In modern robot manipulators, the robot controllers are required to provide the capability to overcome unmodeled dynamics, variable payloads, friction torques, torque disturbances, parameter variations, and measurement noises. Full knowledge of these surrounding conditions seems to be impossible in the majority of robotic applications. In addition, fullor limited-state measurements of robotic systems prevent the implementation of some important controllers to some important industrial environment. Mostly, Model-based observers, which are the topic of this article, are considered very well adapted for state estimation and allow, in most cases, a stability proof and a methodology to tune the observer gains, which guarantee a stable closed loop operation. Time delays if not considered might deteriorate the performance of a designed observer and, consequently, leads to degradation of an observer-based control strategy. Although delay systems are still resistant to many control policies, the delay properties are also surprising since several studies have shown that voluntary introduction of delays can also benefit the control system [1]. Time delays, when exist, could be constant [2] in system states or time-varying and unknown [3] and may exist in the output measurements with/without being time-constant [4- 7].

The intended observer design here is based on the observer developed by Germani et al. [8,9] in which the observer consists of a chain of observation algorithms reconstructing the system state at different delayed time instants (chain observer). This chain observer was based on the theory of state observers for systems without time delays that was developed by the same author in [10-12]. The observer design in [8-12] is based on transformation of a general nonlinear system into an observable form, where a Luenberger-like observer is designed to ensure the stability of the estimate error and the error dynamics perturbations are dominated by sufficiently large constant gains. This approach was partially extended in [13] for the design of a variable structure observer with a Luenberger-like term existing-without restoring to a transformation process. Once again, our work is extended to consider the design of a chain (predictive) type observer that is of variable structure and also having a Luenberger-like term. Also, we consider the transformation process as was introduced in [10-12]. The operation of this observer will be planned to reconstruct the velocity and acceleration signals from position measurement. Thus the operation of the joint acceleration feedback control will be based on an observed acceleration signals rather than using differentiation techniques or direct acceleration measurements.

The usual design procedure of a variable structure observer accompanied with a Luenberger-like term is twofold: 1) the sliding observer term takes care of the modeling errors, and 2) the role of the Luenberger-like term is mostly confined to ensuring system asymptotic stability via locating the poles of the observer dynamics in the left-half of the complex loci plane. It will be shown later that values of the gain matrix, K, would also have a great impact on the overall observation process. In the following, we extend the design procedure to derive necessary and sufficient conditions for the observer operation under the assumption that the modeling errors are composed of bounded and Lipschitz terms. The Luenberger-like observation term, in addition to providing the system stability, overpowers the Lipschitz nonlinearity while the sliding term is devoted to canceling the bounded one. Such a design would combine the two observation terms into a more interactive operation, and broaden the range of engineering applications.

The independent joint motion control of articulated manipulators comprises a separate PD or PID controller for each link with the nonlinear dynamic coupling effects handled as disturbances. This type of control has been widely applied to (gear-driven) industrial robots on a commercial scale. Ease of implementation, robustness, simplicity and fault tolerance are all advantages of the independent joint motion control over the centralized one [13,14]. In direct derive industrial robots; the performance of the conventional PD/PID controllers degrades because of the nonlinear coupling of the links which directly applies upon each joint.

Using the acceleration in a feedback loop for controlling robot arms can significantly influence the system performance of direct-drive robot arms [14-21]. An acceleration feedback enhances the disturbance rejection of an independent joint controller, which was originally presented in [17]. In this early study, it was assumed that a perfect measurement of position, velocity and acceleration signals is available. Such a perfect measurement, in a real industrial robotic application, does not exist [18]. Researchers have come to the fact that a successful implementation of an acceleration feedback is eventually digital [19], typically microprocessor based, and can be realized in two alternative techniques: i) signal differenttiation [17], and ii) observer design [20-23]. Xu and Han [13] introduced a different approach for practical implementation of acceleration sensing and modeling in the design of an independent joint motion control of robots with acceleration feedback. It is worth noting that the differentiation techniques inevitably produce noisy signals, which degrade the performance of the controller. Moreover, filtration of these signals leads to phase delay that further degrades the signal quality and limits the performance of the closed loop. The authors further developed a new estimator, which is called Newton Predictor Enhanced Kalman Filter, in [16]. The experimental investigations showed that this estimator provides a wide bandwidth and a small phase lag of the estimated acceleration while attenuating noises. While in [22] the authors introduced a new observer which uses only motor position sensing, together with accelerometers suitably mounted on the links of the robot arm. This configuration made the error dynamics on the estimated state independent from the dynamic parameters of the robot links, and can be tuned with standard decentralized linear techniques.

Obtaining acceleration signals without differentiation necessitates the use of observers instead of direct acceleration measurement. Observing the joint angular accelerations, which is a portion of this paper’s topic, is usually of the predictive type in order to compensate for the time delay that is likely to occur during implementation. On the other hand, a trade-off between the knowledge about the physical system and the number of the system states that may be reconstructed. However, it is quite possible that only the joint position will be available for a measurement from a shaft encoder. Imperfect actuator dynamics and measurements are direct reasons for occurring time delays. It is highly expected that an acceleration feedback control law would use a rather large gains. Therefore, attention should be paid to high frequency unstructured uncertainties such as time delay, flexure and resonance, measurement imperfection, violation of rigid body and continuous time assumptions [23].

Discontinuity in control laws leads to chattering in the torque control inputs, which is a highly undesirable and will result in unnecessary wear and tear on the actuator components. To cope up with this problem the modification of sliding mode control with continuous approximation of discontinuous control law was proposed, where the nonlinearitiy is approximated by high gain feedback in the boundary layer [24,25]. This eliminates chattering to some extent, but also the invariance properties associated with ideal sliding will be lost. A continuous SMC with PI disturbance estimation that was proposed in [26] requires only average inertia matrix and therefore partially solves this problem.

High-gain Luenberger-like and variable structure observers have been both proven effective when applied to robotic and non-robotic systems [13,25-27]. Our observer will combine the two observers in one generic form in order to capitalize on their benefits. The contribution in this paper is based on the contributions made recently by Germani et al. [8,9] and Dalla Mora et al. [10,11], which can all be regarded as extensions of the original work of Ciccarella et al. [12], Gauthier et al. [28], and Garcia and D’Attellis [29]. The approach that was adapted in [8-12,28,29] is to transform (1) into an observable form, where a Luenberger-like observer is designed to ensure the stability of the estimate error and the error dynamics perturbations are dominated by sufficiently large constant gains. This paper relies on the work of the above-mentioned authors and extends it to include a sliding observation term. The design approach of the sliding term exploits the significant contributions of Walcott and Zak [30] and Koshkouei and Zinober [31].

For perfectly rigid robotic manipulators, the inverse dynamics equations of input torques are:

where

where is the joint rotational angles, is the symmetric, positive-definite inertia matrix, is the centripetal and Coriolis terms, is a vector containing the gravity terms, is a constant diagonal matrix of bounded dynamic coefficients of friction, and is a vector of static friction terms, and D is an unknown bounded disturbance. The conventional acceleration feedback loop can be introduced into the robotic system in the form:

where is a diagonal matrix with a positive constant. and are vectors of the desired torques and angular accelerations, respectively.

Neglecting any imperfection and phase lagging in joint actuator dynamics, it follows that:

Equation (1) by the aid of Equations (3) and (4) becomes

where I is the unity matrix. It is now obvious that the acceleration control law (3) reformulate the system dynamics as in Equation (5) regardless of how large the acceleration control gain is chosen.

For purposes of acceleration control and observer codesign, it would much appropriate to rewrite Equation:

Note here that if, then the acceleration feedback will be able to cut down the nonlinearities by. It is also worth noting that when and the system dynamics can be eventually decoupled into a set of linear, time-invariant double integrator

This control criterion will be compared and tested against the same control criterion where condition (7) does not apply and observers are designed to reconstruct the acceleration signals.

The observer and the controller have been applied to the 6-DOF PUMA 560 robot that is shown in Figure 1. The third-order PUMA model used in this study was derived in details in [33]. In this model, the actuator dynamics and the manipulator rigid links and joints were considered. The joint positions are the only measurement and the observer uses the controller output and the meas-

urements to construct the acceleration. For simulation purposes, we only consider the first three major joints while the other three minor joints are considered locked. The third-order nonlinear set of coupled differential equations of the PUMA arm is:

where are the position, velocity, and acceleration of ith joint. u is the armature voltage. Equation (62) can be rewritten as follows:

Applying acceleration feedback with gains as presented in section II, one obtains

Equation (61) in a state space variable x is:

where

,

.

Equation (65) in view of Equation (8) becomes

whereand have obvious definitions and the bounded disturbances, , are temporarily ignored.

The undelayed output functions are

The state transformation in Equation (12), based on the locality of the system observation, is given by:

and the resulting Jacobian (14) will be time invariant. The desired acceleration signals for the simulation purposes are:

The observer design parameters in Equations (32)-(39) have been chosen such that the gain matrix of a resulting high-gain observer is:

Another set of parameters in Equations (32)-(39) has been chosen such that the gain matrix of a resulting lowgain observer is:

The two solutions in Equations (69) and (70) provides the linear part of Equation (24), respectively, with the following eigenvalues:

Initial conditions of the real and observed states were assigned the following values:

When the time delay is neglected, the observed acceleration of joint 3 is shown in Figure 2. It is shown that there is a slight overshoot at the beginning of motion followed by excellent tracking performance throughout the trajectory.

High and low-gain observers were simulated at different sets of values of the acceleration control gains. The results are shown in Figures 3 and 4. The acceleration errors (Figures 3 and 4) are marginally affected by changing the acceleration controller gains. The higher the acceleration gains the lower the peaks of the estimated acceleration errors. This fact holds true for the two kinds of observers. But our predictions have shown that the high-gain observer is of lower steady state error than the low-gain one. On the other hand, comparing responses in Figure 2 reveals two important facts: i) the low-gain observer exhibits overshoot peaks much higher than the ones provided by the high-gain observer, and ii) the

high-gain observer provides faster convergence than the low-gain observer. This puts a robot engineer in place where a choice should be made among solutions of either high-gain or low-gain or moderate-gain observers. Moreover, an attention should be paid to the unmodeled dynamics which can be handled by the sliding term in the observer, if they are bounded. Also, the actuators saturation limits should be paid attention whenever high-gain observers are used. They can easily drive the system unstable if they are not handled carefully.

The effects of time delay on the estimated acceleration errors are shown in Figures 3 and 4 low and high-gain observer used, respectively. The time delay was modeled at the sensor, i.e., there is a lag in receiving the sensor information. For (Δ ≤ 30 ms) nominal values of time

delay less than or equal 30 ms and (K_a = 100) low acceleration gains (Figure 3), the proposed controller and observer behavior remains acceptable. It is also noticeable that the effect of time delay vanishes with (K_a = 5000) higher acceleration controller gains as in Figure 4 for Δ ≤ 30 ms. Using low-gain observer and low acceleration gains makes the effect of time delay much more influential at degrading the observation process.

Now, the worst case scenario for the operation of the observer (24) that is designed not to handle time delays is to use it with low observer gains and low/moderate acceleration controller gains with relatively large output delays. These operating conditions considerably deteriorate the performance of such observer. It is desired now to investigate the performance features of the sequential observer under these worst case scenario operating conditions. Figures 5 and 6 show that the sequential observer is capable of converging to the real values. Comparison Figures 5 and 6 reveals that the larger the time delay the slower is the convergence of the sequential observer to the real values.

In this paper, an acceleration controller and sequential observer are derived and successfully implemented in simulation on a Puma 560 robot. The controller is based on joint acceleration feedback. The joint accelerations are constructed using the joint position from encoder measurement. The procedures to derive the Luenberger like observer gains, the sliding term and the sequential term are outlined in the paper. Simulation results show that the combination of sequential observer and acceleration controller are robust to delay changes. Moreover, it is also shown that measurement time delay below certain

level have slight effect on tracking. However, the tracking performance tends to degrade with large time delay where the sequential observer finds most of its impact.