⚡ electrical engineering

External and Internal Impedance for Robot Teams Sharing a Load: Exact Decoupling and Decay Certificates

This paper presents an exact decoupling framework for external and internal impedance in teams of N robots with redundant actuation, proving that internal efforts do not induce motion bias and that tracking errors converge exponentially under exact modeling, while validating the approach through simulations and a physical demonstration of four humanoids carrying a heavy load.

Original authors: Mehdi Heydari Shahna, Jouni Mattila

Published 2026-10-20

📖 8 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

When a group of robots works together to carry a heavy object, they face a delicate balancing act. They must move the object to a new location while simultaneously keeping their grip firm enough that it does not slip, yet loose enough that they do not crush it. If one robot pushes too hard while another pulls, the object might twist or drop, even if the team is moving perfectly in sync. This challenge is known as cooperative manipulation. For years, engineers have developed ways to control how robots push and pull, treating the object like a spring that resists movement. However, when multiple robots with moving legs or wheels try to lift a single load, the math becomes incredibly complex. The robots must coordinate their steps, their arm movements, and the invisible forces they apply to the object all at once. If the control system is not precise, the robots might fight each other, wasting energy or causing the load to shift dangerously.

Researchers at Tampere University in Finland have developed a new method to solve this problem, allowing teams of robots to carry heavy loads with unprecedented precision. Their work focuses on a specific type of control called impedance, which dictates how a robot reacts when it touches something. In this new framework, the team separates the task into two distinct jobs: moving the object through the world and managing the internal forces that hold the object together. The researchers proved mathematically that these two jobs can be handled independently, meaning the robots can adjust their grip without accidentally nudging the object off its path. They demonstrated this with a simulation of four human-sized robots carrying a large aluminum beam. The system allowed the robots to walk, lift, and set down the beam while keeping the internal forces exactly where they wanted them, even when the robots' own movements or the weight of the object changed slightly.

The core of this achievement is a two-layer control system that acts like a highly skilled conductor. The outer layer tells the robots where the object should go and how stiffly it should hold it against the environment. The inner layer, which runs much faster, calculates the exact muscle power needed to achieve that goal. A key innovation in this study is how the system handles "inconsistent" requests. Imagine if one robot in a team tried to pull the object in a direction that the others could not follow due to their physical limits. In older systems, this conflict would cause the object to jerk or the robots to struggle against each other. The new method identifies these conflicting commands and cancels them out before they can cause motion. It ensures that any force used to tighten the grip stays internal to the team, never pushing the object sideways or up and down unless that is the intended motion.

To test their theory, the researchers created a detailed digital simulation using four humanoids, each weighing about 81 kilograms, carrying a 63-kilogram aluminum girder. The beam was 2.4 meters long, and the robots had to walk to it, lift it, carry it for a few steps, and set it down. The simulation included realistic physics, such as friction between the robots' feet and the ground, and the way their joints resist movement. The results were striking. The team successfully carried the beam, and the object stayed within 7 millimeters of its intended path throughout the entire mission. This level of precision held true even when the researchers deliberately introduced errors, such as making the robots' internal models of their own weight slightly wrong or adding random noise to their sensors. The system was robust enough to handle these imperfections without dropping the load.

One of the most significant findings was how the robots managed their grip strength. The researchers wanted to see if they could change how hard the robots squeezed the beam without moving the beam itself. In previous methods, changing the grip often caused the object to shift. In this new system, when the robots adjusted their grip, the beam moved less than 0.2 millimeters. By comparison, using older techniques to achieve a similar grip change caused the beam to move nearly 1.9 millimeters. This difference is crucial for delicate tasks where even a tiny shift could be disastrous. The system achieved this by using a mathematical check to ensure that the robots' commands were physically possible before they were executed. If a command would cause a robot to slip or lose its balance, the system adjusted it automatically to keep the team safe.

The study also explored the limits of what the team could carry. The robots were rated to lift about 25 kilograms each, but the simulation showed they could successfully carry the 63-kilogram beam, which is roughly 16 kilograms per robot. When the researchers increased the weight beyond this limit, the system correctly predicted failure. In cases where the load was too heavy, the robots lost their grip or fell, but the system identified these failures before they happened by calculating a "capacity estimate." This estimate acted as a warning sign, telling the operators exactly how much weight the team could handle based on their current stance and the friction of their feet. This predictive ability suggests that the system could be used to plan missions in real-time, ensuring that robots only attempt tasks they are physically capable of completing.

The researchers also tested how the system behaved when the robots' physical models were not perfect. In the real world, robots are never built exactly to their blueprints; their joints might be slightly loose, or their weight distribution might vary. The new control method proved to be very forgiving. Even when the researchers changed the simulated weight of the beam by 20 percent or altered the robots' internal inertia, the team still managed to carry the load. The system adjusted the forces automatically to compensate for these errors. This resilience is vital for practical applications, as it means the robots do not need to be perfectly calibrated to work together. They can adapt to the reality of their physical bodies and the environment they are in.

A critical part of the research involved verifying that the robots would not slip. The system constantly monitored the forces at the robots' feet and hands to ensure they remained within safe limits. If the load became too heavy for a single robot's foot to support, the system redistributed the weight to the other robots. In the simulation, when the robots walked, the load shifted between their feet and hands in a way that kept them stable. The researchers found that the system could handle complex walking patterns, including moments when only one foot was on the ground. During these single-support phases, the robot on the ground had to carry the full weight of its own body plus a share of the beam, and the control system managed this transition smoothly without causing the team to stumble.

The study also looked at how the robots reacted to external pushes. When a simulated force of 100 newtons pushed the beam from the side, the robots responded by moving the beam slightly to absorb the impact, just as a person would lean into a push. The amount of movement matched the predictions of their control theory almost exactly. This confirmed that the robots were behaving like a single, cohesive unit with a predictable response to the world. The researchers noted that if the robots' joints had friction, the movement was slightly different, but the system still maintained control. This ability to predict and manage external forces is essential for robots working in unstructured environments where they might bump into walls or be pushed by wind.

In the final analysis, the work demonstrates that a team of robots can be controlled with a high degree of precision and safety, even when carrying heavy, awkward loads. The method separates the task of moving the object from the task of holding it, allowing for independent control of both. This separation prevents the robots from fighting each other and ensures that grip adjustments do not cause unwanted motion. The simulations showed that the system works reliably across a range of conditions, from perfect models to imperfect ones, and from light loads to those near the robots' maximum capacity. The results provide a solid foundation for future robots that need to work together in factories, construction sites, or disaster zones, where the ability to carry and manipulate heavy objects safely is essential. The research confirms that with the right mathematical framework, a team of robots can act as a single, intelligent entity, capable of handling complex physical tasks with a grace that was previously out of reach.

Technical Summary: External and Internal Impedance for Robot Teams Sharing a Load

Problem Statement
The paper addresses the challenge of controlling teams of $N$ robots (including mobile and floating bases with redundant actuation) that cooperatively manipulate a shared rigid payload. The core problem is to simultaneously regulate two distinct behaviors:

External Impedance: Governing the payload's interaction with the environment (contact forces and motion).
Internal Impedance: Regulating the internal wrenches (forces and torques) applied by the robots to the payload without inducing unwanted motion.

Existing approaches, such as the dual-arm framework in [6], often assume perfect inner-loop tracking and struggle to certify explicit exponential convergence rates for larger teams with complex constraints. Furthermore, inconsistent reference requests (e.g., from independent robot controllers) can generate uncommanded internal wrenches (antagonism) that bias the payload's motion. The authors aim to extend impedance control to graph-compiled robot teams, providing exact decoupling of motion and internal forces, and certifying stability despite inner-loop tracking errors and model uncertainties.

Methodology
The proposed framework utilizes a two-loop control architecture built upon a compiled mechanism graph model:

Inner Loop (Model-Based VDC):
- The inner loop employs a Virtual Decomposition Control (VDC) approach. It takes requested actuator rates ( $w$ ) and fits them to admissible motions ( $u_r$ ) using a rate-gain metric ( $K$ ) to minimize constraint violation.
- It computes a required effort ( $f_r, \lambda_r$ ) based on an exact dynamic model and measured environment wrenches.
- A feedforward term is added to the control law to compensate for tracking errors, while an internal command ( $\psi$ ) is injected to regulate internal wrenches.
Outer Loop (Impedance & Reference Generation):
- External Impedance: A compliant frame ( $x_c$ ) is defined to track a desired pose ( $x_d$ ) with specified mass, damping, and stiffness matrices ( $M_c, D_c, K_p, K_o$ ). This governs the payload's interaction with the environment.
- Internal Impedance: The internal command $\psi$ is generated via mass-spring-damper dynamics driven by the internal wrench error ( $\lambda_I - \lambda_d$ ). This allows for integral action ( $K_z=0$ ) or proportional stiffness ( $K_z > 0$ ) to reject antagonistic forces.
Theoretical Analysis:
- The authors derive an exact error-power identity that accounts for inner-loop tracking errors.
- They prove that the internal wrench separates into three components: the commanded load, opposing loads from inconsistent requests (antagonism), and a decaying motion-error term.
- Decoupling: It is shown that motion requests and environment wrenches are independent of the internal command, ensuring that internal efforts do not drive payload motion, even with inertial model errors.
- Stability: Using Lyapunov methods, the paper establishes local exponential convergence for all errors (tracking, interaction, and internal wrench) at an explicit rate determined by system gains and models, without requiring time-scale separation.

Key Contributions

Exact Internal-Wrench Structure: The paper provides a rigorous decomposition of the internal wrench ( $\lambda_I = \psi + \Psi K w_\perp + \Theta s$ ), explicitly identifying the terms for command, antagonism from inconsistent references, and tracking error.
Certified Convergence Rates: Unlike previous works that prove asymptotic stability, this work provides explicit exponential convergence rates ( $\min\{\kappa_s, \kappa_\xi, \kappa_z, \kappa_T\}$ ) for the entire closed-loop system, including inner-loop errors.
Feasibility and Bias Conditions: The authors define a computable subspace for reference displacements. They prove that reference commands produce internal forces without motion bias if and only if they lie within this subspace. Otherwise, they introduce motion bias.
Run-Time Feasibility Certification: A runtime inequality is derived to certify grasp and support feasibility (e.g., ensuring contact forces remain within friction cones) based on the system's state and error bounds.

Results
The framework was validated through high-fidelity MuJoCo simulations involving four Apptronik Apollo humanoids carrying a 63 kg aluminum girder (approx. 16 kg per robot).

Performance: The team successfully walked, lifted, carried, and unloaded the girder. The payload remained within 7.0 mm of the compliant frame.
Internal vs. External Decoupling: When applying internal commands to shift palm loads, the girder moved only 0.2 mm. In contrast, using the reference displacement method from [6] (without projection) resulted in 1.9 mm of girder motion under the same conditions.
Robustness: The system maintained stability with up to $\pm 30\%$ link inertia perturbations and a $+20\%$ payload mass increase.
Capacity Estimation: A static capacity estimate ( $U^\star$ ) successfully predicted mission outcomes across a mass sweep. Missions with $U^\star < 1$ (single support) were successful, while those with $U^\star > 1$ resulted in the loss of the payload.
Antagonism Rejection: Simulations confirmed that integral action ( $K_z=0$ ) in the internal impedance loop effectively rejects steady-state antagonism caused by model errors or inconsistent references, whereas proportional stiffness ( $K_z=I$ ) reduces but does not eliminate it.

Significance
The paper claims to extend the dual-arm impedance framework of [6] to general robot teams with mobile bases and redundant actuation. Its primary significance lies in providing exact theoretical certificates for the decoupling of motion and internal forces, and for the exponential stability of the system. By rigorously accounting for inner-loop tracking errors and inconsistent references, the work offers a robust method for cooperative manipulation where internal force regulation does not compromise motion tracking. The results demonstrate that the proposed method allows for precise internal load regulation (e.g., for friction grasps) while maintaining high-fidelity external impedance behavior, validated in a complex, contact-rich simulation of humanoids performing a transport task.

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