Proceedings of the 35th Chinese Control Conference July 27-29, 2016, Chengdu, China Gain Scheduling based Backstepping Control for Motion Balance Adjusting of a Power-line Inspection Robot DIAN Songyi1, HOANG Son1, 2, PU Ming1, LIU Junyong1, CHEN Lin1 School of Electrical Engineering and Information, Sichuan University, Chengdu 610065, China E-mail: scudiansy@scu.edu.cn Vietnam National University of Forestry, Hanoi, Vietnam E-mail: hoangsonbk83@yahoo.com.vn Abstract: This paper presents a gain scheduling backstepping control (GSBC) for motion balance adjusting of a power-line inspection (PLI) robot, which is an underactuted mechanical system of two degrees of freedom with one control input First, a dynamic model of the motion balance adjusting of the PLI robot is constructed Second, this model is linearized at a nominal operating point to overcome the computation infeasibility of the conventional backstepping technique Finally, to extend the operation area of the closed-loop system further from the nominal operating point, an equilibrium manifold linearization model (EML model) is developed using a scheduling variable, and then the GSBC scheme is designed based on the EML model The robust stability of the closed-loop system is ensured by the Lyapunov theorem Simulation results show that the GSBC gives much better performance than that of the backstepping control, these results illustrate the feasibility of the proposed control scheme Key Words: Gain scheduling backstepping, balance adjusting, power line inspection robot Introduction In recent years, the study on underactuated nonlinear systems has caught extensive attention in theory and practical applications [1-5] The underactuated nonlinear system is a system having fewer actuators than its degrees of freedom [6] Some well-known underactuated systems with two degrees of freedom and one actuation have been considered including the inverted pendulum system [7, 8], the ball and beam system [9-11] The underactuated systems have some advantages such as light weight and low energy consumption, whereas the control design of these systems is more complex than that addressed in fully actuated systems To control underactuated systems, there are several main methods such as the sliding mode control (SMC) [12, 13], the energy method [8, 14], and the backstepping control (BC) [15-17] Among these control methods, the advantage of the BC is the high robustness of a closed-loop system, which is a popular strategy to design the controller for underactuated nonlinear systems However, in some cases, the dynamic model may consist of cross terms or high nonlinearity functions; therefore, it is necessary to consider the computational feasibility with backstepping procedures To resolve this problem, the linearization technique is applied for the sake of having an approximate linear model of the nonlinear plant, which guarantees the simplicity needed to perform backstepping procedures However, the linearization technique is the conversion of a linear approximation into a nonlinear function at a given point, thus a controller design based on linearization model only achieves efficiency when a system operates around a nominal operating point So as to resolve this problem, the researchers focused on expanding the linearization model to an equilibrium manifold model (EML model) using a scheduling variable In this paper, a gain scheduling backstepping control (GSBC) is proposed for the motion balance adjusting of the PLI robot [18] The motion balance adjusting process of the PLI robot in the second step of the Line-leading mode is an underactuated nonlinear system In this step, the PLI robot moves to the power cable and simultaneously keeps balance by adjusting the self-balance mechanism, as illustrated in Fig The proposed GSBC is advantaged in the following aspects The nonlinear dynamic model of the PLI robot is first converted into a linear model at a nominal operating point The BC which is designed based on this linear model can overcome computation infeasibility of the conventional backstepping method Next, in order to expand the operation area of the closed-loop system, the linear model of the plant is transformed to an EML model using a scheduling variable Consequently, the proposed GSBC scheme based on this EML model both extends the operational area of the closed-loop system and overcomes the computational infeasibility of the initial nonlinear model The closed-loop stability of the controlled system is guaranteed by the Lyapunov theorem The simulation results demonstrate the efficiency and feasibility of the proposed GSBC scheme Pulley Two bundled cables Motion arms Insulated access cable Self-balance adjusting mechanism Counter-weight box Fig.1: Line-loading mode of the PLI robot This paper is divided into the following sections The preliminary knowledge of the motion balance adjusting of 441 the PLI robot is discussed in Section In Section 3, the GSBC for the motion balance adjusting of the PLI robot is designed The simulation results are presented in Section Finally, Section provides concluding remarks K P Preliminary knowledge 2.1 where g is the gravitational acceleration In the Table 1, we have (4) m1h1 m2 h20 , so the P can be re-written as (5) P m2 g > d cos T1 l sin T2 sin T1 @ Dynamic model for the motion balance adjusting of the PLI robot h1 X1 h20 Substituting Eq (2) and Eq (5) into Eq (1), it yields êơ m1h12 m2 (d (h20 l sin T ) ) º¼ T1 m1 (kg ) m2 (kg ) Y1 2 º m l 2T m1h12T12 m2 êơ d (h20 l sin T ) ¼ T1 2 ,(2) 2 m1 gh1 sin T1 (3) m2 g > d cos T1 (h20 l sin T )sin T1 @ , Z1 2m2 l (h20 l sin T )(cos T )T1T O1 Z1 X l sin T m1 (2) O1 T m2 (1) d Z2 Y2 X1 m2 gd sin T1 m2 gl sin T cos T1 , u2 X2 O2 Y2 O2 Y1 h20 X1 d T1 O1 Y2 Y2 Contact between pulley and cable 2.2 Z2 l sin T Z2 x (2) h1 (m) h20 (m) l (m) x2 x3 x4 @ T T êơT1 T1 T2 T2 ẳ (8) [2m2 l (h20 l sin x3 )(cos x3 ) x2 x4 x2 Table 1: The parameters of the PLI robot* m2 (kg) > x1 The state-space equations are x1 x2 (c) model in the X1O1Y1 plane Fig.2: Balance adjustment parameters of the PLI robot m1 (kg) Analysis of the motion balance adjusting of the PLI robot To investigate the motion balance adjusting of the PLI robot, we define the state variable vector as follows O2 O2 (7) m2 gd cos T sin T1 Remark 1: The Eq (6) and Eq (7) are dynamic equations of the motion balance adjusting of the PLI robot (1) The center of mass (COM) of the robot body (2) The COM of the counter-weight box (a) 3D model (b) model in the X1O1Z1 plane h1 m2 d 22T cos T m2 d 22T 22 sin T cos T m2 d 2T12 (h220 d sin T ) cos T l active joint (1) (6) d (m) x3 63 27 0.18 0.42 0.5 0.5 *m1 is the mass of the robot body, m2 is the mass of the counter-weight box, l is the length of actuator bar (Fig.2(b)), d is the height of the T-shaped base (Fig.2(a, c)) m2 ggd sin x1 m2 gl sin x3 cos x1 ] ª m1h12 + m2 d (h20 l sin x3 ) ẳ x4 (9) [u2 m2 l (h20 l sin x3 )(cos x3 ) x22 m2 ggl cos x3 sin x1 ] m2 l Remark 2: When the steady-state input equals zero u2 u2(0) , the state-space equation (9) has an equilibrium point at the origin as T T x > x1 x2 x3 x4 @ >0 0 0@ (10) x4 Consider the motion balance adjusting of the PLI robot in Fig.2 Let T1 (t ) be the angle between the robot and the cable, let T (t ) be the angle of the active joint Let h1 be the distance between the cable and the center of mass (COM) of the body, let h2 h20 l sin T2 be the distance from the cable to the COM of the counter-weight box Let u2 (t ) be the torque acting on the active joint T (t ) We use the Euler-Lagrange equation to obtain the motion equations of the motion balance adjusting of the PLI robot The Lagrangian equation is given as follows [19] d ª wL º wL Ui i 1, , m , (1) ằ ô dt wTi ẳ wTi Consider the equation system as follows x2 [2m2 l (h20 l sin x3 )(cos x3 ) x2 x4 0 m2 gd sin x1 m2 gl sin x3 cos x1 ] ª m1h12 + m2 d (h20 l sin x3 ) ẳ x4 [u2(0) m2 l (h20 l sin x3 )(cos x3 ) x22 where U i is the external torque acting on the ith generalized coordinate L K P , with K and P are the kinetic and potential energy of the motion balance adjusting of the PLI robot, respectively: 442 m2 gl cos x3 sin x1 ] m2 l (11) d tan x1 d (substituting l and d in Table into l this equation, it yields S / d x1 d S / ), then the roots of the Eq (11) are x2 V2 If 1 d x3 x4 Đ d ã arcsin ă tan x1 â l Đ [2m2 l (h20 l sin x3 )(cos x3 ) x2 x4 ã ă (24) sin sin cos ] m gd x m gl x x 2 ¸ x e2 ă ă ê m h + m d (h l sin x ) 2d 20 ẳ ăơ 1 â To realize V2 o The problem of concern here is to find the value of x3 , making V2 satisfying (12) V2 § § d ·· tan x1 ¸ ¸ u2(0) m2 gl (sin x1 ) cos ă arcsin ă â l ạạ â Remark 3: From Eq (12), it is shown that with x1 satisfying S / d x1 d S / , the Eq (11) has respective Controller design 3.1 Problem formulation In this paper, the backstepping technique is utilized to design the stabilizing controller for the balance of the PLI robot Step1.1: We consider a control problem with output x1 Let (13) e1 x1 x1d With a21 and a23 3.2 e1 ( x2 x1dd ) e1 (e2 k1e1 ) x2 x2 a21 x1 a23 x3 x3 x4 x4 a41 x1 b4 u2 m2 gd ˈ a41 m1h12 m2 (d h20 ) (26) g ˈ b4 l , m2 l 2 4m22l h20 m2 gl m1h12 m2 (d h20 ) m h m (d h ) 1 20 Controller design of GSBC operating point x >0 0 0@ Therefore, to expand T the operating area of the closed-loop system, this linearization model will be expanded about an EML model using a scheduling variable A Equilibrium manifold linearization model To begin with, the state-space equation of the plant is described in Eq (9) The linearization of Eq (9) about its equilibrium manifold yields the parameterized linearization family [20, 21] d > x x(D )@ A(D )[x x(D )] B(D )[U U (D )] (27) dt § wf · § wf ã Where A(D ) ă and B(D ) ă ; w x â ạx(D ),U (D ) © wU ¹x(D ),U (D ) (19) (20) The now V1 x1 The linearization model in Eq (26) will not be accurate when the closed-loop system operates outside the nominal To realize x2 o x2d , we get a new error x2 x2d , x2 x22d (25) Remark 4: From Eq (26), we can see that BC is designed based on the linearization model of the PLI robot, which can easily overcome the computation infeasibility Where x1d is a desired goal of the balance angle T1 Taking the derivative of (13) as (14) e1 x1 x1dd x2 x11d A Lyapunov function candidate is chosen as V1 e1 , V1 e1e1 e1 ( x2 x11d ) (15) It can be seen that (15) is negative infinity if we choose (16) x2 k1e1 x1d , k1 ! Step 1.2: We choose virtual control as (17) x2d k1e1 x11d , (18) x2d k1e1 x11d e2 e2 k1e12 k2 e22 , k2 ! From equation (24), we can see that finding x3 can lead to computation infeasibility In order to resolve this problem, state-space equation (9) is linearized to make use of the backstepping technique Next, the linearization model of the PLI robot in Eq (9) is at a nominal operating point x [0 0 0]T , it yields yields x3 and u2(0) Therefore, the state equation (9) has equilibrium manifold k1e12 e1e2 (21) To realize e2 o and e1 o , a Lyapunov function candidate is chosen as V2 V1 e22 , (22) then V2 k1e12 e1e2 e2 ( x2 x22d ) (23) A(D ) and B(D ) are the parameterized plant linearization family matrices; x(D ) and U (D ) are the parameterized steady-state variable and the parameterized input; and D is a scheduling variable We choose the x1d of the PLI robot as a scheduling variable, then D x1d , Substituting x2 in state-space equation (9) into Eq (23), it yields x(D ) > x1d U (D ) u2 (D ) 443 x3D 0@ , T m2 gl sin x1d (cos x3D ) , (28) (29) ê0 ô a* ô 21 «0 « * ¬« a 41 A(D ) * a 23 0 * a43 0º »» , and B(D ) 1» » ¼» >0 To realize x2 o x2d , we get a new error e2 x2 x2d , 0 b4 @ (30) T e2 x3D a*21 m1h12 m2 d h20 d tan x1d , V2 ª d (tan x1d ) º « 2m2 l h20 d tan x1d ằu l2 ôơ ằẳ > 2m2lh20 m2 gd sin x1d m2 gd cos x1d tan x1d @ a*23 ª m h m d h d tan x 20 1d 1 ẳ d (tan x1d ) l2 , 2 m1h1 m2 d (h20 d tan x1d ) and a*41 e3 x3 x3d , (45) (46) e3 x3 x3d x4 x33d To realize e3 o , e2 o and e1 o , we design Lyapunov function as 1 2 V3 V2 e32 (e1 e2 e3 ) , (47) 2 then V3 V2 e3e3 (48) From Eq (41), Eq (44), and Eq (46), the derivative of V3 can be obtained as * (49) V3 (ei ) k1e12 k2 e22 e3 (a23 e2 x4 x3d ) Choosing * x4 k3e3 a23 e2 x3d , k3 ! (50) Then we would have V3 k1e12 k2 e22 k3e32 d (51) Step 2.4: We choose the virtual control as * x4d k3e3 a23 e2 x3d , k3 ! (52) To realize x4 o x4d , we get a new error (53) e4 x4 x4d , d (tan x1d ) l2 l Now, the EML model Eq (27) can be re-written as x1 x2 x2 * * a21 ( x1 x1d ) a23 ( x3 x3D ) x3 x4 x4 * * ( x3 x3D ) b4 u2 u2 (D ) a41 ( x1 x1d ) a43 (31) B Controller design Theorem: When the EML model of the PLI robot is defined by Eq (31), the GSBC scheme based on this EML model is designed The robust stability of the closed-loop system is ensured, the operational area of the closed-loop control system is extended, and the computational infeasibility of initial nonlinear model is overcome as well Proof: Step 2.1: Starting with the first equation of Eq (31), we define an error (32) e1 x1 x1d , e4 (33) e1 x1 x1dd x2 x11d To realize e1 o , a Lyapunov function candidate is chosen as V1 e1 , (34) V1 e1e1 e1 ( x2 x11d ) (35) If we choose x2 x1d k1e1ˈk1 ! , then V1 x2 k e 1 x2 x1dd k1e1 , k1e1 x11d , we choose virtual control as x2d k1e1 x1d Step 2.2: To realize * * º k1e12 e2 êơe1 a21 ( x1 x1d ) a23 ( x3 x3D ) x2d ¼ (41) To realize x3 o x3d , we get a new error dg (sin x1d )(tan x1d ) , l2 g (cos x1d ) a ( x1 x1d ) a ( x3 x3D ) x2d (39) If we choose * * ª k2 e2 e1 a21 x3 ( x1 x1d ) a23 x3D x2d º¼ , k2 ! 0, (42) * ¬ a23 then V2 k1e12 k2 e22 d (43) Step 2.3: We choose the virtual control as * * ªk2 e2 e1 a21 x3d ( x1 x1d ) a23 x3D x2d º¼ , k2 ! (44) * ¬ a23 m2 gl (cos x1d ) * a43 (38) * 23 To realize e2 o and e1 o , we design Lyapunov function as V2 V1 e22 (40) From Eq (35), Eq (37), and Eq (39), the derivative of V2 can be obtained as Where § d · arcsin ă tan x1d , l â > m2 gd cos x1d m2 gd (sin x1d )(tan x1d ) @ x2 x2d * 21 To realize e4 o , e3 o , e2 o and e1 o , we design Lyapunov function as 1 2 2 V4 V3 e42 (e1 e2 e3 e4 ) , (55) 2 then V4 V3 e4 e4 (56) From Eq (49), Eq (52) and Eq (54), the derivative of V4 is shown as follows * V4 (ei ) k1e12 k2 e22 k3e32 e4 [e3 a41 ( x1 x1d ) (57) * a43 ( x3 x3D ) b4u2 b4u2 (D ) x4d ] (36) that * * a41 ( x1 x1d ) a43 ( x3 x3D ) b4 u2 u2 (D ) x4d (54) is (37) To realize V4 d , we choose 444 400 * * ª k4 e4 e3 a41 ( x1 x1d ) a43 ( x3 x3D ) x4d ẳ b4 (58) u2 (D ), k4 ! Substituting Eq 58 into Eq 57, we would have V4 k1e12 k2 e22 k3e32 k4 e42 d (59) 200 From Eq (59), we can see that GSBC law u2 (t ) in Eq (58) is a stabilizing function to be determined for the closed-loop system The theorem is proved 100 -100 -200 The physical parameters of the PLI robot are listed in Table I The parameters for BC and GSBC are initially chosen as k1 k2 k3 k4 The control goal is to stabilize the PLI robot at the desired balance position Two cases of different conditions are addressed as follow: Case 1: The initial condition is T x(0) >0.3 0.3145 0@ , the desired balance position T1 (rad/s) T1 (rad) T1 (rad/s) 10 Time (s) 10 T ((rad/s) 10 Time (s) Time (s) 10 10 BC GSBC 10 BC GSBC Time (s) 400 -2 BC GSBC 10 Fig.7: Simulation results of case 2: T (top) and T (bottom) BC GSBC -2 T (rad/s) T (rad) T -0.5 BC GSBC 2 0.5 Fig.3: Simulation results of case 1: T1 (top) and T1 (bottom) 0.5 0 0@ , the Fig.6: Simulation results of case 2: T1 (top) and T1 (bottom) BC GSBC >0 Desired angle BC GSBC 0.5 -0.5 0 -0.5 10 0.2 rad -0.2 0.2 Desired angle BC GSBC 0.2 -1 Time (s) desired balance position is x1d T (rad) T1 (rad) 0.4 Case 2: The initial condition is x(0) rad -0.2 Fig.5: Simulation results of case 1: control signals Simulation Results is x1d BC GSBC 300 u2 (N.m) u2 BC GSBC 300 10 u2 (N.m) Fig.4: Simulation results of case 1: T (top) and T (bottom) 200 100 -100 -200 Time (s) Fig.8: Simulation results of case 2: control signals 445 10 In case 1, the control objective is to keep the robot balanced at the origin x1d rad The simulation results indicate that, all the two schemes managed to asymptotically balancing the PLI robot at this point (Fig 3) In this case, the results corresponding to the BC and GSBC controllers similarly because the linear model and EML model are same * * * * , and a43 (i.e., a21 a21 , a23 a23 , a41 a41 ) Fig and Fig denote the angles, the angular velocities and the control signals of the active joint, respectively Simulation results in case indicate that due to the high nonlinear dynamic model, the BC does not meet the control requirement, and there is a steady-state error while GSBC can asymptotically balance the robot at the desired position (Fig 6) Fig and Fig denote the angles, the angular velocities and the control signals of the active joint, respectively [9] [10] [11] [12] [13] [14] Conclusion In this paper, the control problem of the motion balance adjusting of the PLI robot is addressed by a proposed GSBC Based on the knowledge about dynamic equations of the motion balance adjusting of the PLI robot, an EML model is formulated The GSBC is designed based on the equilibrium manifold linearization model both extending the operation area of the closed-loop system and overcoming the computation infeasibility of the initial nonlinear dynamic model The closed-loop stability of the controlled system is guaranteed by the Lyapunov theorem Simulation results illustrate the superiority of the proposed GSBC scheme [15] [16] 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addressed by a proposed GSBC Based on the knowledge about dynamic equations of the motion balance adjusting of. .. family matrices; x(D ) and U (D ) are the parameterized steady-state variable and the parameterized input; and D is a scheduling variable We choose the x1d of the PLI robot as a scheduling variable,... PLI robot* m2 (kg) > x1 The state-space equations are x1 x2 (c) model in the X1O1Y1 plane Fig.2: Balance adjustment parameters of the PLI robot m1 (kg) Analysis of the motion balance adjusting of