Periodic Steady-State Control of a Handkerchief-Spinning Task Using a Parallel Anti-Parallelogram Tendon-driven Wrist

1. Problem Statement

The paper addresses the challenge of dynamic manipulation of highly flexible objects, specifically the traditional Chinese task of spinning a handkerchief. This task requires generating stable, periodic steady-state motions under complex conditions involving:

• Nonlinear dynamics: Governed by frictional contacts, boundary constraints, and the object's elasticity.
• Hardware limitations: Conventional robotic wrists (serial motor stacks) suffer from high inertia and limited dexterity, making them unsuitable for the rapid, low-inertia, omnidirectional rotations required.
• Control complexity: The transition from a static state to a stable spinning state is difficult to model due to the lack of explicit friction models and the chaotic nature of flexible fabric dynamics.

2. Methodology

The authors propose a holistic framework integrating hardware design, control-oriented modeling, and a hierarchical control strategy.

A. Hardware: Parallel Anti-Parallelogram Tendon-driven Wrist

• Design Inspiration: Modeled after the human wrist's parallel carpal bone structure and antagonistic musculotendinous units.
• Mechanism: Utilizes a parallel anti-parallelogram structure driven by four tendons.
  • Geometry: Two orthogonal anti-parallelogram modules create a 2-DOF wrist. The geometry is optimized to approximate a circular rolling surface, enabling 90° omnidirectional rotation.
  • Inertia Reduction: The design reduces the length-to-width ratio by ~50% compared to previous designs (e.g., Quaternion Joint) and places motors proximally (rear-mounted), shifting actuation away from the distal end to minimize effective inertia.
  • Sensing: Features two decoupled joint-angle encoders on orthogonal subchains, providing independent roll–pitch feedback without cross-coupling.

B. Modeling: Control-Oriented Particle-Spring Model

• Abstraction: The handkerchief is modeled as an octagonal mesh of particles connected by anisotropic spring constraints.
  • Springs: Use modified Hooke's law with distinct stiffness for tension ($k_s$) and compression ($k_c$) to simulate fabric folding vs. stretching.
  • Dynamics: Includes gravity, spring forces, damping, and collision handling with a deformable rigid triangular surface.
• Friction Approximation: Instead of modeling complex friction forces, the contact point is driven by a prescribed velocity along a circular trajectory. This avoids numerical instability while capturing the energy injection needed for spinning.
• Evaluation Metric: The "unfolding ratio" is calculated by projecting the particle positions onto the rotation plane and comparing the enclosed area to the fully unfolded reference area.

C. Control Strategy: High-Low Level Hierarchical Control

• High-Level Control (Initiation):
  • Focuses on the rest-to-steady-state transition.
  • Uses an acceleration-based modulation strategy: Both the rotation radius ($R$) and angular speed ($V$) are ramped up linearly over time.
  • Theoretical Basis: Analyzed using Poincaré maps. The strategy ensures cycle consistency, energy balance, and phase locking. It drives the system toward a stable periodic orbit (attractor) where the maximum Floquet multiplier is $<1$ (specifically $\approx 0.7998$), guaranteeing asymptotic stability.
• Low-Level Control (Maintenance):
  • Once steady spinning is achieved, the controller switches to direct periodic motor commands.
  • This simplifies the control loop, leveraging the wrist's decoupled sensing and high dynamic responsiveness to maintain the spin with minimal computational overhead.

3. Key Contributions

1. Novel Wrist Mechanism: A compact, low-inertia, tendon-driven wrist capable of 90° omnidirectional rotation with decoupled sensing, specifically tailored for high-dynamic flexible object manipulation.
2. Control-Oriented Modeling: A particle-spring model that abstracts friction into a tractable velocity-driven constraint, enabling systematic strategy evaluation without complex friction modeling.
3. Robust Initiation Strategy: Identification and validation of a linear ramp strategy (simultaneous increase in radius and speed) that guarantees a stable transition to periodic steady-state spinning, outperforming directional-throw or sinusoidal energy injection strategies.
4. Hierarchical Control Framework: A unified system that separates the complex initiation phase (kinematic control) from the maintenance phase (periodic motor control).

4. Experimental Results

• Simulation: The acceleration-based strategy achieved a 99% unfolding ratio within 3 seconds. Poincaré analysis confirmed the stability of the periodic orbit (Floquet multiplier $\lambda_{max} \approx 0.7998$).
• Hardware Experiments:
  • Unfolding Performance: The system achieved an unfolding ratio of ~99% with minimal fluctuation (Coefficient of Variation $\approx 0.7\%$).
  • Tracking Accuracy: Fingertip tracking error was RMSE = 2.88 mm ($R^2 = 0.9898$) during high-dynamic spinning around arbitrary axes.
  • Comparison: Alternative strategies (directional-throw and periodic-energy injection) failed to achieve full unfolding (max 77.5% and 22.9% respectively) and exhibited instability (Floquet multipliers $>1$).

5. Significance

This work bridges the gap between bionic hardware design and nonlinear control theory for flexible object manipulation.

• Hardware Impact: Demonstrates that mimicking human skeletal-muscular architecture (parallel tendon-driven) yields superior dynamic performance compared to traditional serial robotic wrists.
• Control Impact: Proves that complex, friction-dominated tasks can be solved by simplifying the control objective (focusing on periodicity and energy balance) rather than attempting to model every physical interaction explicitly.
• Application: Provides a robust recipe for robots to manipulate highly flexible objects (fabrics, cables) in dynamic environments, with potential applications in service robotics, textile handling, and human-robot interaction.