot p_H=J_H\dot q,\qquad \ddot p_H=J_H\ddot q+\dot J_H\dot q. $$ Substituting constrained dynamics gives $$ \ddot p_H= \underbrace{J_H(A_cf_0-M^{-1}J_c^\top W^{-1}\gamma) +\dot J_H\dot q}_{\text{task drift acceleration}} +J_HA_cBu . $$ At a fixed state, $J_HA_cB$ maps an input change to a task-acceleration change. Its singular values require input and output scaling before they can be compared meaningfully. With bounded inputs, examine the image of the actual admissible input set, not only its linear span. The changing-Jacobian term is physical. A point at radius $0.5\ {\rm m}$ rotating at constant $2\ {\rm rad\,s^{-1}}$ has acceleration magnitude $2\ {\rm m\,s^{-2}}$, although angular acceleration is zero. In three dimensions, $$ \ddot p_H=\ddot c+\alpha\times(Rr_H) +\omega\times\bigl(\omega\times(Rr_H)\bigr). $$ This includes tangential and centripetal acceleration. It also makes clear why a zero clubhead-velocity direction need not be a zero clubhead-acceleration direction. Finally, impact control concerns the endpoint of a trajectory. Available time, state evolution, actuator bandwidth and limits determine whether an instantaneous acceleration direction can produce a useful change before contact. ### Energy Flow Analysis For stationary ideal constraints, $$ \mathcal P_c=\dot q^\top J_c^\top\lambda =\lambda^\top J_c\dot q=0. $$ This is total reaction power on the modeled constrained system. A joint or grip can transfer power between connected bodies while contributing zero net power to their combination. Large internal loads can also matter for tissue stress and muscle recruitment, neither of which follows from the zero-net-power identity. For time-independent kinetic and potential energy, write $$ M\ddot q+C(q,\dot q)\dot q+\nabla V+d=Bu+J_c^\top\lambda, $$ with $C$ consistent with the kinetic-energy identity and $d$ the declared dissipative generalized load. Then $$ \frac{d}{dt}(T+V)=\dot q^\top Bu-\dot q^\top d $$ under stationary ideal constraints. This accounting distinguishes actuator power, storage and dissipation; it does not infer metabolic expenditure. For moving constraints, $J_c\dot q=-\Phi_t$, so $$ \mathcal P_c=-\lambda^\top\Phi_t $$ need not vanish. A $2\ {\rm kg}$ mass prescribed to follow $x=t^2$ in SI units has $v=2\ {\rm m\,s^{-1}}$ and $a=2\ {\rm m\,s^{-2}}$ at $t=1\ {\rm s}$. With an applied force of $3\ {\rm N}$, the guide supplies another $1\ {\rm N}$. Applied and guide powers are $6$ and $2\ {\rm W}$, adding to the $8\ {\rm W}$ kinetic-energy rate. Ideal virtual-work behavior is compatible with actual power supplied by a moving support. ## Advanced Topics ### Configuration-Dependent Null Space Dimension Within a regular fixed constraint description, rank is locally constant and so is nullity. At a rank change, a larger linearized kernel may reflect a singular description, a genuine mechanism singularity, or both. Examine feasible paths and higher-order constraints before assigning extra physical mobility. A contact change is different again. A foot lifting, a grip slipping or impact beginning changes the active equations. New contact can require an impulse and a velocity reset. A smooth fixed-mode acceleration solve does not describe that reset. Model each mode and its transition conditions explicitly. ### Velocity-Level vs. Acceleration-Level Null Spaces For prescribed moving holonomic constraints, choose a particular velocity $b(q,t)$ satisfying $J_cb=-\Phi_t$ and write $$ \dot q=b+N\nu,\qquad \ddot q=\dot b+\dot N\nu+N\dot\nu. $$ The dots here are total derivatives along the trajectory, including explicit time dependence. Projecting the dynamics gives $$ (N^\top MN)\dot\nu =N^\top\!\left[f-M(\dot b+\dot N\nu)\right]. $$ The choice of $b$ is not unique; adding a tangent component changes the speed representation, not the set of physical velocities. For a genuinely nonholonomic velocity constraint $H(q)\dot q=0$, the kernel still describes admissible velocities, but it need not be tangent to any lower-dimensional configuration manifold. Do not use a holonomic coordinate reduction without checking integrability. Also distinguish a null-space basis from a generalized-speed map. If a representation uses $\dot q=E(q)v$, the relevant constraint matrix is $H=J_cE$, and the velocity-coordinate mass matrix and force covectors must be consistent with $v$. Reactions then enter that equation through $H^\top\lambda$. Quaternion coordinate rates, for example, are not interchangeable with a three-component angular velocity. Tedrake's multibody treatment explicitly separates configuration rates and generalized velocities [@TedrakeMultibody]. ### Lie Bracket Relationships For smooth control-affine vector fields, brackets describe directions produced through state-dependent combinations of flows. They help analyze accessibility when instantaneous input directions are insufficient. They must be computed on the consistent state manifold, using the complete drift, including curvature and reconstruction. Accessibility, local controllability, small-time local controllability and reachability under bounded unilateral inputs are distinct properties. With drift, a bracket-rank calculation alone does not generally establish all of them. The two-mass example already demonstrates the more basic point: deficient instantaneous acceleration rank is not a necessary obstruction to state controllability. Conversely, full linear controllability does not eliminate force and deadline constraints. ## Conclusions The constraint Jacobian organizes a sequence of questions. Its kernel describes locally admissible velocity under regular stationary constraints. The mass metric and differentiated constraint determine acceleration and reaction. An actuator model limits the accelerations that can be changed. Integration over time determines reachable task states, and a declared objective defines what counts as improvement. For golf and humanoid modeling, the value of this framework is the ability to test those connections. A claim about a grip, posture or coordination pattern should survive its own geometry, dynamics, force limits and task conditions before it becomes an instruction to a player. ## Limitations The analytic results assume smooth local models with independent constraint rows and positive-definite inertia unless explicitly extended. The examples are constructed. The planar grasp has no fitted human inertias, muscle data, ground mechanics or measured swing trajectory. Its rank and task redundancy establish properties of that construction only. ::: {.callout-warning} ## Critics' Comments **Does this prove an effortless swing?** No. Zero ideal reaction power is not zero muscle force or metabolic cost. The relevant optimization and physiological measurements remain separate. **Does a large null space mean better control?** No. It means more first-order compatible velocity directions in a regular description. Input limits, inertia, task geometry and time determine how those directions can be used. **Can the examples validate a golfer model?** They can catch mathematical and implementation errors. Empirical validation requires measurements, uncertainty estimates and out-of-sample comparisons for the intended physical model. **Can a pseudoinverse solve a singularity?** It can select an algebraic solution under specified conditions. It cannot decide whether a rank loss represents physical mobility, inconsistent constraints or a change in contact mode. ::: ## Appendix ### Proof: Orthogonality of Constraint Forces An admissible virtual displacement at fixed time obeys $J_c\delta q=0$. Thus $$ (J_c^\top\lambda)^\top\delta q =\lambda^\top J_c\delta q=0. $$ This is a force–displacement duality statement. With mixed coordinates, interpreting it as an ordinary geometric angle requires a coordinate metric. Actual velocities for moving constraints need not satisfy the homogeneous virtual-displacement equation, which is why their reaction power can be nonzero. ## Derivation of the Null Space Projector For a full-column-rank $N$, minimize $\|w-Na\|_2^2$ over $a$. The normal equations give $a=(N^\top N)^{-1}N^\top w$, hence $P_E=N(N^\top N)^{-1}N^\top$. For the mass metric, minimize $(w-Na)^\top M(w-Na)$. This gives $$ P_v=N(N^\top MN)^{-1}N^\top M . $$ An equivalent constrained minimization over $J_cz=0$ gives $z=w-M^{-1}J_c^\top W^{-1}J_cw$. Equating the two forms proves the stated mass projector and, after right multiplication by $M^{-1}$, the identity for $A_c$. Consequently a force of the form $J_c^\top\mu$ has zero reduced force $N^\top J_c^\top\mu$ and zero $A_c$ acceleration contribution. This does not say the actual reaction is absent or that total acceleration is tangent: the curvature term remains. ## Summary Table of Key Equations | Question | Relation | Conditions and Meaning | |:--|:--|:--| | Compatible velocity | $J_c\dot q=0$ | Stationary holonomic constraint; regularity identifies the tangent space. | | Moving constraint | $J_c\dot q=-\Phi_t$ | Affine velocity set; its direction space is $\ker J_c$. | | Compatible acceleration | $J_c\ddot q=-\gamma$ | Includes curvature and prescribed time dependence. | | Reduced motion | $\dot q=N\nu$ | Stationary case; $\nu$ denotes speeds unless a chart is supplied. | | Reduced dynamics | $M_r\dot\nu=N^\top f-N^\top M\dot N\nu$ | Full-column-rank tangent frame and positive-definite inertia. | | Force-to-acceleration map | $A_c=N(N^\top MN)^{-1}N^\top$ | Fixed-state force contribution; total acceleration also includes constraint curvature. | | Task acceleration | $\ddot p_H=J_H\ddot q+\dot J_H\dot q$ | Differentiable position task; changing-Jacobian term retained. | | Reaction power | $\mathcal P_c=-\lambda^\top\Phi_t$ | Ideal holonomic constraint; zero for stationary constraints. | ## References The [annotated bibliography](null-space-constraint-jacobian-bibliography.html) maps the cited passages to the claims they support and distinguishes source reading from the independent examples. It does not treat conceptual relationships as verified citation links between publications. ::: {#refs} ::: ## Related Articles ::: {.callout-note title="See Also"} - **[Screw Theory Reference](screw-theory-reference.html)** — Wrenches, twists and reciprocal systems. - **[Tangent Hyperplanes I: Geometry](tangent-hyperplanes-series/part-1-geometry.html)** — Tangent spaces as kernels of constraint gradients. - **[Superposition in Affine Control Systems](superposition.html)** — The fixed-state force-to-acceleration map. - **[Force and Mobility Ellipsoids](force-mobility-matrices.html)** — Interpreting directional mobility with a declared metric. - **[Degrees of Freedom and the Curse of Dimensionality](degrees-of-freedom-and-dimensionality.html)** — Configuration dimension and effective task dimension. - **[Tangent-Space Series](../pages/tangent-hyperplanes.html)** — Reading order and provenance. :::