Output-Feedback Adaptive SP-SD-Type Control with an Extended Continuous Adaptation Algorithm for the Global Regulation of Robot Manipulators with Bounded Inputs

In this work, an output-feedback adaptive SP-SD-type control scheme for the global position stabilization of robot manipulators with bounded inputs is proposed. Compared with the output-feedback adaptive approaches previously developed in a bounded-input context, the proposed velocity-free feedback controller guarantees the adaptive regulation objective globally (i.e. for any initial condition), avoiding discontinuities throughout the scheme, preventing the inputs from reaching their natural saturation bounds and imposing no saturation-avoidance restrictions on the choice of the P and D control gains. Moreover, through its extended structure, the adaptation algorithm may be configured to evolve either in parallel (independently) or interconnected to the velocity estimation (motion dissipation) auxiliary dynamics, giving an additional degree of design flexibility. Furthermore, the proposed scheme is not restricted to the use of a specific saturation function to achieve the required boundedness, but may involve any one within a set of smooth and non-smooth (Lipschitz-continuous) bounded passive functions that include the hyperbolic tangent and the conventional saturation as particular cases. Experimental results on a 3-degree-of-freedom manipulator corroborate the efficiency of the proposed scheme.


Introduction
Since the publication of [1], the Proportional-Derivative with gravity compensation (PDgc) controller [2] has proved to be a useful technique for the regulation of robot manipulators. In its original form, such a control technique achieves global stabilization under ideal conditions, for instance unconstrained input, measurability of all the system (state) variables and exact knowledge of the system parameters. Unfortunately, in actual applications, such underlying assumptions are not generally satisfied, giving rise to unexpected or undesirable effects, such as input saturation and those related to such a nonlinear phenomenon [3], noisy responses and/or deteriorated performance [4], or steadystate errors [5]. However, such inconveniences have not necessarily rendered the PDgc technique useless. Inspired by this control method, researchers have developed alternative (nonlinear or dynamic) PDgc-based approaches that deal with the limitations of the actuator capabilities and/or of the available system data, while keeping the natural energy properties of the original PDgc controller, which are the definition of a unique arbitrarily-located closed-loop equilibrium configuration and motion dissipation. For instance, extensions of the PDgc controller that cope with the input saturation phenomenon have been developed under various analytical frameworks in [6, 7, 8, 9, 10 and 11]. Indeed, assuming the availability of the exact value of all the system parameters and accurate measurements of all the link positions and velocities, a bounded PDgc-based approach was proposed in [6] and [7]. In these works, the P and D terms (at every joint) are each explicitly bounded through specific saturation functions; a continuously differentiable one, or more precisely the hyperbolic tangent function, is used in [6] and the conventional nonsmooth one in [7]. In view of their structure, these types of algorithms have been denoted SP-SD controllers in [12]. Two alternative schemes that prove to be simpler and/or give rise to improved closed-loop performance were recently proposed in [8]. The first approach includes both the P and D actions (at every joint) within a single saturation function, while in the second one all the terms of the controller (P, D and gravity compensation) are covered by one such function, with the P terms internally embedded within an additional saturation. The exclusive use of a single saturation (at every joint) including all the terms of the controller was further achieved through desired gravity compensation in [13]. Moreover, velocityfree versions of the SP-SD controllers in [7] and [6] (still depending on the exact values of the system parameters) are obtained through the design methodologies developed in [9] and [10]. In [9] global regulation is proven to be achieved when each velocity measurement is replaced by the dirty derivative [14] of the respective position in the SP-SD controller of [7]. A similar replacement in a more general form of the SP-SD controller is proven to achieve global regulation through the design procedure proposed in [10] (where an alternative type of dirty derivative, which involves a saturation function in the auxiliary dynamics that gives rise to the estimated velocity, results from the application of the proposed methodology). Furthermore, an outputfeedback dynamic controller with a structure similar to that resulting from the methodology in [10], but which considers a single saturation function (at every joint) where both the position errors and velocity estimation states are involved, was proposed in [11] (where a dissipative linear term on the auxiliary state is added to the saturating velocity error dynamics involved for the dirty derivative calculation). Extensions of this approach to the elastic-joint case were further developed in [15].
Furthermore, SP-SD-type adaptive algorithms that give rise to bounded controllers, while alleviating the system parameter dependence of the gravity compensation term, have been developed in [16, 17, and 18]. In [16] global regulation is aimed for, through a discontinuous scheme that switches among two different control laws, under the consideration of state and output feedback. Both considered control laws keep an SP-SD structure similar to that of [7]; the first one avoids gravity compensation taking high-valued control gains (by means of which the closed-loop trajectories are lead close to the desired position) and the second one considers adaptive gravity compensation terms that are kept bounded by means of discontinuous auxiliary dynamics. Each velocity measurement is replaced by the dirty derivative of the corresponding position in the output-feedback version of the algorithm. Unfortunately, a precise criterion to determine the switching moment (from the first control law to the second one) is not furnished for either of the developed schemes.
In [17] semi-global regulation is proven to be achieved through a state feedback scheme that keeps the same structure as the SP-SD controller of [6] but additionally considers adaptive gravity compensation. The adaptation algorithm is defined in terms of discontinuous auxiliary dynamics, by means of which the parameter estimators are prevented from taking values beyond some prespecified limit, which consequently keeps the adaptive gravity compensation terms bounded. This approach was further extended in [19] where the control objective is defined in task coordinates and the kinematic parameters, in addition to those involved in the system dynamics, are considered to be uncertain too.
In [18] a controller that keeps the SP-SD structure of [6] is proposed, where each velocity measurement is replaced by the dirty derivative of the corresponding position and an adaptive gravity compensation term with initialcondition-dependent bounds is considered. Based on the proof of the main result, semi-global regulation is claimed to be achieved.
Let us note that, by the way the SP and SD terms are defined in the adaptive schemes mentioned above, the bound of the control signal at every link turns out to be defined in terms of the sum of the P and D control gains (and of an additional term involving the bounds of the parameter estimators). This limits the choice of such gains if the natural actuator bounds (or arbitrary input bounds) are to be avoided. This, in turn, restricts the closed-loop region of attraction in the semi-global stabilization cases. On the other hand, as far as the authors are aware, the semi-global and/or discontinuous approaches developed in [18] and [16] are the only output-feedback bounded adaptive algorithms proposed in the literature. Moreover, a continuous adaptive scheme with continuous auxiliary dynamics, which achieves the global regulation objective, avoiding input saturation and disregarding velocity measurements in the feedback, is still missing in the literature and consequently remains an open problem. These arguments have motivated the present work, which aims to fill in the aforementioned gap.
It is worth adding that recent works have focused on the global regulation problem in the bounded-input context through nonlinear PID-type controllers. This is the case for instance of [20], [21], [22] where state-feedback and output-feedback schemes were presented, and [23] where a controller with the same structure as the state-feedback algorithm presented in [22] was previously proposed. Such PID-type algorithms are not only independent of the exact knowledge of the system parameters, but also disregard the structure of the system dynamics (or of any of its components). However, in a bounded-input context, the design of an output-feedback adaptive scheme that solves the regulation problem globally, avoiding input saturation, and being free of discontinuities, remains an open analytical challenge. Moreover, as will be corroborated in subsequent sections of this work, regulation towards a suitable configuration permits the output-feedback adaptive scheme to provide an estimation (exact under ideal conditions) of the system parameters (involved in the gravity-force vector), which is not the case for other types of controllers.
In this work, an output-feedback adaptive SP-SD-type control scheme for the global regulation of robot manipulators with saturating inputs is proposed. Through its extended structure, the adaptation algorithm may be configured to evolve either in parallel (independently) or interconnected to the velocity estimation (motion dissipation) auxiliary dynamics, giving an additional degree of design flexibility. With respect to the previous output-feedback adaptive approaches developed in a bounded-input context, the proposed velocity-free feedback controller guarantees the adaptive regulation objective globally (i.e. for any initial condition), avoiding discontinuities throughout the scheme, preventing the inputs from attaining their natural saturation bounds and imposing no saturation-avoidance restriction on the choice of the P and D control gains. Furthermore, contrarily to the adaptive schemes of the previously cited studies, the approach proposed in this work is not restricted to involving a specific saturation function to achieve the required boundedness, but may involve any one within a set of smooth and non-smooth (Lipschitz-continuous) bounded passive functions that include the hyperbolic tangent and the conventional saturation as particular cases. Experimental results on a 3-degree-of-freedom manipulator corroborate the proposed contribution.

Preliminaries
Let us consider the general n-degree-of-freedom (n-DOF) serial rigid robot manipulator dynamics with viscous friction [26,27]: are, respectively, the position (generalized coordinates), velocity and acceleration vectors. is a piecewise continuous function with bounded discontinuities but well defined at i  , are, respectively, the vectors of Coriolis and centrifugal, viscous friction, gravity and external input generalized forces, with n n F R   being a positive definite constant diagonal matrix whose entries 0 i f  , , , 1 i n   , are the viscous friction coefficients. Some well-known properties characterizing the terms of such a dynamical model are recalled here (see for instance [2,Chap. 4] and see further [2, Chap. 14] and [28] concerning Property 6 below).

Property 6
The gravity vector can be rewritten as is a constant vector whose elements depend exclusively on the system parameters and is a continuous matrix function, whose elements depend exclusively on the configuration variables and do not involve any of the system parameters. Equivalently, the potential energy function of the robot can be rewritten as is a continuous row vector function whose elements depend exclusively on the configuration variables and do not involve any of the system parameters. Actually, Property 7 Consider the gravity vector ( , ) Let us suppose that the absolute value of each input i  ( th i element of the input vector  ) is constrained to be smaller than a given saturation bound 0 In other words, letting i u represent the control signal (controller output) relative to the th i degree of freedom, we have: Let us note from (1) and (2)   . Thus, the following assumption turns out to be crucial within the analytical setting considered in this work: The control schemes proposed in this work involve special functions fitting the following definition.

Definition 1 Given a positive constant M , a non-decreasing Lipschitz-continuous function
: Functions meeting Definition 1 satisfy the following: be a generalized saturation with bound M and k be a positive constant. Then The proposed output-feedback adaptive control scheme is defined as    (3),  is a constant that may arbitrarily take any real value and  is a (sufficiently small) positive constant. A block diagram of the proposed output-feedback adaptive control scheme is shown in Fig. 1. Remark 2 Note that the simplest version of the proposed control scheme arises by taking 0   . However, the term extending the adaptation dynamics in (7a) has been included for the sake of generality, since an analogue term was considered in a previous approach [18].
Furthermore, the  -term in (7a) has a natural influence in the closed-loop responses which could be used for performance adjustment purposes. This aspect is not explored in this work.

Experimental results
In order to experimentally corroborate the efficiency of the proposed scheme, referred to as the SP-SDc-ga controller, real-time control implementations were carried out on a 3-DOF manipulator. The experimental setup, shown in Fig. 2, is a 3-revolute-joint anthropomorphic arm located at the Benemerita Universidad Autonoma de Puebla, Mexico. The actuators are direct-drive brushless motors (from Parker Compumotors) operated in torque mode, so they act as a torque source and accept an analogue voltage as a reference of torque signal. Position information is obtained from incremental encoders located on the motors. The setup includes a Pentium host computer and a system of electronic instrumentation, based on the motion control board MFIO3A, manufactured by Precision Microdynamics. The robot software is in open architecture, whose platform is based in C language to run the control algorithm in real time. The control routine registers data generated during the first 2000 samples at a default sample time of 2.5 s T  ms, but s T can be changed to higher values in accordance to the desired experimental duration. The experiments carried out in the context of this work, whose results are presented below, were run taking 0.12 s T  s. A more detailed technical description of this robot is given in [30].
For the considered experimental manipulator, Properties 5 and 6 are satisfied with  For comparison purposes, additional experiments were run implementing the output-feedback adaptive algorithm proposed in [18], referred to as the L00 controller (choice made in terms of the analogue nature of the compared algorithms: output-feedback adaptive developed in a bounded input context; comparison of controllers of a different nature loses coherence), i.e.,  Observe that the regulation objective was achieved preventing input saturation and avoiding steady-state position errors. Furthermore, note that despite the presence of a small overshoot, through the SP-SDc-ga algorithm shorter stabilization times took place in both position error and parameter estimation responses. Let us further note that at 240s, where the experimental data registration was stopped, the parameter estimations were still evolving. This is a consequence of the slow evolution of the adaptation subsystem dynamics, due to the relatively small value of  in the proposed scheme and the analogue coefficients  and  in the L00 controller. Nevertheless, the slow evolution of the adaptation subsystem dynamics did not have any influence on the position responses, which had been stabilized during the initial seconds of the experiment. The subsequent parameter estimator evolution was expected to reduce the difference among the estimations obtained through each implemented controller. 7 One can verify from ( ) G q in (16) that, for the considered manipulator, the desired configurations that satisfy the condition stated by Corollary 1 are those such that    In this work, an output-feedback adaptive control scheme for the global regulation of robot manipulators with bounded inputs was proposed. With respect to the previous output-feedback adaptive approaches developed in a bounded-input context, the proposed velocity-free feedback controller guarantees the adaptive regulation objective: globally, avoiding discontinuities throughout the scheme, preventing the inputs from reaching their natural saturation limits and imposing no saturation-avoidance restriction on the control gains. Moreover, the developed scheme is not restricted to the use of a specific saturation function to achieve the required boundedness, but may rather involve any one within a set of smooth and non-smooth (Lipschitzcontinuous) bounded passive functions that include the hyperbolic tangent and the conventional saturation as particular cases. The efficiency of the proposed scheme was corroborated through experimental tests on a 3-DOF manipulator. Good results were obtained, which were observed to improve those gotten through an algorithm that was previously developed in an analogue analytical context.