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.

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