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Abstract—This paper presents an output feedback proportional–derivative (PD)-type controller for the trajec- tory tracking control of robotic manipulators.
IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 1, FEBRUARY 2011

187

PD Output Feedback Control Design for Industrial Robotic Manipulators Shafiqul Islam and Peter X. Liu

Abstract—This paper presents an output feedback proportional–derivative (PD)-type controller for the trajectory tracking control of robotic manipulators. In the first part of the paper, we propose a PD-like output-feedback control law. The design comprises a PD term with nominal robot dynamics, where the unknown velocity signals are estimated from the output of the linear estimator. Using Lyapunov analysis, we characterize the asymptotic property of all the signals in the closed-loop error model dynamics. This property sets the bound on the tracking error trajectory of the closed-loop system. In the second part, we remove the nominal model dynamics from the control design to formulate a model-independent PD-type output feedback approach. Using an asymptotic analysis for the singularly perturbed closed-loop model, we guarantee that all the signals under the proposed PD output feedback design are bounded and their bounds can be made arbitrarily small by using observer–controller gains. Implementation of results demonstrate the potential application of the proposed method on real systems. Index Terms—Industrial robot controller, output feedback, perturbation, robotics.

I. INTRODUCTION OST advanced industrial manipulators (e.g., those from CRS Robotics Ltd. [1]–[3], Applied AI, Inc., etc.) are controlled by using only proportional–derivative (PD) or proportional–integral–derivative (PID) controllers. Such a PDtype robot controller can be traced back to the literature [9], [10]. The design comprises PD control plus gravity compensation and the PD control plus desired gravity compensation term. Authors in these papers show that a PD-type robot controller can be used to asymptotically stabilize the joint positions of rigid robot manipulators. However, the design is based on the knowledge of the gravitational loading vector of the robot dynamics, where uncertain parameters of manipulator dynamics are essential to meet the desired control objective. Adaptive form of these controllers can be found in [12] and [14]. These designs require a priori known structure of the gravitational loading vector. Also, the parameters of these controllers require to satisfy complex design inequalities that are very difficult to meet in real-time operation. In fact, in the presence of the gravitational loading vector, PD design may cause a steady-state tracking error, which

M

Manuscript received April 30, 2009; revised August 18, 2009; accepted October 11, 2009. Date of publication January 26, 2010; date of current version January 12, 2011. Recommended by Technical Editor Y. Li. This work was supported in part by the Natural Science and Engineering Research Council, Canada, and in part by the Canada Research Chairs Program. This paper was presented in part at the IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Singapore, July 14–17, 2009. The authors are with the Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada (e-mail: sislam@sce. carleton.ca; [email protected]). Digital Object Identifier 10.1109/TMECH.2009.2038374

can be reduced via tuning the proportional and derivative gains or via adding an integral control action. In [5], [16], and [18], authors show the local stability proof of PID-based robot controller. The control design does not require the knowledge of the robot dynamics. It is proven that the control design parameters, depending on the robot manipulator dynamics, can be selected to ensure desired tracking objective. Motivated by the controller proposed in [5], [16], and [18], a stable position controller, so-called P I 2 D controller, without using gravitational loading vector of the robot dynamics was proposed in [17]. To remove the requirement of the gravitational loading vector from the controller structure, a new type of control strategy was introduced in [15] and [19]. The control structure combined an integral term developed by using a bounded nonlinear function of the position error with a PD controller term. The controller was developed via combining a saturated, proportional, and differential feedback term with a proportional–integral component formulated by using a linear sum of velocity and saturated position errors. The basic idea is to use the energy-shaping technique and the passivity theory. The method can achieve desired tracking if the designer can choose suitable design parameters via using Lyapunov second method. As a matter of fact, the controller parameters of these controllers are required to satisfy complex inequalities that may not attract industrials for the real-world application. Rocco proposed Lyapunov-based PID controller in [11], where the local stability property of the closed-loop system was established. The author shows that the PID design is simply a PD controller design. This is because the integral gains used in PID design are smaller than PD control gains. However, the implementation of PD or PID controller is difficult, since it requires velocity signals in addition to joint position signals. The practical problem is that advanced robotic systems do not provide velocity sensors due to the constraints of weight and cost. In order to obtain velocity signals, the common practical approach is to differentiate the position measurements obtained from encoders or resolvers (joint position measurement is also contaminated by noise [1]–[3]), which often leads to severe noise. As a consequence, the performances of the PD and PID controller are limited as, in practice, the noise is amplified with the increase of the values of controller gains. To attenuate the noise amplification, the cutoff frequency of the filter may be chosen via trial-and-error method on the basis of the derivative control gains, the amount of measurement noise associated with the sensor, and the filtered derivative action of the velocity signals [1], [2]. While it may work for some applications, such a simple approximation is often inadequate for cases where velocities are very low and/or very high. Moreover, the quantization effect of the noisy velocity signals usually produces undesir-

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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 16, NO. 1, FEBRUARY 2011

able oscillations in the joint, which may render the controlled system unstable. Most importantly, there is no theoretical proof for such an ad hoc solution that has been used in existing industrial robot controllers. In this paper, we propose a linear estimator to reproduce the unknown velocity signal for PDbased industrial robot controllers. We are mainly interested in designing an observer controller that are free from nonlinear robot dynamics and uncertain parameters. In the first design, we comprise a PD term with nominal robot model dynamics, where the unknown velocity signals are estimated by the output of the linear estimator. The Lyapunov method is utilized to establish necessary conditions that guarantee the asymptotic stability of the closed-loop error model dynamics. This condition sets the bound on the tracking error trajectory of the closed-loop system. In the second design, we remove the nominal model dynamics from the control design to formulate a model-independent PDtype output feedback approach. This design has two steps. In the first step, we develop a PD controller as a state feedback (position velocity) approach. Based on using Lyapunov method, we obtain the bound on the tracking error signals for the closedloop error model. It is shown in our analysis that the tracking error bound can be made arbitrarily small by increasing the minimal eigenvalue of the control gains. In the second step, for a given set of initial conditions, we estimate the region of interest of the PD-based state feedback design. Then, we saturate the control outside the estimated region to obtain bounded control input. Afterward, we use linear estimator to generate unknown velocity signals to develop a PD-based output feedback control law. The idea of introducing saturated control is to protect the controlled plant from the effect of exponentially fast observer dynamics in the output feedback design. The bounded control allows the designer to increase the speed of the observer dynamics without sacrificing the transient tracking performance. Using an asymptotic analysis of the singularly perturbed closed-loop model, we prove that the observer and tracking error variables are ultimately bounded and their bounds can be made to a small neighborhood of the origin by using observer–controller gains. The rest of the paper is organized as follows. Section II describes the system model and its dynamical properties. A new PD-like output-feedback control algorithm for robot manipulators is designed in this section. We then design model-free PD output feedback control approach. This section also provides the error bound analysis of the proposed method via using Lyapunovlike energy function. To validate our theoretical arguments, the method is implemented and evaluated on a 2-DOF robot system in Section III. Section IV concludes the paper. II. SYSTEM MODEL AND OUTPUT FEEDBACK DESIGN

e˙ 1 = e2 , e˙ 2 = φ1 (e) + φ2 (e1 )τ − q¨d

(2)

where e1 = e = q − qd , e2 = e˙ = q˙ − q˙d , φ1 (e) = −M (e1 + qd )−1 [C(e1 + qd , e2 + q˙d )(e2 + q˙d ) + G(e1 + qd )], and φ2 (e1 ) = M (e1 + qd )−1 . The objective of this paper is to design a PD-based output feedback controller that ensures the boundedness of all signals in the closed-loop model such that the output of the system tracks a given desired trajectory qd (t), which satisfies the following assumption: A1 . The reference trajectory, its first and second derivatives are bounded as Qd = [qd , q˙d , q¨d ]T ∈ Ωd ⊂ 3n with compact set Ωd . The following dynamic properties will also be used in convergence analysis [4], [6]–[8], [20]–[22]: 1) M (q) ∈ Rn ×n is a symmetric, bounded, and positive definite matrix that satisfies M (q) ≤ MM and M −1 (q) ≤ MM I , where MM and MM I are bounded positive constant; 2) the matrix M˙ (q) − 2C(q, q) ˙ is skew-symmetric; and 3) the norm of the gravity and centripetral˙ coriolis forces are upper bounded as C(q, q) ˙ ≤ CM q and C(q, q˙d ) ≤ kcd q˙d  ≤ kc , where CM , kc , and kcd are bounded positive constant. Let us first state the main results for PD-like output-feedback method in the following Theorem. Theorem 1: Consider the closed-loop system composed of the system (2) with the following observer–controller qd + algorithm τ (e1 , eˆ2 , Qd ) = −Kp e1 − Kd eˆ2 + M (e1 + qd )¨ C(e1 + qd , q˙d )q˙d + G(q) with eˆ˙ 1 = eˆ2 + (H1 /)˜ e1 and eˆ˙ 2 = e1 , where e˜1 = e1 − eˆ1 , e˜2 = e2 − eˆ2 , eˆ1 and eˆ2 de(H2 /2 )˜ note the estimated values of e1 and e2 , respectively, H1 and H2 are positive definite matrices,  is chosen as a small positive parameter, and Kp ∈ n ×n and Kd ∈ n ×n are constant positive matrices. If d be the positive number such that d ∈ (0, 1), and if there exists a continuous interval  ∈ (0, ∗ ), such that ∗ satisfies ∗ (d) = α1 α2 /(α1 γ + (1/4d(1 − d)) [(1 − d)β1 + dβ2 ]) and the bound on ∗ (d) is given by ∗ = α1 α2 /(β1 β2 + α1 γ), where α1 , α2 , γ, β1 , and β2 are nonnegative constants, then the origin of the closed-loop system is asymptotically stable for all 0 <  < ∗ . Proof: To prove Theorem 1, let us first define the closed-loop tracking error model as e˙ 1 = e2 e˙ 2 = φ1 (e) + φ2 (e1 )τ (e1 , eˆ2 , Qd ) − q¨d .

(3)

Then, we formulate the observer error model dynamics as standard singularly perturbed system η˙ 1 = η2 − H1 η1 η˙ 2 =  [−¨ qd + φ1 (e) + φ2 (e1 )τ (e1 , eˆ2 , Qd )] − H2 η1 (4)

Let us first consider the equation of motion, for an n-link rigid robot [4], [10], [13], [21], [22], which is given by M (q)¨ q + C(q, q) ˙ q˙ + G(q) = τ

coriolis and centrifugal loading vector, and G(q) ∈ n is the gravitational loading vector. We now represent the robot model (1) in state-space form as follows:

(1)

where q ∈ n is the joint position vector, q˙ ∈ n is the joint velocity vector, q¨ ∈ n is the joint acceleration vector, τ ∈ n is the input torque, M (q) ∈ n ×n is the symmetric and uniformly positive-definite inertia matrix, C(q, q) ˙ q˙ ∈ n is the

where the observer error dynamics is replaced by scaled estimation error to form a singularly perturbed system as η1 = e˜1 and η2 = e˜2 . We can see that the model (4) is in a standard singularly perturbed form as e˙ = f (e, η, 0) and η˙ = f (e, η, ). Our aim is to analyze the property of the singularly perturbed system from those of the reduced and the boundary layer models. To begin with this analysis, we first find the reduced and boundary layer models by setting  = 0 in (4)

ISLAM AND LIU: PD OUTPUT FEEDBACK CONTROL DESIGN FOR INDUSTRIAL ROBOTIC MANIPULATORS

as 0 = η˜2 − H1 η˜1 and 0 = −H2 η˜1 , i.e., η1 = 0 and η2 = 0. This also says that e˜1 = 0 and e˜2 = 0. This means that e1 = eˆ1 and e2 = eˆ2 are the equilibrium point of (4). We now use the control law τ (e1 , eˆ2 , Qd ), and then, add and subtract C(e1 + qd , e2 )q˙d to simplify the tracking error model as e˙ 2 = M −1 [−C(e1 + qd , e2 )e2 − Kd e2 + Kd e˜2 − 2C(e1 + qd , q˙d )e2 − Kp e1 ]. Applying e˜2 = 0 in e˙ 2 , one obtains the reduced model as e˙ 1 = e2 and e˙ 2 = M −1 [−C(e1 + qd , e2 )e2 − Kd e2 − 2C(e1 + qd , q˙d )e2 − Kp e1 ], which has an equilibrium at (e1 , e2 ) = (0, 0). The boundary layer system can be defined in faster time scale ζ as dη1 (ζ)/dζ = η˜2 (ζ) − H1 η˜1 (ζ) and dη2 (ζ)/dζ = −H2 η˜1 (ζ), where ζ = t/. The boundary layer model has the following compact form as dη(ζ)/dζ = A0 η˜ with   −H1 In ×n A0 = −H2 0n ×n  T and η(ζ) = η1T (ζ), η2T (ζ) . This implies that boundary layer or fast model has an equilibrium at (η1 , η2 ) = (0, 0). At this point, we claim the following Lemma.  Lemma 1: The equilibrium point of the reduced or slow model and boundary or fast model are asymptotically stable. Proof: To prove the Lemma 1, we use the following controller Lyapunov function Vc for the reduced model and observer Lyapunov function V0 for the boundary layer system as Vc = 1/2(eT1 Kp e1 ) + 1/2(eT2 M e2 ) and V0 = 1/2(η T P0 η). First, we take the derivative of Vc with respect to time along with the reduced model as V˙ c = e˙ T1 Kp e1 + 1/2(eT2 M˙ e2 ) + eT2 M e˙ 2 . Using the property 2 and e˙ 2 , one can simplify the derivative of Vc as V˙ c = eT2 [−Kd e2 − 2C(e1 + qd , q˙d )e2 ]. In view of the property 3, we can further simplify V˙ c as V˙ c ≤ −eT2 Kdc e2 < 0, where Kdc = 2kc + kd with a positive constant derivative control gain kd . By invoking the LaSalles invariance principle, the only solution of e˙ 1 = e2 and e˙ 2 = M −1 [−C(e1 + qd , e2 )e2 − Kd e2 − 2C(e1 + qd , q˙d )e2 − Kp e1 ] evolving in the set {(e1 , e2 ) |e2 = 0} are e˙ 1 = 0 and 0 = −[M −1 Kp ]e1 . Then, we have e1 = 0. Let us now take derivative V0 along ˙ As the matrix A0 is with the fast model as V˙ 0 = 2η T P0 η. Hurwitz, then there exists a positive definite matrix P0 such that AT0 P0 + P0 A0 = −Q with the given positive-definite design matrix Q. Then, we can write V˙ 0 as V˙ 0 ≤ −η T Qη ≤ −λm in (Q)η2 < 0, which guarantees that the equilibrium point η = 0 of the fast model is asymptotically stable. That concludes the proof of Lemma 1.  Note that the Lemma 1 is established for the reduced and boundary layer models by assuming that  = 0. However, in practice, it may not be possible to apply  = 0 in observer dynamics. Therefore, our aim is to establish the stability property of singularly perturbed models (3) and (4), such that there exists a bound on  for which the derivative of the Lyapunov function is negative. To explore that, let us consider the combined Lyapunov  function candidate as  Vco = (1 − d)/2 eT2 M e2 + eT1 Kp e1 + d/2(η T P0 η). Now, we take derivative along with perturbed model, and then, add and subtract C(e1 + qd , e2 )q˙d to simplify V˙ co as V˙ co = (1 − d)/2(eT2 M˙ e2 ) + (1 − d)eT2 [f (e, 0, 0) + f (0, η, )] + d/ (η T P0 η) + dη T P0 [g(0, η, ) + g(e, 0, 0)], where g(0, η, ) =

189

Kd M −1 η2 , f (e, 0, 0) = [−C(e1 + qd , e2 )e2 − Kd e2 − 2C (e1 + qd , q˙d )e2 ], f (0, η, ) = Kd η2 , and g(e, 0, 0) = M −1 [−C (e1 + qd , e2 )e2 − Kd e2 − Kp e1 − 2C(e1 + qd , q˙d )e2 ]. Applying AT0 P0 + P0 A0 = −Q together with the properties of 2 and 3, V˙ co becomes     0 0 0 0 T T ˙ Vco ≤ −(1 − d)e e + (1 − d)e η 0 Kdc 0 Kd d − η T Qη + dη T P0 [g(0, η, ) + g(e, 0, 0)] .  This implies that, for a small value of , the closed-loop trajectory enters into a positively invariant set over a finite time and remains there for all future time. The time and size of this set can be made very small by using small values of . Our aim is now to find the bound on . In view of the properties 1 and 3, the bound on the modeling error terms g(0, η, ) and g(e, 0, 0) can be defined as g(0, η, ) ≤ g η and g(e, 0, 0) ≤ gc e with positive constants g and gc . Applying these inequalities, we can simplify V˙ co as V˙ co ≤ −(1 − d)α1 ψ 2 (e) + (1 − d)β1 ψ(e)ψ(η) − d/(α2 ψ 2 (η)) + dγψ 2 (η) + dβ2 ψ(e) ψ(η), where ψ(e) = e, ψ(η) = η,    0 0   α1 =   0 Kdc     0 0    β1 =  0 Kd  α2 = Q, β2 = gc P0 , and γ = g P0 . Then, V˙ co can be written in the following compact form:     ψ(e) T ψ(e) V˙ co ≤ − Tp d ψ(η) ψ(η) with



⎢ Tp d = ⎣

(1 − d)α1

⎤ 1 1 − (1 − d)β1 − dβ2 ⎥ 2 2 ⎦.

α 2 −γ d 

1 1 − (1 − d)β1 − dβ2 2 2 This implies that V˙ co < 0 if there exists a positive constant ∗ (d) given by ∗ (d) = α1 α2 /(α1 γ + 1/(4d(1 − d))[(1 − d)β1 + dβ2 ]). For maximum value of d = β1 /(β1 + β2 ), one can find the bound on ∗ (d) as ∗ = α1 α2 /(β1 β2 + α1 γ), where α1 , α2 , γ, β1 , and β2 are nonnegative constants. Then, V˙ co can be ˜ 2 , where ζ˜ = [ψ(e) ψ(η)]T written as V˙ co ≤ −λm in (Tp d )ζ and λm in (·) is the minimum eigenvalue of the positive definite matrix Tpd . This implies that, for  < ∗ , the origin of the singularly perturbed systems (3) and (4) is asymptotically stable. That concludes the proof of Theorem 1.  Remark 1: In view of the observer and the closed-loop robot dynamics, we can notice that the speed of the observer dynamics is much higher than the closed-loop robot dynamics. Then, we can say that the observer has a faster dynamics than the robotic system and PD controller. As the observer dynamics is faster and free from robot dynamics, then the perturbation in the model parameters will be negligible. It is worth noting to mention that if the initial error estimates and inertial parameters become very large, then the output feedback control input given by Theorem 1 may become

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very large value during transient phase. To protect the plant from the large transient control action, we introduce saturated control technique where the input is bounded over the domain of interest provided by the state feedback controller. We consider that Ωc 1 is an estimate of the region of attraction of the state feedback controller τ (e1 , e2 , Qd ) such that the controller is bounded over the set Ωc r 1 as τ s (e1 , e2 , Qd ) = τm ax Sat(τ (e1 , e2 , Qd )/τm ax ) = τ (e1 , e2 , Qd ) ∀e(0) ∈ Ωc o 1 ∀e ∈ Ωc 1 and τm ax is taken over the set Ωc r 1 , where Ωc o 1 , Ωc r 1 , and Ωc 1 are the compact sets with cr 1 > c1 ≥ co1 , Qd ∈ Ωd , τm ax = max|τ (e1 , e2 , Qd )| and Sat(·) is bounded smooth saturation function. We now replace the state vectors e in the control law by the output of the linear estimator eˆ. The bounded output feedback controller can also be achieved by saturating outside the region Ωc 1 as τ s (e1 , eˆ2 , Qd ) = e ∈ Ωc 1 ∀ˆ e(0) τm ax Sat(τ (e1 , eˆ2 , Qd )/τm ax ) = τ (e1 , eˆ2 , Qd ) ∀ˆ ∈ Ωc o 1 , Qd ∈ Ωd , one has |τ (e1 , eˆ2 , Qd )| ≤ τm ax ∀t ≥ 0, and therefore, τ (e1 , eˆ2 , Qd ) = τ s (e1 , eˆ2 , Qd ). Note that, in practice, the maximum bounds on the control input is given by the manufacturer. Then, we can write tracking error model as e˙ 1 = e2 and e˙ 2 = φ1 (e) + φ2 (e1 )τ s (e1 , eˆ2 , Qd ) − q¨d . The observer error dynamics has the following form: η˙ 1 = e˜2 − e and η˙ 2 = [−¨ qd + φ1 (e) + φ2 (e1 )τ s (e1 , eˆ2 , Qd )] − (H1 /)˜ H2 η1 . We now analyze the property of the closed-loop observer–controller error model dynamics. First, we add and subtract τ s (e, Qd , 0, 0), and then use τ s (e, Qd , 0, 0) = τ (e, Qd , 0, 0) to simplify the derivative of Vco as V˙ co = (1 − d)1/2(eT2 M˙ e2 ) + d/(η T P0 η) + (1 − d)eT2 [τ s (e, Qd , η, ) − τ s (e, Qd , 0, 0) + M (−C(e1 + qd , e2 )e2 − Kd e2 − 2C (e1 + qd , q˙d )e2 )] + dη T P0 [ξ(0, η, ) + ξ(e, 0, 0)], where ξ(e, 0, 0) = M −1 [−2C(e1 + qd , q˙d )e2 − Kp e1 − Kd e2 − C(e1 + q˙d , e2 ) e2 ] and ξ(0, η, ) = Kd M −1 η2 . Applying AT0 P0 + P0 A0 = −Q and the property 1 and the property 3, V˙ co can be further simplified as   0 0 V˙ co ≤ −(1 − d)eT e + (1 − d)eT2 [τ s (e, Qd , η, ) 0 Kdc d − τ s (e, Qd , 0, 0)] − η T Qη + dη T P0 [ξ(0, η, )  + ξ(e, 0, 0)]. s

s

Since τ (e, Qd , 0, 0) and τ (e, Qd , η, ) are continuous function with respect to their arguments, then, using properties 1 and 3, one has τ s (e, Qd , η, ) − τ s (e, Qd , 0, 0) ≤ β1 η, ξ(0, η, ) ≤ δ1 η, and ξ(e, 0, 0) ≤ δ2 e, where β1 , δ1 , and δ2 are bounded positive constants. Then, with these inequalities, V˙ co can be simplified as V˙ co ≤ −(1 − d)α1 ψ 2 (e) + (1 − d)β1 ψ(e)ψ(η) − d/(α2 ψ 2 (η)) + dγP0 ψ 2 (η) + dβ2 ψ(e)ψ(η), where ψ(e) = e, ψ(η) = η,    0 0   α1 =    0 K dc β2 = δ2 P0 , and γ = δ1 P0 . Then, V˙ co can be rewritten in the following compact form as     ψ(e) T ψ(e) V˙ co ≤ − Up d ψ(η) ψ(η)

with



⎤ 1 1 − (1 − d)β1 − dβ2 ⎢ ⎥ 2 2 Up d = ⎣ ⎦.

α 1 1 2 −γ − (1 − d)β1 − dβ2 d 2 2  In view of V˙ co , we can conclude that V˙ co < 0 if there exists a positive constant ∗ (d), such that ∗ (d) = α1 α2 /(α1 γ + 1/(4d(1 − d)) [(1 − d)β1 + dβ2 ]) and, for the maximum value of d = β1 /(β1 + β2 ), the upper bound on ∗ (d) can be calculated as follows: ∗ = α1 α2 /(β1 β2 + α1 γ), where α1 , α2 , γ, β1 , and β2 are positive constants. Then, for every ˜ 2 , where , the bound on V˙ co becomes V˙ co ≤ −λm in (Upd )ζ T ζ˜ = [ψ(e), ψ(η)] and λm in (·) is the minimum eigenvalue of the positive definite matrix Upd . This implies that the origin of the singularly perturbed systems (3) and (4) is asymptotically stable. That concludes the proof of Theorem 1 when the control input is bounded by a saturated function.  Let us now remove the nominal robot dynamics from the controller to develop model-free controller–observer algorithm as (1 − d)α1

τ (e1 , eˆ2 ) = −K(ˆ e2 + λe1 )

(5)

n ×n

with λ > 0, K ∈  , and eˆ2 designed as eˆ˙ 1 = eˆ2 + (H1 /)˜ e1 2 ˙ and eˆ2 = (H2 / )˜ e1 . In view of the controller and observer structure, we can notice that the design is independent of the system dynamics and parameters. Then, the control input with linear observer can be written as τ (e1 , eˆ2 ) = −Kλe1 − Ke2 + K e˜2 . Now, substitute τ (e1 , eˆ2 ) to formulate the closed-loop tracking error model as e˙ 1 = e2 and e˙ 2 = M −1 [−C(e1 + qd , e2 + q˙d )e2 − C(e1 + qd , e2 + q˙d )q˙d − Kλe1 − Ke2 + K e˜2 ] − q¨d . We then construct observer error dynamics as e˜˙ = A0 e˜ + Bfo (Qd , e, e˜), where   0n ×n B= In ×n and fo (Qd , e, e˜) = [M −1 (−C(e1 + qd , e2 + q˙d )e2 − C(e1 + qd, e2 + q˙d )q˙d − Kλe1 − Ke2 + K e˜2 ) − q¨d ]. Now, represent the observer–controller closed-loop error dynamics in singularly perturbed form, where we transform the observer error in the newly defined variables η as e˙ = fs (Qd , e, , η)

(6)

η˙ = A0 η + Bfo (Qd , e, , η) −1

(7)

where fs (Qd , e, , η) is defined as [e2 ; M (−C(e1 + qd , e2 + q˙d )e2 − C(e1 + qd , e2 + q˙d )q˙d − Kλe1 − Ke2 + Kη2 ) − q¨d ] and fo (Qd , e, , η) = M −1 (−C(e1 + qd , e2 + q˙d )e2 −C(e1 + qd , e2 + q˙d )q˙d − Kλe1 − Ke2 + Kη2 ) − q¨d , with η1 = e˜1 , η2 = e˜2 , and a small positive parameter . The terms fs (Qd , e, , η) and fo (Qd , e, , η) represent the effects of the nonlinear dynamics of the manipulator and the effects due to the observer errors, respectively. The aforementioned model is in a standard singularly perturbed form. Note that if  = 0 in (7), then we have η = 0. Using  = 0 and η = 0, one has the following closed-loop tracking error dynamics under PD-based state feedback control law defined as   fs1 (8) e˙ = fs (Qd , e, 0, 0) = fs2

ISLAM AND LIU: PD OUTPUT FEEDBACK CONTROL DESIGN FOR INDUSTRIAL ROBOTIC MANIPULATORS

where fs1 = e2 and fs2 = [M −1 (−C(e1 + qd , e2 + q˙d )e2 − C(e1 + qd , e2 + q˙d )q˙d − Kλe1 − Ke2 ) − q¨d ]. Equation (8) represents the trajectory tracking error dynamics driven by the nonlinear terms defining the dynamics of the robot manipulator. Error Bound Analysis Under PD State Feedback Control Design: Let us now derive the property of the closed-loop error signals under the position- and velocity-based PD feedback controller given by τ (e1 , e2 ) = −Kλe1 − Ke2 . The filtered version of this PD controller are usually used in advanced industrial robot control algorithm, see for example, [1]–[3]. In our subsequent development, we remove this assumption via replacing the derivative signals by the output of the linear observer. Then, we derive a condition under which all the signals in the PD output feedback control law will be bounded. We begin with the analysis by finding an error boundedness property of the closed-loop system under PD-based state feedback system given by (8). To find the tracking error bound of the PD control law, let us consider the following Lyapunov function candidate: Vr =

1 T S MS 2

(9)

with S = e2 + λe1 and λ = diag[λ1 , λ2 ,. . ., λn ] with positive constants λ1 , λ2 ,. . ., λn . Take the derivative of (9) along the solution of closed-loop system (8). Then, using assumption A1 as well as the property 2, V˙ r becomes V˙ r ≤ −λm in (K)S2 + S T ko , where [λC(e1 + qd + λe2 ) + [λC(e1 + qd , e1 )q˙d − C(e1 + qd , q˙d )q˙d ] + M (−¨ qd , e1 ) − C(e1 + qd , q˙d )]e2  ≤ ko ∀(e, Qd ) ∈ Ωc × Ωd with ko > 0 and e ∈ Ωc = {e|Vr ≤c}. Now, applying S2 ≥ Vr /λm ax (αo ) and S ≤ Vr /λm in (αo ), we have √ V˙ r ≤ −ψo Vr + υo Vr with   M11 M12 αo = 0.5 M21 M22  ψo = Ξ/λm ax (αo ), Ξ = λm in (K), and υo = ko / λm in (αo ). Then, we can find the ultimate bound on the error trajectory as Vr (t) ≤ Vr (0)e−γ o t + βo /γo (1 − e−γ o t ) for any αo > 0, ψo > αo /2, γo = (ψo − (αo /2)), and βo = υo2 /2αo . Using Vr ≥ 4ko2 λ2m ax (αo )/λm in (αo )Ξ2 , the bound on V˙ r can be simplified further as V˙ r ≤ −κVr

(10)

where κ = ψo /2. Thus, the solution of (10) can be written as Vr (t) ≤ Vr (0)e−κt . This implies that the trajectory starting in a region where Vr (0) > c will continue to decrease, until the trajectory enters into the set Ωc , where Ωc = {e | Vr ≤ c}. Now using Vr = 4ko2 λ2m ax (αo )/λm in (αo )Ξ2 , we have V˙ r ≤ 0, where c = 2ko2 λ2m ax (αo )/λm in (αo )Ξ2 , which implies that the solution of (8) starting in the set Ωc = {e | Vr ≤ c} will remain there ∀t ≥ 0, as V˙ r is negative on the boundary Vr = c. The boundary of the set Ωc defines the maximum errors that one can expect from the PD state feedback controller. Note from the relationship c = 2ko2 λ2m ax (αo )/λm in (αo )Ξ that the set Ωc can be made arbitrarily small by increasing the minimal eigenvalue of the control gain K. Based on our aforementioned analysis, let us now state the following Theorem.

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Theorem 2: Consider that the closed-loop control system (8) composed of the nonlinear robot dynamics (2) and the control law τ (e1 , e2 ) = −Kλe1 − Ke2 . Then, for any given initial error states e(0), there exists a controller gain K, such that the tracking error signals are ultimately bounded by a bound that can be made arbitrarily small closed to origin by increasing the minimal eigenvalue of the control gains K. Error Bound Under PD Output Feedback Controller: In this part of the paper, we aim to show that the state variables of the closed-loop system under PD output feedback controller are bounded by a bound that can be made arbitrarily small by using observer–controller design parameter  and K. More precisely, we will show in our subsequent analysis that, for the given set of initial conditions of interest, there exists controller–observer control gains K and ∗1 , such that for every 0 <  < ∗1 , all the state variables of the closed-loop system under PD output feedback are bounded by a bound that can be made very small via using small value of . In other words, the performance under PD state feedback design can be recovered asymptotically by the PD output feedback design. To proceed with that, we consider that all the initial error states are bounded that belongs to the compact set Ωb = {e(0) | Vr (0) ≤ b}, where e1 (0) ∈ Ω1o and e2 (0) ∈ Ω2o . The initial sets Ω1o and Ω2o can be chosen arbitrarily large enough to cover any given bounded initial errors. We also consider that the bounded initial state estimates eˆ(0) = eˆ0 belongs to the compact set Ωb . Now, for the given initial conditions of interest, we consider that the compact set Ωr is the domain of interest for the PD state feedback controller, such that Ωr = {e | Vr ≤ cr } with cr ≥ b . To ensure the bounded PD output feedback control, we saturate the input outside the domain of interest Ωr . The bounded control requires to prevent the system being affected from the large transient control phenomenon from high-speed observer error variable in the face of the large inertia parameters and initial error states. Once the short transient period is over, the observer error variable becomes small and saturation function becomes idle. Note that the bounded control mechanism will be used only when the initial conditions of interest become very large. We now state our main results for PD output feedback design where model-free linear observer incorporates to estimate the unknown velocity signals. Theorem 3: Consider the observer–controller closed-loop control system defined by (6) and (7). Then, for any given compact set (e(0), eˆ(0)) ∈ Ωb , there exists observer–controller gains ∗1 and K, such that for every 0 <  < ∗1 , the state variables (e, η) of the closed-loop system are bounded by a bound that can be made arbitrarily small closed to the origin by choosing observer–controller design parameters. Proof: The proof of the aforementioned Theorem 3 has two parts. In the first part, we show that there exists a transient time period T1 () ∈ [0, T2 ], during which the fast variable η approaches a function of the order O(), while the slow variables remain in the subset Ωb T 2 of the domain of attraction Ωr . In the second part, we will prove the boundedness property of the signal e(t) for all t ∈ [T1 (), T3 ], where T1 () ∈ (0, T2 /2] and T3 ≥ T2 is the first time e(t) exists from the set Ωb T 2 .  Part 1: Let us first prove that there is a short transient period under which the observer error variables η decay to O(), for

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small value of  > 0. During this transient period, the state variables e remain within a domain of interest Ωr . In this period, the fast variables converge faster than the exponential mode of the form 1/(exp−at/ ), for any a > 0. We start by proving that there exist a finite time T2 independent of  such that every trajectory e starting in Ωb will be remained in Ωr , for all t ∈ [0, T2 ]. To prove this argument, let us take the following Lyapunov function candidate as Vr = 1/2(S T M S). Take derivative along the solution of the trajectory (6) and simplify the derivative as V˙ r = −S T KS − S T [−C(e1 + qd , q˙d )e2 − C(e1 + qd , q˙d ) qd − λe2 ) q˙d + λC(e1 + qd , e1 )q˙d + λC(e1 + qd , e1 )e2 − M (¨ e ∈ Ωr , and the + Kη2 ]∀(e(0), eˆ(0)) ∈ Ωb ∀e ∈ Ωr and ∀ˆ second term of V˙ r satisfied the inequality as [λC(e1 + qd + λe2 ) + Kη2 +[λC qd , e1 )q˙d − C(e1 + qd , q˙d )q˙d ] + M (−¨ (e1 + qd , e1 ) − C(e1 + qd , q˙d )]e2  ≤ υ1o ∀(e, Qd ) ∈ Ωr × Ωd 2 and ∀η ∈ Ω r with υ1o > 0. Applying S ≥ Vr /λm ax (αo ) ˙ and S ≤ √ Vr /λm in (αo ), Vr can then be simplified as V˙ r ≤ Vr , where Ξ = λm in (K), ψo = Ξ/λm ax (αo ) −ψo Vr + υ1o  and υ1o = k1o / λm in (αo ). Using √ Young’s inequality, the sec2 ond term can be written as υ1o Vr ≤ (αo /2)Vr + (1/2αo )υ1o , ˙ > 0. Then, V can be further simplified as with α0 r 2 V˙ r ≤ −γo1 Vr + β1 , with ψo > αo /2, β1 = υ1o /2αo , and γo1 = (ψo − αo /2). Now, the solution for V˙ r can be derived as Vr (t) ≤ Vr (0)e−γ o 1 t + β1 /γo1 (1 − e−γ o 1 t ). Since Vr (0) ≤ b < cr , then there always exists a finite time T2 , independent of , such that Vr (t) ≤ b1 < cr , with b1 > b, b1 < cr , and Ωb T 2 = {e|Vr ≤ b1 }, for all t ∈ [0, T2 ]. We now turn our attention to the fast equation and analyze its solution over the interval t ∈ [0, T2 ]. We show that there exists a very short time interval [0, T1 ()] for lim→0 T1 () = 0, such that the observer error variable η approaches to the order of O(). To show that, let us take the following Lyapunov function candidate for the observer error model (7): V b = η T Po η

(11)

where Po = PoT > 0 is the solution of the Lyapunov equation Po A0 + AT0 Po = −I. Take derivative (11) along the trajectory (7), one has V˙ b = 2/(η T Po A0 η) + 2/ (η T P0 Bfo (Qd , e, , η)). Now, we can use the following boundedness property of the function fo (Qd , e, , η) as fo (Qd , e, , η) ≤ k3 ∀e ∈ Ωr , where k3 is a positive constant. Applying Po A0 + AT0 Po = −I and above bound kc , one has V˙ b ≤ − 1 η2 + 2k3 P η, with B = I. Using η2 ≥ ˙ Vb /λm ax (Po ) and η2 ≤ Vb /λm in (Po ), equation  Vb can be ˙ simplified as Vb ≤ −Vb /λm ax (Po ) + 2k3 P  Vb /λm in (Po ), where P  = λm ax (Po ). From V˙ b , we now analyze the property of the fast variable η over the interval t ∈ [0, T1 ()] for three cases when Vb > 2 β, Vb = 2 β, and Vb < 2 β. We first consider if Vb > 2 β, then V˙ b can be simplified as γ V˙ b ≤ − Vb (12)  for γ = 1/2λm ax (Po ) and the inequality Vb > 2 β exists if Vb /2λm ax (Po ) > 2k3√P  Vb /λm in (Po ) holds, i.e., 1/2λm ax (Po )Vb > 2(P k3 V b / λm in (Po )) and Vb > 2 β, where β = 16P 3 k32 . Now from (12), one gets the solution for Vb as Vb < Vb (0) exp(−γ t/) , for Vb > 2 β. Let us

now choose a bounded initial condition for the initial estimates as eˆ(0) ∈ Ωb , then the corresponding scaled initial state estimation error is also bounded by η(0) = (e(0) − eˆ(0))/ ≤ ka / with positive constant ka , where ka depends on the size of the set Ωb . Then, Vb can be further simplified as Vb (η(t, )) ≤ Vb (0) exp(−γ t/) ≤ ko /2 exp(−γ t/) , where ko = ka2 /2γ. We now calculate the transient peaking time T1 () when Vb = 2 β. Let ∗1 > 0 be small so that for all 0 <  < ∗1 ,the time T1 () is calculated when Vb = 2 β as T1 () = /γ(ln ko /β4 ). Then, at time T1 (), Vb = 2 β and from (12), we have V˙ b ≤ −γβ. Therefore, Vb (η(t)) will continue to decrease and for the time t > T1 (), the inequality Vb ≤ 2 β holds. We now examine the property of η when Vb < 2 β. Note that the time T1 is a function of the observer gain  and T1 () tends to zero when  → 0. This implies that we can choose T1 () small enough such that, during the short transient time T1 (), the estimated state variables approach to the true state. This also implies that we can choose T1 () small enough such that T1 () ∈ (0, T2 ]. That is, there is a time T1 () ∈ (0, T2 ], such that for all t ∈ [T1 (), T3 ], Vb < 2 β. Since the transient time T1 () is a function of , then one can make T1 () → 0 as  → 0. Using Lyapunov equation (11), we conclude that as Vb < 2 β ∀t ∈ [T1 (), T3 ], then η is of order O() in the same interval. Part 2: In this part, we study slow model over the time interval t ∈ [T1 (), T3 ]. The system can be viewed as a perturbation of the closed-loop system under state feedback control, with the perturbation term of the order η. It is shown in our previous analysis that the fast observer error variables η decay to O(), i.e., Vb ≤ 2 β ∀t ∈ [T1 (), T3 ]. Using this fact, we now show that there exists a positive constant, such that for all 0 <  < ∗1 , the error signals (e, η) are bounded ∀t ≥ T1 (). To proceed with that, we consider the Lyapunov function Vr defined in (9). First take the derivative of (9) along the trajectory of (6), one has V˙ r = ∂Vr /∂e(fs (Qd , e, , η)). By adding and subtracting ∂Vr /∂e(fs (Qd , e, 0, 0)), we have V˙ r = ∂Vr/∂e[fs (Qd , e, , η) − fs (Qd , e, 0, 0)] + ∂Vr /∂e(fs (Qd , e, 0, 0)). The second term of V˙ r represents the time derivative of V˙ r along the solution of the closed-loop system (8) under PD state feedback control. Then, we can simplify V˙ r as V˙ r ≤ −κVr + ∂Vr /∂e [fs (Qd , e, , η) − fs (Qd , e, 0, 0)]. The second part of V˙ r has the following bound as ∂Vr /∂e[fs (Qd , e, , η) − fs (Qd , e, 0, 0)] ≤ k2 η ∀e ∈ Ωr , with k2 > 0. With this inη ≤ equality, V˙ r leads to V˙ r ≤ −κVr + k2 η. Applying   Vb /λm in (Po ), V˙ r becomes V˙ r ≤ −κVr + k2 Vb /λm in (Po ). Using Lyapunov function Vr of (9) andVb ≤ 2 β, we have V˙ r ≤ −κeT QL e + k2o , where k2o = k2 β/λm in (Po ) and   0.5λ2 M 0.5λM QL = . 0.5I 0.5M From V˙ r , we can see that on the boundary of Vr = cr , we have V˙ r < 0 when cr > k2o /κ. Since  is small and cr is strictly chosen to be greater than b, we can then conclude that the set Ωr is a positively invariant. Note that, for the given set of initial conditions of interest, we can calculate the bound on  as  = Λ/k2o , with Λ = κλm in (QL )e2 . Thus, the state trajectory e is trapped inside the set Ωr × Ω with Ω = {η | W (η) ≤ 2 β}.

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Then, we conclude that all the state variables of (6) and (7) are bounded ∀t > 0 and η is of O() ∀t ≥ T1 (). Therefore, if  → 0, then the error bounded set Ωr under output feedback converge to the bounded set under the state-feedback-based PD control law Ωc . To show that, let us integrate V˙ r from t = 0 T to t = T yields, Vr (T ) − Vr (0) ≤ − 0 κλm in (QL )e2 dt + γ1 , where γ1 = T k2o . Using the Lyapunov equation (9), we T can write 0 κλm in (QL )e2 dt ≤ e(0)T QL e(0) − V (T ) + γ1 . Using T = ∞ and V (∞) ≥ 0, then the tracking error bound T can be calculated as 0 κλm in (QL )e2 dt ≤ e(0)T QL e(0) + γ1 . If  → 0, then the error bound under output feedback converge to the error bound under state feedback, as defined by (10). III. DESIGN SYNTHESIS AND IMPLEMENTATION RESULTS We now show the design process of the proposed output feedback control algorithm on a 2-DOF serial-link manipulator [7], [8], [20], [22]. The equation of motion for this robot system is defined as:         q¨1 c11 c12 q˙1 τ1 m11 m12 + = m21 m22 q¨2 c21 c22 q˙2 τ2

Fig. 1. (a) and (b) Desired (dash) and output trajectory (in radians) (solid). (c) and (d) Control inputs (in newton-meters) for joints 1 and 2 under Theorem 1.

with m11 = (θ1 + 2θ2 + 2θ2 cos q2 ), m12 = (θ2 + θ2 cos q2 ), m21 = (θ2 + θ2 cos q2 ), m22 = θ2 , c11 = −2q˙2 θ2 sin q2 , c12 = −q˙2 θ2 sin q2 , c21 = q˙1 θ2 sin q2 , and c22 = 0, where θ1 = m1 l2 = 8, θ2 = m2 l2 = 9, l = l1 = l2 is the link lengths, and m1 and m2 are the masses of link 1 and link 2, respectively. The robot operates in the horizontal plane, so the gravitational force vector is G = 0. To generate the reference trajectory for the robot to follow, a square wave with a period of 8 s and an amplitude of ±1 rad prefiltered with a critically damped second-order linear filter using a bandwidth of ωn = 2.0 rad/s. More specifically, our main target is to use a desired trajectory that usually uses in industrial robotic systems. In practice, the step reference inputs are not preferred, as such initial jump may reduce the lifetime for the bearing. However, if one uses the step reference trajectory, then very small step sizes are required. To obtain desired control signal to follow the desired path, let us take the coefficients of the PD controller as: λ = 2, Kp = 200I2×2 , and Kd = 250I2×2 . The initial position and velocity errors are chosen as e(0) = 2. With these design parameters, our first aim is to show the design and implementation process of Theorem 1 on the given system. Note that one may choose different control design parameters, and then follow the following design steps to calculate the bound on . The following design steps are also applicable for any rigid robot systems. To begin with the design process, we first calculate β1 using the following inequality τ s (e, Qd , η, ) − τ s (e, Qd , 0, 0) ≤ β1 η, with β1 = 250. Applying the property 3, we calculate the maximum bound on C(q, q˙d ) as kc = 39.6117, where kc = kcd q˙d  with q˙d  = 1.4671, sin q2 = 1 and cos q2 = 1. Using q˙d  and kc , α1 is calculated as α1 = 329.2234 with kd = 250. Now, find the constant β2 and γ. To calculate these constants, one requires to choose α2 = Q = I and Hi = 5I2×2 , with i = 1, 2. Then, we use Lyapunov equation to determine the value of P as

P = 1.1101. Using the property 1 and property 3 along with given initial condition of interest, we then calculate the value of δ1 and δ2 as: δ1 = 215.2778 and δ2 = 329.9937. By knowing P , δ1 , and δ2 , we now determine the values of β2 and γ as β2 = 366.3260 and γ = 238.9799. Using constants β1 , β2 , α1 , and α2 , the value of d∗ is obtained as d∗ = 0.4056. The choice of d∗ results in bound on ∗ = 0.0019 ≈ 0.002. Applying the aforementioned-defined observer–controller design parameters, we then implement Theorem 1 on the given robotic system. The results are given in Fig. 1. We now increase the initial conditions from zero to e(0) = 2, but we keep all other design parameters same, as used for our last evaluation. Due to the presence of initial error estimates, the design may exhibit large transient control input. To guard the plant from such transient control action, we saturate the control input over the domain of interest for the given initial conditions. In practice, one requires to use the predefined saturation level provided by the manufacturer. Therefore, we consider that the saturation levels for the control inputs are predefined by manufacturer as τ1m ax = 100 N·m and τ2m ax = 100 N·m. For the given set of initial interest, this level can also be determined using simulation as well as Lyapunov-function estimation technique introduced in this paper. We then apply Theorem 1 on the same system with the same set of design parameters that were used in our last evaluation. The tested results are shown in Fig. 2. Due to the presence of the initial error estimates, the output trajectory peaks before convergence to the desired one. The presented results are obtained by assuming that the control system is operating under ideal operating condition. Let us now examine the tracking performance of Theorem 1 on the given system under nonideal operating conditions. To do that, we intentionally add a band-limited white noise w(t) into the output measurement q(t) and the input τ (t) in the model.

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Fig. 3.

Input and output disturbance levels of w(t).

Fig. 2. (a) and (b) Desired (dash) and output trajectory (in radians) (solid). (c) and (d) Control inputs (in newton-meters) for joints 1 and 2 using with Theorem 1 with initial conditions.

This is due to the fact that signal obtained from encoders also corrupted by noise, see for example, [2], [3]. For our evaluation, we use the level for w(t), as depicted in Fig. 3. Then, we apply the output feedback design stated by Theorem 1 on the model under the same set of design parameters, as used for our last evaluation. The implemented results are shown in Fig. 4. Even with the input and output disturbances, the performance under output feedback design meets the desired tracking objectives. To examine the robustness property of Theorem 1, we now introduce a more complicated situation. In this test, we aim to change the plant parameter of the joint 1 and joint 2 when the manipulator tracks the desired task with various loads. We consider that the plant is initially operating under the parameters θ1 = 8 and θ2 = 8. Then, at 12 s, the parameters are changed from θ1 = 8 and θ2 = 8 to θ1 = 4 and θ2 = 4, respectively. Again, at 31 s, the parameters θ are changed from θ1 = 4 and θ2 = 4 to θ1 = 2 and θ2 = 2, respectively. Therefore, in the whole process, there are three dynamics changes with relatively large modeling error uncertainty. Then, we implement Theorem 1 on the given system with the same design constants, as used for our last implementation. The tested results are depicted in Figs. 5 and 6 with the given saturation level for two control inputs as τ1m ax = 500 N·m and τ2m ax = 200 N·m. The saturation mechanism under dynamics changes is introduced to protect the plant from the effect of large control action at the time of dynamics changes, as we notice that the control effort becomes very large during dynamics changes. We notice from our implementation results that the transient tracking errors increase at the time of the dynamics changes. However, such transient tracking errors can be reduced by increasing the controller gains. Now, our aim is to show the design and implementation process of Theorems 2

Fig. 4. (a) and (b) Desired (dash) and output trajectory (in radians) (solid). (c) and (d) Control inputs (in newton-meters) for joints 1 and 2 using with output feedback design of Theorem 1 with input and output disturbance noise w(t).

and 3 on the given robot system. To do that, we first consider e(0) = 2. The parameters θ1 and θ2 are chosen as θ1 = 1 and θ2 = 2, respectively. For the given initial conditions of interest, we define the region of interest as follows. Using property 1 and property 3, we calculate the bound for ko = 67.245104 and Ωb = {e(0) | Vr (0) ≤ 104}. Then, we define cr = 105 as the region of interest for e as Ωr = {e | Vr ≤ 105}. Then, using c = 4ko2 λ2m ax (αo )/λm in (αo )Ξ2 , we calculate the bound on the error trajectory under PD state feedback design as e(t) ≤ 237.72573/λm in (K). Note that by increasing the minimal eigenvalue of the control gain K, one can make e(t) under PD state feedback control be very small. For

ISLAM AND LIU: PD OUTPUT FEEDBACK CONTROL DESIGN FOR INDUSTRIAL ROBOTIC MANIPULATORS

Fig. 5. (a) and (b) Desired (dash) and output trajectory (in radians) (solid). (c) and (d) Control inputs (in newton-meters) for joints 1 and 2 with Theorem 1 under dynamics changes.

Fig. 6. (a) and (b) Tracking errors for joints 1 and 2 with Theorem 1 under dynamics changes.

the given estimated region of interest, we now find the bound on the observer design parameter . We start by considering that, during the transient period, the fast variable η approaches to the domain of interest Ωcr before converging to a very small value of the set Ω . Define H1 = I2×2 and H2 = I2×2 to solve the Lyapunov equation for P  = 1.8090. Using CM , MM I , and η ∈ Ωcr , one calculates the maximum bound on fo (Qd , e, , η) as k3 = 3.891317 × 104 . By knowing k3 and P , β is calculated as β = 1.4342577 × 1011 . Using K = 60I2×2 and λm in (Po ) = 0.6910, we now calculate k2o = 27.335359 × 106 . Then, we determine the maximum bound on the error trajectory as e2 ≤ cr /λm in (Q). We then calculate cr = 75.6533396 × 104 . This implies that cr can

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Fig. 7. (a) and (b) Desired (dash) and output trajectory (in radians) (solid). (c) and (d) Control inputs (in newton-meters) for joints 1 and 2 under output feedback design of Theorem 3 without using the input and output disturbance noise.

be made very small by using small value of . For our implementation, let us use the value of  as  = 0.002. Using with these design parameters, we then apply the PD-based output feedback control law (5) on the given model. The conducted results are depicted in Fig. 7. By comparing the results obtained under state (figure is not shown for space limitation) and output feedback feedback based design (see Fig. 7), we observe that the control performance under PD output feedback recovers the performance achieve under PD state feedback approach. We now examine the convergence property of Theorem 3 for the case when the control system is operating under nonideal condition. To create such environment, we add a band-limited white noise w(t), as given in Fig. 3, to the control input τ (t) and output measurement q(t). Then, we apply PD-based output feedback approach on the given system. The tested results are pictured in Fig. 8. This result confirms the desired tracking objective, as the output tracking errors under Theorem 3 recovers the tracking errors achieved with Theorem 2. Let us inspect the transient tracking performance of Theorem 3 with respect to dynamics changes. To induce such dynamics changes, we first operate the system with the parameters θ1 = 2 and θ2 = 2. Then, at 12 s, the parameters θ are changed from θ1 = 2 and θ2 = 2 to θ1 = 4 and θ2 = 4, respectively. Again, at 31 s, the parameters θ are changed from θ1 = 4 and θ2 = 4 to θ1 = 2 and θ2 = 2, respectively. In view of our whole dynamical setup, we can notice that there exists modeling error uncertainty at 12 as well as at 31 s. Therefore, there are three dynamics changes in the whole process. Then, we implement Theorem 3 on the given system with the same design constants as used for our last implementation. The tested results are depicted in Fig. 9 (black solid line) with the given saturation level for two control inputs as τ1m ax = 50 N·m and τ2m ax = 50 N·m. In view of Fig. 7 and the black solid line of the Fig. 9, we can notice

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Fig. 8. (a) and (b) Desired (dash) and output trajectory (in radians) (solid). (c) and (d) Control inputs (in newton-meters) for joints 1 and 2 under output feedback design of Theorem 3 using the input and output disturbance noise w(t).

Fig. 9. (a) and (b) Tracking errors for joints 1 and 2 under PD- and CE-based CAOFB design under dynamics changes.

that the control performance under dynamics changes exhibits relatively large transient tracking errors (see after at 12 and 31 s) than the tracking performance achieved under the control system designed for the given initial conditions of interest. As the proposed output feedback method is semiglobal, then the designer can tune the controller parameters to cover larger initial conditions and initial dynamics of interest that ensures good transient tracking performance. We now compare the tracking performance of Theorem 3 with certainty equivalent principle (CE)-based classical adaptive output feedback (CAOFB) law under dynamical changes. For fair comparison, we consider

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the same setup for dynamical changes that used for PD output feedback controller design. The controller–observer algorithm ˆ = Y (ˆ e, q˙d , q¨d )θˆ − KP e1 − [6] can be designed as τ (ˆ e, Qd , θ) ˙ˆ ˆ where Y (ˆ e, q˙d , q¨d )S, e, q˙d , q¨d )θˆ = KD eˆ2 , with θ = −ΓY T (ˆ n ×n ˙ ˆ ˆ ˆ , KD ∈ n ×n , M (q)¨ qd + C(q, qˆr )q˙d + G(q), KP ∈  ˙ ˙ ˆ S = eˆ2 + λe1 , qˆr = (qˆ2 − λe1 ), λ = λ0 /(1 + e1 ), λ0 > 0, ˆ , C(·), ˆ ˆ define the estimates of the M (·), C(·), and M and G(·) and G(·), respectively. To construct the nonlinear regressor model Y (ˆ e, q˙d , q¨d ) [21], one requires to know the structure of the system. The unknown velocity signal in aforementioned CE-based CAOFB design is now replaced by the output of the e1 and eˆ˙ 2 = (H2 /2 )˜ e1 . linear observer as eˆ˙ 1 = eˆ2 + (H1 /)˜ For comparison, we consider the same set of observer–controller design parameters that were used for PD-based output design of Theorem 3 as KP = 60I2×2 , KD = 60I2×2 , λ0 = 2, H1 = I2×2 , H2 = I2×2 , and  = 0.002. The learning gains Γ are chosen as Γ = 15I2×2 to ensure faster parameter learning. The value of Γ is chosen via using trial-and-error search technique. The trial-and-error procedure is time-intensive, and the existing CAOFB method provides little guidance to parameter selection or design strategy. Then, we apply the CE-based CAOFB design on the given system. The tested results are depicted in Fig. 9 (dash line). By comparing solid and dash line of the Fig. 9, we notice that the tracking errors under CAOFB are larger than the tracking errors obtained with PD output feedback design. The reason for showing poor tracking performance of CAOFB design, with respect to parameter errors because of the assumption that uncertain parameters θ, are assumed to be appeared linearly with respect to unknown nonlinear functions Y (ˆ e, q˙d , q¨d ). We observe from our implementation results that if the initial conditions and parameter errors become large then the transient tracking performance will also become unacceptably large values. Remark 2: We notice from our various evaluations that, in the face of large-scale modeling error uncertainties, the single model-based design, either PD or classical adaptive control, demands large control gain and control saturation level in order to achieve good tracking performance. Remark 3: In all theorems reported in this paper, one can show that the tracking error and its time derivative can be made arbitrarily small by increasing the minimal eigenvalues of the control gains. IV. CONCLUSION In this paper, we have shown that the model-free linearobserver-based PD output-feedback control design can be used to solve the output feedback stabilization and tracking control problem for industrial robotic systems. The Lyapunov method has been utilized to characterize the observer–controller error bounds, where a detailed analysis has been given to show that all the signals in the closed-loop system are bounded. In our first design, the stability condition sets the lower bound on the observer speed, which reflects the uncertainties in initial conditions and initial parameters. In the second design, we show that the error bounded set can be made arbitrarily small by using observer–controller design parameters. The key feature of the

ISLAM AND LIU: PD OUTPUT FEEDBACK CONTROL DESIGN FOR INDUSTRIAL ROBOTIC MANIPULATORS

proposed design is that the control strategy can be applied directly to industrial robotic systems that are operating under the filtered version of the PD controller via replacing the velocity signals with the output of the observer. To deal with the problem associated with high control gain, a multiple-models-based gain scheduling approach will be developed in our future work [20]. The implementation of proposed Theorems on an industrial robotic system will also be focused on our future work. ACKNOWLEDGMENT

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[19] S. Arimoto, T. Naniwa, V. Parra-Vega, and L. Whitcomb, “A quasi natural potential and its role in design of hyper-stable PID servo-loop for robotic systems,” in Proc. CAI Pacific Symp. Control Ind. Autom. Appl., Hong Kong, 1994, pp. 110–117. [20] S. Islam and P. X. Liu, “Adaptive sliding mode control for robotic system using multiple model parameters,” presented at the IEEE/ASME AIM, Singapore, Jul. 14–17, 2009. [21] W.-S. Lu and Q.-H. Meng, “Regressor formulation of robot dynamics: Computation and applications,” IEEE Trans. Robot. Autom., vol. 9, no. 3, pp. 323–333, Jun. 1993. [22] X. Lu and H. Schwartz, “A revised adaptive fuzzy sliding mode controller for robotic manipulators,” Int. J. Model., Identification Control, vol. 4, no. 2, pp. 127–133, 2008.

The authors would like to thank the Technical Editor and all the Reviewers for their valuable and constructive comments. REFERENCES [1] A. Tayebi, S. Abdul, M. B. Zarembi, and Y. Ye, “Robust iterative learning control design: Application to a robot manipulator,” IEEE/ASME Trans. Mechatronics, vol. 13, no. 5, pp. 608–613, 2008. [2] A. Tayebi and S. Islam, “Adaptive iterative learning control for robot manipulators: Experimental results,” Control Eng. Practice, vol. 14, pp. 43– 851, 2008. [3] A. Tayebi and S. Islam, “Experimental evaluation of an adaptive iterative learning control scheme on a 5-DOF robot manipulators,” in Proc. IEEE Int. Conf. Control Appl., Taipei, Taiwan, Sep. 2–4, pp. 1007–1011, 2004. [4] C.-Ho Choi and Nojun Kwak, “Robust control of robot manipulator by model-based disturbance attenuation,” IEEE/ASME Trans. Mechatronics, vol. 8, no. 4, pp. 511–513, 2003. [5] J. T. Wen and S. Murphy, “PID control for robot manipulators,” Rensselaer Polytechnic Institute, Troy, NY, CIRSSE Document 54, 1990. [6] H. Berghuis, R. Ortega, and H. Nijmeijer, “A robust adaptive robot controller,” IEEE Trans. Robot. Autom., vol. 9, no. 6, pp. 825–830, 1993. [7] H. Schwartz and S. Islam, “An evaluation of adaptive robot control via velocity estimated feedback,” presented at the Int. Conf. Control Appl., Montreal, QC, Canada, May 30–June 1, 2007. [8] H. Schwartz, “Model reference adaptive control for robotic manipulators without velocity measurements,” Int. J. Adaptive Control Signal Process, vol. 8, pp. 279–285, 1994. [9] M. Takegaki and S. Arimoto, “A new feedback method for dynamic control of manipulators,” Trans. ASME, J. Dynam. Syst., Meas., Control, vol. 103, pp. 119–125, 1981. [10] M. W. Spong and M. Vidyasagar, Robot Dynamics and Control. New York: Wiley, 1989. [11] P. Rocco, “Stability of PID control for industrial robot arms,” IEEE Trans. Robot. Autom., vol. 12, no. 4, pp. 606–614, Aug. 1996. [12] P. Tomei, “Adaptive PD controller for robot manipulators,” IEEE Trans. Robot. Autom., vol. 7, no. 4, pp. 565–570, Aug. 1991. [13] Q.-H. Meng and W.-S. Lu, “A unified approach to stable adaptive force/position control of robot manipulators,” in Proc. Amer. Control Conf., 1994, pp. 200–201. [14] R. Kelly, “Comments on adaptive PD controller for robot manipulator,” IEEE Trans. Robot. Autom., vol. 9, no. 1, pp. 117–119, Feb. 1993. [15] R. Kelly, “Global positioning of robot manipulators via PD control plus a class of nonlinear integral actions,” IEEE Trans. Autom. Control, vol. 43, no. 7, pp. 934–938, Jul. 1998. [16] R. Kelly, “A tuning procedure for stable PID control of robot manipulators,” Robotica, vol. 13, pp. 141–148, 1995. [17] R. Ortega, A. Loria, and R. Kelly, “A semiglobally stable output feedback P I 2 D regulator for robot manipulators,” IEEE Trans. Robot. Autom., vol. 40, no. 8, pp. 1432–1436, Aug. 1995. [18] S. Arimoto and F. Miazaki, “Stability and robustness of PID feedback control for robot manipulators of sensory capability,” in Proc. First Int. Symp. Robotic Research, 1984, pp. 783–799.

Shafiqul Islam received the B.Sc. degree in electrical and electronic engineering from Chittagong University of Engineering and Technology, Chittagong, Bangladesh, in 1998, and the M.Sc. degree in control engineering from Lakehead University, Thunder Bay, ON, Canada, in 2004. He was with the Department of Electrical and Electronic Engineering, Chittagong University of Engineering and Technology, where, from 1999 to 2001, he was a Full-Time Lecturer and an Assistant Professor. From 2002 to 2007, he was on leave from Chittagong University of Engineering and Technology. He is currently with Carleton University, Ottawa, ON. His research interests include dynamics and control of mechatronic and robotic systems, hybrid and intelligent systems, and bilateral telemanipulation over the Internet and haptics with biomedical applications. Mr. Islam was the recipient of the 2009–2010 Koningstein Award for Excellence in Science and Engineering from Carleton University for outstanding graduate research work.

Peter X. Liu received the B.Sc. and M.Sc. degrees from Northern Jiaotong University, Beijing, China, in 1992 and 1995, respectively, and the Ph.D. degree from the University of Alberta, Edmonton, AB, Canada, in 2002. Since July 2002, he has been with the Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, Canada, where he is currently a Canada Research Chair Professor. He was an Associate Editor for Intelligent Service Robotics, the International Journal of Robotics and Automation, Control, and Intelligent Systems, and the International Journal of Advanced Media and Communication. His research interests include interactive networked systems and teleoperation, haptics, micromanipulation, robotics, intelligent systems, context-aware intelligent networks, and their applications to biomedical engineering. He has authored or coauthored more than 150 research articles. Dr. Liu is a member of the Professional Engineers of Ontario. He was as an Associate Editor for several journals including the IEEE/ASME TRANSACTIONS ON MECHATRONICS and the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING. He was the recipient of the 2007 Carleton Research Achievement Award, the 2006 Province of Ontario Early Researcher Award, the 2006 Carty Research Fellowship, the Best Conference Paper Award of the 2006 IEEE International Conference on Mechatronics and Automation, and the 2003 Province of Ontario Distinguished Researcher Award. He was the General Chair of the 2008 IEEE International Workshop on Haptic Audio Visual Environments and Their Applications and the 2005 IEEE International Conference on Mechatronics and Automation.