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Abstract—In this brief, a dynamic model of a mobile wheeled inverted pendulum (MWIP) system is improved considering fric- tion forces, and a nonlinear ...
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Nonlinear Disturbance Observer-Based Dynamic Surface Control of Mobile Wheeled Inverted Pendulum Jian Huang, Member, IEEE, Songhyok Ri, Lei Liu, Yongji Wang, Jiyong Kim, and Gyongchol Pak

Abstract— In this brief, a dynamic model of a mobile wheeled inverted pendulum (MWIP) system is improved considering friction forces, and a nonlinear disturbance observer (NDO)-based dynamic surface controller is investigated to control the MWIP system. Using a coordinate transformation, this non-Class-I type underactuated system is presented as a semistrict feedback form, which is convenient for dynamic surface controller design. A dynamic surface controller together with an NDO is designed to stabilize the underactuated plant. The proposed approach can compensate the external disturbances and the model uncertainties to improve the system performance significantly. The stability of the closed-loop MWIP system is proved by Lyapunov theorem. Experiment results are presented to illustrate the feasibility and efficiency of the proposed method. Index Terms— Dynamic surface control (DSC), mobile wheeled inverted pendulum (MWIP), nonlinear disturbance observer (NDO), robust control, underactuated mechanical system.

I. I NTRODUCTION

I

N RECENT years, many approaches have been applied in the control of mobile wheeled inverted pendulum (MWIP), including the feedback linearization methods [1], fuzzy control methods [2], neural network-based methods [3], optimized adaptive control methods [4], and robust control approaches [5], [6]. The backstepping control methods are also applied for controlling the MWIP systems, in which backstepping is often used in conjunction with other control strategies [7]. An alternative control design method called multiple sliding surface (MSS) control was developed. However, designing an MSS controller may lead to an explosion of terms problem. Manuscript received November 14, 2014; revised February 3, 2015; accepted February 7, 2015. Manuscript received in final form February 11, 2015. This work was supported in part by the International Science and Technology Cooperation Program of China through the Precision Manufacturing Technology and Equipment for Metal Parts under Grant 2012DFG70640, in part by the Program for New Century Excellent Talents in University under Grant NCET-12-0214, and in part by the National Natural Science Foundation of China under Grant 61473130. Recommended by Associate Editor N. K. Kazantzis. J. Huang, L. Liu, and Y. Wang are with the Key Laboratory of Ministry of Education for Image Processing and Intelligent Control, School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]; liulei@mail. hust.edu.cn; [email protected]). S. Ri is with the School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China, and also with the Department of Control Science, University of Science, Pyongyang, D.P.R. of Korea (e-mail: [email protected]). J. Kim and G. Pak are with the Department of Control Science, University of Science, Pyongyang, D.P.R. of Korea (e-mail: kimjiyong@ 163.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2015.2404897

To avoid the drawback of the MSS controller mentioned above, a robust nonlinear control technique called Dynamic surface control (DSC) was developed in [8]. The DSC design requires the strict or semistrict feedback form of the model [8], [9]. Recently, some researchers applied the DSC technique into the control of underactuated mechanical systems, including the underactuated marine vessels [10] and the inertia wheel pendulum [11]. Because the dynamics of a Class-I underactuated mechanical system may be transformed into a cascade nonlinear system in strict feedback form (according to [12, Lemma 1]), most of aforementioned studies discussed only the Class-I underactuated mechanical system as defined in [12, Definition 3.9.1]. Unfortunately, the MWIP system does not belong to the Class-I underactuated mechanical system. Shojaei and Shahri [13] proposed a dynamic surface controller considering the actuator dynamics for trajectory tracking of uncertain nonholonomic wheeled mobile robots. In their study, however, the authors considered only two of vehicle planar motions (yaw rotation and forward movement). The balancing control problem of the wheeled mobile robot was not discussed. Therefore, the dynamic model in their study is not an underactuated mechanical system. It should be pointed out that the balance of an MWIP system is the prerequisite of its motion control tasks. Owing to this, in this brief, we focus on the balancing control of the MWIP system considering the degrees of freedom of yaw and tilt motion. To facilitate the design of DSC for the MWIP system, we transform the dynamics of an MWIP system into a cascade nonlinear system in semistrict feedback form using a new global change of coordinates. To the best our knowledge, it might be the first attempt of dealing with the DSC design for the non-Class-I type underactuated systems. It is found that using a disturbance observer can further improve the robustness of DSC controller. Chen [14] proposed a nonlinear disturbance observer (NDO) to cope with the disturbance of nonlinear system. An NDO was proposed in [15] considering both the constant and varying disturbances. In this brief, we proposed a dynamic surface controller with NDO (DSCNDO) for the balance control of an MWIP system. The introduction of NDO enhances the robustness of closedloop MWIP system to model errors and external disturbances. Moreover, the explosion term problem is also avoided in the controller design. The rest of this brief is organized as follows. In Section II, an improved dynamic model of an MWIP system is proposed considering friction forces and an NDO is obtained. The detailed design procedure and stability analysis of DSCNDO control strategy is given in Sections III and IV, respectively.

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TABLE I N OTATIONS

Therefore, the dynamic model of the MWIP system is given by ⎧ m 11 ψ¨ + m 12 cos(θ )θ¨ ⎪ ⎪ ⎪ ⎪ ⎪ = m 12 sin(θ )(θ˙ 2 + α˙ 2 ) − 2Dw ψ˙ ⎪ ⎪ ⎪ + 2Db (θ˙ − ψ) ˙ + u r + u l + τext1 ⎪ ⎪ ⎪ ⎪m cos(θ )ψ¨ + m θ¨ ⎪ 22 ⎨ 12 = Ibl sin(θ ) cos(θ )α˙ 2 +G b sin(θ ) (1) ⎪ ˙ − ψ) ˙ − u r − u l + τext2 ⎪ ( θ − 2D b ⎪ ⎪ ⎪ ⎪ (Ibl sin2 (θ ) + m 33 )α¨ ⎪ ⎪ ⎪ ⎪ = −2Ibl sin(θ ) cos(θ )α˙ θ˙ − m 12 sin(θ )α˙ ψ˙ ⎪ ⎪ ⎪ ⎩ − 2b2 (D + D )α˙ + b (u − u ) + τ b w l ext3 r r r2 where 1 (ψr + ψl ). 2 Parameters m 11 , m 12 , m 22 , m 33 , Ibl , and G b satisfy ⎧ m 11 = (m b + 2m w )r 2 + 2Iwa ⎪ ⎪ ⎪ ⎪ m 12 = m b lr ⎪ ⎪ ⎨m = m l 2 + I 22 b yyb G b = m b gl ⎪ ⎪ ⎪ ⎪ Ibl = Izzb + m bl 2 ⎪ ⎪ 2 ⎩ m 33 = 2Iwd + 2b (Iwa + m w r 2 ) r2 ψ=

(2)

(3)

and τext = [τext1 τext2 τext3 ]T are used to denote external disturbances. B. Nonlinear Disturbance Observer Design This section illustrates the design procedure of an NDO in the MWIP system. To simplify the denotation, we rewrite (1) as vector form Fig. 1.

M(q)q¨ + N(q, q) ˙ + F(q) ˙ = u + τext

MWIP.

(4)

where Section V verified the proposed methods by experiments. Finally, we conclude our results in Section VI. In the rest of this brief, (ˆ·) denotes a nominal value of (·). II. S YSTEM F ORMULATION A. MWIP System Dynamic Model Fig. 1 shows the structure of an MWIP system, where ψr and ψl are the rotation angles of the right and left wheels, respectively, and θ is the inclination angle of the body. α is the yaw angle of the MWIP system. To describe the parameters of the MWIP system, some notations should be clarified first (Fig. 1), which are listed in Table I. Based on Euler–Lagrange equations, Pathak et al. [1] derived a dynamic model of this system. However, they only considered the kinetic energy and potential energy of the whole system. In fact, the energy of the MWIP system itself in the motion process due to factors such as friction will dissipate. Thus, we can improve their model by considering the dissipation energy of the whole system 1 1 1 D = Dw ψ˙ r2 + Dw ψ˙ l2 + Db [(θ˙ − ψ˙ r )2 + (θ˙ − ψ˙ l )2 ]. 2 2 2

q = [q1 q2 q3 ]T = [ψ θ α]T. Consider that M(q) and N(q, q) ˙ are the corresponding additive uncertainties presented in the model of the MWIP. That is, we have ˆ M(q) = M(q) + M(q) ˆ N(q, q) ˙ = N (q, q) ˙ + N(q, q). ˙

(5)

It is assumed that model uncertainties and external disturbances are all bounded. This makes that the lumped disturbance vector is bounded and can be given by ⎡ ⎤  d1    ⎣ ⎦ (6) τd  ≤   d2  = τd max .  d3  The effect of all dynamic uncertainties and external disturbances is lumped into a single disturbance vector τd . From (4), it can be seen that ˆ M(q) q¨ + Nˆ (q, q) ˙ = u + τd .

(7)

To estimate the lumped disturbance τd , the NDO is designed as ˆ ˙ τˆd + L(q, q)( ˙ M(q) q¨ + Nˆ (q, q) ˙ − u) τ˙ˆ d = −L(q, q)

(8)

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where L(q, q) ˙ is the observer gain matrix to be determined. Defining τ˜d = τd − τˆd as the disturbance tracking error and using (8), it is observed that we have ˙ τ˜d τ˙ˆd = L(q, q)

(9)

˙ τ˜d . τ˙˜d = τ˙d − L(q, q)

(10)

or, equivalently

Let us define an auxiliary variable z = [z 1 z 2 z 3 ]T = ˆ ˙ where (d/dt) p(q, q) ˙ = L(q, q) ˙ M(q) q. ¨ τˆd − p(q, q), Substituting it to (8), the observer can be represented by z˙ = L(q, q){ ˙ Nˆ (q, q) ˙ − u − p(q, q) ˙ − z} τˆd = z + p(q, q). ˙

(11)

The disturbance observer gain matrix L(q, q) ˙ and vector p(q, q) ˙ are given by L(q, q) ˙ = L(q) = X −1 Mˆ −1 (q) (12) p(q, q) ˙ = p(q) ˙ = X −1 q˙ where X −1 is a invertible matrix to be determined. Lemma 1: Consider the dynamic model of the MWIP system described by (7) in which the rate of change of lumped disturbance is bounded. The disturbance observer is given in (11) with the disturbance observer gain matrix L(q) and the disturbance observer auxiliary vector p(q) ˙ defined in (12). The disturbance tracking error is globally uniformly ultimately bounded if X −1 =

1 (ξ + 2βσ2 )I3 2

where

⎧ ξ = max mˆ 12 , Ibl |q˙2 | ⎪ ⎪ ⎫ ⎧  ⎪ ⎪   ⎨ ⎪ ⎪ ⎪ ⎪  ⎬ ⎨ J1 + J2 + 4mˆ 212 J3 ⎪ , I σ = max + m ˆ 2 bl 33 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎩ ⎭ ⎩

(13)

ˆ Thus, from [15, eq. (17)] the relation σ2 =  M(q) holds. Then, from (14)–(17), the first condition of [15, Th. 2] is then satisfied. According to [15, Th. 2], the disturbance tracking errors are globally uniformly ultimately bounded by ⎧ ⎨|τ˜d1 | ≤ ζ1 |τ˜d2 | ≤ ζ2 ⎩ |τ˜d3 | ≤ ζ3 . This completes the proof. Remark 1: Unlike the theoretical analysis in [15], the matrix X given by (13) is not constant because ξ is a function of q˙2 . This loosens the bounded condition of the NDO by removing the assumption that the velocity vector should lie in a bounded set [15, eq. (3)]. At the same time, the velocity vector of a real system is normally bounded since the kinematic energy cannot be infinite. For simplicity, in the practical controller design we still use a constant matrix X based on an assumption that there is a maximum absolution value |q˙2 |max . III. C ONTROLLER D ESIGN

(14)

J1 = mˆ 211 + 2mˆ 212 + mˆ 222  2 J2 = mˆ 211 − mˆ 222 2  J3 = mˆ 11 + mˆ 22 . β is the minimum convergence rate of the disturbance tracking error, In ∈ R n×n is the identity matrix. Proof: The proof is similar to that of [15, Th. 2]. First, it is obvious from (13) that the matrix X −1 is invertible. Second, from [15, Th. 3] it can be seen that inequality ˙ˆ X + X T − X T M(q)X ≥

Thus, ξ is chosen to be max{mˆ 12 , Ibl }|q˙2 |. To achieve a tradeoff between the accuracy of the estimations and the noise amplification, an optimal Y can be chosen to be Yoptimal = (1/2)(ξ + 2βσ2 )I3 [15, eq. (58)]. This leads to (13) which ensures that inequality (15) holds. Third, we have  ˆ ˆ  M(q) = λmax ( Mˆ T (q) M(q)) ⎫ ⎧   ⎪ ⎪  ⎪ ⎪ 2 ⎬ ⎨ J1 + J2 + 4mˆ 12 J3 = max , Ibl + mˆ 33 . (17) ⎪ ⎪ 2 ⎪ ⎪ ⎭ ⎩

(15)

is equivalent to inequality Y + Y T − ξ I − Y T Y ≥ 0, where Y = X −1 , is a positive definite and symmetric ˙ˆ matrix and ξ is an upper bound of  M(q). Note that we have  ˙ˆ ˙ ˆ  M(q) = λmax ( M˙ˆ T (q) M(q)) = max{mˆ 12 |q˙2 |, Ibl |q˙2 |} ≤ max{mˆ 12 , Ibl }|q˙2 |. (16)

Let us introduce the following variables: ⎧ x 1 = (mˆ 11 + mˆ 12 cos(x 2 ))x 4 ⎪ ⎪ ⎪ ⎪ + (mˆ 22 + mˆ 12 cos(x 2 ))x 3 ⎪ ⎪ ⎨ x2 = θ x 3 = θ˙ = x˙2 ⎪ ⎪ ⎪ ⎪ x = ψ˙ ⎪ ⎪ ⎩ 4 x 5 = α. ˙

(18)

For convenience of the mathematical derivation, we introduce the following notations in advance: ⎧ Mc12 = mˆ 12 cos(x 2 ) ⎪ ⎪ ⎪ ⎪ Ms12 = mˆ 12 sin(x 2 ) ⎪ ⎪ ⎪ ⎪ ˆ 22 cos(x 2 ) M ⎪ c22 = m ⎪ ⎪ ⎨ G s12 = Gˆ b sin(x 2 ) (19) Ibl = Iˆzzb + mˆ blˆ2 ⎪ ⎪ ⎪ ⎪ A = mˆ 11 mˆ 22 − mˆ 212 cos2 (x 2 ) ⎪ ⎪ ⎪ ⎪ ⎪ A¯ = mˆ 11 mˆ 22 − mˆ 212 ⎪ ⎪ ⎩ B = mˆ 33 + (mˆ b lˆ2 + Iˆzzb ) sin2 (x 2 ). Adding the first equation of (1) to the second, the MWIP system model can then be rewritten

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as   ⎧ mˆ 11 x˙4 + Mc12 x˙3 = Ms12 x 32 + x 52 + u r + u l + τd1 ⎪ ⎪ ⎪ ⎪ ⎪ (mˆ 11 + Mc12 )x˙4 + (mˆ 22 + Mc12 )x˙3 ⎪ ⎪   ⎪ ⎪ ⎨ = Ms12 x 2 + x 2 + Ibl sin(x 2 ) cos(x 2 )x 2 3 5 5 (20) ˆ ⎪ + G b sin(x 2 ) + τd1 + τd2 ⎪ ⎪ ⎪ ⎪(Ibl sin2 (x 2 ) + mˆ 33 )x˙5 = −2Ibl sin(x 2 ) cos(x 2 )x 3 x 5 ⎪ ⎪ ⎪ ⎪ ⎩ ˆ − Ms12 x 4 x 5 + brˆ (u r − u l ) + τd3. Lemma 2: For the MWIP system (20), the global change of coordinates (18) transforms the dynamics of the system into a cascade nonlinear system in semistrict feedback form   x˙1 = −Ms12 x 3 x 4 − x 52 + Ibl sin(x 2 ) cos(x 2 )x 52 (21a) ⎧ + G s12 +τd1 + τd2 x ⎪ 2 d ⎪ ⎪ X˙ 1 = dt ⎪ α ⎪ ⎪ ⎪     1   1  ⎪ ⎪ x 1 , X 1T g11 x 1 , X 1T g12 f 11 x 1 , X 1T ⎪ ⎪ ⎪   +     X2 = ⎪ 1 x , X T g1 x , X T ⎪ f 12 x 1 , X 1T g21 ⎪ 1 1 22 1 1 ⎪ ⎪   ⎪ ⎪ x4 ⎨ d X˙ 2 = dt (21b) x5 ⎪         ⎪ ⎪ ⎪ f 21 x 1 , X 1T , X 2T 1 τd , X 1T , X 2T ⎪ ⎪   +   ⎪ = ⎪ T T ⎪ f 22 x 1 , X 1 , X 2 2 τd , X 1T , X 2T ⎪ ⎪ ⎪  2    2 ⎪ ⎪ ⎪ x 1 , X 1T , X 2T g11 x 1 , X 1T , X 2T g12 ⎪ ur ⎪ ⎪ +     ⎪ 2 x , X T , X T g2 x , X T , X T ⎩ ul g21 1 1 2 22 1 1 2 where ⎧   x1 f 11 x 1 , X 1T = mˆ +M ⎪ ⎪ 22 c12 ⎪ ⎪   ⎪ mˆ 11 +Mc12 ⎪ 1 T ⎪ g11 x 1 , X 1 = − mˆ +M ⎪ ⎪ 22 c12 ⎪   ⎪ ⎪ 1 x , XT = 0 ⎪ g ⎪ 12 1 1 ⎪ ⎪ ⎪   ⎪ ⎪ f 12 x 1 , X 1T = 0 ⎪ ⎪ ⎪ ⎪   ⎪ 1 x , XT = 0 ⎪ g21 ⎪ 1 1 ⎪ ⎪ ⎪   ⎪ 1 x , XT = 1 ⎪ g22 ⎪ 1 ⎪ 1 ⎪ ⎪   ⎪ ⎪ ⎪ f 21 x 1 , X 1T , X 2T ⎪ ⎪ ⎪   ⎪  ⎪ ⎪ x 1 −(mˆ 11 +Mc12 ) x 4 2 1 2 ⎪ ⎪ = A mˆ 22 Ms12 + x5 ⎪ mˆ 22 +Mc12 ⎪ ⎪  ⎪ ⎨ −Ms12 Ibl cos2 (x 2 )x 52 −Mc12 G s12 ⎪   ⎪ 2 x , X T , X T = mˆ 22 +Mc12 ⎪ g11 ⎪ 1 1 2 ⎪ A ⎪ ⎪   mˆ 22 +Mc12 ⎪ ⎪ 2 T T ⎪ g12 x 1 , X 1 , X 2 = ⎪ A ⎪ ⎪     ⎪ ⎪ T , XT = 1 m ⎪  , X τ ˆ ⎪ 1 d 22 τd1 − Mc12 τd2 1 2 A ⎪ ⎪ ⎪ ⎪   ⎪ 2Ibl cos(x 2 ) 1 T T ⎪ ⎪ ⎪ f 22 x 1 , X 1 , X 2 = B − mˆ 22 +Mc12 ⎪ ⎪  ⎪ ⎪ ⎪ · [x − ( m ˆ + M )x ] − m ˆ x sin(x 2 )x 5 ⎪ 1 11 c12 4 12 4 ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ g 2 x 1 , X T , X T = bˆ ⎪ 21 1 2 ⎪ rˆ B ⎪ ⎪ ⎪   ˆ ⎪ 2 T T ⎪ g22 x 1 , X 1 , X 2 = − rˆbB ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎩ 2 τd , X 1T , X 2T = τBd3 .

Similar to [9], after coordinate transformation the MWIP system model is represented in a semistrict feedback form as cascade of a outer (21b) and a core (21a) subsystem. Our purpose is to design a control u r and u l forcing x 2 and α to be stabilized around zero. Together with the proposed disturbance observer, for MWIP system (21) we design a new DSCNDO as follows:       u rDSC ur u u¯ = = u DSC + u d = + rd (22) ul u lDSC u ld where 

u rd u ld



⎤ rˆ −mˆ 22 τˆd1 + Mc12 τˆd2 − τˆd3 ⎥ ⎢ 2(mˆ 22 + Mc12 ) 2bˆ ⎥. =⎢ ⎦ ⎣ −mˆ 22 τˆd1 + Mc12 τˆd2 rˆ + τˆd3 2(mˆ 22 + Mc12 ) 2bˆ ⎡

(23)

The pure DSC component of DSCNDO can be obtained through the following procedure. Step 1: Design the virtual control law x¯4 and x¯5 . 1) Define the first dynamic surface         S11 0 x2 x S1 = = . = 2 − S12 α 0 α

(24)

Then, from the first equation of (21b) the derivative of S1 can be expressed as       S˙ x3 x˙ S˙1 = ˙11 = 2 = α˙ x5 S12 ⎡ ⎤ x1 mˆ 11 + Mc12 − x 4 = ⎣ mˆ 22 + Mc12 mˆ 22 + Mc12 ⎦. (25) x5 2) Select the virtual control law x¯4 and x¯5 as ⎡  ⎤   mˆ 22 + Mc12 x1 k S + x¯4 = ⎣mˆ 11 + Mc12 11 11 mˆ 22 + Mc12 ⎦ (26) x¯5 −k12 S12 where k11 > 0, k12 > 0. 3) Input x¯4 and x¯5 to a first-order filter, respectively, then we have # T11 x˙4d + x 4d = x¯4 , x 4d (0) = x¯4 (0), T11 > 0 (27) T12 x˙5d + x 5d = x¯5 , x 5d (0) = x¯ 5 (0), T12 > 0 where T11 > 0 and T12 > 0 are the filter time constants. The filter errors are defined as follows:     e x − x¯4 e = 1 = 4d . (28) e2 x 5d − x¯5 Step 2: Design the actual control law. 1) Define the second dynamic surface     x 4 − x 4d S21 = . S2 = S22 x 5 − x 5d

(29)

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exists a set of the surface gains k11 , k12 , k21, and k22 , the filter time constant T11 and T12 satisfying γ = min(a1 , a2 , a3 , a4 , a5 , a6 ) > 0, ∃γ

(35)

⎧ ⎪ a1 = k11 − mˆ 11mˆ+mˆ 12 ⎪ ⎪ 22 ⎪ ⎪ ⎪ a = k − 1 2 12 ⎪ ⎪ ⎪ ⎪ ⎨a3 = k21 − k11 − mˆ 11 +mˆ 12 2 2mˆ 22 k12 1 ⎪ a = k − − 4 22 ⎪ 2 2 ⎪ ⎪ ⎪ mˆ 12 ⎪ a5 = T111 − 3k211 − mˆ 112m+ ⎪ ⎪ ˆ 22 ⎪ ⎪ ⎩a = 1 + k12 − 1 6 T12 2 2

(36)

where

Fig. 2.

S˙21

Block diagram of the MWIP system with DSCNDO.

Then, from the second equation of (21b), (22), (23), and (27), the derivative of S2 can be expressed as  $ %  x 1 − (mˆ 11 + Mc12 )x 4 2 1 2 mˆ 22 Ms12 = + x5 A mˆ 22 + Mc12  − Ms12 Ibl cos2 (x 2 )x 52 − Mc12 G s12

mˆ 22 + Mc12 (u rDSC + u lDSC ) A 1 x¯4 − x 4d + (mˆ 22 τ˜d1 − Mc12 τ˜d2 ) − (30) A T11   2Ibl cos(x 2 ) 1 − (x 1 − (mˆ 11 + Mc12 )x 4 ) − mˆ 12 x 4 S˙22 = B mˆ 22 + Mc12 bˆ (u rDSC − u lDSC ) × sin(x 2 )x 5 + rˆ B x¯5 − x 5d τ˜d3 − . (31) + B T12 2) Select the control law u DSC as follows: # u rDSC = 12 (u A + u B ) (32) u lDSC = 12 (u A − u B ) +

where )2

mˆ 22 Ms12 (x 1 − (mˆ 11 + Mc12 )x 4 (mˆ 22 + Mc12 )3 Ms12 (−mˆ 22 + Ibl cos2 (x 2 ))x 52 + Mc12 G s12 + mˆ 22 + Mc12 A(x¯4 − x 4d ) Ak21 S21 + − (33) (mˆ 22 + Mc12 )T11 mˆ 22 + Mc12   rˆ 2Ibl cos(x 2 ) = (x 1 − (mˆ 11 + Mc12 )x 4 ) + mˆ 12 x 4 bˆ mˆ 22 + Mc12 rˆ B(x¯5 − x 5d ) rˆ Bk22 S22 × sin(x 2 )x 5 + − (34) ˆ 12 bT bˆ

uA = −

uB

and k21 > 0, k22 > 0. The whole control system block diagram is shown in Fig. 2. IV. S TABILITY A NALYSIS The stability analysis of the whole system is concluded in the following theorem. Theorem 1: Considering (21) with modeling errors, external disturbance, unknown payloads, and frictions, there

such that the NDO-based dynamic surface controller guarantees: Based on the control law (22), (23), and (32), all signals in the closed-loop system are uniformly and ultimately bounded and exponentially converge to a small ball containing the origin. Proof: Choose the following Lyapunov function candidate: 1 1 1 1 V = S1T S1 + S2T S2 + e12 + e22 . (37) 2 2 2 2 From (24)–(26), (28), and (29), the derivative of S11 can be written as mˆ 11 + Mc12 (S21 + e1 ) − k11 S11 (38) S˙11 = − mˆ 22 + Mc12 and S˙12 = x 5 = S22 + e2 + x¯5 = S22 + e2 − k12 S12 .

(39)

Then, from (30)–(32), the derivative of S21 and S22 are given by 1 S˙21 = −k21 S21 + [(mˆ 22 + Mc12 )τ˜d1 − Mc12 τ˜d2 ] A

(40)

and τ˜d3 . (41) S˙22 = −k22 S22 + B From (21a) and (25)–(27), the derivative of e1 is given by e1 G s12 + k11 (S21 + e1 ) − T11 mˆ 11 + Mc12 2 S (mˆ 22 + Mc12 )k11 τd1 + τd2 11 − + mˆ 11 + Mc12 mˆ 11 + Mc12 (mˆ 12 + Ibl cos(x 2 )) sin(x 2 )(S22 + e2 − k12 S12 )2 − mˆ 11 + Mc12 Ms12 (S21 + e1 + k11 S11 )(S21 + e1 ) − mˆ 22 + Mc12 Ms12 k11 S11 (S21 + e1 + k11 S11 ) − . (42) mˆ 11 + Mc12 In addition, from (26)–(28) and (39), the derivative of e2 is given by e2 e˙2 = x˙5d − x¯˙5 = − − k12 S˙12 T12 e2 2 =− − k12 S22 − k12 e2 + k12 S12 . (43) T12 e˙1 = −

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From (38)–(43) and according to the Young,s inequality, it follows that:   mˆ 11 + mˆ 12 2 2 ˙ S11 − (k12 − 1)S12 V ≤ − k11 − mˆ 22   k11 mˆ 11 + mˆ 12 2 − S21 − k21 − 2 2mˆ 22   1 k12 − − k22 − S2 2 2 22   1 3k11 mˆ 11 + mˆ 12 2 − − − e1 T 2 2mˆ 22  11  1 2 mˆ 22 + mˆ 12 2 1 k12 − − + τ˜d1 e + T12 2 2 2 2 A¯ 2 τ˜ 2 τ 2 + τd2 mˆ 12 2 + + ϕ1 (·) (44) τ˜d2 + d3 + d1 2mˆ 33 2mˆ 11 2 A¯ where ϕ1 (·) is a nonnegative continuous function satisfying & & 2 G s12 e1 0 ≤ &&k12 S12 e2 − mˆ 11 + Mc12 2 (mˆ 22 + mˆ 12 )S21 e2 S2 + + 22 + 1 2mˆ 33 mˆ 11 2 A¯ (mˆ 12 + Ibl cos(x 2 )) sin(x 2 )e1 (S22 + e2 − k12 S12 )2 − mˆ 11 + Mc12 2 S e (mˆ 22 + Mc12 )k11 11 1 + mˆ 11 + Mc12 Ms12 (S21 + e1 + k11 S11 )(S21 + e1 )e1 − mˆ 22 + Mc12 & Ms12 k11 S11 (S21 + e1 + k11 S11 )e1 && − & mˆ + M 11

c12

≤ ϕ1 (k11, k12 , S11 , S12 , S21 , S22 , e1 , e2 ).

(45)

Given any p > 0, let us introduce a set  = {(S11 , S12 , S21 , S22 , e1 , e2 ) : V (t) ≤ p}. Apparently, set  is compact in R 6 . Therefore, the continuous function ϕ1 (·) has a maximum, say M on . It follows that:   mˆ 11 + mˆ 12 2 2 ˙ V ≤ − k11 − − (k12 − 1)S12 S11 mˆ 22   k11 mˆ 11 + mˆ 12 2 − − k21 − S21 2 2mˆ 22   1 k12 − − k22 − S2 2 2 22   1 3k11 mˆ 11 + mˆ 12 2 − e1 − − T 2 2mˆ 22   11 1 2 mˆ 22 + mˆ 12 2 1 k12 − e + ζ1 − + T12 2 2 2 2 A¯ +

ζ2 d 2 + d22 mˆ 12 2 + M. ζ2 + 3 + 1 2mˆ 33 2mˆ 11 2 A¯

If the following inequalities are satisfied: ai > 0, i = 1, 2, 3, 4, 5, 6

(46)

V˙ ≤ −2γ V + M1

(47)

then, we have

Fig. 3.

Photograph of the MWIP system. TABLE II E XPERIMENTAL PARAMETERS OF MWIP S YSTEM

where mˆ 22 + mˆ 12 2 mˆ 12 2 ζ1 + ζ2 2 A¯ 2 A¯ ζ2 d 2 + d22 + 3 + 1 2mˆ 33 2mˆ 11 γ = min(a1 , a2 , a3 , a4 , a5 , a6 ) > 0.

M1 = M +

(48) (49)

After solving the differential inequality (47), we have V (t) ≤ V (0)e−2γ t + M1 /(2γ )(1 − e−2γ t ) ∀t ∈ [0, ∞). By the selections of k11, k12 , k21 , and k22 and T11 and T12 , we can make γ > M1 /2 p. This results in V˙ ≤ 0 on V = p. Thus, V ≤ p is an invariant set, i.e., if V (0) ≤ p then V (t) ≤ p for all t ≥ 0. Therefore, V (t) is bounded. This implies that (S11 , S12 , S21 , S22 , e1 , e2 ) are uniformly and ultimately bounded and exponentially converge to a small ball containing the origin. V. E XPERIMENT S TUDY A. Hardware Implementation Fig. 3 shows the overall robot system used in the experiments. A container is fixed on the base of the robot body and a weight of 0.35 kg is placed in the box as test load. The parameters of the MWIP mechanical platform are given in Table II. Fig. 4 shows the corresponding control hardware for the MWIP, which consists of a main control circuit board, a threeaxis gyro and an accelerometer. The main control circuit board is designed based on a 32-b ARM Cortex-M3 microcontroller (LM3S2965, Texas Instruments), the working frequency of which is 25 MHz. The system sampling rate was designed as T = 5 ms.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. HUANG et al.: NDO-BASED DSC OF MWIP

Fig. 4.

7

Control hardware for the MWIP system. TABLE III E XPERIMENT PARAMETERS OF C ASES 1 TO 2

Fig. 6. Balance control results of the MWIP system by employing LQR, DSC, and DSCNDO control strategies with relative large mass and height of the container. (a) Tilt angles. (b) Yaw angles.

Fig. 5. Balance control results of the MWIP system by employing LQR, DSC, and DSCNDO control strategies with nominal parameters. (a) Tilt angles. (b) Yaw angles.

B. Experimental Results The experiments were implemented in the practical MWIP-based autonomous robot system. Three cases (Case 0, Case 1, and Case 2) were studied in the experiments. In the first two cases, the robot was controlled to keep its balance on the flat ground with different group of parameters. While the balance control of robot was studied on a slope in Case 2. The physical parameters of all cases are given in Table III. The parameters of Case 0 were used to represent a nominal case (Table II), from which the DSCNDO is derived. Based on the physical parameters of MWIP-based autonomous robot system, the parameters of proposed DSCNDO are chosen as k11 = k12 = k21 = k22 = 35 T11 = T12 = 0.015, X −1 = 0.0345I3. To investigate whether the proposed DSCNDO achieves better performance in comparison with alternative control

Fig. 7. Balance control results of the MWIP system by employing LQR, DSC, and DSCNDO control strategies with nominal parameters in sloped plane. (a) Tilt angles. (b) Yaw angles.

approaches, an Linear Quadratic Regulator (LQR) controller was also applied for the balance control of the robot. The LQR control coefficients were chosen as K 11 = 10.1196, K 12 = 0.3844, K 13 = 1.0584, K 14 = −0.3177 K 21 = 10.1196, K 22 = 0.3844, K 23 = 1.0584, K 24 = 0.3177 and Q = 5I4 , R = 25I2 . Therefore, the LQR control is given by u LQRr = K 11 θb + K 12 ψ˙ + K 13 θ˙b + K 14 α˙ ˙ u LQRl = K 21 θb + K 22 ψ˙ + K 23 θ˙b + K 24 α. The balance control results of the robot system by employing LQR, DSC, and DSCNDO are shown in Figs. 5–7. The rms errors of the tilt angles and yaw angles are shown in Table IV. From the experiment results of Cases 0 to 2,

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TABLE IV RMS E RRORS OF THE T ILT A NGLES AND YAW A NGLES

it turns out that the control performance and disturbance suppression effect of the proposed DSCNDO is superior to those of conventional LQR and DSC methods. Note that there still exists steady-state error when using DSCNDO, which may be caused by the time-varying model uncertainties or external disturbances in the real robot system. VI. C ONCLUSION Compared with pure DSC and conventional LQR controller, the new controller presents better performance which is verified by experiments. In summary, the major contributions of this brief can be listed as follows. 1) Based on the improved dynamic model of the MWIP system, novel global coordinate transformation was proposed to achieve the DSC control of an MWIP system. In general, it is difficult to design a DSC controller for a non-Class-I type underatcuated mechanical system because DSC requires strict or semistrict feedback form of the dynamic model. 2) To compensate for parametric uncertainties in a real MWIP-based robot system as well as external disturbances, we combined the proposed DSC controller and an NDO. 3) Experimental studies were undertaken by employing LQR, DSC, and DSCNDO for balance control of the MWIP system.

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