⚡ electrical engineering

Tracking Limitations and Rate-Consistent Safety Filtering for Constrained Mechanical Systems

This paper addresses the velocity-tracking limitations caused by incompatible constraint-approach rates in mechanical systems by developing a rate-consistent control barrier function framework that quantifies kinetic errors, ensures exponential convergence under bounded defects, and significantly reduces safety-induced energy dissipation in simulations of hydraulic excavators.

Original authors: Mehdi Heydari Shahna, Jouni Mattila

Published 2026-10-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mehdi Heydari Shahna, Jouni Mattila

Original paper licensed under CC BY 4.0 (https://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 a machine that moves with the precision of a surgeon's hand but the strength of a hydraulic press. To keep such a machine safe, engineers use two different types of digital guardians. One guardian watches the machine's speed, ensuring it never moves too fast toward a wall or a joint limit. The other guardian watches the force the machine exerts, making sure the motors do not push so hard that they break the machine or the environment. In theory, these two guardians should work in perfect harmony. In practice, however, they often speak different languages. The speed guardian might demand the machine slow down immediately, while the force guardian, reacting to the physical weight of the machine, insists on a more gradual deceleration. This mismatch creates a dangerous gap where the machine's actual movement lags behind the safe command, leading to jerky motions, wasted energy, and a higher risk of collision.

Researchers at Tampere University in Finland have identified this specific conflict and developed a new way to synchronize these guardians. Their work focuses on mechanical systems, from simple robotic arms to complex hydraulic excavators, where the laws of physics dictate how quickly a machine can change its speed. The team discovered that when a safety system forces a machine to stop, the physical inertia of the machine means it cannot obey a sudden command instantly. If the safety software demands a stop faster than the machine's physics allow, the machine will overshoot its safe zone, creating a tracking error that no amount of standard tuning can eliminate. This is not a flaw in the machine's design, but a fundamental limitation of how safety constraints interact with physical motion.

To solve this, the researchers created a new framework that ensures the speed guardian and the force guardian agree on the rate at which the machine approaches a limit. Instead of letting the speed guardian issue a command that the force guardian cannot physically fulfill, the new system adjusts the speed command itself. It calculates a path that respects the machine's physical limits while still satisfying the safety requirements. This approach, which the authors call "rate-consistent" control, effectively removes the conflict. By aligning the speed of the safety command with the speed at which the machine can physically react, the system prevents the machine from ever having to make a sudden, impossible correction.

The team tested this idea using two distinct methods. First, they ran detailed mathematical simulations on a six-joint robotic arm and a simplified one-dimensional model. These tests confirmed that without the new method, the machine would consistently overshoot safety limits by a predictable amount, regardless of how powerful the motors were. With the new method, the machine tracked the safe path exactly, with the error dropping to zero in ideal conditions. They then moved to more complex, realistic simulations of two hydraulic excavators, one modeled after a Komatsu machine and another based on a specialized walking excavator known as HEAP. These machines have many moving parts and closed loops of motion, making them difficult to control. In these simulations, the researchers had the machines repeatedly approach their maximum extension limits, a scenario that typically causes large tracking errors and high energy spikes.

The results were striking. In the repeated limit experiments, the new method reduced the initial error when the machine approached a limit by a factor of 3.3 for the Komatsu model and 3.6 for the HEAP model. More importantly, it drastically reduced the amount of energy the safety system had to inject into the machine to correct its path. In the Komatsu simulation, the energy required to maintain safety dropped from over 1,800 joules to less than 0.01 joules. For the HEAP machine, it fell from nearly 500 joules to under 5 joules. This means the machine moved much more smoothly, with the safety system barely needing to intervene. The machines maintained the same minimum distance from their physical limits as before, ensuring safety was not compromised for the sake of smoothness.

The researchers also explored how this system behaves under different conditions, such as changing the weight of the machine's bucket or adjusting the speed of the control loops. They found that the new method remained effective even when the machine carried a heavier load, reducing the error significantly compared to the standard approach. The system also proved robust when the machine encountered unexpected forces or when the sensors provided slightly imperfect data. The key to this success was a mathematical tool called a "kinetic metric," which allowed the researchers to measure the machine's motion in a way that naturally aligned with the forces the motors could produce. This alignment ensured that the safety corrections did not fight against the machine's own momentum.

The study does not claim to have solved every problem in robotic safety. The results are based on computer simulations and mathematical proofs, not physical experiments with real machines in the field. However, the simulations were rigorous, using detailed models of hydraulic systems, friction, and valve limitations that closely mimic real-world behavior. The mathematical proofs established that the new method works under a wide range of conditions, provided the machine's physical properties are known and the safety limits are well-defined. The researchers also noted that the method requires a specific type of controller architecture, one that separates the machine's body movements from its overall motion, but this architecture is already used in many advanced robotic systems.

This work offers a clear path forward for designing safer, more efficient machines. By recognizing that safety commands must respect the physical limits of how fast a machine can change its speed, engineers can build systems that move with fluid precision rather than jerky hesitation. The new framework ensures that the digital guardians of speed and force are no longer at odds, but work together to guide the machine safely to its destination. For industries that rely on heavy machinery, from construction to manufacturing, this could mean equipment that is not only safer but also more productive, as it spends less time correcting errors and more time performing its intended tasks. The research demonstrates that the key to safety is not just in setting limits, but in understanding how the machine moves within those limits.

Technical Summary: Tracking Limitations and Rate-Consistent Safety Filtering for Constrained Mechanical Systems

Problem Statement
The paper addresses a fundamental incompatibility between reference-level and effort-level safety filters in constrained mechanical systems. Control Barrier Functions (CBFs) are typically applied in two layers: a first-order reference filter that constrains commanded velocities to satisfy safety conditions, and a second-order effort filter that enforces safety on the physical actuator efforts. The authors identify that these layers can impose conflicting "constraint-approach rates." Specifically, the reference filter may drive the safety slack to zero in finite time, while the effort filter restricts how quickly the actual motion's barrier quantity can decrease. This rate mismatch forces a velocity-tracking error even under ideal actuation and exact nominal feedforward. The paper seeks to characterize the resulting irreducible tracking limitations and develop a framework to ensure rate consistency between the two layers.

Methodology
The authors employ an energy-based analysis grounded in Virtual Decomposition Control (VDC) and passivity principles.

System Modeling: The mechanical system is modeled with independent coordinates $q$ and speeds $u$ , utilizing a kinetic energy storage function $S = \frac{1}{2}s^\top M s$ , where $s = u_r - u$ is the tracking error and $M$ is the inertia matrix.
Two-Layer Architecture:
- First Layer (Reference): A velocity-level CBF filter generates a required velocity $u_r$ by minimizing deviation from a nominal command subject to first-order barrier constraints.
- Second Layer (Effort): An effort-level CBF filter computes actuator commands $\phi$ to satisfy second-order barrier conditions, correcting the nominal effort $\phi_r$ by a term $\Delta$ .
Kinetic Metric Alignment: A critical design choice is the use of a "kinetic effort metric" where the effort weighting matrix $H$ satisfies $CH^{-1}C^\top = M$ . This aligns the safety correction with the mechanical structure, allowing the safety port power ( $P_f = -s^\top C \Delta$ ) to be directly related to the tracking storage dissipation.
Rate-Consistent Reference Generation: To resolve the rate mismatch, the authors propose replacing the first-order reference filter with an acceleration-level filter. This filter explicitly enforces a rate condition ( $\dot{\tilde{z}}_i + \bar{\beta}\tilde{z}_i \geq 0$ ) on the reference slack, ensuring the reference approaches the constraint no faster than the effort layer can safely brake.

Key Contributions
The paper establishes three primary theoretical results:

Controller-Independent Tracking Limitation (Theorem 1): The authors prove a lower bound on the kinetic tracking error at the moment of constraint activation. This bound depends on the realized reference slack and the barrier rate parameters. Crucially, they show that repeated activations with uniformly positive preceding slack preclude the convergence of the tracking error to zero, regardless of the feedback gain.
Projection-Based Dissipation Characterization (Theorems 2 & 3): Using the kinetic metric, the authors derive an inequality characterizing how safety corrections affect tracking dissipation. They introduce an "exact constraint identity" that separates the effects of reference-rate mismatch, mechanical curvature, and feedforward defects. This reveals that the effort layer only injects energy into the tracking error when the machine moves away from the limit faster than the required motion.
Rate-Consistency and Feasibility (Theorems 4 & 5): The paper provides a constructive condition for rate consistency. By enforcing a common rate $\bar{\beta}$ (where $\bar{\beta} \leq \min \alpha_{2,i}$ ) via an acceleration-level reference filter, the "slack defect" is eliminated. Under this condition, the system preserves exact initial tracking ( $s \equiv 0$ ) in the absence of defects and admits explicit exponential decay bounds in the presence of bounded defects. For two-sided coordinate limits, the authors establish strict feasibility and global continuation of solutions.

Results
The theoretical findings are validated through continuous-time benchmarks and sampled simulations of two hydraulic excavators (Komatsu and HEAP) using the MuJoCo physics engine and RoboCompiler for model generation.

Continuous-Time Benchmarks: Experiments with a six-joint Pinocchio model and a scalar example confirm the analytical lower bounds. The results demonstrate that increasing the VDC feedback gain $k$ does not reduce the onset error in first-order filtering, whereas the rate-consistent approach achieves exact tracking ( $s \equiv 0$ ) in nominal conditions.
Hydraulic Excavator Simulations: In repeated-limit experiments, the proposed rate-consistent layer reduced onset velocity errors by factors of 3.3 (Komatsu) and 3.6 (HEAP) compared to the baseline first-order filter.
Energy Metrics: The most significant improvement was observed in the "positive safety-port energy" (energy injected into the tracking storage by the safety filter). The proposed method reduced this metric by more than two orders of magnitude (e.g., from 1835 J to 0.0063 J for Komatsu) while maintaining comparable minimum clearances.
Robustness: The proposed framework maintained strict feasibility and safety margins under varying gains, payload changes (1.5x bucket mass), and mission scenarios involving soil excavation and obstacle avoidance.

Significance and Claims
The paper claims that the incompatibility between reference and effort safety layers is a structural issue that cannot be resolved solely by tuning feedback gains. The significance lies in:

Quantifying the Cost of Inconsistency: Providing a rigorous, controller-independent lower bound on tracking error caused by rate mismatches.
Unifying Safety and Tracking: Demonstrating that by aligning the reference generation rate with the effort-layer capabilities (rate consistency), one can preserve the "exact tracking" certificate of the underlying VDC controller while maintaining safety.
Practical Implementation: Showing that these theoretical guarantees hold in complex, sampled-data hydraulic systems with closed kinematic chains and force loops, significantly reducing the energy penalty (port injection) associated with safety filtering.

The authors emphasize that their analysis relies on specific mechanical assumptions (e.g., $K=kM$ , kinetic metric alignment) and that the results are specific to the proposed architecture, which integrates RoboCompiler-generated interfaces with VDC and CBF layers. They do not claim universal applicability to all control architectures but provide a specific solution for passivity-based mechanical systems.

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