A Novel Automated Framework for Networked Control of High-DOF Robot Manipulators: A Case Study of IRB140

In numerous cases, the behavior of physical systems could find clearer expression through analytical models. When it comes to robot modeling and analysis, the focal points encompass both its kinematics and dynamics. In this section, the kinematic and dynamic models for the IRB140, a 6 DOF robot manipulator, are formulated, and its workspace is investigated. These models enable precise manipulation of the arm, facilitating control over its movements to attain any viable position and orientation within an unstructured setting. Figure 2 shows the realistic representation of the IRB140.

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Figure 2. The IRB 140 robot manipulator [23]

2.1 Modeling set up

The industrial robot manipulator named IRB 140 has been manufactured by ABB. Their website [23] provides information about the manipulator, along with articles and videos showcasing experiments and the manipulator's application in various companies. Figure 3 offers a distinct perspective of the manipulator, illustrating its degrees of freedom clearly.

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Figure 3. The IRB 140 degrees of freedom [23, 24]

The IRB 140 manipulator consists of six links (excluding the base) and six revolute joints, which are rotary joints controlled by AC motors and depending on an angle θ, thereby resulting in six degrees of freedom. The initial three DOF are situated within the arm, facilitating the determination of the robot's position. Subsequently, the remaining three DOF are situated within the end effectors, enabling the provision of orientation.

Upon closer examination of the robot, it becomes evident that there is some flexibility in how to approach the modeling of joint 4. In fact, opting to model the final three joints might not be the optimal choice due to the nonzero length and mass of the intermediary links (link 4 and 5). To address this, a practical approach is to reinterpret the manipulator configuration, aligning the center point of joint 3 and 4, as well as the center point of joint 5 and 6. Consequently, link 3 and link 5 are then depicted with zero length and mass to accommodate this arrangement. It is important to emphasize the necessity of reducing the computational complexity of the dynamic model without compromising its accuracy for industrial applications. By aligning these joint centers, the model leverages the geometric symmetry of the manipulator to simplify the derivation of kinematic and dynamic equations, particularly in high-DOF systems. This approach also aids in minimizing redundancy in the parameter estimation process, making the model more computationally efficient while maintaining fidelity to the real-world behavior of the manipulator. Figure 4 presents a symbolic depiction of the manipulator according to this interpretation, illustrating the attachment of frames to the links.

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Figure 4. The IRB 140 symbolic representation

Every rotation matrix can be computed as the outcome of combining elemental rotations around the z-axis and the x-axis. These fundamental rotation matrices are expressed in a general form as follows:

$\begin{aligned} & R_{z, \theta}=\left[\begin{array}{ccc}\cos (\theta) & -\sin (\theta) & 0 \\ \sin (\theta) & \cos (\theta) & 0 \\ 0 & 0 & 1\end{array}\right], R_{x, \theta} =\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos (\theta) & -\sin (\theta) \\ 0 & \sin (\theta) & \cos (\theta)\end{array}\right]\end{aligned}$        (1)

Examining Figure 4 makes it relatively simple to compute the rotation matrices for the manipulator. This can be achieved by plugging in the values of q and the fixed rotations for θ. It's important to recognize the simplifications arising from the fact that all constant rotations are in multiples of π/2. The outcome is as follows:

$R_1^0=\left[\begin{array}{ccc}\cos \left(q_1\right) & 0 & -\sin \left(q_1\right) \\ \sin \left(q_1\right) & 0 & \cos \left(q_1\right) \\ 0 & -1 & 0\end{array}\right]$          (2)

$R_2^1=\left[\begin{array}{ccc}\cos \left(q_2-\frac{\pi}{2}\right) & -\sin \left(q_2-\frac{\pi}{2}\right) & 0 \\ \sin \left(q_2-\frac{\pi}{2}\right) & \cos \left(q_2-\frac{\pi}{2}\right) & 0 \\ 0 & 0 & 1\end{array}\right]$       (3)

$R_3^2=\left[\begin{array}{ccc}\cos \left(q_3\right) & 0 & -\sin \left(q_3\right) \\ \sin \left(q_3\right) & 0 & \cos \left(q_3\right) \\ 0 & -1 & 0\end{array}\right]$      (4)

$R_4^3=\left[\begin{array}{ccc}\cos \left(q_4\right) & 0 & \sin \left(q_4\right) \\ \sin \left(q_4\right) & 0 & -\cos \left(q_4\right) \\ 0 & -1 & 0\end{array}\right]$          (5)

$R_5^4=\left[\begin{array}{ccc}\cos \left(q_5+\frac{\pi}{2}\right) & 0 & \sin \left(q_5-\frac{\pi}{2}\right) \\ \sin \left(q_5+\frac{\pi}{2}\right) & 0 & -\cos \left(q_5-\frac{\pi}{2}\right) \\ 0 & 1 & 0\end{array}\right]$      (6)

$R_6^5=\left[\begin{array}{ccc}\cos \left(q_6\right) & -\sin \left(q_6\right) & 0 \\ \sin \left(q_6\right) & \cos \left(q_6\right) & 0 \\ 0 & 0 & 1\end{array}\right]$       (7)

2.2 Parameters estimation

This section outlines the process of estimating the dynamic parameters. It's noted that these estimations are challenging due to the constraints of limited available information. However, they are aimed at achieving a proximity to the actual unknown parameters, allowing simulations to exhibit behavior somewhat consistent with that of an ideal model. For sure, estimating the inertia parameters represents the most challenging task. The intricate shapes of the links combined with the scarcity of available data significantly complicate the process of deriving accurate parameters without resorting to some form of identification. As a justifiable simplification, the links are conceptualized as cylindrical structures with uniform mass distribution, wherein the center of mass for each link corresponds to the geometric center of the cylinder. Figure 5 serves as an illustration of how this simplification can be implemented, using link 2 as an example.

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Figure 5. Simplified modeling example of link 2

The diagram in light blue depicts the rear view of link, with the center of mass denoted by the red dot. The inertia tensor for this cylinder is demonstrated [25] as:

$I=\left[\begin{array}{ccc}\frac{1}{12} m h^2+\frac{1}{4} m r^2 & 0 & 0 \\ 0 & \frac{1}{12} m h^2+\frac{1}{4} m r^2 & 0 \\ 0 & 0 & \frac{1}{2} m r^2\end{array}\right]$         (8)

where, m, r and h represent the mass, the radius and the height of the link. The cross products are consistently zero, resulting in the inertia tensor adopting a diagonal matrix configuration in its principal axis formulation.

Finding the mass, radius, and height of the cylinders involves working within specific limitations. These limitations dictate that the combined mass should be 97kg. Additionally, the cylinders' radius and height must align with the dimensions of the manipulator outlined in Figure 4. Similar to the physical links, the cylinders will exhibit overlapping due to the non-central placement of their centers of mass between frames. These considerations were made under the assumption of uniform mass distribution.

According to the study [24], it is reasonable to assume that the mass density of each link is approximately equal. The links are constructed with a metal shell containing internal components like motors, gearboxes, cables, and belts. Additionally, a significant portion of the total volume is simply air between these components. Through a trial-and-error approach, the masses, radius, and heights were eventually determined to match the physical shape of the manipulator using a uniform mass density.

2.3 Dynamic modeling

Dynamic modeling is crucial for understanding how the robot manipulator responds to external forces, such as gravity, friction, or applied loads. It involves deriving equations of motion that govern the robot's behavior, accounting for its mass distribution, inertial properties, and joint configurations [26]. The proposed process based on Newton-Euler Method relies on Newton's laws of motion and recursive algorithms to calculate the forces and torques acting on each link. It involves propagating the forces and torques from the end-effector back to the base of the robot, taking into account the kinematics of the robot.

2.3.1 Forward recursion

The process of forward recursion delineates the progression of linear and angular motion through the series of links, commencing from link 1 and ending at link 6. Within this framework, a crucial step involves the determination of $b_i$ the rotational axis for each joint i as denoted in frame i. Preceding the initiation of the recursions, these calculations will be promptly executed for all joints, underscoring a significant benefit of the Newton-Euler formulation.

Considering the study [27], angular velocity and acceleration can be calculated as follows:

$\omega_i=\left(R_i^{i-1}\right)^T \omega_{i-1}+b_i \dot{q}_i$            (9)

where,

$b_i=\left(R_i^0\right)^T R_{i-1}^0 z_0$          (10)

$a_i=\left(R_i^{i-1}\right)^T a_{i-1}+b_i \ddot{q}_i+\omega_i \times b_i \dot{q}_i$         (11)

The equation describing the acceleration of the center of mass of link i, as expressed within frame i, transforms into:

$\alpha_{c, i}=\left(R_i^{i-1}\right)^T \alpha_{e, i-1}+\dot{\omega}_i \times r_{i-1, c i}+\omega_i \times\left(\omega_i \times r_{i-1, c i}\right)$         (12)

To determine the acceleration of the end of the link, $r_{i-1, c i}$ is substituted with $r_{i-1, i}$.

$\alpha_{\mathrm{e}, \mathrm{i}}=\left(\mathrm{R}_{\mathrm{i}}^{\mathrm{i}-1}\right)^{\mathrm{T}} \alpha_{\mathrm{e}, \mathrm{i}-1}+\dot{\omega}_{\mathrm{i}} \times \mathrm{r}_{\mathrm{i}-1, \mathrm{i}}+\omega_{\mathrm{i}} \times\left(\omega_{\mathrm{i}} \times \mathrm{r}_{\mathrm{i}-1, \mathrm{i}}\right)$          (13)

The rotational axis within frame 0 is directly provided as follows:

$z_0=\left[\begin{array}{lll}0 & 0 & 1\end{array}\right]^{\mathrm{T}}$          (14)

Subsequently, the computation of the rotation axes for the joints is as follows:

$b_1=\left(R_1^0\right)^T z_0=\left[\begin{array}{lll}0 & -1 & 0\end{array}\right]^T$          (15)

$b_2=\left(R_2^0\right)^T R_1^0 z_0=\left[\begin{array}{lll}0 & 0 & 1\end{array}\right]^T$          (16)

$b_3=\left(R_3^0\right)^T R_2^0 z_0=\left[\begin{array}{lll}0 & -1 & 0\end{array}\right]^T$        (17)

$b_4=\left(R_4^0\right)^T R_3^0 z_0=\left[\begin{array}{lll}0 & 1 & 0\end{array}\right]^T$           (18)

$b_5=\left(R_5^0\right)^T R_4^0 z_0=\left[\begin{array}{lll}0 & 1 & 0\end{array}\right]^T$        (19)

$b_6=\left(R_6^0\right)^T R_5^0 z_0=\left[\begin{array}{lll}0 & 0 & 1\end{array}\right]^T$          (20)

Because of the interrelated nature of kinematics, these rotation axes typically become functions of q, akin to the rotation matrices. Their behavior is contingent upon the specifications of the coordinate frames, thus exerting a direct impact on the effectiveness of the Newton-Euler formulation. Upon examining the frame definitions in Figure 4, it becomes evident that when observing from frame i to frame i-1, the angular velocity ω_i is unaffected by q_i itself; rather, it hinges entirely on the rotational axis. As a result, the rotation axes b_i  remain independent of q.

Link 1

The starting parameters are as follows:

$\omega_0=\alpha_0=\alpha_{\mathrm{c}, 0}=\alpha_{\mathrm{e}, 0}=0$          (21)

According to Eqs. (9-11), angular velocity and acceleration are calculated as follows:

$\omega_1$=${b_1}$ $\dot{{q}_1}$         (22)

$\alpha_1=b_1 \ddot{q}_1+\omega_1 \times b_1 \dot{q}_1$        (23)

According to Eq. (12) and Eq. (13), acceleration of the end of the link and acceleration of the center of mass of the link are calculated as follows:

$\alpha_{e, 1}=\dot{\omega}_1 \times r_{0,1}+\omega_1 \times\left(\omega_1 \times r_{0,1}\right)$          (24)

$\alpha_{c, 1}=\dot{\omega}_1 \times r_{0, c 1}+\omega_1 \times\left(\omega_1 \times r_{0, c 1}\right)$           (25)

Link 2

$\omega_2=\left(R_2^1\right) \omega_1+b_2 \dot{q}_2$          (26)

$\alpha_2=\left(R_2^1\right) \alpha_1+b_2 \ddot{q}_2+\omega_2 \times b_2 \dot{q}_2$          (27)

$\alpha_{e, 2}=\left(R_2^1\right) \alpha_{e, 1}+\dot{\omega}_2 \times r_{1,2}+\omega_2 \times\left(\omega_2 \times r_{1,2}\right)$        (28)

$\alpha_{c, 2}=\left(R_2^1\right) \alpha_{e, 1}+\dot{\omega}_2 \times r_{1, c 2}+\omega_2 \times\left(\omega_2 \times r_{1, c 2}\right)$       (29)

And so on till the last link:

Link 6

$\omega_6=\left(R_6^5\right) \omega_5+b_6 \dot{q}_6$        (30)

$\alpha_6=\left(R_6^5\right) \alpha_5+b_6 \ddot{q}_6+\omega_6 \times b_6 \dot{q}_6$           (31)

$\alpha_{c, 6}=\left(R_5^6\right) \alpha_{e, 5}+\dot{\omega}_6 \times r_{5, c 6}+\omega_6 \times\left(\omega_6 \times r_{5, c 6}\right)$         (32)

2.3.2 Backward recursion

The backward recursion computes the forces and joint torques that influence the links, starting from the first link and concluding at the last one. The primary objective of the Newton-Euler formulation is to establish the joint torques, as these torques constitute the external inputs to the model.

Based on the law of action and reaction, and according to the study [28] the force balance equation expressed in frame i can be stated as:

$\sum_{l i n k} f=m a$          (33)

$f_i=R_{i+1}^i f_{i+1}+m_i \alpha_{c, i}-m_i g_i$        (34)

This principle also applies to torque, where the equation for moment balance can be formulated as follows:

$\sum_{\text {link }} \tau=\mathrm{I} \dot{\omega}+\omega \times \mathrm{I} \omega$          (35)

$\begin{gathered}\tau_i=R_{i+1}^i \tau_{i+i}-f_i \times r_{i-1, c i}+\left(R_{i+1}^i f_{i+1}\right) \times r_{i, c i}+\omega_i \times \left(I_i \omega_i\right)+I_i \alpha_i\end{gathered}$         (36)

It's important to observe that the force equation takes into account the gravitational vector. This vector varies for each link, yet its computation can be readily accomplished using rotation matrices, as exemplified in the subsequent recursions.

The final condition is:

$F_7=\tau_7=0$         (37)

Link 6

The gravity vector transforms into:

$g_6=\left(R_6^0\right)^T g_0$          (38)

where,

$g_0=\left[\begin{array}{ll}0 & 0-g\end{array}\right]^T$       (39)

The force and joint torque applied to the link are determined according to Eq. (34) and Eq. (36) as follows:

$f_6=m_6 \alpha_{c, 6}-m_6 g_6$          (40)

$\tau_6=-f_6 \times r_{5, c 6}+\omega_6 \times\left(I_6 \omega_6\right)+I_6 \alpha_6$          (41)

Link 5

$g_5=\left(R_5^0\right)^T g_0$        (42)

$f_5=R_6^5 f_6$          (43)

$\tau_5=R_6^5 \tau_6-f_5 \times r_{4, c 5}+\omega_5 \times\left(I_5 \omega_5\right)+I_5 \alpha_5$          (44)

And so on till the first link:

Link 1

$g_1=\left(R_1^0\right)^T g_0$             (45)

$f_1=R_2^1 f_1+m_1 \alpha_{c, 1}-m_1 g_1$          (46)

$\begin{gathered}\tau_1=R_2^1 \tau_2-f_1 \times r_{0, c 1}+R_2^1 f_2 \times r_{1, c 1}+\omega_1 \times\left(I_1 \omega_1\right)+I_1 \alpha_1\end{gathered}$         (47)

The findings in this section are compelling and validate the reasons behind the frequent preference for the Newton-Euler formulation in manipulators with numerous degrees of freedom. The recursive algorithm is straightforward to deploy, which minimizes the likelihood of errors during derivation. Anomalies in the model's behavior generally trace back to the setup stages, encompassing aspects like kinematic chain configuration, frame definitions, rotation matrices, vector delineations, and inertia tensors.

It's important to acknowledge that even though link 3 and 5 possess negligible length and mass, they must still be taken into account during the recursions. The Newton-Euler formulation is established on a kinematic chain featuring single degree-of-freedom joints, ensuring that there are always 'n' steps in each recursion for 'n' degrees of freedom. Nevertheless, specific terms within the expressions for link 3 and 5 are nullified.

Another intriguing observation within the Newton-Euler formulation emerges from the terminal joint torque vectors during the reverse recursion process. In this kinematic chain, all joints are characterized by having a solitary degree of freedom. Consequently, the torques exerted are scalar values revolving around the rotation axes calculated through Eqs. (15-20). The remaining two components of the torque vectors can be elucidated as follows: Whenever torque is applied to any of the joints, it inevitably generates torque components around the other axes of the joints due to the interconnected kinematics within the system. While these torque components are not integrated into the dynamic model because they don't induce motion (and thus don't impact q), they still offer valuable insights into the manipulator's physical dynamics. If the joints in the manipulator aren't constructed to endure these specific torque magnitudes, there's a risk of joint failure.

Though the application of the Newton-Euler formulation may appear to be a relatively straightforward process, it is essential to emphasize the inherent complexity within the resulting model. The fundamental principle behind recursion is that solving a larger problem is dependent on resolving smaller instances of the same problem. The backward recursion for link 1 is built upon the backward recursion for link 2, and so forth. Consequently, the backward recursion for link 1 is basically influenced by all the preceding 11 steps of the forward recursion for the same link. This connection leads to the computation of $\tau_1$ resulting in a significantly extensive vector.

As introduced in the first section, classical control theory operates under the assumption that communication between sensors, actuators, and controllers is perfect, where data is transmitted and processed instantaneously and with infinite accuracy. However, in reality, digital communication (and computation) introduces some delay and limited precision due to physical constraints. Despite this, this assumption remains valid when the hardware used for control is significantly faster than the system dynamics. However, the assumption of sufficiently fast hardware can be (extremely) expensive to accommodate in practice, and in some situations, this assumption simply cannot be met. An emerging solution is to develop and use NCS theory to be able to specify under what conditions slower, less expensive, hardware can be used reliably in the sense of still guaranteeing proper closed-loop behavior.

This section presents results which work towards making this option possible by contributing towards the development of NCS theory for controlling high-DOF robot manipulators such as the presented case study the IRB 140.

With the purpose of achieving this objective, the suggested approach consists of three sequential phases:

1. Initially, it's crucial to develop an effective control algorithm that can enable the desired movements of the end-effector while maintaining both safety and operational efficiency. Given the complexity of the resulting model, it becomes evident that robust and adaptive control approaches are indispensable. The chosen control method may differ according to the specific application depending on the robot’s kinematics and dynamics.
2. The following step consists of integrating the network communication in the feedback control system. The process must be done with careful consideration of various factors, including: network delay, jitter (variability in delay), packet loss probability, bandwidth, network topology etc. Moreover, the choosing of the network type may also differ depending on the physical environment where the network will operate, the network size and scalability, and also cost and budget constraints.
3. The final phase involves minimizing the impact of network delay and packet dropout in the closed-loop feedback system. Reducing network delay in the control system typically requires a combination of hardware, software, and network infrastructure optimizations. The specific approach will depend on the application and the criticality of real-time control in the system. It's essential to assess the requirements and constraints of the control system and custom the solutions accordingly. On the other hand, reducing packet dropout in a control system often requires a combination of network design, communication protocols, and error-handling mechanisms. Regular monitoring and maintenance are also keys to ensuring that packet dropout remains within acceptable limits.

4.1 Adaptive PD control with gravity compensation

High-DOF manipulators involve nonlinear coupled dynamics and high computational demands as detailled in the previous section. The proportional-derivative (PD) component provides straightforward implementation for fast error correction, while the adaptive term compensates for unmodeled dynamics and variations in system parameters. Compared to more complex control strategies, the adaptive robust PD approach achieves high performance without excessive computational overhead, making it suitable for real-time applications in high-DOF systems. It is quite noteworthy that the PD scheme employed for set-point control demonstrates its effectiveness even in the broader context of a system model represented by Eq. (48). This assertion can be substantiated through a Lyapunov stability analysis, as detailed in references [29, 30]. The proof relies on the concept of independent joint control, whereby each joint is managed as a distinct single-input/single-output system. Upon integrating PD controllers into the model, the input torque 'u' can be expressed in vector form as follows:

$u=K_p\left(q_{r e f}-q\right)-K_d \dot{q}=-K_p \tilde{q}-K_d \dot{q}$           (66)

where, $\tilde{q}$ represents the error between the desired joint positions and the current joint values, while $K_p$ and $K_d$ stand as diagonal matrices with positive definite values, representing the proportional and derivative gains, respectively.

We can make the assumption that gravitational acceleration remains constant and is a known value, allowing us to explicitly compute g(q) for any given instant. By incorporating g(q) into the input, we can accomplish gravity compensation, resulting in the complete system model being expressed as follows:

$M(q) \ddot{q}+C(q, \dot{q}) \dot{q}+g(q)=u$         (67)

$M(q) \ddot{q}+C(q, \dot{q}) \dot{q}+g(q)=-K_p \tilde{q}-K_d \dot{q}+g(q)$           (68)

$M(q) \ddot{q}+C(q, \dot{q}) \dot{q}=-K_p \tilde{q}-K_d \dot{q}$          (69)

In order to demonstrate that the input torque provided in Eq. (66) accomplishes asymptotic tracking, the following Lyapunov function candidate is examined:

$V=\frac{1}{2} \dot{q}^T M(q) \dot{q}+\frac{1}{2} \tilde{q}^T K_p \tilde{q}$           (70)

In the context of the manipulator, V signifies the overall energy that would be present if the actuators were exchanged for springs characterized by stiffness constants denoted as $K_p$, and positioned at the equilibrium state defined by $q_{\text {ref }}$. Consequently, V maintains a positive value except when the system is precisely at the equilibrium position where $q=q_{\text {ref }}$ and $\dot{\mathrm{q}}=0$, resulting in V being zero at that specific point. Demonstrating that ' V ' diminishes during any motion suggests that the robot is progressing towards this equilibrium position. Given that $q_{\text {ref }}$ remains constant, the derivative of V is expressed as follows:

$\dot{V}=\dot{q}^T M(q) \ddot{q}+\frac{1}{2} \dot{q}^T \dot{M}(q) \dot{q}+\dot{q}^T K_p \tilde{q}$          (71)

By solving for $M(q) \ddot{q}$ in Eq. (65) and then substituting into Eq. (71), we derive the following:

$\begin{gathered}\dot{V}=\dot{q}^T(u-C(q, \dot{q}) \dot{q}-g(q))+\frac{1}{2} \dot{q}^T \dot{M}(q) \dot{q}+\dot{q}^T K_p \tilde{q} \\ =\dot{q}^T\left(u-g(q)+K_p \tilde{q}\right)+\frac{1}{2} \dot{q}^T[\dot{M}(q)-2 C(q, \dot{q})] \dot{q} \\ =\dot{q}^T\left(u-g(q)+K_p \tilde{q}\right)\end{gathered}$        (72)

In this context, $\dot{M}(\mathrm{q})$ is a skew-symmetric matrix that can provide the following outcome:

$\dot{q}^T[\dot{M}(q)-2 C(q, \dot{q})] \dot{q}=0$            (73)

Replacing the input torque from Eq. (66) with u in the Eq. (72) results in:

$\dot{V}=-\dot{q}^T K_d \,\,\,\dot q {\leq} 0$         (74)

The analysis presented above demonstrates that V decreases as long as $\dot{q}$ is not equal to zero.

Furthermore, it is essential to establish that the manipulator cannot attain a state in which $\dot{q}=0$ but $q \neq q_{r e f}$. Assuming $\dot{V}$ equals zero, signifying that $\dot{V}$ remains zero at all moments. Given that $K_d$ is a positive definite value, this infers that $\dot{\mathrm{q}}$ is equal to zero, and consequently, $\ddot{\mathrm{q}}$ is also zero. Upon substituting this condition into the system model Eq. (69), the outcome is as follows:

$0=-K_p \tilde{q}$           (75)

This suggests that $\tilde{q}=0$, and subsequently, La Salle's theorem establishes the global asymptotic stability of the equilibrium position $q=q_{r e f}$.

When incorporating adaptive PID tuning for a robot manipulator, it is essential to take into account the specific characteristics of the manipulator, including its kinematics, dynamics, and the type of variations in operating conditions. Additionally, it is crucial to monitor and adjust the adaptation rates to avoid excessive adjustments to parameters, which may result in instability. For this purpose, the ant colony optimization algorithm (ACO) is a successful evolutionary meta-heuristic algorithm rooted in a graph representation [31], effectively addressing challenging combinatorial optimization problems. ACO fundamentally approaches the problem by framing it as the quest for a minimum-cost path within a graph. Artificial ants traverse this graph, actively seeking favorable paths. Individual ants exhibit straightforward behaviors, often identifying suboptimal paths independently. However, superior paths emerge as a collective outcome of the collaborative efforts of ants within the colony.

With the aim of achieving optimal control performances through the adaptive tuning of controller parameters, the chosen objective function is the integral of time-weighted absolute error (ITAE) criterion [32], which is expressed as follows:

$J=\int_0^{\infty} t|e| d t$            (76)

The optimization problem can be represented as the following:

Minimize J under the constraint of:

$K_p^{\min } \leq K_p \leq K_p^{\max }$          (77)

$K_d^{\min } \leq K_d \leq K_d^{\max }$         (78)

In the optimization process, after completing a tour, each ant revises the pheromones left on the paths it traversed and adjusts the rules according to the following:

$\tau(k)_{i j}=\tau(k-1)_{i j}+\frac{0.01 \theta}{J}$         (79)

Here, $\tau(k)_{i j}$ represents the pheromone value between nests $i$ and $j$ at the $k^{\text {th }}$ iteration, $\theta$ is the general coefficient governing pheromone updating, and $J$ denotes the cost function for the tour undertaken by the ant. The pheromones on the path associated with both the best and worst tours of the ant colony undergo updating according to the following procedure:

$\tau(k)_{i j}^{ {worst }}=\tau(k)_{i j}^{ {worst }}-\frac{0.3 \theta}{J_{ {worst }}}$            (80)

$\tau(k)_{i j}^{ {best }}=\tau(k)_{i j}^{ {best }}+\frac{\theta}{J_{b e s t}}$         (81)

In this context, $\tau^{\text {best }}$ and $\tau^{\text {worst }}$ represent the pheromones on the paths taken by the ant during the tour with the lowest cost value $J_{\text {best }}$ and the tour with the highest cost value $J_{\text {worst }}$ in a single iteration, respectively. Following pheromone evaporation, the ant algorithm discards its historical information and shifts its focus towards exploring new directions, avoiding entrapment in local minima, as follows:

$\tau(k)_{i j}=0 \tau(k)_{i j}^\lambda+\left[\tau(k)_{i j}^{ {best }}+\tau(k)_{i j}^{ {worst }}\right]$          (82)

Here, $\lambda$ denotes the evaporation constant.

4.2 PROFINET based network integration

PROFINET, which stands for Process Field Network, is an industrial Ethernet standard used in automation and control systems. It is designed to facilitate real-time communication and data exchange in industrial environments, providing a foundation for the implementation of NCS. In our case study where a high-DOF robot manipulator is the controlled plant which has demanding performance requirements, the choosing of PROFINET relays mainly on the fact that is offers high-speed communication, enabling efficient data exchange. Additionally, PROFINET’s network protocol ensures deterministic behavior, meaning that communication delays are predictable and can be tightly controlled [33]. Table 1 summarizes the protocols features for the chosen network.

Control network implementation transforms the structure of robot manipulator control, expands its applicability, and facilitates the realization of distributed control. Figure 6 illustrates a standard setup of a control system utilizing PROFINET.

The presented setup comprises two components: a network closed-loop system and a local closed-loop system. The controller initiates bus communication by transmitting the position reference to the actuators. Subsequently, the actuators employ this position reference to implement closed-loop control of the manipulator motors positions. Simultaneously, sensors periodically provide feedback data to the controller which then utilizes it to calculate a new position, initiating a cyclical repetition of the process. It is then evident that the number of sensors, actuators and motors is proportional to the number of DOF.

Table 1. PROFINET Protocols features

6.png

Figure 6. Control system implementation with PROFINET

4.3 Time delay consideration

Transmission of data through industrial networks introduces time delays in control loops due to factors such as network topology, traffic, and distance. In a PROFINET network, minimizing the NCS time delay is essential for achieving faster and more responsive communication among devices. However, in the NCS implementation, there are two types of time delay terms within the network. The first is a feedback time delay term $delay$$_{s-c}$ associated with the data transmission path from sensor to controller. The second is a feedforward time delay term $delay$$_{c-a}$ related to the data transmission path from controller to actuator. Figure 7 illustrates these two network delays.

7.png

Figure 7. NCS-induced delay

Any delay in the controller can be assimilated into either of the two terms without compromising generality. A practical perspective acknowledges that these two transmission delays fluctuate for each data transfer. Consequently, it is anticipated that a controller design approach with adaptive capabilities will help in adjusting to variable time delays. Nevertheless, this alone is generally insufficient for ensuring the preservation of the desired control system performance.

4.3.1 Network delays on S-C path

During each sampling period $T_s$, sensor data are emitted after being processed by sensor task $J_s$, the data wait acquisition by the S-C network task $J_{n e t \_s c}$. The start time of sensor task $n$ in the $k_{t h}$ control cycle is determined as follows:

$S_s^n(k)=S_s^n(0)+k T_s$         (83)

With global clock synchronization, sensors can be configured such that their start times are identical. Knowing that the sync signal cycle can align with the bus cycle, an input shift delay is introduced after the sync event, and inbound data are secured once the delay has expired. This delay must be set to ensure that sufficient time for sensor data processing is reserved, allowing the sensor data to be available prior to the arrival of the S-C frames which evidently will minimize the waiting time for sensor data [34].

Subsequently, sensor data are saved in the network slave controllers but are not transmitted until the following bus cycle if clock synchronization is not used. This raises the potential for an extra waiting delay $T_w^{J_s^n \rightarrow J_{n e t \_s c}}(k)$, which is between 0 and $T_s$. Clock synchronization results in dependable and predictable system behavior. In order to reduce the latency between the sensor task $J_s$ and $J_{\text {net_sc }}$, clock synchronization would be ideal. Taking into account both situations, the bus cycle's start time can be expressed as follows [35]:

$\begin{aligned} S_{n e t_{-} s}(k)=S_s^n(k) & +D_s+(1 \left.-\lambda_{\text {sync}}\right) T_w^{J_s^n \rightarrow J_{n e t \_s c}}(k)\end{aligned}$         (84)

where, $\lambda_{ {sync }}$ is a boolean parameter that indicates the use of clock synchronization, and $D_s$ is the sensor execution time. When clock synchronization is used, $\lambda_{ {sync}}=1$. In both situations, the cyclic communication structure eliminates the need for a local queue on sensors.

All sensor data is gathered in the network master controller at the end of a communication cycle. In order to read the sensor input data and begin calculating actuator set-points using the PD algorithm, the network master controller sends an interrupt to the controller. The controller start time is expressed as follows:

$S_c(k)=S_{n e t_{-} s c}(k)+D_{n e t \_s c}^n+T_w^{J_{n e t} s c \rightarrow J_c}(k)$         (85)

where, $T_w^{J n e t_{-} s c \rightarrow J_c}$ is the time required by the controller to retrieve the data, and $D_{\text {net_sc}}^n$ is the network delay for the data collected at sensor $n$ to reach the controller.

The final S-C network delay can be determined by measuring the time interval between the end of the sensor task $J_s^n$ and the beginning of the controller task $J_c$, i.e.

$\begin{gathered} { delay }_{s c, { network }}^n(k)=S_c(k)-\left(S_s^n(k)+D_s\right) =D_{ {net_{-}sc }}^n+\left(1-\lambda_{s y n c}\right) T_w^{J_s^n \rightarrow J_{\text {net_sc }}}(k) +T_w^{J_{n e t \_s c} \rightarrow J_c}(k)\end{gathered}$        (86)

4.3.2 Network delays on C-A path

The $C-S$ network task $J_{n e t_{-} c a}$ can begin once the controller completes its computation and the outgoing process data have been transferred to the network.

$S_{n e t_{-} c a}(k)=S_c(k)+P_c+T_w^{J_c \rightarrow J_{n e t_{-} c a}}(k)$            (87)

where, $P_c$ the actuation deadline task. Upon receiving the output data from the network via $J_{e_{-} c a}$ the task of actuator $n$ is initiated:

$S_a^n(k)=S_{n e t_{-} c a}(k)+D_{{n e t_{-}c a }}^n$           (88)

where, $D_{ {n e t_{-}c a }}^n$ is the communication delay between the beginning of the bus cycle and the actuator $n$ receiving the output data. After a bus cycle is initiated, the actuators typically get data over the network in less than a full bus cycle. The node index $n$ determines how each actuator endure $D_{ {n e t_{-}c a }}^n$. In contrast, $D_{ {n e t_{-}s c }}^n$ is the same for every sensor in the case of clock-driven S-C delay.

The actuators begin execution as soon as their network slave controllers receive data, with actuation occurring within $D_a$ (actuator execution time). By disregarding the actuators data access time, the C-A network delay can be calculated as:

$\begin{gathered} { delay }_{ {ca,network }}^n(k)=S_a^n(k)-\left(S_c(k)+P_c\right) \\ =D_{ {net_{-}ca }}^n+T_w^{J_c \rightarrow J_{{net_{-}ca }}}(k)\end{gathered}$          (89)