## 2 Materials and Methods
### Reinforcement Learning in Continuous Environments
#### Performance Evaluation

To provide a comprehensive evaluation, it is crucial to compare the proposed DRL-based robot control simulation with existing techniques. This comparison should demonstrate the strengths of the new approach in terms of adaptability, efficiency, and task performance metrics. Benchmark techniques for comparison are noted as follows:

* Traditional Control Methods
* Classical Motion Planning Algorithms
* Other Reinforcement Learning Algorithms
* Learning by Demonstration

To objectively compare techniques, the following key metrics were established:

* Task Completion Rate
* Adaptability to Dynamic Environments
* Path Accuracy (Deviation)
* Task Completion Time
* Energy Efficiency
* Learning Efficiency

Each algorithm is initialized and trained within the simulation environment. The robot performs a series of tasks (pick-and-place and path following) under each control method. These tasks are repeated multiple times (3000 trials) to collect statistically significant data. For each trial, data on task completion rate, path deviation, completion time, and energy consumption are collected. The results are presented in a Table 5 to illustrate each technique's performance. The proposed DRL-based approach outperforms PID and A* in task completion rate, particularly in dynamic environments, due to its ability to adapt actions in real time. DRL-based control methods, particularly with DMP integration, achieve lower path deviation and shorter completion times than traditional control methods (PID) and classical motion planning (A*), which struggle to make real-time adjustments. The DRL-based approach achieves higher energy efficiency, making it suitable for tasks requiring repetitive motion with minimal power use. The comparison confirms that the DRL-based approach with \begin{table}
\begin{tabular}{l l l} \hline \hline  & **Strengths** & **Limitations** \\ \hline \multirow{2}{*}{Deep Q-Networks (DQN)} & Simplicity, & Sample Inefficiency, \\  & Experience Replay, & Action Space Limitation, \\  & Target Network & Stability Issues \\ \hline \multirow{2}{*}{Double DQN} & Reduced Overestimation Bias, & Increased Complexity, \\  & Improved Stability & Still Sample Inefficient \\ \hline \multirow{2}{*}{Basic DMP} & Trajectory Representation, & \\  & Generalization, & Manual Tuning, \\  & Smooth Trajectories & Limited Expressiveness \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Strengths and limitations of different deep reinforcement learning (DRL) and dynamic movement primitives (DMP) algorithms. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Metric**} & **Proposed** & \multirow{2}{*}{**PID Control**} & \multicolumn{2}{c}{**A*} & \multirow{2}{*}{**DDPG**} & \multirow{2}{*}{**PPO**} & \multirow{2}{*}{**LbD Only**} \\  & **DRL-Based** & & & & & \\ \hline Task Completion Rate & 96\% & 69\% & 80\% & 91\% & 92\% & 85\% \\ Adaptability Score & High & Low & Medium & Medium & High & Low \\ Path Accuracy (cm) & 0.4 & 2.0 & 1.5 & 0.7 & 0.6 & 1.0 \\ Task Completion Time (s) & 7.5 & 11.2 & 9.8 & 9.1 & 8.8 & 9.5 \\ Energy Efficiency (J) & N/A & N/A & N/A & N/A & N/A & N/A \\ Learning Efficiency & 3000 episodes & N/A & N/A & 1000 & 700 & 300 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Analysis of results compared with existing techniques. DMP integration offers significant improvements over existing techniques, especially in dynamic and unstructured environments (Figure 9). A performance comparison diagram between DQN- and DMP-based algorithms illustrates their effectiveness in a robotic or control-based task over a set number of training episodes or trials. DMP may reach a steady performance level earlier due to its predefined movement primitives, making it suitable for tasks with known, repeatable trajectories. DQN, on the other hand, may take longer but has greater flexibility to adapt to complex or changing environments, given enough episodes. DQN may potentially outperform DMP in highly dynamic tasks due to its capacity for ongoing learning and adaptability.