Abstract

The singularly perturbed differential systems which describe the dynamics of the manipulator with flexible joints are investigated under the condition of weak dissipation. The method of integral manifolds is used to construct the reduced model of robot. Integral manifolds may be constructed as an asymptotic power series. The simplified model is used to construct the control law for the robot with two flexible joints. The dynamic and control problems for robotic systems are connected with difficulties caused by high dimensions of models and availability of several time scales. Thereby the reduction problem ( the problem of the construction the lower order corrected models) is topical. We investigate the model of n-links robot-manipulator with flexible joints where dissipation is small. The dynamics of such manipulators is described by quasi-oscillating singularly perturbed differential systems, which contain small parameter at the leading derivative. The conditions ensuring the possibility of using classical asymptotic methods are described in the wellknown Tikhonov's theorem. The main of them is the asymptotic stability of the so-called boundary layer system. For investigated class of systems this condition of Tikhonov's theorem is not satisfied. One of the approaches, which allows to reduce the complex multirate dynamic systems, is based on the theory of integral manifolds [1-14]. The conditions of the existence of an attractive slow integral manifold are investigated. This makes it possible to use the slow subsystem, which describes the motion on the manifold, as the simplified model of the manipulator. Similar questions for other classes of quasi-oscillating systems are studied in [10-13]. We consider the dynamic model of n-links robot with flexible joints Fig. 1.

Keywords

• mathematical model
• integral manifolds
• reduction
• asymptotic methods

1. Introduction

The dynamics of manipulator is described by the system [15 – 18] D(q1)q¨1 + c(q1, q˙1) + K(q1 − q2) + B(q˙1 − q˙2) = 0,

Jq¨2 − K(q1 − q2) − B(q˙1 − q˙2) = u, (1)

2. Reduction of the model

Putting q = q1, z = k(q1 − q2) gives us the following system where

We obtained [14] the conditions for the existence of attractive slow integral manifold of system ( 3 ) where x = x1 ! x2 , y = y1 ! y2 , h(t, x, μ) = h1(t, x, μ) ! h2(t, x, μ)

.

Let function u(t, x, μ) is represented in the form u(t, x, μ) = u0(t, x) + μu1(t, x) + . . . .

Integral manifold (4) may be constructed as an asymptotic power series of the small parameter μ with any degree of accuracy. Substituting (5) to the equations q¨ = μz¨ = a1(q, q˙) + A1(q)z + μA3(q)z˙, a2(q, q˙) + A2(q)z + μA4(q)z˙ + M2u, a1(q, q˙) = a2(q, q˙) = −

D−1(q)c(q, q˙), A2(q) = − A4(q) = − (D−1(q) + J−1)Ke, (D−1(q) + J−1)B,

A3(q) = − M2 =

J−1. −

A1(q) = − D−1(q)B,

D−1(q)Ke, x˙1 x˙2 μy˙1 μy˙2 = x2, = = = μy2, a1(x) + A1(x1)y1 + μA3(x1)y2, a2(x) + A2(x1)y1 + μA4(x1)y2 + M2u(t, x, μ.

y = h(t, x, μ), yi =

hi(0)(t, x) + μhi(1)(t, x) + μ2hi(2)(t, x) + . . . , i = 1, 2 ∂h1 + ∂h1 ∂x1 x2 + ∂h1 ∂x2 ∂t ∂h2 + ∂h2 ∂x1 x2 + ∂h2 ∂x2 (a1(x) + A1(x1)h1 + μA3(x1)h2) = h2,

(a1(x) + A1(x1)h1 + μA3(x1)h2) = = a2(x) + A2(x1)h1 + μA4(x1)h2 + M2u(t, x, μ), hi = hi(t, x, μ) ∂t μ( ∂t ∂x1

Using the coordinates x1 = q, x2 = q˙, y1 = z, y2 = z˙ we can rewrite system (2) to the form and equating the coefficients at the same powers of μ we can get hi( j) = hi( j)(t, x) for any j. In particular h(0) = −A2−1(x1)[a2(x) + M2u0(t, x)], 1 h(0) = 2 ∂x1 ∂x2

∂x2 h(1) = A2−1(x1) 1 ∂h(20) + ∂h(20) x + ∂h(20) [a1(x) + A1(x1)h( 10 )] − A4(x1)h(20) − M2u1(t, x) ,

2 h(1) = 2 ∂h( 11 ) + ∂h( 11 ) x + ∂h( 11 ) [a1(x) + A1(x1)h( 10 )] + ∂h( 10 ) [A1(x1)h( 11 ) + A3(x1)h(20)].

2

∂x2 ∂t

∂x1 ∂h( 10 ) + ∂h( 10 ) x + ∂h( 10 ) [a1(x) + A1(x1)h( 10 )], 2 ∂x2 For hi( j), i = 1, 2 from ( 6 ) we have h( j) 1 h( j) 2 = − = s=0 ∂t ∂t ∂x2 3rd International conference Information Technology and Nanotechnology, ITNT-2017

A2−1(x1) M2u j(t, x) − A4(x1)h(2j−1) − Xj−2 ∂h(2s) [A1(x1)h(1j−s−1) + A3(x1)h(2j−s−2)] , ∂x1

∂x2 ∂h(1j) + ∂h(1j) x + ∂h(1j) [a1(x) + A1(x1)h( 10 )] + Xj−1 ∂h(1s) [A1(x1)h(1j−s) + A3(x1)h(2j−1−s)].

2 s=0

∂x2 ∂h(2j−1) ∂t − ∂h(2j−1) ∂x1 x2 − ∂h(2j−1) ∂x2 [a1(x) + A1(x1)h( 10 )] − (2) ( 3 ) (4) (5) ( 6 ) ( 7 ) The system, which describes the motion on the slow integral manifold, is The dimension of this system is half of the dimension of the initial system. The slow subsystem has not fast variables, but nonetheless reliably describes the behavior of full system near the slow integral manifold. This allows to use it as a simplified model of flexible joints robot. The proposed approach to construct the reduced model is used in [11 – 13] to solve the problems of control and estimation for robot with one flexible joint.

3. Example

Let us consider the control problem for the robot with two flexible joints. The dynamics of manipulator is described by system (1), where [18]

D(q1) = θ1 + θ2 + 2θ3 cos ϕ2 θ2 + θ3 cos ϕ2 θ2 + θ3 cos ϕ2 ! θ2 , q1 = ϕ1 ! ϕ2 , q2 = ψ1 ! ψ2 , θ1 = m1lc21 + m2l12 + I1, θ2 = m2lc22 + I2, θ3 = m2l1lc2 , θ4 = m1lc1 , θ5 = m2l1, θ6 = m2lc2 , c(q1, q˙1) = θ3 sin ϕ2 −2ϕ˙ 1ϕϕ˙˙212 − ϕ˙ 22 ! + (θ4 + θ5)g cos ϕ1 + θ6g cos(ϕ1 + ϕ2) ! θ6g cos(ϕ1 + ϕ2) , y = , Using the coordinates x1 = q, x2 = q˙, y1 = z, y2 = z˙ we rewrite system (2) to the form ( 3 ), where a1(x) = a2(x) = (−θ3 sin x2(1)(2x1(2) + x2(2))x2(2) + θΔ4g cos x1(1) + θ6g cos(x1(1) + x2(1)))  −θ2  +  (θ2 + θ3 cos x2(1))  + (θ3 sin x2(1)(x1(2))2 + θ6g cos(x1(1) + x2(1))) Δ

(θ2 + θ3 cos x2(1)) −(θ1 + θ2 + 2θ3 cos x2(1)) ! ,

Δ = θ1θ2 − θ32 cos2 x2(1), −θ2 θ2 + θ3 cos x2(1)

θ2 + θ3 cos x2(1) −(θ1 + θ2 + 2θ3 cos x2(1)) !

, −θ2 (θ2 + θ3 cos x2(1))

J1 J2(θ2 + θ3 cos x2(1)) −(θ1 + θ2 + 2θ3 cos x2(1)) !

, k ΔJ1 J2

−J2(θ2 J1 + Δ) J1 J2(θ2 + θ3 cos x2(1)) J1 J2(θ2 + θ3 cos x2(1)) J1(J2(θ1 + θ2 + 2θ3 cos x2(1)) + Δ) ! , A4(x) = ΔJb1 J2 J1 J2−(θJ22(+θ2θJ31c+osΔ(x)2(1))) J1(J2(θJ11 J+2(θθ22 ++2θθ33ccoossxx2(12(1)))) + Δ) ! , M2 =  −0J11 −0J12  .

The slow integral manifold (4) takes the form (5), where coefficients h(ji,)k are obtained from ( 7 )by using the computer algebra system Maple. In particular h(0) 1,1 h(0) 1,2 = − + =

1 − kS (u( 10 )(θ1θ2 + J2θ1 + J2θ2 − θ32(cos x2(1))2 + 2J2θ3 cos x2(1)) + u(20) J1(θ2 + θ3 cos x2(1)) − J1θ32(x1(2))2 cos x2(1) sin x2(1) − J1θ3 sin x2(1)(θ2(x1(2) + x2(2))2 + J2 x2(2)(2x1(2) + x2(2))) + J1g(θ4 + θ5) cos x1(1)(θ2 + J2) + cos(x1(1) + x2(1))J1gθ6(J2 − θ3 cos x2(1))),

1 − kS (u( 10 ) J2(θ2 + θ3 cos x2(1)) + u(20)(θ1θ2 + J1θ2 − θ32(cos x2(1))2) − J2g(θ4 + θ5) cos x1(1)(θ2 + θ3 cos x2(1)) + 3rd International conference Information Technology and Nanotechnology, ITNT-2017 3rMdaItnhteemrnaattiicoanlaMlcoodnefleinregn/cOe.“VIn.fVoridmilaitnioa,nNT.eVc.hVnoolroogpyaeavnad Nanotechnology 2017” x1(2), 1 S 1

S The reduced system ( 8 ) takes the form (u( 10 )(θ2 + J2) − u(20)(θ2 + θ3 cos x2(1)) − g(θ4 + θ5) cos x1(1)(θ2 + J2) + θ32 cos x2(1) sin x2(1)(x1(2))2 + (u( 10 )(θ2 + θ3 cos x2(1)) − u(20)(θ1 + θ2 + 2θ3 cos x2(1) + J1) − cos x1(1)(θ4 + θ5)g(θ2 + θ3 cos x2(1)) + J2gθ6 cos(x1(1) + x2(1))(θ1 + J1 + θ3 cos x2(1)) + J2θ32 cos x2(1) sin x2(1)(2x1(2) x2(2) + (x2(2))2 + 2(x1(2))2) + J2θ3 sin x2(1)((θ1 + θ2)(x1(2))2 + 2θ2 x1 2

(2) x(2) + θ2(x2(2))2 + J1(x1(2))2)), (J2θ1 + θ1θ2 + J1θ2 + J2θ2 + J1 J2 − θ32(cos x2(1))2 + 2J2θ3 cos x1(1)). 0.8 0.6 0.4 0.2

0

4. Conclusion

Acknowledgements References

x2 –0.8

0 –0.2 –0.4 –0.6

–1 –1.2 –1.4 –1.6 2 4 t 6 8 10

It can be seen that the trajectories of initial system, which is characterized by damped high-frequency oscillations tend to the trajectories of reduced system, and those tend to the required fixed positions.

The application of the integral manifolds method allows us to reduce the dimension and simplify the problem of the control law construction.

The research has been supported by the Russian Foundation for Basic Research and Government of the Samara region(grant 16-41-630524). [1] Sobolev VA. Integral manifolds and decomposition of singularly perturbed systems. Syst. & Control Lett. 1984; 5: 169–279. [2] Voropaeva NV, Sobolev VA. Geometric decomposition of singularly perturbed systems. Fizmatlit: Moscow, 2009. [in Russian] [3] Shchepakina E, Sobolev V, Mortell MP. Singular Perturbations: Introduction to System Order Reduction Methods with Applications. In: Springer Lecture

Notes in Mathematics, Cham: Springer International Publishing, 2014. [4] Voropaeva NV, Sobolev VA. Decomposition of a linear-quadratic optimal control problem with fast and slow variables. Automation and Remote Control 2006; 67( 8 ): 1185-1193. [5] Voropaeva NV. Decomposition of problems of optimal control and estimation for discrete systems with fast and slow variables. Automation and Remote

Control 2008; 69( 6 ): 920-928.

References

[6] Sobolev VA . Singular perturbations in linearly quadratic optimal control problems . Automation and Remote Control 1991 ; 52 ( 2 ): 180 - 189 .

[7] Vidilina OV , Voropaeva NV . The construction of the observers for dynamic systems with fast and slow variables . CEUR Workshop Proceedings , 2016 ; 1638 : 750 - 758 .

[8] Smetannikova EN , Sobolev VA . Regularization of cheap periodic control problems . Automation and Remote Control 2005 ; 66 ( 6 ): 903 - 916 .

[9] Strygin VV , Sobolev VA . Effect of geometric and kinetic parameters and energy dissipation on orientation stability of satellites with double spin . Cosmic Research 1976 ; 14 ( 3 ): 366 - 371 .

[10] Strygin VV , Sobolev VA . Separation of motions by the method of integral manifolds . Nauka: Moscow , 1988 . [in Russian]

[11] Osintsev MS , Sobolev VA . Dimensionality Reduction in Optimal Control and Estimation Problems for Systems of Solid Bodies with Low Dissipation . Automation and Remote Control 2013 ; 74 ( 8 ): 121 - 137 .

[12] Mortell MP , O'Malley R , Pokrovskii A , Sobolev V . Singular Perturbation and Hysteresis . SIAM: Philadelphia, 2005 .

[13] Aksenova NK , Sobolev VA . Control of a one rigit-link manipulator in the case of nonsmuth trajectory . CEUR Workshop Proceedings 2016 ; 1638 : 493 - 497 .

[14] Vidilina OV , Voropaeva NV . Reduction of mathematical model of robot with elastic joints . Vestnik SamGU. Estestvennonauchnaya seriya . 2014 ; 3 ( 114 ): 16 - 29 .[in Russian]

[15] Spong MW . Modeling and control of elastic joint robots . Journal of Dynamic Systems, Measurement and Control 1987 ; 109 : 310 - 319 .

[16] Spong MW , Khorasani K , Kokotovic PV . An integral manifold approach to feedback control of flexible joint robots . IEEE Journal of Robotics and Automation 1987 ; 3 ( 4 ): 291 - 301 .

[17] Moberg S. On modeling and control of flexible manipulators . Linkoping University: Linkoping, 2007 .

[18] Spong MW . On the robust control of robot manipulators . IEEE Trans. Automatic Control 1992 ; 37 ( 11 ): 1782 - 1786 .

3rd International conference Information Technology and Nanotechnology, ITNT-2017 3rd International conference “Information Technology and Nanotechnology 2017 ”