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An approach for motion control and observation of robotic manipulators is presented in this article. It links the design of a joint acceleration controller to the design of a variable structure observer including Luenberger-like observation term. Both the joint acceleration controller and the observer that are introduced in this paper are very likely to use either large or moderate or small gains. Thus the time delay issue of the output measurements is highly taken into consideration in the design of the intended observers. The observer design is therefore based on two different generalized forms of nonlinear systems with/without undelayed outputs. A study to investigate the effects of the gains of the joint acceleration controller on the performance capabilities of the observer is introduced. Also, the effects of the time delay factor on the operation of both the controller and the observer and their own interaction are studied. Then a chain observer design is presented for circumventing the time delay effects. The time delay constant is found to be of vital importance to the robot performance capabilities. Moreover, the results show that the gains of the joint acceleration controller are of significant influence on the operation of the proposed observers.

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 [

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 [

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 [

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 [

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 [

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. [

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.

Now, Equation (6) can be reformed into a generalized (undelayed) nonlinear system:

or into a generalized (delayed) system as follows:

where is the output measurement delay, , is a vector of control inputs, and is the undelayed output, the delayed output is a function of the state at time, where, and are integer exponents. Note that is a matrix whose columns are, and are smooth vector fields of and is a smooth function, where is an integer that allows all the necessary differentiations needed in the paper. A function is said to be of class if it is continuously differentiable times. is a perturbation input map and is a bounded disturbance. It is assumed that the disturbance distribution is bounded and known.

The following notation will be used in this paper. denotes an n-vector of real elements with the associated norm where denotes transposition. and λ_{min} refer to the largest singular value and smallest eigenvalue of a matrix, respectively. A function is said to be Lipschitz function with Lipschitz constant γ if it satisfies .A Lie derivative of a function along a vector field is given by with defined as the Lie derivative of order 0 and .

defined as repeated Lie derivative of order. For simplicity in the remaining of the text, and will denote, and, respectively.

In the following, the design procedure is to derive necessary and sufficient conditions for the operation of a variable structure observer under the assumption that the modeling errors are composed of Lipschitz terms and that the disturbance is unknown but bounded. The Luenberger-like observation term, in addition to providing the system stability, overpowers the Lipschitz nonlinearity by high gains while the sliding term is devoted to canceling disturbances. 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. It will be shown later that values of the acceleration controller gains, , would be of great impact on the overall observation process.

For the undelayed system (8), let

where is the total number of outputs, and for each a vector function exists such that:

where is multi-index such that

and then the square transformation matrix becomes:

Let represents the vector of output derivatives:

where when Thus, exact state reconstruction is allowed due to the invertibility of and the exact knowledge of. Furthermore, a map is said to be an observability map in a set if it is a diffeomorphism in an open set that contains or coincides with. A system that admits an observability map in for a given is said to be drift-observable in.

The nonsingular Jacobian that is associated with the observability map can be obtained as follows:

The inverse map exists in, and is defined by. If the system is drift-observable in and the maps and are uniformly Lipschitz in and, respectively, then the system is said to be uniformly Lipschitz drift-observable in a set. In addition, the system is said to be globally uniformly Lipschitz drift-observable if [

The product of has a useful structure:

where

where, and is the relative degree of the jth output such that [

where is an observability map for the first outputs, and is another one for the remaining so that the last

row blocks of the product are typically zeros. The H matrix is given by:

where the matrix is, and is defined by:

The undelayed system (8) can now be written in r-coordinates as follows:

where

and A, B and C in Equation (20) are block diagonal:

in which the triples are Brunowsky matrices [10,11,14]:

An efficient observer that combines the benefits of using a Luenberger-like observation term in addition to a sliding mode observation term is given by:

This observer in -coordinates is given by:

where

represents a discontinuous function and the term is devoted to enhancing the observer robustness against the disturbance via sliding mode. Recalling Equations (20) and (23), the error difference,

between the true state and the observer estimate will be

where

and

.

For now, the systems considered in this study obey an assumption that the resulting transformation

and

are time invariant. Consequently, becomes time invariant too.

It is more convenient to rewrite (24) in the following form:

where and express system nonlinearities such that

Now, the r-transformed error system (25) is a linear system with continuous-time nonlinearities or uncertainties in the plant. The poof of the exponential convergence to zero observation error is given in [

where and are Lipschitz constants of the uniformly Lipschitz transformations

.

The matrix P for a positive scalar, is the solution of the following Riccati-like inequality

and the gains of the sliding mode term, , is given by

where E is a positive definite matrix, and CB and CF are nonsingular matrices.

The observer design approach is also based on the justification of the following assumptions, which are applicable to the incoming observer design for delayed system:

• A constant exists such that

• The system is drift-observable in, and the map is uniformly Lipschitz together with in, with constants and that satisfy:

• The nonlinearities and are uniformly Lipschitz in with a Lipschitz constants and such that:

where is the constant that appeared in Equation (29).

• The discontinuous function can be defined as [26, 32]

and must be satisfied.

• The maximum singular value of the matrix F satisfies:

The design goal here is to provide appropriate conditions for choosing the constant observation gains and such that exponential convergence of the observation error to zero and existence of the sliding mode are both ensured.

The gain matrix is of block-diagonal structure such that is in the block-companion form [

In (20), the matrix pair is observable and the eigenvalues of can be assigned in the following companion form:

where the vector contains the coefficient of the monic polynomial that has as roots, and is an n-eigenvalues that have to be assigned. When the assigned eigenvalues of are distinct, a Vander-monde matrix can diagonalize this matrix

Remark: Given a set of eigenvalues to be assigned to, the gain vector is readily computed using the equation [

with having non-repeated eigenvalues. The eigenvalues of are chosen such that the norm of is bounded. If the eigenvalues of are assigned such that

with, then

The possibility of choosing

where is obtained via Equations (39)-(41) offers a higher number of degrees of freedom. It makes it possible to optimize the observer performance by proper choice of the eigenvalues for the physical system [

Lemma. For and the Riccatilike inequality (29) provides solution with symmetric positive definite.

The proof of this Lemma is given by Della Mora et al. [9,10]. An interesting result is that the choice of the matrix P such that:

solves the Riccati-like inequality (29) for sufficiently large values of.

Remark [

for a given. Thus the inequality (29) becomes a true Reccati inequality

in which the matrix P is the only unknown.

For the sake of simplicity we will consider the derivation for single-input single-output systems. Differentiating and upon the use of (9) and (14), and making an assumption like the ones in Equations (32) and (33), one gets the following properties:

where the matrices are Brunowski matrices [9,10]:

According to (17), it is assumed that the system has a relative degree n, which implies uniform observability as introduced in [

The delayed state and input representation is such that:

The chain observer that is proposed in this study is an extension of the one developed by Germani et al. [8,9]. The chain observer developed here is based on a variable structure observer rather than a Luenberger-like observer as was done in [8,9]. But the Luenberger-like observation term is also included in our design. The proposed observer design is such that:

The system is initially at the following conditions:

where is any a priori estimate of the state. While represents an estimate of the delayed state that is denoted here as.

Now we would like to express the proposed chain observer in (51) and (52) in the r-coordinates as follows:

where

.

The transformed chain observer (54)-(57) comes out from the coordinate change of the proposed observer in (51) and (52). The validity of this transformation can be easily proven by: i) differentiation of w.r.t. time to give (51), and ii) differentiation of w.r.t. time for taking into consideration (51) in which is substituted. Further details of this proof are found in [8,9].

The exponential convergence of this observer assumes that the function is Lipschitz such that:

A positive real and an integer are chosen such that the Lipschitz coefficient of the function and the time delayare such that:

A positive, a positive, and gain vectors and will exist for the observer (51) and (52) such that for, and hence

where depends on the estimation error in as follows:

in which and are suitable positive constants. When the triple has a uniform observation relative degree equal to, then on can be chosen equal to. The proof of convergence is a slight modification of the proofs which were introduced in [8,9].

The observer and the controller have been applied to the 6-DOF PUMA 560 robot that is shown in

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

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

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 (_{a} = 5000) higher acceleration controller gains as in

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.