Output-feedback proportional-integral-derivative-type control with multiple saturating structure for the global stabilization of robot manipulators with bounded inputs

An output-feedback proportional integral derivative-type control scheme for the global regulation of robot manipulators with constrained inputs is proposed. It guarantees the global stabilization objective—avoiding input saturation—releasing the feedback not only from the exact knowledge of the system structure and parameter values but also from velocity measurements. With respect to previous approaches of the kind, the proposed controller is expressed in a generalized form whence multiple saturating structures may be adopted, thus enlarging the degree of design flexibility. Furthermore, experimental tests on a two-degree-of-freedom direct-drive manipulator corroborate the efficiency of the proposed scheme.


Introduction
The classical proportional-integral-derivative (PID) control law has been frequently implemented in industrial manipulators. [1][2][3] This is mainly due to the practical certainty on the achievement of the regulation goal experienced through its simple linear structure. 4 A simple structure that avoids involving the system model and exact knowledge of the system parameters. 2 Nevertheless, through the classical PID linear structure, it has not been possible to derive a global proof of the closed-loop stability properties experimentally observed. This is why alternative nonlinear versions of the PID controller, mainly oriented to guarantee global regulation, have been developed for instance in Arimoto, 5 Kelly 6 and Santibáñez and Kelly. 7 However, these algorithms implicitly assume that actuators can furnish any required torque value. Unfortunately, this is impossible in practice in view of the saturation nonlinearity that generally relates the controller outputs to the plant inputs in actual feedback systems. Furthermore, disregarding such natural constraints may lead to unexpected behaviors and/or degraded closed-loop performances. 8,9 For this reason, bounded PID-type approaches have been further developed. For instance, semiglobal regulators with different saturating PID-type structures have been proposed in a frictionless setting by Alvarez-Ramrez J and colleagues. 10,11 The closed-loop analysis in these works was carried out using singular perturbation methodology. Through such a methodology, the authors show the existence of some suitable tuning, mainly characterized by the requirement of small enough integral action gains and sufficiently high proportional and derivative ones. Furthermore, the first globally stabilizing bounded PID-type control scheme, that the authors are aware of, was previously achieved by Gorez. 12 The control algorithm developed therein was carried out through the explicit consideration of friction forces on the system open-loop dynamics. The resulting algorithm gives the alternative to include or disregard velocities in the feedback. Nevertheless, the approach presented by Gorez 12 is quite complex. This inspired other researchers to find alternative bounded PID-type structures. Such efforts gave rise, for instance, to the Saturating-Proportional Saturating-Integral Saturating-Derivative (SP-SI-SD) type algorithm. This was developed by Meza et al. 13 via passivity theory and later on by Su et al. 14 through Lyapunov stability analysis. A Saturating-Proportional-Derivative Saturating-Integral (SPD-SI) type controller was further proposed by Santibáñez et al. 15 More recently, a state-feedback PID-type scheme with a generalized saturating structure, that includes both the SP-SI-SD and SPD-SI as particular cases, was presented by Mendoza et al. 16 In particular, Su et al.'s work 14 includes a velocity-free version of the presented controller by involving the dirty derivative. Further concerns on the bounded input problem have led to the additional consideration of the saturation effects of the electronic control devices of practical PID regulators. 17,18 Exponential and/or global asymptotic stabilization conditions were obtained under such natural restrictions for several implementation structures that are common in industrial robots.
The above-cited bounded PID-type approaches give a solution to the formulated problem under input constraints and restricted data. In this direction, output-feedback schemes, like the velocity-free extensions of the algorithms presented by Gorez 12 and Su et al., 14 are particularly important. This is so since they achieve regulation not only without the need for the exact knowledge of the system structure and parameter values but also through the exclusive feedback of the position variables. This proves to be particularly useful when velocity measurements are unavailable, which seems a common practical situation. However, how dirty-derivative-based output-feedback PID-type bounded schemes with alternative saturating structures (different to the SP-SI-SD one) can be designed and analytically supported is not yet clear. Although this has already been treated in the state-feedback context, 16 it remains an unsolved problem within the dirty-derivative-based output-feedback framework, where the analytical complication is considerably higher. A solution to such an open problem is not only motivated by the implicated analytical challenge but also by the nice performance expectations generated by analog saturating structures in gravity-compensation-type state-feedback contexts. 19 These arguments actually constitute the main motivation of the present work, which aims at giving a formal solution to the referred unsolved problem. As a result, an output-feedback PID-type control scheme with generalized saturating structure for the global stabilization of robot manipulators with bounded inputs is contributed here. With respect to previous approaches of the kind, it increases the degree of design flexibility through its generalized form that permits the implementation of multiple saturating structures. The proposed scheme finds potential applications in numerous types of autonomous robot systems, saving these from undesirable behaviors due to actuator saturation, releasing them from the need for speed sensors, and opening new control design possibilities to improve their closed-loop behavior. The result is developed through formal analysis based on Lyapunov stability theory. Furthermore, experimental tests on a two-degree-of-freedom (DOF) direct-drive manipulator support the analytical developments.

Preliminaries
Let X 2 R mÂn and y 2 R n . Throughout this article, X ij stands for the element of X at its i th row and j th column and y i represents the i th element of y. 0 n denotes the origin of R n and I n the n Â n identity matrix. k Á k represents the standard Euclidean norm for vectors, that is, kyk ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P n i¼1 y 2 i q , and induced norm for matrices, that is, kX k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi max fX T X g p where max fX T X g is the maximum eigenvalue of X T X . For a continuous scalar function : R ! R, 0 represents its derivative, when differentiable, D þ its upper right-hand (Dini) derivative, that is, D þ ð&Þ ¼ lim sup h!0 þ ð&þhÞÀ ð&Þ h , with D þ ¼ 0 at points of differentiability (Appendix C.2 in Khalil 20 ), and À1 its inverse, when invertible. Consider the n-DOF serial rigid robot manipulator dynamics with viscous friction 21 where q, _ q, € q 2 R n are, respectively, the position (generalized coordinates), velocity and acceleration vectors. HðqÞ 2 R nÂn is the inertia matrix, which is a continuously differentiable symmetric matrix function being positive definite and bounded on the whole configuration space, that is m I n HðqÞ M I n (2) 8q 2 R n , for some constants 0 < m M . Cðq; _ qÞ _ q is the Coriolis and centrifugal (generalized) force vector, with Cðq; _ qÞ 2 R nÂn (the Coriolis matrix) satisfying 8ðq; _ qÞ 2 R n Â R n , for some constant k C ! 0, 8ðq; _ qÞ 2 R n Â R n . F _ q is the viscous friction force vector, with F 2 R nÂn being a positive definite constant diagonal matrix whose entries f i > 0, i ¼ 1; . . . ; n, are the viscous friction coefficients, such that gðqÞ ¼ rU ðqÞ is the gravity force vector, with U ðqÞ being the gravitational potential energy, or equivalently for any q; q 0 2 R n . From the conservative character of gðqÞ ¼ rUðqÞ, for any q; q 0 2 R n , the inverse relation UðqÞ ¼ Uðq 0 Þ þ R q q 0 g T ðrÞdr is independent of the integration path (Khalil et al., p. 120). 20 Equation (7b) considers integration along the axes. This way, on every axis (that is at every integral in the right-hand side of (7b)), the corresponding coordinate varies (according to the specified integral limits) while the rest of the coordinates remain constant.
This work is addressed to robots whose gravity force term gðqÞ is a continuously differentiable bounded vector function with bounded Jacobian matrix @g @q . Equivalently, manipulators whose gravity force vector components, g i ðqÞ, i ¼ 1; . . . ; n, satisfy 8q 2 R n , for some positive constant B gi , and @g i @q j , j ¼ 1; . . . ; n, exist and are continuous and such that @g i @q j ðqÞ @g @q ðqÞ k g (9) 8q 2 R n , for some positive constant k g , and consequently jg i ðxÞ À g i ðyÞj kgðxÞ À gðyÞk k g kx À yk, 8x; y 2 R n . This is satisfied, for instance, by robot manipulators having only revolute joints (Kelly et al., 21 § 4.3). Finally, 2 R n is the external input force vector, whose elements i , i ¼ 1; . . . ; n, are assumed in this work to be constrained by a given saturation bound T i > 0, that is, j i j T i , i ¼ 1; . . . ; n. More precisely, letting u i represent the control variable (controller output) relative to the i th DOF, we have that where satðÁÞ is the standard saturation function, that is, satð&Þ ¼ signð&Þminfj&j; 1g. From equations (1) and (10), one sees that T i ! B gi (see (8)), 8i 2 f1; . . . ; ng, is a necessary condition for the robot manipulator to be stabilizable at any desired equilibrium configuration q d 2 R n . Thus, the following assumption turns out to be important within the analytical setting considered here.
The control scheme proposed in this work involves functions fulfilling the following definition.
if is strictly increasing then, for any constant a 2 R, ð&Þ ¼ ð& þ aÞ À ðaÞ is a strictly increasing generalized saturation function with bound Proof. The proof of items 1, 2, 5-8 is found in López-Araujo et al., 22  The proposed control scheme Consider the following generalized output-feedback bounded PID-type control law uðq; #; Þ ¼ Às P ðK P qÞ þ s I ðK I Þ À s d ð q; #; Þ where q ¼ q À q d , for any constant desired equilibrium position vector q d 2 R n . ; # 2 R n are the output vector variables of the integral-action dynamics, defined as _ c ¼ À"K À1 P s P ðK P qÞ; ¼ À q þ c (12) and the velocity estimation auxiliary subsystem, defined as Under time parametrization of the system trajectories, the integral-action dynamics in equations (12) adopts the (equivalent) integral-equation form ðtÞ ¼ ð0Þþ qð0Þ À qðtÞ À R t 0 "K À1 P s P ðK P qð&ÞÞd&, for any initial vector values ð0Þ; qð0Þ 2 R n . For any x 2 R n , s P ðxÞ ¼ ð P1 ðx 1 Þ; . . . ; Pn ðx n ÞÞ T and s I ðxÞ ¼ ð I 1 ðx 1 Þ; . . . ; In ðx n ÞÞ T , with Pi ðÁÞ, i ¼ 1; . . . ; n, being linear saturation functions for ðL Pi ; M Pi Þ and Ii ðÁÞ, i ¼ 1; . . . ; n, being strictly increasing generalized saturation functions with bounds M Ii , such that . . . ; n, 8ð q; #; Þ 2 R n Â R n Â R n , for suitable bounds M Pi and M Ii of Pi ðÁÞ and Ii ðÁÞ. K P ; K I ; A; B 2 R nÂn are positive definite diagonal matrices-that is, with as defined through (15a). Finally, " (in equations (12)) is a positive constant satisfying (16a) and (14): PiM being the positive bound of D þ Pi ðÁÞ, in accordance with item 2 of Lemma 1, and m , M , k C , f m , f M , B gi and k g as defined through the system properties expressed by inequalities (2), (3), (6), (8) and (9).
q is not involved in any of the expressions in equations (11)- (13). In fact, _ q is estimated online through the auxiliary subsystem in equations (13), driven by q as input variable. Its output variable # gives the estimated vector value of _ q. As a matter of fact, the auxiliary subsystem in equations (13) gives rise to the so-called dirty derivative of q. This is the derivative of q (or the velocity vector _ q) with each of its components going through a first-order low pass filter. This is commonly done in practice to bound the high-frequency gains, giving rise to a causal (approximated) derivative operator.
Remark 2. In order to preserve the main feature of PID-type controllers, the vector function s d in equation (11) shall not involve any term of the open-loop system dynamics (whether as online or desired compensation) or the exact value of any of its parameters. In general, s d will include a computed-derivative-action term (acting on the estimated velocity vector) and may involve some form of the proportional and/or the integral ones, as illustrated in Appendix 1.

Remark 3.
It is important to note that, depending on the specific choice of the vector function s d , Assumption 1 may be required to be satisfied with some strictly greater than unity. This arises as a requirement to guarantee the feasibility of the simultaneous fulfillment of (15b) and inequalities (14). For instance, in the particular control structure cases presented in Appendix 1, such a feasibility is achieved by requiring ¼ 3, as pointed out in Remark 5. A similar condition on the control input bounds has been required by other approaches where input constraints have been considered. 24 In some saturating PID-type schemes from previous references, a similar or analog condition on the control input bounds remains implicit, by requiring corresponding parameters to be high enough to satisfy conditions coming from the stability analysis and simultaneously low enough to fulfill the input-saturation-avoidance inequalities.

Closed-loop analysis
Consider system (1),(10) taking u ¼ uðq; #; Þ as defined through equations (11)- (13). Let us define the variable transformation . . . ; n (notice that their strictly increasing character renders Ii invertible). Observe that the satisfaction of (15b), under the consideration of (10), shows that Hence, under the consideration of the variable transformation (18), the closed-loop dynamics adopts the (equivalent) form Observe that, by item 1 of Lemma 1, the elements of s I ð Þ, that is, . . . ; n, turn out to be strictly increasing generalized saturation functions.
Proposition 1. Consider the closed-loop system in equations (20), under the satisfaction of inequalities (14), the conditions on the vector function s d stated through expressions (15), and Assumption 1 with suitable value of . Thus, for any positive definite diagonal matrices A, B, K I and K P such that inequalities (16) are fulfilled, and any " satisfying inequality (17), global asymptotic stability of the closed-loop trivial solution ð q; #; ÞðtÞ ð0 n ; 0 n ; 0 n Þ is guaranteed with j i ðtÞj ¼ ju i ðtÞj < T i , i ¼ 1; . . . ; n, 8t ! 0.
Proof. By (19), one sees that, along the system trajectories, j i ðtÞj ¼ ju i ðtÞj < T i , 8t ! 0. This proves that, under the proposed scheme, the input saturation values, T i , are never attained. Now, in order to carry out the stability analysis, the following scalar function is defined Ii ðr i Þdr i and recall that U represents the gravitational potential energy. Note, by recalling equations (2), that the defined scalar function can be rewritten as . . . ; q dn Þ g 2 ðr 2 Þ ¼ g 2 ðq 1 ; r 2 þ q d2 ; q d3 ; . . . ; q dn Þ . . . g n ðr n Þ ¼ g n ðq 1 ; q 2 ; . . . ; q nÀ1 ; r n þ q dn Þ and g 0 is a constant satisfying (observe, from inequality (17) and the definition of 0 , that 0 < 0 " 2 =" 2 1 < 0 < 1). Under this consideration, U c g 0 ð qÞ turns out to be lower-bounded by with 0 < k li ð1 À g 0 Þk Pi À k g and q Ã i ¼ ½L Pi À 2B gi = ð1 À g 0 Þ=k Pi (note that by inequality (21) and the definition of 0 : 0 < ð1 À g 0 Þk Pi À k g and q Ã i > 0); this is proven in Appendix 2 of Mendoza et al. 16 From this, inequality (2) and item 5 of Lemma 1, we have Zavala-Rio et al.

Experimental results
In order to corroborate the efficiency of the proposed scheme, several real-time control tests were implemented on a two-DOF robot manipulator. The experimental setup, shown in Figure 1, is a two-revolute-joint mechanical arm (on a vertical plane) located at the Instituto Tecnolo´gico de la Laguna, Mexico. The robot actuators are direct-drive brushless servomotors operated in torque mode: that is, they act as torque sources and receive an analog voltage as a torque reference signal. Joint positions are obtained using incremental encoders on the motors. In order to get the encoder data and generate reference voltages, the robot includes a motion control board based on a DSP 32-bit floating point microprocessor. The control algorithm is executed at a 2.5 millisecond sampling period on a PChost computer. Further technical information on this robot, as well as its model and parameter values, can be found in Reyes and Kelly. 26 For the experimental manipulator, inequalities (2), (3), (6), (8) and (9) i ¼ 1; 2. The SPD-SI form-with s d ð q; #; Þ ¼ s P ðK P qþ K D #Þ À s P ðK P qÞ-that is uðq; #; Þ ¼ Às P ðK P q þ K D #Þ þ s I ðK I Þ with bound values fulfilling i ¼ 1; 2. The Saturating-Proportional-Integral-Derivative like (SPID-like) form-with s d ð q; #; Þ ¼ s I ðK I ÞÀ s P ðK P qÞ À s 0 ðs I ðK I Þ À s P ðK P qÞ À K D #Þ-that is uðq; #; Þ ¼ s 0 ðÀs P ðK P qÞ À K D # þ s I ðK I ÞÞ with the saturation functions involved in s 0 -that is 0i ðÁÞ, i ¼ 1; 2-being linear saturation functions for ðL 0i ; M 0i Þ, and the involved linear/generalized saturation function parameters satisfying i ¼ 1; 2. And the SP-SID form-with s d ð q; #; Þ ¼ s I ðK I Þ À s I ðK I À K D #Þ-that is uðq; #; Þ ¼ Às P ðK P qÞ þ s I ðÀK D # þ K I Þ Let us note that with these saturation functions, we have 0 . As a consequence, for all the four controllers, inequalities (15a) and (16b) are satisfied with ¼ max i fk Di g (see equations (39)).
For comparison purposes, additional experimental tests were implemented using the output-feedback version of the bounded PID-type controller presented in Su et al. 14 The choice was made taking into account the analog nature of the compared algorithms: globally stabilizing via output feedback developed in a boundedinput context, and the recent appearance of Su et al. 14 That is u ¼ ÀK P Tanhð qÞ À K D Tanhð#Þ À K I TanhðÞ (30a) with being a (sufficiently large) positive constant and TanhðxÞ ¼ ð tanh x 1 ; . . . ; tanh x n Þ T for any x 2 R n . In place of equations (30c), the work of Su et al. 14 defines ðtÞ ¼ 2 qðtÞ þ R t 0 Tanhð qð&ÞÞd&, which imposes the auxiliary variable initial condition ð0Þ ¼ 2 qð0Þ (or, equivalently, c ð0Þ ¼ 0 n in the context of equations (30c)). Instead, equations (30c)-or their (equivalent) time representation ðtÞ ¼ ð0Þþ 2 ½ qðtÞÀ qð0Þ þ R t 0 Tanhð qð&ÞÞd&-keeps the required auxiliary dynamics while permitting any initial condition for (or, equivalently, for c in the context of equations (30c)). This proves to be more appropriate in the global stabilization framework considered in Su et al. 14 (and what is generally expected from an approach developed within such a framework). For the sake of simplicity, this algorithm is subsequently referred to as the S10 controller.
At all the experiments, the desired joint positions were fixed at q d ¼ ð=4; =4Þ T (rad), that is q d1 ¼ =4 rad for the shoulder and q d2 ¼ =4 rad for the elbow. The initial conditions were qð0Þ ¼ 0 2 (the home position), _ qð0Þ ¼ 0 2 and, for the algorithms obtained through the proposed scheme, c ð0Þ was taken so as to have ð0Þ ¼ 0 2 , while c ð0Þ ¼ 0 2 was taken for the S10 controller in view of the way it is presented by Su et al. 14 The control and saturation function parameter values were set so as to achieve pre-specified performance requirements. Two such performance requirements were a priori considered. The first one consisted in getting closed-loop responses with small transient peak values (whether as overshoot or undershoot) within a tolerance margin level of 40% of the desired position value at every link. The second aimed at achieving closed-loop responses with stabilization times shorter than five seconds. Both such requirements were achieved through a single test by each one of the algorithms obtained from the proposed scheme. On the contrary, the tuning procedure presented by Su et al. 14 permitted the S10 controller to achieve the first requirement only through long stabilization times, and the second only with high transient peak values, but not both requirements simultaneously. The resulting control and saturation function parameter values are presented in Table 1. One can corroborate that inequalities (16)- (17) are fulfilled by all the controllers obtained through the proposed scheme, as well as the corresponding saturationavoidance inequalities (26)-(29) (through which (15b) is guaranteed).The considered performance requirements were achieved under an additional control-parameter adjustment procedure that does not only take into account the conditions obtained through the closed-loop analysis (in the eponymous section) but also adopts the spirit of performance-oriented tuning methods. 27 Guidelines are given in Appendix 2. The tunings for the S10 controller are labeled as S10a and S10b for the peak and stabilization-time requirements respectively. This labeling is subsequently used to differentiate from the tests under tunings S10a and S10b. Figures 2 and 3 show the experimental results. One sees from the graphs that in all the experiments the control objective is achieved avoiding input saturation. In order to establish a comparison criterion, a performance index was evaluated for every controller: the integral of the square of the position error (ISE), that is R t f 0 ½ P 2 i¼1 q 2 i ðtÞdt (with t f the final time of the experiment). We further show evaluations of the stabilization time, taken as t s ¼ inf ft e ! 0 : k qðtÞk 0:02kq d k; 8t ! t e g, and the largest transient peak (LTP) at every link, measured as a percentage of the corresponding desired position. For each one of the considered quantifications (ISE, t s , LTP), the lowest estimated value indicates the best evaluated performance. Table 2 shows the resulting evaluations.
One sees from the obtained values that the controller with the lowest ISE index evaluation was the SPD-SI algorithm (indicated by a check mark). On the other hand, the algorithm with the highest ISE index value is the S10 controller (indicating through asterisks the two higher ones, with double asterisk for the highest). The rest of the evaluations give an analog idea on the system performance at every implementation. They confirm that the algorithms obtained from the proposed methodology were able to meet both pre-specified performance requirements through a single test. On the other hand, the S10 controller is the one with the highest number of largest index evaluations.

Conclusions
Up to the submission of the present article, a methodology for the design of output-feedback bounded PID-type controllers for robot manipulators with constrained inputs, leading to multiple saturating structures, was lacking in the literature. For instance, it was not clear how to get a velocity-free version of the SPD-SI state-feedback structure of Santibáñ ez et al. 15 Such an open problem was tackled in this work, leading to a generalized design method and the corresponding closed-loop analysis, developed with the required rigorous formality. The proposed scheme gives rise to bounded PID-type controllers with multiple saturating structures, extending the degree of design flexibility when velocity measurements are not available. For instance, it does not only extend the SPD-SI approach to such a velocity-free context and includes the SP-SI-SD as a particular case, but it also offers the possibility to generate innovative saturating structures as thoroughly shown. In addition, the design and analysis were further addressed so as to include not only smooth, but also nonsmooth (Lipschitz-continuous), saturation functions in the control structure. Further efforts made possible the corroboration of the analytical developments through experimental tests on a two-DOF manipulator, which showed the efficiency of the proposed controller. The contributed approach is thus concluded to find potential applications in numerous types of autonomous robot systems, saving these from undesirable behaviors due the actuator saturations, releasing them from the need for speed sensors, and opening new control design possibilities to improve their closed-loop behavior.

Acknowledgement
Victor Santibáñez thanks Victor de León Gómez for his invaluable help during the experimental essays.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: Marco Mendoza was supported by CONACYT, Mexico. Victor Santibáñ ez was supported by CONACYT (project number 134534) and TNM (Tecnológico Nacional de México), Mexico.