Data-Driven Predictive Control of Nonholonomic Robots Using Bilinear Koopman Realization: Challenges in Replacing Geometry with Data

Core Concepts

While data-driven methods like Extended Dynamic Mode Decomposition (EDMD) show promise in controlling nonholonomic robots, purely data-centric approaches struggle to replace the need for understanding the underlying geometry of these systems, particularly for tasks like precise setpoint stabilization.

Abstract

Bibliographic Information:

Rosenfelder, M., Bold, L., Eschmann, H., Eberhard, P., Worthmann, K., & Ebel, H. (2024). Data-Driven Predictive Control of Nonholonomic Robots Based on a Bilinear Koopman Realization: Data Does Not Replace Geometry. Robotics and Autonomous Systems. (Preprint)

Research Objective:

This research paper investigates the feasibility and effectiveness of using purely data-driven models, specifically those derived from the Extended Dynamic Mode Decomposition (EDMD) method within a Koopman operator framework, for controlling nonholonomic mobile robots in a model predictive control (MPC) setting. The study aims to determine if data-driven models can replace traditional first-principles models for precise control of such robots, particularly for setpoint stabilization tasks.

Methodology:

The researchers employ EDMD to learn surrogate models of both kinematic and second-order dynamics representations of a differential-drive robot from real-world experimental data. They then integrate these learned models into an MPC framework, comparing the performance of controllers using different cost functions: a tailored mixed-exponents cost function derived from the system's sub-Riemannian geometry and conventional quadratic cost functions. The controllers are evaluated both in simulation and through hardware experiments, focusing on their ability to stabilize the robot at a desired setpoint.

Key Findings:

The study demonstrates that EDMD-based surrogate models can enable high-precision predictive control of nonholonomic robots in both simulation and hardware experiments when a cost function respecting the system's sub-Riemannian geometry is used. However, the research highlights a crucial limitation of purely data-driven approaches: they struggle to replace the need for understanding the underlying geometry of nonholonomic systems. Controllers using conventional quadratic cost functions, even when applied to learned models, fail to achieve reliable setpoint stabilization, particularly for maneuvers requiring complex motions like parallel parking.

Main Conclusions:

While data-driven methods like EDMD offer a promising avenue for efficient robot control, relying solely on data without considering the inherent geometric constraints of nonholonomic systems can lead to control failure. The study emphasizes that incorporating geometric insights into the control design, even when using data-driven models, is crucial for achieving reliable and precise control of nonholonomic robots.

Significance:

This research provides valuable insights into the capabilities and limitations of data-driven control methods for nonholonomic robots. It highlights the importance of combining data-driven learning with domain-specific knowledge, particularly geometric understanding, for designing robust and high-performance controllers for these systems.

Limitations and Future Research:

The study focuses on the specific case of a differential-drive robot. Future research could explore the generalizability of these findings to other types of nonholonomic systems with higher degrees of nonholonomy and more complex dynamics. Additionally, investigating methods for automatically incorporating geometric constraints into data-driven control frameworks could further enhance the performance and reliability of such approaches.

Stats

The researchers used a control sampling time of 0.1 seconds for the EDMD-based predictive controller in the kinematic mobile robot simulations.
The prediction horizon for the simulations was set to 60 steps.
For the statistical analysis, 1000 random initial robot poses were used to evaluate the performance of different controller setups.
In the hardware experiments with the kinematic mobile robot, a sampling time of 50 milliseconds was used for data acquisition.
Only five trajectories per input basis were used to train the EDMD model for the hardware experiments, highlighting the data efficiency of the approach.

Quotes

Key Insights Distilled From

Data-Driven Predictive Control of Nonholonomic Robots Based on a Bilinear Koopman Realization: Data Does Not Replace Geometry

by Mario Rosenf... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.07192.pdf

Deeper Inquiries

How can the insights from this research be applied to develop data-driven control methods for more complex robotic systems with higher degrees of freedom and nonholonomic constraints?

This research provides valuable insights that can be extended to develop data-driven control methods for more complex robotic systems:
1. Importance of Geometric Awareness:

Key Takeaway: Even for data-driven methods, understanding the underlying geometry imposed by nonholonomic constraints is crucial. Simply using generic cost functions like quadratic costs, even in lifted observable spaces, can lead to suboptimal or unstable control.
Application to Complex Systems: For robots with higher degrees of freedom and more complex nonholonomic constraints, this understanding becomes even more critical.

Analyze Constraint Structure: Carefully analyze the specific nonholonomic constraints of the system.
Geometrically-Inspired Cost Functions: Design cost functions that respect these constraints. This might involve:

Deriving pseudo-norms or metrics tailored to the system's sub-Riemannian geometry.
Using insights from motion planning algorithms designed for nonholonomic systems.
2. Dictionary Design for EDMD:

Key Takeaway: A well-chosen dictionary of observable functions is essential for accurate EDMD model learning.
Application to Complex Systems:

Exploit System Structure: Leverage any known structure or invariances in the system dynamics (e.g., symmetries, conservation laws) to guide the selection of observables.
Incorporate Domain Knowledge: Incorporate domain expertise to include relevant features that might not be immediately apparent from raw data.
Systematic Dictionary Learning: Explore techniques for automated or semi-automated dictionary learning, potentially using methods like deep learning to discover effective representations.
3. Combining Data-Driven and Model-Based Approaches:

Key Takeaway: This research highlights the limitations of purely data-driven approaches and suggests the potential of hybrid methods.
Application to Complex Systems:

Data-Augmented Control: Use data to learn and refine specific aspects of a model-based controller, such as:

Improving the accuracy of a nominal dynamic model.
Adapting to uncertainties or variations in system parameters.

Constraint Learning: If the exact form of the nonholonomic constraints is unknown, explore using data to learn these constraints.
4. Practical Considerations:

Data Efficiency: For complex systems, acquiring sufficient and representative data can be challenging. Explore methods for:

Active data acquisition to target informative regions of the state space.
Transfer learning to leverage data from similar systems.

Computational Complexity: EDMD-based control involves solving an optimization problem at each control step. Investigate techniques for:

Efficient optimization algorithms.
Model reduction to reduce the complexity of the learned model.

Could incorporating a model of the system's nonholonomic constraints directly into the EDMD learning process improve the performance of the data-driven controller with quadratic cost functions?

Incorporating a model of the system's nonholonomic constraints directly into the EDMD learning process has the potential to improve the performance of the data-driven controller, even with quadratic cost functions. Here's how:
1. Constraint-Aware Dictionary:

Direct Encoding: Instead of relying solely on generic observables, include functions that explicitly represent the nonholonomic constraints in the EDMD dictionary. For example:

For a differential-drive robot, include observables like sin(θ)v - cos(θ)u = 0 (representing the no-sideways-motion constraint).

Benefits: This forces the EDMD model to learn a representation that inherently respects the constraints.
2. Constraint Projection:

Project Onto Feasible Manifold: After each prediction step in the EDMD model, project the predicted state onto the manifold of feasible states defined by the nonholonomic constraints.
Geometrically Consistent Predictions: This ensures that the predictions always satisfy the constraints, even if there are small errors in the learned dynamics.
3. Constrained Optimization:

Formulate Constrained EDMD: Modify the EDMD learning problem itself to incorporate the constraints. This might involve:

Adding penalty terms to the EDMD loss function for violating the constraints.
Using constrained optimization techniques to find the Koopman operator approximation that best fits the data while satisfying the constraints.
Potential Advantages:

Improved Accuracy: By explicitly accounting for the constraints, the EDMD model can learn a more accurate representation of the system's feasible dynamics.
Quadratic Cost Feasibility: It might be possible to use quadratic cost functions effectively, as the constraint enforcement would handle the geometric inconsistencies.
Challenges:

Constraint Complexity: Incorporating complex nonholonomic constraints into the EDMD framework might lead to more challenging optimization problems.
Computational Overhead: Constraint projection or constrained optimization can increase the computational burden of the learning and control algorithms.

How might the increasing availability of large and diverse datasets, coupled with advancements in machine learning techniques, influence the future of data-driven control for complex systems and potentially bridge the gap between purely data-driven and model-based approaches?

The increasing availability of data and advancements in machine learning are poised to significantly influence data-driven control, potentially bridging the gap between purely data-driven and model-based approaches:
1. Improved Model Learning:

Richer Models: Large datasets enable the learning of more complex and expressive models, capturing subtle nonlinearities and interactions within systems.
Deep Learning Integration: Deep learning architectures, particularly recurrent neural networks (RNNs) and transformers, are well-suited for modeling dynamic systems. They can learn complex temporal dependencies and generalize well to unseen data.
2. Data-Driven System Identification:

Accurate Nominal Models: Machine learning can be used to learn accurate nominal models from data, even for systems where deriving first-principles models is challenging.
Uncertainty Quantification: Techniques like Bayesian deep learning and Gaussian processes can provide not only point estimates of system dynamics but also uncertainty bounds, which are crucial for robust control.
3. Hybrid Control Architectures:

Data-Driven Model Adaptation: Machine learning can be used to adapt and refine existing model-based controllers online, using real-time data to account for uncertainties, disturbances, or changes in system behavior.
Learning-Based Control Policy Optimization: Reinforcement learning (RL) algorithms can learn optimal control policies directly from data, potentially surpassing the performance of traditional controllers in complex, high-dimensional environments.
4. Towards Data-Driven Control Design:

Automated Controller Synthesis: Machine learning could potentially automate parts of the control design process, such as:

Tuning controller parameters.
Selecting appropriate control architectures.

Data-Driven Stability Analysis: Emerging techniques aim to use data to directly analyze the stability properties of closed-loop systems, reducing the reliance on analytical methods.
Bridging the Gap:

Data-Augmented Model-Based Control: Data will be used to enhance and complement model-based approaches, leading to more accurate, adaptive, and robust controllers.
Model-Guided Data-Driven Control: Insights from physics-based models and control theory will guide the design of data-driven methods, ensuring stability, safety, and interpretability.
Challenges and Opportunities:

Data Quality and Representativeness: Ensuring high-quality, diverse, and representative datasets for complex systems remains a challenge.
Scalability and Computational Efficiency: Developing scalable and computationally efficient algorithms for learning and control is crucial.
Safety and Verifiability: Addressing safety concerns and ensuring the verifiability of data-driven controllers are paramount, especially for safety-critical applications.
The future of data-driven control lies in a synergistic integration of data-driven and model-based techniques, leveraging the strengths of both approaches to develop more intelligent, adaptable, and high-performance control systems.