Mathematical model of a single-link manipulator is considered. It describes the motion of the manipulator in the case of non-smooth path. Interpolation of the trajectory of motion is used, which makes it possible to reduce the amount of calculations and allows you to take into account the restrictions on the movement of the manipulator. Integral manifold method is used for the system order reduction. As a result, the reduced system of the investigated object is obtained, and the control function for the manipulation robot model in the case of a non-smooth periodic trajectory is constructed.
) 2 − 2 1 sin ( 1 +
1+ 1) − 2 .
This system is singularly perturbed with slow subsystem (
)/2, equation problems, this method was considered in [4-7].
1 ̈1 +
sin 1 + ( ̇1 − ̇ ) + ( 1 − ) = 0, ̈
− ( ̇1 − ̇ ) − ( 1 + ) = ,
The equations of motion of a single-point manipulator have the form [7-8]: image of the single-link manipulator is presented.
1+
1 = 1 1+ , 2 = ̇1, 1 = 1 − , 2 = ̇1,
Then system (
1+ ) 1 − 1) + ( 1 + 1
Variables in the system are changed in the following manner: the right hand side of the last equation the independent subsystem is obtained. ̇1 = 2, ̇2 = − ( ) 1 −
) 2, 1+ 1 +
, 1 ( 1 1 1 1 +
1 1 1
) and
differential 1 2 2 + с ( + ) + (
+ with complex roots 1,2 = −
( 2 1 1 + ) ± √( 1 + 1 ) − 2 2 ( 1 + 4 1
)
3. Integral manifold construction
accuracy of ( 3), 1= 2 + ( 3) и 2 = ( 3)
(
1 = − [
sin( 1) +
0] (
1
+
)
−1
(
Mathematical Modeling / N. Aksenova, V. Sobolev
Movement on the manifold is described by the following equations ̇1 = 2, ̇2 = − sin ( 1 + 2 ) + 0+ 2 1 + ( 3)
1+ 1+ 1+
Manipulator angular displacement q1 is expressed using new variables
1 = 1 + + 1 1,
where 1 = 2 + ( 3). This allows to rewrite the system (
Let ( ) be the required trajectory of the manipulator movement. Slow control function term is in the form
0 = ( 1 + ) + sin 1, где = ̈ − 1( 1 + ) − 2( ̇1 + ̇ ). Using (
Equation (9) gives the possibility to select control function coefficients in such a way that the relevant control affords to achieve the required trajectory. Assume, for instance, 1, 100 , 1, 1 1 , 1, 9.8, 2 , at that 1=3, 2=4, and the required trajectory is of the form = sin , then we obtain the following original variables control law = 2 + 9.8 sin( 1) = 2[− sin − 4( ̇1 − cos ) − 3( 1 − sin )] + 9.8 sin( 1)
The first stage of the control construction is to determine the desired trajectory of motion of the manipulator in the form of some analytically described function. In most cases, the manipulators do not move along smooth trajectories, so that its trajectory is a sectionally smooth line. For smoothing the interpolation of the chosen trajectory is used by polynomials of a certain class approximating the segments of the desired trajectory of the manipulation robot between the node points (for example, lines, arcs, parabolas, etc.). But there is a possibility that there will be a problem associated with the difficulty of calculating a polynomial of high degree. In this regard, to perform interpolation of the trajectory from the given nodal points, it is necessary to choose polynomials of low degrees or to break the trajectory of the manipulator's movement into separate sections.
In Fig. 1 there is a displacement-time diagram in case the required path is written as
, 0 < < − 1 = { ( − 1)4 + ( − 1)2 + 1, − 1 < < + 1
− + 2, + 1 < < 2 Fig. 1. Trajectory .
When the trajectory is substituted in the system of equations of motion of the manipulation robot (
The object of research is a manipulator model describing the manipulator motion in a non-smooth path. The interpolation of the trajectory of motion by polynomials is used that approximates the segments of the desired trajectory of the manipulation
Mathematical Modeling / N. Aksenova, V. Sobolev robot between the nodal points, which makes it possible to reduce the amount and time of calculations, and allows us to take into account the restrictions on the movement of the manipulator. Integral manifold method is used for the system order reduction.
As a result of the work done the reduced system of the object is obtained and the control function for a diagrammatic formulation of the manipulator model motion. It is established that manifold control provides the motion of the system along the trajectory near to the effective one.
This study was supported by the Russian Foundation for Basic Research and Samara region (grant 16-41-630524-p) and the Ministry of Education and Science of the Russian Federation as part of a program of increasing the competitiveness of SSAU in the period 2013–2020.