Page 1 of 12
Transactions on Machine Learning and Artificial Intelligence - Vol. 9, No. 4
Publication Date: August, 25, 2021
DOI:10.14738/tmlai.94.10523. Alpatov, A., Kravets, V., Kravets, V., & Lapkhanov, E. (2021). Representation of the Kinematics of the Natural Trihedral of a Spiral- Helix Trajectory by Quaternion Matrices. Transactions on Machine Learning and Artificial Intelligence, 9(4). 18-29.
Services for Science and Education – United Kingdom
Representation of the Kinematics of the Natural Trihedral of a
Spiral-Helix Trajectory by Quaternion Matrices
Anatolii Alpatov
The Institute of Technical Mechanics of the
National Academy of Sciences of Ukraine and the
State Space Agency of Ukraine, Dnipro, Ukraine
Victor Kravets
Department of Mechanics
Dnipro University of Technology, Dnipro, Ukraine
Volodymyr Kravets
Department of Mechanics
Dnipro State Agrarian and Economic University, Dnipro, Ukraine
Erik Lapkhanov
The Institute of Technical Mechanics of the
National Academy of Sciences of Ukraine and the
State Space Agency of Ukraine, Dnipro, Ukraine
ABSTRACT
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.
Keywords: Quaternion matrices; hodograph; spiral-helix trajectory, natural trihedral,
kinematics, curvature, torsion, Frenet- Serret formulas.
INTRODUCTION
In the process of dynamic design, experimental and natural development, transport crews,
including unmanned vehicles, are required to ensure controllability and maneuverability,
stability and stabilization, orientation and navigation, reliability and safety [1-6]. Dynamic
problems of the class under consideration are investigated in a computational experiment
using mathematical models of nonlinear kinetics, kinematics, orientation, navigation [7-16].
The synthesis of an adequate mathematical model, adapted to computer technologies and
describing spatial transfer and rotation, is an urgent problem of applied mechanics and
Page 3 of 12
20
Transactions on Machine Learning and Artificial Intelligence (TMLAI) Vol 9, Issue 4, August - 2021
Services for Science and Education – United Kingdom
(4)
where:
,
For a given hodograph of the programmed motion, it is required to find the components of the
angular velocity vector of rotation of natural trihedral of the spiral-helix trajectory in its own
relative axes :
, (5)
( ) ( )
( ) ( )
( ) ( )
1 1 33 22 1 22 33 23 32
2 2 3 3 11 2 11 3 3 31 13
3 3 2 2 11 3 11 2 2 12 21
0 0 0
11 1 , ,, r r rr rr r rr rr rr rr
n b
r rr r r r r r rr rr r rr rr rr rr
r r r r rr r rr r r rr r r
+ - + - t = = - = ́ ́ + - + -
+ - + -
! !! ! ! ! ! ! !! ! !! ! ! !! ! !!
! ! ! !! ! !! ! !! ! ! ! ! ! !! ! !! ! ! !! ! !!
! !! ! ! ! ! ! !! ! !! ! ! !! ! !!
1 0123 2
2 3
2 0123 2
2 3
3 0123
2
2
2
1 0123 2 2
2 2 2
2
0 1
1
cos sin ; 2
3
0 1
1
sin cos ; 2
3
0
1
; 2
3
0
1
cos 2 sin ; 2 2
3 6
t
r tt
t t
t t
t
r tt
t t
t t
r h hh h
t
t
t
r tt
t t
t t t
r
é ù
ê ú
= r r r r w -w w
ë û
é ù
ê ú
= r r r r w + w w
ë û
=
é ù -w ê ú
-w = r r r r w- w w
- w
ë û - w
= r
!
!
!
!!
!!
2
2
0123 2 2
2 2 2
3 0123
0
1
sin 2 cos ; 2 2
3 6
0
0 . 2
6
t
t t
t t
t t t
r h hh h
t
é ù -w ê ú
-w
r r r w- w w
- w
ë û - w
!! =
M nb t
n b n b W = t tW + W + W