Koopman operator theory offers a rigorous treatment of dynamics, emerging as a robust alternative for learning-based control in robotics. By representing nonlinear dynamics as a linear, higher-dimensional operator, it provides a fresh lens for modeling complex systems. Its ability to support incremental updates and low computational cost makes it particularly appealing for real-time applications and online learning. This review delves deeply into the foundations, systematically bridging theoretical principles to practical robotic applications. We explain mathematical underpinnings, approximation approaches for inputs, data collection strategies, and lifting function design. We explore how Koopman models unify tasks like model-based control, state estimation, and motion planning. The review surveys cutting-edge research across domains ranging from aerial and legged platforms to manipulators, soft robots, and multi-agent networks. We also present advanced theoretical topics and reflect on open challenges and future research directions. To support adoption, we provide a hands-on tutorial with code athttps://github.com/sunnyshi0310/KoopmanRobo/tree/main.

Soft robotic systems are inherently compliant, giving them unique capabilities not possessed by traditional rigid-bodied robot systems. Many soft systems rely on soft pneumatic actuators. One of the biggest downsides of such actuators is the need for bulky pressure-regulating devices and individual pneumatic supply lines. In this work, a miniaturized pressure-regulating system is developed and embedded into the unused space inside of a soft pneumatic McKibben actuator, enabling the simultaneous pressure control of multiple actuators connected to a single pneumatic supply line. This "valve-embedded" actuator is capable of regulating its internal pressure within 0.05 psi of a desired set point, even under external load. Compared to a conventional McKibben actuator driven by external valves, the valve-embedded actuator is experimentally shown to consistently achieve faster settling times. To showcase the practical application of the valve-embedded actuator on a robotic system, a 0.9m serial-linked robot driven by five independently controlled valve-embedded actuators was assembled, and was shown to achieve an average root mean square error of less than 1.5cm in a waypoint tracking experiment. The miniaturized pressure control system developed in this work is open source and could be embedded in any fluid-driven actuator, enabling more capable and densely actuated pneumatic soft robots.

Proprioception, or the perception of the configuration of one's body, is challenging to achieve with soft robots due to their infinite degrees of freedom and incompatibility with most off-the-shelf sensors. This work explores the use of inertial measurement units (IMUs), sensors that output orientation with respect to the direction of gravity, to achieve soft robot proprioception. A simple method for estimating the shape of a soft continuum robot arm from IMUs mounted along the arm is presented. The approach approximates a soft arm as a serial chain of rigid links, where the orientation of each link is given by the output of an IMU or by spherical linear interpolation of the output of adjacent IMUs. In experiments conducted on a 660mm long real-world soft arm, this approach provided estimates of its end effector position with a median error of less than 10% of the arm's length. This demonstrates the potential of IMUs to serve as inexpensive off-the-shelf sensors for soft robot proprioception.

Controlling soft robots with precision is a challenge due to the difficulty of constructing models that are amenable to model-based control design techniques. Koopman operator theory offers a way to construct explicit dynamical models of soft robots and to control them using established model-based control methods. This approach is data driven, yet yields an explicit control-oriented model rather than just a “black-box” input-output mapping. This work describes a Koopman-based system identification method and its application to model predictive control (MPC) design for soft robots. Three MPC controllers are developed for a pneumatic soft robot arm via the Koopman-based approach, and their performances are evaluated with respect to several real-world trajectory following tasks. In terms of average tracking error, these Koopman-based controllers are more than three times more accurate than a benchmark MPC controller based on a linear state-space model of the same system, demonstrating the utility of the Koopman approach in controlling real soft robots.

This paper presents a soft-robotic platform for exploring the ecological relevance of non-locomotory movements via animal-robot interactions. Coral snakes (genus Micrurus) and their mimics use vigorous, non-locomotory, and arrhythmic thrashing to deter predation. There is variation across snake species in the duration and curvature of anti-predator thrashes, and it is unclear how these aspects of motion interact to contribute to snake survival. In this work, soft robots composed of fiber-reinforced elastomeric enclosures (FREEs) are developed to emulate the anti-predator behaviors of three genera of snake. Curvature and duration of motion are estimated for both live snakes and robots, providing a quantitative assessment of the robots' ability to emulate snake poses. The curvature values of the fabricated soft-robotic head, midsection, and tail segments are found to overlap with those exhibited by live snakes. Soft robot motion durations were less than or equal to those of snakes for all three genera. Additionally, combinations of segments were selected to emulate three specific snake genera with distinct anti-predatory behavior, producing curvature values that aligned well with live snake observations.

Soft robots are challenging to model due in large part to the nonlinear properties of soft materials. Fortunately, this softness makes it possible to safely observe their behavior under random control inputs, making them amenable to large-scale data collection and system identification. This paper implements and evaluates a system identification method based on Koopman operator theory in which models of nonlinear dynamical systems are constructed via linear regression of observed data by exploiting the fact that every nonlinear system has a linear representation in the infinite-dimensional space of real-valued functions called observables. The approach does not suffer from some of the shortcomings of other nonlinear system identification methods, which typically require the manual tuning of training parameters and have limited convergence guarantees. A dynamic model of a pneumatic soft robot arm is constructed via this method, and used to predict the behavior of the real system. The total normalized-root-mean-square error (NRMSE) of its predictions is lower than that of several other identified models including a neural network, NLARX, nonlinear Hammerstein-Wiener, and linear state space model.

Fiber-Reinforced Elastofluidic Enclosures (FREEs), are a subset of pneumatic soft robots with an asymmetric continuously deformable skin that are able to generate a wide range of deformations and forces, including rotation and screw motions. Though these soft robots are able to generate a variety of motions, simultaneously controlling their end effector rotation and position has remained challenging due to the lack of a simple model. This paper presents a model that establishes a relationship between the pressure, torque due to axial loading, and axial rotation to enable a model-driven open-loop control for FREEs. The modeling technique relies on describing force equilibrium between the fiber, fluid, and an elastomer model which is computed via system identification. The model is experimentally tested as these variables are changed, illustrating that it provides good agreement with the real system. To further illustrate the potential of the model, a precision open-loop control experiment of opening a rotational combination lock is presented2.