Since the simplified control method is close to an ideal manipulator and an ideal PID control system, 
and not reflected in the realistic system, we did not include the performance comparison here.
However, the simplified model and control are a good reference for simulation purposes.

### Design Philosophy Comparison

#### `Conventional PID Control Model`
- **Purpose**: Conventional PID control applications
- **Design**: Balanced dynamics for PID control
- **Use Case**: Traditional PID control methods
- **Controller**: Works excellently with typical manipulators
- **Focus**: Controller performance and stability
- **General Applicability**: Good (but needs to be optimised for manufacturer's specific models)

#### `EKF-based PID Control Model`
- **Purpose**: EKF-based trajectory tracking control applications
- **Design**: Ultra-minimal dynamics for state estimation
- **Use Case**: Advanced control with state estimation
- **Controller**: Works excellently with heavily customised manipulators
- **Focus**: State estimation and parameter adaptation
- **General Applicability**: Excellent (optimised for any trajectory tracking)
 

### Performance Characteristics

|           Metric           |      `PID_Conv`    |    `Extended KF`   |           Difference           |
|----------------------------|--------------------|--------------------|--------------------------------|
| **Mass Matrix Condition**  |   1.00 (perfect)   |   1.00 (perfect)   |              Same              |
|       **Gain Ratios**      | 1.00x (identical)  | 1.00x (identical)  |              Same              |
|       **Final Error**      |      0.107 rad     |      0.103 rad     |             Similar            |
|       **Final Velocity**   |     0.014 rad/s    |     0.022 rad/s    |             Similar            |
| **Controller Performance** | Excellent (stable) | Excellent (stable) |              Same              |
|     **EKF Suitability**    |        Good        |      Excellent     | EKF model --> state estimation |


### All Controls Overview

|      Controller      |  Model Used | Performance | Stability | Complexity |    Use Case  |
|----------------------|-------------|-------------|-----------|------------|--------------|
|  **Simplified PID**  | `ur5_model` |     Good    | Excellent |     Low    |   Benchmark  |
| **Conventional PID** | `ur5_model` |  Excellent  | Excellent |    Medium  |      PID     |
|    **Extended KF**   | `ur5_model` |  Excellent  | Excellent |     High   | Adv. control |


### Performance Summary

|       Controller     | Final Error | Final Velocity | Gain Ratios |  Stability  |
|----------------------|-------------|----------------|-------------|-------------|
|  **Simplified PID**  |   0.01 rad  |    0.01 rad/s  |     1:1     |  Excellent  |
| **Conventional PID** |  0.107 rad  |   0.014 rad/s  |     1:1     |  Excellent  |
|    **Extended KF**   |  0.103 rad  |   0.022 rad/s  |     1:1     |  Excellent  |