Dynamic model estimating and designing controller for the 2-DoF planar robot in interaction with cable-driven robot based on adaptive neural network

Abstract

This study intends to investigate the dynamic model estimation and the design of an adaptive neural network based controller for a passive planar robot, performing 2-DoF motion pattern which is in interaction with an actuated cable-driven robot. In fact, the main goal of applying this structure is to use a number of light cables to drive serial robot links and track the desired reference model by the robot’s end-effector. The under study system can be used as a rehabilitation setup which is helpful for those with arm disability. In this way, upon applying sliding mode error dynamics, it is necessary to determine a vector that contains the matrices related to the robot dynamics. However, finding these matrices requires the use of computational approaches such as Newton-Euler or Lagrange. In addition, since the purpose of this paper is to express comprehensive methods, so with increasing the number of links and degrees of freedom of the robot, finding the dynamics of the robot becomes more difficult. Therefore, the Adaptive Neural Network (ANN) with specific inputs has been used for estimation unknown matrices of the system and the controller design has been performed based on it. So, the main idea in using an adaptive controller is the fact there is no pre-knowledge for the dynamic modeling of the system since the human arm could have different dynamic properties. Hence, the controller is formed by an ANN and robust term. In this way, the adaptation laws of the parameters are extracted by Lyapunov approach, and as a result, as aforementioned, the asymptotic stability of the whole of the system is guaranteed. Simulation results certify the efficiency of the proposed method. Finally, using the Roots Mean Square Error (RMSE) criteria, it has been revealed that, in the presence of bounded disturbance with different amplitude, adding the robust term to the controller leads to improve the tracking error about 34% and 62%, respectively.

Keywords

Dynamic model estimation adaptive neural network controller lyapunov approach passive planar robot actuated cable-driven robot and rehabilitation setup

1 Introduction

One of the most important applications of robotic science is in the biological and rehabilitation fields. Due to the anthropomorphic characteristics of serial robots, most of the design presented in the literature and industry are based on this type of manipulators which are intended for rehabilitation purposes [1 –4]. Rehabilitation mechanisms which are purely based on serial manipulators suffers from disadvantages such as the choice of large actuators to move the robot and high-speed movements. As a result, this leads to the design of robots with heavy volume and weight. In addition, the use of serial link structure can lead to the accumulation of errors in the angles of the motors and its transfer to the end-effector of the robot. As a result, the mentioned disadvantages can lead to some drawbacks such as cost and safety concerns and applying actuators with special features [5]. For this reason, some design strategies should be taken into account to overcome the mentioned drawbacks. One of these strategies can be regarded as combining cable-driven robot next to a serial robot and due to the resemblance to the human arm can be regarded as an anthropomorphism robotic arm. Using this method, in which the bones are the rigid part, and the cables with a mass much less than the robot links, which only act as tension to move the robot, play the role of the muscles attached to the bones, desirable characteristics of both robots simultaneously are satisfied, namely achieving a larger workspace, higher load carrying capacity and reliability for performing difficult and complex tasks. To this end, kino-static and dynamic analysis of the under study cable-driven robot, as the actuator part of the mechanisms, should be fully investigated before controlling the mechanism. Using cable robots instead of traditional robots can cover the disadvantages of these robots. One of the advantages that can be achieved with cable robots is the larger workspace compared to these robots. This is due to the increased motion range and compact nature of cable-driven robots [6]. On the other hand, cable-driven robots can be used as tendon-driven robots. In fact, these robots are able to use their cables to transfer motion created at one point to another, much like the muscle system in the human arm that controls finger movements. This type of design has been used in various robots such as artificial hands include the Dexter Robot Arm [7] and the Utah-MIT Dexterous Hand [8]. Another advantage of cable- driven robots is that they can move very quickly. This feature has been used in robots such as Falcon [9]. Also, the parts of the cable-driven robots are compact and can easily be reconfigured [9]. Extracting the problem formulation for a two-degree redundant suspended manipulator by using the relation between the maximum wrench at the end-effector and the tension values has been addressed in [10]. Also, the presented paper in [11] has been dedicated to design and implement a non-redundant cable-driven parallel robots, named T-Bot, which has large workspace with high acceleration capability. As a result, it can be considered for applications like the pick-and-place robots. The expressed results in [12] for a suspended and non-redundant cable-driven parallel robots has been revealed the vibration reduction based on an elasto-dynamic model-based control. Solving a reconfiguration problem for avoiding cable-cable collision in a cable driven parallel mechanisms has been addressed in [13]. The presented method has been derived a benefit of moving one cable’s attachment point on the frame to increase the shortest distance between them. Moreover, [14] has been presented an interactive control approach to prevent the unpredictable behavior of interference between cables. This aim has been done by generating a repulsive force by the controller.

Rehabilitation applications can also be considered for cable-driven robots. In this regard, in [15], using the robots set up with cable, the issue of rehabilitation of human hand nerves has been introduced and by applying cables with tensile forces, it has led to the creation of forces in the desired platform and appropriate results have been obtained. In fact, the designed robot, known as CAREX, uses three cuffs on the shoulder, elbow and forearm, and applies the embedded cables to control the robot with five degree of freedom (DoF). In the proposed example, a robotic arm is used in combination with a cable robot and the position of all three cuffs is assumed to be fixed. However, the method of connecting the seven cables is obtained by solving an optimization problem, and the cable forces are obtained by considering the optimization problem for the cable forces, which has a minimum and maximum value. The paper deals with static, dynamic and controller design issues. In the design of the controller, two control loops have been considered to track the desired forces of the cables by them and to control the human force in interaction with the environment. The presented paper in [16], which is an improvement of the CAREX robot, is known as CAREX-7, in which the design of a seven DoF robot for rehabilitation of the hands of the disabled is considered. In this robot, in addition to the transfer movement of the hand, which was discussed in [15], the rotational movement of the hand has also been considered. Therefore, four cuffs were used on the shoulders, elbows, wrists and hands, and since the robot was seven DoF, it used eight cables to control it. Also, the cuff designed for the shoulder can be adjusted on the human shoulder, and all of these items are located on a fixed chair. Although this robot can be placed in the category of open chain robots driven with cable, but it should be noted that in such cases as cable attachment points, how to route cables, systematic extraction of system Jacobian and comprehensiveness capability has been neglected and has only focused on providing solutions to a known problem. Designing robots similar to the above can also be used to rehabilitate leg muscles. In this regard, [17] has been addressed to the rehabilitation of leg muscles in disabled people and tries to use the proposed structure (C-ALEX) so that people can move in the defined path on the treadmill. In this structure, three cuffs that are located on the back, thighs and legs of the person are used and this structure has two DoF (bending / stretching of the hip and knee joints) by four cables. Also, in [18], an improved model of the [17] is presented, in which the two characteristics of weight and inertia of robots, that are used to rehabilitate leg muscles, are considered as factors that can affect the walking training of a disabled person. In other words, structures composed of rigid links and joints, due to the high weight and inertia of its mechanical components, will have effects on the length and height of the disabled person’s steps while training to walk on the treadmill. Although the presented solutions in the [15 –18] are very widely applicable, but there are following main questions:

If it is necessary to increase the DoF of the robot and use other cuffs, can the proposed methods be generalizable to them?

Has the routing of cables and their non-interference been considered?

Is the location of the cables attached to the cuffs engineered?

So, in [19], the so-called Cable Routing Matrix (CRM) is introduced which indicates the kinematic relation among the cable-driven and the serial robot. Also, using this type of description for cable-driven robots can systematically facilitate how to find the overall Jacobian of the system. In fact, using the CRM matrix, the overall Jacobin of the system can be easily obtained. Jacobian plays a very important role in finding the tensile forces applied by cables to robot links and it is very difficult to find it in systems with different number of links and routing of cables without using methods such as [19]. Additionally, extracting a Wrench Closure Workspace (WCW) for finding feasible wrenches at the end-effector is a main issue. Hence, in [20, 21] analysis of the WCW of the planar parallel cable-driven mechanisms is discussed. Actually in [20], the concept of the WCW is defined as the set of poses of the platform for which any wrench can be generated at the platform by tightening the cables. On the other hand, generating positive tensile forces in the cables and optimizing these forces to track an appropriate trajectory by the end-effector can also be considered as an important issue. Moreover, in the [22] inverse dynamics of multilink cable-driven manipulators with the consideration of joint interaction forces and moments is investigated and to optimize the cables forces a convex optimization problem is used. Also, there are various ways for optimizing cables forces such as using finite-time state-dependent Riccati equation which is used, for instance, in [23] for time- varying non-affine systems.

On the other hand, since it is difficult to obtain the unknown functions and parameters of the dynamic model of robot, it is necessary to use, ’universal model-free solutions.’ In this regard, fuzzy systems and Neural Networks (NNs) techniques are widely applied as general approximators. These systems estimate the unknown functions and parameters of the dynamic model based on mimicking linguistic and reasoning functions and biological neuronal structures of interconnected nodes, respectively. So, it is required to use dynamic modeling and controller designing based on model-free solutions. In this way, the published papers in [24 –29] have been focused on adaptive NN controller and extracting adaptation laws based on tracking error dynamics. Also, [30] has been dedicated to design various algorithms on robots based on classic and intelligent approaches. Although, the solution of using NNs in these articles has been applied to estimate the dynamic model of the robot, but so far the use of this solution has not been used in cable-driven robots.

Upon obtaining the dynamic model of the cable-driven robots, the next step consists in designing an appropriate controller to achieve an optimal performance. So, [5] investigates the control of a cable-driven parallel robot for gait rehabilitation in which uses computed torque method for reducing tracking error based on a PD controller. In [31], dynamic modeling and controller designing for a 6-dof cable-driven parallel robot has been proposed and in this way feedback linearization method has been used. Cable tension modeling based on rigid body static equilibrium and using an adaptive controller based on model identification and reinforcement learning methods has been presented in [32]. Using a PD controller based on Lyapunov approach to control a cable-driven robot has been presented in [33]. In the latter study, for making tensile properties in the cables, internal forces are added to the designed controller. In [34], a semi-adaptive cable-driven rehabilitation device with a tilting working plane is suggested. In [35], a new continuous controller based on fractional-order sliding mode is presented for cable-driven robots where the suggested control platform is not depending on model of the system and does not require system dynamics which benefit from time delay estimation, which is very suitable and easy to use in practical applications. Moreover, in [36], kinematics, statics modeling, determination of the statics workspace and designing a controller based on feedback linearization approach has been discussed.

One of the approaches which have been recently propounded in the literature is the intelligent-based control approach which has stimulated the interest of many robotic researchers. In this way, [37] has been proposed a structure with two controlling loops in which reinforcement learning and NN have been used as key tools. In fact, in the latter paper dynamics modeling and unknown parameters of the system have been estimated by a NN. However, it should be noted that the controlling solution is applied for series robots with rigid links and its applicability for rehabilitation issues is associated with problems. Another approach which is applicable for controlling the system can be found in [38]. In this paper, using combination of a classic and intelligent controller and based on fuzzy systems, a model reference controller has been presented. Also, the adaptation laws of the gains of the controller are obtained based on Lyapunov approach. All the mentioned solutions have been used for serial robots and due to important challenges such as safety in rehabilitation issues, the use of these solutions needs to introduce new concepts. One of the most important issues which may have useful applications in the field of medical engineering is the interaction between serial, or parallel robots with cable robot. Among the applications, one can be regarded as using this idea in the rehabilitation of fingers in physiotherapy centers [39]. Indeed, in [39] an iterative learning controller for a cable-driven hand rehabilitation robot is proposed. Also in [40], an analysis of a model-based identification method of a cable-driven wearable device for arm rehabilitation is presented.

Although in this paper, the kinematics formulation is taken from [19], estimating the complex dynamics of problem as well as the controller design with important features like stability and extracting adaptation laws for problem variants are serious challenges that completely is addressed. In other words, the most important contributions of this paper are the use of neural networks with the proposed structure to estimate the complex dynamic model of the problem and its use to design a stable and appropriate adaptive controller. Also, the use of robust term in the controller structure to eliminate disturbances and adaptive adjustment of robust parameters is another innovation which can be implemented for the proposed Cable-Driven Serial Robot (CDSR) structure. In general, the main advantages of the presented approach can be expressed as follows: i) designing an adaptive controller without need to know the dynamic model of the system ii) applicable to rehabilitation problems, iii) using the appropriate properties of the cable and serial robots simultaneously, iv) decreasing the effect of the disturbance in steady state, v) introducing a novel approach to obtain the overall Jacobian of the system and vi) satisfying asymptotic stability. Accordingly, as the first step, the reasoning of obtaining the Jacobian matrix of the serial-cable mechanism is discussed and it attempts to express novel comprehensive solutions for this type of robots. Then, the dynamic model estimating of the problem based on NN is expressed. In the next step, an adaptive controller based on NN approach is applied to track a defined appropriate trajectory. In the related design, it is attempted to take into account unknown dynamic properties of the system, such as friction, in the disturbance term in system dynamic equations and adaptation laws of the unknown parameters are obtained based on an appropriate Lyapunov function. Hence, by presenting a theorem, asymptotic stability of the whole of the system is guaranteed. The simulation outcomes verify the efficacy of the suggested method.

This paper is organized as follows. Section 2 is devoted to express four parts: dynamic model estimating to remove the complexities of relevant mathematical calculations, designing controller based on NN approach which guarantees the stability of the whole of the CDSR, extracting adaptation laws of the unknown parameters which are done based on Lyapunov function and singularity problem for obtaining tensile forces of the cables that are led to generate positive forces in cables. Section 3 provides simulation results. Finally, the paper in Section 4 concludes by providing the summary of the results and some hints and remarks as ongoing works.

2 Problem statement

One of the researches and innovations that are being done on robots is the use of different methods in launching series-parallel manipulators. One of these methods is cable- driven series- parallel robots. These structures use cables with a mass much less than the robot links, which only act as tension, to move the robot, and as a result can produce relatively high speed and acceleration with relatively small motors. Among the features of this method are achieving a larger workspace, higher load carrying capacity and reliability for performing difficult and complex tasks. Also, since the general method described in this paper also applies to robots with a higher number of links, the importance of using light cables instead of rigid links is much higher. Cable-driven devices use lightweight cables to transmit motion and forces so that the actuators can be mounted away from the joints, which help to achieve low-weight, low-inertia and cost-effectiveness.

Prior to design the appropriate controller for the problem, it is needed to address the dynamic model issues. However, since it is difficult to obtain the unknown functions and parameters of the dynamic model of robot, it is necessary to use, ’universal model-free solutions.’ So, fuzzy systems and neural networks (NNs) techniques which are identified as general approximators can be considered. These systems estimate the unknown functions and parameters of the dynamic model based on mimicking linguistic and reasoning functions and biological neuronal structures of interconnected nodes, respectively. So, the following sections of the paper are dedicated to dynamic modeling and controller designing based on model-free solutions, and in this way, the NNs are considered. In this regard, the adaptation laws of the unknown parameters of the NN controller are obtained based on the presentation of an appropriate Lyapunov function. However, to apply the designed controller, it is necessary to produce positive tension forces in the cables of CDSR. Therefore, it is required to achieve this goal by solving an optimization problem.

2.1 Problem dynamic model estimation

In general, the problem of restraining an n degrees-of-freedom (DoF) system using m cables, falls into three classes as follows [10]:

if m < n + 1 then the system is incompletely restrained.

if m=n + 1 then the system is completely restrained.

if m > n + 1 then the system is redundantly restrained.

In this paper, the redundant state is considered and accordingly the problem consists of a 2-DoF robot (n = 2) by four cables (m = 4). The schematic of the considered scheme is shown in Fig. 1. As depicted schematically in Fig. 1, the 2-DoF robot is driven using four cables with tensile forces. In other words, in the shown structure, four motors are used for expansion and contraction of the cables. In this way, the links of the serial robot are launched. So, in this structure, no motors are considered for the serial robot and they are considered as passive. The start-up and production of torque of these links is done only by the tensile forces of the cables. In fact, from the inherent properties of the cables, it is assumed that cable forces are in tensile mode and have positive values (f_ij ≥ 0, i = 1, . . . , 4 and j = 1). Also, the cable space of the system is expressed by the cable length vector l = [l₁₁ . . . l₄₁] ^T ∈ R⁴ and cable force vector f = [f₁₁ . . . f₄₁] ^T ∈ R⁴. Additionally, the attachment point of segment j (the section of the cable which is between two bodies) of cable i on link k are denoted by A_ijk. The attachment points can be determined by the designer or by solving an optimization problem which the latter problem is beyond the scope of this paper, and for the sake of generality the foregoing parameters are defined for an arbitrary design. The main reason for considering the robot with 2-DoF is simulating arm or foot limb as a two links series planar robot which has two degrees of freedom. Although the mentioned approaches are currently used for the robot with 2-DoF, but in future works, it is intended to generalize the proposed method to the structures of 3-DoF and 6-DoF planar and finally spatial robots.

Fig. 1

Restrained 2-DoF robot by four cables.

In general, based on [41,42, 41,42] the dynamic equations of a CDSR can be expressed as the following: $M (q) \ddot{q} + N (q, \dot{q}) + τ_{d} = - J^{T} (q) f$ (1) in which, $N (q, \dot{q}) = C (q, \dot{q}) + G (q)$ (2) and M,C and G represent the mass-inertia matrix, centrifugal and Coriolis vector and gravitational vector, respectively. Moreover, q = [θ₁, θ₂] ^T is generalized coordinate vector (given that joints are considered as revolute) and τ _d is added bounded disturbance to process. Also, f is denoted to tensions vector of cable-driven robot and J (q) represents the Jacobian of the system as the following [19]: $\dot{l} = J \dot{q}$ (3) However, in order to find the Jacobian matrix of the system, it is necessary to use a new variable which is called task space x defined as follows: $\dot{x} = [^{1} {\dot{r}}_{O o_{1}}^{T}^{1} ω^{T} {_{1}}^{2} {\dot{r}}_{O o_{2}}^{T}^{2} ω^{T}_{2}]$ (4) in which, ^k ω ^T _k and $^{k} ω^{T}_{k}$ refer to the absolute linear and angular velocity of point o_k and body k, respectively. Also, k refers to k-th frame, F_k, which the above computations is done. Finally, the following equations can be expressed: $\dot{l} = V \dot{x}, \dot{x} = W \dot{q} \Rightarrow \dot{l} = (V . W) \dot{q}$ (5) Thus, the corresponding Jacobian is the product of the following matrices: $V = [\begin{matrix} v_{1 x_{1}} & v_{1 θ_{1}} & v_{1 x_{2}} & v_{1 θ_{2}} \\ v_{2 x_{1}} & v_{2 θ_{1}} & v_{2 x_{2}} & v_{2 θ_{2}} \\ v_{3 x_{1}} & v_{3 θ_{1}} & v_{3 x_{2}} & v_{3 θ_{2}} \\ v_{4 x_{1}} & v_{4 θ_{1}} & v_{4 x_{2}} & v_{4 θ_{2}} \end{matrix}] \in R^{4 \times 12}$ (6) $W = [\begin{matrix} w_{1 x_{1}} & 0 \\ w_{1 θ_{1}} & 0 \\ w_{1 x_{2}} & w_{2 x_{2}} \\ w_{1 θ_{1}} & w_{1 θ_{2}} \end{matrix}] \in R^{12 \times 2}$ (7) where, $\begin{matrix} v_{{ix}_{k}} = (\sum_{j = 1}^{s} [c_{ij (k + 1)}^{k} {\hat{l}}_{ij}])^{T}, \\ v_{i θ_{k}} = (\sum_{j = 1}^{s} [c_{ij (k + 1)}^{k} r_{o_{k} A_{ijk}} \times^{k} {\hat{l}}_{ij}])^{T} \end{matrix}$ (8) According to Eqs. (6-7), v_{ax_k} and v_{aθ_k} represent the relationship between the task space velocities of link k and the time derivative of the length of cable i and also, w_{ax_k} and w_{aθ_k} indicate the linear relationship between ${\dot{q}}_{a}$ and the relative linear and angular velocities, respectively, between links k and k - 1.

As can be seen from Eq. (8), to extract the components of the V and W matrices, it is necessary to define the so-called Cable Routing Matrix(CRM). CRM is a matrix which is applied to represent of the cable routing in a multi link cable-driven system and has c_ij(k+1) components with values of -1, 0, 1 and the following properties [19]:

if c_ij(k+1) = -1 then the segment j of cable i begins from link k,

if c_ij(k+1) = +1 then the segment j of cable i ends at link k,

ifc_ij(k+1) = 0 then the segment j of cable i is not connected to link k.

From the above reasoning, referring to Fig. 1, the following CRM can be obtained:

C_{1} = C_{2} = [\begin{matrix} - 1 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}], C_{3} = C_{4} = [\begin{matrix} - 1 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}]

(9) in which the columns of each matrix represent the number of rigid bodies in the robot (links + base) and the rows represent the number of relevant robot segments (the maximum of which is equal to the number of relevant links). Also,

{\hat{l}}_{ij}

is unit vector of l_ij which is defined as follows:

l_{ij} = - r_{{OA}_{ija}} + r_{{OA}_{ijb}}

(10) in which r_{OA_ija} and r_{OA_ijb} are the attachment of each cable. So, the lengths of the considered cables in Fig. 1 is obtained as follows:

length of cable 1 $\begin{matrix} l_{11} & = - r_{{OA}_{110}} + r_{{OA}_{111}} = - [\begin{matrix} x_{1} \\ y_{1} \\ 0 \end{matrix}] + Q_{1} [\begin{matrix} x_{1}' \\ y_{1}' \\ 0 \end{matrix}] + a_{1} \\ = [\begin{matrix} x_{1}' cos θ_{1} - y_{1}' sin θ_{1} - x_{1} + l_{1} cos θ_{1} \\ x_{1}' sin θ_{1} + y_{1}' cos θ_{1} - y_{1} + l_{1} sin θ_{1} \\ 0 \end{matrix}] \\ \Rightarrow {\hat{l}}_{11} = \frac{1}{| l_{11} |} . l_{11} \end{matrix}$ (11)

length of cable 2 $\begin{matrix} l_{21} & = - r_{{OA}_{210}} + r_{{OA}_{211}} = - [\begin{matrix} x_{2} \\ y_{2} \\ 0 \end{matrix}] + Q_{1} [\begin{matrix} x_{2}^{'} \\ y_{2}^{'} \\ 0 \end{matrix}] + a_{1} \\ = [\begin{matrix} x_{2}^{'} cos θ_{1} - y_{2}^{'} sin θ_{1} - x_{2} + l_{1} cos θ_{1} \\ x_{2}^{'} sin θ_{1} + y_{2}^{'} cos θ_{1} - y_{2} + l_{1} sin θ_{1} \\ 0 \end{matrix}] \\ \Rightarrow {\hat{l}}_{21} = \frac{1}{| l_{21} |} . l_{21} \end{matrix}$ (12)

length of cable 3 $\begin{matrix} l_{31} & = - r_{{OA}_{310}} + r_{{OA}_{311}} = - [\begin{matrix} x_{3} \\ y_{3} \\ 0 \end{matrix}] + Q_{1} Q_{2} [\begin{matrix} x_{3}^{'} \\ y_{3}^{'} \\ 0 \end{matrix}] + \\ a_{1} + Q_{1} a_{2} = [\begin{matrix} x_{3}^{'} cos (θ_{1} + θ_{2}) - y_{3}^{'} sin (θ_{1} + θ_{2}) . . . \\ - x_{3} + l_{1} cos θ_{1} + l_{2} cos (θ_{1} + θ_{2}) \\ x_{3}^{'} sin (θ_{1} + θ_{2}) + y_{3}^{'} cos (θ_{1} + θ_{2}) . . . \\ - y_{3} + l_{1} sin θ_{1} + l_{2} sin (θ_{1} + θ_{2}) \\ 0 \end{matrix}] \\ \Rightarrow {\hat{l}}_{31} & = \frac{1}{| l_{31} |} . l_{31} \end{matrix}$ (13)

length of cable 4 $\begin{matrix} l_{41} & = - r_{{OA}_{410}} + r_{{OA}_{411}} = - [\begin{matrix} x_{4} \\ y_{4} \\ 0 \end{matrix}] + Q_{1} Q_{2} [\begin{matrix} x_{4}^{'} \\ y_{4}^{'} \\ 0 \end{matrix}] + \\ a_{1} + Q_{1} a_{2} = [\begin{matrix} x_{4}^{'} cos (θ_{1} + θ_{2}) - y_{4}^{'} sin (θ_{1} + θ_{2}) . . . \\ - x_{4} + l_{1} cos θ_{1} + l_{2} cos (θ_{1} + θ_{2}) \\ x_{4}^{'} sin (θ_{1} + θ_{2}) + y_{4}^{'} cos (θ_{1} + θ_{2}) . . . \\ - y_{4} + l_{1} sin θ_{1} + l_{2} sin (θ_{1} + θ_{2}) \\ 0 \end{matrix}] \\ \Rightarrow {\hat{l}}_{41} & = \frac{1}{| l_{41} |} . l_{41} \end{matrix}$ (14)

in which | . | is referred to the magnitude of its vector argument. It should be noted that two types of coordinates are considered in the above equations, where the coordinates in the base frame are [x_i, y_i, 0] ^T (since the robot is in a planar mode, the third element is considered zero) and the cable end coordinates on another frame are considered as

[x_{i}^{'}, y_{i}^{'}, 0]^{T}

, but it is needed to overwrite all the attachment points coordinates with respect to the base frame and finally, it is rewritten in the frame F_k. Therefore, Q₁, Q₂, a₁, a₂ and p are considered as follows [33]:

\begin{matrix} [a_{1}]_{1} & = [\begin{matrix} l_{1} cos θ_{1} \\ l_{1} sin θ_{1} \\ 0 \end{matrix}], [a_{2}]_{2} = [\begin{matrix} l_{2} cos θ_{2} \\ l_{2} sin θ_{2} \\ 0 \end{matrix}], \\ Q_{1} & = [\begin{matrix} cos θ_{1} & - sin θ_{1} & 0 \\ sin θ_{1} & cos θ_{1} & 0 \\ 0 & 0 & 1 \end{matrix}], Q_{2} = [\begin{matrix} cos θ_{2} & - sin θ_{2} & 0 \\ sin θ_{2} & cos θ_{2} & 0 \\ 0 & 0 & 1 \end{matrix}] \end{matrix}

\begin{matrix} [p]_{1} & = [\begin{matrix} X \\ Y \\ 0 \end{matrix}] = [a_{1}]_{1} + Q_{1} [a_{2}]_{2} \end{matrix}

(15) Also, p is denoted to end-effector coordinate of CDSR. Reaching this step, in order to find the overall Jacobian matrix of the system with the mentioned structure, one should obtain the W matrix. However, in this paper, a general form is expressed for the latter matrix. With some reflection, it can be seen that this matrix is the Jacobian matrix of the serial robots discussed in [43]. Therefore, with some modifications, the W matrix can be matched to the structure described in Eq. (7). Without loss of generality, to simplify the design, the center of the frames can be considered on the joints, and at each time of designing the g intervals are determined according to the end-effector. In fact, in this structure, a one number of joints is subtracted each time and consider the end-effector on the last joint and find the resulting W matrix by putting together the Jacobian matrices. In general, in this method, and given the structure expressed for the CDSR, J′ is defined as follows:

\begin{matrix} J^{'} = [\begin{matrix} J_{1}^{'} \\ J_{2}^{'} \end{matrix}] \end{matrix}

(16) Therefore, based on the outlined concepts and given the type of robot (2-DoF planar robot), finding the J′ matrix is done in two steps:

The first step In the first step, it is assumed that both revolute joints are presented in the design. The goal consists in finding $J_{1}^{'}$ where from the results reveled in [31], one has: $\begin{matrix} J_{1}^{'} = [\begin{matrix} γ_{1} & γ_{2} \\ γ_{1} \times ζ_{1} & γ_{2} \times ζ_{2} \end{matrix}] \end{matrix}$ (17) in which, γ _i (i=1,2) is a unit vector that is referring to axis of the rotation of the joints.

Using Eq. (15), ζ ₁ and ζ ₂ are obtained as follows: $\begin{matrix} ζ_{1} & = \sum_{i = 1}^{2} a_{i} = a_{1} + a_{2} \\ \Rightarrow [ζ_{1}]_{1} = [a_{1}]_{1} + [a_{2}]_{1} \\ = [\begin{matrix} l_{1} cos θ_{1} + l_{2} cos (θ_{1} + θ_{2}) \\ l_{1} sin θ_{1} + l_{2} sin (θ_{1} + θ_{2}) \\ 0 \end{matrix}] \\ ζ_{2} & = \sum_{i = 2}^{2} a_{i} = a_{2} \Rightarrow [ζ_{2}]_{1} = [a_{2}]_{1} = Q_{1} [a_{2}]_{2} \\ = [\begin{matrix} l_{2} cos (θ_{1} + θ_{2}) \\ l_{2} sin (θ_{1} + θ_{2}) \\ 0 \end{matrix}] \end{matrix}$ (18) Also $\begin{matrix} [γ_{1}]_{1} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}], [γ_{2}]_{1} = Q_{1} [γ_{2}]_{2} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] \end{matrix}$ (19) From Eqs. (17-19), it can be deduced that: $\begin{matrix} J_{1}^{'} = [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 1 & 1 \\ - l_{1} sin θ_{1} - l_{2} sin (θ_{1} + θ_{2}) & - l_{2} sin (θ_{1} + θ_{2}) \\ l_{1} cos θ_{1} + l_{2} cos (θ_{1} + θ_{2}) & l_{2} cos (θ_{1} + θ_{2}) \\ 0 & 0 \end{matrix}] \end{matrix}$ (20)

The second step In the next step, in the design of the robot the first revolute joint is only taken into account while the second one is removed and the aim consists in finding $J_{2}^{'}$ . These yields to: $\begin{matrix} J_{2}^{'} = [\begin{matrix} γ_{1} \\ γ_{1} \times ζ_{1} \end{matrix}] \end{matrix}$ (21) where $\begin{matrix} [ζ_{1}]_{1} = [a_{1}]_{1} = [\begin{matrix} l_{1} cos θ_{1} \\ l_{1} sin θ_{1} \\ 0 \end{matrix}], [γ_{1}]_{1} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] \end{matrix}$ (22) Hence, using Eqs. (21) and (22), one has: $\begin{matrix} J_{2}^{'} = [\begin{matrix} 0 \\ 0 \\ 1 \\ - l_{1} sin θ_{1} \\ l_{1} cos θ_{1} \\ 0 \end{matrix}] \end{matrix}$ (23) Finally, referring to Eqs. (15), (20) and (23), it follows that: $\begin{matrix} W = [\begin{matrix} - l_{1} sin θ_{1} & 0 \\ l_{1} cos θ_{1} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 1 & 0 \\ - l_{1} sin θ_{1} - l_{2} sin (θ_{1} + θ_{2}) & - l_{2} sin (θ_{1} + θ_{2}) \\ l_{1} cos θ_{1} + l_{2} cos (θ_{1} + θ_{2}) & l_{2} cos (θ_{1} + θ_{2}) \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 1 & 1 \end{matrix}] \end{matrix}$ (24)

As a result, after obtaining the V and W matrices, the Jcobian of the system can be determined.

By assuming the CDSR dynamics in Eq. (1) as unknown and supposing a desired trajectory for generalized coordinate vector, q_d, the following tracking error is generated: $\begin{matrix} e (t) = q_{d} (t) - q (t) \end{matrix}$ (25) Now, the sliding mode error is defined as follows: $\begin{matrix} r (t) = \dot{e} (t) + Λ e (t) \end{matrix}$ (26) in which, Λ is a symmetric positive definite matrix, usually selected diagonal. The goal is finding the forces of the cables in a tensile mode such that r (t) is bounded. As a result, e (t) is bounded, too. So, $∥ e (t) ∥ \leq \frac{∥ r (t) ∥}{λ_{min} (Λ)}$ and $∥ \dot{e} (t) ∥ \leq ∥ r (t) ∥$ where λ_min (Λ) is minimum singular value of Λ.

By using Eqs. (1-2) and Eq. (25) and also differentiating Eq. (26), the filteres tracking error dynamics is written as follows with respect to the time: $\begin{matrix} M (q) \dot{r} = - N (q, \dot{q}) r + J^{T} (q) f + h (z) + τ_{d} \end{matrix}$ (27) in which $\begin{matrix} h (z) = M (q) ({\ddot{q}}_{d} + Λ \dot{e}) + N (q, \dot{q}) ({\dot{q}}_{d} + Λ e) \end{matrix}$ (28) represents the nonlinear CDSR function which includes the unknown parameters of the system. It should be noted that, for simplification in the above equations, it is avoided writing the ^′t′ parameter.

In order to prove the stability of the design and the asymptotically convergence of the sliding mode error to zero, M, C and G matrices are required. Since finding these matrices using methods such as Newton-Euler and Lagrange suffers from high computational complexity, so the intention is to provide a comprehensive solution which can be used in other CDSRs with higher links and DoF while using it for the mentioned CDSR. Referring to Eq. (28), since the nonlinear CDSR function, h (z) , is unknown, a NN can be used to approximate it. In fact, given the general approximator property of the NNs, it can be expressed there always exists a NN which can applied to estimate h (z) with an accuracy ε_N. In this way, the input vector of the NN, z, can be selected as follows: $\begin{matrix} z = [e^{T} {\dot{e}}^{T} q_{d} {\dot{q}}_{d} {\ddot{q}}_{d}]^{T} \end{matrix}$ (29) It should be noted that the mentioned parameters in Eq. (29) are measurable. Also, it is assumed that the desired trajectory is bounded so that: $\begin{matrix} ∥ q_{d} ∥ \leq q_{B 1}, ∥ {\dot{q}}_{d} ∥ \leq q_{B 2}, ∥ {\ddot{q}}_{d} ∥ \leq q_{B 3} \end{matrix}$ (30) where, q_B1, q_B2 and q_B3 are the upper bounds of the desired trajectory and respective differentials. In this paper, a 2-layer NN is used to fulfill the mentioned goal which has the following general form: $\begin{matrix} \hat{h} (z) = {\hat{B}}^{T} σ ({\hat{A}}^{T} z) \end{matrix}$ (31) where ${\hat{A}}^{T}$ and ${\hat{B}}^{T}$ are the first and second layer weight with dimensions of S₁ × 10 and S₂ × S₁, respectively. Also, S₁ and S₂ are the number of neurons in first and second layer, respectively. In addition, σ (.) represents the activation function in the first layer which is considered as the hyperbolic tangent form: $\begin{matrix} σ ({\hat{A}}^{T} z) = \frac{e^{({\hat{A}}^{T} z)} - e^{(- {\hat{A}}^{T} z)}}{e^{({\hat{A}}^{T} z)} + e^{(- {\hat{A}}^{T} z)}} \end{matrix}$ (32) The structure of the NN estimator is depicted in Fig. 2.

Fig. 2

The NN estimator for CDSR.

2.2 Designing controller based on NNs

Once the dynamic model of the system is properly estimated, it is necessary to address the control objective. Here, the purpose of the control is to properly design the tension forces of the cables so as to lead to the desired trajectory being tracked by the CDSR end-effector. Hence with respect to Eq. (27), the goal consists in finding f such that leads to the stability of the error dynamics. Then, the following control input is considered: $\begin{matrix} - J^{T} (q) f = \hat{h} (z) + K_{v} r - β_{robust} \end{matrix}$ (33) in which, K_v is positive symmetric definite matrix that is generally selected diagonal and β_robust is a robust signal which is applied to overcome the unknown disturbance. By replacing Eq. (33) into Eq. (27), the following is obtained: $\begin{matrix} M (q) \dot{r} & = - N (q, \dot{q}) r - \hat{h} (z) - K_{v} r + β_{robust} \\ + h (z) + τ_{d} \\ \Rightarrow M (q) \dot{r} & = - (N (q, \dot{q}) + K_{v}) r + (h (z) - \hat{h} (z)) \\ + β_{robust} + τ_{d} \end{matrix}$ (34) Finally, by defining $\tilde{h} (z) = h (z) - \hat{h} (z)$ as the estimation function error, the above equation is simplified to the following: $\begin{matrix} M (q) \dot{r} & = - (N (q, \dot{q}) + K_{v}) r + \tilde{h} (z) + β_{robust} + τ_{d} \end{matrix}$ (35) Given that M (q) is bounded by m₁ and m₂ positive constants such that m₁I < M (q) < m₂I and also, the matrix M–2 $N (q, \dot{q})$ is skew-symmetric with a bounded norm for some function n_b (q) as n_b (q)∥ q ∥, selecting K_v plays an important role in the stability of closed loop error dynamics. In other words, in order to reduce the tracking error as small as possible, K_v should be selected large enough.

According to Eq. (35), it can be inferred that if the estimation error of the unknown function tends to zero ( $\tilde{h} \to 0$ ) and the robust signal (β_robust) is designed to overcome the unknown disturbance then, in steady-state the sliding mode error will tend to zero (r → 0) and as a result, the tracking error will asymptotically converge to zero (e → 0).

2.3 Adaptation laws of the controller parameters based on the lyapunov approach

As stated in the previous section, in order to stabilize the error dynamics, the parameters of the problem needs to be tuned so that the estimation function error is tend to zero, and also overcomes the unknown disturbance. So, by supposing the ideal weights of NN as unknown and constant, one has: $\begin{matrix} S = [\begin{matrix} B & 0 \\ 0 & A \end{matrix}] \end{matrix}$ (36) in which, it is assumed that S is bounded. In other words, ∥S ∥ _F ≤ S_Bounded. Also, the weight estimation errors can be defined as follows: $\begin{matrix} \tilde{B} = B - \hat{B}, \tilde{A} = A - \hat{A} \end{matrix}$ (37) As a result, with respect to Eq. (31), $\tilde{h} (z)$ can be simplified as follows: $\begin{matrix} \tilde{h} (z) & = h (z) - \hat{h} (z) = B^{T} σ (A^{T} z) - {\hat{B}}^{T} σ ({\hat{A}}^{T} z) + ε_{N} \\ = {\tilde{B}}^{T} σ ({\hat{A}}^{T} z) + B^{T} [σ (A^{T} z) - σ ({\hat{A}}^{T} z)] + ε_{N} \end{matrix}$ (38) Defining the hidden-layer output error of NN as the following: $\begin{matrix} \tilde{σ} ({\tilde{A}}^{T} z) = σ (A^{T} z) - σ ({\hat{A}}^{T} z) \end{matrix}$ (39) one has: $\begin{matrix} \tilde{h} (z) = {\tilde{B}}^{T} σ ({\hat{A}}^{T} z) + B^{T} [\tilde{σ} ({\tilde{A}}^{T} z)] + ε_{N} \end{matrix}$ (40) Now, by writing the Taylor series expansion for σ ( A ^Tz) as follows: $\begin{matrix} σ (A^{T} z) = \hat{σ} + {\hat{σ}}^{'} {\tilde{A}}^{T} z + O (({\tilde{A}}^{T} z)^{2}) \\ \Rightarrow \tilde{σ} ({\tilde{A}}^{T} z) = {\hat{σ}}^{'} {\tilde{A}}^{T} z + O (({\tilde{A}}^{T} z)^{2}) \end{matrix}$ (41) in which, $\hat{σ} = σ ({\hat{A}}^{T} z)$ and ${\hat{σ}}^{'} = \frac{d σ ({\hat{A}}^{T} z)}{d ({\hat{A}}^{T} z)}$ . Also, $O (({\tilde{A}}^{T} z)^{2})$ expresses the high order term of Taylor series. By substituting Eq. (41) into Eq. (40), one has: $\begin{matrix} \tilde{h} (z) & = {\tilde{B}}^{T} \hat{σ} + B^{T} [{\hat{σ}}^{'} {\tilde{A}}^{T} z + O (({\tilde{A}}^{T} z)^{2}) + ε_{N} \\ = {\tilde{B}}^{T} [\hat{σ} - {\hat{σ}}^{'} {\hat{A}}^{T} z] + {\hat{B}}^{T} {\hat{σ}}^{'} {\tilde{A}}^{T} z + {\tilde{B}}^{T} {\hat{σ}}^{'} A^{T} z \\ + B^{T} O (({\tilde{A}}^{T} z)^{2}) + ε_{N} \end{matrix}$ (42) Hence, using Eqs. (35) and (42), the sliding mode error dynamics are converted as follow: $\begin{matrix} M (q) \dot{r} & = - (N (q, \dot{q}) + K_{v}) r + {\tilde{B}}^{T} [\hat{σ} - {\hat{σ}}^{'} {\hat{A}}^{T} z] \\ + {\hat{B}}^{T} {\hat{σ}}^{'} {\tilde{A}}^{T} z + d (t) + β_{robust} \end{matrix}$ (43) where d (t) represents the disturbance term of the problem as follows: $\begin{matrix} d (t) = {\tilde{B}}^{T} {\hat{σ}}^{'} A^{T} z + B^{T} O (({\tilde{A}}^{T} z)^{2}) + ε_{N} + τ_{d} \end{matrix}$ (44) In order to extract adaptation laws of the parameters, the following Lyapunov function is considered: $\begin{matrix} L = & \frac{1}{2} r^{T} M (q) r + \frac{1}{2} trace ({\tilde{B}}^{T} F^{- 1} {\tilde{B}}^{T}) \\ + \frac{1}{2} trace ({\tilde{A}}^{T} G^{- 1} {\tilde{A}}^{T}) \end{matrix}$ (45) in which, F and G are constant and symmetric positive definite matrices. By differentiating Eq. (45), the following equation is obtained with respect to the time: $\begin{matrix} \dot{L} = & r^{T} M (q) \dot{r} + \frac{1}{2} r^{T} \dot{M} (q) r + trace ({\tilde{B}}^{T} F^{- 1} \dot{\tilde{B}}) \\ + trace ({\tilde{A}}^{T} G^{- 1} \dot{\tilde{A}}) \end{matrix}$ (46) Now, substituting Eq. (43) into Eq. (46) yields: $\begin{matrix} \dot{L} & = r^{T} (- (N (q, \dot{q}) + K_{v}) r + {\tilde{B}}^{T} [\hat{σ} - {\hat{σ}}^{'} {\hat{A}}^{T} z] \\ + {\hat{B}}^{T} {\hat{σ}}^{'} {\tilde{A}}^{T} z + d (t) + β_{robust}) + \frac{1}{2} r^{T} \dot{M} (q) r \\ + trace ({\tilde{B}}^{T} F^{- 1} \dot{\tilde{B}}) + trace ({\tilde{A}}^{T} G^{- 1} \dot{\tilde{A}}) \end{matrix}$ (47) Simplifying the above equation leads to the following: $\begin{matrix} \dot{L} & = - r^{T} K_{v} r + \frac{1}{2} r^{T} (\dot{M} (q) - 2 (N (q, \dot{q})) r \\ + trace ({\tilde{B}}^{T} [F^{- 1} \dot{\tilde{B}} + \hat{σ} r^{T} - {\hat{σ}}^{'} {\hat{A}}^{T} z r^{T}] \\ + trace ({\tilde{A}}^{T} [G^{- 1} \dot{\tilde{A}} + z r^{T} {\hat{B}}^{T} {\hat{σ}}^{'}]) + d (t) + β_{robust} \end{matrix}$ (48) Consequently, in order to asymptotically stability, the $\dot{L}$ should be negative definite. In this way, the updating rules of the parameters are obtained based on the following theorem.

Theorem 1. Consider the CDSR dynamic equations from Eqs. (1-2) and extract the Jacobian matrix from Eqs. (3), (6-8) and (24). Let the control input be chosen by Eqs. (31) and (33) and suppose the desired trajectory and weights of the NN are bounded. Considering the adaptation laws of parameters based on Eqs. (49-51), it is guaranteed that all the signals are bounded. Also, in steady-state the sliding mode error will tend to zero (r → 0) and as a result, the tracking error will asymptotically converge to zero (e → 0). $\begin{matrix} \dot{\hat{B}} = F \hat{σ} r^{T} - F {\hat{σ}}^{'} {\hat{A}}^{T} z r^{T} - μ F ∥ r ∥ \hat{B} \end{matrix}$ (49) $\begin{matrix} \dot{\hat{A}} = G z ({\hat{σ}}^{' T} \hat{B} r)^{T} - μ G ∥ r ∥ \hat{A} \end{matrix}$ (50) $\begin{matrix} β_{robust} (t) = - α (∥ \hat{S} ∥_{F} + S_{Bounded}) r \end{matrix}$ (51) where, μ is a small design parameter and α represents the robust cofficient and is selected as positive.

It should be noted that, to the best of the authors knowledge, the use of control structures and modeling presented in this paper is the first time that has been proposed for the CDSR structure. Therefore, the main contributions of the paper can be summarized as follow:

Providing dynamic model estimation structures and controller design based on neural networks for the CDSR, which in addition to being able to be used in serial robots with a higher number of links, has sufficient accuracy and also proof of stability can be easily implemented in it.

Reducing the effects of external disturbances on the CDSR by using robust terms with adaptive rules for its free parameters.

2.4 Singularity problem and pseudoinverses

As aforementioned, after the design parameters are adaptively obtained, the cable tension forces applied to the links can be obtained by Eq. (8). But due to non-square form of the Jacobian matrix (J (q) ∈ R^2×4), it is not invertible, and pseudo inverse calculation methods should be used. Moreover, it should be noted that the cable forces should be positive. For this reason, multiple approaches have been presented for solving Eq. (33) which falls into three general approaches [5 , 44]:

Jacobian-based method In this approach, the obtained solution minimizes a prescribed suitable (possibly weighted) norm.

Null-space method In this method, a term is added to the previous solution such that pertains to the null-space of J (q).

Task augmentation method In this method, redundancy is reduced/eliminated by considering auxiliary tasks.

Among these three methods, the most widely used is the method of using the null-space of the Jacobian matrix of the robot, which has been used in various papers such as [5 , 35–40].

So, in this paper the null-space method is adopted for solving Eq. (33). Hence, the solution of Eq. (33) is obtained as follows: $\begin{matrix} f & = - [J_{1} (q) (\hat{h} (z) + K_{v} r - β_{robust}) \\ + (I - J_{1} (q) J^{T} (q)) f_{0}] \end{matrix}$ (52) where $\begin{matrix} J_{1} & = (J (J^{T} J)^{- 1}) \end{matrix}$ (53) in which, I is reffered to identity matrix. As it can be observed from Eq. (52), the solution can be divided into two parts, namely, a particular solution and an orthogonal projection of f₀ into the null-space of the Jacobian matrix (N (J (q))). In fact, the solution of Eq. (52) is made equivalent to the solution of the following optimization problem (proof in Appendix 1): $\begin{matrix} min H (f) = \frac{1}{2} (f - f_{0})^{T} (f - f_{0}) \\ s . t . - J^{T} (q) f = \hat{h} (z) + K_{v} r - β_{robust} \end{matrix}$ (54)

Algorithm 1 summarizes the general idea of the proposed control approach.

Algorithm 1 Designing adaptive neural network controller for CDSR.

Data: Plant from Eqs. (1-2) and desired trajectory for q_d.

Goal: Applying Eq. (33) in designing an appropriate controller with desired trajectory tracking performance by the end-effector.

Step1: Calculating the Jacobian of system by using the Eqs. (3), (6-8) and (24).

Step2: Estimating the dynamic model of the plant by adaptive neural network based on Eqs. (49-51)

Step3: Obtaining the cables tension forces and applying to the system by Eq. (52).

Step4: Stop.

The block diagram of this process is demonstrated in Fig. 3.

Fig. 3

The scheme of designing NN controller for the under study CDSR.

3 Simulation outcomes

In this part, in order to peruse the applicability and effectiveness of the presented approach, some simulations are performed on the under study robot illustrated in Fig. 1. In order to indicate the efficiency of the suggested controller, simulation results perform in presence and without bounded disturbance.

3.1 Simulation results without the presence of disturbance

For the sake of simulation, the following parameters are considered for the under study CDSR: $\begin{matrix} l_{1} & = 1 (m), l_{2} = 1 (m), m_{1} = 0.8 (Kg), \\ m_{2} & = 2.3 (Kg), g = 9.8 (\frac{m}{s^{2}}), I_{1 zz} = I_{2 zz} = 10 (\frac{Kg}{m^{3}}) \end{matrix}$ (55) Although the system parameters are given, but the main purpose of the applying an Adaptive Neural Network (ANN) is not to identify system parameters. In fact, the goal of applying the ANN approach consists in estimating the h (z) (shown in Eq. (28)) without knowing M, C and G martices from the outset. Since finding these matrices in a conventional state requires complex mathematical operations, so an ANN with known inputs (shown in Eq. (29)) has been used to estimate h (z) . Here, considering the system parameters is only for running the simulation, and for implementing the presented approaches on a real robot, there is no need to know these parameters.

Also, the desired trajectory q_d (t) have the following components: $\begin{matrix} θ_{1 d} = 0.05 sin (π t), θ_{2 d} = 0.05 cos (π t) \end{matrix}$ (56) Moreover, the NN parameters and the initial values are selected as: $\begin{matrix} S_{1} = 3, S_{2} = 2, {\hat{A}}_{0}^{T} = rand (3, 10), {\hat{B}}_{0}^{T} = rand (2, 3), \\ [\begin{matrix} θ_{10} \\ {\dot{θ}}_{10} \\ θ_{20} \\ {\dot{θ}}_{20} \end{matrix}] = [\begin{matrix} 0.01 \\ 0.01 \\ 0.001 \\ 0.001 \end{matrix}], [\begin{matrix} e_{10} \\ e_{20} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}], f_{0} = [\begin{matrix} 75 \\ 100 \\ 125 \\ 100 \end{matrix}] \end{matrix}$ (57) The number of neurons in the hidden layer of the NN can be calculated based on the optimization of the Mean Squared Error (MSE) and the number of neurons. Also, it is obvious that according to the dimensions of h (z) , the number of neurons in the output layer should be considered as two neurons. Choosing the initial weights of the NN is not considered a challenge. In the simplest case, they can be set to zero, and until the NN has begun to learn according to the stated algorithm, the PD term of the controller is responsible for the stability of the system. Of course, considering weights randomly can be a better choice for practical activities. For this reason, these weights are randomly selected. Also, f₀, which belongs to the null-space of the CDSR Jacobian, is determined by trial and error so that the forces of the cables are only in the tensile state.

In addition, the attachment points of the cables are selected as follows: $\begin{matrix} [\begin{matrix} x_{1}^{'} \\ y_{1}^{'} \end{matrix}] & = [\begin{matrix} \frac{1}{3} \\ \frac{1}{3} \end{matrix}], [\begin{matrix} x_{1} \\ y_{1} \end{matrix}] = [\begin{matrix} 0 \\ 1 \end{matrix}], [\begin{matrix} x_{2}^{'} \\ y_{2}^{'} \end{matrix}] = [\begin{matrix} \frac{2}{3} \\ \frac{2}{3} \end{matrix}], [\begin{matrix} x_{2} \\ y_{2} \end{matrix}] = [\begin{matrix} 1 \\ 0 \end{matrix}], \\ [\begin{matrix} x_{3}^{'} \\ y_{3}^{'} \end{matrix}] & = [\begin{matrix} \frac{4}{3} \\ \frac{4}{3} \end{matrix}], [\begin{matrix} x_{3} \\ y_{3} \end{matrix}] = [\begin{matrix} \frac{1}{3} \\ 2 \end{matrix}], [\begin{matrix} x_{4}^{'} \\ y_{4}^{'} \end{matrix}] = [\begin{matrix} \frac{4}{3} \\ \frac{5}{3} \end{matrix}], [\begin{matrix} x_{4} \\ y_{4} \end{matrix}] = [\begin{matrix} 2 \\ 1 \end{matrix}] \end{matrix}$ (58) Finally, by inspection and using the mentioned conditions, the adaptation laws and sliding mode error parameters are chosen as follows: $\begin{matrix} K_{v} & = [\begin{matrix} 1500 & 0 \\ 0 & 1500 \end{matrix}], Λ = [\begin{matrix} 20 & 0 \\ 0 & 20 \end{matrix}], \\ F & = I_{3 \times 3}, G = I_{10 \times 10}, μ = 0.01 \end{matrix}$ (59) In addition, the only condition for selecting the value of μ is to consider it small. In the case of matrices F and G, they should be selected to be symmetric positive definite. In turn, K_v and Λ are selected diagonal and large enough so that the tracking error converges asymptotically to zero

In order to perform the simulation, the combination of M-File and Simulink environments of the MATLAB software, version 2018-b, is used. The outcomes are indicated in Figs. 4 to 6. The time response of the states of CDSR θ₁, θ₂ and the corresponding reference signal are shown in Figs. 4(a) and (b), respectively. The cables tension forces are illustrated in Fig. 5. As it can be seen from foregoing figures, the desired trajectories are tracked by generalized coordinate vector variables with a very high performance. Moreover, the time response of the end-effector positions along the X and Y axes are indicated in Figs. 6a and 6b, respectively.

Fig. 4

Tracking reference signals by the states of CDSR.

Fig. 5

Cables tension forces based on Eq. (52).

Fig. 6

Tracking reference signals by the End-Effector of CDSR.

As aforementioned, the NN plays a key role in the proposed control structures. In fact, successful updating of NN weights, which depends on the proper selection of the initial parameters and variables defined in Eqs. (49-50), can lead to faster convergence of the tracking error to zero. Therefore, the role of ANN cannot be easily overlooked in this design.

3.2 Simulation results in the presence of bounded disturbance

In this section, unknown disturbance term is added to the dynamic equation of CDSR. Thus, by using Eq. (1), τ_d is considered as sinusoidal and it is applied to the system at t = 10 (sec):

$\begin{matrix} τ_{d} = A_{d} sin π t \end{matrix}$ (60) in which, A_d represents the amplitude of applied disturbance. Now with respect to the previous values for parameters, the robust term with the following characteristics is added to the mentioned design controller. $\begin{matrix} α = 0.7, S_{Bounded} = 1500 \end{matrix}$ (61) The whole design process except the following parameters is the same as without the disturbance: $\begin{matrix} f_{0} = [\begin{matrix} 75 \\ 300 \\ 125 \\ 150 \end{matrix}], K_{v} = [\begin{matrix} 500 & 0 \\ 0 & 500 \end{matrix}] \end{matrix}$ (62) It should also be noted that selecting the larger components for K_v leads to smaller tracking errors, but the tensile forces of the cables will be more oscillating.

The simulation results for A_d = 65 are shown in Figs. (7-12). The time response of the states of CDSR θ₁, θ₂ and the reference signal are shown in Figs. 7 and 8, respectively. The cables tension forces are illustrated in Fig. 9. Additionally, the time response of the end-effector positions along the X and Y axes are indicated in Figs. 10 and 11, respectively. As it can be observed from the latter figures, the robust term leads to an appropriate performance compared the results obtained for the controller without robust term in the presence of unknown bounded disturbance. In fact, it is seen that adding a robust term which is shown in Fig. 12, to the controller can lead to reduction of steady state error of the system.

Fig. 7

The effect of robust term in tracking reference signals by the state of CDSR θ₁.

Fig. 8

The effect of robust term in tracking reference signals by the state of CDSR θ₂.

Fig. 9

Comparison Cables tension forces in presence of disturbance.

Fig. 10

The effect of robust term in tracking reference signals by the End-Effector of CDSR X.

Fig. 11

The effect of robust term in tracking reference signals by the End-Effector of CDSR Y.

Fig. 12

The time response of robust signal.

Also, the comparison of performance improvement by adding the robust term to the controller for different values of the A_d and based on RMSE criteria are shown in Table 1. It is quite obvious that by adding the robust term, the controller has much better performance.

Table 1

Comparison of performance improvement by adding the robust term to the controller for different values of A_d and based on RMSE criteria

Amplitude	Controllerwithoutrobustterm	Controllerwithrobustterm	Performanceimprovement (%)
A_d = 60	e₁ = 0.0074e₂ = 0.0083	e₁ = 0.0026e₂ = 0.0055	e₁ = 35e₂ = 66.2
A_d = 65	e₁ = 0.0077e₂ = 0.0085	e₁ = 0.0026e₂ = 0.0056	e₁ = 33.76e₂ = 65.88
A_d = 70	e₁ = 0.0081e₂ = 0.0087	e₁ = 0.0027e₂ = 0.0056	e₁ = 33.33e₂ = 64.36
A_d = 100	e₁ = 0.0100e₂ = 0.0102	e₁ = 0.0034e₂ = 0.0059	e₁ = 34e₂ = 57.84
A_d = 120	e₁ = 0.0113e₂ = 0.0112	e₁ = 0.0038e₂ = 0.0061	e₁ = 33.62e₂ = 54.46

Moreover, Table 2 provides a functional comparison between the presented concepts in this paper and the closest available references to it. Accordingly, the following criteria are used as comparison parameters:

Designing structural controller and prove of the stability.

Comprehensiveness of the mentioned solutions.

Addressing to optimization issues.

Applying intelligent approaches in estimating unknown dynamics equations.

As it can be seen from this Table, the contents of this paper are complete in terms of covering the important mentioned criteria and complements other references.

Table 2

Comparison of the performed activities in the references with the considered activities in this paper

4 Conclusion

This paper presented a novel structure in using of the main properties of a serial robot in interaction with a cable-driven robot which, in short, is referred to as CDSR. In fact, the present paper provided a dynamic model estimating and controller designing for the planar 2-DoF robot which is in interaction with a cable- driven robot. The proposed structure has very important applications in the issue of rehabilitation. One of these applications is the design of platforms for the rehabilitation of the disabled, such as the hands or feet of them. In fact, by considering the limb as a passive two-link serial robot and designing and building a platform to place it on the injured limb of the disabled person, they can be rehabilitated by holding treatment sessions. Although, in this paper, the injured limb is considered as a passive two-link serial robot that is driven with four light cables, but the disabled person can be considered as multi-link platforms with different degrees of freedom. Therefore, in this case, it is necessary to use more cables to restrain the relevant platform using the proposed structure. However, the key for designing and controlling such platforms is the need to know the dynamics of the issue. This becomes more difficult as the number of links and degrees of freedom increase. So, in this article, an attempt has been made to provide comprehensive solutions to solve the problem ahead. In this regard, NNs have been used as general approximators in estimating the dynamics of the problem, and accordingly, its weights are updated as free parameters based on the introduction of the appropriate Lyapunov function. As a result, the proposed controller has an adaptive property and at the same time has the property of proving asymptotic stability. So, this paper investigated various challenges such as the issue of CDSR stability assurance based on presenting an appropriate Lyapunove function, rejecting of unknown bounded disturbance in a steady state with adding a robust term to the controller and presenting a new method to obtain the overall Jacobian of the CDSR. Hence, the related design was divided in two parts: in presence and without bounded disturbance. The simulation results indicated that, in presence of the bounded disturbance, the controller with robust term has better result in comparison of the controller without robust term based on RMSE criteria. Indeed, the numeric results revealed that, in the presence of bounded disturbance with different amplitude, adding the robust term to the controller led to improve the tracking error of the 2-DoF robot joints about 34% and 62%,, respectively.

Since the most important application of the mentioned approaches is in rehabilitation issues, so it is intended to design and use of the presented platform in future work in the Human and Robot Interaction Laboratory. In fact, the intention is to provide platforms for the rehabilitation of limbs in the disabled humans and applying the proposed control methods. In addition, force control at the end-effector can be considered as another ongoing work. This means that we are looking to control the force at the end-effector for example, the end-effector should manipulate an egg without damaging it. This can be practically tested by adding skin robot at a different part of the serial robot.

Footnotes

Appendix 1

In order to solve Eq. (54), the following Lagrangian function (with multipliers λ ) is first formed as follows: (63) $\begin{matrix} L (f, λ) = H (f) + λ^{T} (- J^{T} (q) f - \hat{h} (z) - K_{v} r + β_{robust}) \end{matrix}$ Then, by examining the necessary and sufficient conditions for the existence of the minimum, the following are extracted:

Necessary conditions

(64) $\begin{matrix} \nabla_{f} L = (\frac{\partial L}{\partial f})^{T} = (f - f_{0}) - J (q) λ = 0 \\ \Rightarrow f = f_{0} + J (q) λ \end{matrix}$ (65) $\begin{matrix} \nabla_{λ} L = (\frac{\partial L}{\partial λ})^{T} = - J^{T} (q) f - \hat{h} (z) - K_{v} r + β_{robust} = 0 \end{matrix}$ By replacing Eq. (64) into Eq. (65), the following is obtained: (66) $\begin{matrix} - J^{T} (q) f_{0} - J^{T} (q) J (q) λ - \hat{h} (z) - K_{v} r + β_{robust} = 0 \\ \Rightarrow λ = (J^{T} J)^{- 1} (- J^{T} f_{0} - \hat{h} (z) - K_{v} r + β_{robust}) \end{matrix}$ Substituting Eq. (66) into Eq. (64), the following is obtained: (67) $\begin{matrix} f = - [J_{1} (q) (\hat{h} (z) + K_{v} r - β_{robust}) + (I - J_{1} (q) J^{T} (q) f_{0}] \end{matrix}$ in which, (68) $\begin{matrix} J_{1} (q) = (J (q) (J^{T} (q) J (q))^{- 1}) \end{matrix}$

Sufficient conditions

(69) $\begin{matrix} {\nabla_{f}}^{2} L = I > 0 \end{matrix}$

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