3.1. Modelling of bilateral teleoperators

Given the absence of friction and other disruptive factors, the Euler–Lagrange equations describe the motion of $n$ -connection robots for the master and slave can be presented as

(1) \begin{align} M_{m}\left(q_{m}\right)\grave {q}_{m}+C_{m}\left(q_{m},\grave {q}_{m}\right)\grave {q}_{m}+g_{m}\left(q_{m}\right) =F_{h}+\tau _{m}\end{align}

(2) \begin{equation}M_{s}\left(q_{s}\right)\grave {q}_{s}+C_{s}\left(q_{s},\grave {q}_{s}\right)\grave {q}_{s}+g_{s}\left(q_{s}\right)=\tau _{s}-F_{e}\end{equation}

where $q_{m},q_{s}\in \mathbb{R}^{n}$ are the joint displacement vectors, $\tau _{m},\tau _{s}\in \mathbb{R}^{n}$ are the control input torques, and $F_{h},F_{e}\in \mathbb{R}^{n}$ are, in turn, the force exerted by the environment and the force used by the human operator. $M_{m},M_{s}\in \mathbb{R}^{n\times n}$ are the inertia matrices, $C_{m}\grave {q}_{m},C_{s}\grave {q}_{s}\in \mathbb{R}^{n}$ are the vectors of centrifugal and Coriolis torque, and $g_{m},g_{s}\in \mathbb{R}^{n}$ are the gravitational torque vectors. The subscripts s and m represent the slave and master robots from {1, 2,….., N}, based on the number of slaves connected with the master.

The inertia matrix $M(q)$ is limited, symmetric, and positive definite as follows:

(3) \begin{equation}\lambda _{m}I_{n\times n}\leq M(q)\leq \lambda _{M}I_{n\times n}\end{equation}

where $\lambda _{m},\lambda _{M}$ are the eigenvalues of the lowest and maximum $M(q)$ correspondingly and $I_{n\times n}\in \mathbb{R}^{n\times n}$ is the identity matrix.

Using the matrix of inertia $M(q)$ as well as the matrix Coriolis/centrifugal and the $C(q,\grave {q})$ , the matrix $\grave {M}(q)-2C(q,\grave {q})$ has a skew symmetry.

The n-link robot’s equations of motion can be linearly parameterised as

(4) \begin{equation}M(q)\grave {q}+C(q,\grave {q})\grave {q}+g(q)=Y(q,\grave {q},\grave {q}){\Theta}\end{equation}

where $Y(q,\grave {q},\grave {q})\in \mathbb{R}^{n\times l}$ is a known function matrix, referred to as a regressor, and ${\Theta} \in \mathbb{R}^{l}$ is a vector with unidentified dimensions.

The following models assume that the human operator and external factors are non-passive:

(5) \begin{equation}F_{h}\left(t\right)=-K_{h0}+K_{h{q_{m}}}+C_{h}\dot{q}_{m}\end{equation}

(6) \begin{equation}F_{si}\left(t\right)=-K_{e0}+K_{e{q_{i}}}+C_{e}\dot{q}_{i}\end{equation}

where $K_{w0}, K_{w}, C_{w}\in \mathbb{R}^{n\times n}, w=\{h,e\}$ .

Let us define $d_{\mathrm{mi}}(\mathrm{t})$ as the slave’s master’s tardiness $i$ and $d_{\mathrm{mi}}(\mathrm{t})$ as the slave’s delay $i$ to the master.

3.1.1. Control objectives

A system transformation is first performed to control the coupled nonlinear dynamics of the closed-loop system. The following synchronisation signals should be defined:

(7) \begin{equation}r_{m}\left(t\right)=\dot{q}_{m}\left(t\right)+A_{m}q_{m}(t)\end{equation}

(8) \begin{equation}r_{i}\left(t\right)=\dot{q}_{i}\left(t\right)+A_{i}q_{i}(t)\end{equation}

Stability and transparency are the primary concerns of bilateral teleportation systems. Our goal is to create control inputs, $e_{m}, e_{x}$ , to ensure system stability and accomplish transparency in a coordinated manner.

where ${\Lambda} _{\mathrm{m}}$ and ${\Lambda} _{i},i=\{1,2,\cdots, N\}$ are diagonal matrices that are positive definite.

Additionally, define

(9) \begin{equation}\grave {q}_{mr}(t)=-{\Lambda} _{m}q_{m}(t), \grave {q}_{ir}(t)=-{\Lambda} _{i}q_{i}(t),\end{equation}

(10) \begin{equation}\mu _{m}(t)=M_{m}\left(q_{m}\right)\overset{\leftharpoonup}{y}_{\mathrm{mor}}(t)+C_{m}\left(q_{m},\grave {q}_{m}\right)\grave {q}_{m}(t)+G_{m}\left(q_{m}\right)q_{m}(t)-F_{h}(t)\end{equation}

(11) \begin{equation}\mu _{i}(t)=M_{i}\left(q_{i}\right)\grave {q}_{ir}(t)+C_{i}\left(q_{i},\grave {\phi }_{i}\right)\grave {q}_{i}(t)+G_{i}\left(q_{i}\right)q_{i}(t)-F_{si}(t)\end{equation}

Using (9)–(11), (1) can be rewritten as

(12) \begin{equation}M_{\mathrm{m}}\left(q_{\mathrm{m}}\right)\grave {r}_{\mathrm{m}}(t)=\tau _{\mathrm{m}}(t)-\mu _{\mathrm{m}}(t)\end{equation}

(13) \begin{equation}M_{i}\left(q_{\mathrm{i}}\right)\grave {r}_{\mathrm{i}}(t)=\tau _{i}(t)-\mu _{i}(t).\end{equation}

Now employ the nonlinear feedback control theory by defining the control signals as follows:

(14) \begin{equation}\tau _{m}(t)=M_{m}\left(q_{m}\right)\left(u_{m}(t)+M_{m}^{-1}\left(q_{m}\right)\mu _{m}(t)\right)\end{equation}

(15) \begin{equation}\tau _{i}(t)=M_{i}\left(q_{i}\right)\left(u_{i}(t)+M_{i}^{-1}\left(q_{i}\right)\mu _{i}(t)\right)\end{equation}

where $u_{m}(.)$ and $u_{1}($ are auxiliary control variables, with which the dynamics of coordination errors can be formulated as follows:

(16) \begin{equation}\dot{e}_{\mathrm{m}}\left(t\right)=\frac{\mathrm{d}}{\mathrm{d}t}\left[k_{\mathrm{m}}q_{\mathrm{m}}\left(t\right)-k_{\mathrm{m}j}\frac{1}{N}\sum _{i=1}^{N}\,\,q_{i}\left(t-d_{\mathrm{mm}}\left(t\right)\right)\right]\end{equation}

(17) \begin{align}= & -{\Lambda} _{\mathrm{m}}\varepsilon _{m}\left(t\right)-\frac{k_{\mathrm{m}}}{N}\sum _{i=1}^{N}\,\,\left({\Lambda} _{m}-{\Lambda} _{i}\right)q_{i}\left(t-d_{im}\left(t\right)\right)-\frac{k_{\mathrm{m}}I}{N}\sum _{i=1}^{N}\,\,\left(1-d_{im}\left(t\right)\right)\times r_{i}\left(t-d_{im}(t)\right)\nonumber\\&+k_{m}r_{m}(t)-\frac{k_{m}I}{N}\sum _{i=1}^{N}\,\,d_{im}(t){\Lambda} _{i}q_{i}\left(t-d_{im}(t)\right)\end{align}

(18) \begin{equation}\dot{e}_{i}(t)=\frac{\mathrm{d}}{\mathrm{d}t}\left[k_{i}q_{i}(t)-k_{im}q_{m}\left(t-d_{ma}(t)\right)\right]\end{equation}

(19) \begin{align} = & -{\Lambda} _{i}e_{i}(t)-k_{im}\left({\Lambda} _{i}-{\Lambda} _{m}\right)q_{m}\left(t-d_{mi}(t)\right)-k_{im}{\Lambda} _{m}\grave {d}_{mi}q_{m}\left(t-d_{mi}\right)\nonumber\\ & +k_{i}r_{i}(t)-k_{im}\left(1-\grave{d}_{mi}(t)\right)r_{m}\left(t-d_{mi}(t)\right) \end{align}

Define

(20) \begin{equation}e\left(t\right)=\left[e_{m}^{T}\left(t\right),\mathbf{e}_{s}^{T}\left(t\right)\right]^{T}\end{equation}

(21) \begin{equation}\mathbf{e}_{s}(t)=\left[e_{1}^{T}(t),\cdots e_{N}^{T}(t)\right]^{T},\boldsymbol{r}(t)=\left[\boldsymbol{r}_{m}^{T}(t),\mathbf{r}_{s}^{T}(t)\right]^{T},\boldsymbol{r}_{m}(t)\end{equation}

(22) \begin{equation}=\left[r_{m}^{T}(t),\cdots, r_{m}^{T}(t)\right]^{T},\boldsymbol{r}_{s}(t)=\left[r_{1}^{T}(t),\cdots, r_{N}^{T}(t)\right]^{T}\end{equation}

(23) \begin{equation}X(t)=\left[e^{T}(t),\boldsymbol{r}^{T}(t)\right]^{T}.\end{equation}

Consequently, it is possible to deduce the robot’s dynamics from the augmentation of the error and synchronisation signals.

Considering Synchronisation error, which is the time lag in the communication channel between the slave and master robots, may be expressed as

(24) \begin{align} e_{m}(t) =q_{x}(t-T)-q_{m}(t)\end{align}

(25) \begin{equation}e_{i}(t)=q_{m}(t-T)-q_{x}(t)\end{equation}

Consequently, bilateral teleoperators’ synchronisation might be described as

(26) \begin{equation}\lim _{t\rightarrow \mathrm{\infty }}\,e_{m}(t)=\lim _{t\rightarrow \mathrm{\infty }}\,e_{i}(t)=0\end{equation}

Achieving force reflection from the remote environment in a contact situation is also crucial.

\begin{equation*}F_{h}=F_{\mathrm{c}}\ \textrm{with}\ \grave {q}_{m}(t)=\leftharpoonup{q}_{x}(t)=\grave {q}_{m}(t)=\grave {q}_{s}(t)=0.\end{equation*}

An adaptive technique is used to estimate parameters while taking modelling uncertainty into account. A nonlinear adaptive feedback controller is built using the new output variables, a linear combination of position and velocity. Create the nonlinear adaptive feedback compensation’s control input torques as

(27) \begin{align} \tau _{m} & =-\grave {M}_{m}\left(q_{m}\right)\lambda \grave {q}_{m}-\grave {C}_{m}\left(q_{m},\grave {q}_{m}\right)\lambda q_{m}+\grave {g}_{m}\left(q_{m}\right)+F_{m}\end{align}

(28) \begin{equation}\tau _{x}=-\grave{M}_{x}\left(q_{s}\right)\lambda \grave {q}_{x}-C_{x}\left(q_{s},\grave {q}_{s}\right)\lambda q_{x}+\grave {g}_{x}\left(q_{x}\right)+F_{x}\end{equation}

where $\hat{M}_{\mathrm{m}},\hat{M}_{x},\hat{C}_{m},\hat{C}_{x},\hat{g}_{m},\hat{g}_{x}$ are estimated parameters as a result of robot manipulators’ modelling uncertainty, $\lambda \in \mathbb{R}^{\mathrm{n}\times \mathrm{n}}$ is a matrix with positive definiteness, and $F_{\mathrm{m}}, F_{n}$ are extra control inputs that help with synchronisation.

(29) \begin{align}M_{m}\left(q_{m}\right)\left(\leftharpoonup{q}_{m}+\lambda \grave {q}_{m}\right)+C_{m}\left(q_{m},\grave {q}_{m}\right)\left(\grave {q}_{m}+\lambda q_{m}\right)= &\; F_{h}+\overset{\leftharpoonup}{M}_{m}\left(q_{m}\right)\lambda \grave {q}_{m}+\overset{\leftharpoonup}{C}_{m}\left(q_{m},\grave {q}_{m}\right)\lambda q_{m}\nonumber\\&-\overset{\leftharpoonup}{g}_{m}\left(q_{m}\right)+F_{m}\end{align}

(30) \begin{align} M_{x}\left(q_{x}\right)\left(\overset{\leftharpoonup}{q}_{x}+\lambda \grave {q}_{x}\right)+C_{x}\left(q_{x},\grave {q}_{x}\right)\left(\grave {q}_{x}+\lambda q_{x}\right) = & -F_{c}+\overset{\leftharpoonup}{M}_{x}\left(q_{x}\right)\lambda \grave {q}_{n}+\overset{\leftharpoonup}{C}_{x}\left(q_{x},\grave {q}_{x}\right)\lambda q_{n}\nonumber\\&-\overset{\leftharpoonup}{g}_{x}\left(q_{s}\right)+F_{x} \end{align}

where $\grave {M}_{i}=M_{i}-\grave {M}_{i},\grave { C}_{i}=C_{i}-\grave {C}_{i},\grave {g}_{i}=g_{i}-\grave {g}_{i}(i=m,s)$ are parameter estimation errors.

(31) \begin{align} & M_{m}\left(q_{m}\right)\grave {r}_{m}+C_{m}\left(q_{m},\grave {q}_{m}\right)r_{m}=F_{h}+F_{m}+Y_{m}\left(q_{m},\grave {q}_{m}\right)\grave {{\Theta} }_{m}\end{align}

(32) \begin{equation}M_{s}\left(q_{s}\right)\grave {r}_{s}+C_{s}\left(q_{s},\grave {q}_{s}\right)r_{s}=-F_{e}+F_{s}+Y_{s}\left(q_{s},\grave {q}_{s}\right)\grave {{\Theta} }_{s}\end{equation}

where $Y_{m}(q_{m},\grave {q}_{m}), Y_{s}(q_{s},\grave {q}_{s})\in \mathbb{R}^{n\times l}$ are regressor matrices, $\grave {{\Theta} }_{m}, \grave {{\Theta} }_{s}\in \mathbb{R}^{l}$ are parameter estimation error vectors, and $r_{m}, r_{s}$ are new output variables defined as

(33) \begin{align} r_{m} =\grave {q}_{m}+\lambda q_{m}\end{align}

(34) \begin{equation}r_{s}=\grave {q}_{s}+\lambda q_{s}\end{equation}

To estimate unknown parameters, parameter update laws are selected as

(35) \begin{align} \grave {\grave {\theta }}_{m} ={\Gamma} Y_{m}^{T}r_{m} \end{align}

(36) \begin{equation}\grave {\grave {\theta }}_{s}={\Lambda} Y_{s}^{T}r_{s}\end{equation}

where ${\Gamma}, {\Lambda} \in \mathbb{R}^{n\times n}$ are positive definite matrices.

PID controllers are used as additional control inputs for synchronisation control. The master and slave synchronisation is the focus of this control input. The term “synchronisation error” may be redefined by using additional output variables as

(37) \begin{align} e_{Tm}(t) =r_{s}(t-T)-r_{m}(t) \end{align}

(38) \begin{equation}e_{rs}(t)=r_{m}(t-T)-r_{s}(t)\end{equation}

To get synchronisation, we create extra control inputs. $F_{m}, F_{s}$ as

(39) \begin{align} F_{m} =Kpe_{Tm}(t)+K_{D}\grave {e}_{rm}(t)\end{align}

(40) \begin{equation}F_{s}=K_{p}e_{Ts}(t)+K_{D}\grave {e}_{rs}(t)\end{equation}

where $K_{p}, K_{D}\in \mathbb{R}^{n\times n}$ are diagonal matrices that are positive definite. To keep things simple, it is assumed that $Kp=k_{p}I_{n\times n}, K_{D}=k_{d}I_{n\times n}$ where $k_{p},k_{d}\in \mathbb{R}, I_{n\times n}\in \mathbb{R}^{n\times n}$ is the identity matrix. These control gain matrices can be optimised by a PID controller based on SQP, which optimises control parameters in a bilateral teleoperation system to minimise time delay and synchronisation error. This entails setting the PID controller’s initial parameters and modifying them repeatedly by figuring out sub-problems in quadratic programming that almost correspond to the nonlinear optimisation task. The time delay and synchronisation error are measured at each iteration, providing the foundation for minimising the objective function. SQP provides accurate management of nonlinear goals and constraints, which results in optimal performance and convergence to control satisfactory parameter values. The flexibility of PID control and the optimisation power of SQP provide improved teleoperation systems, convergence qualities, and performance.

3.2. PID parameters tuning using SQP

A simple control structure is the feedback control structure, in which the automatic controller compares the actual value of the plant output to the reference input (desired value), calculates the deviation, and generates a control signal to reduce the deviation to zero or a small value.

One possible component of the control in the above structure is a PID controller, which is based on three fundamental behaviour types: derivative (D), integral (I), and proportional (P). The proportional action produces a control signal proportionate to the error between the reference signal and the actual output. While the derivative action offers a derivative signal of the mistake, the integral action provides an essential signal of the error. The relation between the control $n(t)$ and error $e(t)$ can be expressed in the following form:

(41) \begin{equation}u(t)=K_{p}\left[\epsilon (t)+\frac{1}{T_{i}}\int _{0}^{t}\,\,\epsilon (\tau )d\tau +T_{d}\frac{d}{dt}\epsilon (t)\right]\end{equation}

$\mathrm{Kp},\text{ Ti}$ , and $\mathrm{Td}$ are the parameters to be tuned. The corresponding transfer function is given by

(42) \begin{equation}K(s)=K_{f}\left(1+\frac{1}{T_{j}s}+T_{d}s\right)\end{equation}

The Matlab Toolbox’s Nonlinear Control Design (NCD) block set is a tool for adjusting PID controller settings and identifying process attributes. It is employed to establish controller settings and retrieve process variables. The NCD block set transforms constraints and simulated system output into an optimisation problem to satisfy time domain performance requirements in a nonlinear Simulink model. In process control, this guarantees that the PID controller’s time will continue to be significant.

(43) \begin{equation}min_{x,y}\,\gamma\ {s}{.}{t}{.}\ \begin{cases} g(x)-w\gamma \leq 0\\ x_{l}\leq x\leq x_{w} \end{cases}\end{equation}

This work presents an approach that uses the SQP optimisation technique to minimise the maximum constraint error in an NCD block set. The variable x represents tumble variables, and tumble variables’ lower and upper limits are defined by the variables $xl$ and $xu$ . Vectorisation is used for the constraint bound and constraint weighting. After resolving the Karush–Kuhn–Tucker (KKT) condition, the SQP algorithm minimises a quadratic approximation of the Lagrangian function over a linear approximation of constraints, which becomes a sub-problem. The suggested SQP-PID controller holds great potential for bilateral teleoperation systems due to its simplicity, strong performance, accurateness, and the availability of efficient yet straightforward tuning methods to minimise synchronisation errors.

4.1. Sequential quadratic programming with KP

In the realm of optimisation algorithms, the SQP method is a powerful tool that seeks to enhance the performance of a given function iteratively. In this particular scenario, the fitness function graph for SQP is intricately tied to a proportional constant, denoted as Kp. As the algorithm embarks on its journey of iterations, the fitness landscape unfolds dynamically. At the outset, the fitness value stands tall at 14 e4, representing an initial state of the system. However, the optimisation process unfolds its prowess, initiating a sudden descent in fitness, a testament to the algorithm’s ability to refine the solution rapidly. Figure 2 illustrates a significant drop to a fitness level of 5 e4, signifying a notable improvement in the system’s performance. Subsequently, the fitness curve gradually descends, steadily progressing towards greater efficiency. Notably, by the 30th iteration, the fitness function gracefully reaches a low of 2 e4, underscoring the SQP algorithm’s persistent fine-tuning and optimisation capabilities. This graph visualises the iterative journey towards an increasingly optimal solution, guided by the influential constant Kp. SQP facilitates accurate adjustment of the proportional constant (KP) to attain the intended system response. KP tuning is essential to ensure the system reacts to faults effectively and doesn’t overrun or cause excessive oscillations. It makes sure that the control system stays stable by optimising KP. This helps to avoid problems like instability or long-lasting oscillations that might occur from improperly adjusted proportional gains.

Figure 2. SQP with $K_{p}$ .

4.2. Sequential quadratic programming with $\boldsymbol{K}_{\boldsymbol{d}}$

The fitness function graph is a visual optimisation narrative in SQP accompanied by a derivative constant (Kd). At the onset, the fitness score loomed at a staggering 2.03e5, portraying a landscape of inefficiencies and deviations. However, the algorithm’s strategic manoeuvres swiftly induced a marked descent, plunging to 2.01e5 within a concise timeframe, indicating an abrupt rectification in the system. This rapid shift was akin to an orchestrated symphony of adjustments, steering towards enhanced efficiency. As the iterations progressed, a deliberate and gradual descent ensued, illustrating a consistent refinement in performance. By the 30th iteration, the fitness function graph exhibited a commendable stature, boasting a refined fitness score of 1.98e5. In Fig. 3, this gradual yet persistent decline epitomised the algorithm’s calibrated precision and adaptability, illustrating its prowess in converging towards optimal solutions methodically. The derivative constant is carefully adjusted using SQP to enhance the system’s response to changes. The derivative component reduces oscillations and improves stability by using the current rate of change to predict future errors. The optimal damping effect can be achieved by optimising the Kd value using SQP while avoiding instability or undue control effort.

Figure 3. SQP with Kd.

4.3. Sequential quadratic programming with $\boldsymbol{K}_{\boldsymbol{PID}}$

In the landscape of SQP with proportional-integral-derivative (PID) control, the fitness function graph is a visual symphony of optimisation. At the outset, the fitness function soared to a formidable 16e4, an imposing peak challenging the algorithm. As the iterations commenced, the PID control strategised and manoeuvred, orchestrating a swift and surprising descent in fitness, a plunge akin to a free fall, steadily carving through obstacles, arriving at 5e4 – a notable accomplishment in a short span. However, the journey was far from over. What followed was a nuanced narrative of perseverance; the graph portrayed a gradual decline in fitness, akin to a meticulously planned descent down a winding mountain trail. At the 30th iteration, the fitness stood at a commendable 1e4, a testament to the precision and adaptability of the SQP with PID, showcasing its capacity to navigate complexities and optimise towards the desired outcome. SQP optimises the KPID controller’s proportional, integral, and derivative gains to maximise response time, stability, and error reduction. SQP successfully controls the dynamics of the system’s nonlinearities, guaranteeing that the KPID controller operates well across various operational environments and disruptions. Figure 4 encapsulates the dynamic tale of optimisation, showcasing the methodical evolution from challenge to triumph through iteration and precision engineering.

Figure 4. SQP with PID.

Figure 5. Master and slave radar position.

4.4. Radar positioning

In radar positioning, the dynamics between master and slave positions unfold fascinatingly. Figure 5 shows that at a mere 0.1 s into the sequence, the master and slave signals surge forth with a steep rise on the graph, a testament to their synchronised initiation. A sudden increase in the master signal indicates the start of the radar activity. The master radar system begins transmitting signals or positional data immediately, as shown by the sudden and abrupt spike on the graph. Mirroring the master’s first surge, the slave signal also indicates a high ascend simultaneously. It is essential to have this synchronised initiation to guarantee that the slave and master systems are operating simultaneously from the beginning. The master, a stalwart beacon, maintains its unwavering presence at a resolute altitude of 1000m.

Meanwhile, the slave, harmonising in the symphony of radar echoes, mirrors this ascent with an equally steep climb, settling distinctly but confidently at a steady 500 m. This visual representation encapsulates the interplay of supremacy and synchronisation, where the master leads the charge. At the same time, the slave follows suit, each asserting their defined position with an eloquent precision that mirrors the artistry of radar technology. Table I represents the comparison of gain level on time and position domain.

Table I presents a performance comparison of the three different gain levels (low gain (Kp), middle gain ( $K_{d}$ ), and high gain ( $K_{PID}$ ) in the time and position domains in the context of control systems. To be more precise, the steady-state error in millimetres ( $10^{-3}m$ ) represents the position domain, and the time constant in milliseconds ( $10^{-3}s$ ) represents the time domain. The system shows a steady-state error of 14.8 millimetres and a time constant of 0.109 milliseconds at low gain (Kp), showing higher position error but slower response. The time constant rises to 0.293 milliseconds with a middle gain ( $K_{d}$ ), but the steady-state error dramatically falls to 3.1 millimetres, indicating a trade-off between response time and positional accuracy. PID control effectively minimises both response time and positional error; with high gain ( $K_{PID}$ ), the system achieves a fast response with a time constant of 0.054 milliseconds and the lowest steady-state error of 1.5 millimetres. This comparison shows how various gain choices affect the dynamic and steady-state performance of the control system.

Table I. Time domain versus position domain.

4.5. Comparative analysis with different fitness

The nature of the optimisation issue, including its complexity, smoothness of the solution space, gradient information availability, and tolerance for local versus global optima, determines the nonlinear optimisation methods and metaheuristic algorithms. In general, metaheuristics are chosen for exploring complex, uncertain, or highly nonlinear optimisation landscapes. In contrast, nonlinear optimisation techniques are preferred when the problem structure permits accurate computations and deterministic convergence.

4.5.1. Other optimisation techniques

To achieve optimal performance for synchronisation of bilateral teleoperation systems against time delay, the performance of tuned controllers is compared with the gains obtained by several optimisation techniques [Reference Shokri-Ghaleh and Alfi23] such as COA, Biogeography-Based Optimization (BBO), Imperialist Competitive Algorithm (ICA), Artificial Bee Colony (ABC), PSO, GA, Ant Colony Optimization with Reinforcement learning (ACOR), Self-adaptive Normalized Differential Evolution (SaNSDE), Adaptive Differential Evolution (JADE), Enhanced Population-Based Incremental Learning (EPSDE), and Cuckoo Search (CS). Optimising synchronisation control parameters for enhanced bilateral teleoperation systems using a PID controller offers several benefits compared to KP and Kd controllers. The PID controller includes an integral component (KI) in addition to the proportional (KP) and derivative (Kd) components, which enhances the system’s ability to eliminate steady-state errors and improve overall accuracy. This results in better tracking stability and performance, particularly in the presence of external forces and time delays. The real-time adaptability of the PID controller to changing conditions leads to smoother and more accurate synchronisation, improving user experience and increasing system reliability. Existing works using only KP and Kd controllers may be highly susceptible to disturbances and variations in system parameters, potentially causing instability or oscillations. Additionally, derivative control can amplify high-frequency noise in the feedback signal, leading to unpredictable and unstable system behaviour. PID controllers are designed to overcome these limitations by integrating the strengths of proportional, integral, and derivative actions. This combination ensures improved performance, stability, and precision in synchronisation control.

4.5.1.1. Comparison of other techniques over the proposed model

This study compared the proposed model with various metaheuristic algorithms to determine the best performance for synchronising bilateral teleoperation systems. The optimisation algorithms included in the list are CS, JADE, EPSDE, SaNSDE, ICA, ABC, PSO, GA, BBO, COA, and ACOR. These methods are appropriate for various optimisation issues and provide varying convergence speeds while balancing exploration and exploitation. They are renowned for their accuracy over a wide range of difficulties, robustness, fast convergence, and simplicity of implementation. In this study, we used a nonlinear optimisation technique, the SQP-PID controller, and compared it with various metaheuristic algorithms. We could not compare it with other nonlinear optimisation techniques because existing works are not concentrated on employing nonlinear optimisation for synchronisation issues in bilateral telecommunication. Most existing studies have used metaheuristic algorithms for Bilateral Teleoperation Systems. Therefore, we compared the proposed model with these metaheuristic algorithms. In the future, this framework will be extended to include a comparison of nonlinear optimisation techniques. Table II compares the limitations of several optimisation techniques over the proposed model.

Table II. Comparison of limitations of other optimisation techniques over the proposed model.

The drawbacks of several optimisation techniques include sensitivity to initial conditions and the need for careful parameter tuning. These techniques can also experience delayed convergence in high-dimensional or complex environments, and they often involve significant processing costs when solving large-scale issues. Additionally, there may be a need for parameter modifications, a risk of premature convergence, and system complexity. These techniques may converge slowly in high-dimensional problems and have limited accuracy in complex scenarios. The proposed model aims to overcome these drawbacks, offering better convergence and performance for the synchronisation control of bilateral teleoperation systems.

4.6. Worst fitness

Table III and Fig. 6 showcase the worst fitness scores for various algorithms, illustrating their performance in tackling specific tasks. In fitness evaluation, lower scores indicate better efficiency, making the algorithms with higher values less optimal for the given task. Among the algorithms assessed, GA displays a fitness score of 0.824778, followed closely by PSO at 0.817223, signifying their relatively poor performance. ABC and BBO demonstrate slightly lower scores of 0.797696 and 0.80379, respectively, showing a marginally improved but subpar performance. However, algorithms like ICA and ACOR reveal higher scores of 0.757233 and 0.783745, indicating a less favourable fitness outcome. Further down the line, CS displays a score of 0.750607, while JADE and EPSDE show even higher scores above 1. The algorithms with the least optimal fitness scores include SaNSDE at 0.995551 and COA at 0.732162, highlighting their notably poorer performance than others evaluated in this context. We can determine the algorithm’s flexibility to changes in initial conditions and issue cases by comparing the worst fitness values across various algorithms. This data shows that, compared to other well-known optimisation methods, the suggested strategy effectively produces better optimisation results across all three types of controllers.

Table III. Worst fitness.

Figure 6. Comparison graph for worst fitness.

4.7. Mean fitness

A performance spectrum emerges in fitness algorithms, revealing distinct characteristics among various methodologies. When assessing these algorithms based on their mean fitness values, it’s apparent that some showcase superior aptitude while others lag. We may ascertain how consistently the algorithm finds acceptable responses by looking at the mean fitness. A lower mean fitness shows that the algorithm is generally good in various scenarios. Among the contenders, the GA demonstrates a middling performance with a mean fitness of 0.776423, trailing closely to the PSO algorithm at 0.773652. Yet, the ABC algorithm must catch up with a mean fitness of 0.745914, hinting at its struggles in optimisation landscapes.

Similarly, the ICA and the BBO exhibit commendable but not outstanding performances, scoring 0.735631 and 0.762197 in mean fitness, respectively. However, outliers like JADE and EPSDE present intriguing contrasts; while they display higher mean fitness values, 0.985966 and 0.925921, respectively, their performance across other metrics might warrant further investigation. Finally, the CS, Scatter Search with Local Descent, and different algorithms reveal consistent patterns, hovering around the 0.74 – 0.76 range for mean fitness, showcasing stability but potentially lacking in exceptional performance. Notably, the COA algorithm (COA) presents a mean fitness of 0.732139, hinting at its struggles to find optimal solutions within the fitness landscape. This spectrum of fitness metrics offers insights into the varied performances of these algorithms, inviting deeper scrutiny into their methodologies and potential areas for improvement. This data shows that, compared to other well-known optimisation techniques, the suggested strategy is remarkably effective at consistently producing better optimisation outcomes for all controllers. Table IV and Fig. 7 present the algorithms’ mean fitness ratings to demonstrate how well different algorithms perform when faced with particular tasks.

Table IV. Mean fitness.

Figure 7. Comparison graph for mean fitness.

4.8. Best fitness

Table V and Fig. 8 showcase the performance metrics of the best fitness algorithms, ranging from the worst to the best fitness scores across different parameters. In the realm of PID algorithms, the least effective in fitness improvement is the ABC algorithm with a score of 0.732150, closely followed by ICA and COA, sharing the same fitness score of 0.732135. Moving along the spectrum towards more effective fitness enhancement, algorithms like PSO and BBO demonstrate slightly higher scores of 0.742116 and 0.738460, respectively, depicting moderate performance. The ACOR and CS algorithms display a comparable fitness level of 0.742116 and 0.742116, respectively, a notch higher than the previous ones. However, the most impressive enhancement in fitness is observed in algorithms like JADE with a score of 0.908048, EPSDE at 0.763318, and SaNSDE scoring 0.904672, showcasing substantial improvements. These algorithms demonstrate significantly better fitness in their respective methodologies, positioning them at the apex of the fitness evaluation among the listed algorithms. The best fitness value indicates the algorithm’s most optimal solution. It displays the algorithm’s optimal result and its peak performance. The algorithms can be evaluated against one another by comparing their best fitness value and finding the best outcomes.

Table V. Best fitness.

Figure 8. Comparison graph for best fitness.

The results highlight the effectiveness of the SQP-PID algorithm’s control structure, which achieves superior convergence speed in position tracking despite starting from a nonzero offset. This implies that even with the initial discrepancy between the master and slave joint angles (0.4 units offset), the control approach enables the slave to converge accurately to the joint angle of the master. Time delays can disrupt synchronisation between the master and slave, causing inaccuracies or instability. However, the control method mitigates this issue, enabling effective tracking despite the delay. It emphasises the robustness of this proposed control structure in dealing with two critical factors: time delay in communication channels and parameter uncertainty with accurate outcomes in handling delays inherent in communication channels. Despite the delay in receiving information or commands from the master, the slave converges accurately to the desired joint angle. This implies that the control approach incorporates mechanisms to compensate for or predict the effects of these delays. Also, the control structure exhibits robustness against such uncertainties, indicating that it can adapt or adjust its strategies to accommodate variations in system parameters. The abovementioned control structure achieves optimal performance, but the control structure keeps the system stable even in the face of unclear or variable system characteristics. This indicates that the control approach is designed to handle parameter variations without compromising its stability or performance. In control systems, like bilateral teleoperation, the application of SQP rationalises effective parameter optimisation under nonlinear constraints. Because of SQP’s capacity to manage intricate dynamics and stringent operating constraints, control performance is robustly maintained, improving system responsiveness and stability. This robustness and adaptability make it a promising approach for ensuring accurate position tracking and stability in such challenging scenarios, as seen in the figure. We may examine the algorithm’s convergence behaviour by tracking the patterns in the worst, mean, and best fitness values over iterations. Convergence towards ideal solutions is indicated by a decreasing difference between the worst and best fitness values over time. This examination aids in comprehending how fast and steadily the algorithm converges to superior solutions.