# Experimental Investigation of Free-Motion Task Implementation on a Serial Metamorphic Manipulator
## 4 Experimental Case Study
### Experimental Validation of Dynamic Modeling  
The next phase of the validation process evaluated the accuracy of the simplified dynamic model by comparing analytical calculations with experimental data. That phase focused on assessing the model's ability to capture the system's fundamental behavior under basic task conditions, including low end-effector speed and the absence of external forces or loads. Analytical calculations executed in MATLAB for a specific anatomy (ID: 1) and the Cartesian path were compared with experimental data, particularly focusing on joint 2, the main contributor to the joint effort. Graphs of joint torques and velocities, as well as torque and velocity deviation for all joints, were used for the investigation. This comparison enabled a detailed assessment of how effectively the theoretical model reflected the core system dynamics. Additionally, the comparison highlighted the limitations of the simplified model by quantifying discrepancies and offering insights into the effects of unmodeled factors such as joint friction, stiffness, parameter uncertainties, and external disturbances.  
In Figures 9 and 10, the torques and velocities calculated by the dynamic model and the ones produced by robot joints are compared. In Figures 9a and 10a the ability of the established dynamic model to effectively capture the general trend in torque and velocity is illustrated. The torque comparison, shown in Figure 9a, reveals that the theoretical calculations were noticeably smoother, largely due to the simplified model used in MATLAB. The experimental data from the robot presented fluctuations, primarily attributed to the PID controller's influence and unmodeled dynamics such as friction, joint compliance, and external disturbances. To extract a more detailed view of the observed discrepancies, the deviation between theoretical and experimental data is illustrated in Figures 9b and 10b. In  
Figure 8: TCP trajectory error (ID: 3, task path: 28). (**a**) TCP trajectory error graphs (desired vs. executed). (**b**) TCP position error in 3D space.  
both graphs, polynomial regression trend lines were employed to analyze the deviation patterns. Joints 1 and 2 produced lower effort values, with their discrepancies exhibiting smaller variations (approximately 0.5 Nm), primarily due to evenly distributed sensor noise over the task's duration. The fluctuations in torque deviations observed for joint 2 correlated with the overall effort trend, suggesting that joint friction had a more significant impact. Additionally, transient effects and controller response contributed to the observed variations. The observed torque peak deviations reached approximately 3.5 Nm, with a median deviation value of 1.5 Nm. Velocity discrepancies displayed similar sensitivity to unmodeled system uncertainties, with trend lines remaining below 0.2 rad/s, which was considered adequate for this analysis.  
These results, although not optimal, offered a reliable base reference for evaluating the experimental data obtained from the corresponding task paths executed by the robot prototype. This evaluation, presented in the remainder of this section, focused on the following three main areas: optimizing joint torque to velocity output, optimizing end-effector translation response, and the joint torque controller output.  
Figure 10: Joint velocities’ investigation for optimized EMI CP anatomy (ID: 1, task path: 16). (**a**) Joint 2 velocity. (**b**) Joints’ velocity deviation.  
Figure 9: Joint torques’ investigation for optimized EMI CP anatomy (ID: 1, task path: 16). (**a**) Joint 2 torque. (**b**) Joints’ torque deviation.