Robust disturbance rejection control of a biped robotic system using high-order extended state observer.

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Robust disturbance rejection control of a biped robotic system using high-order extended state observer$ Nadhynee Martínez-Fonseca a, Luis Ángel Castañeda b, Agustín Uranga c, Alberto Luviano-Juárez b,n, Isaac Chairez a a

Department of Bioprocessess UPIBI-IPN, Av. Acueducto S/N Col., Barrio La Laguna Ticomán, C.P. 07340 México, D.F., Mexico Instituto Politécnico Nacional - UPIITA Av. IPN 2580 Col., Barrio La Laguna Ticomán D.F. México, Mexico c Universidad Politécnica de Cuautitlán Izcalli, Department of Biomedical Engineering, Av Constitución 1000, Cumbria, 54740 Cuautitlán Izcalli, Mexico b

art ic l e i nf o

a b s t r a c t

Article history: Received 7 May 2015 Received in revised form 22 January 2016 Accepted 3 February 2016 This paper was recommended for publication by Dr. Dong Lili

This study addressed the problem of robust control of a biped robot based on disturbance estimation. Active disturbance rejection control was the paradigm used for controlling the biped robot by direct active estimation. A robust controller was developed to implement disturbance cancelation based on a linear extended state observer of high gain class. A robust high-gain scheme was proposed for developing a state estimator of the biped robot despite poor knowledge of the plant and the presence of uncertainties. The estimated states provided by the state estimator were used to implement a feedback controller that was effective in actively rejecting the perturbations as well as forcing the trajectory tracking error to within a small vicinity of the origin. The theoretical convergence of the tracking error was proven using the Lyapunov theory. The controller was implemented by numerical simulations that showed the convergence of the tracking error. A comparison with a high-order sliding-mode-observer-based controller confirmed the superior performance of the controller using the robust observer introduced in this study. Finally, the proposed controller was implemented on an actual biped robot using an embedded hardware-in-the-loop strategy. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Active disturbance rejection Extended state observers Biped robot Nonlinear mechanical systems Disturbance observers

1. Introduction Biped locomotion is considered an active field in robotics and mechatronics [1]. The number of possible applications for biped robots is continuously increasing. Among others, human interactive robots, military exoskeletons, gait cycle rehabilitation systems, and augmented reality systems have benefited from recent advances in designing, constructing, and controlling this type of robotic system. Automatic control design for biped robots has received special attention because of the wide variety of challenges that this task entails. Most of the existing literature in this field has considered the problem of generating a stable dynamical walking cycle. A large set of different control strategies has been successfully proposed and tested on different types of biped robots [2]. ☆ This work was supported by the Secretaría de Investigación y Posgrado (SIPIPN) and CONACyT, México, under research Grants SIP-20150279, SIP-20140274, Conacyt CB-221867. n Corresponding author. Tel.: þ 52 55 57296000x56918. E-mail addresses: [email protected] (N. Martínez-Fonseca), [email protected] (L.Á. Castañeda), [email protected] (A. Uranga), [email protected] (A. Luviano-Juárez), [email protected] (I. Chairez).

A different type of solution emerged recently in which the climbing, running, and one-legged hopping problems have been tackled. These controllers consider different paradigms in which the robot is modeled as a switched system affected by intense perturbations. Moreover, the new class of controllers used to solve these problems is adjusted to control biped robots in rough and/or difficult terrains. Despite the nature of these new problems in control design for biped robots, more general and more robust control laws are demanded. One popular approach to solve this task divides the robot into smaller and simpler subsystems; the result is usually called a decentralized control system [3]. However, many of these studies consider the possibility of using a state feedback control form, which is not a realistic solution. Measuring the position and velocity of biped robots is not impossible, but it is expensive and compromises their mechanical design. Considering the complications in implementing state feedback controls as a feasible solution for biped robots, it is more natural to consider output-based control strategies. Some remarkable solutions have been developed considering this possibility. However, many of them require an exact and unperturbed mathematical representation of the biped robot. Others have used high-gain

http://dx.doi.org/10.1016/j.isatra.2016.02.003 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Martínez-Fonseca N, et al. Robust disturbance rejection control of a biped robotic system using high-order extended state observer. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.003i

2

N. Martínez-Fonseca et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

observers that neglect the partial knowledge over the biped robot model. One remarkable option for controlling biped robots is using sliding mode control [4]. In particular, finite-time estimation of the velocity of biped robot joints has been solved recently [5]. These estimations are injected into a discontinuous controller that also provides finite-time tracking of a reference trajectory that corresponds to a regular dynamical gait cycle. Another popular reported technique uses adaptive schemes (see [6,7], and references therein). These methods have shown excellent results, and, in some cases, robustness is included as part of the solution. Even when the response of these schemes is quite competitive, the computational cost involved may be higher than that of other classic approaches such as proportional integral derivative (PID) controllers. One additional possibility is using control schemes based on active disturbance rejection control (ADRC) [8,9]. This control philosophy has been proposed for solving many different problems (see [10–15]), as the complexity in the control design can be reduced by lumping all the nonlinearities as well as the external disturbances into a generalized term. Indeed, this is a strong motivation to design such a class of robust decentralized controllers. The ARDC method proposes to estimate the lumped term mentioned above. If an accurate estimation is obtained, then it can be used in the control action that implements a disturbance cancelation method. The lumped disturbance input can be estimated by nonlinear and linear extended state observer schemes. Concerning linear methods, in [16], a high-order extended state observer was proposed to solve the problem of estimating highfrequency sinusoidal disturbances in the context of increasing the bandwidth of the estimator. This work offers a comprehensive frequency-domain analysis of the use of high-order extended state observers by means of linear analysis. On the other hand, using the same principle, [17–20] proposed a high-order extended state Luenberger-like observer (defined as a generalized proportional integral observer, or GPI) for the robust trajectory tracking problem for nonlinear differentially flat systems [21]. The observer includes a time-velocity dating, linear model approximation of the perturbation input. Thus, ADRC is a new robust approach that takes advantage of the classic PID control strategy and modern control technology. A perturbation and nonlinearity approximation is generally placed in the feedforward path, and an extended state observer appears in the feedback path. These two parts are designed to obtain a highquality and accurate approximation of unknown disturbance signals without establishing an accurate plant model. Some studies have considered the approximation of the uncertain section of a biped robot [22,23]. These control designs use adaptive algorithms to obtain accurate approximations, but they require exhaustive preliminary adjustments before they can be used. Moreover, they require additional efforts to include new scenarios that were been considered in the pre-adjustment procedure. Therefore, a self-adjusting scheme seems to be a more promising option for approximating the uncertain section of a biped robot. Nevertheless, it demands a more complicated approximation algorithm structure with more robust control strategies. The main contribution of this study is the design of an ADRC output-based multivariable controller that uses the estimation of the lumped uncertainties provided by a linear extended state observer. This new approach to biped robotic systems allows tracking of a reference trajectory that represents a standard gait cycle obtained by a biomechanical study. A new formal analysis based on the Lyapunov stability theory provides the conditions for obtaining the ultimate boundedness of the tracking deviation as

well as the estimation error; this analysis allows the estimation of a larger class of disturbances because it is free of differentiability conditions. The estimation of the lumped uncertainties is solved by a least mean square algorithm, which is also included in the Lyapunov analysis. A set of numerical simulations is presented to demonstrate the effectiveness of the proposed output-based controller, and real experiments using a biped robot prototype are presented in which the proposed controller provides rapid convergence with small steady-state errors. In addition, comparisons with a variable structure observer-based control scheme are provided. The remainder of the paper is organized as follows: the model structure of the biped robot is given in Section 2. Section 3 describes the problem formulation, and Section 4 presents some preliminary properties necessary to solve the ADRC design. Then, in Sections 5 and 6, the main theoretical contributions are presented, including a novel Lyapunov-based analysis of the GPI observer-based control and an alternative gain optimization procedure based on the invariant ellipsoid algorithm. The reference trajectory planning criterion is given in Section 7. The numerical results are shown in Section 8. Section 9 presents the experimental results. In both the numerical and experimental results, a comparison test is provided. Finally, some concluding remarks are presented.

2. Mathematical description of the biped robot The robotic system considered in this study is presented in Fig. 1. This robot can move freely only in the x–y plane; it contains five links and is defined by seven degrees of freedom. The corresponding seven coordinates are shown on the left side of Fig. 1. T These are given by the set q ¼ x0 y0 α β L βR γ L γ R , where the coordinates x0 ; y0 correspond to the position of the torso center of mass; α is the angle formed between the torso and the vertical axis; βL and βR are the angles formed between the upper sections of the left and right legs, respectively, and the torso; and γL and γR are the angles formed between the upper and lower sections of the left and right legs, respectively. The links' lengths are denoted as l0 ; l1 ; l2 , and their masses are denoted as m0 ; m1 ; m2 . The links' centers of mass are located at the distances r 0 ; r 1 ; r 2 from the corresponding joints (see right side of Fig. 1). The model is actuated with four moments given by M ¼ ½M L1 M L2 M R1 M R2 T . Two of these moments act between the torso and either thigh, and the other two act at the knee joints. The walking surface is modeled using a set of external forces, T denoted by F ¼ F Lx F Ly F Rx F Ry , that affects the leg tips. When the leg is touching the ground, the corresponding forces are switched to support the leg. As the leg rises, the forces are zeroed. Using the Euler–Lagrange modeling process, the dynamic equations for the biped system can be derived. These equations can be represented as dq ¼ b q; ; MðqÞ; FðtÞ; τðtÞ dt dt 2

d q 2

ð1Þ

Here M ðqÞ A Rnn ; n ¼ 7, is the inertia matrix, and b q; dq ; MðqÞ; FðtÞ; τðtÞ A Rn is a vector containing the right-hand dt sides of the seven differential equations. The vector τ corresponds to the external torques applied over each articulation (hip, ankles, and knees). The matrix M ðqÞ and the vector b q; dq ; MðqÞ; FðtÞ; τðtÞ dt

are defined in [24].



3

Fig. 1. Biped robot diagram showing state variables as well as the forces involved in its movement.

The right-hand side of the differential equations describing the biped robot can be represented as follows: dq dq b q; ; MðqÞ; F; τ ¼ GðqÞ þ R q; λþτ ð2Þ dt dt where the matrix GðqÞ A Rn is the gravity vector, the matrix R q; dq dt : R2n -Rnp represents the restriction matrix, λ A Rp is the constraints vector (associated with the Lagrange multipliers), and τ A Rn is the input vector.

3. Control challenge in biped robots The objective of the control design for the biped robot is defined as follows. Problem statement: Consider an alternative way of describing the mathematical model of the biped robot (1) affected by any possible external perturbations and considering the presence of unmodeled dynamics (both lumped in the term ξðq; tÞÞ:

2 d q dq 1 1 ¼ M ðqÞ τ ðtÞ þ ξ ðq; tÞ M ðqÞ GðqÞ þ R q; λ ð3Þ 2 dt dt Now, consider ψ ðq; tÞ, a vector that includes some of the dynamical taken as uncertain, such as h model terms i þ Þ λ ξ ðq; tÞ, and some additive disturbance M 1 ðqÞ GðqÞ þ Rðq; dq dt terms due to mechanical coupling in the gear train of the motors, or various unmodeled perturbation terms. Then, the dynamic system (3) can be presented as 2

d q dt

2

¼ M 1 ðqÞτðtÞ þ ψ ðq; tÞ

ð4Þ

Consider the alternative mathematical representation of the biped robot, and define a set of desired articular positions and torso coordinates qn. Then the control challenge is to ensure convergence of the error between the articulation positions and the reference (desired) trajectories in an ultimate boundedness sense (see [25,26] for further information), despite the presence of the set of disturbances and uncertainties lumped in ψ (according to Assumption 1). Formally, the control design must solve the

problem lim supt-1 maxψ A Ψ J qðtÞ qn ðtÞ J r β, where β is a positive scalar that must be proportional to the power of the uncertainties and perturbations affecting the nominal model of the biped robot. A feasible control design must consider that only position measurements can be used, and no velocity data are available.

4. Biped robot as interconnected uncertain second-order systems The model (4) belongs to a class of coupled second-order nonlinear systems with uncertain structure. On the basis of the state variable theory, the biped robot dynamic system can be represented as follows: dxa ðtÞ ¼ xb ðtÞ dt dxb ðtÞ ¼ f ðxÞ þ gðxa ðtÞÞuðtÞ þ ξðx; tÞ dt

ð5Þ

where x ¼ ½xa> ; xb> > ; x A R2n , is the state of the robotic system. . The initial condition of the system (5) is Indeed, xa ¼ q and xb ¼ dq dt given by xð0Þ ¼ x0 ; J x0 J o1.



4

Let X be the set such that x A X R2n . The bounded function u A Rn is referred to as the control function. All the controls belong to the following admissible set: n o ð6Þ U adm ¼ u : J u J 2 r u0 þ u1 J x J 2 ; u0 ; u1 A R þ 2n

n

The nonlinear Lipschitz function f : R -R is composed of nuncertain nonlinear functions that describe the drift term of (5). The matrix function g : Rn -Rnn is invertible and meets the following constraint: 0 o g r J gðÞ J rg þ g ;

g þ ARþ

ð7Þ

As mentioned before, the term ξðx; tÞ was included to consider the effects of perturbations/uncertainties that can also include, for example, parameter variations and modeling errors. In this study, an important assumption regarding ξðx; tÞ is considered: Assumption 1. The class of uncertainties ξðx; tÞ considered in this study satisfies the following inequality: ‖ξðx; tÞ‖2Λξ r γ 0 þ γ 1 J x J 2 ;

γ0; γ1 A R þ ;

8 t Z0 >

0 o Λξ ¼ Λξ A Rnn

Z þ⋯þ

t

τ1 ¼ 0

⋯

Z τp 1 τp ¼ 0

p!ap dτ p ⋯dτ1

ð10Þ

The last equation can be expressed in differential form: a > κ ðtÞ ¼ ρ0 ðtÞ dρi ðtÞ ¼ ρi þ 1 ðtÞ; i ¼ 0; …; p 1 dt dρp ðtÞ ¼ 0 ρj ð0Þ ¼ aj ; j ¼ 0; 1; …; p dt

ð11Þ

Now the problem formulation given above can be rephrased as follows. Given an output reference trajectory xn for the system (5), design an output feedback controller that, regardless of the unknown unmodeled dynamics or external disturbances (both lumped in an additive signal F(x)), forces the states x to track asymptotically the desired reference trajectories, with the tracking error restricted to a small neighborhood near the origin and proportional to a power of the uncertainties and perturbations. The first stage in solving this problem is designing an extended state observer to reconstruct the unmeasurable part of the state.

ð8Þ 5. The high-order extended state observer The extended state observer for the system (5) (with the corresponding feedback controller using the observer states ^ uðtÞ≔uðxðtÞÞ) is proposed as

The system (5) can be rewritten as follows: dxðtÞ ¼ AxðtÞ þ Gðxa ðtÞÞuðtÞ þ Fðx; tÞ dt " " # # 0n 0nn I nn A¼ ; GðxÞ ¼ gðxa Þ 0nn 0nn h i> > > Fðx; tÞ ≔ 0n> f ðxÞ þ ξ ðx; tÞ

ð9Þ

The function F(x) satisfies the following inequality for all t Z 0 : J Fðx; tÞ J 2 r f 0 þ f 1 J x J 2 ; f 0 ; f 1 A R þ . The following extra assumption regarding the function F(x) is needed to propose the controller design. Assumption 2. There is a constant vector a such that the function f ðxÞ þ ξðxðtÞ; tÞ evaluated over the system trajectories, that is, x ¼ xðtÞ, can be represented as f ðxÞ þ ξðx; tÞ ¼ a > κ ðxÞ þ f~ ðx; tÞ, where the vector a A Rðp þ 1Þn is formed with a set of constant parameters ak that must be adjusted to improve the approximation of f ðxÞ þ ξðx; tÞ. According to the structure of the so-called GPI extended state observers, the vector κ A Rp þ 1 is composed as follows: κ ¼ ½1; t; …; t p . This is a regular decomposition of f ðxÞ þ ξðx; tÞ in terms of a finite number of elements that form a basis. In particular, the last set of polynomials is considered in this study (see [27,20] for further details). The term f~ ðx; tÞ is called the modeling error produced by the approximation of f ðxÞ þ ξðx; tÞ by a finite number p of elements in the basis. The so-called nominal model a > κ ðtÞ can be represented as a > κ ðtÞ ¼ a0 þ a1 t þ a2 t 2 þ⋯ þ ap t p . Each term aj A Rn ; j ¼ 0; …; p. This representation has been used in various articles regarding the application of high-order extended observers [17]. However, this approximation method usually requires the differentiability of the term f ðxÞ þ ξðx; tÞ, which is a strong restriction. In this study, an alternative to this problem is proposed in which the function a > κ ðxðtÞÞ can be represented as a chain of integrators. Thus, the approximation presented above states that f ðxÞ þ ξðx; tÞ must be the solution of integration of an uncertain function plus the approximation error. This condition relaxes the constraint that is usually a drawback in this class of extended state observers. Thus, the previous equation is written as Z τ1 Z t Z t a > κ ðtÞ ¼ a0 þ a1 dτ1 þ 2a2 dτ2 dτ1 τ1 ¼ 0

τ1 ¼ 0 τ 2 ¼ 0

^ dxðtÞ ^ þGðxa ðtÞÞuðtÞ þ Bρ^ 0 ðtÞ þ HeðtÞ ¼ AxðtÞ dt dρ^ ðtÞ ¼ Φρ^ ðtÞ þ LeðtÞ dt 2 39 0nn I nn 0nn ⋯ 0nn > > > 60 7> > 6 nn 0nn I nn ⋯ 0nn 7> 6 7= 7 ptimes ⋮ ⋮ ⋮ ⋯ ⋮ Φ¼6 6 7> 60 7> > 4 nn 0nn 0nn ⋯ I nn 5> > > ; 0nn 0nn 0nn ⋯ 0nn

ρ^ 0 ðtÞ ¼ D > ρ^ ðtÞ;

^ eðtÞ ¼ xa ðtÞ C > xðtÞ

ð12Þ

Here the vectors x^ and ρ^ are the estimated states for x and ρ, respectively. The matrix H A R2nn is the state observer gain. The matrix L A Rnðp þ 1Þn is the extended observer gain. The matrix D A Rnðp þ 1Þp is represented as D > ¼ ½I nn 0nn ⋯ 0nn . The matrix Φ A Rnðp þ 1Þnðp þ 1Þ is called the extended state self-matrix. The output matrix C A R2nn is given by C > ¼ ½I nn 0nn . Finally, the matrix B A R2nn is represented as B > ¼ ½0nn I nn . On the basis of the extended state observer proposed in (12), the output-based controller can then be designed using the following structure: ^ uðtÞ ¼ gðxa ðtÞÞ 1 K > σ ðxðtÞ; xn ðtÞÞ þ ρ^ 0 ðtÞ sðxn ðtÞÞ ð13Þ The variable σ is the tracking error, σ ≔ x^ xn . The vector xn A R2n is the reference trajectory and satisfies n

dx ðtÞ ¼ Axn ðtÞ þ Bsðxn ðtÞÞ; dt

xn ð0Þ ¼ xn0 A R2n

ð14Þ

where s : Rn -Rn is a Lipschitz function. The reference trajectory is 2 designed to be bounded 8 t Z 0; that is, J xn ðtÞ J 2 r xnmax , þ n xmax A R . The controller gain matrix K is designed such that the matrix A BK is Hurwitz. To solve this part of the problem, by using Ackermann's approach [28], the matrix K can be obtained as K ¼ ½0nn 0nn 0nn ⋯ I nn S 1 νðAÞ h i S ¼ B BA BA2 ⋯ BAn

νðAÞ ¼ An þ a1 An 1 þ ⋯ þ an I nn

ð15Þ



tracking trajectory error ε ¼ xn xþ n is ultimate n bounded withoan upper f þ 4f 1 ½xmax , αQ ≔λmin P 1=2 Q 0 P 1=2 . bound given by β ¼ 2 0 αQ

Comparison of actual angular velocity of left knee

Angular vel (rad/s)

1.5 1

Proof. The estimation error

0.5

0

1

2

3

4

5

6

ð19Þ

7

Time (s)

The approximation proposed in (11) makes it possible to transform the previous differential equation into

Comparison of actual angular velocity of left knee: Zoom

1.5

Angular vel (rad/s)

Δ satisfies

dΔðtÞ ¼ A HC > ΔðtÞ þ FðxðtÞÞ Bρ^ 0 ðtÞ dt

0

dΔðtÞ ¼ ½A HC > ΔðtÞ þBf~ ðxðtÞ; tÞ þ BD > ρðtÞ ρ^ ðtÞ dt

1

ð20Þ

0.5

On the other hand, the extended state error, defined as

δ ¼ ρ ρ^ , and the tracking error σ satisfy

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Time (s) Comparison of MSE of the whole estimation error

1

Estimation error

5

0.8 0.6

ð21Þ

dσ ðtÞ ^ þGðxa ðtÞÞuðtÞ þ Bρ^ 0 ðtÞ þ HeðtÞ sðxn ðtÞÞ ¼ AxðtÞ dt

ð22Þ

If the control action described by (13) is substituted in the previous differential equation, one gets

0.4 0.2 0

dδðtÞ ¼ ΦδðtÞ LC > ΔðtÞ dt

0

0.5

1

1.5

2

2.5

3

dσ ðtÞ ¼ ðA BKÞσ ðtÞ þ HC > ΔðtÞ dt

ð23Þ

Time (s)

Fig. 2. Top and center: Comparison of actual angular velocity of left knee (green solid line) with that obtained using the proposed observer (blue dashed line) and the ST used as a differentiator (red dashed line). Bottom: Comparison of mean square error (MSE) of the entire estimation error obtained by the proposed observer (blue solid line) and the ST used as a differentiator (red dashed line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

To prove that the equilibrium point of (20) and (21) is stable, the following Lyapunov function candidate is proposed: VðzÞ ¼ z > Pz

ð24Þ >

>

where the vector z A R is constructed as z ¼ ½Δ δ σ . Taking the Lie derivative of V(z) with respect to time, one has dVðtÞ ¼ 2z > ðtÞ P dzðtÞ . dt dt >

Np

>

From (20) and (21), the time derivative of z is The main result of this study is developed as follows.

2

Theorem 1. Consider the state observer given in (12) and the output feedback controller proposed in (13) with the gain adjusted according to (15) for the class of nonlinear uncertain systems (5) fulfilling the condition (7) with incomplete information and affected by perturbations that obey the constraints given in (8). Define the matrices Π ðH; L; KÞ, R, and Q described by 2

A HC > 6 > Π ðH; L; KÞ ¼ 6 4 LC HC > 2

2f 1 I 6 R ¼ Λþ4 0 0

0 0 0

BD >

Φ 02n2n

3

02n2n 7 0nðp þ 1Þnðp þ 1Þ 7 5

ð16Þ

A BK

3 0 0 7 5 4f 1 I

dzðtÞ 6 > ¼6 4 LC dt > HC

BD >

Φ 02n2n

3

2 ~ 3 Bf ðxðtÞ; tÞ 7 6 7 0nðp þ 1Þnðp þ 1Þ 7 5zðtÞ þ 4 0nðp þ 1Þ 5 0n A BK 02n2n

ð25Þ Using this result in the previous differential equation, one gets dVðtÞ ¼ 2z > ðtÞP Π ðH; L; KÞzðtÞ þ2z > ðtÞP ηðtÞ dt

ð26Þ

where η represents all the internal uncertainties and external perturbations. The direct application of the matrix inequality X T Y þY T X r X T NX þ Y T N 1 Y, which is valid for any X; Y A Rrs and any 0 o N ¼ N T A Rss [29], yields

ð17Þ

>

where Λ ¼ Λ A RNp Np ; Λ 4 0, Q ¼ Q 0 ; Q 0 ¼ Q 0> A RNp Np ; Q 0 4 0, and N p ¼ 4n þ nðp þ 1Þ. If there is a positive definite matrix Q0 such that the algebraic Riccati equation RicðPÞ ¼ 0 with RicðPÞ ¼ P Π ðH; L; KÞ þ Π ðH; L; KÞ > P þ PRP þ Q

A HC >

ð18Þ

has a positive definite symmetric solution P A RNp Np , then the

dVðtÞ 1 r z > ðtÞ P Π ðH; L; KÞ þ Π ðH; L; KÞ > P þP ΛP zðtÞ þ η > ðtÞΛ ηðtÞ dt ð27Þ An algebraic manipulation involving the term z > ðtÞQ 0 zðtÞ yields dVðtÞ r z > ðtÞðP Π ðH; L; KÞ þ Π ðH; L; KÞ > P þ P ΛP þ Q 0 ÞzðtÞ dt þ η > ðtÞΛ

1

ηðtÞ z > ðtÞQ 0 zðtÞ

ð28Þ



By taking the upper bound presented in (8), the term η > Λ for all t Z0 is bounded from above by

η > Λ 1 η rf 0 þ 2f 1 J Δ J 2 þ 4f 1 J σ J 2 þ4f 1 J xn J 2

1

η

ð29Þ

Comparison of reference angle proposed for the left knee Angle (rad)

6

where β 0 ¼ f 0 þ4f 1 ½xn þ . Using the assumption regarding the existence of a positive definite solution to the Riccati equation RicðPÞ ¼ 0, the previous differential inclusion is modified to

ð32Þ

The last inequality was obtained using the so-called Rayleigh inequality and the Cholesky decomposition [29]. According to the definition of β, one has the following differential inequality: dVðtÞ r αQ V ðtÞ þ β0 dt

3

4

5

6

7

6 Biped System ST GPI

5.5 5

0

0.1

0.2

0.3

0.4

0.5

0.6

3 2

GPI ST

1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (s) Fig. 3. Top and center: Comparison of reference angle proposed for left knee (green solid line) with the trajectories obtained using the proposed observer as a velocity estimator (blue dashed line) and using the ST as a differentiator (red dashed line). Bottom: Comparison of MSE of the entire tracking error caused by the proposed observer (blue solid line) and the ST used as a differentiator (red dashed line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

to be state-invariant for the system (25) with respect to the uncertainties η if

eq

dV ¼ αQ V eq þ β0 dt

ð34Þ

The solution of the previous differential equation is

β0 ð1 e αQ t Þ αQ

ð35Þ

By applying the comparison principle [25] to the previous equation and using the fact that VðtÞ Z 0, one has VðtÞ re αQ t Vð0Þ þ

2

ð33Þ

Taking the equality in the previous differential inclusion, one has

V eq ðtÞ ¼ e αQ t V eq ð0Þ þ

1

Comparison of MSE of the whole tracking error

ð31Þ

The right-hand side of the previous differential inclusion can be bounded from above as n o dVðtÞ ¼ λmin P 1=2 Q 0 P 1=2 z > ðtÞPzðtÞ þ β 0 dt

0

Time (s) Angle (rad)

dVðtÞ r z > ðtÞQ 0 zðtÞ þ β0 dt

4

Comparison of reference angle proposed for the left knee: Zoom Angle (rad)

ð30Þ

Biped System ST GPI

6

Time (s)

On the basis of this inequality, the differential inclusion presented in (28) is transformed into dVðtÞ r z > ðtÞRicðPÞzðtÞ z > ðtÞQ 0 zðtÞ þ β0 dt

8

β0 ð1 e αQ t Þ αQ

ð36Þ

Taking the upper limit of the previous inequality, the main result presented in the theorem is finally obtained.□

The possibility of finding a possible invariant ellipsoid with the largest matrix P yields the minimal deviation of all the possible trajectories from the origin. This fact is easily understood because of the inequality presented in (37), but it also requires a measure or criterion to define the size of the matrix P. In this case, the trace operator n o is used as the criterion, that is, trfP g-max or

tr P 1 -min. In the theorem presented above, the existence of the invariant set in which the trajectories of z converge has already been proven. According to the definition presented in (37), the configuration matrix P α ¼ αQ P=β satisfies the invariant ellipsoid concept if one considers the inequality (36) and the application of the upper limit when t-1. To obtain the solution of the minimal invariant ellipsoid, the following lemma gives the sufficient conditions for designing the gain controller. Lemma 3. If the tuple (ν1 ; ν2 , αQ,P) is a solution of the optimization problem trfP g-max subject to (8) and

6. Optimization of convergence region The ultimate boundedness (practical stability) condition obtained in the previous theorem is a consequence of the unmatched uncertainties and perturbation effect. The so-called invariant ellipsoid technique [30,31] was applied recently to minimize the size of the region characterized by the ultimate boundedness condition. This method uses the concept of an ellipsoid: Definition 2. (Polyak et al. [31]). The ellipsoid

ϵðPÞ ¼ z A Rn : z > Pz r 1; P 4 0

(1) The initial condition zð0Þ A ϵðPÞ implies zðtÞ A ϵðPÞ for all t Z 0. (2) The initial condition zð0Þ a ϵðPÞ implies zðtÞ-ϵðPÞ as t-1.

ð37Þ

with the center at the origin and the configuration matrix P is said

2 6 6 6 4

PA þA > P þ ν2 P

3

αQ P β0

P ν1

1

β0

Λ

1

7 7 7r0 5

ð38Þ

then the corresponding controller proposed in (13) guarantees that any trajectory of the system (5) converges to a quasi-minimal ellipsoid ϵðPÞ. Proof. Consider the Lyapunov candidate function used in Theorem 1 and given in (24). The ellipsoid ϵðPÞ is invariant for the


1.4 1.38 1.36 0

2

4

6

Ang. vel. (rad/s)

Angle (rad)


0.4 0.2 0 0

2

0.1 0.05 0 2

4

6

0

0

2

4

Angle (rad)

4

6

Time (s)

6

Time (s)

Ang. vel. (rad/s)

2

6

0.5

Time (s)

0.4 0.2 0 -0.2 0

4

Time (s) Ang. vel. (rad/s)

Angle (rad)

Time (s)

-0.05 0

7

0.5

0

0

2

4

6

Time (s)

Fig. 4. Comparison of reference states (green solid lines) and controlled trajectories produced by the PID controller using the states estimated by a set of STs (red dashed lines) as well as the GPI controller using the high-gain observer (blue dashed lines). All the sections of the biped robot are shown: (a) left hip, (b) left knee, (c) right knee, (d) right hip, (e) left ankle, (f) right ankle. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

trajectories of z if and only if dVðtÞ r 0 holds for any z such that dt z > P α z Z 1 and for any uncertainties η satisfying the condition (8). The expression in (26) can be represented as " #> " # zðtÞ zðtÞ dVðtÞ r Ω0 ; ηðtÞ ηðtÞ dt

"

Ω0 ¼

PAþ A > P

P

P

0NN

# ð39Þ

An algebraic manipulation makes it possible to express the inequality (29) as " #> z

η

Ω1

" # z

η

2

4f 1 6 β I NN 6 Ω1 ¼ 6 0 4 0NN

r 1;

3 0NN 7 7 1 1 7 5

β0

Λ

ð40Þ

7. Reference trajectories The reference trajectories proposed in (14) were obtained by a biomechanical analysis of regular gait cycles. The angles observed for the ankles, knees, and hips were reported in several studies [32,33]. This information was collected from several articles and specialized books. These angles are usually reported at very specific moments of the gait cycle. Therefore, these values were interpolated using a nonlinear algorithm based on a least mean square approximation that uses cubic polynomials. This algorithm is already implemented in MATLAB (the interp1 function with a cubic interpolation method). The corresponding function sðÞ was obtained after differentiating the third-order polynomials obtained by this procedure.

8. Numerical results >

The condition z ðtÞP α zðtÞ Z1 can be represented as "

zðtÞ

ηðtÞ

#>

"

Ω2

zðtÞ

ηðtÞ

#

" r 1

Ω2 ¼

Pα

0NN

0NN

0NN

# ð41Þ

According to the S-procedure [29], the previous matrix inequalities imply that (39) is negative if and only if there exist two scalars ν1 and ν2 such that Ω0 o ν1 Ω1 þ ν2 Ω2 ; ν1 ν2 r 0. Finally, the Schur complement provides the inequality in (38).□

A set of numerical simulations was developed in MATLAB for testing the robust observer and the controller. The numerical simulations were executed using the fixed step fourth-order Runge–Kutta method with a simulation step of 0.005 s. The biped robot simulation used the MATLAB model Bipedsim proposed by [24]. This system was used to simulate the dynamics of a biped robot. The simulation was executed using the licensed file provided by the original authors of the study. However, the original controller based on a proportional derivative scheme was modified to include ADRC. The biped robot parameters required by



8

0.3 0.2 0.1 0

0.3 0.2 0.1 0

Y (m) 0.04 0.08 0.12 0.16 0.2 0.24

0

0

0

0.3 0.2 0.1 0

Reference GPI ST 0

0.04 0.08 0.12 0.16 0.2 0.24

X (m)

X (m)

T=0.65 s

T=1.05 s Y (m)

0.3 0.2 0.1 0

0

0.04 0.08 0.12 0.16 0.2 0.24

0.3 0.2 0.1 0

0

0.04 0.08 0.12 0.16 0.2 0.24

X (m)

X (m)

T=2.01 s

T=2.97 s Y (m)

0.3 0.2 0.1 0

T=0.41 s

0.04 0.08 0.12 0.16 0.2 0.24

0.3 0.2 0.1 0

0

0.04 0.08 0.12 0.16 0.2 0.24

X (m)

X (m)

T=3.29 s

T=3.53 s Y (m)

Y (m)

Y (m)

Y (m)

Y (m)

T=0.17 s

0.04 0.08 0.12 0.16 0.2 0.24 X (m)

0.3 0.2 0.1 0

0

0.04 0.08 0.12 0.16 0.2 0.24 X (m)

Fig. 5. Comparison of the reference postural evolution tracking of the simulated biped robot (reference in blue lines) obtained by the proposed observer (red lines) and the ST used as a differentiator (green lines). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

[24] and used for the numerical simulations are the torso length (l0 ¼0.8 m), thigh length (l1 ¼0.5 m), and shank length (l2 ¼0.5 m). The distances with respect to the center of mass of the torso (r0) and the thigh (r1) from the hip are r 0 ¼ 10 =2 and r 1 ¼ l1 =2, respectively. The distance from the knee to the center of mass of the shank is r 2 ¼ l2 =2. The masses of the torso, thigh, and shank are m0 ¼ 5 kg, m1 ¼ 2 kg, and m2 ¼ 1 kg, respectively. In addition, the link inertias of the torso, thigh, and shank are given by 2 2 2 J 0 ¼ ð1=12Þm0 l0 , J 1 ¼ ð1=12Þm1 l1 , and J 2 ¼ ð1=12Þm2 l2 , respectively, in units of kg m2. The gravitational acceleration is g ¼9.81 m/s2. The initial conditions for the positions (velocities) of the system are x0 ð0Þ ¼ 0 m (0.1 m/s), y0 ð0Þ ¼ 1:3749 m ( 0.05 m/s), að0Þ ¼ 0:04 rad ( 0.06 rad/s), bl ð0Þ ¼ 0:11 rad ( 0.6 rad/s), cl ð0Þ ¼ 0:1696 rad ( 0.06 rad/s), and cr ð0Þ ¼ 0:1616 rad (0.0 rad/s). The estimated information obtained by the observer was injected into the controller structure introduced in (13). The control and observer parameters are given by H ¼ ½46 80 60 25 30 25 40T , L ¼ 7:5 ½25 25 25 35 35 35 35T , and K ¼ 6 ½15 15 15 25 25 25 25T . The optimization parameter obtained from the Riccati equation is 2

1:715 6 0:291 6 6 6 0:001 6 6 P α ¼ 6 0:040 6 6 0:043 6 6 4 0:080 0:464

0:291

0:001

0:040

0:043

0:080

1:2526 0:1697

0:1697 1:3312

0:0049 0:1097

0:0008 0:0001

0:0001 0:0063

0:0049 0:0008

0:1097 0:0001

1:1720 0:1064

0:1064 0:8959

0:0214 0:1785

0:0001 0:025

0:0063 0:0816

0:0214 0:3442

0:1785 0:3078

1:1559 0:098

0:464

3

0:025 7 7 7 0:0816 7 7 7 0:3442 7: 7 0:3078 7 7 7 0:098 5 12:5111

The simulation was evaluated in two steps. The first one considers only the observer performance. The observer trajectories converged to a bounded zone around the actual trajectories of (5). This performance was obtained without any knowledge of the drift section obtained from the biped robot model.

Fig. 2 compares the actual velocity measured at the left knee of the biped robot versus two estimated trajectories. The first was produced by the robust observer described in (12), whereas the second was produced by a multivariable super-twisting algorithm (ST) used as a robust differentiator [34,35]. The green solid line corresponds to the actual angular velocity of the biped robot knee. The blue dashed line depicts the corresponding estimated velocity of the same articulation according to the robust observer proposed in this study. The red dashed line describes the velocity obtained by applying the ST. Even when both trajectories seem to reproduce the actual angular velocity of the left knee at the same time, the observer proposed in this study does not use discontinuous functions. This feature yields a relative advantage because no high-frequency oscillations appear in the velocity estimation. This behavior reportedly affects the output-based controller energy [36]. The observer proposed in this study exhibited better velocity estimation for all the articulations of the simulated biped robot. These comparisons are typically made using the mean square error (MSE) for the observation error, which corresponds to MSE ¼ J Δ J . This value was obtained both when the observer (12) was applied and when the ST was used. The results are shown in Fig. 2. The superiority observed in each comparison was confirmed by the MSE value. This value showed that the estimation error obtained by the robust observer was closer to zero than that of the ST, as shown by the ultimate bound obtained after 7 s (0.01 vs. 0.04). The same controller, (13), was implemented using the information provided by either the robust observer or the ST. This strategy was useful for comparing the effect of the estimation quality on the tracking error. Although the controller structure used the estimated velocity, it successfully tracked the reference



9

ST

0.6 0.4 0.2 0

2

Ref

4 Time, (s)

GPI

6

0.2 MSE

trajectory that was based on the regular gait cycle. However, the controller based on the robust observer produced better tracking performance than the one that used the ST-estimated velocities. Fig. 3 shows the tracking comparison reference trajectory proposed for the left knee of the biped robot vs. the two controlled trajectories. Again, the green solid line corresponds to the reference angular trajectory of the biped robot knee. The blue dashed line depicts the corresponding angle of the same articulation based on the robust observer proposed in this study. The red dashed line describes the result of applying ST as the velocity estimator over the controller structure. The MSE of the tracking process (MSEtr), which corresponds to MSE ¼ J Δ þ σ J , was determined. Again, this value was obtained both when the observer (12) was applied and when the ST was used as a velocity estimator to feed the controller structure. The results are shown in Fig. 3. The MSE values confirm that the tracking error of the robust observer approached zero more rapidly than that of the ST (0.3 vs. 0.5 s). The corresponding ultimate bound of the tracking process was 4 times better at a time of 7 s (0.09 vs. 0.41) for the controller that used the estimated velocities and the uncertain section of the system. Fig. 4 compares the reference states and controlled trajectories produced by the PID controller using the states estimated by a set of STs as well as the GPI controller using the proposed observer. In all cases, the GPI controller exhibited better tracking of the reference trajectories. Fig. 5 shows the numerical results in such a way that the anatomical posture can be appreciated. This figure shows the behavior of both the proposed method and the ST. The comparison confirms the enhanced tracking performance of the proposed method introduced in this study. The complete reproduction of the reference gait cycle is more accurate and faster when the robust observer is used.

Angle, [rad]

Fig. 6. Experimental robotic system.

ST

0.1

GPI

0 -0.1

0

2

4 Time, (s)

6

Fig. 7. Upper figure: Comparison of reference angle proposed for the left knee articulation in the experimental biped robot (red solid line) with the trajectory obtained using the proposed observer as a velocity estimator (blue lines) and the ST used as a differentiator (green lines). These trajectories were measured directly from the instrumented robot presented above. Bottom figure shows the mean square error of estimation generated by the GPI observer (red lines) as well as the generated by the ST algorithm (blue lines). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

9. Hardware-in-the-loop implementation The experiment was executed using an adaptation of the robot “Brat” produced by the Lynxmotion company (see Fig. 6). The robot had the following specifications originally:

Number of degrees of freedom per leg: 3. Servo motion control by local closed loop. Total height: 8.5 in. Total width: 6.0 in. Ground clearance: 2.5 in. Weight (without batteries): 22.2 oz.


10


Range of motion per axis: 180°. Voltage input for each motor: 6 VDC. The original servo-motors (HITEC HS-422) in the robot were modified to serve as simple DC motors. Each motor was driven by an external power DC motor driver (L293D). The angle of each articulation was measured by a single-turn potentiometer (Suntan, TSR-3590) of 10 kΩ. The output-based controller was implemented in a hardware-in-the-loop scheme. Simulink/MATLAB was connected to an Arduino UNO board through the Arduino IO Library. Each driver was connected to an optoisolation system based on 4N35 circuits. A pair of Arduino UNO boards was attached to the set of optoisolators. These boards implemented only a data acquisition board emulator. The information acquired by the boards was fed into a Simulink/MATLAB-based simulation system. This simulation system implemented both the high-order extended state observer and the subsequent automatic controller. The gains of the observer and the controller were not adjusted with respect to those reported in the Numerical section. No information from the actual robot was used to design the observer. The controller action produced by the Simulink program was adjusted (normalized to 5.0 V) in order to use the analog output block from the Arduino IO library. This information was combined with an additional digital output used to characterize the movement direction of each articulation. All the experimental conditions and the reference trajectories proposed in this study yield a robot step size of 5 cm. As in the numerical simulation, the controller structure tracked the reference trajectory that was proposed on the basis of the regular gait cycle (Fig. 7). In this case, only the tracking performance of the controller using the estimated velocities from the robust observer was evaluated. The same figure shows the tracking comparison reference trajectory proposed for the left knee of the biped robot versus the closed loop controlled trajectories. The solid line corresponds to the reference biped robot knee angular trajectory. The dashed line depicts the corresponding angle of the same articulation based on the robust observer proposed in this study. The MSE of the tracking process obtained in the actual robot (MSEtr;r ) was calculated. In this case, this value was obtained only when the observer (12) was applied as velocity estimators to feed the controller structure. The result is shown in Fig. 7. Two additional reference trajectories were proposed to evaluate the controller performance. These trajectories forced the biped robot to sit down and stand up. The results of the controller action were similar to those obtained when the regular gait cycle was used as a reference trajectory. These results are not shown to avoid repetition of the already reported results.

10. Conclusions In this paper, we propose an active disturbance rejection multivariable control mixed structure based on a robust observation and output-based controller to adjust the movement of biped robot articulations. This mixed structure was useful for controlling the class of uncertain nonlinear systems represented by the biped robot. Convergence of this controller was ensured by a simple quadratic Lyapunov function. The last proof contributes to pioneering work in the area of high-order extended state observers for ADRC, such as those by [16,20]. The novelty in this work is the fact that the set of disturbance inputs does not need to be differentiable, which allows a larger set of admissible disturbance inputs to be estimated. A regular linear feedback controller was designed to show how the observer states can be used to replace the real variables of an

uncertain system. A basic representation of the biped robot was used to generate a set of numerical simulations that validated the theoretical result achieved in this study. The controller provided the solution to the trajectory tracking problem. As future work, the observer gain may be tuned in the context of low or adaptive gain control, which is an alternative for a class of constraint systems, especially in actuators or when the working conditions may vary with respect to time. On the other hand, the observer can be modified in the context of low-gain control, which may have a remarkable impact on energy saving, a demanding open problem in the active prosthesis field. Even when the proposed controller can handle a wide variety of systems, there are still problems to consider when using linear extended state observers for active disturbance rejection, where Lyapunov analysis can be used and extended. For example, this scheme can be improved for dealing with a set of nonlinearities in the control input such as dead zones and hysteresis. These effects can be reduced if further actions are introduced in the proposed controller, from adaptation schemes to classical procedures. On the other hand, adaptive forms in the gain tuning can be applied for energy optimization, which is relevant for the class of mobile robotic systems representing a biped robot. For this purpose, alternative procedures such as implicit Lyapunov functions can be introduced. These schemes are mentioned in particular because the proposed biped robot control design can be improved by introducing such actions. Because the class of systems to consider can be subject to additive noise effects, a further development would consider these effects in the design process or in the system structure (see [37] for an authoritative analysis of the effects of output noise on high-gain observers). Additional improvement can be obtained if the linear controller action proposed in this study can be combined with some type of variable structure forms such as sliding modes. This mixed strategy can enlarge the type of perturbations that can be handled by our proposed method.

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