Adaptive Finite-time Synergetic Control of Delta Robot based on Radial basis Function Neural Networks

: This paper proposes a novel robust proportional derivative adaptive non-singular synergetic control (PDATS) for the delta robot system. A proposal radial basis function approximation neural networks (RBF) compensates for external disturbances and uncertainty parameters. To counteract the chattering noise of the low-resolution encoder, a second-order sliding mode (SOSM) observer in the feedback loop showed the ability to obtain the angular velocity estimations. The stability of the PDATS approach is proven using the Lyapunov stability theory. Both the simulation and experiment result effectiveness and performances of the PDATS controller in trajectory; pick and place operations of a parallel delta robot. The characteristics of the controller demonstrate that the proposed method can effectively reduce external disturbance and uncertainty parameters of the robot by a convergent finite-time, and provide higher accuracy in comparison with finite-time synergetic control and PD control.


Introduction
Delta parallel robot is uncontrovertibly the most successful commercial parallel robot to date. In general, a parallel manipulator has the advantages of high precision trajectory combined with high speed and acceleration [1], [2]. Many industrial robots employ the type of parallel robot, consists of three arms connected to universal joints at the base. The robot is a multivariable, multi-parameter coupling, multidegree-of-freedom nonlinear system. Many studies showed the remarkable effect of the kinematics [3][4] [5] and dynamics model [6,7] directly on precise trajectory performance. There are many control strategies for the robot. In the studies [8], [9], the authors improved the performance via PD feedforward control based on the nominal dynamics robot system. In [10], A dynamic feedforward decoupling control scheme aimed at a compensated feedforward gravity and an independently adjustable three PID control loop, corresponding to the position, velocity, and acceleration control in simulation. The simulation results showed that the scheme was capable of controlling each motion factors effectively. But it is hard to practically implement because it can meet noise with the loop feedback signals. In addition to this, the closed-loop system did not include the influences of uncertainty parameters and unknown external disturbance.
To solve robust and high speed trajectory problems with the reduced influences of external disturbance, a sliding mode control (SMC) approach based on delta robot dynamics is introduced in [11], which is beneficial to remain the system trajectories on the sliding surface. Another SMC improvement is terminal sliding mode (TSMC) control [12] applied to robotic manipulators for finite-time stability.
Thanks to Lyapunov function analysis, these controllers verified the ability to superior performance. But both SMC and TSMC techniques have a drawback in that they produce the high-frequency oscillations of the controller output known as chattering.
To eliminate chattering phenomenon, another new methodology for modern technologies and complex systems led to the introduction of synergetic control (SC) theory [13]. SC employs a state-space method providing the object motion stability by the Analytical design of Aggregated Regulators (ADAR) method. It simplifies the original nonlinear mathematical model without losing its dynamical properties to make engineering implementation easier. Similar to sliding mode control properties, the advantages of synergetic control are well-suited for order reduction of the controlled system, robustness to external disturbance, and better control of the off-manifold dynamics. In many studies about synergetic literature [14][15][16], the synergetic-based controllers are characterized chattering-free dynamics compared to conventional sliding mode controls. In [17,18] a finite-time synergetic control (FTS) scheme employed the combination between the synergetic theory and a terminal attractor technique for controlling series robot manipulators. The new synergetic approach which allowed to perform well in many areas of synthesis of multiply connected systems of continuous, discontinuous, discrete-time, terminal, and adaptive control for nonlinear dynamic objects such as robots [18] [19], electric drives [20], and flying apparatus [21].
Delta parallel robot is highly nonlinear complexity system, so the design of the suitable control scheme with parametric and non-parametric uncertainties and external disturbance represent a significant challenge. Many pieces of research [22][23][24], Boudjedir and co-workers proposed the nonlinear controller to solve robust and precise trajectory problems using sliding mode or Nonlinear PD controller based on iterative learning. However, the scheme requires time consumption on some desired continuous repetitive or non-repetitive trajectories to adjust the PD parameters to improve precise performance and deal with the practical external disturbance. Additionally, an adaptive computed torque controller [25] was developed for trajectory tracking control of the 2-DOF parallel manipulator, and the adaptive law estimate the dynamics model includes the friction of active joints.
The development of adaptive high-order sliding mode scheme [26] offer the alternative modulation gains to maintain the sliding in the presence of bounded and derivative bounded uncertainties and cancel chattering phenomenon. In [27], the authors present an adaptive sliding mode controller with switching gain which is adjustable real-time online by selecting the appropriate PID type sliding surface.
Utilization of fuzzy logic system estimates the unknown nonlinear behavior of the system and adapted successfully via synergetic control theory in some robots [28].
However, the choice of the linguistic rules and guaranteeing the system stability remain a challenging issue. A RISE (Robust Integral of the Sign Error) controller in [29] with a BSNN feedforward compensation was applied to a delta robot to regulate the trajectory tracking for a Pick and Place application. Since the addition of an intelligent compensation term may reduce the tracking error considerably and might cancel the steady-state error. However, only simulating validations is taken into account. In [30], a novel output feedback controller with a feedforward term The main contributions of this paper lie in the following aspects: (1) A systematic approach is developed to control a class of nonlinear systems with unknown uncertainty parameters in the nonlinear delta manipulator model and DC motor model.
(2) An implement of high order sliding mode observer reduce the chattering of low resolution encoders for angular velocity estimation.   Based on experimentation [31] and real data collection [33], using theoretical results and simulation DC motor model to acquire the dynamic parameters of DC motor servo.

Delta robot manipulator
Consider the Delta robot illustrated in Fig.2, consisting of three symmetric arms constrained in a kinematic manner by universal or spherical joint at the end-effector. The inverse dynamic model of Delta robot developed in the work [8,9] based on the Newton-Euler method with the following simplifying hypotheses: -The rotational inertias of forearm are neglected.
-For analytical purposes, the masses of forearms are optimal separated into portions and places at their extremities: a two-third majority part at its upper extremity and the other part at its lower extremity, which is joined to the traveling plate mass.
-Fiction effects and elasticity are neglected With the above mentioned simplifying hypothesis, the robot can be reduced to only 4 bodies: the travelling plate and the 3 upper arms. At the travelling plate, the total mass is calculated based on the hypothesis as At the upper arm, the center of mass of the virtual arm based on the hypothesis is adjustable as The inertia matrix of the upper arms is a diagonal matrix in joints space, where 1 = 2 = 3 = is given by: There are two kinds of forces act on the travelling plate: the gravity force and the inertia force. They are respectively given by: We can transform into generalized angular coordinates by Jacobian, the description of Jacobian matrix are discussed in more detail in [1] and [3] so According to the virtual work principle, We define is the torque produced by the gravitational force of the arms, so the torque contributions of delta robot applying to the motors is given by The relationships between velocities ̇, ̇ and accelerations ̈, ̈ is described as From (10), we state the inverse dynamic model in function produced by the mass matrix ( ) , the Coriolis and centrifugal matrix ( ,̇) and the gravity contribution as equation below Where: 3 The proposed controller

The finite time synergetic controller (FTS) for Delta manipulator
Rewriting the nominal class of nonlinear dynamics delta robot model (13) as follows: Step 1: The design procedure of the controller starts by choosing a macro-variable which is generally a linear combination of the state variables.
The control law forces the trajectories to operate on the manifold ( ) = 0 and to move toward this manifold exponentially according to the following equations: Step 2: The proposed controller will be designed here such that it will force the states to approach the manifold M smoothly at finite time with a new evolution constraint according to the following constraint equation: Where ̇= [1 1 1 , ̇2 2 2 ,̇3 3 3 ] , and are positive odd numbers which satisfy the condition 1 < < 2, = 1, 2, 3. This constraint will drive the macro variable and its derivative ̇ to zero at finite time Step 3: From the constraint equation and substituting (24) into nonlinear dynamics robot model, one can obtain the resulting control law associated with delta manipulator can be expressed as Lemma 1. [12,34] Suppose that a continuous, positive-definite function satisfies the following inequality Where > 0, 0 < < 1 are constants. Then, for any given 0 , ( ) satisfies the following inequality: With 1 is given by , and we have its derivative as Substituting ̇+ = 0 into the equation above, and then Compare with (28). The synergetic manifold can converge to zero in the finite time, which depends on r, p given by

The PD adaptive finite time nonsingular synergetic controller based on radial basis function neural network (PDAFTS)
The general equations of nonlinear dynamics Delta robot and DC motor servo, that including external disturbance is presented as follows To control the system model as (35), this paper proposes a PD-AFTS controller  [30], and designed on computer as The adaptive law is designed as By the use of the gradient descent, the weight are adapted as In totally, the proposal controller of the nonlinear delta robot is proposed as The system errors and error rate will converge to zero in finite time with the rate convergence depending on the parameter , , and if the control law is selected as (39). The adaptive control law ̂ in (36) is used to estimate the external disturbance and model errors torque that can bring the tracking to the Fig. 3. The RBF neural network in this paper synergetic manifold surface. In the following, we will prove the asymptotic stability of the parallel manipulator system controlled by the PD-AFTS controller Substituting the adaptive law (40) into (56), yields

̇= (−̂( ) ( ) ) ≤ 0 (57)
According to the Lyapunov theory, we prove that the proposed PDAFTS is stable for the tracking control of the delta robot. The equality ̇= 0 is satisfied if and only if =̇= 0. Since the Lyapunov function is positive definite and ̇ is negative definite, it is concluded that the proposed PDAFTS is globally asymptotically stable from the Lyapunov stability criterion.

The velocity second order sliding mode observer
To implement the PDAFTS defined in (43)

Remark 1:
Using the PD controller is very realistic in the experiment, where the PD controller is implemented directly on the microcontroller and stabilizes the robot in the short term. And the complex computation is performed on the computer, then transfer data to guarantee the accuracy and robustness of the robot in the long term control.

Remark 2:
The proposed control guarantees the exponential convergence of the tracking error with the manifold sliding surface and eliminates the chattering phenomenon from the sign function.

Remark 3:
In the developed FTS in (38), the control system has a finite time convergence with the experience of external disturbance. Fortunately, the proposed PDAFTS based on RBF approximation can compensate for the effects of the disturbance torque or unknown parameters.

Remark 4:
It can be noticed from equation (59) The dynamic model of delta robot parameter are selected as Table 1 and Table 2.
The parameters of controllers (43) in detail as (37), (38) and (39)  The initial neural network weights ̂ vector are selected as zeros. The updated weight law is given by (40) and (42)   The desired trajectories for simulation use the information of helix and parabolic characteristics in Fig. 4 and Fig. 5 and generalized in Table 3. The objective of these cases is to validate the speed changes that affect the controllers' stability Table 3. The parameters of helix and parabolic trajectory for delta robot  Table 4 and Table 5 are the lower values than those of the others, and the proposed controller provides the best performance.
The control signal of the controllers is presented in Fig. 9 and In Fig. 10 and Fig. 13, it is noteworthy that the behavior of external torque estimation is similar to the added external torque. The effective action of RBF approximation leads the estimated disturbance torques to converge to the curve of the disturbance torques, which is described as the equation in (64). The accurate compensation to control signal leads to the highest performance among the simulated controller in both study cases.

Experiment 1: Helix and parabolic trajectory without external load
In this section, the two cases experiments were conducted based on the desired trajectories in the simulation section. The experimental results in helix and parabolic trajectories were illustrated in Fig. 16, Fig. 17 and Fig. 20, Fig. 21, respectively. For comparing the simulation results, the tracking performance witnesses an accurate reduction in the RMSE in Table 6 and Table 7 It is noteworthy that the behavior of the proposed controller in Fig. 18 and Fig. 22.
is very similar to the torque produced by the remaining schemes. The speeding up velocity effort to the tracking performance and control signal consolidates the analyses in the simulation section. Although the SMC can guarantee stability, it is obtained clearly at the price of high control chattering in the control signal graph.
Meanwhile, the smoothing of control output from the synergetic algorithm demonstrated the advantage in maintaining the attractiveness of the boundary layer.
The adaptive unknown torque approximation term in the proposed approach reduced quite well the effect of the disturbance, resulting in better dynamics performances than the other four methods. The updated weight in Fig. 19 and Fig.   23 presented the reaction with errors fluctuation and convergence to a constant value as soon as the system becomes more immovable. The power of torque estimations of the parabolic case witnesses a higher value than that of the helix case, so it is possible to track well the fast transition of external disturbances.   The productiveness of RBF networks in the proposed controller is confirmed by adding an object into the end effector of the delta robot in helix operation. The objective is to evaluate the adaptation and robustness of the proposed approach against model parameter variation. In this case, the end effector has added an external object with 0.2 kg. Despite an increasingly small number of errors, the PDAFTSC algorithm successfully overwhelms the external disturbance to achieve similar performance compared to others in the no-load case in Fig. 25 and Table 8.
The control signal tends to widen the boundary when the joints increase the speed.
In addition to this, Fig. 26 demonstrated a similar curve of disturbance torques from the RBF estimators. This action will create accurate signal compensation to catch up with the speed of tracking errors and keep the system in robustness. As a result, the adaptation of the proposed controller has been validated based on disturbance torque estimation and compared to no-load cases. Despite the influence of the external disturbances generated by joint friction or uncertainty parameters, the system with RBF NNs attachment has presented remarkable stability compared with the others.

Conclusions
In this work, a novel adaptive PDAFTS controller using synergetic theorem is the first time developed for dynamic delta robot model. To obtain results, various simulation and experiment are verified. Firstly, the delta robot generated by the combination of DC motor model and delta robot manipulator is developed for simulation. Then, the velocity estimator using SOSM and LPF to observe the angular velocity and reduce the chattering phenomenon, which then provides full states for the proposed controller. The proposed system stability is proved by Luyapunov method. Finally, the practicality of the PDAFTS controller is extracted from the experiment results with the unknown extended load. As a result, the proposed controller performs better than the other controllers in the tracking tasks.
The comparison of the obtained results of the PDAFTS and the other control algorithms show the superiority of the PDAFTS in the presence of disturbance and reduction of chattering phenomenon. Thanks to the RBF neural networks in the proposed technique, the unknown disturbance is estimated effectively and contribute to the control signals, which drive the robot more accuracy and robustness.