⚡ electrical engineering

Passive iFIR filters for data-driven velocity control in robotics

This paper presents a passive, data-driven velocity control method for nonlinear robotic manipulators that utilizes VRFT-based identification of iFIR controllers to achieve superior tracking performance and stability guarantees compared to optimized PID, while allowing for rapid re-learning when system dynamics change.

Original authors: Yi Zhang, Zixing Wang, Fulvio Forni

Published 2026-04-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yi Zhang, Zixing Wang, Fulvio Forni

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are teaching a robot arm to move a cup of coffee from a table to a person's hand without spilling a drop. This is the job of velocity control: telling the robot exactly how fast to move at every moment.

For decades, the standard way to do this has been using a "PID controller." Think of a PID controller like a very experienced but rigid grandfather. He knows the basics: "If you're too slow, speed up. If you're too fast, slow down." He's reliable, easy to set up, and works well for simple tasks. But if the robot arm suddenly picks up a heavy, wobbly load (like a long, flexible wooden strip) or needs to move with extreme precision, the grandfather gets confused. He only has three "knobs" to turn, and he can't adapt quickly enough to the complex, wiggly physics of the new situation.

On the other side of the spectrum, there are modern "AI" controllers (like Deep Reinforcement Learning). These are like genius prodigies who can learn anything. They can figure out how to move that wobbly load perfectly. But there's a catch: they are unpredictable. They might learn a trick that works in the simulation but causes the robot to spin out of control and crash in the real world. They lack "safety guarantees."

The Solution: The "Smart, Safe Coach" (Passive iFIR)

This paper introduces a new method called Passive iFIR. Think of this as a Smart, Safe Coach that sits right in the middle between the rigid grandfather and the reckless genius.

Here is how it works, broken down into simple concepts:

1. The "Memory" Trick (iFIR)

The PID controller only looks at the now (current speed), the recent past (how fast it was a second ago), and the accumulated past (total error over time). It has a very short memory.

The iFIR (Integrating Finite Impulse Response) controller is like a coach with a longer memory. Instead of just looking at the last few seconds, it remembers a whole sequence of past movements (like a 2.5-second history).

Analogy: If you are driving a car, PID is like only looking at the speedometer right now. iFIR is like looking at the speedometer and remembering how you accelerated over the last few seconds to predict where you'll be next. This allows it to smooth out movements and handle "wobbly" loads much better.

2. Learning from "Probing" (Data-Driven)

Usually, to program a robot, you need a perfect mathematical model of how it moves. But robots are messy; friction changes, loads change, and parts wear out.

The Innovation: This method doesn't need a math model. It just needs 3 minutes of "probing" data.
Analogy: Imagine you want to learn how a new car handles. Instead of reading the manual (the math model), you just drive it around a parking lot for 3 minutes, testing how it turns and brakes. The iFIR algorithm watches this data and instantly figures out the best way to control the car. It's like the robot "feels" its way to the solution.

3. The "Safety Net" (Passivity)

This is the most important part. Many AI methods learn by trial and error, which can lead to dangerous instability.

The Innovation: The researchers built a "Safety Net" directly into the math. They force the controller to be Passive.
Analogy: Think of "Passivity" as a rule that says, "The robot can never inject more energy into the system than it takes out." In real life, this means the robot can never suddenly "jerk" or "explode" with energy. It guarantees that no matter how weird the load gets, the robot will remain stable and won't go haywire. It's like putting a governor on a car engine so it can never exceed a safe speed, no matter how hard you press the gas.

The Results: The Race

The team tested this "Smart, Safe Coach" on a Franka Research 3 robot (a high-tech robotic arm) in two scenarios:

Joint Control: Moving a single joint with a long, flexible wooden strip attached (simulating a heavy, wobbly load).
Cartesian Control: Moving the robot's hand (end-effector) through 3D space to follow a precise path.

The Outcome:

Against the "Grandfather" (PID): The iFIR coach was significantly better. In the hardest tests, it reduced tracking errors by up to 74.5%. That's the difference between a robot spilling your coffee and a robot pouring it perfectly.
Against the "Genius" (Unconstrained AI): The iFIR coach was just as good at performance but 100% safe. When they tried to train an iFIR without the safety net, the robot started shaking violently and almost crashed. The safety net was essential.
Adaptability: When they changed the load (swapped the long wood strip for a short one), the iFIR coach could re-learn the new dynamics in minutes and get back to peak performance immediately.

The Big Picture

This paper bridges the gap between old-school reliability and new-school learning.

Old School: Safe, but limited and rigid.
New School: Powerful, but risky and unstable.
This Paper: A method that learns from data like the new school but keeps the safety guarantees of the old school.

It's like giving a robot a "muscle memory" that it can update in minutes whenever its environment changes, all while guaranteeing it will never hurt itself or the people around it. This is a huge step toward making robots that can work safely in dynamic, unpredictable real-world environments like factories, hospitals, or homes.

1. Problem Statement

The paper addresses the challenge of designing velocity controllers for nonlinear robotic manipulators that balance performance, stability guarantees, and data efficiency.

The Gap: Classical PID controllers are simple and stable but lack the degrees of freedom to shape complex closed-loop responses for high-order or nonlinear dynamics. Conversely, modern learning-based control methods (e.g., Deep RL) offer high performance but often lack formal stability guarantees and require massive datasets.
The Goal: Develop a controller that learns directly from a small batch of experimental data (without an explicit robot model) to approximate a desired reference model, while mathematically guaranteeing closed-loop stability even when robot dynamics change or are uncertain.
Specific Context: The study focuses on velocity control (both joint-space and Cartesian-space), which is the standard interface for industrial robots, requiring smoothness, robustness, and inherent damping.

2. Methodology

The proposed solution utilizes Passive Infinite Impulse Response (iFIR) filters synthesized via Virtual Reference Feedback Tuning (VRFT).

A. Controller Structure (iFIR)

The iFIR controller extends the traditional PID structure:

Parallel Architecture: It consists of a MIMO integrator (analogous to the 'I' in PID) in parallel with a Finite Impulse Response (FIR) filter (replacing the 'P' and 'D' terms).
Mathematical Form:
$u(t) = \sum_{k=0}^{m-1} H_k e(t-k) + T_s \Gamma \sum_{k \ge 0} e(t-k)$
Where $H_k$ are FIR coefficients, $\Gamma$ is the integral gain matrix, and $e(t)$ is the error signal.
Advantage: The FIR term provides a richer temporal representation than a simple derivative, allowing for nuanced control actions without the noise sensitivity of pure differentiation.

B. Data-Driven Synthesis (VRFT)

The controller parameters are identified directly from input-output data without modeling the plant:

Probing: The robot is excited with a multi-sine signal for a short duration (e.g., 1–3 minutes) to collect input ( $u$ ) and output ( $y$ ) data.
Virtual Reference: A desired reference model $M_r$ is defined. The algorithm computes a "virtual reference" $r = M_r^{-1}(y)$ to generate an error signal $e = r - y$ .
Optimization: The controller parameters are found by minimizing the least-squares error between the actual control input and the input required to produce the virtual reference, subject to passivity constraints.

C. Stability via Passivity Constraints

To ensure stability, the optimization problem includes constraints that force the controller to be passive (energy-dissipating).

Theorem 1: The authors prove that if the integral gain $\Gamma$ is positive semi-definite, the FIR coefficients decay exponentially, and the frequency response of the FIR part satisfies a positive realness condition (approximated via finite sampling), the resulting controller is passive.
Guarantee: Since the robot and environment are inherently passive, interconnecting them with a passive controller guarantees closed-loop stability, even with model uncertainties or discretization effects (provided the sampling frequency is sufficiently high).

3. Key Contributions

First Experimental Validation: This is the first study to experimentally validate passive iFIR controllers on a real-world robotic manipulator (Franka Research 3).
MIMO Generalization: The method extends previous SISO (Single-Input Single-Output) passive iFIR concepts to MIMO (Multiple-Input Multiple-Output) systems, essential for Cartesian-space control.
Stability-Performance Trade-off: It bridges the gap between classical control and learning by offering a data-driven approach that retains rigorous stability guarantees (unlike "black-box" learning methods).
Rapid Adaptability: The method allows for re-learning the controller in minutes (approx. 100s) when dynamics change (e.g., payload changes), regaining nominal performance.

4. Experimental Results

Experiments were conducted on a Franka Research 3 robot in two scenarios:

A. Joint-Space Velocity Control

Setup: Controlled a single joint with a flexible wood strip attached to simulate high-order, poorly modeled dynamics.
Performance:
- Nominal Case: Passive iFIR reduced tracking error (NRMSE) by 41.0% compared to an optimized PID.
- Dynamics Change: When the load changed (strip length altered) without retraining, iFIR maintained better tracking than PID (28.1% improvement).
- Aggressive Reference: When tested with a demanding second-order reference model, the unconstrained (non-passive) iFIR became unstable, while the passive iFIR and PID remained stable. This highlights the critical role of passivity constraints.

B. Cartesian-Space Velocity Control

Setup: Controlled end-effector velocity in 3D space (SISO and MIMO configurations).
Performance:
- SISO: iFIR outperformed PID by 24.6% (random profile) and 19.7% (trapezoidal profile).
- MIMO (1st Order): iFIR improved performance by 32.0% over PID.
- MIMO (2nd Order/Demanding): Under a high-bandwidth, second-order reference model, iFIR achieved massive improvements: 67.7% reduction in error for random trajectories and 74.5% for trapezoidal profiles.
- Robustness: When operating away from the training configuration, PID exhibited high-frequency oscillations, whereas iFIR degraded more gracefully.

5. Significance and Conclusion

Bridging the Gap: The paper successfully demonstrates that data-driven control does not have to sacrifice stability. By enforcing passivity during the learning process, the controller achieves the performance of complex learning methods with the safety guarantees of classical control.
Practicality: The method requires minimal data (minutes of probing) and low computational resources, making it viable for industrial deployment where model derivation is difficult.
Future Directions: The authors note limitations regarding the manual selection of reference models and the linear nature of the design. Future work aims to automate reference model derivation and develop nonlinear, adaptive iFIR controllers.

In summary, Passive iFIR offers a robust, stable, and high-performance alternative to PID for robotic velocity control, particularly in scenarios involving complex dynamics, payload variations, and demanding motion profiles.