Solid:Effective mass without load. Dotted:Effective mass with load

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1 Adaptive Robust Precision Motion Control of Single-Rod Hydraulic Actuators with Time-Varying Unknown Inertia: A Case Study Fanping Bu and Bin Yao + Ray W. Herrick Laboratory School of Mechanical Engineering Purdue University West Lafayette, IN byao@ecn.purdue.edu Abstract This paper studies the precision motion control of single-rod hydraulic actuators with time-varying unknown inertia loads. The swing motion control of a robot arm driven by a single-rod hydraulic actuator (a scaled down version of industrial backhoe loader arm) is used as a case study. During the swing motion, the eective mass acting on the hydraulic actuator varies greatly with the swing angle and thus time-varying. Furthermore, due to the change of the load of the robot arm in operations, the eective mass is also unknown. A discontinuous projection based adaptive robust controller is constructed to take into account this strong dependence of the eective mass on the fast changing swing angle and the unknown payload of the arm. The resulting ARC controller is able to take into account not only the eect of parametric uncertainties coming from the payload and various hydraulic parameters but also the eect of uncertain nonlinearities such as uncompensated friction forces and external disturbances. The ARC controller guarantees a prescribed output tracking transient performance and nal tracking accuracy while achieving asymptotic output tracking in the presence of parametric uncertainties. In addition, the zero output tracking error dynamics for tracking a large class of time-varying trajectories is shown to be globally uniformly asymptotically stable. Comparative experimental results are presented to show the eectiveness and the high performance nature of the proposed control algorithm. Keywords Electro-Hydraulic System, Motion Control, Adaptive Control, Robust Control, Nonlinearity, Servo Control This work is supported in part by the National Science Foundation under the CAREER grant CMS and in part by a grant from the Purdue Research Foundation.

2 I. Introduction The dynamics of hydraulic systems are highly nonlinear [].The system may be subjected to non-smooth and discontinuous nonlinearities due to control input saturation, directional change of valve opening, friction, and valve overlap. Aside from the nonlinear nature of hydraulic dynamics, hydraulic servosystems also have large extent of model uncertainties. The uncertainties can be classied into two categories: parametric uncertainties and uncertain nonlinearities. Examples of parametric uncertainties include the large changes in load seen by the system in industrial use and the large variations in the hydraulic parameters (e.g., bulk modulus) due to the change of temperature and component wear [2]. Other general uncertainties, such as the external disturbances, leakages, and friction, cannot be modeled exactly and the nonlinear functions that describe them are not known. These kinds of uncertainties are called uncertain nonlinearities. These model uncertainties may cause the controlled system, designed on the nominal model, to be unstable or have a much degraded performance. Nonlinear robust control techniques, which can deliver high performance in spite of both parametric uncertainties and uncertain nonlinearities, are essential for successful operations of high-performance hydraulic systems. In the past, much of the work in the control of hydraulic systems uses linear control theory [3, 4, 5, 6] and feedback linearization techniques [7, 8]. In [9], Alleyne and Hedrick applied the nonlinear adaptive control to the force control of an active suspension driven by a double-rod cylinder. They demonstrated that nonlinear control schemes can achieve a much better performance than conventional linear controllers. They considered the parametric uncertainties of the cylinder only. In [], the adaptive robust control (ARC) approach proposed by Yao and Tomizuka in [, 2, 3, 4] was generalized to provide a rigorous theoretical framework for the high performance robust motion control of a double-rod hydraulic actuator by taking into account the particular nonlinearities and model uncertainties of the electro-hydraulic servo-systems. The presented ARC scheme uses smooth projections [2, 3] to solve the design conicts between adaptive control technique and robust control technique. In [5], the recently proposed discontinuous projection method based ARC design [4] is generalized to solve the practical problems associated with smooth projections []. In [6], the discontinuous projection based ARC Scheme [5] is extended to the precision control of a single-rod hydraulic actuator, in which a constant inertia load is considered and the zero output tracking error dynamics for tracking any non-zero constant desired velocity trajectory is shown to be stable. This paper continues the work done in [6] and will extend the results to the precision control of single-rod hydraulic actuators with time-varying and unknown inertia load. To make the development more understandable and application specic, the swing motion control of a robot arm driven by singlerod hydraulic actuators will be used as a case study. During the swing motion, the eective mass acting on the hydraulic actuator varies drastically with the swing angle and thus time-varying. Furthermore, due to the change of the payload of the robot arm, the eective mass is also unknown. This strong dependence of the eective mass on the changing swing angle and the unknown payload of the robot arm makes the swing motion control a perfect case study for the precision motion control of single-rod actuators with time-varying unknown inertia load. A discontinuous projection based ARC controller will be constructed to take into account the strong dependence of the eective mass on the fast 2

3 changing swing angle and the unknown payload of the arm. The resulting ARC controller is able to take into account not only the eect of the parametric uncertainties coming from the payload and various hydraulic parameters but also the eect of uncertain nonlinearities such as uncompensated friction forces and external disturbances. The ARC controller guarantees a prescribed output tracking transient performance and nal tracking accuracy while achieving asymptotic output tracking in the presence of parametric uncertainties only. Comparative experimental results will be presented to show the eectiveness and the high performance nature of the proposed control algorithm. As pointed out in [6], a particular phenomenon associated with the motion control of single-rod hydraulic actuators is that there exists a one-dimensional internal dynamics, which is due to the fact that the two chambers of a single-rod actuator have dierent areas. Thus, it is of practical interests to consider the stability of the resulting internal dynamics. It is shown in this paper that when tracking a fairly general class of time-varying desired trajectories, the resulting zero output tracking error dynamics is globally uniformly asymptotically stable, which is the rst theoretical result available in the literature on the stability of zero output tracking error dynamics when tracking a non-constant velocity trajectory. The paper is organized as follows. Problem formulation and dynamic models are presented in section II. The proposed nonlinear ARC strategy is given in section III. Experimental set-up and comparative experimental results are presented in section IV. Conclusions are drawn in section V. II. Problem Formulation and Dynamic Models Q2 Q L L2 q P2 A2 A P xl Figure : Swing Motion of a Robot Arm Driven by a Single-Rod Hydraulic Cylinder The system under consideration is depicted in Fig., which represents the swing motion of a 3 degree-of-freedom(dof) hydraulic robot arm with the other two joints xed. The goal is to have the cylinder displacement x L (or the swing angle q) track any feasible desired motion trajectory as closely as possible for precision maneuver of the inertia load of the hydraulic cylinder. Using the joint angle q as the coordinate, the kinetic energy of the swing motion can be expressed as T = 2 (J + J ) _q 2 where J is the swing inertia of the robotic arm and J is the swing inertia due 3

4 to the payload mounted at the end of the robot arm. Since the other two links of the robot arm are xed, J and J are constant; J is unknown since it depends on the unknown payload of the robot arm. In terms of the cylinder displacement x L, the kinetic energy of the swing motion can be expressed as T = 2 (J + J ) 2 x_ L L, where the fact that q is uniquely related to x L has been used, i.e., q(x L ). Using Euler-Lagrange equation, The dynamics of the swing motion can be described by (J + J ) 2 L L = P A? P 2 A 2? b _x L? F fc ( _x L )? (J _x 2 L + f(t; ~ x L ; _x L ) () L where P and P 2 are the pressures of the two cylinder chambers respectively, A and A 2 are the ram areas of the two cylinder chambers respectively, b represents the combined coecient of the modeled damping and viscous friction forces on the load and the cylinder rod, F fc represents the modeled Coulomb friction force, and ~ f(t; x L ; _x L ) represents the lumped modeling error including external disturbances and terms like the unmodeled friction forces. 2 x Solid:Effective mass without load Dotted:Effective mass with load.4 Mass(kg) x L (m) Figure 2: Eective Mass vs Cylinder Displacement As seen in (), the eective mass driven by the cylinder is (J + J ) L, which could change greatly during operations since L is a nonlinear function of the cylinder position x L and varies signicantly; for the system used in the experiments in section IV, the dependence of the eective mass on the cylinder position with and without a payload of 45kg is shown in Fig.2, in which the eective mass varies from 2:3 3 kg to 6:656 3 kg for no-load situation and 5:746 3 kg to :884 4 kg for loaded situation. Furthermore, it is unknown due to the unknown swing inertia J caused by the payload. For simplicity, the swing inertia due to the swing cylinder is neglected since the mass of the swing cylinder is quite small comparing to the inertia of the whole robotic arm 4

5 The actuator (or the cylinder) dynamics can be written as [], rst for the forward chamber: V e P_ =?A _x L? C tm (P? P 2 )? C em (P? P r ) + Q (2) where V = V h + A x L is the total volume of the forward chamber including the hose volume between the forward chamber and the servovalve, V h is the forward chamber volume when x L =, e is the eective bulk modulus, C tm is the coecient of the internal leakage of the cylinder, C em is the coecient of the external leakage for the forward loop. Q is the supplied ow to the cylinder. Q is related to the spool valve displacement of the servovalve, x v, by [] Q = k q x v p P ; P = ( P s? P for x v P? P r x v < where k q is the ow gain coecient for the forward loop, P s is the supply pressure of the pump uid, and P r is the reference pressure in the return tank. chamber is: where V 2 V 2 e (3) Similarly, the actuator dynamics for the return P_ 2 = A 2 _x L + C tm (P? P 2 )? C em2 (P 2? P r )? Q 2 (4) = V h2? A 2 x L is total volume of the return chamber including the hose volume between the return chamber and the servovalve, V h2 is the return chamber volume when x L =, C em2 is the coecient of the external leakage for the return loop, Q 2 is the return ow to the cylinder. Q 2 is related to the spool valve displacement of the servovalve, x v, by [] Q 2 = k q2 x v p P2 ; P 2 = ( P 2? P r for x v P s? P 2 x v < As in [6], to minimize the numerical error and facilitating the gain-tuning process, constant scaling factors are introduced to the pressures and the valve opening respectively; the scaled pressure and valve opening are P = P S c3, P2 = P S c3 2, Ps = P S c3 s, Pr = P S c3 r, and x v = x S c4 v respectively. Dene a set of state variables as x = [x ; x 2 ; x 3 ; x 4 ] T = [x L ; _x L ; P ; P 2 ] T, the entire system, Eqs.()-(3) and (4), with control input u = x v can be expressed in state space form as: (5) _x = x 2 _x 2 = m (x ) + J (x 3? A c x 4? bx 2? F fc )? p q (x )x J _x 3 = h (x )[ es c4k q psc3 (? A V x 2 + g 3 (x 3 ; sign(u))u)? Ctme h V h d m (x ) (x V h 3? x 4 )? C em e (x V h 3? P r )] _x 4 = h 2 (x )[ es c4k q psc3 ( A V 2 x 2? g 4 (x 4 ; sign(u))u) + Ctme (x 3? x 4 )? C em2 e (x h V h 4? P r )] (6) where m (x ) = L ) 2, m (x ) = m (x ) s c3 A, Ac = A 2 A, b = d = + J J ~ f(t; x ; x 2 ), h (x ) = + A h x, h 2 (x ) = b S c3 A, Ffc = F fc(x 2 ) S c3 A k v? A h2 x, k v, p q (x ) L L, = V h2 V h, Ah = A V h, Ah2 = A 2 V h, 5

6 A = A k q S c4 p Sc3, A2 = A 2 k q S c4 p Sc3, and g 3 and g 4 are dened by q ( Ps? x 3 for u g 3 = P ; P = x 3? P r u < ( x4? P g 4 = k qc qp r for u 2 ; P 2 = P s? x 4 u < (7) in which k qc = k q2 k q. Given the desired motion trajectory x Ld (t), the objective is to synthesize a control input u such that the output y = x tracks x Ld (t) as closely as possible in spite of various model uncertainties. III. Adaptive Robust Control Design III. Design Model and Issues to be Addressed In general, the system is subjected to parametric uncertainties due to the variations of J, b, F fc, e, C tm, C em, and C em2. For simplicity, in this paper, we only consider the parametric uncertainties due to the unknown payload swing inertia J, the bulk modulus e, the nominal value of the lumped modeling error d, d n, leakage coecients C tm, C em, and C em2. Other parametric uncertainties can be dealt with in the same way if necessary. In order to use parameter adaptation to reduce parametric uncertainties to improve performance, it is necessary to linearly parametrize the state space equation (6) in terms of a set of unknown parameters. To achieve this, dene the unknown parameter set = [ ; 2 ; 3 ; 4 ; 5 ; 6 ] T as = + J, 2 = d n, 3 = es p c4k q V J h Sc3, 4 = Ctme, V 5 = C em e, and h V 6 = C em2 e. The state space h V h equation (6) can thus be linearly parametrized in terms of as _x = x 2 _x 2 = m (x ) (x 3? x 4Ac? bx 2? F fc (x 2 ))? p q (x )x m (x ) + d(t; ~ x ; x 2 ); _x 3 = h (x )[ 3 (? A x 2 + g 3 (x 3 ; u)u)? 4 (x 3? x 4 )? 5 (x 3? P r )] _x 4 = h 2 (x )[ 3 ( A 2 x 2? g 4 (x 4 ; u)u) + 4 (x 3? x 4 )? 6 (x 4? P r )] ~ d = d(t;x ;x 2 )?d n m (x ) For simplicity, the following notations are used throughout the paper: i represents the i-th component of the vector, and the operation < for two vectors is performed in terms of the corresponding elements of the vectors. Since the extents of the parametric uncertainties and uncertain nonlinearities are normally known, the following practical assumption is made Assumption Parametric uncertainties and uncertain nonlinearities satisfy (8) 2 = f : min < < max g j ~ d(t; x ; x 2 )j d (x ; x 2 ; t) (9) where min = [ min ; : : : ; 6min ] T ; max = [ max ; : : : ; 6max ] T and d (t; x ; x 2 ) are known. Physically, 6

7 > and 3 >. So it is also assumed that min > and 3min >. At this stage, it can be seen that the major diculties in controlling (8) are: (i) The system dynamics are highly nonlinear, either due to the dependence of the eective mass on swing angle or the inherent nonlinearities of hydraulic dynamics such as the nonlinear ow gains (represented by g 3 (x 3 ; sign(u)) and g 4 (x 4 ; sign(u))) and the change of control volumes (represented by h (x ) and h 2 (x )). (ii) The system has severe parametric uncertainties like large variation of the inertial load J, the change of bulk modulus due to entrapped air or temperature, etc. (iii) The system may also have large extent of lumped modeling error (represented by the uncertain nonlinearity d) ~ including external disturbances, unmodeled friction force, etc. (iv) The model uncertainties are mismatched, i.e. both parametric uncertainties and uncertain nonlinearities appear in the dynamic equations which are not directly related to the control input u = x v. (v) As will become clear later, the "relative degree" of the system is three while the system state has a dimension of four. Thus, there exists a one-dimensional internal dynamics after an ARC controller is synthesized. It is thus necessary to consider the stability of the resulting internal dynamics, or at least the stability of the resulting zero output tracking error dynamics when tracking a large class of reference trajectories. To address the above challenges, following general strategies are adopted: a. Physical model based nonlinear analysis and synthesis will be employed to deal with the nonlinear nature of the system dynamics directly, i.e., the subsequent controller design will be based on the nonlinear physical model (8). b. The adaptive robust control (ARC) approach [, 4] will be used to handle the eect of both parametric uncertainties and uncertain nonlinearities eectively for high performance - certain robust feedback will be employed to attenuate the eect of various model uncertainties as much as possible while parameter adaptation will be utilized to reduce model uncertainties for an improved performance. c. Backstepping design [9] via ARC Lyapunov function [3, 4] will be used to overcome the design diculties caused by unmatched model uncertainteis. Following the above general guideline, the proposed ARC design is detailed in the subsequent subsections. III.2 Notations and Discontinuous Projection Mapping 7

8 Let ^ denote the estimate of and ~ the estimation error (i.e., ~ = ^? ). Viewing (9), a simple discontinuous projection can be dened [7, 8] as 8 >< if ^ i = imax and > P roj^ () = if ^ i = imin and < >: otherwise () By using an adaptation law given by _^ = P roj^(?) () where? > is a diagonal matrix and is an adaptation function to be synthesized later. It can be shown [7, 8, ] that for any adaptation function, the projection mapping used in () guarantees (P) ^ 2 = f^ : min ^ max g (P2) ~ T (?? P roj^ (?)? ) (2) ; 8 For simplicity, let ^_x 2 represent the calculable part of _x 2, which is given by ^_x2 = ^ m (x ) (x 3? A c x 4? bx 2? F fc (x 2 ))? p q (x )x ^ 2 m (x ) (3) III.3 ARC Controller Design The design parallels the recursive backstepping design procedure via ARC Lyapunov functions in [4, 3] as follows. Step Noting that the rst equation of (8) does not have any uncertainties, an ARC Lyapunov function can thus be constructed for the rst two equations of (8) directly. Dene a switching-function-like quantity as z 2 = _z + k p z = x 2? x 2eq ; x 2eq = _xd? k p z (4) where z = x? x d (t), in which the reference trajectory x d (t) will be specied later to achieve a guaranteed transient performance and k p is any positive feedback gain. If z 2 converges to a small value or zero, then the output tracking error z will converge to a small value or zero since G s (s) = z (s) z 2 (s) = s+k p is a stable transfer function. So the rest of the design is to make z 2 as small as possible with a guaranteed transient performance. Dene the load pressure as P L = x 3? A c x 4. Dierentiating (4) and noting (8) _z 2 = _x 2? _x 2eq = m (x ) (P L? bx 2? F fc )? p q (x )x m (x ) + ~ d? _x 2eq (5) where _x 2eq = xd? k p _z is calculable. In (5), if we treat P L as the virtual control input to (5), we can synthesize a virtual control law 2 for P L such that z 2 is as small as possible. Since (5) has both 8

9 parametric uncertainties and 2 and uncertain nonlinearity ~ d, the ARC approach proposed in [4] will be generalized to accomplish the objective. The control function 2 consists of two parts given by 2 (x ; x 2 ; ^ ; ^ 2 ; t) = 2a + 2s 2a = bx 2 + F fc + m (x ) ^ ( _x 2eq + p q (x )x 2 2? ^ 2 m (x ) ) (6) in which 2a functions as an adaptive control law used to achieve an improved model compensation through on-line parameter adaptation given by (), and 2s is a robust control law to be synthesized later. If P L were the actual control input, then in () would be 2 = w 2 2 z 2 ; 2 = h m (x ) ( 2a? bx 2? F i T fc ); m (x ) ; ; ; ; (7) where w 2 > is a constant weighting factor. Due to the use of discontinuous projection (), the adaptation law () is discontinuous and thus cannot be used in the control law design at each step as contrary to the tuning function based backstepping adaptive control [9]; backstepping design needs the control function synthesized at each step to be suciently smooth in order to obtain its partial derivatives. To compensate for this loss of information, the robust control law has to be strengthened. So the robust control function 2s consists of two terms given by 2s = 2s + 2s2 2s =? m max min (k 2s + k 2 )z 2 ; k 2 > where m max represent the upper bound of the inertia m (x ), k 2 is a positive scalar, k 2s a positive nonlinear control gain function to be specied later, and 2s2 is a robust control function synthesized as follows. Let z 3 = P L? 2 denote the input discrepancy. Substituting (6) and (8) into (5) while noting (7), (8) _z 2 = m (x ) z 3 + m (x ) 2s + ^ m (x ) ( 2a? bx 2? F fc )? ~ m (x ) ( 2a? bx 2? F fc )?p q (x )x m (x ) + d ~? _x 2eq = m (x ) z 3? m max min m (x ) (k 2 + k 2s )z 2 + m (x ) 2s2? ~ T 2 + d ~ (9) The robust control function 2s2 is now chosen to satisfy the following conditions condition i z 2 [ m (x ) 2s2? ~ T 2 + d] ~ " 2 condition ii z 2 2s2 (2) where " 2 is a design parameter which can be arbitrarily small. Essentially, condition i of (2) shows that 2s2 is synthesized to dominate the model uncertainties coming from both parametric uncertainties ~ and uncertain nonlinearities d, ~ and condition ii is to make sure that 2s2 is dissipating in nature so that it does not interfere with the functionality of the adaptive control part 2a. How to choose 2s2 to satisfy constraints like (2) can be found in [2, 3]. 9

10 For the positive semi-denite (p.s.d.) function V 2 dened by V 2 = 2 w 2z 2 2 (2) from (9), its time derivative is _V 2 = m (x ) w 2 z 2 z 3 + w 2 z 2 ( m (x ) 2s2? ~ T 2 + ~ d)? w 2 min m max m (x ) (k 2 + k 2s )z 2 2 (22) Step 2 As seen from (22), if z 3 =, then, output tracking can be achieved in viewing i of (2). So Step 2 is to synthesize a control input so that z 3 converges to zero or a small value with a guaranteed transient performance as follows. From (8) where z_ 3 = P _ L? _ 2 = 3 [?( A h + A 2 Ac h 2 )x 2 + (h g 3 + A c h 2 g 4 )u]? 4 (x 3? x 4 )(h + A c h 2 )? 5 h (x 3? P r ) + 6 Ac h 2 (x 4? P r )? _ 2c? _ 2u (23) _ 2c _ 2u 2 [? m (x ) (x 3? A c x 4? bx 2? F fc ) ~? m (x ) ~ 2 + ~ d] ^ In (24), _ 2c is calculable and can be used in the design of control functions, but _ 2u cannot due to various uncertainties. Therefore, _ 2u has to be dealt with via certain robust feedback in this step design. Dene the continuous nonlinear load ow mapping function as _^ (24) Q L = h (x )g 3 (x 3 ; sign(u)) + A c h 2 (x 2 )g 4 (x 4 ; sign(u)) u (25) From (23), Q L can be thought as the virtual control input and step 2 is to synthesize a control function 3 for Q L such that P L tracks the desired control function 2 synthesized in Step with a guaranteed transient performance. Consider the augmented p.s.d. function V 3 = V w 3z3 2, where w 3 > is a weighting factor. From (23) and (25) V_ 3 = V _ 2 + w 3 z 3 _z 3 = V _ 2 j 2 +w 3 z 3 3 Q L + 3e? T ~ 2 d 2 ^ (26)

11 where 3e ; 3 are dened by 3e = w 2 m (x ) w 3 z 2 ^? ^ 3 ( A h + A 2 h 2 Ac h 2 )x 2? ^ 4 (h + A c h 2 )(x 3? x 4 )?^ 5 h (x 3? P r ) + ^ 6Ac h 2 (x 4? P r )? _ 2c 2 h w2 m (x ) w 3 z 2 (x 3? A c x 4? bx 2? i 3 F fc 2 m (x 2 3 =?( A h + A 2Ac h 2 )x 2 + 3a?(h + A c h 2 )(x 3? x 4 ) 6 4?h (x 3? P 7 r ) 5 A c h 2 (x 4? P r ) (27) and _ V 2 j 2 is a short-hand notation used to represent _ V 2 when P L = 2 (or z 3 = ), i.e. _V 2 j 2 = V _ 2? m (x ) w 2z 2 z 3 w 2 z 2 ( m (x ) 2s2? ~ T 2 + ~ d)? w 2 min m max m (x ) (k 2 + k 2s )z 2 2 (28) Similar to (6), the control function 3 consists of two parts and the control functions are thus chosen as 3 (x ; x 2 ; x 3 ; x 4 ; ^; t) = 3a + 3s 3a =? ^ 3 3e 3s = 3s + 3s2 3s =? 3min (k 3 + k 3s )z 3 where k 3 > is a constant, k 3s is a positive denite control gain function which will be specied later. Substitute(29) into (26) and noting (27), the time derivative of Lyapunov function V 3 can be expressed by V_ 3 = V _ 2 j 2 +w 3 z 3 ( 3 3s2? ~ T 2 2 d)? w3 3 (k 3min 3 + k 3s )z3 2? 2 3z 3 ^ (3) As in (2), 3s2 is a robust control function satisfying the following two conditions (29) condition i z 3 h 3 3s2? ~ i T 2 2 d " 3 condition ii z 3 3s2 (3) where " 3 is a design parameter. Once the control function 3 for Q L is synthesized as given in (29), the actual control input u can be backed out from the continuous one-to-one nonlinear load ow mapping (25), i.e., determining a u(x ; x 2 ; x 3 ; x 4 ; ^; t) such that: h (x )g 3 (x 3 ; sign(u)) + A c h 2 (x )g 4 (x 4 ; sign(u)) u = 3 (x ; x 2 ; x 3 ; x 4 ; ^; t) (32) Since h ; h 2 ; g 3, and g 4 are all physically positive functions, the sign of control input u is always the

12 same as the load ow rate Q L = 3. Thus, the solution of (32) is u = h (x )g 3 (x 3 ;sign( 3 ))+ A ch2 (x )g 4 (x 4 ;sign( 3 )) 3(x ; x 2 ; x 3 ; x 4 ; ^; t) (33) III.4 Main Performance Results With the above design, the following theoretical results on the performance of the proposed ARC controller can be obtained. Lemma Let the parameter estimates be updated by the adaptation law () in which is chosen as = P 3 j=2 w jz j j (34) If nonlinear control gains k 2s and k 3s are chosen such that k 2s 2 w 2k? 2 k 2 and k 3s w 3 ^ 2 k w 3k? 3 k 2 respectively, then, the control law (33) guarantees that V 3 (t) exp(? V t)v 3 () + " V V [? exp(? V t)] (35) where V = 2minfk 2 ; k 3 g and " V = w 2 " 2 + w 3 " 3 4 Proof. See Appendix A. 2. As seen from (35), the initial value of V 3 aects the transient performance of the system. To further minimize the transient tracking errors, the following trajectory initialization [2, 9] can be used to render V 3 () =. Lemma 2 Let x d (t) be the trajectory generated through a stable 3-rd order lter given by x (3) d + x (2) d + : : : + 3x d = x (3) Ld + x (2) Ld + : : : + 3x Ld (36) If the initials x d (); : : : ; x (3) d () are chosen as x d () = x () _x d () = x 2 () x d () = ^_x 2 () (37) then, z i () = ; i = ; 2; 3 and V 3 () =. 4 Proof: The Lemma can be proved in the same way as in [2]. 2 Theorem Given the reference trajectory x d (t) generated through (36) with initial conditions chosen by (37), the following results hold if the control law (33) with the adaptation law () is applied: 2

13 A. In general, the tracking errors z = [z ; z 2 ; z 3 ] T are bounded. Furthermore, V 3, an index for the bound of the tracking error z, is bound above by V 3 (t) " V V [? exp(? V t)] (38) The output tracking error e y = x? x Ld (t) can be guaranteed to have any prescribed transient performance by suitably selecting certain controller parameters. B If after a nite time t, ~ d =, i.e., the model uncertainties are due to parametric uncertainties only, in addition to the results in A, asymptotic output tracking or zero nal tracking error (i.e., z?! as t?! ) is obtained for any positive gains k and ". 4 Proof : See Appendix B. 2 Remark Results in A of Theorem indicate that the proposed controller has an exponentially converging transient performance with the exponentially converging rate V and the nal tracking error being able to be adjusted via certain controller parameters freely in a known form; it is seen from (38) that V can be made arbitrarily large, and " V, the bound of V () (an index for the nal tracking V errors), can be made arbitrarily small by increasing feedback gains k = [k 2 ; k 3 ] T and/or decreasing controller parameters " = [" 2 ; " 3 ] T. Such a guaranteed transient performance is important for the control of electro-hydraulic systems since execute time of a run is normally short. Theoretically, this result is what a well-designed robust controller can achieve. B of Theorem implies that the parametric uncertainties may be reduced through parameter adaptation and an improved performance is obtained. Theoretically, Result B is what a well-designed adaptive controller can achieve. } III.5 Internal Dynamics and Zero Dynamics Theorem shows that, under the proposed ARC control law, the three transformed system state variables, z = [z ; z 2 ; z 3 ] T 2 R 3, are bounded, from which one can easily prove that the position x, the velocity x 2, and the load pressure P L = x 3? A c x 4 are bounded. However, since the original system (8) has a dimension of 4, there exists a one-dimensional internal dynamics, which reects the physical phenomenon that there may exist innite number sets of pressures in the two chambers of the single-rod actuator to produce the same required load pressure P L = x 3? A c x 4. In other words, the fact that the load pressure P L = x 3? A c x 4 and the valve opening x 5 are bounded does not necessarily imply that the pressures x 3 and x 4 in the two chambers of the single-rod cylinder are bounded. It is thus necessary to check the stability of the resulting internal dynamics to make sure that the two pressures x 3 and x 4 are bounded in implementation in order to have a bounded control input. Although simulation and experimental results seem to verify that the two pressures are indeed bounded when tracking any motion trajectory, it is of both practical and theoretical interests to see if we can prove this fact, which is the focus of this subsection. 3

14 As a rst step toward to the proof of the stability of the resulting internal dynamics, we will consider the stability of the zero output tracking error dynamics [2] when tracking any reference trajectory x d (t). The reason for doing this is that the zero output tracking error dynamics for tracking a desired trajectory x d (t) is the same as the internal dynamics when z(t) = ; 8t. Thus, if the zero output tracking error dynamics is proven to be globally uniformly asymptotically stable, then, it is reasonable to expect that all internal variables are bounded when the proposed ARC law is applied, since z is guaranteed to converge to a small value very quickly by Theorem. In [6], the zero output tracking error dynamics when tracking a non-zero constant velocity reference trajectory was proved to be stable. Here, we consider the general situation of tracking any feasible timevarying reference trajectory x d (t). As in [6], for simplicity, the leakage ows are neglected in the analysis since they are normally relatively much smaller than the control ows needed to track a timevarying trajectory. The results are summarized in the following Theorem. Theorem 2 Assume that the leakage ows can be neglected and the system is absence of uncertain nonlinearities, i.e., 4 = 5 = 6 = and ~ d = in (8). Let _x d (t) be any feasible time-varying reference trajectory that the system can follow, i.e., there exists a bounded state equilibrium trajectory x e (t) = [x e (t); x 2e (t); x 3e (t); x 4e (t)] T 2 L 4 and a bounded control input u e(t) satisfying the system equation (8) and having x d (t) as the output (i.e., x e (t) = x d (t); 8t). When tracking such a feasible time-varying trajectory _x d, the following results hold: A. The desired load pressure trajectory P Ld (t) and the desired load ow trajectory Q Ld (t) for tracking _x d perfectly are unique and given by P Ld (t) = b _x d (t) + F h i fc ( _x d (t)) + m (x d (t)) x d (t) + p q (x d (t)) _x 2 d (t)? 2 m (x d (t)) Q Ld (t) = P_ 3 Ld (t) + A? hd (t) + A 2 c h 2d(t) _x d (t) where h d (t) = h (x d (t)) and h 2d (t) = h 2 (x d (t)). In addition, if Q Ld (t) is zero only at discrete points, any bounded equilibrium state trajectory x e (t) and the bounded equilibrium control input u e (t) having x d (t) as the output satisfy the following algebraic equations (39) x e (t) = x d (t) x 2e (t) = _x d (t) x 3e (t)? A c x 4e (t) = P Ld (t) u e (t) = h d (t)g 3e (t)+ A ch2d (t)g 4e (t) Q Ld(t) (4) and dierential equations _x 3e (t) = h d(t)g 3e (t) _ P Ld (t)+ 3 _x d h d h 2d A 2 ( A cg3e?g 4e ) _x 4e (t) =?h 2dg 4e _ h d g 3e + A ch2d g 4e P Ld + 3 _x d h d h 2d A ( A cg3e?g 4e ) h d g 3e + A ch2d g 4e (4) where g 3e (t) = g 3 (x 3e (t); sign(q Ld (t)) and g 4e (t) = g 4 (x 4e (t); sign(q Ld (t))). 4

15 B. Suppose that the reference trajectory x d (t) is such that the resulting desired load pressure P Ld (t) in (39) is within the physical limit, i.e.,?( A c Ps? P r ) < P Ld (t) < P s? A c Pr. In addition, assume that x d (t) satises the following persistently-exciting-like condition, PE: 9T; c > such that R t+t t j Q Ld () j d c > ; 8t (42) where T and c are any positive constants, and Q Ld (t) is given by (39). Then, the resulting zero output tracking error dynamics (i.e., the dynamics when z = ; 8t) is globally uniformly asymptotically stable with respect to the bounded equilibrium state trajectory x e (t) in (4). } Proof: See Appendix C Remark 2 Roughly speaking, the PE condition (42) is equivalent to the physical situation that the desired valve opening corresponding to the reference trajectory x d (t) is not always, which includes the non-zero constant velocity reference trajectory studied in [5], and any sinusoidal reference trajectories. } Remark 3 Result B of Theorem 2 indicates that all bounded equilibrium state trajectories x e (t) converge asymptotically to a unique trajectory depending on the reference trajectory x d only. } IV. Comparative Experiments IV. Experimental Setup To test the proposed nonlinear ARC strategy and study fundamental problems associated with the control of electro-hydraulic systems, a three-link robot arm (a scaled down version of industrial backhoe loader arm) driven by three single-rod hydraulic cylinders has been set up at the Ray W. Herrick Laboratory of the School of Mechanical Engineering. The three hydraulic cylinders are controlled by two proportional directional control valves and one servovalve manufactured by Parker Hannifan company. The entire set-up is shown in Fig.3. For the purpose of this paper, experiments have been conducted on the swing motion control of the arm (or the rst joint) with the other two joints xed. The schematic of the swing circuit is shown in Fig.4. The swing circuit is driven by a single-rod cylinder (Parker D2HXTS23A) and controlled by a servovalve (Parker BD76AAAN). The cylinder has a built-in Temposonic LDT sensor, which provides the position and velocity information of the cylinder movement. Pressure sensors (Entran's EPXH-X-2.5KP) are installed on the each chamber of the cylinder. Due to the noises in the analog signals, the eective measurement resolution for the position is around mm and for the pressure is :35 5 P a. Since the range of the velocity provided by the Temposonic LDT sensor is too small (less than.9m/sec), backward dierence plus lter is used 5

16 Figure 3: Experimental setup to obtain the needed velocity information at high speed movement. The entire system is controlled by a dspace system and monitored by a host Pentium II PC. Control program is download from the host PC to the dspace system. Through plugged in A/D and D/A boards, all analog measurement signals (the cylinder position, velocity, forward and return chamber pressures, and the supplied pressure) are fed back to the dspace system and the calculated control command is then sent out to the driver of the servovalve. The supply pressure is 6:9 6 P a. Swing cylinder Cylinder position velocity P P2 Valve Command Dspace System 32Channels 6 bit A/D DS23 6Channels 6 bit D/A DS22 CPU Board DS3 PC Interface Board DS8 Pentium II 266 Swing Motion Host Computer Figure 4: Schematic of Experimental setup 6

17 IV.2 System Identication The swing inertia of the robot arm is estimated to be J = 78:69kgm 2. The cylinder physical parameters are A = 2:27?3 m 2, A 2 = :69?3 m 2, V h = 4:995?4 m 3, and V h2 = 9:68?4 m 3. Although tests have been done to identify the friction curve as seen in Fig.5, for simplicity, friction compensation is not used in the following experiments (i.e., b = F fc = in ()) to test the robustness of the proposed algorithm to these hard-to-model terms. By assuming no leakage ows and moving the cylinder at dierent constant velocities, we can back out the static ow mappings (3) and (5) of the servovalve. The estimated ow gains are k q = 3:59 p?8 m 3 =( P asecv ) and k q2 = 3:72 p?8 m 3 =( P asecv ) respectively. Dynamic tests are also performed and reveal that the servovalve has a bandwidth around Hz. The eective bulk modulus is estimated around 2:7 8 P a. 2 Friction Force(lb) Velocity(in/sec) Figure 5: Friction force IV.3 Controller Simplication A simplication is made to the selection of the specic robust control term 2s in (8) and 3s in (29). Let k 2s and k 3s be any nonlinear feedback gains satisfying k 2s m max min k 3s 3min h h 2 w 2k? 2 k 2 + k 2 + 2" 2 (k M k 2 k 2 k d ) i i k 3 + w 3 ^ k2 + 2 w 3k? 3 k 2 + 2" 3 (k M k 2 k 3 k kd 2) (43) in which M = max? min and d is dened in (9). Then, using similar arguments as in [5, 4], one can show that the robust control function of 2s =?k 2s z 2 satises (8) and (2), and 3s =?k 3s z 3 satises (29) and (3). Knowing the above theoretical results, we may implement the needed robust control terms in the following two ways. The rst method is to pick up a set of values for k 2 ; k 3 ; w 2 ; w 3 ; " 2 ; and " 3 to calculate the right hand side of (43). k 2s and k 3s can then be determined so that (43) is satised for a guaranteed global stability and a guaranteed control accuracy. This approach is rigorous and should be the formal approach to choose. However, it increases the complexity of the resulting control law considerably since it may need signicant amount of computation time to calculate the lower bounds. 7

18 As an alternative, a pragmatic approach is to simply choose k 2s and k 3s large enough without worrying about the specic values of k 2 ; k 3 ; w 2 ; w 3 ; " 2 ; and " 3. By doing so, (43) will be satised for certain sets of values of k 2 ; k 3 ; w 2 ; w 3 ; " 2 ; and " 3, at least locally around the desired trajectory to be tracked. In this paper, the second approach is used since it not only reduces the on-line computation time signicantly, but also facilitates the gain tuning process in implementation. IV.4 Comparative Experimental Results Three controllers are tested for comparison: ARC (with time varying inertia compensation) : the controller proposed in this paper with the simplication in IV.3. For simplicity, in the experiments, only two parameters, and 2, are adapted; the rst parameter represents the eect of external inertia load and the second parameter represents the eect of the nominal value of the lumped modeling error. Since the valve dynamics is neglected in the design and its bandwidth is not so high (around Hz), not so large feedback gains are used to avoid instability; the control gains used are k p = k 2s = k 3s = 9. The scaling factors of s c3 = 2:885 6 and s c4 = :8588 are used. Adaptation rates are set at? = diagf:; :5g. ARC2 (with constant inertial compensation) : the controller that is synthesized in [6] with the assumption that the mass driven by the cylinder is constant. Scaling factors are the same as in ARC. The control gains used are k p = k 2s = k 3s = 9. Adaptation rates are set at? = diagf:; :8g. Motion Controller : the state-of-the-art industrial motion controller (Parker's PMC627ANI 2- axis motion controller) that we bought from Parker Hannian company along with the Parker's cylinders and valves used for the experiments. The controller is essentially a PID controller with velocity and acceleration feedforward compensation. Controller gains are obtained by strictly following the gain tuning process stated in the "Servo Tuner User Guide" coming with the motion controller [2]. The tuned gains are SGP = 2; SGI = :5; SGV = 22; SGV F = ; SGAF = :2, which represent the P-gain, I-gain, D-gain, the gain for velocity feedforward compensation, the gain for acceleration feedforward compensation respectively. The three controllers are rst tested for a point-to-point motion trajectory shown in Fig.6, which has a maximum velocity of v max = :m=sec and a maximum acceleration of a max = :2m=sec 2. The tracking errors are shown in Fig.7. As seen, the proposed ARC and ARC2 have a better performance than the Motion Controller in term of both transient and nal tracking errors. Both ARC controllers reduce the nal tracking error almost down to the measurement noise level. However, since ARC employs time varying inertia compensation which captures the nonlinear eect of the swing motion, ARC achieves a better tracking performance than ARC2; the tracking error of ARC is almost within the position measurement noise level all the time while ARC2 still has small transient error at the beginning of travel where the eective mass driven by the cylinder varies signicantly with the change of swing angle as shown in Fig. 2. 8

19 To test the performance robustness of the proposed algorithms to parameter variations, a 45kg load is added at the end of the robot arm, which is equivalent to a swing inertia of J = 44kgm 2. The tracking errors of all three controllers are shown in Fig.8. As seen, due to the fact that the adaptation algorithm of the controller ARC is able to pick up the change of the swing inertia, a better transient tracking performance is achieved in comparison to the constant inertia compensation controller ARC2. Again, both ARC and ARC2 exhibit better performance than Motion Controller. The system is then run for a fast point to point trajectory shown in Fig.6, which has a maximum velocity of v max = :3m=sec and a maximum acceleration of a max = :5m=sec 2. The tracking errors of all three controllers are shown in Fig.9. As seen, the Motion Controller cannot handle such an aggressive movement well and a large tracking error around 5mm exhibits during the constant highspeed movement. In contrast, the tracking error of the proposed ARC during the entire run is kept within 3mm. Furthermore, the tracking error goes back to the measurement noise level of mm very quickly after the short large acceleration and deceleration periods. The transient pressures and the control input of ARC are shown in Fig. and Fig. respectively, which are regular. Again, the proposed ARC performs better than ARC2. To further illustrate the eectiveness of the time-varying inertia compensation, the experiment is run for tracking a Hz sinusoidal desired trajectory (:8sin(t=2)) with 45kg load for 6sec. The errors 4 of ARC and ARC2 are shown in the Fig.2. Again, since ARC uses time-varying inertia compensation and adapts the constant unknown payload instead, it achieves a better tracking performance than ARC2. All these results illustrate the eectiveness of the proposed time-varying inertia compensation ARC algorithm. The transient pressures of ARC when tracking the above sinusoidal trajectory are shown in Fig. 3. As seen, all signals are bounded and within their physical limits, which agrees with the theoretical prediction stated in Theorem 2. Also, the two internal pressures converge to certain periodic trajectories, which agrees with the prediction in Remark 3; the reference trajectory x d (t) is periodic and, as such, the resulting steady state equilibrium pressure trajectories should be periodic. V. Conclusions In this paper, a discontinuous projection based ARC controller is constructed for the precision motion control of single-rod hydraulic actuators with time-varying unknown inertia loads. The controller takes into account the particular nonlinearities associated with the time-varying inertia load and the hydraulic dynamics, and allows parametric uncertainties such as variations of robot payload and various hydraulic parameters as well as uncertain nonlinearities coming from external disturbances, uncompensated friction forces, etc. Zero output tracking error dynamics when tracking a large class of time-varying trajectories is proved to be globally uniformly asymptotically stable, which is the rst theoretical result available on the subject. Extensive comparative experiments have been conducted on the swing motion control of a hydraulic robot arm. Experimental results veried the eectiveness and the high-performance nature of the proposed ARC algorithm; in comparison to a state-of-the-art industrial motion controller, the proposed ARC algorithm achieves more than a magnitude reduction of tracking errors. Furthermore, 9

20 during the constant velocity portion of the motion, the ARC controller reduces the tracking errors almost down to the measurement resolution level. The paper also serves as an illustrating example on how to incorporate the particular nonlinearity of a physical system into the proposed nonlinear ARC framework for an improved performance (e.g. the nonlinear transformation between the constant swing inertia and the time-varying eective mass of the swing hydraulic cylinder in this paper). References [] H. E. Merritt, Hydraulic control systems. New York: Wiley, 967. [2] J. Whatton, Fluid power systems. Prentice Hall, 989. [3] T. C. Tsao and M. Tomizuka, \Robust adaptive and repetitive digital control and application to hydraulic servo for noncircular machining," ASME J. Dynamic Systems, Measurement, and Control, vol. 6, pp. 24{32, 994. [4] P. M. FitzSimons and J. J. Palazzolo, \Part i: Modeling of a one-degree-of-freedom active hydraulic mount; part ii: Control," ASME J. Dynamic Systems, Measurement, and Control, vol. 8, no. 4, pp. 439{448, 996. [5] A. R. Plummer and N. D. Vaughan, \Robust adaptive control for hydraulic servosystems," ASME J. Dynamic System, Measurement and Control, vol. 8, pp. 237{244, 996. [6] J. E. Bobrow and K. Lum, \Adaptive, high bandwidth control of a hydraulic actuator," ASME J. Dynamic Systems, Measurement, and Control, vol. 8, pp. 74{72, 996. [7] R. Vossoughi and M. Donath, \Dynamic feedback linearization for electro-hydraulically actuated control systems," ASME J. Dynamic Systems, Measurement, and Control, vol. 7, no. 4, pp. 468{477, 995. [8] L. D. Re and A. Isidori, \Performance enhancement of nonlinear drives by feedback linearization of linear-bilinear cascade models," IEEE Trans. on Control Systems Technology, vol. 3, no. 3, pp. 299{ 38, 995. [9] A. Alleyne and J. K. Hedrick, \Nonlinear adaptive control of active suspension," IEEE Trans. on Control Systems Technology, vol. 3, no., pp. 94{, 995. [] B. Yao, G. T. C. Chiu, and J. T. Reedy, \Nonlinear adaptive robust control of one-dof electro-hydraulic servo systems," in ASME International Mechanical Engineering Congress and Exposition (IMECE'97), FPST-Vol.4, (Dallas), pp. 9{97, 997. [] B. Yao and M. Tomizuka, \Smooth robust adaptive sliding mode control of robot manipulators with guaranteed transient performance," in Proc. of American Control Conference, pp. 76{8, 994. The full paper appeared in ASME Journal of Dynamic Systems, Measurement and Control, Vol. 8, No.4, pp , 996. [2] B. Yao and M. Tomizuka, \Adaptive robust control of siso nonlinear systems in a semi-strict feedback form," Automatica, vol. 33, no. 5, pp. 893{9, 997. (Part of the paper appeared in Proc. of 995 American Control Conference, pp25-255). 2

21 [3] B. Yao and M. Tomizuka, \Adaptive robust control of mimo nonlinear systems," 997. Submitted to Automatica (revised in 998). Parts of the paper were presented in the IEEE Conf. on Decision and Control, pp , 995, and the IFAC World Congress, Vol. F, pp335-34, 996. [4] B. Yao, \High performance adaptive robust control of nonlinear systems: a general framework and new schemes," in Proc. of IEEE Conference on Decision and Control, pp. 2489{2494, 997. [5] B. Yao, F. Bu, and G. T. C. Chiu, \Nonlinear adaptive robust control of electro-hydraulic servo systems with discontinuous projections," in Proc. of IEEE Conf. on Decision and Control, pp. 2265{227, 998. The revised full paper 'Nonlinear Adaptive Robust Control Of Electro-Hydraulic Systems Driven by Double-rod Actuators' is submitted to International Journal of Control in 999. [6] B. Yao, F. Bu, J. Reedy, and G. Chiu, \Adaptive robust control of single-rod hydraulic actuators: theory and experiments," in Proc. of American Control Conference, 999. (to appear). The revised full paper is submitted to IEEE/ASME Trans. on Mechantronics. [7] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness. Englewood Clis, NJ 7632, USA: Prentice Hall, Inc., 989. [8] G. C. Goodwin and D. Q. Mayne, \A parameter estimation perspective of continuous time model reference adaptive control," Automatica, vol. 23, no., pp. 57{7, 989. [9] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and adaptive control design. New York: Wiley, 995. [2] H. K. Khalil, Nonlinear Systems (Second Edition). Prentice Hall, Inc., 996. [2] C. D. P. H. Co., Servo tuner user guide. Software Mannual, Nov

22 Appendix A Proof of Lemma : Noting (22) and (3) _V 3 = w 2 z 2 ( m (x ) 2s2? ~ T 2 + d) ~? w m max 2 min m (x ) (k 2 + k 2s )z2 2 +w 3 z 3 ( 3 3s2? ~ T 2 ~ d)? w 3 (k 3min 3 + k 3s )z3 2? w 3z 3 2 )_^ (44) If k 2s and k 3s satisfy the conditions in the lemma, by completion of square and noting (34), j w 3 z ^ _^ jj w 3 z 3 P 3 j=2 w jz j j j j w 2 z 2 j k? 2 k j w 3 z 3 j ^ k+ j w 3z 3 j ^ k j w 3z 3 j k? 3 k 2 w2 2 z2 2 k? 2k 2 + w 2 3 z2 3 ^ k2 + 2 k? 3k 2 ) w 2 k 2s z w 3k 3s z 2 3 (45) Thus, noting the condition i of (2) and (3) respectively, (44) becomes _ V 3? P 3 j=2 (w j k j z 2 j + w j" j )? V V 3 + " V (46) which leads to (35) 2 Appendix B Proof of Theorem : From Lemma 2, V 3 () =. From Lemma, (38) is true and thus z = [z ; z 2 ; z 3 ] T is bounded. Furthermore, from (38), the tracking error z (t) is within a ball whose size can be made arbitrarily small by increasing k and/or decreasing " in a known form. From (36) and (37), the trajectory planning error, e d (t) = x d (t)? x Ld (t), can be guaranteed to possess any good transient behavior by suitably choosing the Hurwitz polynomial G d (s) = s 3 + s 2 + : : : + 3 without being aected by k and ". Therefore, any good transient performance of the output tracking error e y = z (t) + e d (t) can be guaranteed by the choice of k and " in a known form. This proves A of Theorem. In the presence of parametric uncertainties only, i.e., d ~ =, noting (34), (45), and conditions ii of (2) and (3) respectively, from (44), we have _V 3? P 3 j=2 [w j z j T j ~ + w j k j z 2 j ] =? P 3 j=2 w j k j z 2 j? T ~ (47) Dene a new p.s.d. function V as V = V ~ T?? ~ (48) Noting P2 of (2), from (47) and (), the derivative of V is _V? P 3 j=2 w jk j z 2 j? ~ T + ~ T?? _^? P 3 j=2 w jk j z 2 j (49) 22