A Novel Hybrid PID-LQR Controller for Sit-To-Stand Assistance Using a CAD-Integrated Simscape Multibody Lower Limb Exoskeleton

1. Problem Statement

The sit-to-stand (STS) transition is a biomechanically complex daily activity requiring precise coordination of lower limb joints against gravity. For individuals with neurological impairments (e.g., spinal cord injury, stroke), this transition is often a barrier to independence. While exoskeletons offer a solution, their effectiveness is limited by control strategies that struggle with:

• Nonlinear Dynamics: The human-exoskeleton system exhibits highly nonlinear, time-varying dynamics.
• Control Trade-offs:
  • PID Controllers: Simple and easy to implement but inherently local (single-variable), often resulting in steady-state errors under load and requiring manual tuning.
  • LQR Controllers: Provide optimal state-feedback and global stability but rely on linearized models, making them sensitive to modeling uncertainties and lacking integral action to eliminate steady-state errors.
• Clinical Safety: There is a critical need for controllers that minimize tracking errors, overshoot, and settling time to ensure user safety and natural movement rhythms.

2. Methodology

The authors developed a comprehensive simulation framework to design, test, and compare three control strategies: a classical PID, a Linear Quadratic Regulator (LQR), and a novel Hybrid PID-LQR.

A. High-Fidelity Modeling (CAD-to-Simscape)

• Geometry & Inertia: A bilateral lower limb exoskeleton was modeled in SolidWorks (using Al 6061-T6 material). Instead of manually deriving Lagrangian equations, the CAD assembly was directly imported into MATLAB/Simulink Simscape Multibody.
• Preservation of Properties: This workflow automatically preserved accurate geometric and inertial properties (mass, inertia tensors, link lengths) for all 9 rigid links and 6 active joints (hip, knee, ankle per side).
• Dynamic Model: The system follows the standard equation of motion: $M(q)\ddot{q} + C(q, \dot{q})\dot{q} + G(q) = \tau$.

B. Reference Trajectory Generation

• Source: Physiologically representative STS trajectories were generated using OpenSim musculoskeletal simulation for a 50th-percentile subject (75 kg, 1.75 m).
• Phases: The 3.0-second motion was decomposed into three biomechanical phases:
  1. Flexion-momentum (0–33%): Trunk flexion and center of mass shift.
  2. Momentum-transfer (34–66%): Seat-off and rapid angular changes.
  3. Extension (67–100%): Progressive extension to upright stance.
• Processing: Trajectories were low-pass filtered (6 Hz) and resampled at 1 kHz.

C. Control Strategies

1. PID: Tuned independently per joint using iterative simulation.
2. LQR: Designed using a linearized state-space model to minimize a quadratic cost function ($J = \int (x^T Qx + u^T Ru) dt$).
3. Hybrid PID-LQR (Proposed): A weighted superposition of the two strategies:
   $$\tau_{Hybrid} = \alpha \tau_{LQR} + (1-\alpha) \tau_{PID}$$
   • Optimization: The blending coefficient $\alpha$ was tuned to minimize a composite performance index (tracking error + control effort).
   • Result: The optimal $\alpha$ was found to be 0.65, favoring the LQR's optimal transient response while retaining PID's integral action for error elimination.

3. Key Contributions

• CAD-Integrated Modeling: Demonstrated a workflow to import SolidWorks assemblies directly into Simscape Multibody, eliminating manual derivation errors and ensuring high-fidelity physical parameters.
• Novel Hybrid Architecture: Proposed a Hybrid PID-LQR controller that synergizes the optimal state-feedback of LQR with the disturbance rejection of PID.
• Systematic Comparative Analysis: Conducted a rigorous multi-metric evaluation (RMSE, MAE, overshoot, settling time) across all three controllers specifically for the STS task.
• OpenSim Integration: Utilized OpenSim to generate physiologically accurate reference trajectories across distinct biomechanical phases.

4. Results

The Hybrid PID-LQR controller significantly outperformed both baseline controllers across all metrics:

• Tracking Accuracy (RMSE):
  • Hip: Reduced from 3.82° (PID) to 1.06° (Hybrid), a 72.3% improvement over PID.
  • Knee: Reduced from 4.67° (PID) to 1.38° (Hybrid), a 70.4% improvement over PID.
  • Ankle: Reduced from 1.93° (PID) to 0.58° (Hybrid).
• Transient Response (Settling Time):
  • The PID controller failed to settle the ankle trajectory within the 3.0s window (2.849s).
  • The Hybrid controller achieved settling times of 0.070s (Hip), 0.040s (Knee), and 0.140s (Ankle).
  • This represents a >90% reduction in settling time compared to PID across all joints.
• Overshoot:
  • The Hybrid controller maintained overshoot between 2.39% and 6.10%, significantly lower than PID (up to 20.32%) and LQR (up to 14.47%).
• Steady-State Error:
  • Unlike LQR, which showed residual offsets (especially at the ankle), the Hybrid controller achieved near-zero steady-state error due to the integral term.

5. Significance and Conclusion

• Clinical Relevance: The results validate that the Hybrid PID-LQR controller meets the stringent safety and performance requirements for clinical deployment. The low overshoot prevents unintended joint forces on the user, while the rapid settling time preserves the natural rhythm of the STS transition.
• Overcoming Limitations: The study successfully resolves the trade-off between PID (steady-state error) and LQR (sensitivity to model uncertainty/lack of integral action).
• Future Work: While the current study is simulation-based with ideal torque actuation, the authors plan to validate the controller on a physical prototype using hardware-in-the-loop testing, incorporate sensor noise and actuator dynamics, and eventually conduct clinical trials with patient populations.

In summary, this paper presents a robust, high-fidelity approach to exoskeleton control, proving that a blended Hybrid PID-LQR strategy offers superior trajectory tracking and stability for rehabilitation robotics compared to traditional standalone methods.