Despite the remarkable acceleration of robotic development through advanced simulation technology, robotic applications are often subject to performance reductions in real-world deployment due to the inherent discrepancy between simulation and reality, often referred to as the "sim-to-real gap". This gap arises from factors like model inaccuracies, environmental variations, and unexpected disturbances. Similarly, model discrepancies caused by system degradation over time or minor changes in the system's configuration also hinder the effectiveness of the developed methodologies. Effectively closing these gaps is critical and remains an open challenge. This work proposes a lightweight conformal mapping framework to transfer control and planning policies from an expert teacher to a degraded less capable learner. The method leverages Schwarz-Christoffel Mapping (SCM) to geometrically map teacher control inputs into the learner's command space, ensuring maneuver consistency. To demonstrate its generality, the framework is applied to two representative types of control and planning methods in a path-tracking task: 1) a discretized motion primitives command transfer and 2) a continuous Model Predictive Control (MPC)-based command transfer. The proposed framework is validated through extensive simulations and real-world experiments, demonstrating its effectiveness in reducing the sim-to-real gap by closely transferring teacher commands to the learner robot.

Normalizing flows are generative models that provide tractable density estimation by transforming a simple base distribution into a complex target distribution. However, this technique cannot directly model data supported on an unknown low-dimensional manifold, a common occurrence in real-world domains such as image data. Recent attempts to remedy this limitation have introduced geometric complications that defeat a central benefit of normalizing flows: exact density estimation. We recover this benefit with Conformal Embedding Flows, a framework for designing flows that learn manifolds with tractable densities. We argue that composing a standard flow with a trainable conformal embedding is the most natural way to model manifold-supported data. To this end, we present a series of conformal building blocks and apply them in experiments with real-world and synthetic data to demonstrate that flows can model manifold-supported distributions without sacrificing tractable likelihoods.

Dimensionality reduction is the process by which a set of data points in a higher dimensional space are mapped to a lower dimension while maintaining certain properties of these points relative to each other. One important property is the preservation of the three angles formed by a triangle consisting of three neighboring points in the high dimensional space. If this property is maintained for those same points in the lower dimensional embedding then the result is a conformal map. However, many of the commonly used nonlinear dimensionality reduction techniques, such as Locally Linear Embedding (LLE) or Laplacian Eigenmaps (LEM), do not produce conformal maps. Post-processing techniques formulated as instances of semi-definite programming (SDP) problems can be applied to the output of either LLE or LEM to produce a conformal map. However, the effectiveness of this approach is limited by the computational complexity of SDP solvers. This paper will propose an alternative post-processing algorithm that produces a conformal map but does not require a solution to a SDP problem and so is more computationally efficient thus allowing it to be applied to a wider selection of datasets. Using this alternative solution, the paper will also propose a new algorithm for 3D object classification. An interesting feature of the 3D classification algorithm is that it is invariant to the scale and the orientation of the surface.