Hybrid Learning-based Control of Closed-Kinematic Chain Mechanism Robot Manipulators

1. Introduction

In many industrial applications, robot manipulators are required to execute identical tasks repeatedly. From an engineering standpoint, exploiting information from previous executions to improve tracking accuracy in subsequent trials is highly desirable. However, classical control approaches offer limited capability for achieving high-precision tracking in such repetitive operations. Consequently, significant research efforts have focused on developing control strategies that explicitly utilize data from prior trials, commonly referred to as iterative learning control (ILC) [1,2,3]

ILC enhances the performance of conventional feedback control by iteratively updating the control input based on historical tracking errors and control signals. Compared with many advanced control schemes, ILC possesses a relatively simple structure and is well suited for real-time implementation, as it requires low computational complexity and minimal a priori knowledge of system dynamics. Since the seminal work in [4], which first introduced ILC for robotic manipulator motion tracking, numerous ILC algorithms and applications have been extensively investigated [5,6,7,8].

Most existing studies share a common framework, with primary differences arising in the analysis of stability, convergence, and performance. Among various learning schemes, PD-type and D-type ILC algorithms are the most widely adopted in practical robotic applications, largely due to the fact that many advanced ILC formulations remain highly theoretical and difficult to implement in real-world systems [9,10,11,12].

Closed-kinematic-chain mechanism (CKCM) manipulators, which feature parallel mechanical structures, have attracted considerable attention due to their ability to achieve high-precision positioning and their inherent avoidance of error accumulation associated with serial joint configurations [13,14,15]. CKCM manipulators have been applied in a wide range of fields, including space-based telerobotic assembly, aircraft simulators for pilot training, and the monitoring of human arm motion for stroke rehabilitation [16]. Despite extensive research on the mechanism design and dynamic modeling of CKCM manipulators, relatively limited effort has been devoted to addressing their motion control problems [17,18,19,20,21,22,23].

In this paper, a novel hybrid learning control strategy combining PID feedback control with an iterative learning control (ILC) scheme is developed for position control of CKCM robot manipulators. The learning controller adopts a PID-type iterative update law for a class of linear time-varying dynamic systems. Uniform convergence of the learning process is established using an exponentially weighted supremum-norm analysis and a contraction mapping framework. The effectiveness of the proposed hybrid learning control scheme is evaluated through computer simulations for the position control of a six-degree-of-freedom (DOF) CKCM manipulator.

This paper is organized as follows. Section 2 introduces the hybrid learning control scheme which integrates a Feedback Controller with a Learning Controller. Section 3 presents the design of the Feedback Controller, while Section 4 describes the development of the Learning Controller. An algorithm providing a design procedure for the learning control scheme is given in Section 5. Section 6 presents and discusses the results of computer simulation studies conducted to evaluate the effectiveness of the hybrid control scheme in tracking several paths. Finally, Section 7 concludes the paper with a summary of the main findings and outlines directions for future research.

2. The Hybrid Learning Control Scheme

Figure 1 presents the hybrid learning control scheme (HLCS) developed to control the motion of CKCM robot manipulators. As shown in the figure, the HLCS mainly consists of two controllers: a Feedback Controller and a Learning Controller. In general, the Feedback Controller ensures the overall stability of the system through appropriately selected control gains, while the Learning Controller aims to improve both the transient and steady-state responses of the manipulator end-effector position after each trial.

Robot kinematics and control deal with two types of variables, Cartesian variables and joint variables. Cartesian variables include positional variables x, y, and z, expressing the position of the end-effector with respect to a reference frame, and three orientational variables, such as Euler angles, expressing the orientation of the end-effector with respect to the same reference frame. For CKCM robot manipulators possessing 6 DOFs, the joint variables are the lengths of the six actuators, $l_1$, $l_2$, $l_3$, $l_4$, $l_5$, and $l_6.$The Feedback Controller takes advantage of the existence of a closed-form solution for inverse kinematics of CKCM robot manipulators and uses it to convert the desired Cartesian variables expressed in vector ${\mathit{x}}_d\left(t\right)$ to the desired joint variables contained in vector ${\mathit{l}}_d\left(t\right)$. The actual joint variables expressed by vector l(t) are measured by joint position sensors, and then the measured actual joint variables are subtracted from the desired joint variables contained in vector ${\mathit{l}}_d\left(t\right)$ to produce the joint variable errors expressed by vector ${\mathit{l}}_e\left(t\right)$. The joint variable errors are then sent to the Feedback Controller, which is a proportional–derivative–integral (PID) controller employing the PID control law to produce an input ${\mathit{\tau }}_f\left(t\right)$ for the manipulator. The controller gains could be set at certain fixed values using standard control techniques, including linearization and pole placement or tuning guidelines to stabilize the closed-loop control system.

To improve the performance of CKCM robot manipulators in repetitive tasks, the Learning Controller utilizes an ILC algorithm based on errors from previous trials to generate a corrective input ${\mathit{u}}_k$, which complements the input ${\mathit{\tau }}_f\left(t\right)$ produced by the Feedback Controller. The total input to the CKCM robot manipulator consists of the feedback input ${\tau }_f\left(t\right)$, the learning input $u_k$, and an auxiliary input ${\mathit{\tau }}_a\left(t\right)$, which compensates for the manipulator dynamics, as will be discussed later in the analysis of the Feedback Controller.

3. The Feedback Controller

This section will address the design of the Feedback Controller. The dynamical equations of a CKCM robot manipulator possessing 6 DOFs are given by

$$\mathit{M}\left(\mathit{l}\right)\ddot{\mathit{l}}\left(t\right)+\mathit{C}\left(\mathit{l},\dot{\mathit{l}}\right)\dot{\mathit{l}}\left(t\right)+\mathit{G}\left(\mathit{l}\right)\mathit{l}\left(t\right)=\mathit{\tau }\left(t\right)$$

(1)

where $\mathit{l}\left(t\right)$ denotes the (6x1) joint variable vector containing the lengths of the 6 actuators of the manipulator, $\mathit{M}\left(\mathit{l}\right)$, $\mathit{C}\left(\mathit{l},\dot{\mathit{l}}\right)$ and $\mathit{G}\left(\mathit{l}\right)$ represent the (6x6) manipulator mass matrix, the (6x1) centrifugal and Coriolis force vector and the (6x1) gravitational force vector, respectively. In addition, $\mathit{\tau }\left(t\right)$ denotes the (6x1) control input vector to the manipulator.

Linearizing (1) about the desired joint variable vector ${\mathit{l}}_d$ by using Taylor Series expansion and neglecting higher order terms, we obtain

$$\mathit{M}\left(t\right)\ddot{\mathit{z}}\left(t\right)+\mathit{C}\left(t\right)\dot{\mathit{z}}\left(t\right)+\mathit{G}\left(t\right)\mathit{z}\left(t\right)+{\mathit{\tau }}_d\left(t\right)=\mathit{\tau }\left(t\right)$$

(2)

where

$$\mathit{z}\left(t\right)=\mathit{l}\left(t\right)-{\mathit{l}}_d\left(t\right)$$

(3)

$$\mathit{M}\left(t\right)=\mathit{M}{\left({\mathit{l}}_d\right)}_{{\mathit{l}}_{\mathit{d}}{\dot{,\mathit{l}}}_{\mathit{d}}}$$

(4)

$$\mathit{C}\left(t\right)={\displaystyle \frac{\partial }{\partial \dot{\mathit{q}}}}{\left[\mathit{C}\left(\mathit{l},\dot{\mathit{l}}\right)\dot{\mathit{l}}\right]}_{{\mathit{l}}_d,{\dot{\mathit{l}}}_d}$$

(5)

$$\mathit{G}\left(t\right)={\displaystyle \frac{\partial }{\partial \mathit{q}}}{[\mathit{M}\left(\mathit{l}\right){\ddot{\mathit{l}}}_{\mathit{d}}+\mathit{C}\left(\mathit{l},\dot{\mathit{l}}\right)\dot{\mathit{l}}+\mathit{G}(\mathit{l}\left)\right]}_{{\mathit{l}}_{\mathit{d}},{\dot{\mathit{l}}}_{\mathit{d}}}$$

(6)

and

$${\mathit{\tau }}_d\left(t\right)=\mathit{M}\left({\mathit{l}}_d\right){\ddot{\mathit{l}}}_d+\mathit{C}\left({\mathit{l}}_d,{\dot{\mathit{l}}}_d\right){\dot{\mathit{l}}}_d+\mathit{G}\left({\mathit{l}}_d\right)$$

(7)

From Figure 1, we get

$$\mathit{\tau }\left(t\right)={\mathit{\tau }}_f\left(t\right)+{\mathit{u}}_k\left(t\right)+{\mathit{\tau }}_a\left(t\right)$$

(8)

where ${\mathit{\tau }}_f\left(t\right)$, the output of the Feedback Controller employing the PID control law is given by

$${\mathit{\tau }}_f\left(t\right)={\mathit{K}}_{\mathit{P}}{\mathit{l}}_{\mathit{e}}\left(t\right)+{\mathit{K}}_{\mathit{D}}{\dot{\mathit{l}}}_{\mathit{e}}\left(t\right)+{\mathit{K}}_{\mathit{I}}\int {\mathit{l}}_{\mathit{e}}\left(t\right)dt$$

(9)

with

$${\mathit{l}}_e\left(t\right)={\mathit{l}}_d\left(t\right)-\mathit{l}\left(t\right)$$

(10)

and ${\mathit{K}}_p$, ${\mathit{K}}_i$ and ${\mathit{K}}_d$ are the controller gain matrices of the PID control law, respectively.

Now substituting (8)-(9) into (2) yields

$$\mathit{M}\left(t\right)\ddot{\mathit{z}}\left(t\right)+\left[\mathit{C}\left(t\right)+{\mathit{K}}_d\right]\dot{\mathit{z}}\left(t\right)+[\mathit{G}\left(t\right)+{\mathit{K}}_p]\mathit{z}\left(t\right){+\mathit{K}}_i\int \mathit{z}\left(t\right)dt={\mathit{u}}_k\left(t\right)$$

(11)

where we let

$${\mathit{\tau }}_a\left(t\right)={\mathit{\tau }}_d\left(t\right)$$

(12)

We also note that

$${\mathit{l}}_e\left(t\right)=-\mathit{z}\left(t\right)$$

(13)

The linearized dynamics model (34) can be semi-globally stabilized by properly selecting the gain matrices ${\mathit{K}}_p$, ${\mathit{K}}_i$ and ${\mathit{K}}_d$ of the PID control law by employing tuning guidelines given in [24].

4. The Learning Controller

Figure 2 illustrates the structure of the Learning Controller consisting of a PID-type learning control law and a large-scale integrated random-access memory (LSI RAM). The learning process is described by

$${\mathit{u}}_{k+1}\left(t\right)={\mathit{u}}_k\left(t\right)+\{\mathit{\Gamma }{\displaystyle \frac{d}{dt}}+\mathit{\Phi }+\mathit{\Psi }\int dt\}{\mathit{l}}_{ek}\left(t\right)$$

(14)

where ${\mathit{u}}_k\left(t\right)$ denotes the output of the learning controller during the $k^{th}$trial, $\mathit{\Phi }$, $\mathit{\Psi }$ and $\mathit{\Gamma }$ are the gain matrices of the PID learning control law, respectively. ${\mathit{l}}_{ek}\left(t\right)$ is the joint error as given in (10) during the $k^{th}$ trial. During the $k^{th}$ trial, the information of ${\mathit{u}}_{k+1}$ is computed using (14) and stored in the lower part of the RAM as a set of densely sampled digital data. After the $k^{th}$ trial, the stored data will be loaded to the upper part of the memory and will be sent to the actuators of the CKCM robot manipulator during the ${\left(k+1\right)}^{th}$ trial. The lower part of the memory is now empty and ready to store new data.

In general, a learning controller deals with a class of linear time-varying systems described by

$$\left\{\begin{array}{c}R\left(t\right)\ddot{\mathit{x}}\left(t\right)+Q\left(t\right)\dot{\mathit{x}}\left(t\right)+P\left(t\right)x\left(t\right)=u\left(t\right)\\ y\left(t\right)=x\left(t\right)\end{array}\right.$$

(15)

where $\mathit{y}\left(t\right), \mathit{x}\left(t\right) \mathrm{a}\mathrm{n}\mathrm{d} \mathit{u}\left(t\right)\mathrm{a}\mathrm{l}\mathrm{l}\in R^n$denote the output vector, the state vector and input vector, , respectively. The system matrices $\mathit{R}\left(t\right), \mathit{Q}\left(t\right), \mathrm{a}\mathrm{n}\mathrm{d} \mathit{P}\left(t\right)\in R^{n\times n}$are assumed to be continuously differentiable and satisfy

$$\left\{\begin{array}{c}R\left(t\right)={\mathit{R}}^T\left(t\right)\ge {\mathit{R}}_0>0\\ Q\left(t\right)\ge Q_0\\ P\left(t\right)\ge {\mathit{P}}_0\end{array}\right.$$

(16)

where ${\mathit{R}}_0, {\mathit{Q}}_0, \mathrm{a}\mathrm{n}\mathrm{d} {\mathit{P}}_0$ are the lower limit of $\mathit{R}\left(t\right), \mathit{Q}\left(t\right), \mathrm{a}\mathrm{n}\mathrm{d} \mathit{P}\left(t\right)$, respectively. Given a desired output ${\mathit{y}}_d\left(t\right)\in R^n$ that is continuously differentiable on $[0, T]$, and an input ${\mathit{u}}_k\left(t\right)\in R^n$ that is continuous on $[0, T]$, we consider a PID-type ILC law described by

$$\left\{\begin{array}{c}R\left(t\right){\ddot{\mathit{x}}}_k\left(t\right)+Q\left(t\right){\dot{\mathit{x}}}_{\mathit{k}}\left(t\right)+P\left(t\right){\mathit{x}}_{\mathit{k}}\left(t\right)={\mathit{u}}_k\left(t\right)\\ {\mathit{y}}_k\left(t\right)={\mathit{x}}_k\left(t\right)\\ {\mathit{e}}_k\left(t\right)={\mathit{y}}_d\left(t\right)-{\mathit{y}}_k\left(t\right)\\ {\mathit{u}}_{k+1}\left(t\right)={\mathit{u}}_k\left(t\right)+\{\Gamma {\displaystyle \frac{d}{dt}}+\Phi +\Psi \int dt\}{\mathit{e}}_k\end{array}\right.$$

(17)

In applying the PID learning algorithm defined in (17), three standard assumptions commonly adopted in ILC are considered:

• The repeatability of the initial conditions is ensured; that is, the initial state ${\mathit{x}}_k\left(0\right)$ of the system can be reset to the same value at the beginning of each operation as follows:

$${\mathit{x}}_k\left(0\right)={\mathit{x}}^0 for k=\mathrm{1,2},\dots$$

(18)

• Each operation ends at a fixed finite time $T>0$.
• The system dynamics represented by $\mathit{R}\left(t\right), \mathit{Q}\left(t\right)$ and $\mathit{P}\left(t\right)$ is assumed to remain invariant throughout the repeated training process.

Now an operator $\mathsf{\Delta }$ for $\mathit{x}\left(t\right)$ is defined as

$$\mathsf{\Delta }\left(\mathit{x}\left(t\right)\right)=\mathsf{\Gamma }{\displaystyle \frac{d\mathit{x}\left(t\right)}{dt}}+\mathsf{\Phi }\mathit{x}\left(\mathrm{t}\right)+\mathsf{\Psi }{\int }_0^t\mathit{x}\left(t\right)dt$$

(19)

and we recall that a $\rho$-vector norm of the state vector $\mathit{x}\left(t\right)$ with element $x_i, i=\mathrm{1,2},3,\dots ,n$ is defined by

$${\left|\mathit{x}\right|}_{\rho }={\left(\sum _{i=1}^n{\left|x_i\right|}^{\rho }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\rho $}\right.}}$$

(20)

which is a positive scalar.

In order to relate the elements of the state vector $\mathit{x}\left(t\right)$ to the gain matrices $\mathsf{\Gamma },\mathsf{\Phi },\mathrm{a}\mathrm{n}\mathrm{d}\mathsf{\Psi }$ of the ILC law, we force the $\rho$-vector norm of $\mathit{x}\left(t\right)$ with an initial condition $\mathit{x}\left(0\right)=0$ to be as follows

$${\left|\mathit{x}\left(t\right)\right|}_{\rho }={\int }_0^t{e}^{-\rho \tau }{\left\{\mathsf{\Delta }\mathit{x}\left(\tau \right)\right\}}^T{\mathsf{\Phi }}^{-1}\left\{\mathsf{\Delta }x\left(\tau \right)\right\}d\tau$$

$$={\int }_0^t{e}^{-\rho \tau }{\left\{\mathsf{\Phi }\mathbf{x}+\mathsf{\Gamma }\mathbf{X}+\mathsf{\Phi }\mathbf{Y}\right\}}^T{\mathsf{\Phi }}^{-1}\left\{\mathsf{\Phi }\mathbf{x}+\mathsf{\Gamma }\mathbf{X}+\mathsf{\Phi }\mathbf{Y}\right\}d\tau$$

$$={\int }_0^t{e}^{-\rho \tau }\left\{{\mathbf{x}}^{\mathrm{T}}\mathsf{\Phi }\mathbf{x}+{\mathbf{X}}^{\mathrm{T}}\mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }\mathbf{X}+{\mathbf{Y}}^{\mathrm{T}}\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathsf{\Psi }\mathbf{Y}\right\}d\tau +{\int }_0^t{e}^{-\rho \tau }\left\{{\mathbf{x}}^{\mathrm{T}}\mathsf{\Gamma }\mathbf{X}+{\mathbf{x}}^{\mathrm{T}}\mathsf{\Psi }\mathbf{Y}+{\mathbf{X}}^{\mathrm{T}}\mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathsf{\Psi }\mathbf{Y}\right\}d\tau$$

(21)

where the mathematical operator $\mathbf{X} \mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{s} {\displaystyle \frac{d}{dt}}\mathit{x},\mathrm{a}\mathrm{n}\mathrm{d} \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l} \mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r} \mathbf{Y} \mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{s}{\int }_0^t\mathit{x}$. By applying integration by parts to (21), this equation can be rewritten as

$${\left|\mathit{x}\left(t\right)\right|}_{\rho }={\int }_0^t{e}^{-\rho \tau }\left\{{\mathbf{x}}^{\mathrm{T}}\left(\mathsf{\Phi }+\mathsf{\rho }\mathsf{\Gamma }\right)\mathrm{x}+{\mathbf{X}}^{\mathrm{T}}\mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }\mathbf{X}+{\mathbf{Y}}^{\mathrm{T}}\left(\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathsf{\Psi }+\mathsf{\rho }\mathsf{\Psi }\right)\mathbf{Y}\right\}d\tau +{\int }_0^t{e}^{-\rho \tau }\left\{2{\mathbf{X}}^{\mathrm{T}}\mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathsf{\Psi }\mathbf{Y}\right\}d\tau +e^{-\rho \tau }\left\{{\mathbf{Y}}^{\mathrm{T}}\mathsf{\Psi }\mathbf{Y}+{\mathbf{x}}^{\mathrm{T}}\mathsf{\Gamma }\mathbf{x}\right\}$$

(22)

We also have from (17)

$$\mathit{R}\left(t\right){\stackrel{⃛}{\mathit{d}}}_k+\mathit{Q}\left(t\right){\ddot{\mathit{d}}}_k+\mathit{P}\left(t\right){\dot{\mathit{d}}}_k=\mathsf{\Delta }\left(e_k\left(t\right)\right)$$

(23)

where we define

$${\dot{\mathit{d}}}_k={\mathit{x}}_{k+1}\left(t\right)-{\mathit{x}}_k\left(t\right)\to {\dot{\mathit{d}}}_k={\mathit{e}}_k\left(t\right)-{\mathit{e}}_{k+1}\left(t\right)$$

(24)

Now, from (24) let us consider

$${\left|{\mathit{e}}_{k+1}\left(t\right)\right|}_{\rho }={\left|{\mathit{e}}_k\left(t\right)-{\dot{\mathit{d}}}_k\left(t\right)\right|}_{\rho }$$

(25)

$$={\left|{\mathit{e}}_k\left(t\right)\right|}_{\rho }+{\left|{\dot{\mathit{d}}}_k\left(t\right)\right|}_{\rho }-2{\int }_0^t{e}^{-\rho \tau }{\left\{\mathsf{\Delta }{\dot{\mathit{d}}}_k\left(\tau \right)\right\}}^T{\mathsf{\Phi }}^{-1}\left\{\mathsf{\Delta }{\mathit{e}}_k\left(\tau \right)\right\}d\tau$$

(26)

$$={\left|{\mathit{e}}_k\left(t\right)\right|}_{\rho }+{\left|{\dot{\mathit{d}}}_k\left(t\right)\right|}_{\rho }-2{\int }_0^t{e}^{-\rho \tau }\left\{{\ddot{\mathit{d}}}_k{\left(\tau \right)}^T\mathsf{\Gamma }+{\dot{\mathit{d}}}_k{\left(\tau \right)}^T\mathsf{\Phi }+d_k^T\mathsf{\Psi }\right\}{\mathsf{\Phi }}^{-1}\left\{\mathit{R}{\stackrel{⃛}{\mathit{d}}}_k+\mathit{Q}{\ddot{\mathit{d}}}_k+\mathit{P}{\dot{\mathit{d}}}_k\right\}d\tau$$

(27)

Substituting ${\left|{\dot{\mathit{d}}}_k\left(t\right)\right|}_{\rho }$from (26) into (27) and simplifying the integral term yield

$${\left|{\mathit{e}}_{k+1}\left(t\right)\right|}_{\rho }={\left|{\mathit{e}}_k\left(t\right)\right|}_{\rho }-{\int }_0^t{\mathit{e}}^{-\rho \tau }\left\{{\ddot{\mathit{d}}}_k^T{\mathit{S}}_2{\ddot{\mathit{d}}}_k+{\dot{\mathit{d}}}_k^T{\mathit{S}}_1{\dot{\mathit{d}}}_k+{\mathit{d}}_k^T{\mathit{S}}_0{\mathit{d}}_k\right\}d\tau -e^{-\rho t}\left\{{\ddot{\mathit{d}}}_k^T{\mathit{K}}_2{\ddot{d}}_k+{\dot{\mathit{d}}}_k^T{\mathit{K}}_1{\dot{\mathit{d}}}_k+{\mathit{d}}_k^T{\mathit{K}}_0{\mathit{d}}_k\right\}+e^{-\rho t}\left\{{\mathit{d}}_k^T{\mathit{L}}_0{\ddot{\mathit{d}}}_k+{\dot{\mathit{d}}}_k^T{\mathit{V}}_0{\ddot{\mathit{d}}}_k+{\mathit{d}}_k^T{\mathit{U}}_0{\dot{\mathit{d}}}_k\right\}+{\int }_0^t{e}^{-\rho \tau }\left\{{\mathit{d}}_k^T{\mathit{L}}_1{\ddot{\mathit{d}}}_k+{\dot{\mathit{d}}}_k^T{\mathit{V}}_1{\ddot{\mathit{d}}}_k+{\mathit{d}}_k^T{\mathit{U}}_1{\dot{\mathit{d}}}_k\right\}d\tau$$

(28)

where

$${\mathit{S}}_2=\rho \mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathit{R}-\mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathit{R}-\mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\dot{\mathit{R}}-\mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }-2\mathit{R}$$

$${\mathit{S}}_1=\rho \left(\mathit{Q}+{\mathit{P}}^T{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }-\mathsf{\Gamma }\right)-\mathsf{\Phi }-\dot{\mathit{Q}}-{\dot{\mathit{P}}}^T{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }-2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{Q}+2\mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathsf{\Psi }$$

$${\mathit{S}}_0=\rho \left(\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{P}-\mathsf{\Psi }\right)-\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathsf{\Psi }-\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\dot{\mathit{P}}$$

$${\mathit{K}}_2=\mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathit{R}$$

$${\mathit{K}}_1=\mathit{Q}+{\mathit{P}}^T{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }-\mathsf{\Gamma }$$

$${\mathit{K}}_0=\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{P}-\mathsf{\Psi }$$

$${\mathit{L}}_1=\rho \left(-2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{R}\right)+2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\dot{\mathit{R}}$$

$${\mathit{V}}_1=\rho \left(-2\mathit{R}\right)+2\dot{\mathit{R}}+2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{R}$$

$${\mathit{U}}_1=\rho \left(-2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{Q}+2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }\right)+2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\dot{\mathit{Q}}-2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }$$

$${\mathit{L}}_0=-2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{R}$$

$${\mathit{V}}_0=-2\mathit{R}$$

$${\mathit{U}}_0=-2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{Q}+2\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }$$

We have now successfully established a relationship between the performance errors ${\mathit{e}}_k\left(t\right)$ and ${\mathit{e}}_{k+1}\left(t\right)$ of the $k^{th}$and $k^{th}$trials, respectively, in terms of the ILC gain matrices and the system matrices of the linear time-varying system, as shown on the right-hand side of (28). Equation (28) can be regarded as a discrete-time difference equation that relates ${\mathit{e}}_k\left(t\right)$ to ${\mathit{e}}_{k+1}\left(t\right)$ when the system is controlled by the PID-ILCS described in (17). Furthermore, the system described by (28) may be regarded as a discrete closed-loop feedback system in the k-domain—rather than the time domain—in which the PID-ILCS functions as a discrete feedback controller.

Based on (28), the central theorem of the proposed PID learning law can now be stated as follows:

Theorem 1: Assume that each component of ${\mathit{u}}_0\left(t\right)$ is continuous and each component of ${\dot{\mathit{y}}}_d\left(t\right)$ is continuously differentiable, namely

$${\mathit{u}}_0\left(t\right)\in \mathbb{C}\left[0,T\right]\mathrm{a}\mathrm{n}\mathrm{d}{\mathit{y}}_d\left(t\right)\in {\mathbb{C}}^2[0,T]$$

(29)

and the gain matrices $\mathsf{\Gamma },\mathsf{\Phi },\mathrm{a}\mathrm{n}\mathrm{d}\mathsf{\Psi }$are symmetric positive definite and satisfy

$$\left\{\begin{array}{c}\left({\lambda }_2-{\displaystyle \frac{1}{2}}{\lambda }_1-{\displaystyle \frac{1}{2}}\right)R>0\\ \left(1+{\displaystyle \frac{1}{2}}{\lambda }_1\right)Q+{\lambda }_2{\mathit{P}}^T-2R-\left({\displaystyle \frac{1}{2}}{\lambda }_1+1\right)\Gamma >0\\ {\lambda }_1\left(-2\mathit{Q}-2\mathit{R}+\mathit{P}+2\mathsf{\Gamma }\right)-\Psi >0\end{array}\right.$$

(30)

where $‖A‖$ denotes the spectral norm of matrix A and the scalars ${\lambda }_1 \mathrm{a}\mathrm{n}\mathrm{d}{\lambda }_2$ are defined as

$$\left\{\begin{array}{c}{\lambda }_1=‖{\mathsf{\Phi }}^{-1}\mathsf{\Psi }‖\\ {\lambda }_2=‖{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }‖\end{array}\right.$$

(31)

then the PID-type learning control described by (17) with the same initial position and velocity, is convergent in the sense that ${\dot{\mathit{x}}}_k\left(t\right)\to {\dot{\mathit{x}}}_d\left(t\right)$ pointwise in $t\in \left[0,T\right]$ and ${\mathit{x}}_k\left(t\right)\to {\mathit{x}}_d\left(t\right)$ uniformly in $t\in \left[0,T\right]$ as $k\to \infty$.

Proof: Applying Schwarz’s inequality to the last six terms of (28), it can be seen that

$${\left|{\mathit{e}}_{k+1}\left(t\right)\right|}_{\rho }\le {\left|{\mathit{e}}_k\left(t\right)\right|}_{\rho }-{\int }_0^t{e}^{-\rho \tau }\left\{{\ddot{\mathit{d}}}_k^T{\mathit{A}}_2{\ddot{\mathit{d}}}_k+{\dot{\mathit{d}}}_k^T{\mathit{A}}_1{\dot{\mathit{d}}}_k+{\mathit{d}}_k^T{\mathit{A}}_0{\mathit{d}}_k\right\}d\tau -e^{-\rho t}\left\{{\ddot{\mathit{d}}}_k^T{\mathit{B}}_2{\ddot{\mathit{d}}}_k+{\dot{\mathit{d}}}_k^T{\mathit{B}}_1{\dot{\mathit{d}}}_k+{\mathit{d}}_k^T{\mathit{B}}_0{\mathit{d}}_k\right\}$$

(32)

where

$${\mathit{A}}_2=\rho \mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathit{R}-\gamma \mathit{\rho }\left(\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{R}+\mathit{R}\right)+\mathit{Z}$$

$${\mathit{A}}_1=\rho \left(\mathit{Q}+{\mathit{P}}^T{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }-\mathsf{\Gamma }\right)-\rho \left({\mathit{\gamma }}^{-1}\mathit{R}+\gamma \left(\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{Q}-\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }\right)\right)+\mathit{Z}$$

$${\mathit{A}}_0=\rho \left(\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{P}-\mathsf{\Psi }\right)-\rho {\gamma }^{-1}\left(\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{R}+\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{Q}-\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }\right)+\mathit{Z}$$

$${\mathit{B}}_2=\mathsf{\Gamma }{\mathsf{\Phi }}^{-1}\mathit{R}-\gamma \left(\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{R}+\mathit{R}\right)$$

$${\mathit{B}}_1=\left(\mathit{Q}+{\mathit{P}}^T{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }-\mathsf{\Gamma }\right)-\left({\gamma }^{-1}\mathit{R}+\gamma \left(\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{Q}-\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathsf{\Gamma }\right)\right)$$

$${\mathit{B}}_0=\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{P}-\mathsf{\Psi }-{\gamma }^{-1}(\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{R}+\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathit{Q}-\mathsf{\Psi }{\mathsf{\Phi }}^{-1}\mathsf{\Gamma })$$

where $0<\gamma <1$ is a constant and $\mathit{Z}$ contains the remaining term of matrices that are independent of $\rho$. If $\rho$ is chosen sufficiently large and $\gamma =1/2$, we can make all terms of the right-hand side of (32) become positive definite using the conditions given in (30). With that, it can be concluded that

$${\left|{\mathit{e}}_{k+1}\left(t\right)\right|}_{\rho }\le {\left|{\mathit{e}}_k\left(t\right)\right|}_{\rho }$$

(33)

which in light of (26) also implies that

$${\ddot{\mathit{d}}}_k\to 0,{\dot{\mathit{d}}}_k\to 0 as k\to \infty$$

(34)

Note that according to the condition (29), ${\mathit{x}}_0\left(t\right)\in {\mathbb{C}}^2\left[0,T\right],$which implies $\mathsf{\Delta }\left({\mathit{e}}_0\left(t\right)\right)\in {\mathbb{C}}^1\left[0,T\right]$ and also ${\mathit{d}}_0\left(t\right)\in {\mathbb{C}}^4\left[0,T\right]$. Hence, with the help of a similar argument in the derivation of (23), we obtain

$${\displaystyle \frac{d}{dt}}(\mathit{R}\left(t\right){\stackrel{⃛}{\mathit{d}}}_k+\mathit{Q}\left(t\right){\ddot{\mathit{d}}}_k+\mathit{P}\left(t\right){\dot{\mathit{d}}}_k)=\mathsf{\Delta }\left({\dot{\mathit{e}}}_k\left(t\right)\right)$$

(35)

Similarly, we have

$${\left|{\dot{\mathit{e}}}_{k+1}\left(t\right)\right|}_{\rho }\le {\left|{\dot{\mathit{e}}}_k\left(t\right)\right|}_{\rho }-{\int }_0^t{e}^{-\rho \tau }\left\{{\stackrel{⃛}{\mathit{d}}}_k^T{\mathit{A}}_2{\stackrel{⃛}{\mathit{d}}}_k+{\ddot{\mathit{d}}}_k^T{A}_1{\ddot{d}}_k+{\dot{\mathit{d}}}_k^T{\mathit{A}}_0{\dot{\mathit{d}}}_k+{\mathit{d}}_k^T\mathit{Z}{\mathit{d}}_k\right\}d\tau -e^{-\rho t}\left\{{\stackrel{⃛}{\mathit{d}}}_k^T{\mathit{B}}_2{\stackrel{⃛}{\mathit{d}}}_k+{\ddot{\mathit{d}}}_k^T{\mathit{B}}_1{\ddot{\mathit{d}}}_k+{\dot{\mathit{d}}}_k^T{\mathit{B}}_0{\dot{\mathit{d}}}_k\right\}$$

(36)

Applying the same argument as in the derivation of (34), it can be concluded that

$${\stackrel{⃛}{\mathit{d}}}_k\to 0,{\ddot{\mathit{d}}}_k\to 0,{\dot{\mathit{d}}}_k\to 0 as k\to \infty$$

(37)

Substituting this result into (23) yields

$$\mathsf{\Delta }\left({\mathit{e}}_k\left(t\right)\right)\to 0 as k\to \infty$$

(38)

From the expression for $\mathsf{\Delta }\left({\mathit{e}}_k\left(t\right)\right)$ and from the definition as in (31), we obtain the integral inequality

$$‖\mathsf{\Phi }{\mathit{e}}_k‖\le ‖\mathsf{\Delta }{\mathit{e}}_k‖+{\displaystyle \frac{d}{dt}}{\lambda }_2‖\mathsf{\Phi }{\mathit{e}}_k‖+{\int }_0^t{\lambda }_1‖\mathsf{\Phi }{\mathit{e}}_k‖d\tau$$

(39)

With the assumption that $‖\mathsf{\Phi }{\mathit{e}}_k‖\le ‖\mathsf{\Phi }{\mathit{e}}_k\left(0\right)‖e^{{\lambda }_{2}t}$, we apply the Bellman-Gronwall Inequalities (Appendix A) to obtain

$$‖\mathsf{\Phi }{\mathit{e}}_k‖(1-{\lambda }_2)\le ‖\mathsf{\Delta }{\mathit{e}}_k‖+{\lambda }_1{\int }_0^t‖\mathsf{\Delta }{\mathit{e}}_k‖e^{{\lambda }_{1}t}d\tau$$

(40)

Then, for any fixed $t\in \left[0,T\right]$, equation (26) implies that

$${\mathit{e}}_k\left(t\right)\to 0 as k\to \infty$$

(41)

Finally, the Ascoli-Arzela’s Theorem (Appendix B) can be invoked in the following way. Observing that ${\dot{\mathit{x}}}_d\left(t\right)-{\dot{\mathit{x}}}_k\left(t\right)$ is uniformly bounded, which implies the uniform boundedness and equicontinuity of ${\mathit{x}}_d\left(t\right)-x_k\left(t\right)$. In addition, since ${\mathit{x}}_d\left(t\right)-{\mathit{x}}_k\left(t\right)$ is pointwise convergent to zero, it follows from Ascoli-Arzela’s Theorem that ${\mathit{x}}_k\left(t\right)$ converges to ${\mathit{x}}_d\left(t\right)$ uniformly in $t\in \left[0,T\right]$. Thus, this completes the proof of Theorem 1.

5. Design Procedure for the Hybrid Learning Control Scheme

This section combines developments of the Feedback Controller in Section 3 and the Learning Controller in Section 4 to outline a design procedure for the HLCS.

First, for the Feedback Controller, we recall that a general 6 DOF CKCM robot manipulator with the dynamics given in (1) can be represented by a linearized model about a desired joint variable vector ${\mathit{l}}_d$ given in (2). After applying a PID feedback control law, the closed-loop system can be represented by (11), which can be semi-globally stabilized by employing tuning guidelines given in [24].

For the Learning Controller, comparing (15) and (2), we recognize that $\mathit{R}\left(t\right)$, $\mathit{Q}\left(t\right)$ and $\mathit{P}\left(t\right)$ in (15) could represent $\mathit{M}\left(t\right),\mathit{C}\left(t\right)$ and $\mathit{G}\left(t\right)$, respectively, and$\mathit{x}\left(t\right)$ in (15) represents $\mathit{l}\left(t\right)$. If Theorem 1 is employed to determine the gain matrices $\mathsf{\Gamma },\mathsf{\Phi },\mathrm{a}\mathrm{n}\mathrm{d}\mathsf{\Psi }$ of the ILC law so that ${\mathit{x}}_k\left(t\right)\to {\mathit{x}}_d\left(t\right)$, then ${\mathit{l}}_k\left(t\right)\to {\mathit{l}}_d\left(t\right)$.

Based on the above discussion, we obtain a design procedure for the HLCS as follows:

Algorithm 1. Design Procedure for HLCS

Step 1. Linearize the dynamics of the CKCM robot manipulator (1) about a desired joint variable vector ${\mathit{l}}_d\left(t\right)$ to obtain $\mathit{M}\left(t\right),\mathit{C}\left(t\right)$ and $\mathit{G}\left(t\right)$ of the linearized model.

Step 2. Use the tuning procedure in [24] to obtain ${\mathit{K}}_p$, ${\mathit{K}}_i$ and ${\mathit{K}}_d$ of the PID control law of the Feedback Controller to globally stabilize the closed-loop feedback system (11).

Step 3. Use Theorem 1 to obtain the gain matrices $\mathsf{\Gamma },\mathsf{\Phi },\mathrm{a}\mathrm{n}\mathrm{d}\mathsf{\Psi }$ of the ILC law so that ${\mathit{x}}_k\left(t\right)\to {\mathit{x}}_d\left(t\right)$ resulting in ${\mathit{l}}_k\left(t\right)\to {\mathit{l}}_d\left(t\right)$.

6. Computer Simulation Study of a 6-DOF CKCM Manipulator

This section presents a computer simulation study conducted to evaluate the effectiveness of the developed HLCS employed to control the motion of a 6-DOF CKCM manipulator. A brief kinematic and dynamical analysis of this manipulator will first be presented, followed by a presentation and discussion of the computer simulation results.

6.1. Inverse Kinematics and Dynamics of the 6-DOF CKCM Manipulator

Figure 3 shows the CKCM manipulator that possesses six DOFs and comprises a stationary base platform and a moving platform coupled together through six links. Each link is a linear actuator consisting of a DC motor and a linear ball screw drive and its ends are connected to the base and moving platforms through ball joints. A robotic end-effector can be attached to the moving platform to carry out tasks. The displacements of the six linear actuators result in the motion of the moving platform and its end-effector.

The manipulator inverse kinematics is defined as the determination of the lengths of the six actuators to produce a desired position and orientation of the moving platform. Appendix C containing a complete kinematic analysis of the above manipulator provides a closed-form solution of the inverse kinematics from (A.21) as

$$l_i^2=x^2+y^2+z^2+r_A^2+r_B^2+2\left(r_{11}{a}_{ix}+r_{12}{a}_{iy}\right)\left(x-b_{ix}\right)+2\left(r_{21}{a}_{ix}+r_{22}{a}_{iy}\right)\left(y-b_{iy}\right)+2\left(r_{31}{a}_{ix}+r_{32}{a}_{iy}\right)z-2\left(x{b}_{ix}+y{b}_{iy}\right)$$

(42)

where $l_i$for i = 1,2,---6 are the lengths of the six actuators, and $r_A$ and $r_B$ denote the radii of the moving and base platforms, respectively. Furthermore, $r_{ij}$ are the elements of the orientation matrix ${}_A{}^B\mathit{R}$ and $x, y, z$ denote the desired position of the moving platform with respect to the base platform. We note that Equation (42) indeed presents the closed-form solution of the inverse kinematics, in the sense that the actuator lengths $l_i$ for i = 1, 2…6 can be determined from the desired Cartesian configuration of the moving platform, specified by x, y, z,$\alpha , \beta , \gamma$.

Appendix D contains a dynamical analysis of the above CKCM manipulator. It employs the Lagrangian formulation to derive the dynamics of the manipulator in terms of the kinematic energy and the potential energy of the manipulator. In particular, Equation (A.31) represents the equations of motion of the manipulator expressing the relationship between the Cartesian force/torque $F_j$ applied to the moving platform and the Cartesian position and orientation of the moving platform ${\varphi }_j$., $\mathrm{f}\mathrm{o}\mathrm{r} j=\mathrm{1,2},\dots ,6$. We also see that (A.31) contains the three terms, the Inertial Term, the Centrifugal and Coriolis Term and the Gravity Term.

6.2. Computer Simulation Results

Table 1 presents the parameters of the 6-DOF CKCM manipulator employed in the computer simulation study. The simulations are conducted using SimMechanics™ (Version 2019), a toolbox of MATLAB/Simulink (Version 2019). This toolbox enables efficient modeling and visualization of real mechanical systems composed of multiple rigid bodies. Moreover, by leveraging the Simulink environment, SimMechanics™ facilitates the simulation of the physical behaviors and motions of the modeled systems.

The SimMechanics library represents physical components as interconnected blocks, explicitly capturing the geometric and kinematic relationships among the robot components. Consequently, this modeling approach does not require sophisticated mathematical formulations of the system dynamics, while providing a convenient and accurate means of deriving the dynamic model of the system. Figure 4 illustrates the SimMechanics™ model of the manipulator. The integration of the plant with the HLCS is illustrated in Figure 5. The orange blocks represent memory units that store commands from previous iterations, the gray blocks correspond to the feedback and learning controllers, and the cyan block denotes the robot manipulator (plant).

In the simulation program, the sampling period is fixed at 1 ms, and the numerical solver employed is the ODE8 (Dormand–Prince) method. The plant parameters and controller gains are listed in Table 1.

To evaluate the tracking performance of the HLCS, its simulation model is obtained and employed to track the manipulator end-effector along two different planar trajectories. The obtained results are then compared with those achieved using a PID feedback controller alone and with those achieved by a hybrid control scheme combining PID feedback with PD-type ILC.

Case 1: Tracking a Straight Line Motion

The straight-line trajectory to be tracked by the end-effector is defined by $x+y-2z+1=0$ (in meters) over a duration of 15 s. The simulation results for this case are presented in Figure 6(a)–6(d). Figure 6(a)–6(c) show the time responses of the leg tracking errors obtained using the PID controller, the PD-type ILC, and the developed PID-HLCS, respectively. Figure 6(d) compares the desired and actual motions achieved by the three control schemes.

As observed from the results, the HLCS yields substantial improvements in both transient and steady-state performance after only two learning trials, whereas the PD-type ILC requires four trials to reach a comparable level of performance. In Figure 6(c), the steady-state tracking error of the HLCS converges to approximately −0.22 cm within 0.6 s, while the responses of the other two controllers fail to converge after the transient phase.

Case 2: Tracking a Circular Motion

The circular trajectory to be tracked by the manipulator end-effector is defined by$x^2+(y-0.2{)}^2+(z+0.5{)}^2={0.2}^2$(in meters) over a duration of 5 s. Similarly, Figure 7(a)–7(c) illustrate the time responses of the leg tracking errors obtained using the PID controller, the PD-type ILC, and the proposed PID-type ILC, respectively. Figure 7(d) compares the desired and actual end-effector motions achieved by the three control schemes.

The circular path to be followed by the manipulator end-effector is described by $x^2+{\left(y-0.2\right)}^2+{\left(z+0.5\right)}^2={0.2}^2$ [in m] in 5s. Similarly, Figure 7a-c represent the time responses of the leg errors from the PID, PD ILC, and the developed HLCS, respectively. Figure 7d compares the actual and desired motions of all three controllers.

Figure 7(b) illustrates the improvement in tracking error achieved by the PD-type ILC; however, its transient response exhibits the largest degradation, as evidenced in the zoomed-in view of Figure 7(d). In contrast, Figure 7(c) demonstrates that the HLCS achieves satisfactory tracking performance after only three learning iterations.

Figure 8 depicts the error convergence across iterations, represented by the ρ-norm of the tracking error as a function of the iteration number. The results clearly indicate that the developed HLCS attains a faster convergence rate than the conventional PD-type ILC for both straight-line and circular trajectory tracking tasks.

7. Conclusions

This paper presents the development of the HLCS, a novel hybrid learning control strategy that integrates feedback control with learning control for the motion control of robot manipulators. The hybrid control scheme is designed for robot manipulators based on the CKCM architecture, which offers several attractive features, including the existence of inverse kinematics solutions, high-precision positioning capability, and inherent avoidance of joint error accumulation commonly associated with manipulators designed based on open kinematic chain mechanisms.

The HLCS mainly consists of two components: a Feedback Controller and a Learning Controller, each serving a distinct control function. The Feedback Controller employs a PID control law whose gains are selected using a linearization method to globally stabilize the closed-loop system. The Learning Controller stores joint error information and the corresponding control inputs during a trial and utilizes this information to improve both transient and steady-state responses of the manipulator motion in subsequent trials. Uniform convergence of the learning process is established through exponentially weighted supremum-norm analysis and a contraction mapping framework. Based on the development of the Feedback and Learning Controllers, a systematic design procedure for the HLCS is proposed.

Computer simulations are conducted to evaluate the control performance of the HLCS in comparison with a conventional PID control scheme without learning capability and a PD control scheme combined with iterative learning control (ILC). All control schemes are implemented to track the same straight and circular paths. Simulation results demonstrate that the HLCS outperforms the other two control schemes in terms of convergence rate and tracking accuracy.

Future work on this novel learning control scheme is recommended as follows:

• Experimental evaluation of the developed HLCS on real CKCM manipulators and comparison with other conventional learning control schemes. [25,26,27,28,29]
• Application of the HLCS to serial robot manipulators, with performance evaluation through both simulation and experimental validation.
• Extension of the hybrid concept proposed in this paper to integrate a feedback controller with intelligent control techniques, such as fuzzy control, neuro-fuzzy control, or neural networks, to enhance real-time control performance of robot manipulators.
• Investigation of the robustness and resilience of the developed HLCS.