Reading a moving body

A body can look convincing while its forces are wrong. A model can balance its forces while its surface looks wrong. Kenoma’s educational book asks how to tell the difference, beginning with quantities that we can calculate and inspect before adding anatomy. The intended reader is a programmer, technical artist, or engineering student who wants to connect geometry, mechanics, computation, and evidence.

This edition develops force, torque, energy, anatomical evidence, and a force-driven schematic elbow with activation and a dumbbell load. Seven original resettable 3D laboratories progress through compliant tendon storage, a reduced continuum compression block, a worked tetrahedral FEM/compliant-solver comparison, and a spatial muscle-under-skin/contact strip at identical poses. An eighth, worker-backed experiment jointly solves deforming tissue, distributed tendons and a moving hinge. Its geometry is schematic; the requested anatomical lifting capstone remains under development. A separate evidence viewer contains ten actual atlas meshes, one recorded normalized human trial, and six published model actuator parameter sets. These are three independent evidence streams, not a calibrated person. Three progressive property lessons additionally measure prescribed tetrahedral deformation, compare isochoric volume with cross-section and exterior area, and expose uneven axial strain in a nonuniform bar. A fourth prescribed property lesson separates local from total volume conservation under uneven finite stretch. Three further property lessons add bulk/shear confinement, passive axial load/hold/release, and force-driven unequal 3D blocks with individually preserved volume. The printed edition contains the same explanations, static diagrams, data figures, worked values and original numerical experiments. Clinical or subject-specific prediction is not claimed or required to complete the teaching examples.

The content is an educational extension. The existing production plans in docs/plans/ remain separate. A teaching experiment does not silently add a simulator requirement to the Bevy MVP, change its canonical storage format, or establish a production algorithm choice.

Reading path: measure first, then add a law

Read force, torque and energy for the rigid-body quantities and system boundary. Then use the three property lessons in order: measure an imposed deformation; construct prescribed volume preservation; calculate uneven strain under a reduced axial law. Their worked examples introduce length, area, volume, stretch, strain, force and stress before a contact or continuum solver is needed. Continue with bulk/shear confinement, then load/hold/release to distinguish recoverable storage from viscous loss in the passive axial bar. Then compare unequal blocks under one force, where a declared incompressible material law determines the local stretches and current areas. This separate-block prerequisite does not complete the requested connected continuous specimen with varying cross section, coupled local axial/lateral deformation and volume; that demonstration is not included or established in this edition.

Next distinguish anatomical evidence from teaching assumptions, and follow the actuator into elbow motion. The later tissue, tendon/compression and continuum chapters add material energies, boundary conditions and force-driven deformation. Prescribed constant volume is not a solved material response, and the axial bar is not a full muscle. The fixed-field projection interlude follows the spatial continuum comparison and precedes interfaces and coupling. Its 27 exact finite weighted-vector statements concern representation at one prescribed state. Spatial solver comparisons and the anatomical research apparatus come after those distinctions; their unfinished scope remains explicit.

Three different kinds of evidence

A physical assumption describes the model: a constant force, a perfectly rigid lever, or an ideal spring. A numerical check compares the implemented arithmetic with a reference or studies its error. A formal proof establishes a precisely stated mathematical claim within its declared domain. Each can be useful; none automatically supplies the other two.

The 115 proof cards in this research edition are generated only after checking fourteen source files with Lean 4.19.0: Mechanics.lean, AnatomicalTransfer.lean, CoupledMechanics.lean, AnatomicalArm.lean, ContinuumProperties.lean, MaterialResponse.lean, MaterialResponseReal.lean, MechanicsReal.lean, ActuatorConstitutiveReal.lean, DissipativeBarReal.lean, SerialSpecimenReal.lean, MixedLogVolume.lean, NonuniformIsochoric.lean and ArchitectureForce.lean. Thirty declarations use integer domains and bundled Std, including five narrow material-response contracts; eighty-five quantify reals with pinned mathlib v4.19.0. These comprise four kinematic identities, five constrained specimen algebra identities, three force/torque/power identities, eleven activation/Gaussian/guarded-series results, eleven local standard-linear-solid energy/relaxation results, twelve homogeneous neo-Hookean serial-block identities, derivatives and ordering results, 27 finite weighted-vector projection results, six declared-gradient/interval/algebraic-cell nonuniform-volume contracts and six scalar architecture-force bookkeeping contracts. The actuator results include real derivatives and conditional nonlinear root existence/uniqueness; the compressible confinement specimen’s constitutive derivatives and equilibrium obligations, continuous actuator-mechanics work balances and numerical refinement remain unfinished. Each card reports its exact theorem, assumptions, source hash, implementation link and transitive axioms. The stated real results do not establish floating-point equivalence, general force equilibrium, anatomical validity or biological correctness. This distinction stays visible beside each claim.

The new coupled fixture adds a deforming quadratic-tetrahedron block, distributed tensile branches and an implicitly moving hinge whose tissue torque comes from those branches. It is an intermediate engineering experiment. The actual atlas arm, connected attachment patches and interior-axis diagnostics are available separately for review; shared anatomical tendons/aponeuroses and tissue contact are not yet accepted or complete. The earlier schematic strip retains its declared one-way coupling and is no longer presented as completion of the requested anatomical capstone.

Coordinates and units

The original seven laboratories use a right-handed frame: x points right, y points up, and z points out of the page. Laboratories 1–5 solve planar motion or homogeneous block deformation and render them in a 3D scene. Laboratories 6–7 add genuine spatial degrees of freedom; each declares its constitutive and coupling limits. The coupled fixture instead uses +Z downward, so its gravitational potential is −mgz. The separate atlas assembly uses +X anatomical left, +Y posterior and +Z superior. Each viewer identifies its own frame. The scene can be viewed from the front or obliquely; changing the camera changes neither state nor results. A 3D rendering does not turn planar equations into a spatial biomechanical simulation.

Store physics in SI units and convert only at input and output. The International System of Units names the metre, kilogram, and second; force has unit newton, energy joule, and torque newton metre. Torque and energy share dimensions but represent different quantities. We write torque in N m to preserve that distinction. BIPM, SI Brochure, §2.3.4

How to use a laboratory

Read the model label and the worked example first. Change one control, predict a result, and inspect the numerical output before interpreting the image. Step through time when needed; use Reset to return all physics parameters and the camera to defaults. The laboratories start paused and stop at a bounded step count. Laboratories 1–5 start a new experiment after a non-excitation parameter edit; excitation edits in Laboratories 4–5 retain activation, velocity and time. Laboratory 6 recomputes a static or single-step comparison rather than integrating a trajectory. Laboratory 7 retains current activation, time, pose, velocity and accumulated hinge work/dissipation across parameter edits, then recomputes the spatial shape deterministically. Its angle control explicitly sets pose with zero velocity and clears a domain halt; entering prescribed hold stops motion at the current pose. Reset and the 90° compression fixture deliberately initialize its state. See each laboratory’s trace policy before comparing experiments. The first four property lessons update static imposed geometry, an axial calculation or a homogeneous material equilibrium without integrating time. The fifth follows a finite SLS load/hold/unload/recovery protocol; each parameter edit explicitly starts a new experiment and clears its internal-state and energy history. The sixth property lesson recomputes a static, force-driven two-block assembly without changing the requested force; it exposes each finite root residual. The coupled fixture also preserves its tissue coordinates, angle, velocity, activation and time when effort or dumbbell mass changes. Its mass control records the external energy added or removed with the mass; it does not restart the trajectory.

Every scene has a text description and static illustration. If WebGL is unavailable, the controls and numerical results still work. All seven mechanics laboratories use original procedural schematic teaching geometry: their cylinders, spheres, blocks, grids and arrows are separate from anatomical measurements. The evidence viewer instead shows actual licensed static atlas surfaces, with their source labels, units and limitations. Neither the atlas nor the human-study summaries calibrates the schematic actuator.

A useful question to carry forward

Imagine a forearm holding a dumbbell. Ask: what system did we isolate, what forces act on it, where do those forces act, and what energy enters or leaves it? These questions remain useful whether the next model is a rigid-body joint, a finite-element tissue mesh, or an animation rig. A more detailed surface cannot rescue an unspecified system boundary.

The property lessons use the same SI convention with prescribed positive stretches or an explicitly reduced small-strain axial law. The first four update a prescribed geometry or static equilibrium; the fifth integrates an internal viscous strain under its stated axial law, and the sixth solves two separate incompressible homogeneous blocks and independently measures their 3D geometry. Independent geometric measurements and midpoint-integration error remain visible; their exact kinematic claims do not prove a constitutive law or qualify the anatomical tissue envelope.

The architecture-force chapter and portable lab connect reference/current area, nominal/Cauchy stress, projection, parallel aggregation and series exchange. Every integrated artifact requires its own complete-book proof receipts, browser/PDF checks and release qualification. The historical nonuniform-volume total was 103 + 6 = 109; this source registry adds six for 115. The preserved standalone qualification does not establish complete-book proof closure, integrated browser/PDF acceptance or anatomical calibration. All example counts, areas, stresses and control ranges are authored.

Read the model before moving it

Every chapter now has an example in the shared Kenoma embedded GUI. The label above each view distinguishes illustrative geometry, prescribed kinematics, analytic bookkeeping, a numerical demonstration, and recorded evidence. Orbiting a model does not increase its physical dimension or validate its parameters. An algebraic statement can be kernel-checked while its browser implementation, mesh approximation and biological interpretation remain separate obligations.

The simple human poser is an independent artistic system: a graph, generated surface and pose rig with gizmo controls and local SQLite scene files. It has no tissue forces or contact response. Educational model adapters share only presentation components with it. The research apparatus retains its original laws, acceptance criteria and failed-state evidence.

Checked cards show their claim, assumptions and limits immediately. Expand a card for its exact statement, source and compilation receipt. The complete Lean appendices are also collapsible on screen; the PDF retains all statements and full sources. A GUI badge never substitutes for the source-bound kernel receipt.

Force changes motion

A force is not a displacement instruction. For constant mass in an inertial frame, the resultant external force determines acceleration. Doubling force doubles acceleration; doubling mass halves it. Newton’s second law and the separation of action-reaction pairs are presented in Dourmashkin’s original course notes, §§7.3-7.4. Newton’s laws

Isolate the body before adding arrows

Consider a small object translated horizontally by an actuator. The schematic laboratory includes one constant horizontal net force. If we imagined gravity and a support, their vertical balance would be an additional assumption; this example simply solves the horizontal component. Do not mistake its missing contact solver for evidence that a support is physically correct.

For net force $F_x$, mass $m>0$, initial position $x_0=0$, and initial velocity $v_0=0$:

Text equation: acceleration equals force divided by mass. Velocity equals acceleration times elapsed time. Position equals half acceleration times elapsed time squared, with both initial conditions zero.

This is an analytic solution, so the force laboratory does not accumulate time-integration error. JavaScript still evaluates floating-point arithmetic. We compare selected outputs with exact worked values using numerical tolerances; no theorem in this edition proves the JavaScript implementation.

Laboratory 1 · Constant force

Analytic planar motion; constant net force; no contact or anatomy.

Start interactive 3D

Static reference diagram. Step, Play or Pulse starts live 3D; if unavailable, the display is explicitly numerical-only.

Mass (kg)

Net force x (N)

Elapsed time (s)

Play Single step Reset defaults Front / oblique view Read current results Copy state

A position marker moves along x. Position scale is fixed over the 0–2 s trajectory for the selected mass and force; grid spacing is reported below. Yellow force and violet velocity arrows use separate scales. Values retain SI units. Axes: x right, y up, z out of the plane.

Worked example

Set mass to 2 kg and force to 4 N. The acceleration is 2 m/s². At 1 s the position is 1 m and velocity 2 m/s. At 2 s the position is 4 m and velocity 4 m/s. A negative force produces negative position and velocity; the object accelerates along the negative x direction.

Try: keep force at 4 N, reset, and increase mass from 2 to 4 kg. At the same simulated time the position is half as large. Then set force to zero: position and velocity stay zero because the initial velocity is zero. A zero force would preserve a nonzero initial velocity in a different experiment.

Equal and opposite forces belong to different bodies

If a tendon pulls a bone, the bone pulls back on the tendon. Those two forces cancel in a combined bone-plus-tendon system. They do not cancel on the isolated bone: only one member of that pair acts on it. This is why a diagram with every arrow drawn on a single object can be misleading.

For one scalar component, a pair $f$ and $-f$ sums to zero. That identity is enough to check bookkeeping once we have assumed an action-reaction pair. It does not prove that an actual biological interface follows our chosen constitutive law.

The same assumed cancellation also holds for real force components, without introducing an integer SI scale:

Work offers a second calculation

A constant horizontal force does work $W=F_x\Delta x$. Starting from rest, the kinetic energy is $K=\tfrac12 m v_x^2$. Substituting the analytic solution gives $W=K$ in this ideal isolated horizontal model. For 4 N over 4 m, both are 16 J.

A negative force over a negative displacement also gives positive work. Force direction alone does not determine work’s sign; the force-displacement relation does. The next chapter keeps track of the application point of that force, while the energy chapter studies what happens when we replace analytic solutions with time steps.

Approximation boundary

This laboratory omits drag, friction, impacts, variable mass, deformation, actuator limits, and all anatomy. Its moving sphere is a marker for position, not a tissue particle whose radius or volume has medical meaning. Positions beyond the viewing window are reported numerically; the physical trajectory is not clamped to fit the image.

A force has an application point

A force that passes through a pivot produces no moment about that pivot. The same force applied farther from the pivot can produce a larger moment. The laboratory introduces the moment of a force about a chosen origin; the definitions and axis convention are verified against Dourmashkin’s rotational-dynamics notes, §17.1. Rotational dynamics

A planar cross product

Let $\mathbf r=(r_x,r_y,0)$ point from the pivot to the application point, and $\mathbf F=(F_x,F_y,0)$ be the force on the lever. The component along z is:

Text equation: torque around z equals x position times y force minus y position times x force. Positive torque is counterclockwise when viewed from positive z toward the origin.

In the schematic example a massless rigid lever of length $L$ supports a point load of mass $m$ at its tip. The angle $\theta$ is measured counterclockwise from the horizontal, not from a clinical elbow coordinate convention. Uniform gravity is $\mathbf F=(0,-mg,0)$, with illustrative $g=9.81$ m/s². Therefore:

This is a posed lever. The slider prescribes its angle; no active muscle, torque controller, angular acceleration, or joint-contact force maintains the pose. A physical static hold would require an opposing moment supplied by some actuator and appropriate support forces.

Laboratory 2 · Force and lever arm

Prescribed pose; massless rigid lever; point load; uniform illustrative gravity.

Start interactive 3D

Static reference diagram. Start interactive 3D to view current results; numerical controls also work without 3D.

Point-load mass (kg)

Lever length (m)

Angle from horizontal (°)

Reset defaults Front / oblique view Read current results Copy state

A rigid bar extends from a fixed pivot to a load. The yellow arrow points downward. The white projection marks the horizontal distance to the gravity line. This is a posed lever, not an active arm. Axes: x right, y up, z out of the plane.

Worked values and the moment arm

At $m=5$ kg, $L=0.30$ m, and $\theta=0$°, gravity is 49.05 N downward and torque is -14.715 N m. At 90° the line of gravity passes through the pivot and the moment is zero within floating-point tolerance. At 120° the torque becomes positive because the load lies on the other side of the pivot’s vertical line.

The perpendicular moment-arm magnitude is $d=|L\cos\theta|$. It is not generally the lever length. Its magnitude tells us how strongly gravity acts about this pivot, while the cross product retains direction.

Try: compare 0°, 60°, and 90°. Predict the torque ratio before moving the slider. At 60° the torque magnitude is half the horizontal value. Move from 89° to 91° and watch the sign change.

The exact worked arithmetic can be represented by 300 mm and -49050 mN. Multiplying those scales gives $10^{-6}$ N m per integer product unit. The following theorem checks the integer cross product; division by one million to obtain SI torque is an explicitly stated interpretation.

Combining forces

Two forces at the same point contribute torques that add. We can either add their force vectors first or add the two moments. The identity is useful when assembling muscle and external-load contributions around a joint.

For the real quantities in the stated cross product, the corresponding identity is checked separately:

It is unsafe to add torques computed around different origins without transforming them. If the new origin is at $\mathbf o$ relative to the old one, the position becomes $\mathbf r-\mathbf o$, and:

Internal forces and couples

Equal and opposite forces can have zero resultant force and a nonzero resultant torque. Think of two forces acting along different parallel lines: they form a couple. Force cancellation alone is not enough to infer moment cancellation.

For a central pair between positions A and B, let the force at A be $k(\mathbf B-\mathbf A)$ and the force at B its negative. Their summed moment vanishes because both forces act along the joining line. The theorem checks this specifically for planar integer coordinates and integer k.

This assumption should be inspected before applying it to a tissue model. A model that exchanges angular momentum through distributed stress or explicit bending moments needs the corresponding moment balance, not a point-force argument with missing terms.

Connecting torque to energy

The point load has gravitational potential $U=mgL\sin\theta$ relative to its horizontal position. Differentiating with respect to angle in radians gives $\tau_z=-dU/d\theta$. A finite-difference numerical test checks this relation for the implemented lever. It does not establish a floating-point theorem. This independent calculation is valuable because it can reveal a sign convention or degree/radian conversion error.

Energy reveals numerical error

Energy accounting can expose a trajectory that looks smooth while gaining energy from its numerical update. The definitions of kinetic energy and work are verified against Dourmashkin’s notes, §§13.2-13.6. Energy and work The numerical experiment below is original code and analysis in this repository; it is a synthetic oscillator experiment, not new biological data.

An ideal oscillator with a reference solution

A mass attached to a linear spring has extension x and velocity v. There is no damping, gravity, contact, or active energy input. Its equation and total mechanical energy are:

Text equation: mass times acceleration equals negative stiffness times extension. Energy equals half mass times velocity squared plus half stiffness times extension squared.

The default parameters are $m=1$ kg, $k=40$ N/m, $x_0=0.20$ m, and $v_0=0$ m/s. Initial energy is 0.80 J. The angular frequency is $\omega=\sqrt{k/m}$, and the reference trajectory is $x(t)=x_0\cos(\omega t)$, $v(t)=-\omega x_0\sin(\omega t)$.

Laboratory 3 · Numerical energy

Ideal undamped linear spring; fixed physics steps; 12 simulated seconds per trajectory; 120–2,400 fixed steps for the supported h.

Start interactive 3D

Static reference diagram. Step, Play or Pulse starts live 3D; if unavailable, the display is explicitly numerical-only.

IntegratorExplicit EulerSymplectic EulerVelocity Verlet

Step size h (s)0.005 s0.01 s0.02 s0.05 s0.10 s

Play Single step Run to 12 s Download trace Reset defaults Front / oblique view Read current results Copy state

A spring connects a wall to a mass marker. Extension is visually exaggerated and fitted to the view. The energy chart shows a solid numerical trace and a dashed initial-energy reference. Read numeric values when the chart rescales. Axes: x right, y up, z out of the plane.

Three ways to take a step

Let h be a positive time step in seconds, and $a_n=-kx_n/m$.

Explicit Euler updates both values from the old state:

Symplectic Euler updates velocity before position:

Velocity Verlet includes the new acceleration:

The methods are implemented separately from rendering. Playback advances fixed simulation steps and reports their index and elapsed time; it never inserts the browser’s frame duration into the physics formula. Play advances approximately one simulated second per wall-clock second. Run to 12 s finishes the current trajectory in bounded frame batches; either running mode can be paused and resumed. Each supported h reaches the same 12-second endpoint (120–2,400 steps), and every physics sample is retained even when the display redraws only once per frame. The readout reports completed/total steps, elapsed/target time, the maximum energy deviation over all samples and the current position error against the analytic reference. Download trace preserves all samples and these measurements. A parameter change starts a new trajectory. Reset reproduces the same sequence for a given method and step size.

Why explicit Euler gains energy

Substitute the explicit update into the quadratic energy and expand. The mixed terms cancel, leaving:

For positive k, m, and h, every nonzero-energy step multiplies energy by a factor greater than one. Making h smaller delays the growth; it does not produce bounded energy over arbitrarily long time with fixed h. This is a paper-and-pencil derivation for the ideal oscillator. No Lean theorem in this edition checks that integrator identity.

For the other two methods, the linear oscillator’s usual bounded regime is $h\sqrt{k/m}<2$. Even inside that regime, neither method preserves the exact physical energy at every step. The observed energy oscillates, and phase error accumulates. A stable picture and an accurate time history are different checks.

Original reproducible experiment

Run each method with the same initial conditions over 12 simulated seconds. Compare h = 0.01, 0.02, and 0.05 s, measuring the maximum absolute relative energy deviation over all sampled steps and the final absolute position error against the analytic reference. No wall-clock speed claim is made. The table below is generated by executing web/mechanics.mjs, the same pure physics module used by the browser.

The final position error can be small by phase coincidence; it should not be used alone to select a method. The energy maximum shows explicit Euler’s growth even when a snapshot happens to cross the reference. Symplectic Euler has a larger oscillating energy deviation than Verlet for these settings. Both still require refinement studies for any intended application.

Try: reset with explicit Euler and h = 0.05 s. Step, play or run to 12 s to see the energy increase. Reset with Verlet at the same h. Compare numeric energy with the dashed 0.80 J reference, then halve h and compare again. The plotted vertical range adapts to the computed energy; read its numeric maximum so that rescaling cannot disguise instability.

A modest formal claim

With a nonnegative exact integer mass, the numerator $m(v_x^2+v_y^2)$ is nonnegative. With positive unit scales and the factor one half, it corresponds to nonnegative kinetic energy. That claim says nothing about conserving energy from one step to the next.

Approximation boundary

The spring is linear at all extensions, including compression, and has no tensile-only constraint. It is therefore not a tendon model. An active muscle adds energy through a contraction law; a dissipative tissue removes energy; contact may exchange or dissipate energy. Before interpreting their energy residuals, include those terms and define the model’s system boundary.

Before adding those material mechanisms, continue to the three property lessons. They measure an imposed shape, construct constant-volume kinematics, and calculate nonuniform strain with one declared axial law. This separates geometric observations from the force and energy models that will later produce them.

Measure deformation before choosing a muscle law

These progressive property lessons separate geometry, constitutive response and equilibrium. A shape that keeps its total volume can still have uneven local compression; a shape that looks muscular has not thereby passed a mechanical test. The anatomical capstone retains its failed cases and its force and geometry gates.

Start here after force, torque and energy. First ask what changed in the shape? Then impose a simple volume relation. Finally introduce one axial force law to ask why different sections strain differently. No contact solver, anatomical mesh or finite-element assembly is needed for these three lessons. The later material chapter adds energy, stress and equilibrium in three dimensions.

The symbol $F$ below is a deformation matrix, not the force vector of Laboratory 1. The bar’s $E$ is a modulus, not energy in joules; $E_G$ denotes Green strain. Stretches and strains have no units: stretch 1.2 means 20% longer, while axial strain 0.01 means 1% extension in the small-strain bar. These static property lessons do not advance time, integrate activation or calculate metabolic energy.

1. Measure a prescribed deformation

Question: if we move the vertices ourselves, how do we measure stretch, shear and volume change? The reference tetrahedron has four vertices; its three edges from vertex zero are stored as the columns of $D_m$. The current edges form $D_s$. Multiplying by $D_m^{-1}$ removes the original edge scale, so $F$ maps a small reference edge to its current edge. You can begin with the numerical stretch and volume readouts before working through this matrix notation.

For reference vertices $X_i$ and current vertices $x_i$, each in metres, construct edge matrices $D_m=[X_1-X_0, X_2-X_0, X_3-X_0]$ and $D_s=[x_1-x_0,x_2-x_0,x_3-x_0]$. A nondegenerate, positively oriented reference is required. The affine deformation gradient, local volume ratio and Green strain are

The controls impose $F=R_z(\theta)U$ with $U=\begin{pmatrix}s_x&k&0\\0&s_y&0\\0&0&s_z\end{pmatrix}$, then add a common translation: $x=FX+t$. Column vectors are used; $U$ acts first, the rigid rotation second. The off-diagonal coefficient $k$ is entered directly; this is not a shear matrix multiplied after a separately defined stretch. The diagonal coefficients are not all material-line stretches when shear is present. The reference X, Y and Z line stretches are $s_x$, $\sqrt{s_y^2+k^2}$ and $s_z$, respectively; the browser independently measures them from the transformed edges. Rotation and translation leave those line lengths unchanged.

Translation cancels from the edges. A rigid rotation has $E_G=0$; simple shear can have $J=1$ while $E_G\ne0$. Volume preservation does not mean absence of strain.

The independent measurement uses the oriented boundary triangles. For any common origin $o$, with outward face ordering,

The browser computes this triangle sum without calling the determinant helper. It reports $V_{\partial}/V_{\partial,0}-J$, in dimensionless units, rather than assuming agreement. Negative signed volume or nonpositive $J$ rejects the candidate and retains the last valid displayed state. This four-node affine test does not certify curved P2 elements or a whole moving tissue envelope.

Property lab 1 · Measure deformation

Declared reduced fixture; SI units; scope and assumptions in the adjacent derivation.

Authored 80 × 60 × 50 mm reference. Gray is reference; blue is prescribed geometry. This lesson imposes kinematics and does not solve force equilibrium.

X diagonal coefficient sx

Y diagonal coefficient sy

Z diagonal coefficient sz

Off-diagonal coefficient k (Uxy)

Rotation (degrees)

Translation X (mm)

Translation Y (mm)

Translation Z (mm)

Reset Try inverted candidate Read current results Copy state

Try and predict. Set all three diagonal coefficients to 1, shear and rotation to 0. The measured $J$ and boundary-volume ratio should both be 1, and Green strain should be zero. Change only translation X to 30 mm: it moves the tetrahedron but changes neither measurement. Next set axial stretch to 1.2: expect $J=1.2$. Restore unit diagonal coefficients and set shear to 0.25: volume stays the same, but Green strain is nonzero. Rotation alone also keeps volume and strain unchanged; the components of $F$ can still change.

The controls prescribe the vertices. This lab measures $F$, $J$, Green strain and an independent boundary volume; it solves no forces, material energy, contact or equilibrium. A valid positive volume is a geometry check, not evidence of a realistic muscle response.

2. Exact isochoric kinematics

Question: what lateral change would keep a rectangular block’s volume exactly fixed while its axial length changes? Isochoric means constant volume. Here we impose the answer; no pressure or material stiffness chooses it.

Prescribe $F=\operatorname{diag}(\lambda,b,b)$ with positive dimensionless stretches. Then $J=\lambda b^2$. In the isochoric construction $b=1/\sqrt\lambda$, so $J=1$ in exact real arithmetic. The interactive controls also allow independent stretches, which need not preserve volume. This is a kinematic construction, not a minimization or a muscle contraction law.

For the rectangular reference, $L=\lambda L_0$, $A=b^2A_0$ and $AL=JV_0$. Cross-section area is the area normal to the axial direction, measured from two transverse edges. Exterior area is the sum of all boundary-triangle areas. They have the same units, square metres, but different mechanical meanings. Neither is automatically physiological cross-sectional area (PCSA). The independent boundary-volume measurement remains visible when the isochoric toggle is selected.

Property lab 2 · Volume and cross-section

Declared reduced fixture; SI units; scope and assumptions in the adjacent derivation.

Authored 80 × 60 × 50 mm reference. Gray is reference; blue is prescribed geometry. This lesson imposes kinematics and does not solve force equilibrium.

Axial stretch

Independent lateral stretch

Stretch relationIsochoric: b = 1 / sqrt(axial)Independent stretches

Reset Read current results Copy state

Try and predict. At axial stretch $\lambda=0.8$, the 0.08 m reference length becomes 0.064 m. The imposed lateral stretch is $b=1/\sqrt{0.8}\approx1.11803$, so the reference cross-section 0.003 m² grows to 0.00375 m². Their product remains 0.00024 m³. Switch to independent stretches and set $b=1$: the same shortening now gives $J=0.8$. Read cross-section area and exterior area separately; they need not change by the same factor.

This volume-preserving motion is prescribed kinematics, not material-driven volume redistribution. The free-side compression example in Laboratory 5 later determines lateral stretch from a material law and boundary conditions, rather than imposing this square-root relation. Neither lesson specifies a complete deforming muscle.

3. Nonuniform cross-section: a reduced axial oracle

Question: can one constant tensile force produce different strain along a bar? Here shape is no longer prescribed at every section. We assume a one-dimensional material law and axial balance, then calculate strain. Positive strain is extension; negative strain is compression. The coordinate $s$ runs along the reference bar; $N/A$ is axial stress in pascals. Larger area or modulus reduces passive strain at the same force.

Take a small-strain bar with length $L$ in metres, positive area $A(s)$ in square metres, constant modulus $E$ in pascals, no distributed axial load, and a tensile-positive resultant $N$ in newtons. Force balance gives $dN/ds=0$. The passive law gives $\epsilon(s)=N/[EA(s)]$, so

A1 is the reference area at s=0 and A2=rA1 is the end area at s=L. The ratio permits either direction of taper: for r<1, A2 is the narrow end; for r>1, A1 is the narrow end. The readout derives narrow/wide areas from the selected geometry.

For linear area taper $A(s)=A_1[1+(r-1)s/L]$, the exact compliance is $L\ln(r)/[EA_1(r-1)]$, with limit $L/(EA_1)$ at $r=1$. Numerical midpoint integration is compared with that analytic integral. Linear area taper is a separate geometric assumption from a radius taper. The exterior surface influences surface traction and contact; it does not replace $A(s)$ in this axial law.

Add a separately declared uniform active-stress offset $a\sigma_0$, still under this reduced small-strain law: $\epsilon=N/(EA)-a\sigma_0/E$. With both ends fixed, $\int\epsilon ds=0$ determines the constant $N$. Two equal-length segments with $A_2=2A_1$ and $a\sigma_0/E=0.01$ give $N=(4/3)A_1a\sigma_0$, narrow strain $+1/300$ and wide strain $-1/300$. Local extension and compression coexist even though total length is fixed. This offset model has no force–length/velocity dependence, transverse equilibrium, pennation, ECM, fluid or anatomical calibration; it is not a complete active muscle model.

Property lab 3 · Uneven axial strain

Declared reduced fixture; SI units; scope and assumptions in the adjacent derivation.

Small-strain constant-modulus bar; no distributed axial load. Strain by axial position, with blue extension and red compression. Area controls an axial resultant, not a complete muscle law.

Reference length (m)

Reference area A1 at s=0 (m²)

End area ratio A2/A1

Modulus E (Pa)

Passive tensile force (N)

Active stress offset (Pa)

Activation fraction

Boundary / load conditionPrescribed passive forceUniform active stress; fixed ends

Area profileLinear area taperTwo equal-length segments

Midpoint cells8163264128

Reset Read current results Copy state

Try and predict. In passive mode, keep the default geometry and modulus and change force from 0.3 to 0.6 N: every strain and the total extension double. Doubling the modulus instead halves them. In active fixed-end mode, choose two equal-length segments and keep area ratio 2, activation 1 and active stress 1,000 Pa at modulus 100,000 Pa. The narrow section extends by $1/300$ and the wide section compresses by $1/300$, while their total length change is zero. Passive force is disabled in that mode because the fixed-end compatibility determines the reaction; active controls are disabled in passive mode.

This bar solves an axial resultant and axial strain under its stated assumptions. It does not determine lateral stretch, cross-section evolution, transverse stress, contact, three-dimensional fibre aggregation or muscle shape. The area profile is an input; neither zero net extension nor the active-stress offset supplies the missing transverse equilibrium. Refining midpoint cells improves the integral of this same reduced law, not its biological scope.

The illustrative 5% small-strain warning uses the true endpoint extrema of this monotone positive-area law, independent of midpoint display resolution. It is a warning about this approximation, not material validation.

The executable oracle is the tapered-bar module. Tests check the logarithmic integral, refinement and the two-segment fixed-end case. Its scope is numerical evidence for this declared reduced law; the kinematic Lean claims above do not prove its mechanics.

Continue with anatomical evidence before assigning these quantities to a named muscle. When you reach tissue and rendered skin, reuse the measurements above while adding a material law and boundary forces. The next material-response lesson adds bulk/shear and unilateral confinement controls. Load/hold/release now adds a bounded passive axial SLS lesson. The two-block specimen then solves unequal 3D stretches and current areas under one force, with an explicit incompressible material constraint. Measured fibre-to-muscle aggregation and force–length/velocity lessons remain pending, as recorded in the coverage chapter.

What the exact claims cover

The freshly checked real-number proof bundle establishes edge translation invariance, determinant composition, the actual diagonal determinant and the positive square-root isochoric construction. The four claims above are compiled from their displayed real definitions against pinned mathlib. The edge and matrix identities do not prove the oriented boundary-triangle oracle, and the isochoric identity does not prove a free-side material equilibrium. Separate numerical and browser tests cover the implemented controls and boundary measurements. Neither class of test validates measured biological parameters or resolves the capstone’s local compression and full nodal stationarity failures.

Uneven stretch and local volume

The previous property lesson measured homogeneous deformations. Now prescribe a smooth deformation along one reference prism: neighbouring material sections can stretch differently while each point preserves its volume. This is an imposed geometric map, with no force balance, passive material response, activation or anatomical prediction. It is a different construction from the later serial specimen, whose two separate homogeneous blocks are joined by a spacer.

A complete gradient, including shear

Let the material coordinate satisfy $0\le S\le L_0$, with $L_0>0$, mean stretch $m>0$ and dimensionless uneven-stretch amplitude $-1<a<1$. The reference section is a regular 16-sided polygon of circumradius 15 mm; the reference length is 120 mm. Define

The transverse coordinates are $y=b(S)Y$ and $z=b(S)Z$, with $b(S)=1/\sqrt{\lambda(S)}$ under local compensation and $b=1$ otherwise. Differentiating the map gives

The actual stretch gradient is $\lambda'=2ma/L_0$, with units of inverse length. Under compensation, $b'=-\lambda'/(2\lambda^{3/2})$. The off-axis terms describe shear; dropping them would misstate the gradient even though its determinant would agree. The displayed engineering strain $\lambda-1$ refers to the centerline, not every surface fibre.

Positivity makes the axial map strictly increasing, and positive transverse scaling makes the map injective. The ordinary change-of-variables argument then gives continuum volume conservation when $J=1$. These differentiation, injectivity and integration arguments are separately audited mathematical reasoning. The Lean declarations below prove properties of the declared matrix, positive stretch on the stated interval and an algebraic straight-cell error formula; they do not formally prove those analytic arguments or the mesh implementation.

Compare local and total conservation

Open the interactive local-volume lab. Change mean stretch, uneven-stretch amplitude, mesh cells and local compensation; reset returns to the declared reference. The full six-contract inventory, complete Lean source and independent audit downloads are bundled with the lab.

At $m=1$, $a=0.4$, local compensation gives endpoint stretches 0.6 and 1.4, while $J=1$ throughout. Disable compensation: the total continuum volume ratio still equals one, but the local Jacobian ranges from 0.6 to 1.4. Expansion and contraction cancel globally. Total volume alone cannot establish local incompressibility.

The closed three-dimensional boundary mesh and its displayed longitudinal section use the same vertices and fixed millimetre scale. Signed boundary-triangle volumes are independently compared with regular-polygon/frustum volumes. Straight cells approximate a curved exact map: their measured volume is slightly excessive under nonuniform compensation and approaches the continuum result as cells are refined. The controls report this nonzero discrepancy explicitly.

For positive endpoint ratio $r=\sqrt{\lambda_1/\lambda_0}$, the independently derived straight-cell ratio is

This identity explains the discrepancy. Its Lean proof concerns the algebraic expression; it does not certify triangle orientation, topology, floating-point arithmetic or convergence of a numerical implementation.

Six scoped Real contracts

Accepted source and remaining dependencies

Independent acceptance applies to exact source 15c624c32378b9c773a3d0c25a65150804b68c90, following mathematical, model, browser and PDF review of 8fd21a3177242fde3dca2a3dffa1c5f4c0350bdd. Their sole difference corrects the browser receipt’s description of the six Real contracts. The complete original contribution and its claim inventory are preserved unchanged. The book registry adds display metadata and copies the exact Lean source; all original 103 proof identities and anatomy references remain unchanged. Source composition and independent acceptance accompany the portable edition.

This lesson supplies no material law or equilibrium solve. New continuous passive-specimen, scalar tension/mass controller and anatomical-capstone sources remain excluded. The next anatomical dependency is displacement-space and material/pressure coupling adequacy at isolated matched-pose fixtures, against retained local compression and full-nodal residual failures, before force rematching or new trajectories. Prescribed local volume conservation does not resolve that dependency or establish anatomical completion.

Connect fibre architecture to force without inventing strength

The unresolved anatomical requirement is a source-bound map from fibre architecture and measured force to the active continuum coefficient. Kenoma currently fits model-reference actuator forces while its calibration states retain unresolved stationarity and compression. That fit must not be renamed physiological specific tension.

After the prescribed local-volume map and before material response, this lesson supplies the bookkeeping prerequisite: distinguish fibre number from fibre area, parallel groups from sections in series, nominal from Cauchy stress, and projected from unprojected PCSA. It does not repair the continuum solve or supply a new anatomical dataset. All lab numbers are authored examples, not calibration inputs.

Retain the failed calibration evidence

At PR11 commit 0bed56a, the fixed-end fits use approximately 3.60, 8.71, and 20.38 MPa for brachialis, short biceps, and long biceps. The source documents corner volume ratios near 0.56–0.62 and free nodal force components of 64.07, 52.06, and 54.22 N at those frozen calibration states. A successful reduced force match is therefore not an equilibrated full-nodal material calibration.

This book integration adds six scoped Real claim registrations and a portable analytic lab. No fit, material coefficient, calibration state, accepted trajectory, reference asset, or failure threshold is changed. Resolving local compression and full displacement/pressure stationarity remains an upstream gate. The analytic examples below cannot overrule it.

Decide which area a measurement describes

For a declared parallel reference bundle with volume V₀ in m³, representative fibre length Lf₀ in m, and contractile volume fraction φ₀, the idealization Acontractile₀ = φ₀ V₀ / Lf₀ gives area in m². It requires a meaningful representative length and fraction. A tapered whole belly, distributed fibre terminations, varying orientations, or nonuniform packing need a spatial integration and load-path model; inserting their total volume into this formula does not establish those conditions.

PCSA definitions must travel with the values. Murray, Buchanan and Delp (2000), Methods and Table 2 computes PCSA from volume divided by optimal fascicle length, then uses the pennation cosine separately in moment capacity. Its elbow architecture study used ten upper extremities from nine cadavers, with exclusions and different sample counts by muscle. In contrast, Ward and Lieber (2005), equation 1 includes the cosine in PCSA. A projection included in an imported area must not be multiplied again.

Use three explicit metadata fields before combining a force and area:

• area convention: fibre-normal or already tendon-projected
• stress basis: contractile area or an effective whole-muscle area
• configuration: reference/optimal or current, with the stated stretch and volume ratio

An effective whole-muscle specific tension may already include noncontractile fractions through its area definition. Multiplying it by φ₀ again double-counts that reduction. Conversely, a single-fibre measurement does not automatically define a whole-muscle effective stress. A smallest fibre diameter is not a measured circular cross-sectional area, and a fibre count fraction is not an area fraction.

Sum parallel fibre groups by area

For group k with Nk fibres, reference fibre area ak₀, nominal axial stress Pk, and current tendon-direction cosine ck, use

A₀ = Σk Nk ak₀
T = Σk Pk Nk ak₀
Ttendon = Σk Pk Nk ak₀ ck

This is a declared independent parallel-force decomposition. It omits passive matrix force, pressure traction, lateral exchange between groups, and curved aponeurosis geometry. It cannot infer recruitment or force–velocity properties from fibre typing alone.

Two synthetic groups with equal counts of 50,000 and areas of 1,000 and 2,000 µm² have area shares 1/3 and 2/3. At nominal stresses of 100 and 300 kPa, total area is 150 mm² and force is 35 N. The count-average stress gives 30 N and is the wrong weighting for this example. Different angles likewise require force-weighted projections, not an unqualified average angle.

Mattiello-Sverzut and Martins (2023) assessed isokinetic elbow performance in 32 volunteers; usable biceps morphology samples came from 26 participants (9, 8, and 9 across the three age groups). Fibre subtypes and smaller diameters describe that morphology subset, not all 32 participants. The study supports keeping type and size distinct; it does not provide a subject-matched spatial packing map for the atlas. Bottinelli et al. (1996) reports type- and temperature-dependent single skinned-fibre mechanics. Its 12 °C load-clamp protocol is not a body-temperature whole-biceps calibration. Neither paper’s parameters are imported here.

Track the same material cut during shortening

Let F be a locally homogeneous orientation-preserving deformation gradient, J = det F > 0, f₀ a reference unit fibre, λ = |F f₀| > 0, and f = F f₀ / λ. For a reference cut normal to f₀ with area A₀, the current oriented area vector is a = J F⁻ᵀ f₀ A₀. Its projection normal to the current fibre is

A⊥ = a · f = J A₀ / λ.

A⊥ is a projected material-cut area. In a shear-free straight bundle it is the familiar orthogonal cross-sectional area. With shear, the actual deformed cut may be oblique and have a different surface area. The formula is not a global statement about an anatomical imaging plane or a curved heterogeneous belly. Spatially varying F requires local integration.

The existing active law in anatomical-material.mjs differentiates a full-stretch potential. With p = aactivation σ₀ fL(λ), its active first Piola tensor is p f ⊗ f₀, and its active Cauchy tensor is (p λ/J) f ⊗ f. Thus

σfibre = p λ / J
Tactive = σfibre A⊥ = p A₀.

Both p and σfibre have units Pa, but they divide force by different areas. The lab calls the scalar p “nominal stress P” and keeps it fixed initially. With A₀ = 80 mm², p = 300 kPa, J = 1 and λ = 0.8, A⊥ grows to 100 mm² and σfibre falls to 240 kPa. Axial force stays 24 N. At 30° pennation the projected component is about 20.78 N. There is no fibre-count increase and no extra bulge multiplier.

Try and predict. With force–length held at 1, lower the stretch while holding J and the reference bundle fixed. Predict the new projected area and Cauchy stress before checking the unchanged axial force. Reset, change only pennation, then compare the axial force with its tendon-directed component. Finally enable the authored force–length curve: explain which assumption changed. Changing packing compares different reference bundles, not acute fibre creation.

Holding Cauchy stress fixed instead would produce a different force as area changes. That is a different constitutive prescription, not a universal strength bonus. Enabling the existing authored fL curve in the lab explicitly changes p and therefore force. In the anatomical model, activation, length, passive stresses, pressure and attachments jointly matter; this calculation is only its active material-cut identity.

Raiteri, Cresswell and Lichtwark (2016) examined human tibialis anterior with three-dimensional ultrasound and found spatial shape and architectural changes during isometric contraction. It motivates regional rather than uniform geometry. It is neither biceps anatomy nor proof of exact local J = 1. The volume controls here are prescribed and have no biological acceptance range.

Follow a variable cross section through a series load path

Under a one-dimensional static section resultant convention, a cell between cuts i and i+1 obeys

Ti+1 − Ti + Qi = 0,

where Qi is net axial force from lateral or external loading on that cell. With no lateral load or inertia, axial force is the same at every cut. A 100–200–100 mm² reference-area sequence carrying 30 N therefore has nominal section stresses of 300–150–300 kPa. A thicker middle section does not manufacture an additional end force.

If one assigns 300 kPa independently to each cut, the resulting 30–60–30 N sequence needs −30 N and +30 N lateral forces in the two intervening cells. Such exchange could belong to a resolved ECM/matrix path, but it must be represented and balanced. It cannot be inferred from the taper alone. The scalar sequence is a necessary force-balance example, not a compatible displacement field, three-dimensional equilibrium, shear-stress solution, or material fit.

Internal paired forces cancel in the whole-body resultant. Their power does not generally vanish: q v₁ + (−q) v₂ = q(v₁−v₂). An ECM coupling can store or dissipate energy when adjacent points move differently. A no-slip tie, compliant bond, frictional slip and damage law are distinct constitutive choices; none is supplied by the arithmetic identity.

Huijing and Baan (2003) found differing proximal/distal tendon forces in maximally active rat extensor muscles with connective tissues retained and controlled relative length changes. This is evidence for additional force paths, not a human-biceps coupling coefficient. Zhang and Gao (2012) used a two-dimensional fibre/endomysium model to study shear transfer and tapered fibre ends. Its model supports the need to declare interface mechanics; it does not calibrate Kenoma’s matrix.

Density and fluid mechanics remain separate inputs

Mass density maps volume to mass, not area directly to force. Ward and Lieber’s fixation/hydration experiment reported preparation-dependent human muscle density and warned of PCSA errors from mismatched assumptions. Their fixed-tissue results are not live biceps density measurements. A later mass-based estimate must retain species, tissue preparation, hydration and the density uncertainty rather than silently treating 1,060 kg/m³ as universal.

Sleboda and Roberts (2020) applied a pressure cuff during isolated bullfrog semimembranosus contractions. The effect on measured force changed with muscle length, and physical ECM analogues helped interpret the result. The experiment does not identify a human biceps pressure law or a unique mechanism. It does show why fluid–matrix interactions cannot be represented by “more blood equals more strength.”

Intracellular/interstitial fluid, blood inside vessels, solid-matrix stress, incompressibility pressure and measured intramuscular pressure are different quantities. The original cat fast/slow muscle pressure and perfusion study varied perfusion conditions and examined isometric endurance. Its experimental scope requires vascular boundary conditions and timescale-dependent physiology; a bulk penalty contains neither. No numerical perfusion, permeability, ECM bonding, blood-volume or edema parameter is imported in this lesson. Exercise pump and chronic growth are not acute material-area transformations.

For a future coupled fluid model, record the conserved constituents, fluid pressure definitions, permeability and boundary conditions, vascular compartments, compressibility assumptions, and the drain/undrain time scale. Validation must distinguish local from global volume, and conservation from accurate geometry. A shared external skin/fat envelope must enclose the assembled tissues and obey its own interface/contact conditions; seven separately expanding bellies do not define it.

Verification and next acceptance gate

The preserved contribution has twelve Node contracts for scalar units and transformations, explicit PCSA/packing conventions, heterogeneous sums, taper equilibrium and lateral force balance. Its repository-contract tests compare the active-stress map with the unchanged material operator, including shear and rigid rotation. Seven SymPy identities separately check declared algebra, including cofactor projection and exchange power. Symbolic simplification and unit tests are not Lean kernel proofs.

The six original claim IDs and theorem names are registered in this book candidate. ArchitectureForce.lean is copied byte-for-byte from the independently qualified contribution. The cards state explicit assumptions and retain the original exclusions; the fresh book checker must compile every registered family before issuing book proof receipts. The historical nonuniform-volume integration was 103 + 6 = 109 declarations. Adding these six architecture-force contracts gives a 115-declaration source inventory across fourteen files: 30 integer and 85 Real declarations. Source registration alone does not establish a freshly checked 115-declaration book, a rendered PDF or hosted release.

The bounded standalone contribution checks in run 37743673184, at exact commit 143ac8863cd8a5be736e2c455cac9605c2e20a1f, compiled all six contracts and passed real Chromium controls at 1280×1000 desktop and 393×852 mobile sizes, including the no-JavaScript fallback. The recorded agent visual review passed all five JPEG quality-85 captures. That evidence applies to the standalone source; it does not qualify this integrated book. Its original automated pending human inspection marker remains historical evidence, unchanged.

The portable lab adds default values for print and no-JavaScript reading while preserving the model and control code. Original contribution sources, primary-source ledger and book assembly binding keep source, test and evidence scopes distinct. Statements in the preserved standalone documents about pending registration describe that earlier stage. Integrated desktop/mobile controls, PDF readability, complete proof closure, owner release review and any publication remain separate gates. The full book build also launches numerical experiments; it is not a render-only check.

The next anatomical implementation gate is still an equilibrated, locally admissible calibration specimen with an adequate displacement/pressure space. Only after that gate should a source-bound architecture dataset be mapped into σ₀ or a replacement law. Every proposed parameter should carry the source, specimen/protocol, area/stress convention, units, uncertainty, transformation and destination equation. Independent attachment, tendon, cartilage, fat and shared-skin validation remain necessary for the full anatomical arm goal.

Let material stiffness and boundaries choose the lateral shape

Question: when a block is shortened between plates, must its volume stay exactly fixed? The preceding isochoric lesson imposed that answer. Here a material law and lateral boundary conditions choose the shape. Compare both conditions at one prescribed height, then change one stiffness at a time.

This is an authored, homogeneous elastic specimen, not measured muscle. The reference width, height and depth are $W_0=0.08$ m, $H_0=0.06$ m and $D_0=0.05$ m; $V_0=0.00024$ m³. Ideal frictionless plates prescribe height stretch $0.5\le h\le1$. The single unknown lateral stretch $b>0$ acts equally in X and Z:

This diagonal homogeneous ansatz rules out bulging, shear localization, buckling and spatially varying fields. Uniform stress makes the lateral traction condition meaningful within this ansatz; this is not a mesh solve or a proof of the unrestricted specimen’s preferred deformation.

An elastic energy, finite stiffness and units

Reuse the existing homogeneous tissue energy, leaving the earlier labs’ equations and calibration unchanged:

Here $\mu$ and $K$ are shear and bulk parameters of this declared energy, each in Pa = J/m³. They characterize the reference response; finite-deformation tangent stiffness need not equal these constants. Controls permit $500\le\mu\le5000$ Pa and $0\le K\le250000$ Pa. These are bounded teaching choices, not fitted biological ranges. The $K=0$ endpoint deliberately removes resistance to pure volume change.

A finite bulk term penalizes $J\ne1$ with finite energy; it does not impose $J=1$. The shear term penalizes distortional shape change and vanishes under positive isotropic scaling. Neither term supplies plasticity, damage, viscosity, fluid transport, fibre architecture or activation.

The constitutive framework has an original-author reference: A. F. Bower, Applied Mechanics of Solids, §3.5 describes hyperelastic stress as an energy derivative and distinguishes nominal stress from force per deformed area. The formula and boundary reduction here are the book’s illustrative model. No experimental response curve is reproduced or calibrated from that source.

Derive the reduced equilibrium

The nominal stress components computed from this energy are

Both lateral directions vary with $b$, hence $\partial U/\partial b=2V_0P_x$; $\partial U/\partial h=V_0P_y$. Finite-difference tests check these derivatives and the reduced plate force. Free sides require zero lateral traction, $P_x=0$. The numerical root bracket is $h\le b\le1/\sqrt h$: the left endpoint has zero shear contribution and nonpositive bulk contribution; the right has $J=1$ and nonnegative shear contribution. Analytic endpoints handle $K=0$ and $h=1$. This bracket supports the implemented root search; no Lean proof of existence, uniqueness, convergence or global optimality is claimed.

Unilateral confinement places smooth rigid walls at the reference X/Z faces. Walls prevent outward expansion but allow inward pull-away: $b\le1$. They apply inward compressive reaction $R\ge0$ and cannot pull the block outward. The reduced wall contract is

At $b=1$, if $P_x<0$, use $b=1$ and $R=-P_x$. Otherwise solve the free-traction root on the restricted bracket $[h,1]$ and use $R=0$. The separate bracket keeps even a finite approximation inside the walls. At weak bulk stiffness, the specimen can shrink away from the walls; forcing $b=1$ in that case would require an unmodeled tensile attachment.

Plate force and pressure follow from the same nominal stress, not from displayed arrows:

The sign convention makes compression positive. Each X wall carries $N_x=RH_0D_0$; each Z wall carries $N_z=RW_0H_0$. Their current-area pressure is $R/(bh)$. Per-wall clearances are $g_x=W_0(1-b)/2$ and $g_z=D_0(1-b)/2$. The reported complementarity work is $2(N_xg_x+N_zg_z)$ in joules, accounting for both walls in each direction.

Property lab 4 · Compression, bulk stiffness and confinement

Homogeneous quasistatic elastic equilibrium; shared prescribed height; free sides versus unilateral rigid walls. Authored parameters, SI units; no muscle calibration.

Front view of the solved blocks at the same height and fixed physical scale. Dashed: reference. Gold: frictionless plates. Red walls: compressive contact; gray walls allow pull-away. Both unshown depths also scale by b. Numerical values are unscaled SI. Finite residual success does not establish physical validity.

Prescribed height stretch h

Shear parameter μ (Pa)

Bulk parameter K (Pa)

Bisection cap (per free root)8: finite approximation3264

Reset Test isochoric wall candidate Read current results Copy state

Predict, change one input, and inspect the residual

At default $h=0.8$, $\mu=1500$ Pa and $K=50000$ Pa, the plate spacing is 0.048 m in both panels. These values come from a fresh reduced solve:

Try and predict. Raise only $K$ to 250000 Pa. The free block approaches $J=1$ without imposing it: $J\approx0.99878057$, while its plate force is about 4.56728 N. Active rigid walls retain $b=1$ and $J=h=0.8$, so the plate force rises to about 202.08871 N. Confinement changes the work required to reach the same height; increasing finite bulk stiffness cannot restore volume while all three stretches are fixed.

Restore $K=50000$ Pa and raise only $\mu$ to 5000 Pa. Read the force, lateral expansion, and separate bulk/shear energy contributions. Height remains shared. Every comparison changes only the named input; no time or activation state is hidden in this static lesson.

Counterexample. Set $K=0$. The solved block has $b=h$, $J=h^3$ and zero stored energy/plate force up to roundoff: at $h=0.8$, $J=0.512$. Isotropic collapse costs no distortional energy in this deliberately degenerate law. It is not a plausible soft-muscle response or plastic failure. At $K=500$ Pa, both solutions shrink laterally to $b\approx0.90043115$, with zero wall reaction and positive clearance. Confinement does not manufacture tensile attachment.

Feasibility check. The “Test isochoric wall candidate” button tries $b=1/\sqrt h$ in the confined geometry. For $h<1$ it has positive $J=1$ but penetrates the lateral walls and is rejected without changing the solved state. Positive volume is necessary but does not establish boundary feasibility.

Numerical check. Select eight bisections. Geometry remains positive and wall-feasible, but the signed free residual generally fails the displayed criterion $|P_x+R|\le10^{-5}$ Pa. Restore 32 or 64 and compare the bracket width and residual. The criterion belongs to this new lesson; it changes none of the existing labs’ acceptance limits. “Residual criterion met” refers only to this reduced lateral equation. Invalid, blank, nonfinite or out-of-domain inputs retain the prior displayed state.

Finally restore $h=1$: $b=1$, $J=1$, energy and force return to zero. This model has no history or irreversible state, so it cannot demonstrate plastic deformation or dissipative load/hold/release. Reset deliberately restores all authored defaults.

Real properties: checked algebra and unfinished formal obligations

The specimen uses real stretches, stresses and energies. Five real-domain algebra claims below are compiled against the pinned mathlib dependencies before this book is built. Constitutive derivatives and equilibrium existence, uniqueness and optimality remain unfinished. The separate integer contracts do not substitute for these real claims or their remaining obligations.

The earlier real determinant and square-root identities support the kinematic volume formula. They do not prove this constitutive law, wall solution, equilibrium existence/uniqueness or global optimality. A release build must compile any future real-domain claim with its supported pinned dependencies before presenting it as checked.

Discrete arithmetic contracts and separate numerical evidence

The five declarations below are freshly compiled with pinned Lean 4.19.0 and bundled Std. They concern exact scaled integers, with positive external SI scaling factors. These are separate implementation arithmetic checks; real stretches, stresses and energies are not their quantified domain. The unfinished derivative and equilibrium obligations above remain separate.

Numerical tests independently derive a stationarity equation in $J$, check energy derivatives and SI conversions, sweep the bounded controls, compare feasible energy samples, and exercise finite caps, wall onset, zero bulk and invalid candidates. Browser tests execute the generated controls, verify exported states and SVG dimensions at a common physical scale, and check rollback, reset, mobile/print and no-JavaScript alternatives. These tests establish behavior for this declared implementation, not global optimality, mesh accuracy, material calibration or biology.

Continue with anatomical evidence before attaching these parameters to a named muscle. Laboratory 5 later reuses the free-side law in a coupled hinge example. The unfinished anatomical capstone, measured fibre-to-muscle aggregation, force–length/velocity lessons retain their separate limits in the coverage chapter.

Load, hold, release: storage and viscous loss

The axial bar lesson made area and elastic strain visible. Keep its passive force balance, area profiles and equilibrium modulus, then add one internal strain that takes time to change. The learning objective is to distinguish creep under held force from stress relaxation under held extension, and account for recoverable storage, signed external work and viscous loss throughout unloading.

The bulk/shear compression lesson remains a separate finite-deformation model. This lesson follows the earlier small-strain axial bar: fixed material reference area $A(x)>0$, homogeneous axial coefficients, no body force, inertia, transverse equilibrium or active stress. Displacement and strain have tensile-positive signs. It uses a standard linear solid: an equilibrium spring in parallel with one Maxwell spring–dashpot branch. Roylance’s MIT teaching notes, §4.2 present this arrangement and distinguish creep from relaxation. COMSOL’s constitutive theory describes the same one-branch topology and viscosity/time-scale relation. These sources motivate the model; the coefficients below are authored demonstrations, not measurements of human tissue.

One law for the image and the ledger

Let $\epsilon$ be total axial strain and $v$ the viscous strain in the dashpot. The Maxwell spring strain is $\epsilon-v$. With fixed $E_0>0$, $E_1\ge0$ and $\eta>0$,

Here $E_0,E_1$ have units Pa and $\eta$ has units Pa s. This adds memory to the passive axial law. Setting $E_1=0$ recovers $\sigma=E_0\epsilon$ and the earlier bar’s same midpoint strain, displacement and stored energy for identical geometry and load.

The stored energy density and viscous loss rate density are

Differentiation and the Maxwell equation give the local balance

For the whole bar, $U=\int_0^L A\psi\,dx$, $\dot D=\int_0^L A d\,dx$ and external power is $N\dot\delta$. Work $W=\int N\dot\delta\,dt$ is signed; unloading can return energy to the apparatus. $D$ is cumulative lost mechanical energy and cannot decrease. Starting from rest gives $W=U+D$, with the observed finite-precision residual displayed separately.

Constant force balance and an internal state

With constant resultant $N$, $\sigma(x)=N/A(x)$. Uniform coefficients and an initially unstrained bar imply $\epsilon=c/A$ and $v=z/A$, where $c,z$ have units m². Define the geometric compliance factor $C_g=\int_0^L dx/A(x)$, with units 1/m. Then

Under imposed force,

Holding force therefore permits increasing extension. Holding extension fixes $c$; then $\dot z=E_1(c-z)/\eta$ and force relaxes. For elapsed hold time $s$,

The relaxation time is $\eta/E_1$ and the force-controlled creep time is $\eta(E_0+E_1)/(E_0E_1)$ when $E_1>0$. With the default equal moduli, these are 1 s and 2 s. Increasing viscosity delays response; it does not imply a monotone change in energy lost during every finite protocol.

The runtime integrates the scalar state exactly for each constant-extension interval and each linear force ramp. It separately integrates boundary work and squared viscous rate; loss is not filled in from the energy residual. Stable exponential differences avoid cancellation near $E_1=0$. Each step clips at protocol boundaries before changing the imposed condition. The bar uses explicit midpoint area quadrature. For a linear taper, the independent factor is $C_g=L\log(r)/(A_1(r-1))$ with limit $L/A_1$ at $r=1$; for two equal-length segments, $C_g=L(1+1/r)/(2A_1)$. The displayed refinement error compares midpoint extension to this exact profile integral.

Run the same finite protocol

Default reference: $L=0.2$ m, $A_1=0.0003$ m², area ratio $r=2$, $E_0=E_1=100{,}000$ Pa, $\eta=100{,}000$ Pa s and 32 midpoint cells. Load ramps from 0 to 0.6 N over 1 s, holds for 2 s, unloads from the actual reaction to zero over 1 s, then recovers at zero force for 3 s. Switching to extension hold captures the extension reached at the end of loading. The unloading ramp begins at the relaxed reaction. Both switches preserve extension and internal state continuously.

During force hold, compare the extension at 1 and 3 s. During extension hold, compare those forces instead. During the latter hold, boundary work stays constant while spring storage decreases and $D$ rises. At 4 s the applied force is zero, yet extension and stored energy remain positive. Recovery consumes that storage through viscous loss. At finite time 7 s a residual extension remains; the figure does not declare full recovery.

Try viscosity 20,000 and 500,000 Pa s, then change the ramp duration while retaining the same force target. Compare the same protocol boundary times and read $U$, $W$, $D$ and the residual separately. Set the Maxwell branch off: there is no viscous loss, force hold no longer creeps, and restoring zero force restores zero strain and storage. Reverse the taper or use two equal-length areas; the strain field and extension change through $A(x)$ while the same constitutive law remains in force.

The scene uses a fixed physical reference scale, explicitly magnifies axial displacement tenfold, and colors strain against a fixed ±5% scale. Thickness is schematic; there is no predicted lateral contraction. Changing a model or protocol parameter deliberately starts a new experiment and clears time, memory and energy history. Invalid edits retain the prior valid state. Reset restores the authored defaults. Step and Play advance physical time; Run completes the bounded protocol in small animation-frame batches. Export includes the initial row and every accepted observation.

Controls restrict positive geometry, equilibrium modulus at least 100,000 Pa, force at most 0.6 N and area ratio at least 0.5. The true narrow-end strain remains at most 4% in these tensile protocols, below the unchanged 5% small-strain warning used in the earlier axial lesson. The runtime rejects inconsistent constitutive state, imposed-force schedules, held extension or energy ledger rather than transferring an arbitrary state to new coefficients.

What the real proofs establish

The eleven declarations below quantify real local constitutive variables. They check storage and loss signs, the storage derivative given path derivatives, power balance assuming the Maxwell equation, the zero-branch elastic limit, and the exact held-strain exponential’s derivative, bounds and weak monotonicity. They do not prove the force-controlled bar reduction, midpoint quadrature, JavaScript arithmetic or exported discrete energy ledger. Independent numerical and actual-control checks cover those implementation obligations separately. No anatomical calibration, force–velocity muscle law or completed capstone follows from this example.

Continue to anatomical evidence before deciding which properties a named tissue would need. The anatomical capstone and its calibration obligations remain separate.

Continue with unequal blocks under one force to connect finite force-driven stretches, current areas and local incompressibility in a separate two-block 3D assembly. The SLS law and energy history above remain unchanged.

One force, unequal stretches, and local volume

Question: can blocks of the same material carry the same force, stretch by different amounts, and each preserve volume? The reduced axial bar already demonstrates uneven strain from an input area profile, including local extension and compression under fixed ends. It predicts no lateral deformation. This lesson adds a different, explicitly three-dimensional specimen and a constitutive law whose force, current area and local volume agree.

Two blocks and an explicit interface

Use two separate, homogeneous rectangular blocks in axial series. Each has a square reference section and reference length $L_0=0.025$ m. Their reference areas are $A_1=0.0003$ m² and $A_2=rA_1$, with default $r=2$. Ideal bilateral axial traction fixtures transmit signed force and permit tangential sliding at the loading faces. They can pull or push; ordinary frictionless compression platens alone could not provide the tensile case. A massless spacer has fixed length 0.012 m. Its positions move with the block ends.

The two blocks may therefore change lateral size independently. The gold fixture connection is excluded from material volume. This is an assembly of separate blocks, not a continuous stepped or tapered solid with an unmodeled material interface. The solved deformation is homogeneous within each block. Buckling, shear localization and the preferred deformation among unrestricted 3D fields are outside this homogeneous model.

With no body load, the intervening massless fixture balances the two opposing axial resultants, so $N_1=N_2=N$. Positive $N$ is tension and negative $N$ is compression. The equilibrium equation uses reference area for nominal stress, and current area for Cauchy stress; these are different stress measures.

Material response and the volume constraint

Each block is an incompressible isotropic neo-Hookean solid with authored shear parameter $\mu=1500$ Pa. Its energy per reference volume is

Incompressibility is a local material constraint. It is not a finite bulk penalty, and constant force does not establish it. The earlier bulk/shear confinement lesson retains its different finite-bulk law and boundary conditions. The material framework and the role of incompressibility pressure are described by A. F. Bower, Applied Mechanics of Solids, §§3.5.4–3.5.6. The following homogeneous reduction and assembly are the model used here.

For axial stretch $\lambda_i>0$ and equal transverse stretches $b_i>0$,

The full Cauchy stress is $\sigma_i=-p_iI+\mu B_i$, where $B_i=F_iF_i^T$. Free lateral faces require $\sigma_{yy}=\sigma_{zz}=0$, giving $p_i=\mu b_i^2=\mu/\lambda_i$. This $p_i$ is an incompressibility multiplier, not measured axial plate pressure. At rest it is $\mu$, while the complete stress tensor is zero. The resulting axial Cauchy and nominal stresses are

The executable root solves this last equation separately for both blocks. It does not assign stretches from a visual taper. The reduced energy and its derivative use the same material law:

Thus $U_i=A_iL_0W(\lambda_i)$, and differentiating with respect to current block length $L_i=L_0\lambda_i$ gives $dU_i/dL_i=A_iP_i=N$. The response is elastic and history-free; no time integration, activation or viscous loss is added.

Solve, measure, and compare

For positive stretch and $\mu>0$,

The force curve is strictly increasing, with zero force at $\lambda=1$. The narrower section has larger nominal tensile stress and larger stretch for $N>0$. With $N<0$ it has more negative stress and smaller stretch: it compresses more. Both blocks have the same material; this difference comes from geometry.

The current geometry is calculated from the solved stretches:

The 3D meshes use these actual lengths, widths and depths without displacement magnification. Gray wire outlines mark the reference blocks; the camera uses a shared physical scale across parameter edits. Front/oblique views change the camera, not the model. Exported vertices drive a separate closed-triangle volume measurement. Measured cross-sectional edges, length and boundary volume are reported beside $J_i$; the volume oracle does not simply return one.

Default forces, areas, stretches and independent measured volumes are generated from the same executable law:

Try and predict. Reverse $N$ from +0.1 to -0.1 N. Both blocks shorten and widen, and the narrower block has the larger compression. Set $N=0$: both stretches, areas and lengths return to their reference values, and stress and energy vanish. Set $r=1$: equal material and area give equal stretches. Reverse the ratio to $r=0.5$: block 2 becomes the narrow section and now strains more. Increase only $\mu$: at the same force the stretches move closer to one. Reference length changes total extension, volume and energy, but does not change local stress or stretch at fixed area and force.

Local-volume counterexample. The test button constructs an unused candidate with unchanged axial stretches but local volume ratios 1.10 and $1-0.10/r$. Their reference-volume-weighted total is still one. Nevertheless, each block violates $J_i=1$, and its free-side traction no longer vanishes. The candidate is rejected and the prior solved mesh and state remain visible. A correct total volume cannot hide a violation of this local material law.

Controls bound $500\le\mu\le5000$ Pa, $0.0003\le A_1\le0.001$ m², $0.5\le r\le4$, $-0.1\le N\le0.1$ N, and $0.01\le L_0\le0.04$ m. Every target force lies within the fixed positive bracket $0.6\le\lambda\le1.75$: at the smallest $A_i\mu=0.075$ N its endpoint forces are approximately -0.163333 and +0.106760 N. Bisection uses a visible 8–64 iteration cap. The force criterion is $|A_iP_i-N|\le10^{-10}$ N for each block. Eight iterations can leave a valid positive-volume shape with unequal approximate resultants; the UI reports each computed resultant and residual, rather than calling it shared-force equilibrium. Restore 48 or 64 and compare. Invalid edits retain the prior valid specimen; Reset restores the authored defaults and reference camera.

Exact real claims and independent implementation checks

Twelve declarations below quantify the actual real homogeneous law. They prove the energy factor, sign and zero state; its derivative; stress identities and signs; strict monotonicity and conditional positive-root uniqueness; area-times- length volume; and same-force ordering for unequal areas in tension and compression. Root existence, numerical bisection, pressure elimination, the matrix/mesh volume measurement and JavaScript arithmetic are not certified by these declarations. The real kinematic identities in the first property lessons remain separate.

Independent numerical checks solve the cubic equation $\lambda^3-[N/(A_i\mu)]\lambda^2-1=0$, sum tetrahedral volumes from the actual mesh, differentiate the stored energy, and verify the complete stress and current-area conversions. Actual browser controls must also verify the displayed 3D geometry, solver status, local-volume rejection and fallbacks. These obligations cannot be replaced by a scope label. The lesson makes no claim about measured muscle, continuous fibre aggregation or the unfinished anatomical capstone.

Anatomy is evidence, not a silhouette

The previous laboratories isolate mechanics. An arm introduces three descriptions that must agree: rigid skeletal geometry, force-producing musculotendon paths, and deformable tissue. The humerus, radius and ulna are distinct bones. Elbow bending and forearm rotation are distinct motions; the biceps’ distal attachment to the radius makes this separation important. Our next laboratory fixes forearm rotation and shoulder posture and replaces the elbow complex with one hinge. Its rods are teaching geometry, not segmented bones.

Name the structures before assigning properties

The anterior volume includes biceps and the deeper brachialis; brachioradialis contributes a separate lateral path. The triceps heads provide extensor paths, and some muscles span the shoulder as well as the elbow. A single red actuator cannot identify their individual contributions. The chapter’s synthetic flexor is deliberately called a synthetic flexor, rather than assigning its invented origin and insertion to a named anatomical muscle.

An attachment region, a segmented surface, a tendon guide point, and a fiber direction field are different data. The surface of a muscle does not tell us how its fascicles run or where a numerical routing point should be placed. An anatomical fascial expansion such as the bicipital aponeurosis also needs an attachment/interface model; drawing another cable is insufficient. The audited anatomy contribution supplies anatomical literature and distinctions, but no imported mesh in this edition establishes those features for our schematic.

Published human numbers with their context

Murray, Buchanan and Delp examined ten human upper-extremity specimens. Measurements included architecture and tendon excursion; optimal fascicle lengths and physiological cross-sectional areas (PCSA) were then estimated. Their Table 2 gives the following study summaries. These are published biological estimates, not this laboratory’s defaults, and no participant-level raw data are redistributed. Original study, Table 2 and §3

Convert 17.7 cm to 0.177 m and 1.2 cm² to 0.00012 m² before calculations. For a uniform architecture idealization, PCSA is volume divided by optimal fiber length. If maximum tendon-directed force is modeled as specific fiber tension times PCSA times cos(pennation), check whether the source already includes that cosine in its area convention. The segmented volume and PCSA do not themselves specify a maximum force: a specific-tension assumption is still needed.

Iivarinen and colleagues measured forearm indentation in nine healthy people, used ultrasound for tissue thickness, and fitted a layered hyperelastic model. Their resting effective moduli were 210 kPa for skin and 1.9 kPa for adipose tissue, equivalent to 210,000 Pa and 1,900 Pa. These are inverse-model values under that study’s loading and layer assumptions. They are not universal tissue constants or an elbow-contact calibration. Neither value is inserted into the schematic hinge. Original abstract and DOI

The distinction matters: a measured displacement, a parameter fitted from that displacement, and a new model’s prediction are three separate objects. Testing the fitted model on its fitting observations is calibration. Testing on a reserved pose or load is a stronger check of generalization.

A dataset must say what it observes

The audited research identifies several useful routes to actual data. The table records research-stage access, not completed integration into this executable.

The following data chapter integrates a specific licensed Multisensor trial after a reviewed passive conversion, preserving normalized values and actual relative timestamps. Ordinary untrusted pickle deserialization is unsuitable; no pickle is bundled or executed here. The acquired atlas, recording and model parameters remain visibly separate from the published study summaries above and from the schematic actuator.

The current MoBL-ARMS package has a noncommercial restriction according to the audited provider page, conflicting with an older catalogue’s MIT label. It is excluded. Apache-2.0 on Kenoma code does not relicense any dataset, model, mesh, scan, or paper figure. Data access and licence sources

Expertise follows the contribution

Murray, Buchanan and Delp’s original architecture experiments are relevant to architecture and moment capacity. Millard, Uchida, Seth and Delp’s actuator comparisons are relevant to musculotendon modeling. Wakeling, Ryan and colleagues’ continuum work is relevant to force/deformation coupling. Hallock’s OpenArm research is relevant to imaging-derived muscle deformation. These are source-based areas of contribution, not endorsements of Kenoma or claims that any author reviewed it.

No audited source supplies a complete, co-registered loaded curl with all bones, fibers, activation histories, skin self-contact and pressure. The book therefore keeps the evidence gap visible while building mechanisms that can later be compared with compatible measurements.

Inspect real anatomy and recorded evidence

An atlas, a recorded human trial and an actuator model answer different questions. This chapter includes actual licensed data bytes, with their source-specific rights, transformations and hashes. None of the three evidence streams is registered to either of the others or to the teaching actuator. Looking more anatomical cannot turn a schematic force law into a validated physiological model.

Ten static anatomical surfaces

Ten BodyParts3D 4.0 right-arm surfaces preserve the original atlas pose: humerus, radius, ulna, brachialis, brachioradialis, both biceps heads and three triceps heads. The official 99%-polygon-reduced release supplies 13,852 vertices and 20,218 triangles. It is a constructed static adult-male reference atlas, rather than scanned geometry for the OpenArm participant. Rendering colors are explanatory. Source axes are x toward anatomical left, y posterior and z superior; original OBJ positions are in millimetres. The JSON divides positions by 1,000 and converts face indices to zero-based, with no topology editing, joint fitting or rigging.

BodyParts3D, © The Database Center for Life Science licensed under CC Attribution 4.0 International. Changes: ten-part selection, SI conversion and original rendering; this book’s print derivative enlarges labels and reflows the legend without changing geometry. The original OBJ comments retain the historical CC BY-SA 2.1 Japan notices; the package also includes the current official archive’s February 2025 CC BY 4.0 grant. Source and rights record

The browser viewer centers and rotates the atlas for display only. It can show bones or individual named parts, and switch anterior, oblique and posterior views. It does not animate a guessed elbow axis. Polygon reduction and edge diagnostics do not establish a mesh suitable for medical contact calculations. No skin, fat, fascia, cartilage or tendon material field is provided.

One recorded trial at a held posture

OpenArm Multisensor 2.0 participant code 2, trial 1b, was recorded with the right elbow held at 90° and the forearm supinated. The retained segment contains 4,403 preprocessed normalized samples over 85.903537 s. Ultrasound measures brachioradialis thickness; sEMG measures biceps brachii electrical activity; wrist-contact force is a net external signal. They are different quantities and, for ultrasound and sEMG, different muscles. The fixed acquisition posture is not a measured angle trajectory. Hallock et al. (2021), DOI 10.1109/TNSRE.2021.3133813; data CC BY 4.0. Changes: selected trial/segment, relative timestamps, display binning and original plot.

The source’s retained Processed values are already trial-calibrated. Zero refers to relaxed calibration and one to its calibration contraction: (signal − relaxed mean)/(MVC mean − relaxed mean). The package does not normalize them again or clamp values outside [0,1]. Their amplitude unit is one, not metres, volts or newtons. Calibration to those physical units is not established, and sEMG is not labeled activation a or individual-muscle force.

The saved timestamps are irregular. The segment averages 51.243524 saved samples/s despite the protocol’s nominal best-effort 1 kHz description. The display uses 172 half-second bins, each [start,end), with sample-weighted arithmetic means and the actual mean sample time; the last bin can be partial. Use original relative timestamps for subsequent analysis. These binned curves do not establish spectral behavior, latency, causality, population uncertainty or a universal force law.

SI atlas JSON, original recorded samples CSV, display bins CSV, trial metadata.

Six source model actuators, with units

Arm26’s six Thelen2003 actuator sets come from the pinned official OpenSim XML. These are model parameters, not direct participant measurements or clinical norms. Three parameters are displayed below; the complete extracted JSON and CSV also retain pennation angle (rad), activation/deactivation time constants (s), units, source paths and coordinate bounds. There is no brachioradialis actuator in Arm26. Do not replace the schematic actuator with these values while retaining its unrelated geometry and then call the result calibrated.

OpenSim Development Team (Reinbolt, Seth, Habib and Hamner), adapted from Kate Holzbaur’s model; CC BY 3.0. The complete source credit and licence remain in unchanged model XML. Changes: explicit numerical extraction into SI-labelled files. Official source revision is 84b487c4e3245359a64381e01f01b9cf4772d457; XML SHA-256 is e2224d0044eb393b05d64926c3fa1682c451a9adc7f510e5517ef9958d3d41b9.

What the checks establish

The exact 2,058,262-byte received package has SHA-256 af02aaeb37d626f448a658183f764dc6cf68b979730c413ba31997f26062d169. Its 992 checks pass again in this repository: source/payload hashes, atlas units and face bounds, six parameter sets against XML, finite strictly ordered recorded times, full bin accounting and independently recomputed means. The passive pickle reader rejects executable globals; no source pickle is bundled or executed. These are data/transformation checks, not biological validation. Archive directory/CRC evidence is preserved; a full BodyParts3D archive hash is deliberately not claimed because only selected members were acquired.

Per-file provenance, licences and attribution, reproduction and limitations. Later anatomical simulation requires registered joint frames, attachment/path maps, material fields, boundary conditions and independent validation data. A single isometric thickness trace cannot supply all of those missing quantities.

Muscle pulls through a changing path

A muscle is not merely a red spring. An actuator model needs an input, internal state, force law, geometry and a clear statement of what it omits. Begin with a line of action because it makes loading understandable. Add a three-dimensional continuum only when the question requires internal deformation, local stress, broad attachments or contact.

The benchmark paper by Millard, Uchida, Seth and Delp compares equilibrium, damped-equilibrium and rigid-tendon models against numerical and biological tests. Its measured speedups belong to its hardware, integrators, tolerances and test actuators. They are not transferable frame-rate promises for an arm demo. Its central practical lesson here is to make tendon assumptions and numerical singularities explicit.

1 Excitation is not instantaneous force

Let u ∈ [0,1] be neural excitation and a ∈ [0,1] be activation. A deliberately simple teaching model is

For constant u and τ during a step, use the exact update

This avoids the overshoot that a large explicit-Euler step can produce. Separate rise and fall constants can be added, with the switch rule specified. This first-order filter is an illustrative activation model; its time constants must be fitted or sourced before making quantitative physiological claims.

Reader exercise. Pulse u while plotting u, a and force. Keep length fixed first. Then move the endpoints without changing u to show that activation alone does not determine force.

2 A Hill-type actuator is a collection of functions

Define fiber length l_f, optimal fiber length l_opt, fiber velocity v_f = l̇_f and maximum isometric force F₀. Adopt positive v_f for lengthening. One model family has

Here f_l is the active force-length curve, f_v the active force-velocity curve, f_pe passive force-length response and β a dimensionless damping coefficient under this normalization. Choose v_scale explicitly, often from optimal length times a maximum-shortening-rate parameter. A model that omits force-velocity effects must not claim to predict the load dependence of shortening speed.

For a toy first lesson, f_l can be a smooth bell curve and f_pe a tension-only polynomial. Label those curves as authored teaching functions. A Gaussian width is a design parameter, not a measured human constant. Later replace the toy law with a specified, tested musculotendon formulation and its published parameter conventions.

Concentric contraction means active shortening; eccentric contraction means active lengthening; isometric means fixed length of the specified quantity. Fixed joint angle does not necessarily imply fixed fiber length when a compliant tendon stretches. A motion label must say whether it refers to whole musculotendon length or fiber length.

3 Tendon and pennation add internal state

For a simple pennated actuator,

The equal-force relation assumes the massless equilibrium idealization of this model. A dynamic tendon or muscle-mass model requires its own momentum equations.

A tendon force law normally depends on normalized tendon length or strain and is tensile. A teaching spring can use F_T = k_T max(0,l_T−l_slack), while a research model should use a specified nonlinear curve, transition smoothing and fitted slack length. A tendon should not become a compressive strut just because its length falls below slack length.

If pennation is modeled using constant thickness h_f,

Therefore

The last term is not generally l̇_f cos α because α also changes. The model becomes poorly conditioned as cos α approaches zero; use a physiologically justified domain and explicit failure reporting.

A rigid-tendon approximation removes tendon stretch from the state. It can be useful, but it changes fiber operating lengths, velocities and stored elastic energy. It is a model simplification, not just a faster integrator. Do not switch tendon assumptions silently when moving from offline calculations to an interactive preview.

muscle_step(path_length, path_speed, excitation, old_state, h):
    advance activation with the specified activation dynamics
    establish admissible fiber-length and pennation bounds
    initialize tendon and fiber state consistently at the starting pose
    evaluate tendon tension from tendon length
    solve the chosen fiber/tendon equilibrium equation
        with safeguarded root finding and the model's velocity convention
    reject or reduce h on loss of admissibility or failed convergence
    return tendon tension, fiber state, stored energy, work and residual

This is a solver contract, not an interchangeable formula for every Hill model. For a damped equilibrium formulation one may solve for fiber velocity and integrate fiber length. A fully implicit formulation may solve the next fiber length and velocity together. State which unknown is being solved and report the normalized force residual.

4 The signed moment arm comes from virtual work

For musculotendon length l_MT(q), positive tension F_T resists increasing length. Virtual work gives

For a rotational q_j, r_j has units of metres. For a translational coordinate, the corresponding derivative has a different interpretation. For coupled coordinates, derivatives must follow the permitted motion, not move one geometrically dependent coordinate independently.

A line segment from an origin point to an insertion point is enough for the first example. Intermediate via points and wrapping surfaces avoid impossible paths through bones, but they introduce branches and possible changes in contact topology. A discontinuous path produces discontinuous moment arms. Plot l(q) and its derivative over the complete supported range before trusting the force curve.

Reader exercise. Move a synthetic insertion point farther from the elbow. Predict how tension changes for the same moment. Then compare a straight path with a wrapped path. Verify the analytic or automatic-differentiation moment arm against centred finite differences of path length.

5 One net moment does not identify individual muscle forces

If several muscles cross the hinge,

One measured net moment usually leaves multiple unknown tensions. A flexor and extensor can both increase their forces while leaving the net moment unchanged. An optimization criterion or motor-control assumption can select one solution, but that assumption is additional information, not a measurement.

The reference environment OpenSim supports musculoskeletal model construction and analyses. It is appropriate for independent comparisons when geometry, parameters, excitations, conventions and solver tolerances are matched. Agreement with another program is verification evidence; it is not automatically validation against a person.

6 A line actuator does not determine muscle shape

A line model supplies tension and length. It does not uniquely determine bulging, fascicle curvature, local strain, intermuscular pressure or skin sliding. An ellipsoid whose radius increases as its length decreases illustrates an approximate volume constraint, but that geometry is an authored visualization.

For a cylindrical illustration with volume V = πr²l, constant volume implies

Show this as a first intuition for lateral expansion, then show why a real muscle with nonuniform architecture cannot be inferred from that formula alone. Blemker, Pinsky and Delp used a three-dimensional biceps model and compared local strains with measurements during low-load elbow flexion; their result motivates spatially varying fiber architecture instead of uniform shortening everywhere.

Three-dimensional active tissue can be coupled in either of two educational modes:

• One-way visualization: a line actuator drives joint mechanics; an explicitly approximate volume-preserving shape follows activation and length. No tissue reaction is claimed to affect the lift.
• Mechanical tissue model: active stress or active strain in a continuum produces tissue forces, contact and bone reactions. Its contribution is included exactly once in the coupled equations.

Adding a line-muscle torque and a second continuum actuator representing the same muscle can count active work twice. The model must declare which component supplies the mechanical action.

7 What would make the model credible

Check force-length and force-velocity curves separately before attempting a curl. Test a slack tendon, an isometric activation transient, passive stretch, constant-velocity shortening, constant-velocity lengthening and a return cycle. Check energy accounting for the passive parts and report active work separately. Vary time step and tolerances without retuning material parameters.

For biological validation, match the intended task and available independent observations: external torque, kinematics, fascicle/tendon behavior or deformation fields, with measurement uncertainty. Electromyography can inform activation timing, but it is not a direct tendon-force measurement. Reserve independent trials from calibration. A visually recognizable biceps contraction is an instructional success; it does not establish the accuracy of its predicted internal force.

Laboratories 4–5 implement the declared simplified line and series laws rather than all functions in the general Hill formulation above. Laboratory 7 adds a solved spatial preferred-shortening model, membrane and sampled bone contact under one-way coupling. The line remains the hinge force owner. This lets the reader separate excitation, actual active/passive force, tendon storage and shape, while preserving the stated force–velocity, pennation and architecture omissions.

Let forces move the elbow

A pose slider answers where a segment is. A forward-dynamics model answers how it accelerates under the forces actually included. Laboratory 4 makes that distinction visible: the same schematic can either move under its actuator and gravity or remain at an explicitly prescribed angle with an external hold moment.

Laboratory 4 · Articulated elbow and activation

Schematic one-hinge forward dynamics or prescribed hold; toy line muscle; rigid tendon; no force–velocity law or tissue/contact mechanics.

Start interactive 3D

Static reference diagram. Step, Play or Pulse starts live 3D; if unavailable, the display is explicitly numerical-only.

Dumbbell mass (kg)

Excitation u (0–1)

Initial / prescribed flexion q (°)

Motion modeForce-driven hingePrescribed static hold

Physics step h (s)0.0025 s0.005 s0.01 s

Play Single step Pulse current state / release Release excitation Download trace Reset defaults Front / oblique view Read current results Copy state

The fixed upper segment joins a rotating forearm and dumbbell. Gold markers locate a synthetic flexor path. The red belly is an illustrative shape with no mechanical force of its own. The yellow arrow marks dumbbell gravity. Axes: x right, y up, z out of the plane.

Define the hinge and gravity once

Use q = 0 for a downward-pointing forearm and positive q for flexion toward the right. The earlier lever’s horizontal-zero angle is θ = q − π/2. A point at distance L from the hinge has position (L sin q, −L cos q). With a dumbbell mass m_d at L and a combined forearm/hand mass m_f at c, let

The point-mass terms are explicit approximations; I_b is an authored residual inertia. Gravitational potential is −C cos q, so the gravity moment is −C sin q. At horizontal, using 5 kg, 0.35 m, 1.5 kg, 0.15 m and g = 9.81 m/s² gives −19.37475 N m. The fixture’s inertia is 0.68125 kg m². These calculations include gravity once: there is no second gravity contribution hidden in a bias force.

Our equation is

It contains active muscle moment, passive muscle moment, gravity, and viscous hinge damping. It contains no tracking motor in forward mode and no tissue-contact force. The supported 0–135° interval is a chosen teaching domain, not a statement about a person’s range of motion. If the next step leaves that interval, playback stops at the last admissible state; it does not invent a hard-stop collision or clamp velocity to zero.

A changing path produces a changing moment arm

Put the synthetic origin at (0,o) and insertion at (s sin q, −s cos q). The straight path has length

The path transmits tension toward its origin. Taking the cross product of insertion position and that force gives the same signed moment rF. For positive flexion, shortening corresponds to dl/dt = −r q̇. Active power delivered to the hinge is τ_A q̇ = −F_A l̇. The numeric tests compare the analytic derivative with centered finite differences, compare it with the independent force cross product, and check this power relation over the supported interior angles. The geometry is a straight synthetic path; wrapping, attachment regions and forearm rotation are absent.

This exact-integer proof checks the multiplication and sign contract once the derivative and rate relations are supplied. It does not prove the trigonometric derivative or its JavaScript implementation. The corresponding real product identity below retains those supplied geometric assumptions. The separate geometric checks remain necessary.

Excitation, activation, and force are different states

The control u is excitation in [0,1]. Activation a follows a first-order filter. With a constant input during one step,

The implementation uses τ = 0.05 s while excitation is at least the current activation, and 0.15 s otherwise. Both are illustrative choices. Release sets u to zero while retaining a, q and q̇. Force therefore decays rather than disappearing instantaneously. Changing excitation preserves the trajectory; changing the load, mode, initial angle or timestep starts a fresh experiment.

The exported trace has one row per step. A same-time excitation change refreshes the current row’s input and instantaneous derived outputs, so an immediate download matches the display. Earlier rows remain intact; time, state and accumulated work do not advance. The current row’s excitation applies to the next step. Intermediate input edits with zero elapsed time are not separate events.

The exact real-arithmetic update is a convex mixture of a and u with weight e^(−h/τ). The first contract below concerns integer-weighted mixture numerators. The separate real claims prove the exponential update’s interval bound and its constant-input derivative. The derivative treats τ as fixed; the implementation selects its rise/fall value for each step. These claims do not prove floating-point equivalence or the physiological validity of the authored time constants.

For this authored force law, define fiber length l_f = l − l_T with a rigid tendon segment of length l_T, normalized fiber length λ = l_f/l_opt, and extension δ = max(0,l_f − l_opt):

The bell curve, tension-only passive law, and their parameters are teaching choices. Force–velocity dependence and pennation are deliberately omitted. This is not an implementation of Millard’s calibrated muscle curves or a prediction of human shortening speed. The rigid tendon removes fiber/tendon redistribution and tendon elastic energy. The original actuator comparison explains why compliant and rigid-tendon formulations answer different questions. Millard et al.

The following real claims prove the Gaussian force bound and its derivative with respect to fiber length. The same force law is used in Laboratory 5; the compliant equilibrium changes its fiber length rather than its constitutive definition.

The red muscle belly is a one-way drawing following path length. Its transverse radius scales as sqrt(l_ref/l_f), an intuition for transverse expansion, not a spatial tissue solution. It adds no second active force, inertia, elastic energy, or contact response. The line actuator is the sole mechanical owner of active force.

Lift, release, or prescribe a hold

Try the pulse. Click “Pulse current state / release.” It keeps the selected load, angle, time step, excitation and mode, continues the current activation and velocity, and releases excitation at the first step boundary at or after 0.30 additional simulated seconds. Use “Reset defaults” first to reproduce the default u = 0.6 forward experiment. Step, Play and Pulse start live 3D automatically; if WebGL fails, a prominent numerical-only notice identifies the static reference diagram. Watch initial gravity-driven lowering before activation builds, subsequent flexion, and continued motion during activation decay. A falling angle is not proof that the muscle is inactive; inspect activation and fiber velocity separately. Pause, single-step, and download the trace to inspect q, a, moments and work.

Try an unassisted load change. Increase the dumbbell mass and replay with the same excitation. The model may lower or leave its domain. No hidden motor raises a weak actuator to the requested pose. Reset clears state, camera, and stale input validation.

Try the prescribed hold. Select “Prescribed static hold,” vary q and excitation, then step activation. The bones stay fixed because the mode prescribes them. The displayed external hold moment is −(τ_A + τ_P + τ_g). It is an explicit support, not muscle-generated motion. Since q̇ = 0, the support and muscle deliver zero skeletal mechanical power in this fixture, while real muscle metabolism is outside the modeled system. Switching modes resets the experiment rather than injecting an unexplained constraint impulse.

Mechanical work gives an independent diagnostic

The mechanical energy includes hinge kinetic energy, gravitational potential, and the passive fiber spring:

For this continuous model, E − E₀ − W_A + D = 0. Activation is an externally driven state; its chemical energy is not part of E. Negative active work during active lengthening can occur and is retained with its sign. Gravity is already in E and is not added as external work a second time.

The implementation advances q, q̇, W_A and D using classical RK4 with the exact activation state evaluated at each stage. The passive-force kink can affect observed convergence; no blanket fourth-order result is claimed for every trajectory. An original 0.60 s pulse/release experiment produces the table below from the same pure module used by the browser. The residual is a numerical balance diagnostic, not biological validation.

The smallest-step trajectory supplies a numerical comparison, not a measured reference. Tests also check static gravitational moment, inertia, activation transients, path derivatives, signed power, replay, and the explicit stop/hold behaviors. Actual tendon compliance, co-contraction and coupled tissue forces belong to later stages.

From line forces to tissue and rendered skin

Laboratory 4 moves a rigid skeleton. Its red ellipsoid cannot tell us where tissue compresses. The next layer must introduce material deformation, interfaces and contact before it supplies forces or pressure. This chapter integrates the audited simulation and anatomy research into the progression. Laboratory 5 implements a bounded affine-block/contact proxy; Laboratories 6–7 implement the spatial FEM comparison and a separate schematic skin/fascia/contact model described in later chapters. Their stated limits distinguish working examples from anatomical prediction.

Deformation needs a reference configuration

The earlier property measurements already defined reference/current positions, dimensionless strain and volume ratio. There, the controls chose the vertices. Here the new question is what material energy and boundary forces should determine those positions. Reuse the measurements as checks; positive volume or a convincing outline is not a constitutive law.

Write a material point X in the reference shape and its current position x(X). The deformation gradient F = ∂x/∂X describes local changes; J = det F measures local volume ratio. A rigid rotation has F = R and J = 1, while stretch and shear change the local geometry. Volume preservation alone does not establish an appropriate tissue response or fiber architecture.

For a tetrahedral finite element, construct F from current and reference edge matrices, assign a material energy density Ψ(F, fibers), integrate that energy over the reference volume, and differentiate to obtain nodal forces. A six-edge spring network on the same tetrahedron is a different model. It does not become FEM by sharing the topology. Fiber directions, large deformation, incompressibility, boundary conditions, and inversion behavior each need an explicit choice.

First Piola–Kirchhoff stress P and Cauchy stress σ both have Pa units. They act in different configuration descriptions and satisfy σ = J⁻¹ P Fᵀ under the usual mapping. Confusing those measures is a configuration error, not a unit conversion. Local stress cannot be read from a skinning weight or inferred from a single silhouette.

An active continuum may add fiber-directed stress or use an active-strain decomposition. If it represents the same synthetic flexor as the line actuator, choose a single force owner: either its tissue reactions drive the joint, or the line drives the joint and the tissue is an explicitly one-way visualization. Adding both actuator moments for the same muscle doubles active work. Ryan and colleagues’ original finite-element compression study examines how transverse loading changes muscle work and force; its simplified contracting blocks do not validate a human anterior elbow fold. Original study

Skin, fascia and interfaces answer separate questions

The skin envelope, subcutaneous layer, deep fascia, muscle connective tissue and tendons differ in geometry, attachments, and loading. An attachment condition controls motion relative to a bone or neighboring layer. A sliding interface permits tangential relative motion under its stated law. Contact controls whether two surfaces may occupy the same region. A generic stiffness number cannot represent all three.

Material evidence must keep species, orientation, strain range, activation and loading context. The audited Takaza study is passive porcine tensile testing at different orientations; it is not transverse human-arm compression evidence. The forearm indentation estimates introduced earlier have their own layer and inverse-identification context. A shear-wave or effective indentation modulus should not silently become a universal large-strain Young’s modulus.

The concave anterior elbow crease and convex posterior tissues have different deformation demands. Posterior fat-pad gliding is not an anterior skin-contact measurement. Similarly, a rigid-body joint reaction is a resultant, whereas cartilage or skin pressure requires a local contact area and a constitutive/contact model. Report “not modeled” for those absent outputs, rather than zero.

Normal contact, friction and coupling

For a chosen gap g_n and compressive normal multiplier λ_n, ideal unilateral contact satisfies g_n ≥ 0, λ_n ≥ 0 and g_n λ_n = 0. A penalty method tolerates penetration and estimates force from a specified law; it requires a penalty/timestep sensitivity study. A constraint solver has its own iteration residual and tolerance. Friction adds a tangential law and dissipation and should not be confused with the normal condition.

Define a transferred force f as the force on the bone and −f on tissue. Match action/reaction, moment about a common origin, and power at the same interface velocity. If a model already includes the forearm’s tissue mass in its segment inertia, adding massive tissue proxies requires redistributing that mass explicitly.

Near-volume preservation can cause lateral expansion when tissue flattens. It cannot by itself prevent surface intersection. Test compression, sliding, attachment, and volume separately before a coupled crease. Mesh refinement, timestep refinement and contact-tolerance studies answer different questions; none can replace an independent measured deformation field.

Faster methods change the model or its solution

PBD projects geometric constraints after a prediction step. Its stiffness can vary with timestep and iteration count. XPBD introduces compliance scaled by timestep and accumulated multipliers to reduce that coupling, while finite iterations, nonlinearities and contact can still produce error. Macklin et al.’s original XPBD paper

Linear blend skinning (LBS) instead evaluates a weighted sum of bone transforms. It has no material energy, activation or contact reaction. Two equally weighted rotations about a common axis illustrate its volume artifact: in the rotated plane, averaging R(+φ) and R(−φ) yields cos φ times the identity. At φ = 60°, that plane’s area scales by cos² φ = 0.25. This is an authored exact graphics counterexample, not a measurement of elbow skin.

Dual-quaternion skinning improves rotational interpolation and addresses familiar LBS deformation artifacts, but it does not introduce contact or a tissue constitutive law. Kavan and colleagues’ original paper analyzes that approximation. Neither technique supplies pressure by itself. Original paper

Reduced deformation bases, cached simulations and learned mappings can save computation by removing degrees of freedom or reusing prior solutions. Their domain needs testing on unseen poses and contact states. Compare the same geometry, load, material, boundary conditions and error measure before attributing a difference purely to solver speed. A medical/CAD analysis needs parameter identification, convergence and independently measured outputs; a game/film deformation may legitimately optimize visible shape and latency. No universal speed/accuracy ranking follows from the category name.

Keep the pose in one place

If a detailed render surface follows a tissue proxy, embed it directly in the posed proxy, or add a displacement residual measured against its baseline at the same current pose and coordinate frame. Adding an unposed rest-relative displacement to an already skinned vertex applies part of the pose twice. Test this with a pure bone-motion sweep: a zero tissue residual must leave exactly the baseline surface.

For the spatial capstone, synchronize the actual q(t) from forward dynamics across the naive skinning and tissue views. Keep the base mesh and skeletal transforms identical. Show geometric distortion separately from detected intersection, contact gap separately from pressure, and each force owner separately from its visual shape. Laboratory 4 supplies the first trajectory and actuator telemetry for that comparison; Laboratory 7 adds the spatial volume, membrane, fascia-tether and sampled bone-contact solve with a synchronized baseline and measured defects.

The separate artistic poser

The chapter’s embedded poser uses a stick graph and a surface bound in the neutral pose. It blends rigid bone transformations with dual quaternions while retaining mesh topology. This is geometric skinning, not a constitutive material model. Kavan and colleagues describe approximate dual-quaternion blending and distinguish geometric from physically based skinning in their original paper (2008, §§1–2). Author-hosted paper.

Kenoma’s implementation has its own weights, bind mesh and limitations: extreme bends can crease or bulge and surfaces may interpenetrate. It does not promise volume preservation, nonintersection or anatomical accuracy. Its visual result must not be used as a muscle-force, pressure or physiological measurement. Versioned rig contract.

Store tendon energy and resist compression

Laboratory 4 used a rigid tendon and an illustrative belly. Laboratory 5 adds two physical mechanisms while preserving that earlier lesson: a massless series tendon/fiber equilibrium and a quasistatic tissue block whose reaction changes the hinge acceleration. The right-hand drawing is a synchronized graphics baseline at the same q, not another dynamics simulation.

Laboratory 5 · Tendon, tissue and the same-pose skinning baseline

Schematic series actuator; quasistatic affine hyperelastic block with ideal plate contact; tissue reaction drives the hinge. No anatomical calibration, spatial FEM or separate skin/fascia law.

Start interactive 3D

Static reference diagram. Step, Play or Pulse starts live 3D; if unavailable, the display is explicitly numerical-only.

Both schematic arms have the same angle. Left: red active/passive fiber and gold tensile tendon; enlarged teal tissue block between gold frictionless plates. Right: the identical bind block receives uniform 50/50 linear blend skinning and has no force model. Insets are enlarged eight times in world units; read SI values for actual dimensions.

Dumbbell mass (kg)

Excitation u (0–1)

Initial / prescribed flexion q (°)

Motion modeForce-driven hingePrescribed static hold

Physics step h (s)0.0025 s0.005 s0.01 s

Tendon assumptionCompliant tensile tendonRigid tendon

Tendon stiffness (N/m)15,00030,00060,000

Tissue/plate contactEnabled; force feeds hingeDisabled; show penetration

Tissue bulk modulus (Pa)50,0005,0000; remove volume resistance

Play Single step Pulse current state / release Release excitation Download trace Reset defaults Front / oblique view Read current results Copy state

Both schematic arms have the same angle. Left: red active/passive fiber and gold tensile tendon; enlarged teal tissue block between gold frictionless plates. Right: the identical bind block receives uniform 50/50 linear blend skinning and has no force model. Insets are enlarged eight times in world units; read SI values for actual dimensions. Axes: x right, y up, z out of the plane.

The tissue block’s boundary is the rendered surface. There is no separate skin membrane, fascia layer, fiber architecture field or anatomical attachment map yet. The plate gap is an authored function of angle, rather than a distance measured from elbow meshes. This bounded proxy makes tendon stretch, elastic storage, compression and skinning volume loss measurable before anatomical detail is introduced.

One-dimensional contraction can redistribute length at fixed endpoints

Let l be the synthetic musculotendon path from Laboratory 4, l_f the fiber length and l_T the tendon length. There is zero pennation and l = l_f + l_T. An active force source in parallel with a passive fiber spring supplies

This is the same bell-shaped active force–length law, passive spring, activation and hinge geometry as Laboratory 4. The compliant tendon changes fiber length, so its effect on active tension is solved inside the equilibrium rather than attributed to a changed actuator law. The rigid-tendon/contact-off limit recovers Laboratory 4 at identical angle, activation and velocity. Neither lesson implements a measured force–velocity law or calibrated Hill model. Original actuator comparison

Set $c_T=1/k_T$ (zero for the rigid switch), $L=l-l_{Ts}$ and solve for the real fiber length $f$:

The supported tendon stiffnesses 15,000–60,000 N/m retain a unique root in $[0,L]$. The active slope satisfies $|\partial F_A/\partial f|\leq F_0\sqrt{2/e}/(wl_0)$, about 8,578 N/m at full activation. Therefore $H'(f)\geq1-c_TF_0\sqrt{2/e}/(wl_0)>0$ on either passive branch. For the supported pose/activation domain, $H(0)\leq0$ and $H(L)\geq0$; the executable bracket checks the lower end before bisection. A nonpositive-root candidate is explicitly inadmissible. The real claims below prove the exact Gaussian slope bound, strict monotonicity including the passive kink, and root uniqueness under the same sufficient guard. A separate continuity result proves bracket existence given $L\geq0$ and $H(0)\leq0$; it does not certify that every application supplies those assumptions.

The implementation bisects the monotone scalar equation, then reports $l_T=l-f$, $F_T=F_A+F_P$ and the independent tendon-force residual $k_T(l_T-l_{Ts})-F_T$. At zero activation and a short path, the tendon carries zero force; it is not a compressive strut. The rigid switch sets $f=L$ and stores no tendon energy. The supported fiber domain remains $f\geq0.04$ m; leaving it stops the teaching solve. Material coefficients and that admissibility bound are unchanged.

Write $F_{A,f}=\partial F_A/\partial f$ and $F_{A,a}=F_0f_l(f)$. With $d=1+c_T(F_{A,f}+k_M\mathbf1_{f>l_0})>0$, differentiating the equilibrium gives