CONTROL OF A ONE RIGIT-LINK MANIPULATOR IN THE CASE OF NONSMOOTH TRAJECTORY

Abstract

Mathematical model of a single-link manipulator is considered. It describes the motion of the manipulator along a given path which cam be nonsmooth. Integral manifolds method is used for the system order reduction and design of control laws. The simplest model of the manipulator consists of one link. In this case the equations of the manipulator motions are

Keywords

• single-link manipulator
• integral manifold
• singular perturbances

-

1 ̈1 +

sin 1 + ( ̇1 − ̇ ) + ( 1 − ) = 0, ̈ − ( ̇1 − ̇ ) − ( 1 + ) = , ( 1 ) where: –is the motor inertia; 1 –is the link inertia; –is the link mass, –is the link length; –is the damping coefficient; –is the stiffness; 1 –is the link angle;

–is the r angle, and is the torque input which is the controller.

The use of the small parameter =

and new variables 1 = 1 1+ , 2 = ̇1, 1 = 1 − , 2 = ̇1, 1 √ ( 2 ) ( 3 ) ( 4 ) yields the system ̇1 = 2, ̇2 =

sin ( 1 +

1+ ) 1 − 1) + ( 1 1+ + 1 ) 2 − 2 This system is singularly perturbed with slow subsystem ( 3 ) and fast subsystem ( 4 ). Neglecting all terms of order ( 2) in the right hand side of the last equation the independent subsystem is obtained. ̇1 = 2, ̇2 = − ( +

) 1 − Solutions of which are characterized by high frequency ( 1 polynomial 2 2 + с ( with complex zeros 1 1 1 + 1 2 and relatively slow decay

( 1 ), since this differential system has a characteristic 1 1 + ) + (

+ 1 1 1

) 1

( 2 1 1,2 = − + ) ± √( +

) − 2 2 1 1 1 2

( 4 1 1 + 1 ) 2 Since the real part of these numbers is negative, slow invariant manifold can be used for the analysis of the manipulator model under consideration noting that this manifold is attractive and the reducibility principle holds [1]. Note that the Jacobian matrix of fast subsystem ( 4 ) for = 0 has eigenvalues on the imaginary axis with nonvanishing imaginary parts. This means that we have so called critical case [1, 6]. A similar case and some other critical cases have been investigated in [7-12]. Slow integral manifold The terms of ( 2) of the fast subsystem ( 4 ) leads us to conclude that the slow integral manifold may be found in the form 1 = 2 + ( 3) and 2 = ( 3), where Here we used the representation = 0 + 2 1 + ( 3).

0] ( 1 + 1 −1

) The flow on this manifold is described by equations: ̇1 = 2, ̇2 = − Note that due to ( 2 ) 1 is expressed through new variables 1 = 1 + where

+ 1 1, 1 = 2 + ( 3) Control function what allows presenting the system ( 5 ) on the slow integral manifold as ̈1 − 2 + 1 ̈ = − not use a fast term added to the control input to make the fast dynamics asymptotically stable to guarantee the fast decay of fast variables 1, 2. We use the slow component of the control function which is written as a sum. 0 = ( 1 + ) +

sin 1, where order ( 2)

= ̈ − 1( 1 + ) − 2( ̇1 + ̇ ).

Setting = 0, using ( 7 ) and the definition of 0 and we obtain to an accuracy of ̈1 − ̈ + 2( ̈1 + ̈ ) + 1( 1 + ) = 0 ( 5 ) ( 6 ) ( 7 ) ( 8 ) for the difference 1 − , since 1 = 1 + ( 3) on the slow integral manifold. Equation ( 8 ) gives the possibility to choose coefficients in the control function in such a way that the corresponding control function gives the possibility of realizing a desired trajectory. Assume, for example [4, 5], 1, 100 , 1, 11 , 1, 9.8, 2 . Setting 1=3, 2=4 for the desired trajectory = sin we obtain the following control law for the original variables = 2 + 9.8 sin( 1) = 2[− sin − 4( ̇1 − cos ) − 3( 1 − sin )] + 9.8 sin( 1) It is illustrated in fig. 2 that the trajectory of controlled single-link manipulator tends to the desired trajectory as t increases. In many cases manipulators describe nonsmooth paths, polygonal lines, for example. It is impossible to use the integral manifolds method to construct approximations of slow integral manifold as an expansions in powers of the small parameter. One possibility is the use of polynomial smoothing. It is illustrated in fig. 3, which contains the response of the controlled single-link manipulator, that the trajectory tends to the desired polygonal trajectory to the approximation of which is

, 0 < < − 1 = { ( − 1)4 + ( − 1)2 + 1, − 1 < < + 1

− + 2, + 1 < < 2 Conclusion The manipulator model describing the manipulator motion in a nonsmooth path is considered. Integral manifold method is used for the system order reduction. Acknowledgment This work is supported in part by the Russian Foundation for Basic Research (grant 14-01-97018-p) and the Ministry of Education and Science of the Russian Federation under the Competitiveness Enhancement Program of Samara University (2013–2020).

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