Data-Driven Kinematic Model for the End-Effector Pose Control of a Manipulator Robot

Assuming that the observed real behavior of the end-effector can be approximated by: where $ϵ\left(t\right)\dot{q}\left(t\right)$ are unknown effects caused by the uncertain knowledge of the Jacobian matrix. Let us transform (5) into the discrete domain: where $\Delta {\chi }^{*}\left(k\right)={\delta }_t{\dot{\chi }}^{*}\left(k\right)$, $\Delta q\left(k\right)={\delta }_t\dot{q}\left(k\right)$, and ${\delta }_t$ is the sampling time. Also, $\Delta {\chi }^{*}\left(k\right)={\chi }^{*}(k+1)-{\chi }^{*}\left(k\right)$ and $\Delta q\left(k\right)=q(k+1)-q\left(k\right)$. Considering these changes, (6) becomes:

Since $J_A^{*}\left(k\right)$ is the true Jacobian and $ϵ\left(k\right)$ is an uncertainty term, ${\widehat{J}}_A\left(k\right)=J_A^{*}\left(k\right)+ϵ\left(k\right)$ describes our estimated Jacobian matrix, with a vanishing term $ϵ\left(k\right)$ according to (4). Therefore, (7) becomes:

In a general sense, (8) is valid when the orientation representation is minimal and in such a form that its update can be done through discrete summations in a Euler-like integration.

The pose errors at step $k+1$ become: where ${\theta }_d^{\prime }={\left[{\theta }_{w_d},-{\theta }_{v_d}\right]}^T$ is the conjugate of ${\theta }_d$. Note that computing the errors in terms of quaternion products requires the complete quaternion $\theta$; therefore, ${\theta }_v$ can be used to reconstruct $\theta$ and then normalize it to ensure its unitary property. In this case, the orientation error $e_{\theta }(k+1)$ accounts for the rotation necessary for $\theta (k+1)$ to minimize $e_{\theta }(k+1)$, which happens when $e_{\theta }(k+1)\to q_I$, where $q_I={\left[1,0,0,0\right]}^T$ is the quaternion identity. By substituting (9) and adding $p_d\left(k\right)-p_d\left(k\right)=0$ to (14), it is straightforward to obtain the position error update: where $e_p\left(k\right)=p\left(k\right)-p_d\left(k\right)$ is the position error at iteration k and $\Delta p_d\left(k\right)=p_d\left(k\right)-p_d(k+1)=-{\delta }_t{\dot{p}}_d\left(k\right)$ accounts for changes in the desired trajectory. The orientation update may be tricky, as the orientation error requires the full quaternion product (15) and the orientation-state update is done with only the vector part of the quaternion (13). Additionally, it requires multiplying by ${\theta }_d\left(k\right)\otimes {\theta }_d^{\prime }\left(k\right)=q_I$ to identify $e_{\theta }\left(k\right)$ and $\Delta {\theta }_d\left(k\right)$; consequently, even combining (15) with (12) is problematic, because it ends up with a product like $J_Q\left(k\right)\Delta q\left(k\right)\otimes {\theta }_d^{\prime }(k+1)$, where $J_Q\left(k\right)\Delta q\left(k\right)$ is not a true unit quaternion.

Now, consider the following error function, with errors $(k+1)$ and k: where ${\alpha }_1$ and ${\alpha }_2$ are positive constants. By substituting (16) and (17) into (30), we obtain:

Note that, since $e_{v_{\theta }\left(k\right)}$ can be integrated like $e_{p\left(k\right)}$, we can stack $\chi ={\left[p^T,{\theta }_v^T\right]}^T$, $e={\left[e_p^T,e_{v_{\theta }}^T\right]}^T$, ${\widehat{J}}_A={\left[{\widehat{J}}_{\nu },{\widehat{J}}_o\right]}^T$, and $s={\left[s_p^T,s_o^T\right]}^T$. Now, let us use $s(k+1)$ as a constraint for the following constrained quadratic program: which leads to the following objective function:

Then, solve the following partial derivatives: where (35) and (36) build the following system: where $a={\displaystyle \frac{{\alpha }_1}{{\alpha }_1+{\alpha }_2}}$. From (37), solving first for $\lambda \left(k\right)$:

Then, solve for $\Delta q\left(k\right)$: where ${\widehat{J}}_A^{†}={\widehat{J}}_A^T{\left({\widehat{J}}_A{\widehat{J}}_A^T\right)}^{-1}$ is the Moore–Penrose right pseudoinverse of ${\widehat{J}}_A$ and $\widehat{N}=I-{\widehat{J}}_A^{†}{\widehat{J}}_A$ is a null-space projector that introduce joint velocities $\Delta z\left(k\right)\in {\mathbb{R}}^n$ that have a null effect on $\dot{\chi }\left(k\right)$. Note that (39) effectively minimizes the joint-velocities (33) under the constraint defined by (30). The minimization describes the dynamics of the end-effector task: which is then mapped to the joint-space through ${\widehat{J}}_A^{†}$. As is typically the case with most manipulator’s non-square Jacobians, ${\widehat{J}}_A^{†}$ has a solution as long as it is full-column-rank, namely, $\rho \left({\widehat{J}}_A^{†}\right)=m$. In order to ensure a solution always exist, this work uses a damped ${\widehat{J}}_{\varrho A}^{†}$, with a regularization factor $\varrho$:

Since the Jacobian is estimated, an insufficiently defined direction of the Jacobian that correspond to the coordinates of $\dot{\chi }$ can be compensated by the action of an adaptable scaling factor for the whole mapping ${\mathbb{R}}^n\to {\mathbb{R}}^m$ of (40); this is defined as follows: with ${\varsigma }_i>0$. Then, let $C_k\in {\mathbb{R}}^{6\times 6}$ be a diagonal matrix with:

Let us establish a behavior to the task dynamics, which adapts to the changes of the estimated Jacobian while updating. Thus, by using the damped pseudoinverse (41), and (43), (40) in (39), $\Delta q\left(k\right)$ must be redefined as: where with ${\alpha }^{*}\left(k\right)=\alpha \left(k\right){\displaystyle \frac{{\alpha }_1}{{\alpha }_1+{\alpha }_2}}$. Also, $\alpha \left(k\right)=C^{-1}\left(k\right)$ if $\parallel \Delta {\widehat{J}}_A\left(k\right)\parallel \approx 0$, otherwise $\alpha \left(k\right)=1$. Finally, the updated joints’ positions are:

While the term $\widehat{N}\left(k\right)\Delta z\left(k\right)$ does not contribute to the main task (45), it contributes null-space joint-excitation ${\dot{q}}_{null}\left(k\right)=\widehat{N}\left(k\right)\Delta z\left(k\right)$ to (4), which keeps the update going in case the range-space joint-velocity ${\dot{q}}_{range}\left(k\right)={\widehat{J}}_{\varrho A}^{†}\left(k\right)u\left(k\right)$ is insufficient (e.g., if the robot moves slowly).

The closed-loop configuration is depicted in Figure 1. The PJM algorithm estimates the Jacobian matrix derived from the association between the output signals (end-effector velocities) and input signals (joint’s velocities) since the estimation error $ϵ\left(k\right)\to 0$. The closed-loop controller starts when the control errors in (14) and (15) are generated between the current end-effector pose and the desired set point. The proposed control law considers the minimization of an objective function, treated as a constrained quasi-Lagrangian function, with the error function $s(k+1)$ from (30) as a constraint, which contains the control parameters ${\alpha }_1$, ${\alpha }_2$ to scale the convergence of the control error $e\left(k\right)$. During the learning phase, the control law in (45) is adjusted for each end-effector’s coordinate independently of each other, considering the dynamic system from the equivalent Jacobian matrix with adaptive gain $c_i\left(k\right)$ in (42) scaling the i-th row of the Jacobian matrix.

The numerical method used to solve the equations in the simulations presented in this article is the Euler step integration method, with a step size of 0.01. This method was employed to integrate the first-order kinematic system and to numerically compute the end-effector pose for feedback to the estimator. Using the same integration method, consistency is ensured between system integration and Jacobian updates. To account for potential numerical effects, such as integration errors or step size sensitivity, the selected step size was validated to ensure stability and convergence in the simulations. In real-world applications, the Euler step size should be adjusted based on the precision and noise characteristics of the measurement system to balance computational efficiency and precision.

This section demonstrates the Lyapunov stability analysis, involving the system dynamics during the model estimation—learning phase—and the convergence of the error after concluding the learning phase, which shows that a lack of model knowledge, i.e., $\parallel ϵ\left(k\right)>0\parallel$, is tolerated for large errors, as the control effort is strongly directed at reducing $\parallel e\left(k\right)\parallel$; then, as $e\left(k\right)\to 0$, the influence of $ϵ\left(k\right)$ becomes more significant relative to $e\left(k\right)$ and the system’s stability depends on the residual dynamics. The interplay between $e\left(k\right)$ and $ϵ\left(k\right)$ and its impact on the stability of the system is addressed in detail in Appendix A, from which the parameters in Table 1 were established.

When the manipulator is fixed on its base, it has limited mobility and reach, resulting in a reduced set of compatible position and orientation coordinates that can be assigned. Figure 3 presents the pose commanded for the UR5 robot during 250 s of the simulation. Subfigure (a) displays the position coordinates (x, y, z) compared to the desired values ($x_d$, $y_d$, $z_d$) over time. It is evident that the position commands were dynamically adjusted, reflecting significant changes in the z-axis around the 75 s and 150 s marks. The x and y coordinates follow a smoother trajectory, although slight deviations from the desired positions can be observed, particularly in the y-axis after 200 s. Subfigure (b) depicts the commanded orientation of the end-effector, represented as a quaternion (${\theta }_{\omega }$, ${\theta }_x$, ${\theta }_y$, ${\theta }_z$). The orientation follows a more complex behavior with abrupt changes, particularly in ${\theta }_x$ and ${\theta }_z$, around the same time intervals as the positional changes. These variations indicate rotational adjustments made to maintain the desired end-effector pose, with the quaternion components stabilizing towards the end of the simulation.

Figure 4 illustrates the absolute errors of position (14) and orientation (15) of the UR5 end-effector throughout the simulation. Subfigure (a) shows the position errors for the coordinates x, y, and z. Initially, there are noticeable fluctuations, particularly in the z-axis, but these errors quickly converge to values close to zero, indicating an effective stabilization of the system. Small deviations can still be observed during moments where the commanded pose changes, particularly around the 50, 100, and 150 s marks, reflecting the system’s response to new target positions. Subfigure (b) presents the orientation errors, represented as quaternion components ($e_{{\theta }_{\omega }},e_{{\theta }_x},e_{{\theta }_y},e_{{\theta }_z}$). Similarly to position errors, orientation errors stabilize over time. In particular, the quaternion component $e_{{\theta }_{\omega }}$, associated with the real part, remains close to one, indicating a stable rotation. The imaginary components ($e_{{\theta }_x},e_{{\theta }_y},e_{{\theta }_z}$) exhibit transient variations but converge to near-zero values, ensuring that the end-effector maintains its desired orientation as commanded. The system remains stable until significant changes in the commanded orientation are introduced.

Figure 5 illustrates the end-effector velocities of the UR5 robot, derived from the time derivative of the pose coordinates captured in the simulation. Subfigure (a) displays the linear velocities along the x-, y-, and z-axes. Initially, there are notable fluctuations, particularly in the x direction, where the velocity reaches peaks of approximately ±1.5 m/s during the transitions. These fluctuations indicate the system’s response to the commanded poses, which subsequently stabilize as the motion becomes smoother over time. Subfigure (b) presents the angular velocities represented by the quaternion rates (${\dot{\theta }}_x,{\dot{\theta }}_y,{\dot{\theta }}_z$). Similarly to linear velocities, the angular rates exhibit initial peaks, particularly around the transition phases. As the simulation progresses, these rates stabilize and converge toward zero, indicating that the end-effector effectively maintains its orientation without excessive rotation. This behavior underscores the efficiency of the control system in achieving stable movement in both linear and angular dimensions.

Figure 6a illustrates the evolution of the adaptive gains for each element of the end-effector’s pose. These gains start from an initial value near 0.5 and evolve dynamically as the controller adapts to changes in the task, reaching values between 0.4 and 2. In particular, the gains associated with the orientation components ($c_{{\theta }_x}$, $c_{{\theta }_y}$, and $c_{{\theta }_z}$) show significant variations when the orientation commands change, indicating the controller’s response to orientation uncertainties. The position gains ($c_x$, $c_y$, and $c_z$) exhibit more gradual variations, reflecting smaller positional uncertainties throughout the trajectory. Figure 6b shows the joint velocities of the UR5 robot as the control output. The velocities remain continuous and stable, with values ranging between −2 and 2 rad/s for all joints. Peaks in velocity can be observed at the beginning of the task and during key moments when the desired pose coordinates are altered. These peaks are consistent across all joints, indicating synchronized movement. Despite these peaks, the joint velocities remain well within the operational safety limits, ensuring smooth robot motion without abrupt changes. These peaks can be handled with smooth transition functions, which take values from 0 to 1; these would be applied at the moments when the target pose changes are generated, as done in Toro-Arcila et al. [29]. The simulation for this experiment is presented in a video available at https://drive.google.com/file/d/12vOkFRL-_AVmSgVu3xISPHUkiIzejoGR/view?usp=sharing (accessed on 30 October 2024).

When the manipulator has an omnidirectional mobile base, it gains an additional 3 DOF because of its holonomic constraint, which makes it a redundant manipulator. Also, its mobility and reach are increased; therefore, it has a wider set of feasible coordinates for the end-effector to reach.

Figure 7 shows the evolution in time of the commanded position and orientation of the end-effector during the task (200 s of simulation). In Figure 7a, the position coordinates ($x_d$, $y_d$, $z_d$) are interactively modified at specific moments during the simulation. The component $x_d$ shows a significant change at around 150 s, while $z_d$ decreases slightly and $y_d$ remains relatively stable throughout the simulation, with minimal fluctuations. In Figure 7b, the commanded orientation, expressed as quaternion components (${\theta }_{\omega }$, ${\theta }_x$, ${\theta }_y$, ${\theta }_z$), also undergoes dynamic changes. Significant variations occur in ${\theta }_x$ and ${\theta }_y$, especially during the first 50 s, reflecting adjustments in the desired orientation. Approaching the end of the simulation, the quaternion values stabilize, indicating a steady desired orientation after the commanded modifications.

Figure 8 illustrates the evolution in pose errors in time for the KY end-effector. In subfigure (a), the position errors ($e_x$, $e_y$, $e_z$) initially exhibit significant discrepancies, especially in the z-axis, but rapidly converge to near-zero values as the system adapts. This rapid reduction in error demonstrates the effectiveness of the control algorithm in aligning the actual position with the commanded values. However, small oscillations can be observed around $t=100$ s, particularly on the z-axis, which correspond to minor fluctuations likely caused by dynamic disturbances or changes in the commanded pose. Subfigure (b) depicts the orientation errors represented by the imaginary parts of the unit quaternion components ($e_{{\theta }_{\omega }}$, $e_{{\theta }_x}$, $e_{{\theta }_y}$, $e_{{\theta }_z}$). The quaternion errors follow a similar trend, with a rapid reduction in discrepancies over time. Although the errors converge close to zero, there are slight variations, especially in $e_{{\theta }_x}$ and $e_{{\theta }_y}$, reflecting the challenge of maintaining precise orientation under dynamic conditions. These variations remain within acceptable limits, indicating the overall robustness of the system in handling orientation changes. Both subfigures confirm that the control approach ensures the convergence of the pose errors over time, with the position and orientation errors stabilizing near zero until the introduction of new commanded pose coordinates, where a temporary increase in error occurs before reconvergence.

Figure 9 presents the linear and angular velocity profiles of the end-effector. Subfigure (a) shows the linear velocities $\dot{x}$, $\dot{y}$, and $\dot{z}$ in the Cartesian coordinate system, while Subfigure (b) displays the angular velocity rates ${\dot{\theta }}_x$, ${\dot{\theta }}_y$, and ${\dot{\theta }}_z$, expressed in terms of quaternions. These velocities were obtained by numerically differentiating the pose data captured from the simulation environment. The linear velocity plot shows significant variations in the early stages, followed by stabilization, which aligns with the expected transient behavior of the system. Similarly, the quaternion rates in Subfigure (b) illustrate the dynamic changes in orientation over time. Both sets of velocity serve as feedback for the adaptive control algorithm, enabling the system to correct deviations in real time by adjusting the control inputs. This feedback loop ensures precise motion control of the end-effector throughout the simulation period.

Figure 10a depicts the evolution of the adaptive gains for the controller, applied to the individual elements of the complete pose. The adaptive gains, denoted as $c_x$, $c_y$, $c_z$ for the Cartesian coordinates and $c_{{\theta }_x}$, $c_{{\theta }_y}$, $c_{{\theta }_z}$ for the quaternions, vary within the range of $0.8$ to $2.0$, particularly under conditions of maximum uncertainty. These gains adjust dynamically in response to changes in the system’s state, converging to stable values when the desired pose coordinates remain constant. The plot illustrates how the controller adapts to ensure optimal performance over time. Figure 10b illustrates the joint angular velocities ${\dot{q}}_1$ to ${\dot{q}}_8$, representing the control output for each of the manipulator’s joints. The velocities remain within feasible and safe operational limits and exhibit smooth transitions without abrupt discontinuities. These control signals ensure that the manipulator operates within a stable range, maintaining the safety and efficiency of the system. The combined data from both plots demonstrate the robustness of the adaptive control strategy in handling both position and orientation tracking. The simulation for this experiment is presented in a video available at https://drive.google.com/file/d/1bhFaYJZR0S9HDmpxmqKqhFtpigMTwi0H/view?usp=sharing (accessed on 30 October 2024).

When the omnidirectional mobile base is equipped with a dual system of manipulators, there are two end-effectors to control and additional degrees of freedom (DOF), totaling 13 DOF. Of these, three are shared (omnidirectional mobile base), while each manipulator contributes five exclusive DOF for pose control. In this case, it becomes important to assign feasible pose coordinates to both end-effectors, which must remain near their mobile base.

Figure 11 and Figure 12 present the commanded pose coordinates for the left and right end-effectors, which were modified during simulation time (110 s). Figure 11a shows the commanded position trajectories for the left tip of the manipulator in Cartesian space, with position coordinates $x_l$, $y_l$, $z_l$, as well as their desired references $x_{ld}$, $y_{ld}$, $z_{ld}$. The plot highlights the convergence of the actual position coordinates to the desired values over time. It is evident that the system rapidly stabilizes the $x_l$ coordinate, while minor oscillations can be observed in the $y_l$ and $z_l$ coordinates before reaching steady state. These variations reflect the system’s response to the initial command and subsequent adjustments, achieving precise position control under dynamic conditions. Figure 11b presents the commanded orientation based on quaternions for the left tip, including the actual components of the quaternions ${\theta }_{l_{\omega }}$, ${\theta }_{l_x}$, ${\theta }_{l_y}$, and ${\theta }_{l_z}$, along with their desired references ${\theta }_{l_{{\omega }_d}}$, ${\theta }_{l_{x_d}}$, ${\theta }_{l_{y_d}}$, and ${\theta }_{l_{z_d}}$. The system demonstrates smooth transitions between the desired orientations, with notable jumps in the quaternion components corresponding to large rotational adjustments. The data in both plots demonstrate the manipulator’s ability to achieve accurate position and orientation tracking, even when subjected to complex commands. This precise control is crucial for tasks that require high levels of dexterity and adaptability in robotic manipulation.

The results presented in Figure 12 illustrate the evolution of the commanded pose for the right end-effector, including both position and orientation, over a simulation period of 110 s. The figure is divided into two subplots: position (Figure 12a) and orientation (Figure 12b). In position subplot (a), the actual and commanded Cartesian coordinates, x, y, and z, of the right end-effector are displayed over time. Solid red, green, and blue lines represent the actual values of $x_r$, $y_r$, and $z_r$, while the corresponding dashed lines represent the desired trajectories, $x_{rd}$, $y_{rd}$, and $z_{rd}$. The coordinate x (red line) shows a rapid rise at the beginning of the simulation, reaching the desired value of approximately 2 m within the first 10 s. This initial transient phase indicates that the controller successfully manages the acceleration and deceleration necessary to achieve the target. After reaching the desired value, the position x stabilizes with minimal overshoot and remains steady until the end of the simulation, showing no significant deviation from the reference trajectory. The coordinates y (green line) and z (blue line) exhibit a more complex behavior. The coordinate y shows an initial negative deviation before converging to zero. Around the 80 s mark, a minor disturbance occurs, likely due to changes in the commanded orientation, which is quickly corrected, indicating robust control. The coordinate z remains largely stable, fluctuating around the desired value with only small deviations, particularly between 40 and 60 s. These deviations are due to changes in the commanded orientation but are effectively controlled to maintain the end-effector within the desired values. In orientation subplot (b), the four quaternion components (${\theta }_{r_{\omega }}$, ${\theta }_{r_x}$, ${\theta }_{r_y}$, and ${\theta }_{r_z}$) are shown. Initially, all quaternion components deviate significantly from their desired values, reflecting the misalignment of the end-effector at the start of the simulation. However, as time passes, the controller adjusts the orientation smoothly. Between 10 and 30 s, there is a noticeable shift in all components, with the system gradually aligning with the target orientation. The most prominent change occurs in ${\theta }_{r_y}$ (yellow line) and ${\theta }_{r_z}$ (cyan line), which undergo significant transitions before settling in their desired values. By the 40 s mark, the quaternion components stabilize, indicating that the end-effector has successfully aligned with the commanded orientation. The transient behavior observed in the position and orientation plots suggests that the control strategy employed in the KYD robot is effective in handling both position and orientation. The system demonstrates a quick response to initial errors, as evidenced by the rapid convergence to the desired pose, while also showing resilience to disturbances encountered mid-simulation, particularly in the y and z coordinates. Overall, the results suggest that the controller is capable of achieving precise and stable control over the end-effector’s position and orientation, making it a robust candidate for tasks requiring high precision in dynamic environments.

Figure 13 and Figure 14 display the pose errors for the left and right end-effectors, respectively, including position and orientation errors. The graphs reveal a consistent trend of error convergence towards zero over time, signifying effective control despite initial deviations. In Figure 13a, the position errors for the left end-effector, represented by $e_{x_l}$ (red), $e_{y_l}$ (green), and $e_{z_l}$ (blue), show a rapid reduction to near zero values within the first 10 s of the simulation. The largest initial error is observed in $e_{x_l}$, where the deviation reaches nearly −2 m at the beginning but quickly stabilizes. Similarly, $e_{y_l}$ and $e_{z_l}$ also converge, demonstrating smooth error correction over time. Small oscillations are noticeable around 50 s, particularly in $e_{y_l}$ and $e_{z_l}$, which are a consequence of changes in the desired position. However, these deviations are promptly corrected, leading to stable performance in the later stages of the simulation. The quaternion errors that represent the orientation of the left end-effector are shown in Figure 13b. Initially, significant deviations are observed in all quaternion components ($e_{{\theta }_{l_{\omega }}}$, $e_{{\theta }_{l_x}}$, $e_{{\theta }_{l_y}}$, $e_{{\theta }_{l_z}}$). These errors gradually decrease, with the largest deviations in $e_{{\theta }_{l_{\omega }}}$ (black) and $e_{{\theta }_{l_y}}$ (yellow). Between 20 and 60 s, fluctuations in $e_{{\theta }_{l_x}}$ and $e_{{\theta }_{l_y}}$ indicate small orientation adjustments. By the end of the simulation, the quaternion errors have reduced substantially, reflecting that the left end-effector aligns closely with the desired orientation.

In Figure 14a, the position errors for the right end-effector follow a similar trend. The $e_{x_r}$ (red) error starts at approximately −2 m, the largest deviation among the three axes, but rapidly decreases within 10 s. The errors $e_{y_r}$ (green) and $e_{z_r}$ (blue) also exhibit a fast convergence, with only minor deviations throughout the simulation. The convergence of position errors demonstrates the effectiveness of the controller in maintaining the desired end-effector positions. The orientation errors for the right end-effector, depicted in Figure 14b, show a pattern similar to that of the left end-effector. The quaternion errors initially present significant deviations, especially in $e_{r_{{\theta }_{\omega }}}$ (black) and $e_{r_{{\theta }_y}}$ (yellow). Although small fluctuations are observed between 40 and 80 s, particularly in $e_{r_{{\theta }_x}}$ and $e_{r_{{\theta }_y}}$, the system exhibits stable behavior in the final phase of the simulation, with all quaternion errors converging toward zero.

In both cases, the results clearly demonstrate that the controller effectively reduces both position and orientation errors over time, ensuring accurate tracking of the end-effector poses. The system shows rapid error convergence during the first 10 s and remains robust against disturbances (changes in the desired values of the position and orientation of both end-effectors), particularly in the middle stages of the simulation. This performance highlights the controller’s ability to adapt to changing reference coordinates while maintaining high precision and stability.

In Figure 15 and Figure 16, the velocity profiles of the left and right end-effectors are presented, detailing their linear and angular components over time. These velocity profiles were obtained from the simulation environment and serve as critical feedback for the adaptation algorithm used in the control strategy. In Figure 15, the linear velocities of the left end-effector along the axes x, y, and z are shown in subplot (a). It can be observed that the velocities stabilize after an initial transient phase, converging close to zero after approximately 10 s. The x-axis velocity (${\dot{x}}_l$) exhibits minor oscillations initially but rapidly attenuates to near zero values. Similarly, the y-axis velocity (${\dot{y}}_l$) stabilizes, while the z-axis velocity (${\dot{z}}_l$) maintains minimal fluctuations throughout the simulation. Subplot (b) of Figure 15 depicts the angular velocities of the left end-effector, represented as quaternion rates ${\dot{\theta }}_{l_x}$, ${\dot{\theta }}_{l_y}$ and ${\dot{\theta }}_{l_z}$. These quaternion rates capture the rotational dynamics, showing an initial adjustment period where the angular velocities exhibit greater variance, particularly in the ${\dot{\theta }}_{l_z}$ component. After approximately 10 s, the angular velocities stabilize, with ${\dot{\theta }}_{l_z}$ showing minor oscillations, while ${\dot{\theta }}_{l_x}$ and ${\dot{\theta }}_{l_y}$ converge towards zero.

Similarly, Figure 16 presents the linear and angular velocities of the right end-effector. Subplot (a) shows the linear velocities along the x-, y-, and z-axes. The velocity of the x-axis (${\dot{x}}_r$) of the right end-effector follows a trend similar to that of the left end-effector, showing initial oscillations before stabilizing to near zero after 10 s. The velocity of the y-axis (${\dot{y}}_r$) also stabilizes, although with slightly higher transient peaks compared to the left end-effector. The velocity of the z-axis (${\dot{z}}_r$) remains stable, with minor deviations throughout the simulation. Subplot (b) in Figure 16 displays the quaternion rates ${\dot{\theta }}_{r_x}$, ${\dot{\theta }}_{r_y}$, and ${\dot{\theta }}_{r_z}$, corresponding to the angular velocities of the right end-effector. Similarly to the left end-effector, the quaternion rates exhibit an initial adjustment phase, particularly in ${\dot{\theta }}_{r_x}$ and ${\dot{\theta }}_{r_z}$. The angular velocities then stabilize, with all three components converging to near-zero values after approximately 10 s.

Overall, the velocity profiles for both the left and right end-effectors indicate effective stabilization of both linear and angular velocities over time, highlighting the robustness of the control algorithm in adapting to dynamic conditions. The transient oscillations observed in the first 10 s are typical of systems with high sensitivity to initial conditions, but the convergence to near-zero velocities demonstrates that the system successfully achieves stable motion control of both end-effectors.

Figure 17 illustrates the adaptive control gains and joint velocities for both left and right end-effectors, providing insight into the controller’s performance under varying values of the desired pose. In Figure 17a,b, the adaptive gains for the left and right pose controllers are presented, respectively. The adaptive gains for the left end-effector, shown in Figure 17a, vary within a range of 0.6 to 2, with noticeable adjustments occurring principally when the system experiences greater changes in the desired values of the pose. The gain values ($C_{l_x}$, $C_{l_y}$, $C_{l_z}$, $C_{l_{{\theta }_x}}$, $C_{l_{{\theta }_y}}$ and $C_{l_{{\theta }_z}}$) initially fluctuate before stabilizing once the system reaches a steady state, and the desired pose coordinates remain unchanged. Similarly, in Figure 17b, the adaptive gains for the right end-effector ($C_{r_x}$, $C_{r_y}$, $C_{r_z}$, $C_{r_{{\theta }_x}}$, $C_{r_{{\theta }_y}}$ and $C_{r_{{\theta }_z}}$) follow a similar pattern, ranging from 0.5 to 2. These gains exhibit an increase in response to changes in the desired values of the pose and then stabilize as the system converges. Figure 17c depicts the joint angular velocities, representing the control signals generated by the system to drive the manipulator. All these angular velocities (${\dot{q}}_{b_1}$, ${\dot{q}}_{b_2}$, ${\dot{q}}_{b_3}$, ${\dot{q}}_{l_1}$, …, ${\dot{q}}_{l_5}$, ${\dot{q}}_{r_1}$, …, ${\dot{q}}_{r_5}$) are continuous and remain within feasible operational ranges, ensuring safe and stable motion control. The velocities show small oscillations in the transient phase, followed by stabilization once the adaptive control mechanism compensates for the changes in the desired values of the pose. The absence of abrupt changes in joint velocities further demonstrates the robustness of the control strategy in maintaining smooth and controlled motion under variable conditions. The simulation for this experiment is presented in a video available at https://drive.google.com/file/d/1Yhp-rIkspLm5suoJIVKH6w81mnRh2ACN/view?usp=sharing (accessed on 30 October 2024).