Elsevier

European Journal of Control

Volume 67, September 2022, 100715
European Journal of Control

Experimental study on a novel simultaneous control and identification of a 3-DOF delta robot using model reference adaptive control

https://doi.org/10.1016/j.ejcon.2022.100715Get rights and content

Highlights

  • Introducing a novel simultaneous control and identification algorithm by using MRAC.

  • Guaranteeing the simultaneous control and identification of MIMO time-varying systems.

  • Online identification of unknown parameters of the second-order kinematic equation.

  • Simulation and practical implementation on a 3-DOf Delta robot.

  • Demonstrating a reliable performance in tracking desired path for pick-and-place purposes.

Abstract

Delta robot is one of the most widely used parallel robots in the robotics industry. Therefore, it is vital to employ appropriate methods to control Delta robot in order to perform a specific task. Modeling parallel robots is a complex and time consuming task where model-based methods which are mainly dependent to accurate modeling, lose their performance in confronting with model uncertainties and disturbances. In this paper, a novel simultaneous control-and-identification structure is proposed for Multi-Input Multi-Output (MIMO) time-varying systems with unknown parameters which guarantees closed-loop stability and simultaneous identification in presence of model uncertainties and disturbances. Therefore, the unknown parameters of system can be acquired without any prior knowledge from the outset; thus, the suggested method becomes more attractive for robotics system having sophisticated models. For the purpose of this paper, the proposed algorithm is practically implemented on a 3 Degrees-of-Freedom (DOF) Delta parallel robot. The result of simulation and practical implementation are compared with three well known methods which have been previously implemented on the robot, namely PID, adaptive method, and sliding mode control algorithm. In this regard, the results both in simulation and practical implementation reveal remarkable performance for the suggested method in compare with other aforementioned algorithms.

Introduction

There are two classifications of manipulators based on their kinematic topology, namely, serial manipulators and parallel manipulators. Serial manipulators are comprised of one open-loop kinematic chain. In contrast, parallel manipulators are comprised of one base platform, one moving platform, referred to as End-effector (EE) which are connected to each other with several limbs. In fact, the kinematic arrangement of each limb of a parallel robot can be obtained from a type synthesis performed for a prescribed motion pattern of the EE. [2]. Compared to the serial manipulators, the structure of parallel manipulators enables higher rigidity and load-carrying capacity, in addition to better EE precision [2], [45]. This class of manipulators can achieve higher accelerations while maintaining better precision than their serial manipulator counterparts. Moreover, in contrast to the serial robots whose all the joints are actuated, in parallel manipulators, usually, each kinematic chain has only one actuator mounted on the fixed base platform. This fact leads to reduced weights for the manipulator links; therefore, these manipulators can perform high precision pick-and-place tasks for heavier objects. Due to the existence of unactuated joints on the one hand, and the presence of more than one closed-loop kinematic chains on the other, the most recognizable drawback of parallel manipulators can be regarded as, generally, more complex kinematics/dynamics analysis than serial manipulators [2]. The original Delta parallel robot, which can be regarded as one of the most successful commercial parallel robot to date, has three translational Degrees-Of-Freedoms (DOF) which makes this parallel mechanism very applicable in the automation of the industries, especially in those that both precision and speed are of great essence; ranging from delicate surgeries [4], [27] to high-precision machining [24], [48] and high-speed pick-and-place applications [9], [33].

The design and implementation of an appropriate control algorithm are very important in order to accomplish the foregoing tasks perfectly. A number of researches have been dedicated to investigation of the Delta parallel robot control. In [6], three controllers, namely sliding mode controller, adaptive controller algorithm and PID controller have been designed and evaluated. In [14], similar to the LQG method, a state space H and PID controller has been implemented on the Delta parallel robot. In [35], two model-based control schemes, namely H and Computed Torque Method (CTM), have been designed and investigated for the Delta parallel manipulator. In [46], a model-based predictive control has been used to control the Delta parallel robot based on the manipulator simplified dynamics model. In [30], a Type-2 fuzzy logic controller has been proposed to control the end-effector position of the Delta robot.

The aforementioned papers utilized model-based control algorithms; therefore, the determination of an accurate model of the plant is a prerequisite to achieving a precise control. The main drawback of this approach is that the derived models are not always accurate, for instance, due to unmodeled dynamics or the deviation of the actual system parameters from those used in the simulations. In robotics, one of the approaches to achieve a better model-based controller consists in determining the dynamics equations of motion of the robot. Not only derivation of the governing dynamics equations of a parallel manipulator, especially for the over-constraint 3-DOF Delta parallel robot, is a cumbersome task, but also requires an accurate estimation of the inertial parameters to determine the precise dynamics model [1], [31], which is a very complicated task and demands a high computational cost, if possible. In some researches, the dynamics model has been simplified to alleviate this problem [40], but the model would not be accurate to achieve the high precision and high velocity capabilities of the parallel manipulators. Additionally, in practice, the plant may be subjected to external disturbances, reducing the effects of the modeled dynamics.

In order to address the dilemma associated with model-based controllers, as discussed in the rearmost paragraph, the applications of system identification methods in control contexts has been extended significantly. However, the employment of system identification methods during the control process introduces some problems like instability, multi variability, etc., in the control task. In this regard, adaptive controllers with specific characteristics like robustness, estimation of the parameters, and sufficient excitation have been proposed to enable these concepts [38], [47]. As a result, thanks to adaptability and robustness of robust adaptive control methods [11], [13], [18], [26], [37], these approaches have became more attractive; however, most of these methods still require determination of a general formulation of the system model [3], [15], [21].

This fact leads one to the idea of designing an algorithm to increase the knowledge of the plant as the system is being controlled for tracking the desired objective [44]. In the literature, this method is regarded as indirect adaptive control and also based on the specification of this approach it can be referred as simultaneous identification-and-control [5]. One of the ideas, according to the literature, has been the proposition of learning-based algorithms [10], [43], or artificial neural networks [7], [8], [51]. For instance, in [29], an integral Q-learning algorithm has been suggested in order to solve a Linear Quadratic Regulation problem. In [41], control of a 3-DoF Delta parallel robot has been investigated using an iterative learning controller compared to other classical control algorithms. In [19], in order to control the motion and the trajectory tracking and estimate the dynamics of the system an adaptive control approach has been employed. In [16], an innovative vision-based robust method has been proposed to control a Delta parallel robot based on a linear camera-space approach. One of the most significant drawbacks of the latter approaches is the need for cumbersome calculations and huge amount of data which makes them inappropriate for real-time control purposes.

As an alternative structure for simultaneous identification-and-control, the so-called Self-Tuning Regulators (STR) [5] can be regarded which uses an adaptation rule in order to generate an estimation of the unknown parameters in an on-line manner. In addition, STRs have a valuable characteristic; they are flexible in choosing the control methodology [20]. In previous researches, such as [22], the stability of the closed-loop system while tracking the desired behavior has been proven; however, in STRs, the convergence rate of the identified parameters and initial conditions of the identification process has a remarkable effect on control procedure [5].

Generally, initial design of the controller is highly dependent on the structure of the system model. In this regard, Model Reference Adaptive Control (MRAC) has been proposed as an example of model free controllers in order to achieve the control objective without the aforementioned prerequisite. In this control strategy, the unknown system is being forced to track a specified reference system, while updating unknown parameters according to the adaptation rule. By opting an appropriate reference system, this approach would result in achieving the desired objective by a much simpler linear closed-loop system. Furthermore, in the literature, MRAC strategy has been opted to design various controllers with improved performance or desired characteristics. For instance, [12] shows an interesting advantage of MRAC where a new control strategy has been proposed based on a fractional order model reference adaptive controller to acquire sufficiently smooth trajectories for varying reference tracking problem of SCARA robot.

In [25], it has been shown that less prior knowledge of the system is required in using MRAC algorithms for affine systems. This fact will result in the efficient application of this algorithm in systems with uncertain dynamics and operating points. Additionally, unlike the direct MRAC algorithm where the control gains are directly determined based on the tracking errors, in the indirect MRAC, the adaptation rule of the control gains is based on time-varying estimates of the system parameters. In this regard, the indirect MRAC algorithm can be used by merging this algorithm with a continuous-time Recursive Least Squares (RLS) method as the adaptive rule to identify the plant and the control algorithm parameters simultaneously [20]. Under the certainty equivalence principle, parameter estimations of the control adaptation rule are handled as if they are the true parameters at all times [42]; afterwards, the control gains can be derived based on the estimates. One of the advantages of this method is that parameter estimation and the control process perform simultaneously. However, the parameter convergence requires sufficient excitations [15], [39], [40]. In parallel manipulators, providing sufficient excitations is a problematic task. In the case of insufficient excitations, the singularity problem happens amidst the identification process. In the literature, this scenario is alluded as estimation wind up in which the estimation gain increases exponentially while the outputs of the estimator are sensitive to any perturbations in the regressors [32].

In order to ameliorate the oscillations of parameter identification by RLS, in [28], Damped Least Squares (DLS) method has been proposed. Similar methods were used by some other researchers to address this problem [17], [52]. Following the proposition of the DLS idea, this algorithm has been improved by introducing the Generalized DLS (GDLS) method [50] where the approach methodically emends the estimation wind-up problem in a closed-loop control procedure. In some other researches such as [34], [53], this method has been employed in order to control a system with slowly varying parameters. Moreover, in [23], [49], the Singular Value Decomposition (SVD) approach has been adopted in order to solve the singularity in the LS method; however, the estimation wind-up problem caused by these singularities has not been addressed appropriately. Moreover, in [36], in order to solve singularity problem in the RLS method, a SVD-DLS algorithm is introduced. Similar to the RLS equations, estimation wind-up would occur in the adaptation rules if the system is subjected to insufficient excitations in the control structure.

In order to address the aforementioned drawbacks, a novel method has been introduced in this paper. Thus, a simultaneous control-and-identification structure by employing indirect MRAC and adaptation rules is proposed to identify the system parameters and meanwhile, tracking the desired path for control purposes. Moreover, by utilizing sufficiently rich reference signal, the problem of insufficient excitation condition is solved. Furthermore, the suggested method is implemented on a 3-DOF Delta robot without any prior knowledge of its model.

The remainder of the paper is organized as follows. In Section 2, the basic algorithm has been thoroughly explained. By proposing a MIMO time-varying system in state space, one can guarantee the simultaneous identification and control of a system. In Section 3.1, the technical specifications of the under study robot are introduced. Besides, in order to control the 3-DOF Delta robot, the second-order kinematics relations is employed in Section 3.2. In Section 3.3, the simulation results have been demonstrated to show the superiority of the proposed novel method in comparison with the other well-known methods. Finally, in Section 3.4, the implementation results of the 3-DOF Delta robot are provided, revealing practical verification for the proposed method.

Section snippets

Simultaneous control and identification using MRAC

In this section, the simultaneous control-and-identification structure is explained by taking into account the slowly variation of the parameters of the system. The basic model of a linear system in state space is considered as:x˙=Ax+Buwhere xRn, uRm, ARn×n, BRn×m, and (A,B) is a controllable pair. In addition, it is assumed that all of the states of the system are measurable. In order to introduce identification procedure, system Eq. (1) is rewritten as follows:x˙=[AB][xu]=θφθ=[AB]φ=[xu]

Simultaneous control and identification of delta robot using MRAC

In this section, the specifications of the under study robot is introduced. Moreover, it will be shown that by adopting simple linear model rather than a complex one, the 3-DOF Delta robot will be controlled. In addition, the results of implementation of the proposed algorithm on Delta robot in simulation and in practice will be discussed.

Conclusion

In this paper, the structure of a novel simultaneous control-and-identification method for MIMO time-varying systems was suggested which was practically implemented on a 3-DOF Delta robot for pick-and-place purposes. It was shown by the Lyapunov theorem, the whole procedure remains UUB stable and the identification error stays bounded. By defining appropriate indices, the results of implementing proposed method on the Delta robot were compared with three well-known method namely, PID, adaptive

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The work presented in this paper and all the associated tests done were conducted at Human and Robot Interaction Laboratory (TaarLab). We would like to thank all of the members of Taarlab for their supprting. Furthermore, this research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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