**Appendix 2 | Human Collision Physics Model**

**Introduction**

As in other biomechanics-type simulation programs, such as Articulated Total Body [1], the Virtual CRASH human model is a multi-ellipsoid system connected by joints. The human model is divided into 14 parts (head, neck, torso, hips, thighs, calves, feet, upper arm and forearm); each of these parts is allowed to interact with vehicle contact planes, ground planes, and each other. Each segment has its own size and weight, which is estimated from the height and weight specified for the whole person. In Virtual CRASH 3, one has the option to choose from a variety of predefined poses, but one also has the freedom create custom poses by manipulating the joints directly. In what follows below, we describe the steps involved in the multibody human collision physics model. The same collision model is selected with the “default-auto” contact model. Note, for any two bodies that can collide, if either one has the “default-auto” model selected, the default-auto model will used; however, the Kudlich-Slibar model will used only if both objects have “kudlich-slibar” selected as the preferred contact model.

**The Multi-ellipsoid System**

Virtual CRASH defines mechanical linkages between the adjacent segments; each linkage has its own specific type of connection (see below) and joint articulation (for example, the elbow joint can be rotated throughout 360°, but only about 165°). We define the shape of the segment by its ellipsoid, which is defined by the expression:

where

- a, b, c - the semi-principal axes,
- x, y, z – position on surface,
- n – exponent defining shape. n>2 defines a hyper-ellipsoid shape.

The exponent, n in equation (69), defines the segment’s overall shape, the effect of the value of n can be seen in the following example. The figure below illustrates an example of an ellipsoid where n = 2:

Here we illustrate an example where n=30:

Segments (ellipsoid) have the following properties:

**Overall size**: the ellipsoid sizes for each segment are determined based reasonable approximations for typical humans. A similar approach was used in Articulated Total Body;**Moments-of-inertia**: Moments-of-inertia are specified about all three axes of an ellipsoid. Currently these data are converted from a proportional distribution of sizes and weights from a specified height and weight of a person;**Coefficients-of-friction**: the user specifies two friction coefficients, one for the contact ellipsoid with the vehicle, the second for the contact with the ground, ellipsoid or other ellipsoid;**Coefficients-of-restitution**: similar to friction, the coefficients-of-restitution are also taken into account. The user specifies two coefficients: one for the contact with the vehicle, one for contact with the ground or other ellipsoids.

As mentioned above, in Virtual CRASH 3 neighboring segments within the human model are connected by a so-called “ragdoll joint.”

This joint has three articulation angles whose limits and resistive spring joint torque parameters can be individually set [2].

**Simulation of Motion**

The human body model motion is simulated using numerical integration techniques. At each simulation time-step, the following steps are performed:

- Calculation of contacts (positions and normal vectors)
- Calculation of the determinant
- Determining the conditions and calculation of forces
- Applications forces on the ellipsoids
- Solve the equations of motion

Symbolically, these steps are:

where

**\(J\)**- Jacobian,**\(v\)**- velocity,**\(c\)**- constant,**\(F\)**- force**\(\lambda\)**- result**\(M\)**- matrix moments of inertia,**\(dt\)**- integration step.

The Jacobi determinant (also Jacobian) is a determinant of a square matrix n × n, used to determine the behavior of the function on n variables around a given point. The matrix then contains partial derivatives of various functions to suit individual variables. Each condition has a matrix of three rows, which contains the first the collision plane orientation, and the remaining two represent the direction of the tangential forces.

**Equations of motion**

The force and torque on each segment, i, is given by:

where

**\(m_i\)**- weight of i-th element ellipsoid**\(X^*_i\)**- acceleration of gravity of the ith element of ellipsoid in the solid coordinate system,**\(F^*_i\)**- external force acting on the ith element of the ellipsoid, expressed in a fixed coordinate system,**\(M^*_i\)**- external operating moments, expressed in the internal coordinate system ellipsoid**\(\Theta_i\)**- matrix of inertia moments of ith element ellipsoid expressed in the internal coordinate system,**\(\dot{\omega}^*_i\)**- angular acceleration of the ith element in the internal coordinate system ellipsoid**\(\omega^*_i\)**- angular speed of the ith element in the internal coordinate system of the ellipsoid.

**Ellipsoid Impacts**

Ellipsoid impacts are divided into two types. The first impact is ellipsoid versus contact plane, i.e. with the vehicle body or a flat road (or polygon, etc.). The second type is ellipsoid versus ellipsoid. Though multi-body contact forces are ultimately derived from the force-based constrained dynamics model described above, the model effectively behaves as an infinitely stiff force versus deflection model given by:

where

**\(F_A\)**- the normal force during the compression phase,**\(F_S\)**- the normal force in the restitution phase,**\(S\)**- coefficient of stiffness (spring constant),**\(\epsilon\)**- spring restitution,**\(\lambda\)**- the depth of overlap.

Note for two objects undergoing collision under the “default-auto” model, the friction value used for the collision will be the minimum value of the friction values specified for either object. The restitution value used for the collision will be maximum value of the restitution values specified for the two objects. Note objects can have different friction and restitution values set for ground contact (“friction-ground” and “restitution-ground”), and for contact with other objects (“friction-body” and “restitution-body”). This makes it possible for object versus object contact to have different dynamics than object versus ground. Just as with the Kudlich-Slibar model, the default-auto model’s coefficient-of-restitution is equal to the ratio of final to initial relative velocity vectors projected on the normal collision axis for each point of contact.

*Ellipsoid versus contact plane*

As mentioned previously, ellipsoid versus plane impacts apply both to the ground plane or polygon, and the plane of the vehicle body. In a contact between an ellipsoid and vehicle mesh, the vehicle mesh is not treated as deformable. The depth of overlap is given by the distance to the intersection point level (= point of impact) from "top" submerged portion of an ellipsoid.

Again, the constrained dynamics model effectively behaves as an infinitely stiff force deflection model:

where

**\(F_n\)**- is the normal contact force,**\(F_t\)**- tangential contact force,**\(\lambda\)**- is the depth of overlap,**\(\mu\)**- is the coefficient of friction,

*Ellipsoid versus ellipsoid*

This type of contact is more complicated because it involves the mutual intersection of two ellipsoids.

In the limit of infinite stiffness, the effective behavior of the force response is described by:

where

**\(F_n\)**- normal contact force,**\(F_t\)**- tangential contact force,**\(\lambda\)**- depth of overlap,**\(\mu_C\)**- coefficient of friction,

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