⚡ electrical engineering

Adaptive PD Gains for Energy-Conscious Control in Physical Human-Robot Interaction

This paper proposes a novel adaptive PD controller that ensures safe physical human-robot interaction by dynamically limiting a robot's kinetic and potential energy without requiring external force or torque sensors, while providing a formal stability proof and experimental validation on the TALOS robot.

Original authors: Danyal Saqib, Francisco Andrade Chavez, Marie Charbonneau

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Danyal Saqib, Francisco Andrade Chavez, Marie Charbonneau

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

The Big Picture: The "Energy Budget" Robot

Imagine a robot arm as a very strong, heavy person trying to hug you. If that person is too stiff and moves too fast, a hug could accidentally hurt you.

In the world of robots, safety is usually handled by two main methods:

Force Sensors: Like having a person wear a pressure-sensitive suit to feel exactly how hard they are pushing. (The paper notes this is expensive and not available on all robots).
Torque Sensors: Like having sensors in every joint to measure the twisting force. (Also not available on every robot).

This paper proposes a third, smarter way: Energy Limiting. Instead of measuring the push or the twist, the robot simply keeps a running tally of its own "energy budget." If the robot gets too energetic (too fast or too tense), it automatically slows down and relaxes to stay within its safe budget.

The Problem with Old Robots

Traditional robots often act like a stiff spring. If you push them, they push back hard. If they are moving fast and hit something, they transfer a lot of energy instantly, which can cause injury.

To stop this, engineers usually program robots to just "stop" if they hit something. But the paper argues that stopping isn't always safe. Imagine a robot arm pinning a person against a wall; if the robot just freezes, the person is still trapped. They need the robot to be "soft" and compliant, allowing the person to push it away safely.

The Solution: The "Adaptive PD Controller"

The authors created a new type of controller (a set of rules the robot follows) called an Adaptive PD Controller.

Think of a standard robot controller like a car with a fixed gas pedal and a fixed brake. No matter what happens, it reacts the same way.

The new controller is like a smart driver who adjusts the car based on the fuel tank:

The "P" (Proportional) part: This is like the robot's "stiffness" or how much it wants to hold its position.
The "D" (Derivative) part: This is like the robot's "damping" or how much it resists moving quickly (like shock absorbers).

How it works:
The robot constantly calculates its total energy (a mix of how fast it's moving and how much it's straining against its target position). It has a pre-set Energy Limit (like a speed limit or a fuel cap).

When things are calm: If the robot is moving slowly and safely, the controller acts like a normal, stiff robot. It follows its instructions perfectly.
When energy gets high: If the robot starts moving too fast or gets pushed hard, the controller senses the energy is getting close to the limit.
- It softens the "P" (stiffness): The robot stops fighting so hard to hold its position. If you push it, it lets go easily instead of pushing back.
- It boosts the "D" (damping): If the robot is moving too fast because someone pushed it, the controller adds "brakes" to slow the motion down without making it stiff.

The "Dissipative Functions" (The Dimmer Switch)

The paper introduces two mathematical "dimmer switches" (called dissipative functions, $r$ and $r_T$ ).

Imagine a dimmer switch for the robot's stiffness. When the energy is low, the switch is at 100% (full stiffness). As the energy gets closer to the limit, the switch slowly turns down to 0%.
This ensures the robot never suddenly "breaks" or becomes unstable; it just gradually becomes softer and safer as it gets closer to its energy limit.

The Safety Check (Stability Proof)

The authors didn't just guess that this would work; they did the math to prove it. They used a method called Lyapunov stability (think of it as a mathematical safety net).

They proved that as long as the robot follows their specific rules for adjusting the "dimmer switch," the robot will never go into a wild, uncontrollable spin. It guarantees that the robot will always settle down safely, even if the gains (stiffness/brakes) are changing constantly.

The Experiments: Testing the Theory

The team tested this on a real humanoid robot named TALOS (built by PAL Robotics) and in a computer simulation.

The "Hug" Test (External Force): They pushed the robot arm while it was trying to hold a position.
- Old Robot: Fought back hard, stayed stiff, and exceeded the energy safety limit.
- New Robot: Immediately softened up, let the arm be pushed away easily, and stayed well within the energy limit. It acted like a compliant, safe partner.
The "Sudden Move" Test: They told the robot to jump to a new position instantly.
- Old Robot: Moved too fast, creating a huge spike in energy.
- New Robot: Moved more slowly and smoothly, keeping its energy under the limit the whole time.

The Bottom Line

This paper presents a way to make robots safer without needing expensive, specialized sensors on every joint. By simply tracking the robot's total energy and automatically "softening" its movements when that energy gets too high, the robot becomes naturally compliant. It acts like a cautious driver who slows down as they approach a dangerous curve, ensuring that if a collision happens, the energy transferred is low enough to be safe for a human.

The authors conclude that this method is simple to implement, mathematically proven to be stable, and works effectively on real hardware, making it a promising step toward safer human-robot interaction.

Technical Summary: Adaptive PD Gains for Energy-Conscious Control in Physical Human-Robot Interaction

Problem Statement
Physical Human-Robot Interaction (pHRI) requires robust safety mechanisms to prevent injury during collisions. While compliant control strategies (impedance and admittance control) are common, they face significant limitations: they often rely on external force/torque sensors or joint torque estimation, are coordinate-dependent, and lack explicit safety guarantees regarding energy transfer limits as mandated by standards like ISO/TS 15066. Existing energy-based control approaches offer a coordinate-independent alternative but often suffer from complex implementations, the need for "virtual energy tanks," or a lack of explicit stability proofs. Furthermore, many current schemes are reactive, modifying control only after safety limits are breached, rather than proactively managing energy.

Methodology
The authors propose an adaptive Proportional-Derivative (PD) controller designed to limit a robot's total mechanical energy (kinetic and potential) below a specified threshold ( $L_{lim}$ ). The controller modifies standard PD gains based on the system's instantaneous energy state.

System Modeling: The robot dynamics are modeled using standard rigid-body equations. The total energy $L$ is defined as the sum of kinetic energy ( $T$ ) and controlled joint potential energy ( $U_q$ ).
Dissipative Functions: Two novel dissipative functions are introduced to scale the controller gains:
- $r$ (Energy-based scaling): A function of total energy $L$ that scales the proportional gain ( $K_p$ ). As $L$ approaches the limit $L_{lim}$ , $r$ approaches zero, reducing stiffness to lower potential energy.
- $r_T$ (Kinetic-based scaling): A function of kinetic energy $T$ used to scale the derivative gain ( $K_d$ ). This term increases damping when kinetic energy is high to oppose external pushes (user-added energy) without violating the energy limit.
Gain Adaptation:
- The proportional gain $K_p$ is adapted to dissipate as total energy rises, ensuring the potential energy component remains within budget.
- The derivative gain $K_d$ is adapted to increase damping in response to high kinetic energy, preventing unstable "flailing" while allowing the robot to yield to external forces.
Stability Enforcement: A Lyapunov stability analysis is conducted. The authors derive a condition where the time derivative of the Lyapunov function must be negative definite. To satisfy this, an algorithm (Algorithm 1) is proposed to enforce a step-size constraint on the rate of change of the proportional gain ( $\dot{K}_p$ ), ensuring the system remains stable even as gains adapt.

Key Contributions

Adaptive PD Framework: A simple, familiar PD control structure that inherently limits both kinetic and potential energy, avoiding the complexity of energy tanks or passivity-based controllers found in other literature.
Dual Dissipative Mechanism: The introduction of two distinct functions ( $r$ and $r_T$ ) allows for independent management of potential energy (via stiffness reduction) and kinetic energy (via damping injection), specifically addressing the issue of energy injection from human interaction.
Stability Proof: A formal Lyapunov-based stability analysis is provided, along with a practical algorithm to enforce the derived stability condition during operation.
Proactive Safety: Unlike reactive collision detection schemes, this controller proactively shapes the robot's impedance as energy approaches the limit, ensuring compliance before safety thresholds are breached.

Experimental Results
The controller was validated on a TALOS humanoid robot (PAL Robotics) in both Gazebo simulation and hardware experiments, comparing performance against a traditional non-adaptive PD controller.

Sinusoidal Tracking: When system energy was well below the limit, the adaptive controller behaved nearly identically to the traditional PD controller, demonstrating that nominal tuning is preserved during safe operation.
Step Response: In a step-input scenario, the traditional controller violated the energy limit ($1.5$ J in simulation), whereas the adaptive controller successfully capped the energy at the limit by reducing gains. This resulted in a slower settling time (3.21s vs 2.09s) but guaranteed safety.
External Wrench/Force Response:
- Simulation: When an external wrench was applied, the traditional controller exhibited stiff behavior with high deflection and energy violation. The adaptive controller reduced stiffness and increased damping, resulting in compliant behavior (higher deflection of ~1.0 rad) while keeping total energy under the limit.
- Hardware: Similar results were observed on the physical TALOS robot. The adaptive controller limited total energy to below the $0.15$ J hardware limit, whereas the traditional controller exceeded it ($0.17$ J). The adaptive approach yielded a softer interaction (0.47 rad deflection vs 0.185 rad) and a more gradual recovery.

Significance and Claims
The paper claims that the proposed controller offers a practical and implementable solution for safe pHRI. By leveraging the simplicity of PD control, it lowers the barrier to adopting energy-based safety constraints. The authors emphasize that the controller:

Provides explicit energy limiting, aligning with ISO/TS 15066 safety standards.
Naturally induces compliant behavior (softness) when energy limits are approached, reducing injury risk.
Maintains stability through a mathematically proven condition.
Is modular, allowing the use of standard tuning techniques for nominal gains while adding safety layers only when necessary.

The authors modestly note limitations, including the requirement for accurate dynamic modeling, the fact that energy limitation does not guarantee safety against puncture risks (pressure/area dependent), and the need for careful tuning of parameters ( $\alpha, \beta, K_{dmax}$ ) to balance resistance and compliance. Future work is suggested to include broader collision scenarios, inertia-based budget allocation, and friction compensation.