# Independent Double Wishbone reference v1

The 2026-10-09 revised request authorizes mathematical verification without an
external published reproduction. The earlier three-source incompatibility audit
remains unchanged. This reference is authored for VehicleLab's stated model.

## Acceptance declared before evaluation

For all 101 points per geometry, spaced 1 mm across −50 to +50 mm:

| Quantity                                                    | Maximum absolute acceptance | Rationale                                                                                                                                                        |
| ----------------------------------------------------------- | --------------------------: | ---------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| Hardpoints, wheel centre, element length/compression        |                      2e-9 m | Existing production position residual is 1e-11 m; allowance covers Cartesian closure conditioning and independent bisection, without altering production limits. |
| Rigid-link and prescribed-travel residual                   |                     2e-10 m | Retains the existing rigid-link regression bound.                                                                                                                |
| Arm angle                                                   |                    2e-8 rad | Angle reconstructed from Cartesian radial vectors; no production angular iteration reused.                                                                       |
| Spring/damper signed MR versus analytical derivative        |                        2e-7 | Production uses a 10 µm centred derivative; bound allows finite-difference truncation in smooth tested geometries.                                               |
| Analytical MR versus reference-only 1 µm centred derivative |                        2e-8 | Separate reference derivative check, accounting for cancellation and truncation.                                                                                 |
| Reflection coordinate/element agreement                     |            2e-9 m / 2e-7 MR | Same bounds applied after exact y reflection.                                                                                                                    |
| Repeatability                                               |                           0 | Identical input must reproduce identical state.                                                                                                                  |

Maximum and RMS errors use every point. Relative errors use max(abs(reference),
0.01 m) for lengths/compression and max(abs(reference),0.01) for MR and are
informational; no division by a zero compression. No threshold is chosen from
observed results. Feasibility/rejection cases use independent domain conditions
and preserve all existing solver admission/singularity tests.

## Compatible planar specialization

All wishbone hinge axes are parallel to chassis x. Their end points have identical
y and z, and ball joints lie in x=0. Therefore the upright's longitudinal direction
is exactly chassis +x; its local vertical is the normalized lower-to-upper ball
vector u=(dy,dz)/h, and local lateral is v=(uz,−uy). This equals the production
heading construction for this specialization, without copying its pose routine.
The wheel centre remains at its entered design origin, with rigid upright-local
offsets β along v and η along u. It is **not assumed to coincide with a ball joint**.
Spring/damper mount x coordinates remain constant while their y–z coordinates
rotate rigidly with the lower arm; length includes all three Cartesian components.

## Different position calculation

Let lower/upper hinge centres be O and H, design arm radii r and R, and upright
ball separation h. Parameterize lower-ball vertical coordinate t directly:

`Bz=t; By=Oy + side*sqrt(r²−(t−Oz)²)`.

The upper ball A is the selected intersection of a circle centred at H with radius
R and a circle centred at B with radius h. Standard Cartesian circle-intersection
distances determine A without hinge-angle Newton iteration. The signed triangle
orientation at design identifies the branch; it is retained throughout travel.
The wheel is `W = B + β*v + η*u`. Bisection in t imposes Wz=q, within a declared
monotone interval around design. Circle feasibility, branch orientation, a
nonzero derivative and a bracket are checked explicitly. No extrapolation is used.

## Independent differential calculation

For t derivatives, `B' = (−(t−Oz)/(By−Oy), 1)`. Differentiate the two upper-ball
circle equations:

`(A−H)·A' = 0`

`(A−B)·A' = (A−B)·B'`.

A two-by-two Cartesian linear solve gives A'. Then u'=(A'−B')/h,
v'=(u'z,−u'y) and W'=B'+β*v'+η*u'. Division by W'z converts every derivative to
prescribed wheel-centre jounce. This is an analytical position derivative, not
production's angular numerical Jacobian or centred length derivative.
Lower-arm mount coordinates use the design radial/tangential basis relative to O.
Its current radial vector is B−O; differentiate that linear basis construction
directly. For fixed upper mount S and moving lower mount M,
`MR = −(M−S)·dM/dq / |M−S|`.

## Scope and independence

The fixtures reproduce the exact generic front and rear hardpoints and add one
parallel-arm geometry used only by validation. Separate mounts yield different,
nonconstant element ratios. Production defaults are not edited. An additional
closed-form parallel-arm trajectory checks the circle reference itself.
Mirroring and reference-only finite differences provide distinct cross-checks.

The reference imports only domain **types**. It calls no production solver,
residual, pose, rotation, heading or derivative helper. The evaluator alone calls
production to obtain observed values. No Quarter/Half/Full vehicle simulation is
needed for VL-DW-001–007. VL-DW-008 points to the existing separately classified
Full-Car force/energy/power integration suite and does not inflate independent
standalone coverage.

This exact reference covers planar, parallel-x hinges with a well-conditioned
selected branch. Existing invariant/skew-axis tests cover additional 3D behavior;
they are not an exact independent reference for arbitrary spatial geometry.
No steering, toe, compliance, hydraulic state, OEM correlation, published Double
Wishbone reproduction or published Full-Car transient validation is claimed.