Optimization of Fuzzy-PD Control for a 3-DOF Robotics Manipulator Using a Back-Propagation Neural Network

Robotics manipulators play an important role within a human daily-life activity. Due to their flexible deployments in various domains, they offer a wide range of usage in different applications. These include industry production, surgical intervention, space explorations and military operations. In real-world applications, the robotics arm should essentially trace a defined manifold, which are required to determine an accurate position control [1]. Robotic arms have multiple joints and links, making their kinematics complex. Accurate motion planning is required to ensure the end effector (gripper) reaches its intended position and orientation. Inverse kinematics, which involves finding joint angles to achieve a desired end effector pose, can be particularly challenging. Controlling the robotic arm's movements requires considering its dynamic behavior, including inertia, gravity, and friction. Developing robust control algorithms to account for these factors ensures smooth and accurate motion execution. Controlling a robotic arm can be challenging due to several factors like:

1. High dimensionality: Robotic arms typically have multiple joints and links, resulting in a high-dimensional control space. Coordinating and controlling each joint simultaneously require complex algorithms and precise coordination.
2. Nonlinearity and complex dynamics: The dynamics of robotic arms are nonlinear and can be highly complex, involving interactions between joint angles, torques, and end effector positions. Designing control strategies to account for these complexities can be difficult.
3. Uncertainty and variability: Real-world environments are often unpredictable and uncertain. Variations in the robot's mechanical components, changing load conditions, or environmental disturbances can affect the arm's performance. Dealing with uncertainty requires robust control techniques.

The traditional Proportion-Integration-Differentiation (PID) control approach has evolved and been combined with a developed control method, with several applications in robotic controls [2, 3]. Because of adjusting control improvement consumes time and it depends on user’s experience, an advanced control system should be linked with PID to give an intelligent control gain adjustment. Xiong and Liu [4] studied that traditional PID can be combined with Neural Network so that it learns the best fitting parameters. Fuzzy Logic is usually used to optimize the control gains of classical PID due to the benefits of fuzzification mechanism [5-9]. Fuzzification, defuzzification, and fuzzy reasoning are common components of Fuzzy Logic [10-14]. As the link between traditional crisp and fuzzy arithmetic, the Membership Function (MF) is considered the main part in this procedure. According to the main design of the Fuzzy Logic concept, the fundamental of logical function is to discover the best membership function and fuzzy inference approaches. Therefore, there is not a conventional procedure for transforming people's experience or understanding the fuzzy rules or add something to keep consistency of a fuzzy logic system while constructing such system. In addition to building dataset while constructing fuzzy Logic system. This is the fundamental challenge of fuzzy logic system design. Moreover, researchers define membership function clearly without enhancing the effectiveness of reasonability [14-18]. When designing a membership function, the following issues should be taken into consideration:

Which number of membership function should be used?

Which membership function form should be chosen?

How much every neighboring membership function should intersect?

It is critical to develop a technique of membership function of digital training and calibrating, such that we could successfully get an ideal and optimum system's efficiency.

The acquiring and training of membership function with fuzzy rules are, in fact, the most essential and challenging tasks in designing the architecture of Fuzzy Logic methods for actual control systems [19-21]. There is not entirely dependable and methodical approach that has been discovered yet. In fact, this is a significant limitation of Fuzzy Logic control when deploy it in a practical application. As a result, for experiments and simulation, the technique that is suggested in this research is depending on the PD controller. Hence, we present a novel method FUZZY-PD based on Back-Propagation Neural Network, such that it is used to improve the effectiveness of F-PD even more. Consequently, three criteria are specified for the whole formulation of membership function. These are: Amount of Membership Function (QMF), Shapes of Membership Function (SMF), and Overlapping Rate of Membership Function (ORMF). These functions, however, are associated with the three issues listed above.

The relationship between the control efficiency and overlaying rate will be investigated in this research.

Our goal is to get the optimal overlaying rate, where Neural Network with Back-Propagation are used to optimize the membership function variables. In addition, simulations and practical tests with a 3-DOF robot manipulators are used to show the control strategy. The rest of the paper is structured as follows. Section 2 describes the robotics manipulator with 3-chain links. Section 3 presents the standard F-PD design procedure. Section 4 explains the Neural Network with Back-Propagation technology. Section 5 illustrates our proposed model and experimental results. Finally, Section 6 shows conclusions with our future directions. Upon comparing the suggested Fuzzy-PD based on Back-Propagation Neural Network approach with the conventional F-PD technique, we observed that the former demonstrates improved efficiency, completing the task 25% faster. On the other hand, the latter technique displays an overshoot and successfully achieves a 50% reduction in overshoot.

F-PD models, also known as Fuzzy-PID models, are an extension of traditional PID (Proportional-Integral-Derivative) controllers used in control systems. PID controllers are widely employed in various industrial and engineering applications to regulate processes by adjusting a control signal based on the error between the desired setpoint and the actual process variable. F-PD models integrate fuzzy logic principles into the PID control framework, enhancing the controller's adaptability and performance in dealing with non-linear and uncertain systems. By incorporating fuzzy logic, which allows the representation of linguistic variables and expert knowledge, F-PD models can handle imprecise and uncertain information more effectively.

Fuzzy logic is a mathematical approach that deals with uncertainty by permitting degrees of membership instead of strict binary values. Unlike classical logic, which relies on "true" or "false" conditions, fuzzy logic allows intermediate truth values between 0 and 1, representing the degree of membership of an element in a particular set.

In fuzzy logic, variables are defined linguistically using fuzzy sets, and control rules are expressed in the form of IF-THEN statements based on these fuzzy sets. These rules enable the control system to make decisions or adjust actions based on the input conditions, even in the presence of imprecise or ambiguous data.

Importance of F-PD Models and Fuzzy Logic

F-PD models and fuzzy logic concepts are of significant importance for various research and engineering applications due to the following reasons:

1. Handling uncertainty: Real-world systems often encounter uncertainties, variations, and imprecise data. Fuzzy logic allows modeling and control of such systems by providing a more tolerant and adaptive approach to uncertainty, improving the system's robustness and stability.
2. Dealing with non-linearity: F-PD models excel at handling non-linear systems. Traditional PID controllers have limitations when confronted with complex non-linear behaviors, but by integrating fuzzy logic, F-PD models can effectively control such systems.
3. Expert knowledge integration: Fuzzy logic enables the incorporation of human expertise and linguistic terms into control strategies. This is particularly valuable when dealing with systems for which expert knowledge is readily available but challenging to express in precise mathematical terms.
4. Adaptability and flexibility: Fuzzy logic-based control systems can adapt to changing conditions and environments. The ability to adjust control rules based on varying circumstances makes them suitable for dynamic and unpredictable scenarios.
5. Reducing overshoots and oscillations: F-PD models can lead to smoother control responses, minimizing overshoots and oscillations, especially during transients and sudden changes in the system.

In conclusion, F-PD models and fuzzy logic concepts offer an advanced and robust approach to control and decision-making in uncertain and complex systems. Their ability to handle uncertainty, non-linearity, and expert knowledge integration makes them essential tools for various research and engineering endeavors, leading to more efficient and effective control systems in real-world applications.

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Figure 2. F-PD controller

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Figure 3. Fuzzy logic system

Figure 2 presents the configuration of classic F-PD, such that $q_d$ represents the standard desired input and the error $e=$ $q-q_d$ with its derivation $\dot{e}=\dot{q}-\dot{q}_d$ are selected as Fuzzy Logic inputs. The fuzzy controller's sequential coefficients $\Delta K_P$ and $\Delta K_D$ are used for the outputs of PD controller. The main steps of fuzzy logic system are shown in Figure 3. For the inputs, scaling and fuzzy rules are initially performed to convert the better product quality into the fuzzy inputs described inside the fuzzy range between -3 to 3 in this research. The results are computed using the Center of Mass defuzzification methodology (COM) and the scalable scheme of the fuzzy outputs $\Delta K_P$ and $\Delta K_D$, that are also described inside a range between -3 to 3 in this research. At fuzzification, the membership function layout will undoubtedly and immediately affect the outcome of fuzzy theory in addition to, as a response, the effectiveness of PD control.

To understand the connection between membership function and PD control performance, NMF, SMF, and ORMF are established to statistically describe the numbers of membership functions, geometry of membership functions, and overlaying rate of membership functions, respectively. For simplicity and clarity of this research, we consider the overlaying rate among the forementioned functions. NMF has seven fuzzy values. These can be described as LP, MP, SP, Z, SN, MN, LN, which indicate Large Positive, Medium Positive, Small Positive, Zero, Small Negative, Medium Negative, and Large Negative respectively. The following are the Gaussian functions that determine the appropriate membership functions for LN and LP:

$f_i\left(x, \sigma_i, \mu_i\right)=e^{\frac{-\left(x-\mu_i\right)^2}{2 \sigma_i^2}}$      (9)

where, i=1, 2, 3, …7.

Represents a Gaussian distribution function, denoted as f_i. It is a probability density function used to model continuous random variables. The function depends on three parameters: x, which is the input variable; σ_i, representing the standard deviation and controlling the spread of the distribution; and μ_i, which denotes the mean or peak value of the distribution. The Gaussian function is symmetric around the mean and decreases as x moves away from the mean. A smaller σ_i results in a steeper curve with less variability, while a larger σ_ileads to a wider curve with higher variability. This function finds widespread applications in various fields, including statistics, signal processing, and machine learning, due to its ability to accurately model real-world data distributions.

The Gaussian function's prediction and conventional variance are represented by σ_i and μ_i. The equivalent membership functions for other fuzzy terms are created as triangular functions:

$f_i\left(x, a_i, b_i, c_i\right)=\left\{\begin{array}{cc}0 & x \leq a_i \\ \frac{x-a_i}{b_i-a_i} & a_i \leq x \leq b_i \\ \frac{a_i-x}{c_i-b_i} & b_i \leq x \leq c_i \\ 0 & c_i \leq x\end{array}\right.$      (10)

If I=2, 3, …. 6, a_i and c_i are the zero points, and b_i is the membership function's peak point. The deriving process is formed by the (If-Then) parallel fuzzy rules, which describe how to transfer input parameters to an output vector. The rule can be described as follows:

If x₁ is A and x₂ is B, then y is C.

Where x₁ and x₂ represent the input, y represents the result, and A, B, and C represent the fuzzy rules. The fuzzy sets for the traditional Fuzzy-PD are shown below in Tables 2 and 3.

Table 2. Fuzzy rules for $\Delta K_P$

Table 3. Fuzzy rules for $\Delta K_D$

The Mamdani fuzzification approach is chosen depending on the basis functions and fuzzy sets. Since the divining strategy for $\Delta K_P$ and $\Delta K_D$ are similar, just the procedure to find $\Delta K_P$ is described in the next section. In Eq. (4), the fuzzy output of $\Delta K_P$ is determined as follows:

$f p(x)=\sqrt{\left(f_i^E \Lambda f_j^{E C} \Lambda f_k^{\Delta K_P}\right)}$      (11)

The Eq. (11) represents a fuzzy logic function used to compute a final performance value, denoted as $f_p(x)$. This function involves the combination of three intermediate fuzzy membership values: $f_i^E, f_i^E C$, and $f_k^{\Delta K_P}$. These membership values are obtained through fuzzy logic operations, such as the AND operation (represented by the symbol $\wedge$ ), which allows the system to reason with multiple input conditions.

The interpretation of the fuzzy logic function is as follows: $f_i^E, f_i^E C$, and $f_k^{\Delta K_P}$ are fuzzy membership values representing the degree of membership of the input $x$ in three different fuzzy sets $E, E C$, and $\Delta K_P$ respectively. The function combines these memberships using the AND operation, reflecting the minimum membership value of the three sets. The result is then taken as the square root, yielding the final performance value $f_p(\mathrm{x})$.

Fuzzy logic is valuable in situations where precise mathematical relationships are challenging to define, and linguistic variables and expert knowledge are more appropriate for making decisions. The $f_p(\mathrm{x})$ function with fuzzy logic enables the integration of multiple input conditions and uncertainties, making it suitable for applications in control systems, decision-making processes, and various fields where uncertainty and vagueness are prevalent.

Where i, j, and k are integer values between 1 to 7, in which they are referring to the seven fuzzy rules. With the use of COM defuzzification technique, the fuzzy outputs can be described as follows:

$\Delta K^{f p}=\frac{\int_{x=-3}^{x=3} x \cdot f_p(x) d x}{\int_{x=-3}^{x=3} f_p(x) d x}$      (12)

The Eq. (12) is used to determine the change in kinetic energy, denoted as ΔK^fp, using integration with respect to a fuzzy logic performance function f_p(x). This calculation involves two definite integrals, one for the product of x and the performance function f_p(x) and another for f_p(x) alone, both integrated over the range from x=−3 to x=3.

The interpretation of this expression is as follows:

The numerator $\int_{x=-3}^{x=3} x \cdot f_p(x) d x$ calculates the weighted average of $x$ using the fuzzy logic performance function $f_p(x)$ as the weighting factor. It represents the contribution of different $x$ values to the overall kinetic energy. The denominator $\int_{x=-3}^{x=3} f_p(x) d x$ calculates the overall degree of membership (or the total weighting) of the fuzzy logic performance function $f_p(x)$ within the range from $x=-3$ to $x=3$.

By dividing the numerator by the denominator, the expression yields the change in kinetic energy ΔK^fp based on the fuzzy logic performance function f_p(x). This approach allows for the incorporation of fuzzy logic principles into the calculation of kinetic energy changes, which can be particularly useful in scenarios where precise mathematical relationships are uncertain or difficult to define, and fuzzy logic offers a more flexible and adaptable solution.

Based on the scalability procedure, the precise outputs are eventually designated as shows in Eq. (13):

$\begin{aligned} \Delta K_P=\frac{\Delta K_{\max }^P+}{2} & K_{\min }^P  +k_p\left(\Delta K^{f p}-\frac{\Delta K_{\max }^{f P}+K_{\min }^{f P}}{2}\right)\end{aligned}$      (13)

Knowing that

$k_p=\frac{\Delta K_{\max }^P-K_{\min }^P}{\Delta K_{\max }^{f P}-K_{\min }^{f P}}$ which is representing as an output scaling factor.

$\Delta K_{\max }^P$ is the highest level of crisp output.

$\Delta K_{\min }^P$ is the smallest amount of crisp output.

$\Delta K_{\max }^{f p}$ is the highest level of Fuzzy output.

$\Delta K_{\min }^{f p}$ is the smallest amount of Fuzzy output.

Therefore, the usual PD controller cab be presented as follows:

$\tau_s=K_P e+K_D \dot{e}$      (14)

This controller represents a control law for a proportionalderivative (PD) controller, where $\tau_s$ is the control signal or output, $K_P$ is the proportional gain, $K_D$ is the derivative gain, $e$ is the error, and $\dot{e}$ is the derivative of the error with respect to time.

The interpretation of this control law is as follows: In a PD controller, the control signal $\tau_s$ is determined by two components. The first component, $K_P e$, is the proportional term, where the control signal is directly proportional to the error $e$ between the desired setpoint and the actual process variable. The proportional gain $K_P$ determines the sensitivity of the controller to the error, influencing how quickly the controller responds to deviations from the setpoint.

The second component, $K_D \dot{e}$, is the derivative term, where the control signal is proportional to the rate of change of the error (e). The derivative gain K_D controls the damping effect of the controller and helps to stabilize the system by reducing overshoot and oscillations.

By combining the proportional and derivative terms, the PD controller provides a balance between fast response to errors (proportional term) and stability (derivative term). This control law is widely used in various control systems and applications, such as in robotics, process control, and motion control, to achieve accurate and stable control of dynamic systems.

where:

$\begin{aligned} & K_P=K_{P 0}+\Delta K_P \\ & K_D=K_{D 0}+\Delta K_D\end{aligned}$

$K_{D 0}$ and $K_{P 0}$ are represented the initial parameters chosen for $K_D$ and $K_P$ respectively.

The characteristic variables $\left(a_i, b_i, c_i, \eta_i, \sigma_i\right)$ are mostly chosen by many researchers and, therefore, set via the fuzzy inference procedure during the aforementioned Fuzzy Logic system design.

Due to the membership function is in control of converting crisp input to fuzzy input, parameters' choice $\left(a_i, b_i, c_i, \eta_i, \sigma_i\right)$ will immediately affect the overlaying rate of each two neighboring membership functions, hence the effectiveness of PD control. The membership function has a total of 19 factors that are specified as $\left(a_i, b_i, c_i, \eta_i, \sigma_i\right)$.

For $i=1,2,3, \ldots, 7$, there are two limitations that are set to facilitate the optimization technique as follows:

1. All membership function's magnitude is located at the appropriate class point, which implies that every member has a set value at the associated integer number of the fuzzy range between -3 and 3.

$c_i$ has zeros on the right side of $\mathrm{LN}$ and $\mathrm{MN}$, while $a_i$ has ones on the left side of MP and LP. Both are variables which can be changed, while the other zeros are constants.

Although only $\left(a_5, a_6, c_2, c_3\right)$ are selected as the ORMF influencing criteria under the aforementioned two conditions, the challenge of this research is to attain ORMF improvement by improving the values of $\left(a_5, a_6, c_2, c_3\right)$. Next section will explain in detail the ORMF optimization method.

Simulations using MATLAB and practical trials on a 3-DOF robot arm are used to illustrate the efficiency and advantages of the suggested control technique. The robot contains three rotating joints, as shown in Figure 1. Table 4 presents the parameters of the manipulator. These include: inertia moment, length and mass.

To establish a sinusoidal reference trajectory, a 3-DOF robotic arm's end-effector is used. The following are the definitions of the reference trajectories equations.

$\left.\begin{array}{c}x=0.4 \\ y=0.02 t-0.2 \\ 0.13-0.12 \sin (16 y+3.2)\end{array}\right\} \quad 0<t \leq 20$

The position $q=\left[q_1, q_2, q_3\right]^T$ of the reference joints acquired by using inverse kinematics to the end-effector's trajectory. For modeling and tests purposes, the manipulator's starting position is set to

$q_0=\left[q_{01}, q_{02}, q_{03}\right]^T=[0.4,1,0.6]^T$

The initial Proportional-Derivative (PD) controller gains are typically determined through a process called "tuning." The goal of tuning is to find suitable values for the proportional gain (KP) and derivative gain (KD) that result in stable and satisfactory control performance for the specific robot system. There are several methods for tuning the PD controller gains. One of these methods which is used in this paper is Manual Tuning. In manual tuning, an experienced control engineer adjusts the gains iteratively based on intuition and knowledge of the system. The engineer observes the system's response to different gains and makes adjustments until the desired performance is achieved. Manual tuning is often quick and straightforward but may require expertise in control theory and system dynamics.

The initial gains of the PD controller will be as:

$K_{P O}=\left[\begin{array}{ccc}K_{P O}^1 & 0 & 0 \\ 0 & K_{P O}^2 & 0 \\ 0 & 0 & K_{P O}^3\end{array}\right]=\left[\begin{array}{lll}8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8\end{array}\right]$

$K_{D O}=\left[\begin{array}{ccc}K_{D O}^1 & 0 & 0 \\ 0 & K_{D O}^2 & 0 \\ 0 & 0 & K_{D O}^3\end{array}\right]=\left[\begin{array}{ccc}2 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 6\end{array}\right]$

The membership function is shown in Figure 6.

Table 4. Parameters of 3-DOF robot

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Figure 6. Membership function before the optimization

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Figure 7. Parameters optimization

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Figure 8. Joint 1 parameter value

Simulations are run in MATLAB using the 3-DOF manipulator's kinematic characteristics, the end-effector's intended reference path as well as the initial variables for PD, fuzzy logic, and neural networks with back-propagation. Figure 7 describes the improvement of the four essential factors. When the input error increases, the critical factors of the membership function will raise the ORMF under NN modification, hence increasing the F-PD controller's output and forcing the system as a whole to respond quickly.

Figures 8, 9, and 10 depict the changes in each joint's critical parameters over the course of the simulation.

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Figure 9. Joint 2 parameter value

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Figure 10. Joint 3 parameter value

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Figure 11. Trajectory tracking

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Figure 12. Trajectory tracking error

The simulation results of Fuzzy-PD based Back-Propagation Neural Network are compared with the outcomes of the classic F-PD control method to demonstrate the benefit of the proposed approach Neural Network with Back-Propagation BF PD in this article. Figures 11 and 12 demonstrate the path tracking and tracking errors for the two systems, respectively.

As seen in the previous two figures, the suggested Fuzzy-PD Based on Back-Propagation Neural Network approach is quicker, with a 25% shorter time to convergence than the typical F-PD control approach. When the robot instantly adapts to the desired trajectory, the suggested approach has an overrun of 0.006m, whereas the F-PD technique has an overshoot less than 0.01m. When compared to the conventional F-PD technique, the Fuzzy-PD Based on Back-Propagation Neural Network reduces overshoot by 49.1%.

A 49.1% reduction in overshoot and quicker response time in robot control is highly significant due to its profound impact on the robot's performance and efficiency. Overshoot reduction ensures that the robot's movements are more precise and stable, minimizing the risk of unintended collisions or errors in task execution. By reducing overshoot, the robot can reach its target position more accurately and reliably, leading to improved overall performance and safety.

Additionally, the quicker response time is crucial for time-sensitive tasks or dynamic environments. A faster response allows the robot to adapt swiftly to changing conditions, increasing its agility and responsiveness. In applications like industrial automation or autonomous navigation, quicker response times can lead to higher productivity, reduced cycle times, and better decision-making.

The combination of reduced overshoot and quicker response time significantly enhances the robot's capabilities and effectiveness, making it more reliable and efficient in accomplishing complex tasks. This approach can have a substantial impact on various industries, providing a competitive advantage and advancing the state-of-the-art in robot control technology. The comparison between PD, PI, PID, Neural Network, Fuzzy-PD, and suggested controller according to the overshoot and time response is shown in Table 5.

Table 5. Overshoot and time response values with different controllers

The limitations of "Fuzzy-PD Control for a Robotics Manipulator using a Back-Propagation Neural Network" can include the following:

1. Data Availability and Quality: The effectiveness of the Back-Propagation Neural Network relies on having a sufficiently large and diverse dataset for training. Obtaining such data can be challenging in real-world scenarios, and the quality of the data can also impact the controller's performance.
2. Computational Complexity: Back-Propagation Neural Networks, especially when used for optimization tasks, can be computationally demanding. The time and resources required for training and implementing the network may limit real-time applicability, which is crucial for robotic systems.
3. Generalization: The proposed Fuzzy-PD control optimized using the neural network may work well for the specific robotic manipulator and environment considered in the study. However, its ability to generalize to other robotic systems or different environments might be limited.
4. Overfitting: Neural networks can be prone to overfitting, where the model becomes too specialized to the training data and fails to perform well on unseen data. Overfitting could reduce the controller's reliability and generalization capabilities.
5. Tuning Hyperparameters: Neural networks have various hyperparameters that need to be carefully tuned to achieve optimal performance. Finding the right combination of hyperparameters can be time-consuming and may require expertise.
6. Robustness: The Fuzzy-PD control approach using a back-propagation neural network may not be robust to changes in the robot's dynamics or external disturbances. A lack of robustness could lead to instability and performance issues in real-world scenarios.
7. Sensitivity to Noise: Neural networks can be sensitive to noisy input data, which might lead to inaccurate control signals and affect the manipulator's precision and accuracy.