HYDRAULIC systems are favored in industry requiring

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 6, NO. 4, AUGUST 20 707 Integrated Direct/Indirect Adaptive Robust Control of Hydraulic Manipulators With Valve Deadband Amit Mohanty, Student Member, IEEE, andbinyao, Senior Member, IEEE Abstract In this paper, an integrated direct indirect adaptive robust control (DIARC) algorithm is proposed for an electrohydraulic manipulator with unknown valve deadband to improve the achievable output-tracking performance. The controller design for such a system is nontrivial due to various factors, such as nonsmooth static and Coulomb friction model uncertainties arising from the use of a simplified proportional-flow-valve model, and other unknown disturbances present in the system. Furthermore, when an unknown input-valve deadband is present, the controller performance can deteriorate if it is not taken care of explicitly. This paper recognizes the fact that though unknown valve deadband nonlinearity is not globally linearly parameterizable, it can still be linearly parameterized during most of the working ranges. Hence, by using an indirect estimation algorithm with online condition monitoring, accurate estimates of the unknown deadband parameters are obtained for an improved control performance. Comparative experimental results for motion control of an electrohydraulic manipulator with two different valves having deadband illustrate the effectiveness of the proposed algorithm. Index Terms Adaptive control, electro-hydraulic system, motion control, robust control. I. INTRODUCTION HYDRAULIC systems are favored in industry requiring large actuation force [] [4]. In [5], the direct-adaptive robust-control (DARC) technique proposed in [6] was applied for precision-motion control of an electro-hydraulic manipulator. However, being a tuning function-based Lyapunov design, it does not provide the freedom to choose parameter estimation law independent of controller design. The usual gradient-type of parameter estimation law, as used in the DARC design, may not have good convergence properties as the indirect estimation laws (e.g., the least-squares method). In [7], an indirect adaptive robust controller (IARC) design was proposed to overcome the poor parameter-estimation property of the DARC algorithm. The modular design in the IARC algorithm allowed the use of Manuscript received June 7, 2009; revised September 7, 2009; accepted October 9, 2009. Date of publication July, 200; date of current version May, 20. Recommended by Technical Editor N. Chaillet. This work was supported in part by the National Science Foundation under Grant CMS-060056. A. Mohanty is with the Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: amitmohanty@gmail.com). B. Yao is with the Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA, and also a Chang Jiang Chair Professor at the State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 30027, China (e-mail: byao@purdue.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org Digital Object Identifier 0.09/TMECH.200.205037 least-squares type of estimation law to have better parameterconvergence properties and accuracies. It also achieved a complete separation of controller and estimator designs. The outputtracking performance was guaranteed by use of a robust-control law, which was independent of the parameter estimation module. Hence, when the driving signal is not persistently exciting (PE), the estimator module could be turned off without affecting the stability of the overall closed-loop system. This switching of adaptation algorithm ensures that the good parameter estimates obtained by PE signals are not corrupted by non-pe signals. The better estimation properties of such an algorithm was effectively demonstrated in [8]. To overcome the lack of fast-integration-type-model compensation in IARC design, an integrated direct indirect adaptive robust control (DIARC) approach was proposed for a class of single-input single-output (SISO) uncertain nonlinear systems transformable to semistrict parametric feedback form [9], and it has been applied for the control of linear motors [0] and parallel manipulators driven by pneumatic muscles []. In this paper, such an integrated ARC-design framework will be employed for the precision-motion control of an electro-hydraulic robotic arm driven by single-rod hydraulic actuators with valve deadband. In addition to various unmatched model uncertainties considered in [8], this paper also explicitly takes into account the effect of unknown valve deadband. During the past decade, various researchers have tried to solve this problem by estimating the deadband parameters using direct adaptive control algorithms [2], [3]. As pointed out in [4], the unknown deadband nonlinearity cannot be linearly parameterized globally. As such, when direct adaptive control approach is used, it becomes impossible to accurately estimate the deadband parameters even when persistent exciting conditions are satisfied. Departing from these direct estimation laws, this paper will employ indirect estimation design with explicit condition monitoring of signal excitations in the proposed modified DIARC algorithm. The proposed controller design fully utilizes the fact that even though unknown deadband nonlinearity cannot be globally linearly parameterized, it can be linearly parameterized during most of the working range. This approach ensures complete adaptive compensation in the presence of parametric uncertainty only and, thus, is able to achieve asymptotic-output tracking or zero steady-state tracking errors. II. DYNAMIC MODEL AND PROBLEM FORMULATION In this paper, we consider a 3 DOF hydraulic robot arm shown in Fig.. For clarity of presentation, only the swing-joint motion is considered while the other two joints (boom and stick) are kept fixed. 083-4435/$26.00 200 IEEE

708 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 6, NO. 4, AUGUST 20 Fig.. 3 DOF hydraulic manipulator. A. Manipulator Dynamics AsshowninFig.,x s (q) represents the cylinder displacement and q is the angular displacement of the swing arm. P and P 2 are the pressures in the cylinder forward and return chamber, respectively, A and A 2 are the RAM areas of the forward and return chambers, respectively. The applied torque on the swing arm can be written as μ(p A P 2 A 2 ), where μ(q) = x s / q is the first-order partial derivative of cylinder displacement x s with respect to the swing angle q.letb be the combined-damping and viscous-friction coefficient, A be the magnitude of modeled Coulomb friction force, and (D n + D ) be the lumped-modeling error including external disturbances and terms like the unmodeled-friction forces and D n be its nominal value. As shown in [8], the dynamics of the swing motion can be described by () J q = μ(p A P 2 A 2 ) B q AS( q)+d n + D () where J is the moment of inertia of the robotic arm and external payload and S( ) represents a smooth approximation of signum function. B. Pressure Dynamics As shown in Fig., and are the total volumes of the forward and return chambers, respectively, and Q and Q 2 are the supply-flow rate to the forward chamber and return-flow rate of the return chamber, respectively. Let β e be the effective-bulk modulus of hydraulic liquid. The pressure dynamics in both the chambers can be written as [5] (q)β e P = μa q + Q +(D 2n + D 2 ) (2) (q)β e P 2 =+μa 2 q Q 2 (D 22n + D 22 ) (3) where (D 2n + D 2 ) and (D 22n + D 22 ) are the modeling errors associated with forward and return chamber flows, respectively. These uncertainties can be attributed to the nonexact proportional nature of hydraulic valves and the presence of leakages in the hydraulic circuit. D 2n and D 22n are the nominal values of the uncertainties, and D 2 and D 22 are the deviations from the nominal values. C. Deadband and Flow Characteristics Two different valves will be used in the experiments conducted in this paper. One is a servo valve and another one is a Fig. 2. Deadband and inverse deadband functions. (a) Deadband. (b) Inverse deadband. proportional-directional-control (PDC) valve, which has a quite large deadband due to the closed-centered valve configuration. The servo valve has no deadband, so an artificial deadband would be introduced during the experiment while using servocontrol valve. Ignoring the relatively much faster valve dynamics, the actual orifice opening x n due to the spool movement can be modeled as a static mapping of control-voltage command to the valve x v with deadband. Without loss of generality, we assume the slope of the static map beyond deadband region to be unity and any nonunity gain between x n and x v can be taken care of by flow constants k q and k q2, which will be defined later in (7) and (8). As shown in Fig. 2(a), x v x + n, if x v x + n x n = D(x v )= 0, if x n x v x + n x v + x n, if x v x n where x n and x + n are the deadband widths for the positive and negative spool displacements, respectively. Let us define two-set indicator functions as follows: { {, if 0, if 0 ℵ + ( ) =, ℵ ( ) = 0, else 0, else. (5) Using the set-indicator functions ℵ + ( ) and ℵ ( ), the inverse deadband function can be written as follows: (4) x v = ID(x n )=x n + x + n ℵ + (x n )+x n ℵ (x n ). (6) Assuming proportional-flow characteristics of the valves, we can write the supply flow Q and return flow Q 2 as [5] { Ps P, x n 0 Q = k q ΔP x n, ΔP = P P ref, x n < 0 { P2 P ref, x n 0 Q 2 = k q2 ΔP2 x n, ΔP 2 = P s P 2, x n < 0 where k q and k q2 are the flow-gain coefficients, P s and P ref are the supply and tank reference pressures, respectively. (7) (8)

MOHANTY AND YAO: INTEGRATED DIRECT/INDIRECT ADAPTIVE ROBUST CONTROL OF HYDRAULIC MANIPULATORS 709 In general, the considered system is subjected to parametric uncertainties due to the variations of J, B, A, D n, D 2n, D 22n, β e, x + n, and x n. Define a set of the parameters as: θ = J, θ 2 = B, θ 3 = A, θ 4 = D n, θ 5 = βe, θ 6 = x + n, θ 7 = x n, θ 8 = D 2n, and θ 9 = D 22n. Define a set of state variables as x =[x x 2 x 3 x 4 ] T =[q q P A P 2 A 2 ] T. Using these state variables, the system described by () (3), (7), and (8) can be expressed as x = x 2 (9) θ ẋ 2 = μ(x 3 x 4 ) θ 2 x 2 θ 3 S(x 2 )+θ 4 + D (0) θ 5 ẋ 3 = A2 μx 2 + g A x n + θ 8 A + D 2 A () θ 5 ẋ 4 =+ A2 2 μx 2 g 2 A 2 x n θ 9 A 2 D 22 A 2 (2) where g and g 2 are two variables introduced for the convenience of notation g = k q ΔP, g 2 = k q2 ΔP2. (3) Given the desired motion trajectory q d (t) with bounded derivatives up to the third order, the control objective is ) to synthesize a control input x v such that the output q tracks q d as closely as possible in spite of various model uncertainties and disturbances and 2) to design a parameter-estimation algorithm such that accurate parameter estimates are obtained. III. INTEGRATED-DIRECT/INDIRECT-ADAPTIVE ROBUST-CONTROLLER DESIGN A. Assumptions and Notations The following nomenclature is used throughout this paper: is used to denote the estimate of, is used to denote the estimation error of, e.g., θ = θ θ. Since the extents of the parametric uncertainties and uncertain nonlinearities are normally known, the following practical assumptions are made. Assumption : The unknown parameter vector θ lies within a known bounded-convex set Ω θ. Without loss of generality, it is assumed that θ Ω θ, θ imin θ i θ imax, i =,...,9, where θ imin and θ imax are some known constants, respectively. Assumption 2: All the uncertain nonlinearities are bounded, i.e., D i Ω d = { D : D δi,i=, 2, 22}, where δ i (t) is a known bounded function. Assumption 3: The actual valve-orifice opening x n is not measured. B. Projection-Type Adaptation Law With Rate Limits One of the key elements of the DIARC design is to use the practical available a prior information to construct the projection-type adaptation law for a controlled learning process. For this purpose, we define the following projection mapping Projˆθ( ) [6], [6]: ζ, θ Ωθ or n Ṱ ζ 0 θ Projˆθ(ζ) ( = I Γ nˆθn ) Ṱ θ n Ṱ ζ, θ Ωθ & n Ṱ ζ>0 θ Γnˆθ θ (4) where ζ R p and Γ(t) R p p are any time-varying positive definite-symmetric matrix, Ω θ and Ω θ denote the interior and the boundary of Ω θ, respectively, and nˆθ represents the outward unit normal vector at θ Ω θ. We also define the following saturation function sat θm ( ), ζ θ M sat θm (ζ) =s 0 ζ, s 0 = θ M ζ, ζ > θ M. (5) Lemma ([7], [9]): Suppose that the parameter estimate θ is updated using the following parameter estimator structure: ( θ =sat θm Proj θ (Γτ) ), θ(0) Ωθ (6) where Γ=Γ T > 0 is the adaptive-rate matrix and τ is the adaptation function (both of them to be designed later). With this adaptation-law structure, the following desirable properties hold: P: The parameter estimates are always within the known bounded set Ω θ, i.e., θ(t) Ω θ, t. Thus, from Assumption, θ imin θ i (t) θ imax, i =,...,9 t. P2: θ ( T Γ Proj θ(γτ) τ ) 0 τ. P3: The parameter update rate θ is uniformly bounded by θ(t) θm t. C. DIARC-Control Law As pointed out in Lemma, with the use of projection-type adaptation law with rate limit (6), the parameter estimates and their derivatives are bounded with known bounds, regardless of how the actual estimation function τ and adaptation-rate matrix Γ are designed. In this section, this property will be utilized to synthesize a robust-control law for the system (9) (2) that achieves a guaranteed transient and final tracking error independent of the parameter-estimator design. As the system (9) (2) has unmatched model uncertainties, back-stepping design [7] is used as follows: Step : Define a variable z 2 as z 2 = z + k z = x 2 x 2eq, x 2eq =ẋd k z (7) where z = x q d (t) is the output tracking error and k is any positive feedback gain. Since G s (s) =z (s)/z 2 (s) = /(s + k ) is a stable-transfer function, making z 2 small or converging to zero is equivalent to making z small or converging to zero. Differentiating (7) and noting (0), we obtain θ ż 2 = μ(x 3 x 4 ) θ 2 x 2 θ 3 S(x 2 )+θ 4 θ ẋ 2eq + D. (8)

70 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 6, NO. 4, AUGUST 20 Define the actuator load as F L = x 3 x 4. In (8), treating F L as the virtual control, the following virtual DIARC law α 2 for F L is designed α 2 = α 2a + α 2s = α 2a + α 2a2 + α 2s + α 2s2 (9) α 2a = μ [ θ ẋ 2eq + θ 2 x 2 + θ 3 S(x 2 ) θ 4 ] (20) α 2s = μ k 2s z 2,α 2s2 = μ k 2s2 (x,x 2,t)z 2. (2) In (9), α 2a is the usual model compensation with the physical parameter estimates θ, α 2s represents the negative feedback term with gain k 2s > 0 to stabilize the nominal system (i.e., when θ = D =0). Define the input discrepancy z 3 as z 3 = F L α 2 and substituting (9) (2) in (8), the z 2 dynamics can be written as θ ż 2 = μ(α 2s2 + α 2a2 )+μz 3 θ 2 x 2 θ 3 S(x 2 )+ θ 4 θ ẋ 2eq + D. (22) Now, we lump all the uncertainties together and categorize it into a constant or low-frequency component d and a high-frequency component Δ as follows: d + Δ =+ θ ẋ 2eq + θ 2 x 2 + θ 3 S(x 2 ) θ 4 + D. (23) As done in [9], the low-frequency component d can be compensated using α 2a2 as follows: α 2a2 = μ d (24) where d is the estimate of d, which is determined using the following gradient type of estimation law { 0, if d (γ =Proj d z 2 )= d = d M and d (t)z 2 > 0 γ z 2, else (25) where γ > 0, d (0) d M and d M is any preset bound. The nonlinear feedback gain k 2s2 (x,x 2,t) is chosen large enough to satisfy the following inequality: z 2 [μα 2s2 d + Δ ] ε 2 (26) where d = d d and ε 2 > 0 is a design parameter. Remark : One example of k 2s2 so that the robustperformance condition (26) is satisfied, is given by the following: k 2s2 [ θ M ẋ 2eq + θ 2M x 2 + θ 3M S(x 2 ) 4ε 2 + θ 4M + d M + δ ] 2 (27) where θ im = θ imax θ imin. Other smooth or continuous examples of α 2s2 can be found in [6]. Substituting α 2a2 = μ d in (22), we obtain θ ż 2 = k 2s z 2 + μα 2s2 + μz 3 [ d d D ] = k 2s z 2 + μz 3 +(μα 2s2 d + Δ ). (28) Step 2: Let x nd be the control input that stabilizes the system with no deadband. If we knew the actual deadband function D( ), then we can use ID( ) as defined in (6) to find out x v. But, as there is uncertainty in ID( ), the following estimated inverse function would be used: x v = ÎD(x nd)=x nd + θ 6 ℵ + (x nd )+ θ 7 ℵ (x nd ). (29) Substituting (29) in (4), we can write x n = D(x v )=D(ÎD(x nd)) = x nd +Δ d (t) (30) where Δ d (t) is the error due to uncertainty in deadband and is defined as follows: θ 6, x v θ 6 Δ d (t) = x nd, θ 7 <x v <θ 6 (3) θ 7, x v θ 7. From (), (2), and (30), the following z 3 dynamics is obtained: ( ) θ 5 ż 3 = θ 5 F A A 2 A 2 L θ 5 α 2 = θ 8 + θ 9 + A2 2 μx 2 + gx n + gδ d (t) θ 5 α 2c θ 5 α 2u + D 2 (32) where D 2 =( A V D2 + A 2 V D22 2 ), g = g A + g 2 A 2 and α 2c and α 2u represent the calculable and incalculable part of α 2 = α 2c + α 2u as follows: α 2c = α 2 x 2 + α 2 ẋ 2 + α 2 θ + α 2 x x 2 θ t, α 2u = α 2 ẋ 2 x 2 (33) in which ẋ 2 =ẋ 2 ẋ 2 and ẋ 2 represents the estimate of ẋ 2 given by ẋ 2 = θ (μx 3 θ 2 x 2 θ 3 S f (x 2 )+ θ 4 ). (34) With notation u = gx nd, we design the following DIARC control law for u: u = u a + u s = u a + u a2 + u s + u s2 (35) ( A 2 ) u a = + A2 2 μx 2 μz 2 + V θ 5 α 2c θ A 8 2 V θ A 2 9 (36) u s = k 3s z 3,u s2 = k 3s2 (x, t)z 3. (37) In (35), u a is the usual model compensation with the physical parameter estimates θ, u 2s represents the negative feedback term with gain k 3s > 0 to stabilize the nominal system (i.e., when θ = D 2 =0). With (35) (37), we can rewrite (32) as follows: θ 5 ż 3 =(u a2 + u s2 ) μz 2 θ 8 A θ 9 A 2 + gδ d (t) + θ 5 α 2c θ 5 α 2u + D 2. (38) Similar to step, all the uncertainties are lumped together and categorized it into a constant or low-frequency component d 2 and a high-frequency component Δ 2 as follows: d 2 + Δ 2 = θ 5 α 2c θ 5 α 2u A θ8 A 2 θ9 + gδ d + D 2. (39)

MOHANTY AND YAO: INTEGRATED DIRECT/INDIRECT ADAPTIVE ROBUST CONTROL OF HYDRAULIC MANIPULATORS 7 The low-frequency component d 2 can be compensated using u a2 = d 2, where d 2 is the estimate of d 2. The estimate d 2 is determined by the following gradient type of estimation law: { 0, if d 2 = d 2 = d 2M and d 2 z 3 > 0,γ 2 > 0 (40) γ 2 z 3, else where d 2 (0) d 2M and d 2M are any preset bound. The nonlinear feedback gain k 3s2 is chosen large enough to satisfy the following inequality: z 3 [u s2 d 2 + Δ 2 ] ε 3 (4) where d 2 = d 2 d 2 and ε 3 > 0 is a design parameter. With this control law, the z 3 dynamics can be written as θ 5 ż 3 = k 3s z 2 3 μz 2 +(u s2 d 2 + Δ 2 ) (42) Denoting x nd = g u, the control input x v can be solved from the following nonlinear mapping: x v = g u + θ 6 ℵ + (x nd )+ θ 7 ℵ (x nd ). (43) Theorem (Guaranteed transient performance): Under Assumptions 3, consider the DIARC-control law (43) and ratelimited-adaptation law (6), in which τ and Γ(t) could be any adaptation function and adaptation-rate matrix, respectively. In general, all the signals in the resulting closed-loop system are bounded. Furthermore, the positive definite function V =/2(θ z 2 2 )+/2(θ 5 z 2 3 ), an index for the controller performance is bounded earlier by V (t) e λt V (0) + ε λ [ e λt ] (44) where λ = 2min {k 2s /θ max,k 3s /θ 5max } and ε =(ε 2 + ε 3 ). Proof (Outline): Differentiating V (t) and noting (26), (28), (4), and (42), we can obtain V = θ z 2 ż 2 + θ 5 z 3 ż 3 k 2s z2 2 k 3s z3 2 +[ε 2 + ε 3 ] [ λ 2 θ z2 2 + ] 2 θ 5z3 2 + ε λv + ε (45) which leads to (44) by Comparison Lemma. As θ and θ are guaranteed to be bounded by P and P3 of Lemma, respectively, one can follow standard back-stepping proofs to show that all the control functions (9), (35), (43) and states x are bounded for any desired trajectory q d (t) with bounded third-order derivatives. IV. PARAMETER ESTIMATION In this section, the estimation function τ and the adaptationrate matrix Γ are designed so that an improved final-tracking accuracy asymptotic tracking can be obtained in the presence of parametric uncertainties only (i.e., D = D 2 = D 22 = 0). Before we design any parameter-estimation module, we note the following two difficulties in designing an efficient-parameter estimator for the considered electro-hydraulic system. ) As the measurement of ẋ 2, ẋ 3, and ẋ 4 are not available, we cannot directly estimate the unknown parameters θ from (0) to (2). To overcome this difficulty, we need to resort to X-swapping scheme [7, p. 248]. 2) From (3), we observe that the error due to uncertainty in deadband parameters Δ d (t) cannot be expressed globally linearly with respect to uncertainty in the deadband parameters θ 6 and θ 7. However, for most of the working range (e.g., x v < θ 7 and x v θ 6 ), Δ d (t) is linear w.r.t. θ 6 and θ 7.Although the value of the deadband parameters θ 6 and θ 7 are not known, from Assumption, we know the value of θ 6max and θ 7max. Hence, there exists a known region x v A B = {x v : x v θ 6max or x v θ 7max } where Δ d (t) is linear w.r.t. the parametric uncertainty θ 6 and θ 7. Noting aforementioned facts, the condition x A B is explicitly monitored and the parameters in pressure channel were estimated only when x A B. A. Static-Parametric Model for Manipulator Dynamics Assuming that there is no nonlinear uncertainty (i.e., D =0), the manipulator dynamics (0) can be written as θ ẋ 2 = μf L θ 2 x 2 θ 3 S(x 2 )+θ 4. (46) Let us define the following two filters: = τ + μf L, (0) = 0 (47) Ψ T = τ Ψ T + F (x 2 ) T, Ψ (0) = [x 2 (0) 0 0 0] (48) where F (x 2 ) T =[τ x 2 x 2 S(x 2 ) ] and τ > 0 is the break frequency of the filters defined earlier. Let us define a variable Y =, which is available from (47). Defining ɛ = +(Ψ T Λ )θ 234, Λ =[x 2 0 0 0] T, and θ 234 =[θ θ 2 θ 3 θ 4 ] T, we can write Y = (Ψ T Λ )θ 234 + ɛ. (49) Differentiating ɛ, we get ε = τ ε. From the initial conditions of the filters (47) and (48), it can be verified that ε (0) = 0. Hence, ε (t) =0 t. Now, we can rewrite (49) as Y = (Ψ T Λ )θ 234. The prediction of Y is defined as Ŷ =(Ψ T Λ ) θ 234. The prediction error ɛ = Ŷ Y can be written as ɛ =(Λ T Ψ T ) θ 234 =Ω T θ 234 (50) where Ω T =(Λ T Ψ T ). Thus, we obtained the staticparametric model (50) that is linear with respect to the parameter-estimation error θ 234. B. Static-Parametric Model for Pressure Dynamics We define the following time intervals during which the estimation process will be done: [t K ji,t K jf]={t : x v (t) K, K {A, B} j N} (5) where N is the set of positive integers and x v (t) enters region K {A, B} at time t = t K ji and leaves it at time t = tk jf.assuming that there is no nonlinear uncertainties (i.e., D 2 =0),

72 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 6, NO. 4, AUGUST 20 Fig. 3. Three different regions A = {x v : x v θ 7max }, B = {x v : x v θ 6max }, and C = {x v : θ 7max <x v <θ 6max }. The parameter adaptation is done only when x v A B. the pressure dynamics from () can be written as ( θ 5 P = g x v A ) μ q + θ { 8 g θ 6, if x v B g θ 7, if x v A. (52) Let us define F A2 and F B 2 as F A2 =[τ 2 P, 0, g, V ] T R 4 (53) F B 2 =[τ 2 P, g, 0, V ] T R 4. (54) We define following set of low-pass filters to get a staticparametric model: [ K 2 = τ 2 K 2 + g x v A ] μ q, K 2 (t K V ji)=0 (55) Ψ T K 2 = τ 2 Ψ T K 2 + FK T 2, Ψ T K 2(t K ji)=[p 0 0 0] (56) where K {A, B}. As done in the previous section, let us define a variable Y 2 = 2, its prediction Ŷ2 =(Λ T 2 Ψ T K 2 ) θ 5678 where θ 5678 =[θ 5 θ 6 θ 7 θ 8 ] T and Λ 2 = [P 0 0 0] T. Then, the prediction error ɛ 2 = Ŷ2 Y 2 can be written as follows ɛ 2 =(Λ T 2 Ψ T K 2 ) θ 5678 ɛ 2, where ε 2 = K 2 (Λ T K 2 ΨT K 2 )θ 5678. Following a similar procedure as the previous section, it can be shown that ɛ 2 (t) =0and, hence, the prediction error ɛ 2 can be written as ɛ 2 =(Λ T 2 Ψ T K 2) θ 5678 =Ω T 2 θ 5678 (57) where Ω T 2 =(Λ T 2 Ψ T K 2 ). Thus, we obtained the staticparametric model (57) that is linear with respect to the parameter-estimation error θ 5678. Following exact similar procedure, we can obtain the static parametric model for P 2 -pressure dynamics as ɛ 22 =Ω T 22 θ 5679. (58) With the static-parametric models (50), (57), and (58), the leastsquares estimation algorithm with exponential forgetting factor and covariance resetting [8] is used in this paper. For such an estimation algorithm, the adaptation-rate matrix Γ i is given by Γ i = α i Γ i Γ iω i Ω T i Γ i +ν i Ω T i Γ iω i, Γ i (t + r )=ρ im I,ν 0 (59) where Γ i (0) = Γ T i (0) > 0 and the adaptation function τ i is given by τ i = +ν i Ω T i Γ Ω i ɛ i (60) iω i where i = {, 2, 22}. In (59), α i 0 is the forgetting factor, t r is the covariance resetting time, i.e., the time when λ min (Γ i (t)) = ρ i, where ρ i is a preset lower limit for Γ i (t) satisfying 0 <ρ i <ρ im and ν i 0 is the normalizing factor (see Fig. 3). As previously mentioned in this section, the parameter estimation for pressure channel is conducted only when x A B. Hence, as x v leaves the region A B and enters the region C, the rate of change of adaptation rate matrices Γ 2 and Γ 22 and adaptation function τ 2 and τ 22 are set to zero and thus, leading to θ5 = θ6 = θ7 = θ8 = θ9 =0. Theorem 2: In the presence of parametric uncertainties only (i.e., D = D 2 = D 22 =0), by using the control law (43), and adaptation law (6) with least-squares type adaptation function (60) and adaptation rate matrix (59), if the following PE condition is satisfied for i = {, 2, 22} t+t t Ω i (σ)ω T i (σ)dσ κ i I p, for some κ i,t >0 (6) then, θ 0, i.e., θ converges to its true value θ. Furthermore, along with the robust performance of Theorem, an improved final tracking asymptotic tracking is also obtained, i.e., z,z 2,z 3 0 as t. Proof (Outline): Following standard techniques in adaptive control, it can be shown that if D = D 2 = D 22 =0and the PE condition given by (6) is satisfied, then θ 0, ast and θ L 2 [0, ). Defining a positive definite function as V n = 2 θ z2 2 + 2 θ 5z3 2 + 2 γ d + 2 γ 2 d 2, we can write V n = (k 2s + k 2s2 )z2 2 (k 3s + k 3s2 ) z2 2 +[ẋ 2eq θ + x 2 θ2 + S(x 2 ) θ 3 θ 4 ]z 2 + [ θ5 α 2c θ 5 α 2u θ 8 A θ 9 A ] 2 + gδ d (t) z 3 + d γ (Proj d (γ z 2 ) z 2 )+ d 2 γ 2 (Proj d2 (γ 2 z 3 ) z 3 ). (62) Using P2 of Lemma, we see that the last two terms of (62) are less than zero. From (3), it can be verified that Δ d (t) θ 6 ℵ + (x v )+ θ 7 ℵ (x v ). Using aforementioned facts, we rewrite (62) as follows: V n (k 2s + k 2s2 )z2 2 (k 3s + k 3s2 ) z2 2 + ζ (63) where ζ is a term linear with respect to parametric uncertainty θ. Using the fact that θ L 2 [0, ), we assert that ζ L 2 [0, ). Hence, from (63), z 2,z 3 L 2 [0, ). Using Barbalat s Lemma and noticing that z 2 and z 3 are uniformly continuous, we conclude z,z 2,z 3 0 as t.

MOHANTY AND YAO: INTEGRATED DIRECT/INDIRECT ADAPTIVE ROBUST CONTROL OF HYDRAULIC MANIPULATORS 73 Fig. 4. Desired trajectory for the swing joint. TABLE I PERFORMANCE COMPARISON OF IARC AND DIARC: SERVO VALVE Fig. 5. Swing joint-tracking error for the servo valve. V. COMPARATIVE EXPERIMENTAL RESULTS The IARC and DIARC controllers were tested for swing joint control for an electro-hydraulic manipulator, while other two joints were kept fixed. The desired trajectory q d (t) is shown in Fig. 4. The initial values of the unknown parameters were kept same for both the controllers. The complete details of experimental test-bed are omitted for brevity and can be found in [8]. Electro-hydraulic system parameters are as follows: P s =7MPa,P ref =MPa, A =3.4 in 2,A 2 =.66 in 2 in 3 k q =0.82 lb/in 2 sv, k in 3 q2 =0.89 lb/in 2 sv β =2.7e8 Pa, θ min = [60, 0, 0, 0, 2e8, 0, 0, 0.000, 0.000] θ max = [250, 00, 00, 00, 4e8, 2, 2, 0.000, 0.000]. Fig. 6. Estimate of θ and θ 2 for the servo valve. A. Servo Valve The servo valve used in this experiment has a bandwidth around 8 9 Hz and a maximum flow rate of 0 gallon/min. As the servo valve possesses no deadband, an artificial deadband of [.5,.5]V was introduced during the experiment. The valve dynamics was neglected in this experiment as it significantly increases the complexities in real-time controller implementation. Therefore, it was necessary to keep the closedloop bandwidth less than the bandwidth of the servo valve. Keeping this limitation in mind, the feedback gains were chosen to be k =0,k 2s + k 2s2 =0, and k 3s + k 3s2 =0 0 8, so that the closed-loop bandwidth of the system was less than the bandwidth of the servo valve. For DIARC controller, the integral gains were set at γ =20and γ 2 =7 0 7. Fig. 5. shows the tracking errors for IARC and DIARC algorithm with both using the servo valve. Table I compares the transient performance and the final tracking error of two algorithms. The following comparative observations were made: Fig. 7. Estimate of θ 3 and θ 4 for the servo valve. ) during the transient, the maximum absolute error for DIARC is about five times less than IARC; 2) during the transient, the mean absolute error for DIARC is about ten times less than IARC; thus a faster rate of error decay is achieved for DIARC algorithm as compared to IARC; and 3) a better final tracking error is achieved for DIARC. Figs. 6 0 show that the better tracking performance of DIARC is achieved without sacrificing the accuracy of estimated parameters. The dotted line in Figs. 8 and 9 are the true values of estimated parameters.

74 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 6, NO. 4, AUGUST 20 TABLE II PERFORMANCE COMPARISON OF IARC AND DIARC: PDC VALVE Fig. 8. Deadband estimates for the servo valve: θ 6,θ 7. Fig. 2. Estimate of θ and θ 2 for the PDC valve. Fig. 9. Estimate of β e = θ 5 for the servo valve (in Pa). Fig. 3. Estimate of θ 3 and θ 4 for the PDC valve. Fig. 0. Fig.. Estimate of θ 8 and θ 9 for the servo valve. Swing joint-tracking error for PDC valve. B. Proportional-Directional Control Valve Fig. shows the output-tracking performance of both the controller for a PDC valve. The PDC valve used in this experiment has a bandwidth around 5 6 Hz, which is approximately half the bandwidth of the servo valve. Therefore, the feedback gains for IARC and DIARC were set at k = k 2s + k 2s2 =7, k 3s + k 3s2 =7 0 8 for the PDC valve, which are less than the feedback gains used for the servo-control valve. For DIARC controller, the integral gains were set at γ =5and γ 2 =5 0 9, which are less than the integral gains used for the servo valve. The reduced feedback proportional and integral gains caused overall degraded controller performance for PDC valve as compared to the servo valve (for both the controllers). However, comparing the performance of IARC and DIARC algorithm for PDC valve, we observe that by using DIARC algorithm, significant improvement in transient performance and final tracking accuracy is achieved (see Table II). Figs. 2 4

MOHANTY AND YAO: INTEGRATED DIRECT/INDIRECT ADAPTIVE ROBUST CONTROL OF HYDRAULIC MANIPULATORS 75 Fig. 4. Deadband estimates for the PDC valve: θ 6,θ 7. show that the better-tracking performance of DIARC is achieved without sacrificing the accuracy of estimated parameters. VI. CONCLUSION In this paper, DIARC was proposed for high-performancemotion control of an electro-hydraulic systems driven by singlerod actuators with input valve deadband. The proposed controller uses least-squares type indirect parameter adaptation algorithm with certain condition monitoring to estimate accurate deadband parameters. Furthermore, a fast dynamic compensation similar to the DARC design is employed to improve the tracking performance. Experimental results are presented to illustrate the effectiveness of the proposed algorithm. REFERENCES [] E. Papadopoulos, B. Mu, and R. Frenette, On modeling, identification, and control of a heavy-duty electrohydraulic harvester manipulator, IEEE/ASME Trans. 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Syst., 2002, pp. 3020 3025. [] X. Zhu, G. Tao, B. Yao, and J. Cao, Integrated direct/indirect adaptive robust posture trajectory control of a parallel manipulator driven by pneumatic muscles, IEEE Trans. Control Syst. Technol., vol. 7, no. 3, pp. 576 588, May 2009. [2] X.-S. Wang, C.-Y. Su, and H. Hong, Robust adaptive control of a class of nonlinear systems with unknown dead-zone, Automatica, vol. 40, pp. 407 43, 2004. [3] S. Ibrir, W. F. Xie, and C. Y. Su, Adaptive tracking of nonlinear systems with non-symmetric dead-zone input, Automatica,vol.43,no.3,pp.522 530, 2007. [4] C. Hu, B. Yao, and Q. Wang, Integrated direct/indirect adaptive robust control of a class of nonlinear systems preceded by unknown dead-zone nonlinearity, in Proc. 48th IEEE Conf. Decis. Control, Shanghai, China, Dec. 6 8, 2009, pp. 6626 663. [5] H. E. Merritt, Hydraulic Control Systems. New York: Wiley, 967. [6] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 989. [7] M. Krstic, I. Kanellakppoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 995. [8] I. D. Landau, R. Lozano, and M. M Saad, Adaptive Control. NewYork: Springer-Verlag, 998. Amit Mohanty received the B.Tech degree from the Indian Institute of Technology, Kharagpur, India, in 2003, and the M.S. degree from Southern Illinois University, Carbondale, in 2004, both in mechanical engineering. He is currently working toward the Ph.D. degree in mechanical engineering in the Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN. His current research interests include adaptive and robust control of mechatronic systems, nonlinear observer design, fault detection, diagnostics, and adaptive fault-tolerant control. Bin Yao received the B.Eng. degree in applied mechanics from Beijing University of Aeronautics and Astronautics, Beijing, China, in 987, the M.Eng. degree in electrical engineering from Nanyang Technological University, Singapore, in 992, and the Ph.D. degree in mechanical engineering from the University of California, Berkeley, in 996. In 996, he joined the School of Mechanical Engineering, Purdue University, West Lafayette, IN, where he has been a Professor since 2007. Dr. Yao was the recipient of a Faculty Early Career Development (CAREER) Award from the National Science Foundation in 998, a Joint Research Fund for Outstanding Overseas Chinese Young Scholars from the National Natural Science Foundation of China in 2005, the O. Hugo Schuck Best Paper (Theory) Award from the American Automatic Control Council in 2004, and the Outstanding Young Investigator Award of ASME Dynamic Systems and Control Division (DSCD) in 2007. He was honored with the Kuang-piu Professor Award in 2005 and the Chang Jiang Chair Professor in 200 from Zhejiang University, China. He is currently the Chair of the ASME DSCD Mechatronics Technical Committee.