Models and numerical algorithms for tire dynamics and hydraulics for realtime vehicle simulation has been developed and given a design to be interoperable and interchangeable with other model approaches and simulation tools.

# Tire dynamics

Tire dynamics is fundamental to the simulation of off-road vehicle. For heavy machinery, the tire deformation are large, the geometry of the ground is irregular, and the soil deformation is significant and complex. Simple formulae are not sufficient in this case and one needs to model and simulate the dynamics of the tire itself with discrete or finite element models. The focus here is to formulate a model which can capture as many of the relevant tire deformation modes as possible, with the minimum number of discrete elements.

The model prototype shown in the pictures uses discrete-element represented as rigid bodies which integrates directly with the rest of the vehicle model. The current model uses spheres which are connected to each other and to the axle with stiff nonlinear translational and torsional forces. It already works in real time with over 100 elements. By contrast, common models use radial forces only. Having two layers of elements per wheel allows to capture camber and slip deformation as well as compression. We intend to reduce the number of elements by using computational geometry to interpolate the shape of the tire between the elements and provide a smooth ride. A simple nonlinear soil model is also being developed.

# Hydraulics

A significant portion of the dynamics of heavy machinery is due to the fluid power systems, and it is necessary to simulate that for effective operator training in virtual environments. But fluid power systems are rarely simulated with the same models and methods as multibody systems which describe the vehicle. Coupling different simulation tools often leads to instabilities and we therefore developed a model of hydraulic systems which is compatible with the equations of motion and numerical methods for multibody systems.

This is based on analytic system dynamics in which each subsystem is represented with a Lagrangian function and pseudo potentials, respectively, \(\mathcal{L}_i(q_i, \dot{q}_i)\) and \(\mathfrak{R}_i(q_i,\dot{q}_i)\), where \(q_i\) and \(\dot{q}_i\) are the generalized coordinates and velocities of each subsystem \(i=1,2,\ldots,n\). Couplings between subsystems usually correspond to boundary conditions of other types of kinematic constraints, i.e., constraints of the form \(g(q_i, \dot{q}_i, q_j, \dot{q_j})\geq 0 \) for the general case. The correspondance here is that hydraulic pipes are like point particles in one dimension, couplings are kinematic constraints, and connectors introduce dissipation. Hydraulic pistons then impose boundary conditions which couple rigid bodies to the fluid. The contact mechanics used in the rigid body system now enters as check, relief, and load sensing valves. The models were previously investigated within a Master’s thesis project, but a full implementation is now being carried out. As multibody systems are naturally represented with the Lagrangian formulation, the entire system dynamics is determined via D’Alembert’s principle, namely:

\[ \delta \int_0^T \text{d}s \mathcal{L}(q,\dot{q}) + \int_0^T \text{d }s\delta q \frac{\partial \mathfrak{R}(q,\dot{q})}{\partial \dot{q}} \geq 0 \]

Using discrete-time variational mechanics, it is possible to construct a time stepping method which is both fast and stable.