An Extended Generalized Prandtl-Ishlinskii Hysteresis Model for I2RIS Robot

Technical Summary: An Extended Generalized Prandtl-Ishlinskii Hysteresis Model for I2RIS Robot

Problem Statement
Retinal microsurgery requires extreme precision due to the constrained anatomical spaces of the human retina. While robotic systems like the Improved Integrated Robotic Intraocular Snake (I2RIS) have been developed to assist surgeons by adding dexterity (yaw and pitch bending) and tremor cancellation, they face significant control challenges. Specifically, the I2RIS, being a flexible, tendon-driven continuum robot, exhibits complex hysteresis behaviors. Experimental data reveals that the robot suffers from multi-stage hysteresis, particularly within regions of monotonic input changes. This phenomenon, caused by cable-driven actuation mechanisms, material hysteresis, and friction, creates multiple distinct "dead zones" or backlash stages that conventional models fail to capture accurately. The inability to model these transitions precisely hinders accurate position control, posing risks to patient safety.

Methodology
To address the limitations of existing models in capturing multi-stage hysteresis, the authors propose an Extended Generalized Prandtl-Ishlinskii (EGPI) model.

1. Model Architecture:

   • The standard Generalized Prandtl-Ishlinskii (GPI) model uses a weighted sum of play operators with envelope functions to model asymmetric hysteresis. However, the standard GPI assumes a single stage of backlash per monotonic phase.
   • The proposed EGPI model introduces a novel switching mechanism. It treats the presence of multiple hysteresis stages as a switching behavior between different GPI sub-models.
   • The system utilizes two distinct sets of play operators ($H_{r,1}$ and $H_{r,2}$) associated with different pairs of envelope functions ($\gamma_{l,1}, \gamma_{r,1}$ and $\gamma_{l,2}, \gamma_{r,2}$).
   • A switching logic is defined based on flag points ($v_{f,1}$ and $v_{f,2}$) determined from experimental data. The model transitions from one sub-model to another when the input velocity ($\dot{v}$) and input value ($v$) cross these thresholds. This allows the model to represent piecewise monotonic patterns within a single input direction.

2. Parameterization and Optimization:

   • For the I2RIS application, the envelope functions are simplified to linear functions to reduce computational cost while capturing the linearity observed in the data.
   • The model includes a magnitude regulator ($\kappa$) to adjust the backlash magnitude in the second stage.
   • Parameters (including envelope coefficients, density function parameters, backlash ranges, and switching flags) are optimized using the Levenberg-Marquardt (LM) method to minimize the squared error between the model output and experimental measurements.

3. Experimental Validation:

   • Experiments were conducted on the I2RIS robot using a setup with DC servo motors and ArUco markers tracked by microscopes to measure pitch and yaw bending angles.
   • Four datasets were collected covering positive and negative input directions for both yaw and pitch motions.
   • The EGPI model's performance was compared against the conventional GPI model using Root Mean Square Error (RMSE), Normalized Root Mean Square Error (NRMSE), and Maximum Absolute Error (MAE).

Key Contributions

• Novel EGPI Model: The design of an EGPI model capable of describing input-output relationships that exhibit multi-stage hysteresis behavior within regions of monotonic input changes.
• Characterization of I2RIS: The successful characterization of the I2RIS robot's multi-stage hysteresis, specifically mapping motor encoder values to pitch and yaw bending angles, revealing distinct dead zones (S1, S2, S3) in the data.
• Optimization Framework: The application of the Levenberg-Marquardt method to optimize the EGPI parameters using experimental I2RIS data.

Results
Experimental validation demonstrates that the EGPI model significantly outperforms the conventional GPI model across all tested motion directions:

• Accuracy: The EGPI model shows a much closer fit to the measured data, particularly in the decreasing regions of the input where multi-stage hysteresis is most prominent.
• Error Metrics:
  • For Yaw motion (Positive Input): The EGPI model achieved an RMSE of 0.36° and MAE of 1.07°, compared to the GPI model's RMSE of 2.23° and MAE of 4.88°.
  • For Pitch motion (Positive Input): The EGPI model achieved an RMSE of 0.58°, significantly lower than the GPI model's 2.51°.
• Visual Fit: The EGPI model (orange line in results) tracks the experimental data (black dots) with minimal fluctuation, whereas the GPI model (green line) exhibits prominent error fluctuations, failing to capture the transitions between hysteresis stages.

Significance and Claims
The paper claims that the EGPI model highlights the potential for modeling multi-stage hysteresis in minimally invasive flexible robots. By introducing a switching mechanism, the model overcomes the limitation of the standard GPI model, which is restricted to single-stage backlash representation. The authors state that this approach provides a more reliable representation of the I2RIS hysteresis across various motion scenarios, which is a critical step toward improving control accuracy and patient safety in robotic retinal surgery.

Future Work
The authors note that future work will focus on:

1. Introducing mathematical constraints to ensure continuity across the switching points between sub-models.
2. Developing an analytical inverse of the EGPI model to map the I2RIS joint space to the actuation space.
3. Utilizing the inverse model to explore compensation strategies for precise control of the robot's motion toward desired joint angles.