Rapid Vibration Suppression and Trajectory Tracking of a Serial Manipulator with Multi-Flexible Links

Technical Summary: Rapid Vibration Suppression and Trajectory Tracking of a Serial Manipulator with Multi-Flexible Links

Problem Statement
Flexible robotic manipulators (FRMs) offer significant advantages in lightweight design and energy efficiency but suffer from structural flexibility that induces persistent vibrations, degrades end-effector accuracy, and accelerates fatigue. These challenges are particularly acute in multi-link serial manipulators, where increased length amplifies flexibility. Existing control strategies often rely on finite-dimensional Ordinary Differential Equation (ODE) approximations (e.g., Assumed Mode Method or Finite Element Method), which inevitably lead to "spillover effects" where unmodeled high-frequency modes destabilize the system. Furthermore, many boundary control methods only inject damping at specific locations, limiting the speed of vibration suppression. The paper addresses the need for a control framework that handles the infinite-dimensional nature of these systems directly, suppresses vibrations rapidly using only boundary measurements, and ensures accurate trajectory tracking for an $n$-degree-of-freedom serial flexible manipulator robot (nDSFMR).

Methodology
The authors propose a backstepping output-feedback control framework based on an infinite-dimensional Partial Differential Equation (PDE) model.

1. Modeling:

   • The dynamics of the nDSFMR are derived using Hamilton's principle.
   • Each flexible link is modeled as a slender Timoshenko beam coupled with an ODE representing the rigid joint dynamics.
   • Through Riemann transformation and variable substitution, the coupled beam-ODE system is reformulated into a $2 \times 2$ coupled hyperbolic PDE system with boundary dynamics. This canonical form facilitates control design.

2. Control Design:

   • State-Feedback: A backstepping transformation is employed to map the original system into a target system with desired stability properties. This involves designing kernel functions that satisfy specific hyperbolic PDEs to equivalently inject distributed damping along the entire flexible structure using only boundary actuation.
   • Observer Design: Since distributed states (e.g., internal beam deformation) are inaccessible, an observer is constructed using only boundary measurements: the joint angle ($\theta_i$) and the base strain ($E_{bi}$). The observer utilizes output error injections and backstepping transformations to estimate the unmeasured states.
   • Output-Feedback: The final controller combines the state-feedback law with the observer estimates, creating a practical output-feedback loop that does not require full state measurement.

3. Stability Analysis:

   • Theoretical proofs (via Lyapunov functions) demonstrate that the closed-loop system is uniformly ultimately bounded.
   • The observer error system is proven to be exponentially stable.
   • The approach ensures that the tracking error and vibration states converge to a bounded region, with the bound adjustable via design parameters.

Key Contributions
The paper highlights three primary contributions:

1. Physics-Consistent Infinite-Dimensional Modeling: Unlike existing works relying on ODE approximations, this work establishes a PDE-level model for the flexible manipulator. The control design is performed directly on the PDE, effectively eliminating spillover effects and enabling high-accuracy control.
2. Spatially Distributed Damping via Boundary Actuation: While traditional boundary control injects damping at specific points, the proposed backstepping method achieves an equivalent spatially distributed damping profile along the entire flexible structure using only joint-level actuation. This allows for arbitrarily fast vibration suppression and rapid convergence of the end-effector to the desired trajectory.
3. Experimental Validation: The theoretical framework is validated on a physical 2-degree-of-freedom serial flexible manipulator (2DSFMR). The study demonstrates the effectiveness of the observer in reconstructing states from limited sensors and the controller's ability to outperform traditional methods.

Experimental Results
Experiments were conducted on a Quanser 2DSFMR platform, comparing the proposed backstepping output-feedback controller against a baseline Linear Quadratic Regulator (LQR) with feedforward control.

• Observer Performance: The observer successfully estimated the slope of transverse displacement at various points along the link under sinusoidal, square, and sawtooth reference trajectories, even with large initial estimation errors. Statistical analysis showed low Root-Mean-Square Error (RMSE) and Mean Absolute Error (MAE).
• Vibration Suppression: The proposed controller achieved faster vibration suppression compared to the LQR+feedforward approach across all trajectory types.
• Trajectory Tracking: The end-effector tracking error converged to zero more rapidly with the proposed method, particularly under discontinuous trajectories (square and sawtooth waves). The angular tracking error and tip displacement were significantly reduced.

Significance and Claims
The paper claims that the proposed backstepping output-feedback framework offers a superior alternative to finite-dimensional control strategies for flexible manipulators. By operating at the PDE level, it avoids spillover instability. The ability to inject distributed damping via boundary actuation alone enables rapid stabilization, addressing the trade-off between lightweight design and vibration control. The experimental results confirm that the method achieves both fast vibration suppression and high-precision trajectory tracking using only joint angle and base strain measurements, making it viable for practical implementation in high-precision industrial and space applications. The authors note that future work will focus on incorporating adaptive control to handle sudden payload uncertainties and integrating optimal motion planning to smooth reference trajectories.