2. Physics of occupant protection

A physics-informed occupant safety model is only as trustworthy as the physics it encodes. Before any neural network can predict injury risk, the learning problem must be anchored to the real mechanical chain that connects a vehicle deceleration pulse to tissue-level damage. This chapter builds that chain: occupant kinematics, restraint loading, energy absorption, and the injury criteria that translate physical response into a scalar risk. Each of these becomes a constraint — a hard boundary, a monotonicity requirement, or a physically motivated loss term — that shapes how a model may learn.

Reference frames and occupant kinematics

Every downstream quantity depends on a consistent coordinate convention, so we fix one at the outset. Use the vehicle-fixed frame: +x points forward (direction of travel), +y to the occupant's left, +z upward. Accelerations are expressed in g (1 g = 9.81 m/s²), and a frontal crash produces a negative x deceleration of the vehicle while the unrestrained occupant continues forward at initial velocity. Displacements of the occupant relative to the vehicle are therefore positive in +x during forward excursion. This convention is held throughout the chapter and in the accompanying figure.

In a frontal impact the occupant does not decelerate with the vehicle; inertia carries the torso forward until a restraint engages. The kinematics are governed by the coupled equations of the occupant mass and the restraint force:

m_occ * a_occ(t) = -F_restraint(t) - F_contact(t)
x_rel(t) = ∫∫ (a_vehicle(t) - a_occ(t)) dt²

where x_rel is the occupant-to-vehicle relative displacement (the "ride-down" distance available before hard contact). The key physical insight is that the occupant's velocity change Δv is fixed by the crash severity, but the time over which it is delivered is set by the restraint. A longer, smoother deceleration reduces peak acceleration and hence injury risk. This is the ride-down principle, and it is the first constraint a model must respect: for a given Δv, predicted injury must be non-increasing as ride-down distance increases, all else equal.

Restraint loading and force paths

Restraints convert occupant momentum into controlled force. A three-point belt loads the occupant through the lap and shoulder paths; an airbag distributes load over a larger contact area and adds a compliant cushion. The belt force F_belt is limited by the force-limiter (a torsion bar or pyrotechnic element) to a design threshold, typically 3–6 kN in modern frontal systems, while the pretensioner removes slack within the first 10–20 ms. The physical constraint here is a force cap: the restraint cannot deliver more than its limiter setting, so any model predicting belt force above that ceiling is unphysical.

Load path geometry matters as much as magnitude. The shoulder belt routes force from the clavicle and sternum toward the buckle and anchor; the lap belt loads the pelvis. Off-axis loading, such as a belt slipping off the shoulder, redirects force into the neck and increases injury risk. For learning, this means restraint force should be treated as a vector with a known application point, not a scalar, and the model's inputs should include belt geometry (anchor positions, D-ring location) when the task involves out-of-position or oblique scenarios.

Energy absorption and the crash pulse

The vehicle front structure absorbs kinetic energy through controlled deformation. The crash pulse — vehicle deceleration versus time — is the boundary condition the occupant experiences. Its shape, not just its peak, determines occupant response. A pulse with a long, low-amplitude plateau delivers the same Δv as a short, high spike but produces far lower occupant acceleration. Energy absorbed by the structure is:

E_abs = ∫ F_struct dx = ½ m_veh (v_initial² - v_final²)

For a 50 km/h frontal barrier test, Δv ≈ 13.9 m/s and the structure must absorb roughly ½ · m_veh · Δv² of energy. The occupant, however, only "sees" the pulse after the restraint engages. The constraint for a physics-informed model is that the predicted occupant acceleration must be consistent with the integral of the crash pulse minus the restraint force, i.e., the model cannot invent energy. A useful diagnostic: integrate the predicted occupant acceleration over the event and compare to the known Δv; a mismatch beyond a few percent signals a physically inconsistent prediction.

Injury criteria as differentiable constraints

Injury criteria compress a time history into a risk scalar. The most common are summarized below, with the physical quantity each captures and the typical threshold used in regulation and consumer testing.

HIC is defined as HIC = max[(t2 - t1) · (1/(t2-t1) ∫ a dt)^2.5] over any 15 ms window, with a in g. The 2.5 exponent makes HIC strongly superlinear in acceleration, so a model that slightly over-predicts peak head acceleration will over-predict HIC dramatically. This is a strong argument for constraining the model's acceleration output directly rather than only its HIC output. Nij combines neck axial force F_z and bending moment M_y normalized by critical values: Nij = F_z/F_crit + M_y/M_crit, with the interaction surface defining the safe region. Chest deflection is rate-dependent: the same compression delivered faster produces higher injury risk, so a model must include loading rate as an input or as a temporal feature.

Encoding physics as learning constraints

These physical facts translate into concrete constraints on a learning system. Three are especially useful:

• Hard output bounds. Clamp or penalize predictions outside physical ranges (e.g., belt force ≤ limiter setting, HIC ≥ 0, Nij ≥ 0). A soft penalty λ · max(0, ŷ - y_max)² added to the loss keeps gradients well-behaved.
• Monotonicity. Injury risk must be non-decreasing in peak acceleration, chest compression, and neck load. Enforce this with monotone network layers or by adding a penalty on violations of ∂ŷ/∂x ≥ 0 for the relevant inputs.
• Consistency residuals. Penalize disagreement between the model's predicted occupant acceleration and the integral relation to the crash pulse and restraint force. This is the physics-informed loss in its most direct form.

Worked example: checking a predicted response

Consider a frontal sled case with Δv = 13.9 m/s, a 4 kN force limiter, and a 50th-percentile male occupant. Suppose a model predicts peak occupant chest acceleration of 45 g at 60 ms and a belt force of 5.2 kN. The belt force exceeds the 4 kN limiter, so the prediction is unphysical — the model has likely learned a spurious correlation rather than the force cap. The corrective action is to add the limiter as a hard bound and retrain, then re-check that the integrated occupant acceleration matches Δv within tolerance. This kind of check is the practical payoff of treating physics as a constraint: it turns an opaque prediction into a testable claim.

The figure summarizes the chain and the constraints that attach to each stage. The practical takeaway is that physics-informed learning for occupant safety is not a single loss term but a set of structural commitments: bounded outputs, monotone relationships, and consistency residuals that tie the model's predictions back to the crash pulse and restraint hardware. When these are in place, the model's predictions become checkable against the same physical reasoning an engineer would apply by hand — which is exactly what makes them usable in a safety-critical workflow.