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Model-Based Control Using Koopman Operators

Ian Abraham, Gerardo De La Torre, Todd D. Murphey

arXiv:1709.01568v1cs.ROmath.OC

TL;DR

Complex dynamical systems can be difficult to model from first principles, and derived models may have limited long-term predictive use. This paper uses Koopman operator theory to generate linearizable data-driven models for model-based robotic control. Its reported results span simulated pendulum systems and SPRK robot experiments, while noting a limitation in representing behavior consistently across operating terrains with limited information.

  • Problem

    First-principles models can be demanding to derive for complex systems, may omit external disturbances, and leave open questions for data-driven control, including stability and training data.

    Source: §I · p. 1

  • Method

    The paper uses Koopman operator theory to derive a linearizable data-driven model and formulates open- and closed-loop controllers for robotic systems.

    Source: §I · p. 1

  • Results

    Open-loop performance improves as basis complexity increases, while closed-loop experiments show the model capturing nonlinear dynamics in simulated cart- and VTOL-pendulum systems and the SPRK robot.

    Source: §VI · p. 8

  • Takeaways & Limitations

    Basis-function choice and complexity matter for representing particular robotic systems, and greater basis complexity improves the reported open-loop performance.

    Source: B. Simulation: Inversion and… · p. 5, §VI · p. 8

    A representation performing consistently across operating terrains appears infeasible with the limited information used in the SPRK setting.

    Source: 2) Closed-Loop Trajectory Tracking · p. 7

Abstract

from arXiv · show

This paper explores the application of Koopman operator theory to the control of robotic systems. The operator is introduced as a method to generate data-driven models that have utility for model-based control methods. We then motivate the use of the Koopman operator towards augmenting model-based control. Specifically, we illustrate how the operator can be used to obtain a linearizable data-driven model for an unknown dynamical process that is useful for model-based control synthesis. Simulated results show that with increasing complexity in the choice of the basis functions, a closed-loop controller is able to invert and stabilize a cart- and VTOL-pendulum systems. Furthermore, the specification of the basis function are shown to be of importance when generating a Koopman operator for specific robotic systems. Experimental results with the Sphero SPRK robot explore the utility of the Koopman operator in a reduced state representation setting where increased complexity in the basis function improve open- and closed-loop controller performance in various terrains, including sand.

I. INTRODUCTION

First-principles modeling can be demanding for complex systems, and models may poorly predict long-term behavior when they omit external disturbances. The paper applies Koopman operator theory to data-driven robotic models and outlines how its linear operator representation can support modeling and control.

I. INTRODUCTION

  • First-principles modeling is demanding when state interactions are complex, and derived models may poorly predict over long horizons because they omit external disturbances.Despite these limits, system representations remain central to model-based engineering and scientific applications.Source: §I · p. 1
  • Robust and adaptive control mitigate model uncertainty –, while uncertain-system planning incorporates explicit uncertainty models –, which can limit general utility.The text also identifies open questions for data-driven control, including training data, stability, convergence, computational complexity, and mechanical-property conservation.Source: §I · p. 1
  • The paper investigates Koopman operator theory for robotic control using finite-dimensional approximations computed from data without prior system information.The Koopman operator is introduced as an infinite-dimensional linear operator that exactly captures nonlinear-system behavior.Source: §I · p. 1
  • The paper develops a linearizable data-driven model for model-based control and examines basis-function choices in simulated pendulum systems and Sphero SPRK experiments.It formulates open- and closed-loop controllers using the proposed model.Source: §I · p. 1

II. KOOPMAN OPERATOR

  • The Koopman operator is linear on a vector space of observables even for nonlinear systems, but its infinite-dimensional form is approximated by a finite-dimensional operator using data.In this representation, the operator propagates system outputs forward rather than mapping system states to states.Source: §II · p. 2

A. Approximating a Koopman Operator

The approximate Koopman operator is fit from recorded data by least squares. Its approximation quality and computational cost depend on the basis dimension and on the amount and distribution of data.

A. Approximating a Koopman Operator

  • The matrix K advances the basis-function representation Ψ by one time step, with r(x_k) denoting approximation error.The operator is estimated from recorded trajectory data using a least-squares approach and a Moore–Penrose pseudoinverse.Source: A. Approximating a Koopman Operator · p. 2
  • The least-squares formulation uses P recorded data points, and the definitions can also accommodate multiple trajectories or missing samples when consecutive state pairs are equally spaced in time.The paper notes that another optimization could avoid even that spacing requirement.Source: A. Approximating a Koopman Operator · p. 2
  • Increasing the dimension of Ψ generally improves approximation but raises computational burden; the number and state-space distribution of data points also strongly affect K.Source: A. Approximating a Koopman Operator · p. 2

B. Approximating Dynamical Systems

The paper forms a data-driven system model by including the system state among the basis functions and propagating selected quantities with the approximate Koopman operator.

B. Approximating Dynamical Systems

III. CONTROL SYNTHESIS: OPEN- AND CLOSED-LOOP CONTROLLERS

The paper formulates open- and closed-loop model-based controllers around a data-driven Koopman model. With differentiable basis functions, the approximate dynamics can be linearized using data alone.

III. CONTROL SYNTHESIS: OPEN- AND CLOSED-LOOP CONTROLLERS

  • The control-synthesis section formulates open- and closed-loop optimal control using a linearizable Koopman approximation.Source: §III · p. 3

A. Koopman Operator Linearization

  • Choosing a differentiable basis function allows the approximate dynamics to be linearized, with control inputs incorporated by augmenting Ψ with the inputs.The augmented basis is Ψ(x, u) = [x^T, u^T, ψ_1(x, u), …, ψ_N(x, u)].Source: A. Koopman Operator Linearization · p. 3
  • Linearizable equations of motion can be computed solely from data for use in model-based control methods.Source: A. Koopman Operator Linearization · p. 3

B. Optimal Control Problem

The section formulates trajectory optimization as minimizing a state- and control-weighted objective subject to modeled dynamics. It describes open-loop projection-based planning and closed-loop Sequential Action Control (SAC) for generating control actions.

B. Optimal Control Problem

  • Trajectory optimization minimizes a discrete-time objective over state and control trajectories, weighted by positive-definite matrices P and R.The effectiveness of synthesized optimal control depends substantially on the accuracy of the modeled dynamics.Source: B. Optimal Control Problem · p. 3
  • Open-loop projection-based optimization precomputes trajectory and control actions, then sends commands at a fixed rate without using odometry to update them.Odometry is collected for post-processing only.Source: Open-Loop · p. 3
  • Closed-loop SAC uses the Koopman operator for discrete-time optimal control calculations, while the section notes that other MPC techniques can also be used.SAC forward-simulates an open-loop trajectory over a horizon for a control-affine system.Source: Closed-Loop · p. 3
  • SAC selects the time and duration of control application using objective sensitivity and an outward line search for sufficient cost descent.The control duration is found in discrete time.Source: Closed-Loop · p. 3

IV. EXPERIMENTS USING SPHERO SPRK

The experiments use Koopman-derived state-space models with open-loop and closed-loop trajectory optimization on the Sphero SPRK robot. They also examine control in a sand environment.

IV. EXPERIMENTS USING SPHERO SPRK

  • The experimental setup explores Koopman-based model-predictive control on the SPRK, including an additional set of experiments in sand.The section motivates and explores the operator's utility for robotic control.Source: §IV · p. 4
  • The SPRK experiments apply trajectory optimization in both open-loop and closed-loop feedback forms.The experiments also examine control in a sand environment.Source: §IV · p. 4
  • The tracked states are position and velocity, the inputs are desired velocities, and the objective weights remain fixed across both experiment types.P = diag([60, 60, 0.1, 0.1]) and R = diag().Source: §IV · p. 4

A. SPRK

The paper uses a reduced-state model for the SPRK and constructs approximate Koopman operators with polynomial basis functions. In double-pendulum simulations, prediction accuracy improves with basis complexity but depends on mechanical energy and relative angle.

A. SPRK

  • The SPRK's coupled internal mechanism, spherical casing, and proprietary low-level control motivate using a reduced-state model instead of a high-dimensional one.The robot's embedded software uses gyro-accelerometers to balance it while rolling.Source: A. SPRK · p. 4

B. SPRK Koopman Operator

  • For the SPRK, the approximate Koopman operator uses position, velocity, and commanded velocity with polynomial basis functions; training data came from operation below 1 m/s.Higher-order position dependence is excluded to prevent possible overfitting of position-based external disturbances.Source: B. SPRK Koopman Operator · p. 4

A. Simulation: Mechanical Energy

  • The double-pendulum simulation uses randomized initial conditions around stable equilibrium, collects 2,000 data points at 100 Hz, and scales polynomial basis functions by expected state maxima.The modeled pendulums each have 1 kg mass and 1 m length; data outside the defined initial-condition range are excluded from operator approximation.Source: A. Simulation: Mechanical Energy · pp. 4–5
  • Increasing polynomial-basis complexity improves simulated trajectory prediction, while prediction error tends to rise with total mechanical energy.The authors attribute deterioration at larger relative angles to polynomial approximation of transcendental double-pendulum dynamics; polynomial approximation is accurate when relative angles and total energy are small.Source: A. Simulation: Mechanical Energy · p. 5, Figure 2 · p. 5

B. Simulation: Inversion and Stabilization of Pendulum Systems

Simulations examine how Koopman basis choices affect control of cart-pendulum and VTOL-pendulum systems. Basis complexity can improve control, but the useful specification depends on the system.

B. Simulation: Inversion and Stabilization of Pendulum Systems

  • The simulations study basis-function effects for inverting cart-pendulum and VTOL-pendulum systems, including systems with components in SO(n) for n > 1.The cart-pendulum operators are trained on the initial 20 seconds with the nominal model.Source: B. Simulation: Inversion and… · p. 5, Figure 3 · p. 6
  • For the cart-pendulum, the Koopman state includes angle, position, their velocities, input, a constant, and additional basis functions.The simulation compares polynomial and Fourier bases using the same initial condition and nominal model.Source: B. Simulation: Inversion and… · p. 5
  • For the cart-pendulum, Fourier Q = 1 suffices to invert and stabilize, while Q = 2 yields the same Koopman matrix; polynomial-basis control improves with complexity.The polynomial controller produces the desired energy-pumping cart motion.Source: B. Simulation: Inversion and… · p. 5, Figure 3 · p. 6
  • The VTOL simulation assumes its model is known while the interaction between the VTOL and pendulum remains unknown.The simulated goal is to generate a Koopman operator describing that interaction.Source: B. Simulation: Inversion and… · p. 6
  • For the VTOL-pendulum, increasing basis complexity refines control; polynomial Q = 2 makes swing-up attempts more successful, and Fourier functions yield swing-up and inversion.Koopman operators are generated from the first 20 seconds of data.Source: B. Simulation: Inversion and… · p. 6

1) Open-Loop Trajectory Optimization:

Open-loop tracking errors decrease as the Koopman basis complexity Q increases. Closed-loop SPRK tracking improves with increasing basis complexity, while limited data and terrain-dependent nonlinearities constrain the model’s stability and consistency.

1) Open-Loop Trajectory Optimization:

  • A first-order Fourier basis captures enough features for the controller to invert and stabilize the pendulum.The operators are trained on residual modeling error from 20 seconds of attempted inversion using the nominal model.Source: Figure 4 · p. 7
  • Integrated open-loop tracking errors decrease as Q increases, suggesting a better approximate representation of the system dynamics.The reference trajectory was made aggressive enough to excite internal nonlinearities that the minimal state representation cannot fully capture.Source: Figure 5 · p. 7

2) Closed-Loop Trajectory Tracking:

  • The closed-loop reference uses rx = 0.7, ry = 0.4, and v = 0.9, with the nominal A and B defined as a fully controllable double integrator.The optimal control signal is updated at 20 Hz.Source: 2) Closed-Loop Trajectory Tracking · p. 7
  • Higher operator order exposes terrain-dependent internal nonlinearities, but consistent performance across terrains appears infeasible with limited information and no additional structure.The paper names global Lie group structure or mechanical properties such as symmetries as possible added structure.Source: 2) Closed-Loop Trajectory Tracking · p. 7

VI. CONCLUSION

The conclusion reports that increasing basis complexity improves open-loop performance and that Koopman-based closed-loop experiments capture nonlinear dynamics in simulated pendulum systems and the SPRK robot. It identifies basis selection, stability, and numerical analysis as directions for further work.

VI. CONCLUSION

  • SPRK tracking on tarp and sand improves with increasing basis complexity, although the third-order sand operator lacks enough data for a stable model.Despite that instability, the third-order operator performed better than the nominal model.Source: Figure 6 · p. 8
  • Closed-loop experiments show the Koopman operator capturing nonlinear dynamics in simulated cart- and VTOL-pendulum systems and the SPRK robot.The conclusion also reports improved open-loop performance as basis complexity increases.Source: §VI · p. 8
  • Future work includes analyzing basis choice for systems with distinct structure, the relationship between available states and approximation accuracy, stability, and numerical optimization.Source: §VI · p. 8

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