Purpose. To improve both reliability and engineering accuracy of 3D modeling of a vehicle movement. methodology. Orts of natural trihedral of spiral-helix supporting trajectory are expressed vectorially by means of time derivatives of a radius vector of a program movement. Quaternion matrices represent the complicated operations of vector algebra. Spatial orientation or program rotation of a natural trihedral is described with the help of Gibbs vector, Rodriguez-Hamilton vector, matrix of direction cosines depending upon kinematic parameters of the program movement.findings. A hodograph of program transfer of a vehicle is represented in the class of spiral-helix supporting trajectories in terms of earth reference. Unit vectors of a natural trihedral of the spiral-helix supporting trajectory have been obtained depending on the time derivatives of the program motion hodograph.Originality. The program transfer -radius vector hodograph and program rotation (orientation) -Gibbs vector or quaternion on Rodriguez-Hamilton parameters are represented vectorially making it possible to model spatial orientation problems and problems of control of dynamic systems (i.e. vehicles) in the form of quaternion matrices.Practical value. Calculation formulas are represented in the ordered, compacted matrix form adapted directly to computer technology. The algorithm helps solve a wide range of problems of dynamic design of vehicles.

The spiral-helix trajectory of the transport vehicle programmed motion in the form of a hodograph in the stationary frame of reference is considered. A relative frame of reference associated with the natural trihedral of the trajectory is introduced. The formulas of curvature and torsion of the trajectory, the unit vector of the natural trihedral, the components of the angular velocity of rotation of the natural trihedral in the proper axes and in the stationary frame of reference are set in the quaternionic matrices. The results are verified using the Frenet-Serret formulas. The mathematical apparatus of quaternion matrices is tested with the aim of adapting spatial, nonlinear problems of dynamic design of transport vehicles to a computational experiment.