International IRATJ

Robotics & Automation Journal
Review Article
Volume 2 Issue 1 - 2017
About Mathematical Models of System Dynamics with Geometric Constraints in Problems of Stability and Stabilization by Incomplete State Information
Krasinskiy Alexandr Ya*, Esfira M Krasinkaya and Anastasiya N Ilyina
Moscow Aviation Institute, Moscow State technical university BMSTU, Russia
Received: September 12, 2016 | Published: February 06, 2017
*Corresponding author: Krasinskiy Alexandr Ya, Moscow Aviation Institute, Moscow State Food University Moscow, Russia, Email:
Citation: Alexandr KY, Krasinkaya EM, Ilyina AN (2017) About Mathematical Models of System Dynamics with Geometric Constraints in Problems of Stability and Stabilization by Incomplete State Information. Int Rob Auto J 2(1): 00007. DOI: 10.15406/iratj.2017.06.00007

Abstract

This article is devoted to constructing equations of motion for Holonomic systems using Lagrangian variables. We introduce the form of equations which allow for detailed analyses of both linear and non-linear terms of perturbed motion equations. Steady motion stability of systems with redundant coordinates is only possible in critical cases; e.g. when the characteristic equation has roots whose real parts are zero. In this case, it is imperative to analyze the nonlinear terms of the characteristic equation to solve stability issues. We suggest a rigorous method of solving stability problem for systems with geometric constraints. The method is based on analytical mechanics, theory of critical cases, nonlinear stability theory, and N.N. Krasovsky’s procedure of solving linear-quadratic problems. This grants the ability to make reasonable conclusions regarding system stability and calculations and calculate coefficients of the linear stabilizing control.

Keywords: Geometric constraints; Homonymic systems; Lagrangian variables; Mechatronic Systems; Routh variables; Shulgins equations

Introduction

Currently, mainstream classes of technical devices consist of controlled mechanical systems with microcontrollers in the control loop. It is possible for such a system to implement, in real time, control algorithms of almost any complexity. The considerations of control problems for modern technical facilities are comprised of three main steps. Firstly, the program control must be defined. This de-termines an object’s mode of operation (the dynamics of concrete actuating devices should be taken into account in the framework of the adopted model). Secondly, as perturbations inevitably exist, control is required to stabilize the system. The control can be constructed as a linear function of the state variables. However, the physical state of the system is not wholly determinable by direct observation; it is often technically impossible or inefficient, as each applied sensor raises the cost of the system. Therefore, thirdly, it is required to construct a system that provides an estimate of the system’s internal state. This is one of the most important stages to solve many problems in control theory and provides an essential foundation for many practical applications. Suitable and accurate nonlinear mathematical models have fundamental importance at every stage of scrutiny regarding modern technical problems, especially concerning stability and stabilization problems for systems lacking complete state information. Such models are the only reliable basis for the development of rigorous, effective, and more simplistic methods for practical applications. For example, it allows for the possibility of using the characteristics of object’s proper motion (without application of additional actions) to reduce the amount of operating actuators and measurement information.

Analytical mechanics are one of the most formalized and developed areas of applied mathematics, and as such, it provides numerous alternatives for the prevailing choices of modeling methods [1-3]. Generalized coordinates are traditionally used in the study of mechanical systems dynamics. These generalized coordinates are the parameters (in a minimum number) that uniquely define the system’s configuration. While there are many choices for system coordinates, it is possible that a useful set of coordinates may be wholly or partially dependent, which means that they are linked by one or more constraint equations. In this case, we cannot directly use Lagrange’s equations of motion, as it requires all available coordinates to be independent. This is a commonly encountered problem for multilink manipulators and many other Mechatronic systems in the presence of geometric constraints [4,5].

It is efficient to describe the configuration of a Mechatronic system with m geometric constraints by utilizing n + m pa-rameters taken in the number exceeding the minimum required number n. These m parameters are then called redundant coordinates. Detailed analyses of different forms of equations of motion were performed [5] for systems with geometric constraints. It was shown that the application of equations of motion in M.F. Shul’gin’s form [6] can, quite simply, give rigorous nonlinear mathematical models for systems with redundant coordinates. However, the simplicity of the model is not merely dependent on the chosen coordinates and the form of applied equations of motion.

The type of available variables (Lagrange, Hamilton, or Routh’s) [1] are also critically important. The application of Routh’s variables simplifies [7-9] the determination of stabilizing control coefficients, yet remain very disadvantage [8] in the case of asymptotic stabilization by all variables in case of incomplete state information. Knowing the perturbations of momenta corresponding to cyclic coordinates is imperative for constructing the control in the case Routh’s variables, but they cannot be determined by direct measurements. Lagrangian variables are useful in terms of reducing the amount of measuring information as generalized velocities can be determined directly by measuring devices. But the application of Lagrangian variables substantially complicates the determination of stabilizing control coefficients and makes it difficult to undertake an analysis of equations. In this case, the number of characteristic equation zero roots is more than the number of geometric constraints. Thus, the structure of nonlinear closed loop system (in the special form of critical cases theory [10,11]) is clear in Routh’s variables but is difficult for analyses in Lagrangian variables. As the first step (determination of the stabilizing control coefficients in Routh’s variables) was studied in detail [8,9,12], the main idea of this article is constructing the nonlinear model of systems dynamics in Lagrangian variables in the presence of cyclic coordinates. Then a theorem on solvability of the stabilization problem for stationary motion is proved in case of incomplete state information.

MF Shulgins Equations

Suppose configuration of a mechanical system is described by parameter q 1 ;;;;;  q n+m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaCaaabeqcfasaaiaaigdaaaqcfaOaai4oaiaacUdacaGG7aGaai4o aiaacUdaqaaaaaaaaaWdbiaacckapaGaamyCamaaCaaabeqcfasaai aad6gacqGHRaWkcaWGTbaaaaaa@422E@  where n is the number of the system degrees of freedom. Let the system configuration is limited by m independent relations (geometric constraints).

F k ( q 1 ,... q n+m )=0;   k =  1,m ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeOram aaBaaajuaibaGaam4AaaqcfayabaWaaeWaaeaacaWGXbWaaSbaaKqb GeaacaaIXaaabeaajuaGcaGGSaGaaiOlaiaac6cacaGGUaGaamyCam aaBaaajuaibaGaamOBaiabgUcaRiaad2gaaKqbagqaaaGaayjkaiaa wMcaaiabg2da9iaaicdacaGG7aaeaaaaaaaaa8qacaGGGcGaaiiOai aacckacaGGRbGaaiiOaiabg2da9iaacckadaqdaaqaaiaaigdacaGG SaGaamyBaaaaaaa@5147@ (1)
det [ F 1 ,.. F m ] [ q n+1 ,...,. q n+m ]    0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeizai aabwgacaqG0bWaaSaaaeaacqGHciITdaWadaqaaiaadAeadaWgaaqc fasaaiaaigdaaKqbagqaaiaacYcacaGGUaGaaiOlaiaadAeadaWgaa qcfasaaiaad2gaaKqbagqaaaGaay5waiaaw2faaaqaaiabgkGi2oaa dmaabaGaamyCamaaBaaajuaibaGaamOBaiabgUcaRiaaigdaaKqbag qaaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaaiOlaiaadghadaWg aaqcfasaaiaad6gacqGHRaWkcaWGTbaajuaGbeaaaiaawUfacaGLDb aaaaaeaaaaaaaaa8qacaGGGcGaaiiOaiabgcMi5kaacckacaaIWaaa aa@5A4D@

Without loss of generality coordinates q n+1 ......  q n+m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaCaaabeqcfasaaiaad6gacqGHRaWkcaaIXaaaaKqbakaac6cacaGG UaGaaiOlaiaac6cacaGGUaGaaiOlaabaaaaaaaaapeGaaiiOa8aaca WGXbWaaWbaaeqajuaibaGaamOBaiabgUcaRiaad2gaaaaaaa@4474@ can be considered as redundant. Introduce some vectors (the prime denotes transposition):

r  =( q 1 ,..., q n ) s  =( q n+1 ,..., q n+m ) q ( r , s ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabKOCay aafaaeaaaaaaaaa8qacaqIGcGaaKypamaabmaabaGaaKyCamaaBaaa juaibaGaaKymaaqcfayabaGaaKilaiaaj6cacaqIUaGaaKOlaiaajY cacaqIXbWaaSbaaKqbGeaacaqIUbaabeaaaKqbakaawIcacaGLPaaa caqISaGaaKiiaiqajohagaqbaiaajckacaqI9aWaaeWaaeaacaqIXb WaaSbaaKqbGeaacaqIUbGaaK4kaiaajgdaaeqaaKqbakaajYcacaqI UaGaaKOlaiaaj6cacaqISaGaaKyCamaaBaaajuaibaGaaKOBaiaajU cacaqITbaajuaGbeaaaiaawIcacaGLPaaacaqISaGaaKiOaiqajgha gaqbaiaaj2dacaqIGcWaaeWaaeaaceqIYbGbauaacaqISaGabK4Cay aafaaacaGLOaGaayzkaaGaaKilaaaa@6055@
F =( F 1 ,..., F m ); MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabKOray aafaGaaKypamaabmaabaGaaKOramaaBaaajuaibaGaaKymaaqcfaya baGaaKilaiaaj6cacaqIUaGaaKOlaiaajYcacaqIgbWaaSbaaKqbGe aacaqITbaabeaaaKqbakaawIcacaGLPaaacaqI7aaaaa@4321@

Suppose the system is affected by potential forces with energy_ and no potential forces Qr; Qs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyuai ablYJi6iaadkhacaGG7aaeaaaaaaaaa8qacaGGGcGaamyuaiablYJi 6iaadohaaaa@3E74@ corresponded to coordinates; s and L = Τ_ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWI8iIocaWGmbGaaiiOaiabg2da9iaacckacqWI8iIocqqH KoavcqWIHwYvcaGGFbaaaa@412C@ is Lagrange function. The general view Of the kinetic energy is:

  T ˜ = T ˜ 2 + T ˜ 1 + T ˜ 0 = 1 2 q ˙ ' a ˜ ( q ) q ˙ + d ˜ ' ( q ) q ˙ + T 0 ( q ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabKivay aaiaGaaKypaiqajsfagaacamaaBaaajuaibaGaaKOmaaqcfayabaGa aK4kaiqajsfagaacamaaBaaajuaibaGaaKymaaqcfayabaGaaK4kai qajsfagaacamaaBaaajuaibaGaaKimaaqabaqcfaOaaKypamaalaaa baGaaKymaaqaaiaajkdaaaGabKyCayaacaWaaWbaaeqabaGaaK4jaa aaceqIHbGbaGaadaqadaqaaiaajghaaiaawIcacaGLPaaaceqIXbGb aiaacaqIRaGabKizayaaiaWaaWbaaeqabaGaaK4jaaaadaqadaqaai aajghaaiaawIcacaGLPaaaceqIXbGbaiaacaqIRaGaaKivamaaBaaa juaibaGaaKimaaqabaqcfa4aaeWaaeaacaqIXbaacaGLOaGaayzkaa aaaa@55AE@  (2)
a ˜ ( q )=( a ˜ rr a ˜ rs a ˜ sr a ˜ ss ) ;   d ˜ ( q ) = ( d ˜ r d ˜ s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabKyyay aaiaWaaeWaaeaacaqIXbaacaGLOaGaayzkaaGaaKypamaabmaabaWa aSbaaeaafaqabeGacaaabaGabKyyayaaiaWaaSbaaKqbGeaacaqIYb GaaKOCaaqcfayabaaabaGabKyyayaaiaWaaSbaaKqbGeaacaqIYbGa aK4CaaqcfayabaaabaGabKyyayaaiaWaaSbaaKqbGeaacaqIZbGaaK OCaaqcfayabaaabaGabKyyayaaiaWaaSbaaKqbGeaacaqIZbGaaK4C aaqcfayabaaaaaqabaaacaGLOaGaayzkaaaeaaaaaaaaa8qacaqIGc GaaK4oaiaajckacaqIGcGabKizayaaiaWaaeWaaeaacaqIXbaacaGL OaGaayzkaaGaaKiOaiaaj2dacaqIGcWaaeWaaeaafaqabeGabaaaba GabKizayaaiaWaaSbaaKqbGeaacaqIYbaajuaGbeaaaeaaceqIKbGb aGaadaWgaaqcfasaaiaajohaaKqbagqaaaaaaiaawIcacaGLPaaaaa a@5E21@

Kinematic (Holonomic) constraints can be obtained by differentiating the geometric constraint equations (1) with respect to time:
F r r ˙ + F s s ˙ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcaWGgbaabaGaeyOaIyRabmOCayaafaaaaiqadkhagaGa aiabgUcaRmaalaaabaGaeyOaIyRaamOraaqaaiabgkGi2kqadohaga qbaaaaceWGZbGbaiaacqGH9aqpcaaIWaaaaa@447C@  (3)

The depended velocities vector can be expressed from (3):
s ˙ = B( q ) r ˙   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabm4Cay aacaGaeyypa0deaaaaaaaaa8qacaGGGcGaamOqamaabmaabaGaamyC aaGaayjkaiaawMcaaiqadkhagaGaaiaacckaaaa@3F39@ ; B( q )= ( F s 1 ) 1  .  ( F r 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGcbWaaeWaaeaacaWGXbaacaGLOaGaayzkaaGaeyypa0Ja eyOeI0YaaeWaaeaadaWcaaqaaiabgkGi2kaadAeaaeaacqGHciITca WGZbWaaWbaaKqbGeqabaGaaGymaaaaaaaajuaGcaGLOaGaayzkaaWa aWbaaeqajuaibaGaeyOeI0IaaGymaaaajuaGcaGGGcGaaiOlaiaacc kacaGGGcWaaeWaaeaadaWcaaqaaiabgkGi2kaadAeaaeaacqGHciIT caWGYbWaaWbaaKqbGeqabaGaaGymaaaaaaaajuaGcaGLOaGaayzkaa aaaa@5202@ (4)

Let or ( q, ˙ r ), Qs ( q, ˙ r ), L ( q, ˙ r ) = T( q, ˙ r )Π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aaqaaaaaaaaaWdbiaadghacaGGSaGaaeiiaiaacMTacaqGGaGaamOC aaWdaiaawIcacaGLPaaapeGaaiilaiaabccacaWGrbGaam4Caiaabc capaWaaeWaaeaapeGaamyCaiaacYcacaqGGaGaaiy2ciaabccacaWG YbaapaGaayjkaiaawMcaa8qacaGGSaGaaeiiaiaadYeacaqGGaWdam aabmaabaWdbiaadghacaGGSaGaaeiiaiaacMTacaqGGaGaamOCaaWd aiaawIcacaGLPaaapeGaaeiiaiabg2da9iaabccacaWGubWdamaabm aabaWdbiaadghacaGGSaGaaeiiaiaacMTacaqGGaGaamOCaaWdaiaa wIcacaGLPaaapeGaeyOeI0IaeuiOdafaaa@5EC4@  denote the no potential forces and Lagrange function after eliminating the dependent velocities using (4).

T( q, r ˙ ) =  T 2  + T 1 + T 0  = 1 2 r ˙ ' a( q ) r ˙ + d ' ( q ) r ˙ + T 0 ( q ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aabmaabaGaamyCaiaacYcaceWGYbGbaiaaaiaawIcacaGLPaaaqaaa aaaaaaWdbiaacckacqGH9aqpcaGGGcGaamivamaaBaaajuaibaGaaG OmaiaacckaaKqbagqaaiabgUcaRiaadsfadaWgaaqcfasaaiaaigda aKqbagqaaiabgUcaRiaadsfadaWgaaqcfasaaiaaicdaaKqbagqaai aacckacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiqadkhagaGa amaaCaaabeqaaiaacEcaaaGaamyyamaabmaabaGaamyCaaGaayjkai aawMcaaiqadkhagaGaaiabgUcaRiaadsgadaahaaqabeaacaGGNaaa amaabmaabaGaamyCaaGaayjkaiaawMcaaiqadkhagaGaaiabgUcaRi aadsfadaWgaaqcfasaaiaaicdaaKqbagqaamaabmaabaGaamyCaaGa ayjkaiaawMcaaaaa@5F00@ , (5)
a( q ) =  a ˜ rr  + 2 a ˜ rs B +  B ' a ˜ ss B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyyam aabmaabaGaamyCaaGaayjkaiaawMcaaabaaaaaaaaapeGaaiiOa8aa cqGH9aqppeGaaiiOaiqadggagaacamaaBaaajuaibaGaamOCaiaadk haaKqbagqaaiaacckacqGHRaWkcaGGGcGaaGOmaiabgwSixlqadgga gaacamaaBaaajuaibaGaamOCaiaadohaaKqbagqaaiaadkeacaGGGc Gaey4kaSIaaiiOaiaadkeadaahaaqabeaacaGGNaaaaiqadggagaac amaaBaaajuaibaGaam4CaiaadohaaeqaaKqbakaadkeaaaa@5535@ , d ' ( q ) =  d ˜ ' r + d ˜ ' s B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamizam aaCaaabeqaaiaacEcaaaWaaeWaaeaacaWGXbaacaGLOaGaayzkaaae aaaaaaaaa8qacaGGGcGaeyypa0JaaiiOaiqadsgagaacamaaCaaabe qaaiaacEcaaaWaaSbaaKqbGeaacaWGYbaajuaGbeaacqGHRaWkceWG KbGbaGaadaahaaqabeaacaGGNaaaamaaBaaajuaibaGaam4Caaqcfa yabaGaamOqaaaa@4703@ .

Then the equations in M.F. Shul’gin’s from [5], [6] in Lagrangian variables can be written as

d dt L r ˙    L r  =   Q r +  B ' ( q ) ( L s  +  Q s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbaabaGaamizaiaadshaaaWaaSaaaeaacqGHciITcaWGmbaa baGaeyOaIyRabmOCayaacaaaaabaaaaaaaaapeGaaiiOaiabgkHiTi aacckadaWcaaqaaiabgkGi2kaadYeaaeaacqGHciITcaWGYbaaaiaa cckacqGH9aqpcaGGGcGaaiiOaiaadgfadaWgaaqaaKqbGiaadkhaju aGcaGGGcGaey4kaSIaaiiOaaqabaGaamOqamaaCaaabeqaaiaacEca aaWaaeWaaeaacaWGXbaacaGLOaGaayzkaaGaaiiOamaabmaabaWaaS aaaeaacqGHciITcaWGmbaabaGaeyOaIyRaam4CaaaacaGGGcGaey4k aSIaaiiOaiaadgfadaWgaaqcfasaaiaadohaaKqbagqaaaGaayjkai aawMcaaaaa@6185@ ;
s ˙  = B( q )  r ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabm4Cay aacaaeaaaaaaaaa8qacaGGGcGaeyypa0JaaiiOaiaadkeadaqadaqa aiaadghaaiaawIcacaGLPaaacaGGGcGabmOCayaacaaaaa@405D@ (6)

M.F. Shul’gin’s Equations for Systems with Cyclic Coordinates

Suppose the coordinates _ are cyclic in terms [6]

F β  = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcaWGgbaabaGaeyOaIyRaeqOSdigaaabaaaaaaaaapeGa aiiOaiabg2da9iaacckacaaIWaaaaa@3FF4@ , B β  = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcaWGcbaabaGaeyOaIyRaeqOSdigaaabaaaaaaaaapeGa aiiOaiabg2da9iaacckacaaIWaaaaa@3FF0@ , T β  = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcaWGubaabaGaeyOaIyRaeqOSdigaaabaaaaaaaaapeGa aiiOaiabg2da9iaacckacaaIWaaaaa@4002@ , β  = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITcqGHpis1aeaacqGHciITcqaHYoGyaaaeaaaaaaaaa8qa caGGGcGaeyypa0JaaiiOaiaaicdaaaa@40BB@ , Q β  = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyuam aaBaaajuaibaGaeqOSdigabeaajuaGqaaaaaaaaaWdbiaacckacqGH 9aqpcaGGGcGaaGimaaaa@3E00@ .
s ˙  =   ( F s ' ) 1  ( F α ' )  α ˙  =  B α ( α,s ) α ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGZbGbaiaacaGGGcGaeyypa0JaaiiOaiabgkHiTiaaccka daqadaqaamaalaaabaGaeyOaIyRaamOraaqaaiabgkGi2kaadohada ahaaqabeaacaGGNaaaaaaaaiaawIcacaGLPaaadaahaaqabKqbGeaa cqGHsislcaaIXaaaaKqbakaacckadaqadaqaamaalaaabaGaeyOaIy RaamOraaqaaiabgkGi2kabeg7aHnaaCaaabeqaaiaacEcaaaaaaaGa ayjkaiaawMcaaiaacckacuaHXoqygaGaaiaacckacqGH9aqpcaGGGc GaamOqamaaBaaajuaibaGaeqySdegajuaGbeaadaqadaqaaiabeg7a HjaacYcacaWGZbaacaGLOaGaayzkaaGafqySdeMbaiaaaaa@5F14@ . (7)

As cyclic velocities are not included in the equations of the constraints, using (7), we obtain the kinetic energy (2) in the form

T ( α,s. α ˙ ,β ) =  1 2  ( α ˙ '      β ˙ ' )   ( a αα a ˙ αβ a βα a ββ )    ( α ˙ β ˙ )  + ( d ' α d ' β )  +  T 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmivay aauaWaaeWaaeaacqaHXoqycaGGSaGaam4Caiaac6cacuaHXoqygaGa aiaacYcacqaHYoGyaiaawIcacaGLPaaaqaaaaaaaaaWdbiaacckacq GH9aqpcaGGGcWaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGGcWaaeWa aeaacuaHXoqygaGaamaaCaaabeqaaiaacEcaaaGaaiiOaiaacckaca GGGcGaaiiOaiqbek7aIzaacaWaaWbaaeqabaGaai4jaaaaaiaawIca caGLPaaacaGGGcGaaiiOaiaacckadaqadaqaauaabeqaciaaaeaace WGHbGbaqbadaWgaaqcfasaaiabeg7aHjabeg7aHbqcfayabaaabaGa bmyyayaacaWaaSbaaKqbGeaacqaHXoqycqaHYoGyaKqbagqaaaqaai qadggagaafamaaBaaajuaibaGaeqOSdiMaeqySdegajuaGbeaaaeaa ceWGHbGbaqbadaWgaaqcfasaaiabek7aIjabek7aIbqcfayabaaaaa GaayjkaiaawMcaaiaacckacaGGGcGaaiiOaiaacckadaqadaqaauaa beqaceaaaeaacuaHXoqygaGaaaqaaiqbek7aIzaacaaaaaGaayjkai aawMcaaiaacckacaGGGcGaey4kaSIaaiiOamaabmaabaqbaeqabeGa aaqaaiqadsgagaafamaaCaaabeqaaiaacEcaaaWaaSbaaKqbGeaacq aHXoqyaKqbagqaaaqaaiqadsgagaafamaaCaaabeqaaiaacEcaaaWa aSbaaKqbGeaacqaHYoGyaKqbagqaaaaaaiaawIcacaGLPaaacaGGGc GaaiiOaiabgUcaRiaacckacaWGubWaaSbaaKqbGeaacaaIWaaajuaG beaaaaa@8A32@
a ˜ ( α,s ) =  ( a ˜ αα a ˜ αβ a ˜ αβ a ˜ βα a ˜ ββ a ˜ βs a ˜ sα a ˜ sβ a ˜ ss )     MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyyay aaiaWaaeWaaeaacqaHXoqycaGGSaGaam4CaaGaayjkaiaawMcaaaba aaaaaaaapeGaaiiOaiabg2da9iaacckacaGGGcWaaeWaaeaafaqabe WadaaabaGabmyyayaaiaWaaSbaaKqbGeaacqaHXoqycqaHXoqyaKqb agqaaaqaaiqadggagaacamaaBaaajuaibaGaeqySdeMaeqOSdigaju aGbeaaaeaaceWGHbGbaGaadaWgaaqcfasaaiabeg7aHjabek7aIbqc fayabaaabaGabmyyayaaiaWaaSbaaKazfaY=baqcfa4aaSbaaKqbGe aacqaHYoGycqaHXoqyaeqaaaqcfayabaaabaGabmyyayaaiaWaaSba aKqbGeaacqaHYoGycqaHYoGyaKqbagqaaaqaaiqadggagaacamaaBa aajuaibaGaeqOSdiMaam4CaaqabaaajuaGbaGabmyyayaaiaWaaSba aKqbGeaacaWGZbGaeqySdegajuaGbeaaaeaaceWGHbGbaGaadaWgaa qcfasaaiaadohacqaHYoGyaKqbagqaaaqaaiqadggagaacamaaBaaa juaibaGaam4CaiaadohaaeqaaaaaaKqbakaawIcacaGLPaaacaGGGc GaaiiOaiaacckacaGGGcaaaa@73A6@ , d ˜  = ( d ˜ α d ˜ β d ˜ s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmizay aaiaaeaaaaaaaaa8qacaGGGcGaeyypa0JaaiiOamaabmaabaqbaeqa bmqaaaqaaiqadsgagaacamaaBaaajuaibaGaeqySdegajuaGbeaaae aaceWGKbGbaGaadaWgaaqcfasaaiabek7aIbqcfayabaaabaGabmiz ayaaiaWaaSbaaKqbGeaacaWGZbaajuaGbeaaaaaacaGLOaGaayzkaa aaaa@4639@
a ˜ αα  =  a ˜ αα  +2 a ˜ αs B α  + B ' α a ss B α ;   a αβ  = a ˜ αβ  +  B α ' a ˜ sβ ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyyay aaiaWaaSbaaKqbGeaacqaHXoqycqaHXoqyaKqbagqaaabaaaaaaaaa peGaaiiOaiabg2da9iaacckaceWGHbGbaGaadaWgaaqcfasaaiabeg 7aHjabeg7aHbqabaqcfaOaaiiOaiabgUcaRiaaikdaceWGHbGbaGaa daWgaaqcfasaaiabeg7aHjaadohaaKqbagqaaiaadkeadaWgaaqcfa saaiabeg7aHbqabaqcfaOaaiiOaiabgUcaRiaadkeadaahaaqabeaa caGGNaaaamaaBaaajuaibaGaeqySdegajuaGbeaacaWGHbWaaSbaaK qbGeaacaWGZbGaam4CaaqcfayabaGaamOqamaaBaaajuaibaGaeqyS degajuaGbeaacaGG7aGaaiiOaiaacckaceWGHbGbaqbadaWgaaqcfa saaiabeg7aHjabek7aIbqcfayabaGaaiiOaiabg2da9iqadggagaac amaaBaaajuaibaGaeqySdeMaeqOSdigabeaajuaGcaGGGcGaey4kaS IaaiiOaiaadkeadaqhaaqcfasaaiabeg7aHbqcfayaaiaacEcaaaGa bmyyayaaiaWaaSbaaKqbGeaacaWGZbGaeqOSdigabeaajuaGcaGG7a aaaa@7730@
a βα  =  a αβ  ;   a ββ  = a ββ ;   d α =   d ˜ s B α ;   d β ' = d ˜ β ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyyay aauaWaaSbaaKqbGeaacqaHYoGycqaHXoqyaKqbagqaaabaaaaaaaaa peGaaiiOaiabg2da9iaacckaceWGHbGbaqbadaWgaaqcfasaaiabeg 7aHjabek7aIbqabaqcfaOaaiiOaiaacUdacaGGGcGaaiiOaiqacgga gaafamaaBaaabaqcfaIaeqOSdiMaeqOSdiwcfaOaaiiOaaqabaGaey ypa0JabmyyayaauaWaaSbaaKqbGeaacqaHYoGycqaHYoGyaKqbagqa aiaacUdacaGGGcGaaiiOaiqadsgagaafamaaBaaajuaibaGaeqySde gajuaGbeaacqGH9aqpcaGGGcGaaiiOaiqadsgagaacamaaBaaajuai baGaam4CaaqcfayabaGaamOqamaaBaaajuaibaGaeqySdegabeaaju aGcaGG7aGaaiiOaiaacckaceWGKbGbaqbadaqhaaqcfasaaiabek7a IbqaaiaacEcaaaqcfaOaeyypa0JabmizayaaiaWaa0baaKqbGeaacq aHYoGyaeaacaGGNaaaaaaa@7076@

Lagrange’s function is ˇ L( α,s, ˙ α, ˙ β ) = ˇ T  Π. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGhlGaaeiiaiaadYeapaWaaeWaaeaapeGaeqySdeMaaiil aiaadohacaGGSaGaaeiiaiaacMTacaqGGaGaeqySdeMaaiilaiaabc cacaGGzlGaaeiiaiabek7aIbWdaiaawIcacaGLPaaapeGaaeiiaiab g2da9iaabccacaGGhlGaaeiiaiaadsfacaqGGaGaeyOeI0Iaaeiiai abfc6aqjaac6caaaa@51D5@ Then theequations (6) can be written as

d dt   L α ˙ L α  =  Q α  +  B α ' ( L s  +  Q s ); MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaacaGGGcWaaSaa aeaacqGHciITceWGmbGbaqbaaeaacqGHciITcuaHXoqygaGaaaaacq GHsisldaWcaaqaaiabgkGi2kqadYeagaafaaqaaiabgkGi2kabeg7a HbaacaGGGcGaeyypa0JaaiiOaiaadgfadaWgaaqcfasaaiabeg7aHb qcfayabaGaaiiOaiabgUcaRiaacckacaWGcbWaa0baaKqbGeaacqaH XoqyaKqbagaacaGGNaaaamaabmaabaWaaSaaaeaacqGHciITceWGmb GbaqbaaeaacqGHciITjuaicaWGZbaaaKqbakaacckacqGHRaWkcaGG GcGaamyuamaaBaaajuaibaGaam4CaaqcfayabaaacaGLOaGaayzkaa Gaai4oaaaa@61B9@
d dt   L β ˙  = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbaabaGaamizaiaadshaaaaeaaaaaaaaa8qacaGGGcWaaSaa aeaacqGHciITceWGmbGbaqbaaeaacqGHciITcuaHYoGygaGaaaaaca GGGcGaeyypa0JaaiiOaiaaicdaaaa@441D@ ; (8)
s ˙  =  B α ( α,s ) α ˙ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGZbGbaiaacaGGGcGaeyypa0JaaiiOaiaadkeadaWgaaqc fasaaiabeg7aHbqcfayabaWaaeWaaeaacqaHXoqycaGGSaGaai4Caa GaayjkaiaawMcaaiqbeg7aHzaacaGaaiOlaaaa@455F@
As obviously follows from (8) the system has cyclic integrals and holonomic systems always [1,2,6] have steady motions

α ˙  = 0;     α =  α 0   = const; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacuaHXoqygaGaaiaacckacqGH9aqpcaGGGcGaaGimaiaacUda caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiabeg7aHjaacckacqGH9a qpcaGGGcGaeqySde2aaSbaaeaajuaicaaIWaqcfaOaaiiOaaqabaGa aiiOaiabg2da9iaacckacaWGJbGaam4Baiaad6gacaWGZbGaamiDai aacUdaaaa@54DB@
s =  s 0  = const;         β ˙  =  c β  = const. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGZbGaaiiOaiabg2da9iaacckacaWGZbWaaSbaaKqbGeaa caaIWaaabeaajuaGcaGGGcGaeyypa0JaaiiOaiaadogacaWGVbGaam OBaiaadohacaWG0bGaai4oaiaacckacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiaacckacaGGGcGafqOSdiMbaiaacaGGGcGaeyypa0Jaai iOaiaadogadaWgaaqcfasaaiabek7aIbqcfayabaGaaiiOaiabg2da 9iaacckacaWGJbGaam4Baiaad6gacaWGZbGaamiDaiaac6caaaa@6084@  (9)

Equations of Perturbed Motion in the Neighborhood of a Steady Motion

Introduce some initial perturbations:

α =  α 0  + x;   s =  s 0  + y;    β ˙  =  c β  + ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycaGGGcGaeyypa0JaaiiOaiabeg7aHnaaBaaajuai baGaaGimaaqabaqcfaOaaiiOaiabgUcaRiaacckacaWG4bGaai4oai aacckacaGGGcGaaiiOaiaadohacaGGGcGaeyypa0JaaiiOaiaadoha daWgaaqcfasaaiaaicdaaKqbagqaaiaacckacqGHRaWkcaGGGcGaam yEaiaacUdacaGGGcGaaiiOaiaacckacuaHYoGygaGaaiaacckacqGH 9aqpcaGGGcGaam4yamaaBaaajuaibaGaeqOSdigajuaGbeaacaGGGc Gaey4kaSIaaiiOaiabeM8a3baa@6396@

And assume that no potential forces corresponding to the position coordinates have the following structure:

Q ˜ α  =  f αα α ˙  +  f αs s ˙  +  g αα α +  g αs s +  Q ˜ α ( 2 ) ; Q ˜ s  =  f sα α ˙  +  f ss s ˙  +  g sα α +  g ss s +  Q ˜ s ( 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGqa aaaaaaaaWdbiqadgfagaacamaaDaaajuaibaGaeqySdegabaGaey4f IOcaaKqbakaacckacqGH9aqpcaGGGcGaamOzamaaBaaajuaibaGaeq ySdeMaeqySdegajuaGbeaacuaHXoqygaGaaiaacckacqGHRaWkcaGG GcGaamOzamaaBaaajuaibaGaeqySdeMaam4CaaqabaqcfaOabm4Cay aacaGaaiiOaiabgUcaRiaacckacaWGNbWaaSbaaKqbGeaacqaHXoqy cqaHXoqyaKqbagqaaiabeg7aHjaacckacqGHRaWkcaGGGcGaam4zam aaBaaabaWaaSbaaKqbGeaacqaHXoqycaWGZbaajuaGbeaaaeqaaiaa dohacaGGGcGaey4kaSIaaiiOaiqadgfagaacamaaDaaajuaibaGaeq ySdegabaqcfa4aaeWaaKqbGeaacaaIYaaacaGLOaGaayzkaaaaaKqb akaacUdaaeaaceWGrbGbaGaadaqhaaqcfasaaiaadohaaeaacqGHxi IkaaqcfaOaaiiOaiabg2da9iaacckacaWGMbWaaSbaaKqbGeaacaWG ZbGaeqySdegajuaGbeaacuaHXoqygaGaaiaacckacqGHRaWkcaGGGc GaamOzamaaBaaajuaibaGaam4CaiaadohaaeqaaKqbakqadohagaGa aiaacckacqGHRaWkcaGGGcGaam4zamaaBaaajuaibaGaam4Caiabeg 7aHbqcfayabaGaeqySdeMaaiiOaiabgUcaRiaacckacaWGNbWaaSba aKqbGeaajuaGdaWgaaqcfasaaiaadohacaWGZbaabeaaaKqbagqaai aadohacaGGGcGaey4kaSIaaiiOaiqadgfagaacamaaDaaajuaibaGa am4CaaqaaKqbaoaabmaajuaibaGaaGOmaaGaayjkaiaawMcaaaaaju aGcaGGUaaabaaaaaa@9A9D@

Then, the vector-matrix equations perturbed motion with separated first approximation can be written as

A 1 χ ¨  +  A 2 ω ˙  + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbWaaSbaaKqbGeaacaaIXaaajuaGbeaacuaHhpWygaWa aiaacckacqGHRaWkcaGGGcGaamyqamaaBaaajuaibaGaaGOmaaqcfa yabaGafqyYdCNbaiaacaGGGcGaey4kaScaaa@4428@
+ ( c β '   ψ 1  +  D α    F α ) χ ˙  + ( c β '   ψ 2  +  D β    F β )ω+ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHRaWkcaGGGcWaaeWaaeaacaWGJbWaa0baaKqbGeaacqaH YoGyaeaacaGGNaaaaiaacckajuaGcqaHipqEdaWgaaqcfasaaiaaig daaeqaaKqbakaacckacqGHRaWkcaGGGcGaamiramaaBaaajuaibaGa eqySdegabeaajuaGcaGGGcGaeyOeI0IaaiiOaiaadAeadaWgaaqcfa saaiabeg7aHbqabaaajuaGcaGLOaGaayzkaaGafq4XdmMbaiaacaGG GcGaey4kaSIaaiiOamaabmaabaGaam4yamaaDaaajuaibaGaeqOSdi gabaGaai4jaaaajuaGcaGGGcGaeqiYdK3aaSbaaKqbGeaacaaIYaaa juaGbeaacaGGGcGaey4kaSIaaiiOaiaadseadaWgaaqcfasaaiabek 7aIbqabaqcfaOaaiiOaiabgkHiTiaacckacaWGgbWaaSbaaKqbGeaa cqaHYoGyaeqaaaqcfaOaayjkaiaawMcaaiabeM8a3jabgUcaRaaa@6E68@
+ [ C 3   ( B κχ q μ ) 0 ( W q κ ) 0 + B α ' ( 0 ) C 4  + c β ' φ 2 2 c β ]y =  X α ( 2 ) ( x,y, x ˙ ,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHRaWkcaGGGcWaamWaaeaacaWGdbWaaSbaaKqbGeaacaaI ZaaabeaajuaGcaGGGcWaaeWaaeaadaWcaaqaaiabgkGi2kaadkeada WgaaqcfasaaiabeQ7aRjabeE8aJbqcfayabaaabaGaeyOaIyRaamyC amaaBaaajuaibaGaeqiVd0gajuaGbeaaaaaacaGLOaGaayzkaaWaaS baaeaacaaIWaaabeaadaqadaqaamaalaaabaGaeyOaIyRaam4vaaqa aiabgkGi2kaadghadaWgaaqaaiabeQ7aRbqabaaaaaGaayjkaiaawM caamaaBaaabaGaaGimaaqabaGaey4kaSIaamOqamaaDaaajuaibaGa eqySdegabaGaai4jaaaajuaGdaqadaqaaiaaicdaaiaawIcacaGLPa aacaWGdbWaaSbaaKqbGeaacaaI0aaabeaajuaGcaGGGcGaey4kaSIa am4yamaaDaaajuaibaGaeqOSdigabaGaai4jaaaajuaGcqaHgpGAda qhaaqcfasaaiaaikdaaeaacaaIYaaaaKqbakaadogadaWgaaqcfasa aiabek7aIbqabaaajuaGcaGLBbGaayzxaaGaamyEaiaacckacqGH9a qpcaGGGcGaamiwamaaDaaajuaibaGaeqySdegabaqcfa4aaeWaaKqb GeaacaaIYaaacaGLOaGaayzkaaaaaKqbaoaabmaabaGaamiEaiaacY cacaWG5bGaaiilaiqadIhagaGaaiaacYcacqaHjpWDaiaawIcacaGL Paaaaaa@7EF3@
y ˙  =  B α ( 0 ) χ ˙ + B α ( 1 ) ( χ,y ) χ ˙ ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aacaaeaaaaaaaaa8qacaGGGcGaeyypa0JaaiiOaiaadkeadaWgaaqc fasaaiabeg7aHbqcfayabaWaaeWaaeaacaaIWaaacaGLOaGaayzkaa Gafq4XdmMbaiaacqGHRaWkcaWGcbWaa0baaKqbGeaacqaHXoqyaeaa juaGdaqadaqcfasaaiaaigdaaiaawIcacaGLPaaaaaqcfa4aaeWaae aacqaHhpWycaGGSaGaamyEaaGaayjkaiaawMcaaiqbeE8aJzaacaGa ai4oaaaa@50D2@
A 3 x ¨  +  A 4 ω ˙  + ( c β ' ψ 3  +  D ˜ β ) x ˙  =  X β ( 2 ) ( x,y, x ˙ ,ω ); MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbWaaSbaaKqbGeaacaaIZaaabeaajuaGceWG4bGbamaa caGGGcGaey4kaSIaaiiOaiaadgeadaWgaaqcfasaaiaaisdaaKqbag qaaiqbeM8a3zaacaGaaiiOaiabgUcaRiaacckadaqadaqaaiaadoga daqhaaqcfasaaiabek7aIbqcfayaaiaacEcaaaGaeqiYdK3aaSbaaK qbGeaacaaIZaaabeaajuaGcaGGGcGaey4kaSIaaiiOaiqadseagaac amaaBaaajuaibaGaeqOSdigabeaaaKqbakaawIcacaGLPaaaceWG4b GbaiaacaGGGcGaeyypa0JaaiiOaiaadIfadaqhaaqcfasaaiabek7a IbqaaKqbaoaabmaajuaibaGaaGOmaaGaayjkaiaawMcaaaaajuaGda qadaqaaiaadIhacaGGSaGaamyEaiaacYcaceWG4bGbaiaacaGGSaGa eqyYdChacaGLOaGaayzkaaGaai4oaaaa@67F0@
B α ( 1 ) ( x,y ) =  B α ( α 0  + x, s 0  + y )   B α ( 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGcbWaa0baaKqbGeaacqaHXoqyaeaajuaGdaqadaqcfasa aiaaigdaaiaawIcacaGLPaaaaaqcfa4aaeWaaeaacaWG4bGaaiilai aadMhaaiaawIcacaGLPaaacaGGGcGaeyypa0JaaiiOaiaadkeadaWg aaqcfasaaiabeg7aHbqcfayabaWaaeWaaeaacqaHXoqydaWgaaqaai aaicdaaeqaaiaacckacqGHRaWkcaGGGcGaamiEaiaacYcacaWGZbWa aSbaaKqbGeaacaaIWaaabeaajuaGcaGGGcGaey4kaSIaaiiOaiaadM haaiaawIcacaGLPaaacaGGGcGaeyOeI0IaaiiOaiaadkeadaWgaaqc fasaaiabeg7aHbqabaqcfa4aaeWaaeaacaaIWaaacaGLOaGaayzkaa aaaa@5FF9@ ; (10)
A 1  =  a γδ ( 0 ),  A 2  =  aγσ ( 0 ),  A 3  =  a σδ ( 0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbWaaSbaaKqbGeaacaaIXaaabeaajuaGcaGGGcGaeyyp a0JaaiiOamaafeaabaGaamyyamaaBaaajuaibaGaeq4SdCMaeqiTdq gajuaGbeaaaiaawMa7amaafiaabaWaaeWaaeaacaaIWaaacaGLOaGa ayzkaaaacaGLkWoacaGGSaGaaiiOaiaadgeadaWgaaqcfasaaiaaik daaKqbagqaaiaacckacqGH9aqpcaGGGcWaauqaaeaacaWGHbqcfaIa eq4SdCMaeq4Wdmxcfa4aauGaaeaadaqadaqaaiaaicdaaiaawIcaca GLPaaaaiaawQa7aaGaayzcSdGaaiilaiaacckacaWGbbWaaSbaaKqb GeaacaaIZaGaaiiOaaqcfayabaGaeyypa0JaaiiOamaafeaabaGaam yyamaaBaaajuaibaGaeq4WdmNaeqiTdqgajuaGbeaadaqbcaqaamaa bmaabaGaaGimaaGaayjkaiaawMcaaaGaayPcSdaacaGLjWoacaGGSa aaaa@6BA2@
A 4  =  a σγ ( 0 );      C 1 = ( 2 W q x q γ ) 0 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbWaaSbaaKqbGeaacaaI0aaabeaajuaGcaGGGcGaeyyp a0JaaiiOamaafeaabaGaamyyamaaBaaajuaibaGaeq4WdmNaeq4SdC gajuaGbeaaaiaawMa7amaafiaabaWaaeWaaeaacaaIWaaacaGLOaGa ayzkaaaacaGLkWoacaGG7aGaaiiOaiaacckacaGGGcGaaiiOaiaacc kacaWGdbWaaSbaaKqbGeaacaaIXaaabeaajuaGcqGH9aqpdaqadaqa amaalaaabaGaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGcaWGxb aabaGaeyOaIyRaamyCamaaBaaajuaibaGaamiEaaqcfayabaGaeyOa IyRaamyCamaaBaaajuaibaGaeq4SdCgajuaGbeaaaaaacaGLOaGaay zkaaWaaSbaaKqbGeaacaaIWaaabeaajuaGcaGGNaaaaa@6199@
C 2  =  ( 2 W q x q μ ) 0 ,   C 4 = ( 2 W q μ q κ ) 0 ,   C 3  =  C 2 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWaaSbaaKqbGeaacaaIYaaajuaGbeaacaGGGcGaeyyp a0JaaiiOamaabmaabaWaaSaaaeaacqGHciITdaahaaqcfasabeaaca aIYaaaaKqbakaadEfaaeaacqGHciITcaWGXbWaaSbaaKqbGeaacaWG 4baajuaGbeaacqGHciITcaWGXbWaaSbaaKqbGeaacqaH8oqBaKqbag qaaaaaaiaawIcacaGLPaaadaWgaaqcfasaaiaaicdaaeqaaKqbakaa cYcacaGGGcGaaiiOaiaadoeadaWgaaqcfasaaiaaisdaaeqaaKqbak abg2da9maabmaabaWaaSaaaeaacqGHciITdaahaaqabKqbGeaacaaI YaaaaKqbakaadEfaaeaacqGHciITcaWGXbWaaSbaaKqbGeaacqaH8o qBaKqbagqaaiabgkGi2kaadghadaWgaaqcfasaaiabeQ7aRbqcfaya baaaaaGaayjkaiaawMcaamaaBaaajuaibaGaaGimaaqcfayabaGaai ilaiaacckacaGGGcGaam4qamaaBaaajuaibaGaaG4maaqabaGaaiiO aKqbakabg2da9iaacckacaGGdbWaa0baaKqbGeaacaaIYaaabaGaai 4jaaaaaaa@6EF1@ ;
D α  =  d γ q x  +  d γ q μ B μx    d x q x    B μγ   d x q γ 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGebWaaSbaaKqbGeaacqaHXoqyaKqbagqaaiaacckacqGH 9aqpcaGGGcWaauqaaeaadaWcaaqaaiabgkGi2kaadsgadaWgaaqcfa saaiabeo7aNbqcfayabaaabaGaeyOaIyRaamyCamaaBaaajuaibaGa amiEaaqabaaaaaqcfaOaayzcSdGaaiiOaiabgUcaRiaacckadaWcaa qaaiabgkGi2kaadsgadaWgaaqcfasaaiabeo7aNbqcfayabaaabaGa eyOaIyRaamyCamaaBaaajuaibaGaeqiVd0gajuaGbeaaaaGaamOqam aaBaaajuaibaGaeqiVd0MaamiEaaqcfayabaGaaiiOaiabgkHiTiaa cckadaWcaaqaaiabgkGi2kaadsgadaWgaaqcfasaaiaadIhaaKqbag qaaaqaaiabgkGi2kaadghadaWgaaqcfasaaiaadIhaaKqbagqaaaaa caGGGcGaeyOeI0IaaiiOaiaackeadaWgaaqcfasaaiabeY7aTjabeo 7aNbqcfayabaGaaiiOamaafiaabaWaaSaaaeaacqGHciITcaWGKbWa aSbaaKqbGeaacaWG4baajuaGbeaaaeaacqGHciITcaWGXbWaaSbaaK qbGeaacqaHZoWzaKqbagqaaaaaaiaawQa7aiaaicdacaGGGcGaaiil aaaa@7B77@
D β  =  ( d σ q γ  +  B μγ d σ q μ )0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGebWaaSbaaKqbGeaacqaHYoGyaKqbagqaaiaacckacqGH 9aqpcaGGGcWaauqaaeaacqGHsisldaqbcaqaamaabmaabaWaaSaaae aacqGHciITcaWGKbWaaSbaaKqbGeaacqaHdpWCaKqbagqaaaqaaiab gkGi2kaadghadaWgaaqcfasaaiabeo7aNbqabaaaaKqbakaacckacq GHRaWkcaGGGcGaamOqamaaBaaajuaibaGaeqiVd0Maeq4SdCgajuaG beaadaWcaaqaaiabgkGi2kaadsgadaWgaaqcfasaaiabeo8aZbqcfa yabaaabaGaeyOaIyRaamyCamaaBaaajuaibaGaeqiVd0gajuaGbeaa aaaacaGLOaGaayzkaaaacaGLkWoaaiaawMa7aiaaicdacaGGGcGaai ilaaaa@6165@
F α = f αα + f αs B α ( 0 )( f sα + f ss B α ( 0 ) ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWaaSbaaKqbGeaacqaHXoqyaKqbagqaaiabg2da9iaa dAgadaWgaaqcfasaaiabeg7aHjabeg7aHbqcfayabaGaey4kaSIaam OzamaaBaaajuaibaGaeqySdeMaam4CaaqabaqcfaOaamOqamaaBaaa juaibaGaeqySdegajuaGbeaadaqadaqaaiaaicdaaiaawIcacaGLPa aadaqadaqaaiaadAgadaWgaaqcfasaaiaadohacqaHXoqyaKqbagqa aiabgUcaRiaadAgadaWgaaqcfasaaiaadohacaWGZbaabeaajuaGca WGcbWaaSbaaKqbGeaacqaHXoqyaeqaaKqbaoaabmaabaGaaGimaaGa ayjkaiaawMcaaaGaayjkaiaawMcaaiaacYcaaaa@5B76@
F β = f αβ + B α ' ( 0 ) f sβ ;     W=  T 0 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGgbWaaSbaaKqbGeaacqaHYoGyaKqbagqaaiabg2da9iaa dAgadaWgaaqcfasaaiabeg7aHjabek7aIbqcfayabaGaey4kaSIaam OqamaaDaaajuaibaGaeqySdegajuaGbaGaai4jaaaadaqadaqaaiaa icdaaiaawIcacaGLPaaacaWGMbWaaSbaaKqbGeaacaWGZbGaeqOSdi gajuaGbeaacaGG7aGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaWG xbGaeyypa0JaaiiOaiabg+GivlabgkHiTiaadsfadaWgaaqcfasaai aaicdaaKqbagqaaiaacUdaaaa@5A96@
G α =  g αα + B α ' ( 0 ) g sα ,       G s =  g αs + B α ' ( 0 ) g ss , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4ram aaBaaajuaibaGaeqySdegajuaGbeaacqGH9aqpqaaaaaaaaaWdbiaa cckapaGaam4zamaaBaaajuaibaGaeqySdeMaeqySdegajuaGbeaacq GHRaWkcaWGcbWaa0baaKqbGeaacqaHXoqyaeaacaGGNaaaaKqbaoaa bmaabaGaaGimaaGaayjkaiaawMcaaiaadEgadaWgaaqcfasaaiaado hacqaHXoqyaKqbagqaaiaacYcapeGaaiiOaiaacckacaGGGcGaaiiO aiaacckacaGGGcGaam4ramaaBaaajuaibaGaam4CaaqabaqcfaOaey ypa0JaaiiOaiaadEgadaWgaaqcfasaaiabeg7aHjaadohaaeqaaKqb akabgUcaRiaadkeadaqhaaqcfasaaiabeg7aHbqaaiaacEcaaaqcfa 4aaeWaaeaacaaIWaaacaGLOaGaayzkaaGaam4zamaaBaaajuaibaGa am4CaiaadohaaeqaaiaacYcaaaa@6852@
ψ 1 = a γα q x + a γα q μ B μχ a χα q γ B μ γ a χσ q μ 0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHipqEdaWgaaqcfasaaiaaigdaaeqaaKqbakabg2da9maa feaabaWaaSaaaeaacqGHciITcaWGHbWaaSbaaKqbGeaacqaHZoWzcq aHXoqyaKqbagqaaaqaaiabgkGi2kaadghadaWgaaqcfasaaiaadIha aKqbagqaaaaaaiaawMa7aiabgUcaRmaalaaabaGaeyOaIyRaamyyam aaBaaajuaibaGaeq4SdCMaeqySdegajuaGbeaaaeaacqGHciITcaWG XbWaaSbaaKqbGeaacqaH8oqBaKqbagqaaaaacaWGcbWaaSbaaKqbGe aacqaH8oqBcqaHhpWyaKqbagqaaiabgkHiTmaalaaabaGaeyOaIyRa amyyamaaBaaajuaibaGaeq4XdmMaeqySdegajuaGbeaaaeaacqGHci ITcaWGXbWaaSbaaKqbGeaacqaHZoWzaKqbagqaaaaacqGHsisldaqb caqaaiaadkeadaWgaaqcfasaaiabeY7aTbqabaGaeq4SdCwcfa4aaS aaaeaacqGHciITcaWGHbWaaSbaaKqbGeaacqaHhpWycqaHdpWCaKqb agqaaaqaaiabgkGi2kaadghadaWgaaqcfasaaiabeY7aTbqcfayaba aaaaGaayPcSdGaaGimaiaacYcaaaa@79AC@  
ψ 2 =   ψ γσ ( 0 ),    ψ γσ =  ( a γα q γ B μγ a χα q μ ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHipqEdaWgaaqcfasaaiaaikdaaeqaaKqbakabg2da9iaa cckacaGGGcWaauqaaeaacqaHipqEdaWgaaqcfasaaiabeo7aNjabeo 8aZbqcfayabaaacaGLjWoadaqbcaqaamaabmaabaGaaGimaaGaayjk aiaawMcaaaGaayPcSdGaaiilaiaacckacaGGGcGaaiiOaiabeI8a5n aaBaaajuaibaGaeq4SdCMaeq4WdmhajuaGbeaacqGH9aqpcaGGGcGa eyOeI0IaaiiOamaabmaabaWaaSaaaeaacqGHciITcaWGHbWaaSbaaK qbGeaacqaHZoWzcqaHXoqyaKqbagqaaaqaaiabgkGi2kaadghadaWg aaqcfasaaiabeo7aNbqcfayabaaaaiaadkeadaWgaaqcfasaaiabeY 7aTjabeo7aNbqcfayabaWaaSaaaeaacqGHciITcaWGHbWaaSbaaKqb GeaacqaHhpWycqaHXoqyaKqbagqaaaqaaiabgkGi2kaadghadaWgaa qcfasaaiabeY7aTbqabaaaaaqcfaOaayjkaiaawMcaaiaacckacaGG Saaaaa@7688@
ψ 1 2 =  1 2 ψ γσ q μ 0 ,       ψ 2 2 =  1 2 ψ γσ q μ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHipqEfaqabeGabaaabaGaaGymaaqaaiaaikdaaaGaeyyp a0JaaiiOamaalaaabaGaaGymaaqaaiaaikdaaaWaauqaaeaadaqbca qaamaalaaabaGaeyOaIyRaeqiYdK3aaSbaaKqbGeaacqaHZoWzcqaH dpWCaeqaaaqcfayaaiabgkGi2kaadghadaWgaaqcfasaaiabeY7aTb qcfayabaaaaaGaayPcSdaacaGLjWoacaaIWaGaaiiOaiaacYcacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaeqiYdKxbae qabiqaaaqaaiaaikdaaeaacaaIYaaaaiabg2da9iaacckadaWcaaqa aiaaigdaaeaacaaIYaaaamaafeaabaWaauGaaeaadaWcaaqaaiabgk Gi2kabeI8a5naaBaaajuaibaGaeq4SdCMaeq4WdmhabeaaaKqbagaa cqGHciITcaWGXbWaaSbaaKqbGeaacqaH8oqBaeqaaaaaaKqbakaawQ a7aaGaayzcSdGaaGimaaaa@6F43@
ψ 3 =  a γσ q χ +  a γσ q μ B μχ 0 ,      D ˜ β  =  d χ q χ  +  d χ q μ B μx 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHipqEdaWgaaqcfasaaiaaiodaaKqbagqaaiabg2da9iaa cckadaqbbaqaamaafiaabaWaaSaaaeaacqGHciITcaWGHbWaaSbaaK qbGeaacqaHZoWzcqaHdpWCaKqbagqaaaqaaiabgkGi2kaadghadaWg aaqcfasaaiabeE8aJbqcfayabaaaaiabgUcaRiaacckadaWcaaqaai abgkGi2kaadggadaWgaaqcfasaaiabeo7aNjabeo8aZbqcfayabaaa baGaeyOaIyRaamyCamaaBaaajuaibaGaeqiVd0gajuaGbeaaaaGaam OqamaaBaaajuaibaGaeqiVd0Maeq4XdmgajuaGbeaaaiaawQa7aaGa ayzcSdWaaSbaaKqbGeaacaaIWaaabeaajuaGcaGGSaGaaiiOaiaacc kacaGGGcGaaiiOaiaacckaceWGebGbaGaadaWgaaqcfasaaiabek7a IbqabaqcfaOaaiiOaiabg2da9iaacckadaqbbaqaamaafiaabaWaaS aaaeaacqGHciITcaWGKbWaaSbaaKqbGeaacqaHhpWyaKqbagqaaaqa aiabgkGi2kaadghadaWgaaqcfasaaiabeE8aJbqabaaaaKqbakaacc kacqGHRaWkcaGGGcWaaSaaaeaacqGHciITcaWGKbWaaSbaaKqbGeaa cqaHhpWyaKqbagqaaaqaaiabgkGi2kaadghadaWgaaqcfasaaiabeY 7aTbqcfayabaaaaiaadkeadaWgaaqcfasaaiabeY7aTjaadIhaaeqa aaqcfaOaayPcSdaacaGLjWoadaWgaaqcfasaaiaaicdaaKqbagqaai aacYcaaaa@8CFE@

Repeated indexes are summed over the values:

γ,δ,χ=1,...,l;  ξ,α,r=l+1,...,n; μ,k= n+1,..., n + m. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGcq aHZoWzcaGGSaGaeqiTdqMaaiilaiabeE8aJjabg2da9iaaigdacaGG SaGaaiOlaiaac6cacaGGUaGaaiilaiaadYgacaGG7aaeaaaaaaaaa8 qacaGGGcGaaiiOaiabe67a4jaacYcacqaHXoqycaGGSaGaaiOCaiab g2da9iaadYgacqGHRaWkcaaIXaGaaiilaiaac6cacaGGUaGaaiOlai aacYcacaWGUbGaai4oaaqaaiabeY7aTjaacYcacaGGRbGaeyypa0Ja aiiOaiaad6gacqGHRaWkcaaIXaGaaiilaiaac6cacaGGUaGaaiOlai aacYcacaGGGcGaamOBaiaacckacqGHRaWkcaGGGcGaamyBaiaac6ca aaaa@671B@  

 Determine variables corresponding to zero roots of the characteristic equation via the linear substitution [5]:

z = y   B α ( 0 )x. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG6bGaaiiOaiabg2da9iaacckacaWG5bGaaiiOaiabgkHi TiaacckacaWGcbWaaSbaaKqbGeaacqaHXoqyaKqbagqaamaabmaaba GaaGimaaGaayjkaiaawMcaaiaadIhacaGGUaaaaa@4659@  (11)

The system (10) can be written as

A 1 x ¨  +  A 2 ω ˙  + ( Φ α    F α ) x ˙  + ( Γ  F β )ω+Κx + Sz =  Χ ( 2 ) ( x,z, x ˙ ,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbWaaSbaaeaacaaIXaaabeaaceWG4bGbamaacaGGGcGa ey4kaSIaaiiOaiaadgeadaWgaaqaaiaaikdaaeqaaiqbeM8a3zaaca GaaiiOaiabgUcaRiaacckadaqadaqaaiabfA6agnaaBaaabaGaeqyS degabeaacaGGGcGaeyOeI0IaaiiOaiaadAeadaWgaaqaaiabeg7aHb qabaaacaGLOaGaayzkaaGabmiEayaacaGaaiiOaiabgUcaRiaaccka daqadaqaaiabfo5ahjabgkHiTiaacckacaWGgbWaaSbaaeaacqaHYo GyaeqaaaGaayjkaiaawMcaaiabeM8a3jabgUcaRiabfQ5aljaadIha caGGGcGaey4kaSIaaiiOaiaacofacaGG6bGaaiiOaiabg2da9iaacc kacqqHNoWqdaahaaqabeaadaqadaqaaiaaikdaaiaawIcacaGLPaaa aaWaaeWaaeaacaWG4bGaaiilaiaadQhacaGGSaGabmiEayaacaGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@730F@ ; (12)
A 3 x ¨  + A4 ω ˙  +  Φ β x ˙ ,      z ˙  =  Β α ( 1 ) ( x,z ) x ˙ ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbWaaSbaaKqbGeaacaaIZaaajuaGbeaaceWG4bGbamaa caGGGcGaey4kaSIaaiiOaiaadgeajuaicaaI0aqcfaOafqyYdCNbai aacaGGGcGaey4kaSIaaiiOaiabfA6agnaaBaaajuaibaGaeqOSdiga juaGbeaaceWG4bGbaiaacaGGSaGaaiiOaiaacckacaGGGcGaaiiOai aacckaceWG6bGbaiaacaGGGcGaeyypa0JaaiiOaiabfk5acnaaDaaa juaibaGaeqySdegabaqcfa4aaeWaaKqbGeaacaaIXaaacaGLOaGaay zkaaaaaKqbaoaabmaabaGaamiEaiaacYcacaWG6baacaGLOaGaayzk aaGabmiEayaacaGaai4oaaaa@6106@
Φ α  =  c β ' Ψ 1  + D   α ,     Φ β  =  c β ' Ψ3 +  D ˜ β , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeuOPdy 0aaSbaaKqbGeaacqaHXoqyaKqbagqaaabaaaaaaaaapeGaaiiOaiab g2da9iaacckacaWGJbWaa0baaKqbGeaacqaHYoGyaeaacaGGNaaaaK qbakabfI6aznaaBaaajuaibaGaaGymaaqcfayabaGaaiiOaiabgUca RiaacckacaWGebGaaiiOamaaBaaajuaibaGaeqySdegabeaajuaGca GGSaGaaiiOaiaacckacaGGGcGaaiiOaiabfA6agnaaBaaajuaibaGa eqOSdigajuaGbeaacaGGGcGaeyypa0JaaiiOaiaadogadaqhaaqcfa saaiabek7aIbqaaiaacEcaaaqcfaOaeuiQdKvcfaIaaG4maiaaccka juaGcqGHRaWkcaGGGcGabmirayaaiaWaaSbaaKqbGeaacqaHYoGyae qaaKqbakaacYcaaaa@6784@
Γ =   c β ' Ψ 2  +  D β ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHtoWrcaGGGcGaeyypa0JaaiiOaiaacckacaWGJbWaa0ba aKqbGeaacqaHYoGyaeaacaGGNaaaaKqbakabfI6aznaaBaaajuaiba GaaGOmaaqabaqcfaOaaiiOaiabgUcaRiaacckacaWGebWaaSbaaKqb GeaacqaHYoGyaeqaaKqbakaacUdaaaa@4AE8@
Κ =  C 1  +  Β α ' ( 0 ) C 2  +  C 3 Β α ( 0 ) +  Β α ' ( 0 ) C 4 Β α ( 0 )+ + C Β  +  c β ' Ψ 2 1 c β  +  c β ' Ψ 2 2 c β Β( 0 )   G α   G s Β α ( 0 ) ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGqa aaaaaaaaWdbiabfQ5aljaacckacqGH9aqpcaGGGcGaam4qamaaBaaa juaibaGaaGymaaqabaqcfaOaaiiOaiabgUcaRiaacckacqqHsoGqda qhaaqcfasaaiabeg7aHbqaaiaacEcaaaqcfa4aaeWaaeaacaaIWaaa caGLOaGaayzkaaGaam4qamaaBaaajuaibaGaaGOmaaqabaqcfaOaai iOaiabgUcaRiaacckacaWGdbWaaSbaaKqbGeaacaaIZaaabeaajuaG cqqHsoGqdaWgaaqcfasaaiabeg7aHbqcfayabaWaaeWaaeaacaaIWa aacaGLOaGaayzkaaGaaiiOaiabgUcaRiaacckacqqHsoGqdaqhaaqc fasaaiabeg7aHbqaaiaacEcaaaqcfa4aaeWaaeaacaaIWaaacaGLOa GaayzkaaGaam4qamaaBaaajuaibaGaaGinaaqabaqcfaOaeuOKdi0a aSbaaKqbGeaacqaHXoqyaKqbagqaamaabmaabaGaaGimaaGaayjkai aawMcaaiabgUcaRaqaaiabgUcaRiaadoeadaahaaqabKqbGeaacqqH soGqaaqcfaOaaiiOaiabgUcaRiaacckacaWGJbWaa0baaKqbGeaacq aHYoGyaeaacaGGNaaaaKqbakabfI6aznaaDaaajuaibaGaaGOmaaqa aiaaigdaaaqcfaOaam4yamaaBaaajuaibaGaeqOSdigajuaGbeaaca GGGcGaey4kaSIaaiiOaiaadogadaqhaaqcfasaaiabek7aIbqaaiaa cEcaaaqcfaOaeuiQdK1aa0baaKqbGeaacaaIYaaabaGaaGOmaaaaju aGcaWGJbWaaSbaaKqbGeaacqaHYoGyaeqaaKqbakabfk5acnaabmaa baGaaGimaaGaayjkaiaawMcaaiaacckacqGHsislcaGGGcGaam4ram aaBaaajuaibaGaeqySdegabeaajuaGcaGGGcGaeyOeI0Iaam4ramaa BaaajuaibaGaam4CaaqcfayabaGaeuOKdi0aaSbaaKqbGeaacqaHXo qyaKqbagqaamaabmaabaGaaGimaaGaayjkaiaawMcaaiaacckacaGG 7aaaaaa@A0FD@
C Β  =  ( Β μγ q x ) 0 ( W q μ ) 0  +  ( Β κγ q μ ) 0 ( W q κ ) 0   Β α ( 0 ) ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWaaWbaaeqajuaibaGaeuOKdieaaKqbakaacckacqGH 9aqpcaGGGcWaaeWaaeaadaWcaaqaaiabgkGi2kabfk5acnaaBaaaju aibaGaeqiVd0Maeq4SdCgajuaGbeaaaeaacqGHciITcaWGXbWaaSba aKqbGeaacaWG4baabeaaaaaajuaGcaGLOaGaayzkaaWaaSbaaeaaca aIWaaabeaadaqadaqaamaalaaabaGaeyOaIyRaam4vaaqaaiabgkGi 2kaadghadaWgaaqcfasaaiabeY7aTbqcfayabaaaaaGaayjkaiaawM caamaaBaaabaGaaGimaaqabaGaaiiOaiabgUcaRiaacckadaqadaqa amaalaaabaGaeyOaIyRaeuOKdi0aaSbaaKqbGeaacqaH6oWAcqaHZo WzaKqbagqaaaqaaiabgkGi2kaadghadaWgaaqcfasaaiabeY7aTbqc fayabaaaaaGaayjkaiaawMcaamaaBaaabaGaaGimaaqabaWaaeWaae aadaWcaaqaaiabgkGi2kaadEfaaeaacqGHciITcaWGXbWaaSbaaKqb GeaacqaH6oWAaeqaaaaaaKqbakaawIcacaGLPaaadaWgaaqaaiaaic daaeqaaiaacckacqqHsoGqdaWgaaqcfasaaiabeg7aHbqcfayabaWa aeWaaeaacaaIWaaacaGLOaGaayzkaaGaaiiOaiaacUdaaaa@789F@
It is necessary to pay attention on the matrix CB. Its components depend on second-order derivatives of the geometric constraint equations (1).

Stabilization of a Steady Motion for Systems with Incomplete State Information

Consider the case when the control action u acts on by all components of cyclic coordinate’s vector. The normal form of the perturbed motion equations is:

ξ ˙ =Νξ + Vu + Ζz +  Ξ ( 2 ) ( ξ,z ), z ˙  =  Β α ( 1 ) ( x,z ) x 1  ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacuaH+oaEgaGaaiabg2da9iabf25aojabe67a4jaacckacqGH RaWkcaGGGcGaaiOvaiaacwhacaGGGcGaey4kaSIaaiiOaiabfA5aAj aadQhacaGGGcGaey4kaSIaaiiOaiabf65aynaaCaaabeqcfasaaKqb aoaabmaajuaibaGaaGOmaaGaayjkaiaawMcaaaaajuaGdaqadaqaai abe67a4jaacYcacaWG6baacaGLOaGaayzkaaGaaiilaiqadQhagaGa aiaacckacqGH9aqpcaGGGcGaeuOKdi0aa0baaKqbGeaacqaHXoqyae aajuaGdaqadaqcfasaaiaaigdaaiaawIcacaGLPaaaaaqcfa4aaeWa aeaacaWG4bGaaiilaiaadQhaaiaawIcacaGLPaaacaWG4bWaaSbaaK qbGeaacaaIXaaabeaajuaGcaGGGcGaai4oaaaa@6970@
ξ  ( x , x 1 , ω ),     Ν = ( Ι 0 Α 1 Κ ˜ Α 1 Ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOVdG NbauaaqaaaaaaaaaWdbiaacckadaqadaqaaiqadIhagaqbaiaacYca ceWG4bGbauaadaWgaaqcfasaaiaaigdaaeqaaKqbakaacYcacuaHjp WDgaqbaaGaayjkaiaawMcaaiaacYcacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiabf25aojaacckacqGH9aqpcaGGGcWaaeWaaeaafaqabe GacaaabaGaeuyMdKeabaGaaGimaaqaaiabgkHiTiabfg5abnaaCaaa beqcfasaaiabgkHiTiaaigdaaaqcfaOafuOMdSKbaGaaaeaacqGHsi slcqqHroqqdaahaaqcfasabeaacqGHsislcaaIXaaaaKqbakabfg6a sbaaaiaawIcacaGLPaaaaaa@5D9A@ ;
Ρ = ( Φ α F α Γ F β Φ β 0 ) ,      Κ ˜  = ( Κ 0 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHHoGucaGGGcGaeyypa0JaaiiOamaabmaabaqbaeqabiGa aaqaaiabfA6agnaaBaaajuaibaGaeqySdegajuaGbeaacqGHsislca WGgbWaaSbaaKqbGeaacqaHXoqyaKqbagqaaaqaaiabfo5ahjabgkHi TiaadAeadaWgaaqcfasaaiabek7aIbqabaaajuaGbaGaeuOPdy0aaS baaKqbGeaacqaHYoGyaeqaaaqcfayaaiaaicdaaaaacaGLOaGaayzk aaGaaiiOaiaacYcacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiqbfQ 5alzaaiaGaaiiOaiabg2da9iaacckadaqadaqaauaabeqaceaaaeaa cqqHAoWsaeaacaaIWaaaaaGaayjkaiaawMcaaiaacckacaGGSaaaaa@6175@
V = ( 0 Α 1 Μ ) ,  Μ = ( 0 Ε η1 ) ,  Α = ( Α 1 Α 2 Α 3 Α 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGwbGaaiiOaiabg2da9iaacckadaqadaqaauaabeqaceaa aeaacaaIWaaabaGaeuyKde0aaWbaaKqbGeqabaGaeyOeI0IaaGymaa aajuaGcqqHCoqtaaaacaGLOaGaayzkaaGaaiiOaiaacYcacaGGGcGa aiiOaiabfY5anjaacckacqGH9aqpcaGGGcWaaeWaaeaafaqabeGaba aabaGaaGimaaqaaiabfw5afnaaBaaajuaibaGaeq4TdGMaeyOeI0Ia aGymaaqabaaaaaqcfaOaayjkaiaawMcaaiaacckacaGGSaGaaiiOai aacckacqqHroqqcaGGGcGaeyypa0JaaiiOamaabmaabaqbaeqabiGa aaqaaiabfg5abnaaBaaajuaibaGaaGymaaqabaaajuaGbaGaeuyKde 0aaSbaaKqbGeaacaaIYaaajuaGbeaaaeaacqqHroqqdaWgaaqcfasa aiaaiodaaKqbagqaaaqaaiabfg5abnaaBaaajuaibaGaaGinaaqcfa yabaaaaaGaayjkaiaawMcaaaaa@698D@
Ζ = ( 0 Α 1 S ˜ ) ,   Ι = ( 0 Ε ι ) ,    S ˜  = ( S 0 )  , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHwoGwcaGGGcGaeyypa0JaaiiOamaabmaabaqbaeqabiqa aaqaaiaaicdaaeaacqqHroqqdaahaaqcfasabeaacqGHsislcaaIXa aaaKqbakqadofagaacaaaaaiaawIcacaGLPaaacaGGGcGaaiilaiaa cckacaGGGcGaaiiOaiabfM5ajjaacckacqGH9aqpcaGGGcWaaeWaae aafaqabeqacaaabaGaaGimaaqaaiabfw5afnaaBaaajuaibaGaeqyU dKgabeaaaaaajuaGcaGLOaGaayzkaaGaaiiOaiaacYcacaGGGcGaai iOaiaacckaceWGtbGbaGaacaGGGcGaeyypa0JaaiiOamaabmaabaqb aeqabiqaaaqaaiaadofaaeaacaaIWaaaaaGaayjkaiaawMcaaiaacc kacaGGGcGaaiilaaaa@6264@
Ξ ( 2 )  = ( 0 Α 1 Ξ ˜ ( 2 ) )  ,    Ξ ˜ ( 2 ) = ( Χ ( 2 ) 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHEoawdaahaaqcfasabeaajuaGdaqadaqcfasaaiaaikda aiaawIcacaGLPaaaaaqcfaOaaiiOaiabg2da9iaacckadaqadaqaau aabeqaceaaaeaacaaIWaaabaGaeuyKde0aaWbaaeqajuaibaGaeyOe I0IaaGymaaaajuaGcuqHEoawgaacamaaCaaabeqcfasaaKqbaoaabm aajuaibaGaaGOmaaGaayjkaiaawMcaaaaaaaaajuaGcaGLOaGaayzk aaGaaiiOaiaacckacaGGSaGaaiiOaiaacckacaGGGcGafuONdGLbaG aadaahaaqabKqbGeaajuaGdaqadaqcfasaaiaaikdaaiaawIcacaGL PaaaaaqcfaOaeyypa0JaaiiOamaabmaabaqbaeqabiqaaaqaaiabfE 6adnaaCaaajuaibeqaaKqbaoaabmaajuaibaGaaGOmaaGaayjkaiaa wMcaaaaaaKqbagaacaaIWaaaaaGaayjkaiaawMcaaaaa@60A4@ .

The simplest variants of matrices of coefficients for linear approximation of measurement vector are σi =Σiξ: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHdpWCcaWGPbGaaeiiaiabg2da9iabfo6atjaadMgacqaH +oaEcaGG6aaaaa@3FF1@
Σ 1  = ( Ε ι 0 0 ) ;     Σ 2  = ( 0 Ε ι 0 )  ;    Σ 3  = ( 0 0 Ε ηι ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHJoWudaWgaaqcfasaaiaaigdaaeqaaiaacckajuaGcqGH 9aqpcaGGGcWaaeWaaeaafaqabeqadaaabaGaeuyLdu0aaSbaaKqbGe aacqaH5oqAaeqaaaqcfayaaiaaicdaaeaacaaIWaaaaaGaayjkaiaa wMcaaiaacckacaGG7aGaaiiOaiaacckacaGGGcGaaiiOaiabfo6atn aaBaaajuaibaGaaGOmaaqabaqcfaOaaiiOaiabg2da9iaacckadaqa daqaauaabeqabmaaaeaacaaIWaaabaGaeuyLdu0aaSbaaKqbGeaacq aH5oqAaeqaaaqcfayaaiaaicdaaaaacaGLOaGaayzkaaGaaiiOaiaa cckacaGG7aGaaiiOaiaacckacaGGGcGaeu4Odm1aaSbaaKqbGeaaca aIZaaajuaGbeaacaGGGcGaeyypa0JaaiiOamaabmaabaqbaeqabeWa aaqaaiaaicdaaeaacaaIWaaabaGaeuyLdu0aaSbaaKqbGeaacqaH3o aAcqGHsislcqaH5oqAaeqaaaaaaKqbakaawIcacaGLPaaaaaa@6E64@ ;

The observing system is:
ξ ^ ˙  = Ν ξ ^  + Vu   L κ ( Σ κ ξ ^    σ κ ) ,   ξ ˜  = ( x ^ ' , x ^ ' 1, ω ^ ' ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacuaH+oaEgaqcgaGaaiaacckacqGH9aqpcaGGGcGaeuyNd4Ka fqOVdGNbaKaacaGGGcGaey4kaSIaaiiOaiaacAfacaGG1bGaaiiOai abgkHiTiaacckacaGGmbWaaSbaaKqbGeaacqaH6oWAaKqbagqaamaa bmaabaGaeu4Odm1aaSbaaKqbGeaacqaH6oWAaKqbagqaaiqbe67a4z aajaGaaiiOaiabgkHiTiaacckacqaHdpWCdaWgaaqcfasaaiabeQ7a RbqcfayabaaacaGLOaGaayzkaaGaaiiOaiaacYcacaGGGcGaaiiOai qbe67a4zaaiaGaaiiOaiabg2da9iaacckadaqadaqaaiqadIhagaqc amaaCaaabeqaaiaacEcaaaGaaiilaiqadIhagaqcamaaCaaabeqaai aacEcaaaWaaSbaaeaajuaicaaIXaqcfaOaaiilaaqabaGafqyYdCNb aKaadaahaaqabeaacaGGNaaaaaGaayjkaiaawMcaaaaa@6DBB@ ;

It provides an estimate of the phase state of system (13). The vector ˆ ξk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH+oaEcaWGRbaaaa@3957@  is obtained from the measurement σk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHdpWCcaWGRbaaaa@3957@  by the solution of the stabilization dual problem

μ ˙ κ  =  Ν   μ κ  +  Σ κ ν κ ,     ν κ  =  L κ μ κ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacuaH8oqBgaGaamaaBaaajuaibaGaeqOUdSgajuaGbeaacaGG GcGaeyypa0JaaiiOaiqbf25aozaafaGaaiiOaiabeY7aTnaaBaaaju aibaGaeqOUdSgajuaGbeaacaGGGcGaey4kaSIaaiiOaiqbfo6atzaa faWaaSbaaKqbGeaacqaH6oWAaKqbagqaaiabe27aUnaaBaaajuaiba GaeqOUdSgajuaGbeaacaGGSaGaaiiOaiaacckacaGGGcGaaiiOaiab e27aUnaaBaaajuaibaGaeqOUdSgajuaGbeaacaGGGcGaeyypa0Jaai iOaiqadYeagaqbamaaBaaajuaibaGaeqOUdSgajuaGbeaacqaH8oqB daWgaaqcfasaaiabeQ7aRbqcfayabaGaaiOlaaaa@65F6@  (15)
Theorem 1: If the following conditions

rank( V  ΝV   Ν 2 V... Ν η+ι1 V )  =  η+ι  ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGYbGaamyyaiaad6gacaWGRbWaaeWaaeaacaWGwbGaaiiO aiaacckacqqHDoGtcaWGwbGaaiiOaiaacckacqqHDoGtdaahaaqcfa sabeaacaaIYaaaaKqbakaadAfacaGGUaGaaiOlaiaac6cacqqHDoGt daahaaqabKqbGeaacqaH3oaAcqGHRaWkcqaH5oqAcqGHsislcaaIXa aaaKqbakaadAfaaiaawIcacaGLPaaacaGGGcGaaiiOaiabg2da9iaa cckacaGGGcGaeq4TdGMaey4kaSIaeqyUdKMaaiiOaiaacckacaGG7a aaaa@5FA1@  (16)
rank( Σ κ    Ν Σ κ    Ν 2 V... Ν η+ι1 Σ κ )  =  η+ι ;  κ = 1,2,3 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGYbGaamyyaiaad6gacaWGRbWaaeWaaeaacuqHJoWugaqb amaaBaaajuaibaGaeqOUdSgajuaGbeaacaGGGcGaaiiOaiqbf25aoz aafaGafu4OdmLbauaadaWgaaqcfasaaiabeQ7aRbqcfayabaGaaiiO aiaacckacqqHDoGtdaahaaqcfasabeaacaaIYaaaaKqbakaadAfaca GGUaGaaiOlaiaac6cacqqHDoGtdaahaaqabKqbGeaacqaH3oaAcqGH RaWkcqaH5oqAcqGHsislcaaIXaaaaKqbakqbfo6atzaafaWaaSbaaK qbGeaacqaH6oWAaKqbagqaaaGaayjkaiaawMcaaiaacckacaGGGcGa eyypa0JaaiiOaiaacckacqaH3oaAcqGHRaWkcqaH5oqAcaGGGcGaai 4oaiaacckacaGGGcGaeqOUdSMaaiiOaiabg2da9iaacckacaaIXaGa aiilaiaaikdacaGGSaGaaG4maiaacckacaGG7aaaaa@7514@  (17)
Are met for system (17) then there is a linear control

u κ  =  Λ κ ξ ^ κ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyDam aaBaaajuaibaGaeqOUdSgajuaGbeaaqaaaaaaaaaWdbiaacckacqGH 9aqpcaGGGcGaeu4MdW0aaSbaaKqbGeaacqaH6oWAaKqbagqaaiqbe6 7a4zaajaWaaSbaaKqbGeaacqaH6oWAaKqbagqaaaaa@45E1@

Stabilizing steady motion (13) to asymptotic stability with respect to all variables. It should be noted that in Routh’s variables the third column of the matrix of measurements is always consists of zeros. Obviously, any linear combination of columns is possible for matrix of measurements in Lagrangian variables Σ = ( aΣ1+bΣ2+cΣ3 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHJoWucaqGGaGaeyypa0Jaaeiia8aadaqadaqaa8qacaWG HbGaeu4OdmLaaGymaiabgUcaRiaadkgacqqHJoWucaaIYaGaey4kaS Iaam4yaiabfo6atjaaiodaa8aacaGLOaGaayzkaaWdbiaac6caaaa@4826@ Therefore, the use of Lagrangian variables can provide further capacity to full the conditions of observability compared with variables Routh’s.

Conclusion

This work develops results [5,8,9,12]. Mathematical model of a system with geometric constraints was constructed using Shul’gin’s equations of motion and Lagrangian variables. Besides, we suggested that some of coordinates are cyclic. The equations were presented in general form. The advantages of this form of equations for solving stability and stabilization problems were discussed. Then the theorem about solvability of the stabilization problem for steady motions was formulated in case of systems with incomplete state information. This result can be applied for solving stability and stabilization problems for many mechanical systems such as multilink manipulators or other systems with geometric constraints.

References

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