⚡ electrical engineering

Linear or Bilinear: A Criterion for Koopman Rollouts in Sampling-Based Predictive Control

This paper establishes a criterion for selecting between linear and bilinear Koopman models in MPPI control, demonstrating that bilinear lifts are structurally necessary for configuration-dependent input gains and serve as a robust default that outperforms linear counterparts on velocity-driven systems while maintaining efficiency for force-driven platforms.

Original authors: Kangmin Lee, Sanghyun Kim

Published 2026-08-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kangmin Lee, Sanghyun Kim

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

Robots that move through the real world face a constant struggle: they need to predict what will happen next to make good decisions, but the physics of their own bodies are often too messy to write down on paper. Friction, flexible joints, and the way motors respond to commands are difficult to derive by hand, and even a carefully calibrated model often fails when the robot is actually deployed. To solve this, engineers have turned to data-driven methods, teaching computers to learn the rules of motion directly from observing the robot's movements. One powerful approach involves a technique called Model Predictive Path Integral control, which works by simulating thousands of possible future actions at once to find the best path. For this to work in real time, the computer needs a model that is both incredibly accurate and fast enough to run thousands of times in a split second. For years, the standard solution has been to use a simplified, linear version of these learned models, a choice inherited from older control methods that required mathematical simplicity.

However, a new study by researchers Kangmin Lee and Sanghyun Kim at Kyung Hee University challenges this long-held assumption. They discovered that the decision to use a simple linear model versus a slightly more complex one depends entirely on the specific type of robot and how it moves. The researchers found that for robots where the connection between the control command and the resulting movement changes significantly as the robot turns or reorients itself, the simple linear model is fundamentally broken. It is not just a matter of the model being slightly inaccurate; the structure of the math itself prevents it from ever getting the prediction right, no matter how much data is fed into it. In these cases, the researchers proved that a "bilinear" model, which allows the relationship between command and movement to shift with the robot's position, is not just an option but a necessity.

The core of the discovery lies in how different robots translate a command into motion. Consider a wheeled vehicle or a drone that receives instructions in its own local frame of reference, such as "move forward" or "move sideways." To know where this command will take the vehicle in the real world, the system must constantly rotate that instruction based on the vehicle's current heading. If the vehicle is facing north, "forward" means north; if it turns to face east, "forward" means east. This rotation factor changes continuously as the robot moves. The researchers showed that a standard linear model tries to use a single, fixed rule to handle this changing rotation. Because the rotation sweeps through a full range of values, a single fixed rule cannot possibly match the reality of the situation. It is like trying to use a single, straight ruler to measure the circumference of a circle; the tool is simply the wrong shape for the job. In contrast, the bilinear model allows the rule itself to change as the robot's state changes, effectively letting the ruler bend to fit the curve.

To prove this, the team tested their theory on six different robotic systems, ranging from a simple unicycle to a flying drone and a soft, flexible arm. In the case of the unicycle and the drone, where the commands are velocity-based and the direction of movement depends heavily on the robot's orientation, the linear model failed completely. In fifty separate parking trials, the linear controller drove the unicycle off course every single time, unable to correct its path because its internal prediction of how the steering worked was structurally wrong. The bilinear model, however, succeeded in 84 percent of those trials, performing nearly as well as a perfect, theoretical "oracle" that knows the exact physics of the system. Similarly, the drone, which was asked to fly a figure-eight pattern while constantly changing its yaw, drifted away from the path when controlled by the linear model but tracked the route with high precision using the bilinear approach.

The study also clarified why previous research had reported success with linear models. The researchers found that for robots driven by force or acceleration, such as a quadrotor drone that adjusts its thrust to move, the connection between the command and the movement is much more stable. In these systems, the input gain does not vary wildly, meaning the simple linear model is sufficient and the more complex bilinear version offers no significant advantage in accuracy, only a slight increase in computational cost. This distinction explains why linear models worked well for some past experiments but failed for others: the researchers had unknowingly been testing in a regime where the simpler math was adequate. By establishing a clear criterion based on how much the input gain varies over the operating region, the paper provides a definitive guide for engineers: if the robot's movement direction changes significantly with its orientation, the bilinear model is required; otherwise, the linear model will suffice.

Beyond just choosing the right model, the researchers also developed a way to make the system aware of its own uncertainty. They created a method to calculate a safety margin, or an "error tube," around every prediction the model makes. This tube grows larger when the model is less certain and shrinks when it is confident, allowing the controller to penalize risky paths during its decision-making process. This feature ensures that even when the model is not perfect, the robot remains stable and safe. The study confirms that this bilinear approach adds very little extra cost to the computation, requiring only a small, fixed increase in processing power compared to the linear version. The result is a robust, data-driven control strategy that adapts to the specific physical nature of the robot, ensuring that the mathematical tools used to guide the machine are always matched to the reality of its movement.

Technical Summary: Linear or Bilinear: A Criterion for Koopman Rollouts in Sampling-Based Predictive Control

Problem Statement
Sampling-based Model Predictive Path Integral (MPPI) control relies on rolling out thousands of sampled input sequences through a learned dynamics model. While Koopman operator theory offers a computationally efficient, linear-in-lifted-space framework for this task, existing Koopman-MPPI implementations strictly enforce a linear predictor structure (constant input matrix). This linearity is a convention inherited from convex Koopman MPC, where it is required to maintain a convex optimization problem. However, sampling-based control imposes no convexity constraints. Consequently, it remains unclear when a linear Koopman rollout is sufficient and when the constant input matrix assumption structurally fails to capture system dynamics, particularly for systems where the input gain varies significantly with the state (e.g., velocity-commanded wheeled vehicles or drones).

Methodology
The authors propose a rigorous criterion to determine when a bilinear Koopman model is necessary versus when a linear model suffices.

The Gain-Variation Criterion:
For a control-affine system $\dot{x} = f(x) + G(x)u$ with output $y=h(x)$ , the first-order response is governed by the input gain $\Phi(x) = \nabla h(x)G(x)$ . The paper defines the variation of this gain over the operating region $\mathcal{X}$ as $\nu_\Phi = \sup_{x,x' \in \mathcal{X}} \|\Phi(x) - \Phi(x')\|$ .
- Linear Sufficiency: If $\nu_\Phi$ is small (or zero, as in second-order systems where input enters at $O(\Delta t^2)$ ), a constant input matrix can approximate the dynamics, and linear and bilinear predictions coincide.
- Bilinear Necessity: If $\nu_\Phi$ is large (e.g., body-frame velocity commands where the gain rotates with the robot's heading), a single constant matrix cannot reproduce the input response across the operating region. The paper proves that no state dictionary (linear or deep) can absorb this coupling under a linear readout; the error is structural, not a matter of capacity.
Bilinear Koopman MPPI Controller:
The proposed controller utilizes a bilinear predictor:
$z_{t+1} \approx Az_t + B_0 u_t + \sum_{i=1}^m u_{t,i} (B_i z_t)$
where $z$ is the lifted state. This formulation allows the input matrix to vary affinely with the lifted state.
- Computational Cost: A bilinear step requires $m+1$ matrix-vector products (where $m$ is the number of inputs), remaining in the same computational regime as the linear model (which requires 1 product).
- Error-Aware Rollout: The controller incorporates a "tube cost" into the MPPI sampling objective. Using a certified proportional error bound ( $\|r(x,u)\| \le c_x \|\Psi(x)\| + c_u \|u\|$ ), an online error tube $e_t$ is propagated for each sample. This tube penalizes trajectories that deviate significantly from the certified region, providing a practical-stability guarantee via free-energy analysis.

Key Contributions

C1 (Testable Criterion): The paper establishes that a linear rollout incurs an irreducible first-order input-response error of at least $\frac{1}{2}\nu_\Phi \|u\| \Delta t$ when gain variation is high. Bilinear Koopman MPPI achieves near-oracle performance in these regimes where linear methods fail.
C2 (Structural Failure): It demonstrates that the failure of linear Koopman models in high-variation regimes is structural. Enlarging the dictionary size or using deep linear networks cannot compensate for the missing state-input coupling term $\Phi(x)u$ .
C3 (Error-Aware Efficiency): The authors present a bilinear rollout that maintains matrix-vector computational costs while integrating an online error tube derived from certified identification bounds, ensuring practical stability.
C4 (Unifying Boundary): The criterion explains prior successes of linear Koopman control in force- and acceleration-driven systems (where gain variation is low) and identifies the specific regime (velocity-commanded platforms) where bilinearity is required.

Experimental Results
The authors validate the framework on six systems:

High Gain Variation (Velocity-Commanded):
- Nonholonomic Unicycle: In parking tasks, the linear controller failed completely (0% success rate), driving off along the initial heading. The bilinear controller achieved 84% success, approaching the oracle (92%). Open-loop prediction error was 42–48 times lower for the bilinear model.
- Aerial Drone: Tracking a figure-8 path with body-frame velocity commands, the linear controller drifted significantly (1.4 m error), while the bilinear controller matched oracle accuracy (0.05 m error). The bilinear model was 135 times more accurate in open-loop prediction.
Low Gain Variation (Force/Acceleration-Commanded):
- Quadrotor (Thrust-driven) & Pendulum: In these systems, the input gain is either constant or enters at second order. Here, linear and bilinear rollouts performed nearly identically. The bilinear model offered a parameter-efficiency advantage (requiring fewer neurons for the same accuracy) but did not show structural superiority.
- Soft Arm & Torque Arm: Similar equivalence was observed, confirming that linear models suffice when curvature or inertia variations are slow.

Significance and Claims
The paper claims to resolve a long-standing convention in Koopman-based control by replacing the "linear-by-default" assumption with a system-dependent criterion.

Design Rule: The authors conclude that bilinear Koopman rollouts should be the default for sampling-based control. They are necessary for systems with configuration-dependent gains (like velocity-commanded robots) and harmless (though slightly more parameter-heavy) for systems where gains are constant.
Scope: The work highlights that previous linear Koopman successes were limited to low-variation regimes (force/acceleration control). The proposed bilinear approach extends the applicability of Koopman MPC to velocity-commanded platforms where linear models structurally fail.
Limitations: The authors note that the stability guarantee relies on certified error bounds which are currently approximated empirically for deep dictionaries. Furthermore, the criterion assumes smooth kinematics and does not currently apply to contact-rich hybrid dynamics (e.g., legged locomotion).

In summary, the paper provides a theoretical and empirical justification for using bilinear Koopman models as a robust, computationally efficient, and structurally correct alternative to linear models in sampling-based predictive control, particularly for systems where the input gain varies significantly with the state.

Technical Summary: Linear or Bilinear: A Criterion for Koopman Rollouts in Sampling-Based Predictive Control

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