Dimensionless Policies based on the Buckingham $π$ Theorem: Is This a Good Way to Generalize Numerical Results?

TL;DR

This work addresses the challenge of generalizing numerically learned motion-control policies across different physical systems. It exploits the Buckingham $\pi$ theorem to cast policies in dimensionless form, reducing parameter dependence and enabling exact transfer between dimensionally similar contexts through well-defined input/output scaling. The authors derive transformation frameworks, demonstrate the theory on pendulum swing-up and car motion on a slippery surface via dynamic-programming results, and connect these ideas to closed-form solutions such as LQR and computed-torque. The findings suggest a promising transfer-learning tool for numerical methods (e.g., DP and RL), with regime-based extensions to relax strict similarity and potential applicability to data-efficient learning across heterogeneous robotic systems.

Abstract

The answer to the question posed in the title is yes if the context (the list of variables defining the motion control problem) is dimensionally similar. This article explores the use of the Buckingham $π$ theorem as a tool to encode the control policies of physical systems into a more generic form of knowledge that can be reused in various situations. This approach can be interpreted as enforcing invariance to the scaling of the fundamental units in an algorithm learning a control policy. First, we show, by restating the solution to a motion control problem using dimensionless variables, that (1) the policy mapping involves a reduced number of parameters and (2) control policies generated numerically for a specific system can be transferred exactly to a subset of dimensionally similar systems by scaling the input and output variables appropriately. Those two generic theoretical results are then demonstrated, with numerically generated optimal controllers, for the classic motion control problem of swinging up a torque-limited inverted pendulum and positioning a vehicle in slippery conditions. We also discuss the concept of regime, a region in the space of context variables, that can help to relax the similarity condition. Furthermore, we discuss how applying dimensional scaling of the input and output of a context-specific black-box policy is equivalent to substituting new system parameters in an analytical equation under some conditions, using a linear quadratic regulator (LQR) and a computed torque controller as examples. It remains to be seen how practical this approach can be to generalize policies for more complex high-dimensional problems, but the early results show that it is a promising transfer learning tool for numerical approaches like dynamic programming and reinforcement learning.

Figures (29)

• Figure S1: Shared dimensionless policy for inverted pendulums: Under some conditions various dynamic systems will share the same optimal policy up to scaling factors that can be found based on a dimensional analysis.
• Figure S2: The policy $\pi$ is a feedback law that also includes problem parameters as additional arguments.
• Figure S3: A feedback law $f$ is a slice of the higher dimensional policy mapping $\pi$ in a specific context.
• Figure S4: Isolating the dimensionless knowledge in a policy allow its exact transfer to any dimensionally similar motion control problem.
• Figure S5: Example of dimensionally similar contexts subsets that are lines in a plane ($m=2$ and $d=1$). Context $c_a$ is dimensionally similar to $c_b$ but not to $c_c$ or $c_d$.

Theorems & Definitions (12)

• Definition 1
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• Proposition 1
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