4.1. Deterministic GA optimization approach

A deterministic GA-based PID tuning approach is introduced, where the controller gains are optimized using the nominal dynamic model of the P2S-TDCM without considering parameter uncertainties. The controller gains are optimized through a genetic algorithm (GA) to address non-linearities and external disturbances. The P2S-TDCM consists of two sections, each governed by coupled angles of curvature, $\mathbf{q}_{i}$ and the system dynamics as defined by the control effort (τ), as shown in Eq. (15). Thus, GA optimization minimizes the sum of the squares of Cartesian errors, control efforts, and curvature angles. In addition, the constraints ( $C_{1}^{j}$ and $C_{2}^{j}$ ) ensure that the angles of curvature remain within the permissible lower and upper bounds ( $q_{l}$ and $q_{u}$ ) inspired by ref. [Reference Bhandari, Raj, Pathak and Yang34]. These parameters are weighted using $w_{1}$ , $w_{2}$ , and penalties ( $w_{p}$ ) for constraint violations. The fitness function, described in Eq. (16), balances these elements, with constraints detailed in Eq. (17). Equal importance is assigned to weights $w_{1}$ and $w_{2}$ , while a higher penalty factor enforces physical constraints effectively.

(16) \begin{equation}\overline{f}\left(g\right)=w_{1}\int _{0}^{t}\sum _{i=1}^{n-1}\left(e_{i}^{2}+\tau _{i}^{2}\right)dt+w_{2}\sum _{j=1}^{n}\left| q_{j}\right| +w_{p}\left(\sum _{j=1}^{n}\left(C_{1}^{j}+C_{2}^{j}\right)\right)\end{equation}

(17) \begin{equation} C_{1}^{j} :\!= \begin{cases} \quad 0, & q_{i}-q_{l}\geq 0\\ | q_{i}-q_{l}| , & q_{i}-q_{l}\lt 0 \end{cases} \ \text{and}\ C_{2}^{\,j} \ :\!= \begin{cases} \quad0, & -q_{i}+q_{u}\geq 0\\ \left| -q_{i}-q_{u}\right| , & -q_{i}+q_{u}\lt 0 \end{cases}\end{equation}

The Pseudocodes 1 and 2 present the simulated system using specified PID gains (referred as the Main function) and the function to evaluate constraints and fitness function (as described by Eqs. (16) and (17)). The GA algorithm to tune gains (detailed in the Appendix).

Table II.

System geometric and material parameters.

Table III.

Simulation parameters of GA optimization.

Figure 4.

Cartesian model-based control with GA-based optimization under external forces.

Figure 4 illustrates the interaction between the main function (Pseudocode-1), the fitness function (Pseudocode-2), and the GA algorithm. This figure depicts the process of minimizing position errors and GA tuning by optimizing the fitness function. The fitness function is defined by Cartesian space error, control effort, curvature angle, and constraints, detailed in Pseudocode 2. This constraint restricts the curvature angle to the desired range. The system dynamics incorporate inertia, Coriolis, gravitational forces, and viscous damping. Gravitational forces include potential energy and tendon elongation, modeled as a spring connecting anchor points on disks. Viscous damping accounts for tendon elongation rates, while joint positions and velocities are updated via numerical integration. The feedback loop modifies control inputs to maintain desired behavior under external disturbances. GA optimizes PID parameters through selection, crossover, and mutation, minimizing trajectory errors and control effort. Tables II and III summarize system and GA simulation parameters, respectively.

4.2. Robust controller: GA-based random parameter uncertainty and disturbance observer

Although deterministic GA optimization yields effective gains for the nominal model, its performance can degrade under uncertainties such as variations in mass, modulus, damping, and external loads. To overcome this, a two-fold robust strategy is adopted – offline robustness via GA-based parameter randomization and online robustness through an integrated DOB.

In the robust GA tuning phase, the model parameters are randomized within bounded ranges during fitness evaluation to ensure that the optimized control gains perform well across a range of parameter variations. Specifically, the vertebra mass is varied by ±2% and backbone bending modulus by ±5% accounting for manufacturing tolerances and material uncertainties, as detailed in Table IV. The randomization is embedded within the GA fitness evaluation function so that each candidate solution is tested on a randomly perturbed model before its performance is scored. The fitness function and constraints remain identical to Eqs. (16) and (17), but they are now evaluated across randomized parameter realizations. This approach enhances robustness and convergence stability, yielding a conservative yet reliable controller. The overall process is outlined in Pseudocode 3. However, the GA-based optimization alone cannot handle unknown external disturbances; hence, an online compensation layer is added using a DOB.

To enhance robustness against external disturbances, an online DOB is integrated into the feedback loop. The DOB is inspired by refs. [Reference Ohishi27, Reference Chen, Ballance, Gawthrop and O’Reilly28] estimates the lumped disturbance vector, which includes unmeasured external tip forces, payload variations, and residual modeling errors, and compensates them in real-time.

Table IV.

Random parameter uncertainty for GA tuning.

Pseudocode 3:

Robust fitness evaluation under parameter uncertainty

External disturbance is represented as an external force wrench $\mathbf{F}_{ex}$ , expressed as generalized tip forces $\mathbf{J}^{T}(q)\mathbf{F}_{ex}$ . These forces are modeled as a lumped disturbance term $\mathbf{d}$ in the manipulator dynamics as expressed in Eq. (18), where $\mathbf{M}(q)$ is the inertia matrix, $\mathbf{C}(q,\dot{q})$ is the Coriolis and centrifugal terms, $\mathbf{G}(q)$ represents the gravity vector, $\boldsymbol{\tau }$ is the control torque, and $\mathbf{d}$ is the lumped disturbance. Following the formulation of ref. [Reference Chen, Ballance, Gawthrop and O’Reilly28], the DOB introduces an auxiliary state z, whose dynamics are expressed by Eq. (19), where $\boldsymbol{{L}}(q,\dot{q})$ is a diagonal and positive definite observer gain matrix and $\hat{\mathbf{d}}=(\mathbf{z}-\boldsymbol{\tau })$ . The estimated disturbance is then obtained as $\hat{\mathbf{d}}=\mathbf{z}-\boldsymbol{\tau }$ . Using this estimate with Eqs. (19) and (18), the disturbance estimation error dynamics are obtained, as expressed by Eq. (20), where observer error, $\mathbf{e}(t)=\mathbf{d}-\hat{\mathbf{d}}$ , which guarantees exponential convergence of $\hat{\mathbf{d}}$ to $\mathbf{d}$ under a positive definite observer gain matrix $\mathbf{L}(q,\dot{q})$ . For the two-section P2S-TDCM, the gain matrix is $\mathbf{L}(\boldsymbol{{q}})=\text{diag}(\alpha _{1},\alpha _{2})\mathbf{M}^{-\mathbf{1}}(q)$ , with positive scalar observer bandwidths $\alpha _{1},\alpha _{2}\gt 0$ . Since initially no prior disturbance information is available, $\dot{\mathbf{d}}\approx 0$ is assumed, implying that disturbances vary slowly compared to observer dynamics. Accordingly, the control law is revised as $\boldsymbol{\tau }^{\mathbf{*}}=\boldsymbol{\tau }_{\mathrm{GA}-\mathrm{PID}}-\hat{\mathbf{d}}$ . thereby ensuring real-time disturbance compensation. The DOB effectively handles constant and slowly varying external wrenches, such as those arising during vegetable detachment or payload manipulation. However, its ability to reject high-frequency or impulsive disturbances is limited by the observer bandwidth and sensor noise characteristics.

(18) \begin{equation}\mathbf{M}\left(q\right)\ddot{\boldsymbol{{q}}}+\mathbf{C}\left(q,\dot{q}\right)\dot{\boldsymbol{{q}}}+\mathbf{G}\left(q\right)+\left(\frac{\boldsymbol{\partial} \boldsymbol{\mathcal{D}}}{\boldsymbol{\partial} \dot{\mathbf{q}}_{\mathbf{i}}}\right)=\boldsymbol{\tau }+\mathbf{d}\end{equation}

(19) \begin{equation}\dot{z}=-\mathbf{L}(q,\dot{q})z+\mathbf{L}\left(q,\dot{q}\right)\left[\mathbf{M}\left(q\right)\ddot{\boldsymbol{{q}}}+\mathbf{C}\left(q,\dot{q}\right)\dot{\boldsymbol{{q}}}+\mathbf{G}\left(q\right)+\left(\frac{\boldsymbol{\partial} \boldsymbol{\mathcal{D}}}{\boldsymbol{\partial} \dot{\mathbf{q}}_{\mathbf{i}}}\right)-\boldsymbol{\tau }\right]\end{equation}

(20) \begin{equation}\dot{\boldsymbol{{e}}}\left(\boldsymbol{{t}}\right)+\mathbf{L}\left(q,\dot{q}\right)\mathbf{e}\left(\boldsymbol{{t}}\right)=0\end{equation}

4.3. Lyapunov stability analysis

A Lyapunov-based analysis verifies the stability of the PID-controlled continuum manipulator under the influence of external forces. The task-space tracking error is defined as $e(t)=\mu _{d}(t)-\mu (t)$ , where $\mu _{d}(t)$ is the desired trajectory and $\mu (t)$ is the actual end-effector position. Similarly, the task-space velocity error is defined as $\dot{e}(t)=\dot{\mu }_{d}(t)-\dot{\mu }(t)$ . For the considered TDCM, the Lyapunov candidate function ( $V$ ) is derived incorporating proportional, integral, and derivative error terms as shown in Eq. (21), where $M(\mu )$ is the positive definite task-space inertia matrix. This Lyapunov function is positive definite, and its time derivative is shown in Eq. (22). Substituting the control input from the GA-tuned PID controller and incorporating the system dynamics, the corresponding $\dot{V}$ can be simplified as shown in Eq. (23), where $K_{d}$ is a derivative gain and positive definite. Where d is the bounded external disturbances, considered an external force, and gains are positive definite. If the disturbance is small, this implies uniform ultimate boundedness (UUB), and expression (23) can be written as Eq. (24).

(21) \begin{equation}V\left(\mu ,\dot{\mu },\eta \right)=\frac{1}{2}\dot{e}^{T}M\left(\mu \right)\dot{e}+\frac{1}{2}e^{T}K_{p}e+\frac{1}{2}\left(\int edt\right)^{T}K_{I}\left(\int edt\right)\end{equation}

(22) \begin{equation}\dot{V}=e^{T}K_{p}\dot{e}+\dot{e}^{T}M\left(\mu \right)\ddot{e}+\frac{1}{2}\dot{e}^{T}\dot{M}\left(\mu \right)\dot{e}+\left(\int edt\right)^{T}K_{I}e\end{equation}

(23) \begin{equation}\dot{V}=-\dot{e}^{T}K_{d}\dot{e}+\left\| d\right\| \left\| \dot{e}\right\|\end{equation}

(24) \begin{equation}\dot{V}\leq -\dot{e}^{T}K_{d}\dot{e}\Rightarrow e\left(t\right)\rightarrow 0\end{equation}

From the above argument, if the dynamic model of the system as shown in Eq. (15) is represented under the control law presented in Section 4.1, then the Lyapunov function is non-increasing over time. As it is positive definite and radially unbounded, the system is globally stable in Lyapunov. Under the tuning performed via GA, the controller ensures asymptotic convergence of the error to zero, even under external disturbances.

5.1. Setup and trajectory generation

Trajectory tracking is a critical aspect of robotics, especially in environments with external disturbances. The manipulator is tasked with a circular trajectory (Eqs. (25) and (26)) with the trajectory parameter summarized in Table V. The simulations are conducted over 60 s with a fixed integration step size of $dt-0.001s$ using a fourth-order Runge–Kutta scheme. Initial conditions are set with zero curvature angles and tendon lengths corresponding to the neutral configuration. To ensure smooth motion within the TDCM’s curved workspace, the desired trajectory is generated using a quintic polynomial interpolation (as shown in Eq. (27)), which minimizes oscillations between adjacent points. Additionally, the tolerance set between the projected path and workspace points is less than below $10^{-6}$ m, inspired by ref. [Reference Pachouri and Pathak32].

(25) \begin{equation}x_{d}=x_{c}+r\sin \left(\phi \right)\end{equation}

(26) \begin{equation}y_{d}=y_{c}+2r\cos (\phi )\end{equation}

Figure 5 illustrates the position, velocity, and acceleration profiles of the first and second sections. Minor oscillations (<5% of the trajectory scale) are observed when the manipulator transitions across quadrants, caused by sign changes in curvature angles. Notably, the first section exhibits no x-axis translation, as shown by $x_{1}(t)$ , $\dot{x}_{1}(t)$ , $\ddot{x}_{1}(t)$ , which aligns with the physics of the manipulator. The simulation incorporates inertial, Coriolis, gravitational, and damping effects, with two independent PID controllers regulating each section.

Table V.

Trajectory parameters.

Figure 5.

Desired position, velocity, and acceleration data for desired circular trajectory (a) First and (b) Second section.

(27) \begin{equation}p\left(t\right)=a_{0}\hspace{0pt}+a_{1}\hspace{0pt}t+a_{2}\hspace{0pt}t^{2}+a_{3}\hspace{0pt}t^{3}+a_{4}\hspace{0pt}t^{4}+a_{5}\hspace{0pt}t^{5}\end{equation}

5.2. Deterministic GA result

This section presents the results of GA-optimized PID control for a deterministic approach, where system parameters are assumed to be fixed and known. The GA tuning framework follows the methodology described in Section 4. Initial controller gains are selected from a random population ( $P_{1}$ ) of size $(N\times k)$ , where $N=50$ represents the candidate solutions (sets of PID gains) per generation, and $k=6$ , corresponds to the two PID controllers for the TDCM. Gain values are constrained within [0, 10], balancing diversity and convergence rate [Reference Bhandari, Raj, Pathak and Yang34]. The optimization minimizes Cartesian tracking errors and control efforts while enforcing curvature limits. The numerical integration process begins with extracting the desired trajectory data (position, velocity, and acceleration). Then, the Jacobian matrix computes the desired joint velocities. The system’s dynamic parameters, such as inertial, Coriolis, gravitational, and damping effects, are calculated. The simulation incorporates inertial, Coriolis, gravitational, and damping effects, with two independent PID controllers regulating each section. To optimize the PID gains, a GA minimizes the sum of squared errors and control efforts while penalizing deviations from upper and lower bounds on the curvature angles. The GA iteratively refines controller gains via selection, crossover, and mutation, ensuring stable and accurate tracking under varying conditions. Results are reported for two cases: (i) Trajectory tracking without external forces and (ii) Trajectory tracking under an external force. Table VII presents the optimal controller gains for no-load and external load scenarios. This deterministic optimization provides a baseline for later robustness studies with parameter uncertainties.

5.2.1. Trajectory tracking without external forces

The section presents the trajectory tracking of the proposed TDCM without considering any external forces at the manipulator’s tip. Figure 6(a) illustrates the three-dimensional configuration of the manipulator as it follows a circular trajectory without any external load. Figures 6(b) and 6(c) provide the $x-y$ and $x-z$ views, respectively, of the configuration shown in Figure 6(a). The controller gains are optimized using GA-based parameter tuning, as presented in Table VII. The results confirm the controller’s effectiveness in achieving planar movement in each section. This configuration enables the manipulator tip to actuate spatially as desired when actuated orthogonally. Figure 6(d) compares the tip coordinates of the desired trajectory based on numerical simulations and the experimental setup, demonstrating strong agreement between them.

Figure 6.

Trajectory tracking without external load (a) Three-dimensional view, (b) X-Y plane view, (c) X-Z plane view, and (d) Comparison of desired, simulated and actual tip coordinates.

The tip trajectory shown in Figure 6(d) is computed from the cable lengths recorded by the draw-wire sensors from the experimental setup. These lengths, representing actuator space, are input into the forward mapping to determine the corresponding task-space parameters, as described in refs. [Reference Escande, Chettibi, Merzouki, Coelen and Pathak31, Reference Escande, Merzouki, Pathak and Coelen35]. Figure 7 demonstrates the mapping between the actuator, configuration, and task spaces. A potential source of error arises from using the constant-curvature model to compute the tip trajectory based on the draw-wire sensor data. This model may not fully capture all the dynamics involved, which may lead to certain inaccuracies in the calculated trajectory.

Figure 7.

Mapping between actuator, configuration, and task space for TDCMs.

Figure 8 illustrates the tendon lengths based on kinematic and dynamic models, without considering external loads. This comparison highlights the differences in response between the two simulation approaches. In the kinematic simulation (Figure 8(a)), oscillations are observed, particularly in the tendon length of the second section. These oscillations are comparatively lower in the dynamic simulation (Figure 8(b)), indicating improved stability in the model’s response. Additionally, the kinematic model shows flattening behavior in specific tendon lengths, such as $L_{3}$ near the 120^th step and $L_{4}$ near the 370^th step, indicating the kinematic model’s limitations to counter the manipulator’s self-weight. This flattening effect is absent in the dynamic simulation, which continuously accounts for gravitational effects, maintaining smooth periodicity. Quantitatively, the dynamic model outperforms the kinematic model by reducing tendon length oscillations by approximately 12.5%, as indicated by the lower standard deviation. This demonstrates the dynamic model’s improved capability to capture the manipulator’s behavior, effectively minimizing internal disturbances, and resulting in smoother tendon actuation.

Figure 8.

Tendon length variation during circular trajectory without external load (a) Inverse kinematic model and (b) Dynamic model.

5.2.2. Trajectory tracking with external force

This section presents the manipulator’s dynamic model and control strategy under external force ( $\mathbf{F}_{ex}=[\! -100]^{T}$ N) applied in the negative x-direction during trajectory tracking. The controller gains are re-tuned considering the external force acting on the manipulator tip. The motivation for such a loading condition stems from its relevance to the target application of open-air vegetable harvesting, as illustrated in Figure 3. During the picking process, the manipulator primarily experiences a downward external force after detaching a vegetable from the plant, which guided the choice of a constant external tip force.

Figure 9(a) shows the desired tip trajectory, tip deviation under external load and the subsequent correction using updated gains. As illustrated in Table VIII, the corrected trajectory condenses root mean square error (RMSE) values from 3.74 to 2.85 mm, 6.42 to 3.67 mm, and 1.96 to 1.29 mm. This corresponds to improvements of 23.8, 39.88, and 34.11% concerning the deviated trajectory. Figure 9(b) compares tendon lengths for the desired trajectory, deviations, and corrections from the updated control parameters. The first section experiences higher moments, countering tip forces alongside inertial and Coriolis forces from the second section. Although full recovery is constrained by structural flexibility and tendon tension limits, notable improvements are achieved, validating the dynamic control strategies to mitigate external disturbances.

Figure 9.

(a) Tip trajectory under external force and (b) Tendon length variations.

5.3. Robust control: GA evaluation with parameter uncertainty and disturbance observer for external forces

To assess robustness via a GA-based controller, the parameter uncertainties are introduced to the closed-loop system by randomly varying the vertebra mass, $(m_{di}+m_{bi})$ by ±2% and the backbone Young’s modulus by ±5%. The gains obtained from the deterministic GA under external forces are benchmarked by initializing them for GA tuning under parameter uncertainty and external force. The same GA simulation parameters are applied to both GA approaches for fair comparison. The process to tune with random parameter uncertainty is detailed in Pseudocode 3. In addition to parameter randomization, the robust control configuration integrates an online DOB to compensate for external wrenches, thereby enhancing real-time robustness beyond what can be achieved through offline GA tuning alone.

Figure 10 presents the best, median, and worst fitness values across 10 randomly seeded runs of the GA for the deterministic and robust approaches. The results show that the robust GA-PID achieves lower fitness values in fewer generations, with the worst robust case outperforming even the best deterministic case after the 11^th generation. This result demonstrates the superiority of the robust approach, as further confirmed by Table VI, which reports lower fitness values for the robust case presented for sampled generations. Additionally, the shaded regions for the robust approach converge and narrow earlier than those of the deterministic case. Note that this is not a detailed statistical study, as the focus here is on demonstrating control effectiveness, with extended statistical validation left for future work. Table VII presents the gains obtained from robust control.

Table VI.

Median fitness values for the sampled generation for deterministic and robust approaches.

Table VII.

Controller gains: deterministic and robust approaches.

Figure 10.

Best, median, and worst fitness values for sampled generations (deterministic vs robust).

After optimization, numerical experiments are conducted on a circular trajectory. Parameter uncertainty is introduced by increasing vertebra mass by 2% and backbone Young’s modulus by 5% combined with a DOB under external wrenches [1 0 0] ^T N. This combined strategy enhances tracking accuracy even when both parameter variations and external forces are present. System behavior regarding trajectory tracking and RMSE is analyzed for a robust approach over deterministic, demonstrating the ability to maintain improved tracking despite parameter variations and external load under a selected range.

Figure 11 illustrates the task-space trajectory tracking performance of the manipulator, showing the desired circular trajectory, deviation under external force, and corrected under deterministic and robust control. As summarized in Table VIII, both controllers significantly reduce RMSE compared to the deviated case, with the robust controller improving tracking accuracy by 24.21% in $x$ , 20.21% in $y$ , and 4.65% in $z$ comparison to deterministic. Figure 12(a–c) shows detailed tracking along individual axes, with shaded regions highlighting improved error reduction. Most error accumulation occurs in the first quadrant, where the manipulator moves against gravity. Figure 12(d) compares the task-space tracking errors along the in $x$ , $y$ , and $z$ directions for the deterministic and robust control approaches. The robust controller achieves visibly lower error amplitudes and smoother convergence, consistent with the reduced RMSE values. This demonstrates its improved disturbance-handling capability and reduced sensitivity to parameter variations.

Parameter uncertainties are introduced in the closed-loop system along with the external tip force of [1 0.5 0] ^T N. Such uncertainties may arise from inaccuracies in disk manufacturing, machining tolerances, or imprecise estimation of backbone flexible properties. While many factors could contribute to uncertainty (damping, stiffness, friction, sensor noise), this study focuses on vertebra mass and the backbone Young’s modulus. In the robust configuration, the DOB mitigates the effect of unmeasured external wrenches, while the GA-based randomization handles model parameter uncertainty. Thus, the robust controller addresses both major sources of performance degradation.

Table VIII.

RMSE in task-space under external load.

Figure 11.

Trajectory tracking under external tip force for deterministic and robust approaches.

Figure 12.

Trajectory tracking deterministic and robust approaches in (a) $x$ , (b) $y$ , (c) $z$ direction and (d) Task-space tracking errors in the $x$ , $y$ , and $z$ direction under external tip force.

Figure 13 compares deterministic and robust approaches in terms of RMSE and sensitivity for parameter variations. For vertebra mass variation (shown in Figure 13(a)), the deterministic approach shows steep slopes, indicating higher sensitivity to mass perturbations. Specifically, increasing vertebra mass has more adverse effect than reducing it, as reflected by the sharper slope of 5.02 compared to −3.43. This is because added mass increases inertial loading, resulting in amplified trajectory deviation. The robust controller significantly reduces slope magnitudes (–0.72 and 2.36 for –2% and + 2% vertebra mass variations, respectively), reflecting improved robustness. Table IX further shows that the robust controller achieves 43.39%, 18.04%, and 35.87% reductions in RMSE norms across different variations.

Figure 13.

Error norm for deterministic and robust approach under parameter uncertainty and external tip force: (a) Vertebra mass variation and (b) Backbone Young’s modulus variation.

For backbone modulus variation (shown in Figure 13(b)), RMSE decreases when backbone Young’s modulus increases by + 5%, due to reduced deformation. The robust controller again produces flatter slopes (–2.940 and 0.290), indicating reduced sensitivity. An interesting observation from the backbone modulus variation is that the RMSE decreases as the Young’s modulus increases, indicating that a stiffer backbone enhances trajectory tracking accuracy by reducing structural deformation. Future work will include tendon wrench analysis and extended tendon–backbone interaction studies. While detailed statistical analysis is not performed, the comparative trends clearly demonstrate the superiority of the robust controller. The combined influence of GA-based robustness (for parameter variability) and DOB-based compensation (for external disturbances) yields consistently lower RMSE and reduced sensitivity across all tested variations.

Table IX.

Norm of RMSE for deterministic and robust approaches under external load.

6.1. Deployment

The motor counts inputted into the hardware are computed from the respective simulation tendon lengths, as expressed in Eq. (28), where C, $L_{i}$ , $L_{0}$ , and $r_{s}$ denotes the motor counts per revolution (hardware-specific value), the instantaneous tendon length, the initial tendon length and the spool radius, respectively. The DWS simultaneously provides feedback on the $i^{th}$ instantaneous tendon length ( $L_{i}^{DWS}$ ), calculated using Eq. (29), where CPR, $\mathrm{E}_{\mathrm{C}}$ , $\mathrm{L}_{0}$ , and $\mathrm{R}_{\mathrm{s}}$ are the encoder counts recorded from DWS, the counts per revolution of the DWS encoder (hardware-specific), the initial cable length and the spool radius of the DWS. Details of these parameters are provided in Table I.

(28) \begin{equation}M_{ref}=\frac{\mathrm{C}}{2\pi }\left(\frac{L_{i}-L_{0}}{r_{s}}\right)\end{equation}

(29) \begin{equation}L_{i}^{DWS}=L_{0}+2\pi \mathrm{R}_{\mathrm{s}}\left(\frac{\mathrm{CPR}}{\mathrm{E}_{\mathrm{CPR}}}\right)\end{equation}

6.2. Deterministic control under no-load conditions

The proposed dynamic-model-based control is evaluated under deterministic no-load conditions to validate its improvements over kinematic control. Figure 14(a) portrays nine sequential frames during which the manipulator follows the desired trajectory. Based on the tendon length obtained from the DWS, Figure 14(b) compares it with numerical simulations. The dynamic-model-based control strategy produces tendon length variations consistent with the kinematic-based approach [Reference Pachouri and Pathak32, Reference Pachouri and Pathak33], but with reduced oscillations, owing to its ability to capture inertia, Coriolis, and acceleration effects. As the circular trajectory progresses through four quadrants, the backbone curvature changes sign, which is observed in tendon lengths during experiments.

Table X.

Tendon-level tracking error metrics: RMSE and maximum absolute error.

Figure 14.

Trajectory tracking without external load (a) Frames, (b) Tendon length, (c) Tendon length tracking errors and (d) Error distribution.

Figure 15.

Trajectory tracking with external force applied at the manipulator tip using the gains from the case when no external force is acting (a) Frames, (b) Comparison of tendon lengths between numerical simulation and experiment, (c) Tendon length tracking errors and (d) Error distribution.

Figures 14(c) and 14(d) illustrate tendon length error and error distribution, respectively. The norm of the tip position error is reduced to 2.75 mm, representing a significant improvement compared to the 4.72 mm error norm achieved with kinematic control [Reference Pachouri and Pathak32]. These results confirm that the deterministic dynamic-model-based control strategy effectively enhances trajectory tracking. Additionally, the RMSE and absolute maximum errors shown in Table X confirm that the tendon length tracking remains within acceptable bounds, with RMSE values of [1.01, 0.51, 0.69, 0.43] mm and absolute maximum errors of [2.25, 1.72, 2.47, 0.81] mm for $[e_{L1},e_{L2},e_{L3},e_{L4}]$ , demonstrating accurate and stable tendon regulation under deterministic no-load conditions.

6.3. Deterministic performance under external load

The performance of the proposed controller is further evaluated under external loading conditions, first using the GA-tuned gains from the no-load case, then with gains re-tuned to compensate for the external load explicitly. Figure 15(a) shows nine sequential frames of trajectory execution, while Figure 15(b) presents tendon length profiles from simulations and experiments. Figure 15(c) shows the corresponding tendon length errors, and Figure 15(d) presents the tendon length error distribution when gains from the no-load case are used.

Figure 16.

Trajectory tracking under external force with re-tuned GA-based controller gains: (a) Sequential nine frames of the manipulator, (b) Tendon length comparison between simulations and experiments, (c) Tendon length tracking errors and (d) Error distribution.

It is observed that tendon length errors remain bounded within ±3 mm, with RMSE values of [1.33, 1.30, 1.36, 2.32] mm and maximum absolute errors of [2.78, 2.58, 3.14, 3.24] mm for $[e_{L1},e_{L2},e_{L3},e_{L4}]$ with marginal offsets occur during sign changes of backbone curvature, likely due to hysteresis and frictional effects.

The controller gains are re-tuned to improve performance using GA optimization, incorporating the external tip force in the simulation model. The results for this scenario are shown in Figure 16, where Figure 16(a) illustrates the trajectory execution frames, Figure 16(b) compares tendon length variations, Figure 16(c) presents tendon length errors, and Figure 16(d) presents the tendon length error distribution. With re-tuned gains, tendon length exhibits consistent accuracy, with RMSE values of [1.34, 1.29, 1.36, 2.32] mm and maximum absolute errors of [3.17, 2.80, 3.39, 3.23] mm. Moreover, task-space accuracy improves, with RMSE reduced by 23.80% in the X-axis, 39.88% in the Y-axis, and 34.11% in the Z-axis, as shown in Table VIII. These results confirm that re-tuning for external load conditions enhances trajectory tracking robustness and compensates effectively for load-induced disturbances.

6.4. Discussion

This section discusses the results from numerical simulations and experimental trials to assess the performance of the proposed dynamic-model-based GA-PID controller under different scenarios: (i) deterministic no-load case, (ii) deterministic case with external load using no-load tuned gains, and (iii) deterministic case with external load using re-tuned gains. The aim is to highlight how load conditions and controller tuning strategies influence tendon-level accuracy, tip tracking, and robustness.

Table X presents the tendon-level error metrics across the three experimental cases, quantifying the mismatch between simulation and experimental tendon profiles. RMSE indicates the average fidelity in reproducing behavior, while the maximum absolute error captures the worst-case deviation between model and physical response. In the no-load case, RMSE values remain low (0.43–1.01 mm) with maximum deviations under 2.5 mm, demonstrating strong agreement between simulation and experiment. Discrepancies increase under external load without re-tuning, with RMSE values rising to 2.32 mm and maximum deviations above 3 mm, highlighting the effect of unmodeled non-linearities such as friction and hysteresis. When controller gains are re-tuned for the load case, tendon-level RMSE between simulations and experiment remains similar, but task-space results (Table IX) show improvement in trajectory accuracy. This indicates that while tendon-level simulation-experiment agreement is already high, re-tuning enhances global tip tracking robustness.

Table XI.

Comparison of tip positional accuracy of two-section CMs.

Table XI compares the tip positional errors of the proposed TDCM with other two-section manipulators reported in the literature. As the manipulators compared in Table XI have different backbone lengths, the results are presented as a representative comparison rather than a normalized one. The P2S-TDCM with GA-tuned PID control achieves tip errors of 3.15 mm (no-load) and 7.53 mm (under load). This performance surpasses earlier soft continuum designs such as Soft CM (∼35.56 mm) [Reference Mazare, Tolu and Taghizadeh36] and is competitive with TDCM-1 [Reference Aner, Ibrahim and Omar37] and TDCM-2 [Reference Pekris, Williams, Atkins, Georgilas and Bailey38]. While TDCM-1 demonstrates slightly lower error (2.11 mm), it is restricted to planar (2D) trajectories and validated in simulation, whereas the proposed TDCM achieves 3D trajectory tracking and is experimentally validated. Compared to TDCM-2, which employs a feedforward controller and achieves an error of 8.28 mm, the proposed TDCM offers comparable no-load performance but greater robustness under load conditions. Additionally, the TDCM-3 [Reference Qin, Chen and Ming39] employs a sliding-mode controller, reports a positional error of 24.1 mm for 3D circular tracking, and shows higher tracking error than the proposed approach. However, a direct one-to-one comparison with these studies is not feasible due to differences in manipulator architecture, backbone length, actuation configuration, and testing conditions; thus, this table outlines, based on attributes such as robot type, backbone length, loading condition, trajectory, control method, and reported positional error. The following are the key observations from the numerical simulation and experiments conducted in this article:

• • The GA-optimized dynamic controller reduced the norm of tip position error to 2.75 mm, a significant improvement over the 4.72 mm error with inverse kinematic control under no external load, while also reducing tendon oscillations by ∼ 12.5% for smoother stability.

• • Lyapunov-based analysis confirmed global asymptotic stability without external forces and UUB under bounded disturbances, ensuring reliable tracking.

• • The control strategy improved RMSE in task space by 23.80% ( $x$ ), 39.88% ( $y$ ), and 34.11% ( $z$ ) after correction; experimental results kept tip errors below 5%.

• • Deterministic GA ensured baseline accuracy, while robust GA-DOB controller reduced sensitivity to parameter variations and disturbances, achieving lower RMSE and flatter error slopes (e.g., slopes reduced from −3.43 and 5.02 in deterministic to −0.72 and 2.36 in robust for mass variation).

• • The DOB enhanced the manipulator’s recovery under external wrenches by estimating and compensating for unmodeled loads in real-time, further stabilizing the transient response.

Experimental validation is limited to a single constant tip force, demonstrating stability under bounded loads. Future work will extend the DOB-assisted robust control to varying or distributed force conditions for adaptive real-time compensation. While robust GA simulations suggest improved performance across a broader range of force conditions, experimental confirmation under varying or distributed forces remains future work.