2. Head-Torso Kinematic Chain Reconstruction from Crash Pulse Data

Why the kinematic chain must be reconstructed before any Nij judgement

A crash pulse recorded at the vehicle floor or seat base is not the loading that reaches the neck. Between the accelerometer and the cervical spine sit the seat cushion, the seat back, the torso, and the head–neck joint. Each element filters, delays, and rotates the signal. When a neck injury case is analysed, the first defensible question is therefore not "how large was Nij?" but "at what instant did the head begin to move relative to the torso in a way the restraint was not designed to control?" Answering that requires rebuilding the chain: floor/seat pulse → torso response → head response, expressed in a common reference frame and on a common time base.

This case uses a two-channel accelerometer set (seat-base and head-mounted) plus a torso angular-rate sensor. The deliverable is a phase-annotated relative-motion trace that isolates the first divergence between measured head–torso kinematics and the response predicted by the restraint (belt + seat back) alone.

Inputs, sign conventions, and channel inventory

Before any arithmetic, fix the reference frame. All channels below are expressed in the vehicle-forward, right-hand convention: +X forward, +Y to the vehicle's right, +Z up; positive rotation follows the right-hand rule about each axis. A frontal impact therefore produces a predominantly −X acceleration at the seat base. If your data logger uses ISO 6487 sign conventions or a head-mounted sensor with an inverted axis, convert once, document it, and never mix frames inside one comparison.

Two practical constraints dominate the reconstruction. First, the head and seat-base sensors are not co-located, so a rigid-body transformation is required to bring them into one frame; the lever arm between the seat-base origin and the head CG must be measured or taken from the occupant package drawing. Second, the seat-base signal contains the seat structure's own resonance (typically 15–40 Hz for a production seat), which must be separated from the occupant response band (roughly 0–30 Hz for the head–neck system) before the two traces are compared.

Reconstruction procedure

1. Time-align the channels. Cross-correlate ACC_SB_X with ACC_HD_X over the first 20 ms of the pulse. A rigid chain should show a small, positive lag (head lags seat base). A negative lag or a lag larger than the geometric travel time indicates a channel delay or a mislabelled sensor; correct it before proceeding.
2. Low-pass filter consistently. Apply a zero-phase Butterworth filter (4th order, 30 Hz cutoff) to all six channels. Filtering only one side of a comparison is the most common source of false "divergence".
3. Integrate to velocity and displacement. Integrate ACC_SB_X and ACC_HD_X once for velocity, twice for displacement, with a high-pass stage (0.5 Hz) to remove integration drift. The relative head–torso displacement in X is then d_rel(t) = d_head(t) − d_torso(t), where d_torso is derived from the seat-base trace plus the seat-back rotation contribution from GYR_TR_Y.
4. Build the relative rotation term. The head–torso pitch angle is θ_rel(t) = ∫(GYR_HD_Y − GYR_TR_Y) dt. This is the quantity that most directly exposes a restraint that is controlling the torso but not the head.
5. Compute the expected restraint response. For the belt-and-seat-back system, the expected head displacement follows the torso displacement with a bounded lag and a bounded relative angle. Fit a first-order lag model to the pre-divergence window (typically 0–40 ms) and extrapolate; the residual between measured and extrapolated head motion is the divergence signal.
6. Locate the first divergence. Define divergence as the first sample where the residual exceeds a threshold (e.g. 5 mm displacement or 3° relative angle) and remains above it for at least 5 ms. Report the timestamp, the phase (loading, restraint engagement, rebound), and the channel that first crossed.

Reading the reconstructed chain

The figure below shows the reconstructed chain for a representative frontal case. The seat-base pulse rises first, the torso follows with the expected lag, and the head tracks the torso until roughly 55 ms. At that point the relative pitch angle departs from the extrapolated restraint response while the torso trace remains smooth — the signature of a head that has left the controlled envelope before the belt has fully engaged the torso.

Interpreting the divergence phase

Three patterns recur in this reconstruction and each points to a different root-cause family:

• Early divergence (< 40 ms), torso trace smooth. The head is moving before the belt has loaded the torso. This is consistent with a loose belt, a slack retractor, or a head position outside the belt's controlled envelope. The root-cause question shifts to restraint geometry and pre-impact posture.
• Late divergence (> 70 ms), torso trace also departs. The whole chain is leaving the controlled response together. This points to seat-back deflection or recliner yield rather than a head-specific mechanism.
• Divergence in θ_rel without divergence in d_rel. The head is rotating relative to the torso while translating with it — the classic signature of a neck that is being loaded in bending before the torso has been decelerated. This is the phase that most directly feeds the Nij calculation in the next case.

Verification and common failure symptoms

A reconstruction is acceptable when three checks pass. First, the cross-correlation lag between ACC_SB_X and ACC_HD_X is positive and within the geometric travel time (typically 2–8 ms for a mid-size occupant). Second, the integrated seat-base velocity returns to approximately zero after the pulse, confirming that the high-pass stage removed drift without distorting the loading phase. Third, the extrapolated restraint response fits the pre-divergence window with a residual below the noise floor; a poor fit means the lag model is wrong, not that the occupant diverged early.

Typical failure symptoms and their causes: a negative cross-correlation lag usually means a channel is time-stamped from a different trigger; a θ_rel that drifts monotonically after the pulse indicates an uncorrected gyro bias; a d_rel that grows without bound indicates integration drift from an insufficient high-pass cutoff. Each of these produces a false divergence and must be resolved before the phase annotation is trusted.