Enabling Dynamic Tracking in Vision-Language-Action Models via Time-Discrete and Time-Continuous Velocity Feedforward

1. Problem Statement

Vision-Language-Action (VLA) models have shown promise for general robot manipulation but face significant challenges when deployed on rigid industrial robots (e.g., Franka Emika, Kuka, UR).

• The Trade-off: There is an inherent conflict between compliance (safety during contact) and responsiveness (tracking speed).
• The Root Cause: Standard Behavior Cloning (BC) approaches predict discrete end-effector positions at low frequencies (e.g., 15–60 Hz) without velocity ($\dot{x}_d$) or acceleration ($\ddot{x}_d$) feedforward terms.
• The Consequence: Low-level impedance/admittance controllers rely solely on position error ($e = x_d - x$) and stiffness ($K_d$) to track targets. To minimize tracking lag, stiffness must be high, which reduces compliance and risks damaging the robot/environment during contact. Conversely, lowering stiffness for safety increases tracking lag, limiting execution speed and throughput.
• Current Limitations: Existing VLA implementations often set velocity feedforward to zero or rely on zeroth-order control laws, accepting inherent tracking latency that restricts industrial viability.

2. Methodology

The authors propose integrating explicit velocity feedforward into the VLA policy to resolve the tracking-compliance trade-off. They introduce two distinct methods to extract velocity targets from VLA predictions, both of which are model-agnostic (compatible with any standard action-chunking architecture).

A. Method 1: Time-Discrete Velocity Approximation (Finite Difference)

• Concept: A post-processing bridge for existing discrete VLA models.
• Mechanism: The policy predicts discrete positions ($x_t$). During inference, velocity is approximated using finite differences: $\dot{x}_t \approx (x_t - x_{t-1}) / \Delta t$.
• Characteristics:
  • Provides piecewise-constant velocity profiles.
  • Acts as a highly effective bridge for models trained on standard position-only data (provided the low-level controller is velocity-aware).
  • Requires no changes to the neural network architecture.

B. Method 2: Time-Continuous Cubic B-Spline Action Space

• Concept: A native continuous trajectory representation.
• Mechanism: Instead of predicting discrete poses, the VLA predicts Cubic B-Spline control points ($P$).
• Mathematical Foundation:
  • The trajectory is defined as $x_d(t) = \sum B_{i,k}(t)P_i$, where $B_{i,k}$ are basis functions.
  • Using cubic splines ($k=3$) ensures $C^2$ continuity (continuous position, velocity, and acceleration).
  • The low-level controller analytically derives $\dot{x}_d(t)$ and $\ddot{x}_d(t)$ by differentiating the basis functions at the native high-frequency control rate.
• Training: Raw teleoperation data is converted offline into B-Spline control points via Least Squares optimization to create a standard imitation learning dataset.

C. Control Architecture

• Controller: An active Cartesian admittance controller is used, solving for acceleration: $\ddot{x}_{cmd} = \ddot{x}_d - \Lambda_d^{-1}(D_d\dot{e} + K_de - F_{ext})$.
• Data Collection: Teleoperation devices record high-frequency positions ($>500$ Hz). Velocities are computed via low-pass filtered finite differences. Acceleration is set to zero ($\ddot{x}_d=0$) to avoid noise amplification, though the framework supports it.
• Inference: To handle discontinuities at action chunk boundaries, Real-Time Chunking (RTC) is employed to smooth transitions.

3. Key Contributions

1. Demonstration of Velocity Feedforward: Proved that explicitly providing velocity feedforward to low-level controllers allows rigid robots to execute faster motions without sacrificing the compliance needed for contact-rich tasks.
2. Discrete Approximation Method: Proposed a finite-difference approach that effectively upgrades existing discrete VLA architectures to work with velocity-aware controllers.
3. Continuous B-Spline Action Space: Introduced a novel action space predicting spline control points to guarantee $C^2$ continuous trajectories, enabling analytical evaluation of higher-order derivatives at high frequencies.
4. Model Agnosticism: Both methods require modifications only to teleoperation, data processing, and the low-level controller, leaving the core VLA neural network architecture unchanged.

4. Experimental Results

The methods were evaluated on a contact-rich "cube-in-hole" task using Universal Robots (UR) with a strict 1mm tolerance, requiring both fast transfer and delicate insertion.

A. Teleoperation Performance

• Setup: Human operators performed tasks under different controller configurations (Position-only vs. Position + Velocity) at various frequencies (5Hz to 500Hz).
• Result: The velocity-aware controller significantly reduced task completion times across all frequencies ($p < 0.001$).
  • Operators could execute large transfer motions faster and more accurately without fighting the controller's tracking lag.
  • This establishes a higher upper bound for data quality provided to the VLA model.

B. Policy Execution (Rollout)

• Finite Difference Method:
  • Fastest Execution: Mean time of 6.05s (vs. 7.35s baseline).
  • Significance: Significantly outperformed the baseline ($p < 0.001$).
• Cubic B-Spline Method:
  • Execution Speed: Mean time of 7.45s (slightly slower than baseline, not statistically significant).
  • Success Rate: Achieved the highest overall success rate (79.20%).
  • Analysis: The B-Spline method exhibited higher variance. The authors hypothesize that residual "jerky" motions (caused by Real-Time Chunking interactions) inadvertently helped the robot "wiggle" the cube into the hole during the insertion phase, improving success rates despite slower average speeds.
• Ablation Study: A policy trained on velocity-aware data but executed with a position-only controller performed worst (10.01s mean time), highlighting the critical need to match the inference controller with the data collection strategy.

5. Significance and Conclusion

• Industrial Viability: This work bridges the gap between research-grade compliant robots (like ALOHA) and rigid industrial arms. It enables high-throughput, safe contact manipulation on standard industrial hardware.
• Efficiency: By resolving the tracking-compliance trade-off, robots can move faster during transfer phases while maintaining safety during contact phases.
• Future Work: The authors plan to integrate full acceleration feedforward ($\ddot{x}_d$). The guaranteed $C^2$ continuity of the B-Spline approach is identified as a promising foundation for stable, high-speed, compliant manipulation with higher-order derivatives.

In summary, the paper demonstrates that velocity feedforward is not optional but essential for deploying VLA models on rigid industrial robots, offering a simple finite-difference fix for immediate gains and a sophisticated B-Spline framework for future high-performance control.