Stiffness Mapping of Deformable Objects Through Supervised Embedding and Gaussian Process Regression

Abstract

The stiffness map of a deformable object stores information about that object's surface compliance. Thus, through a stiffness map, we gain insight into the physical properties of that object. Depending on the object, an understanding of stiffness has applications ranging from localizing tumors for surgery to grasping policies in manipulation. However, generating a stiffness map or stiffness mapping is challenging as even with information about the thickness of the underlying material, the object's physical geometry, and the composition of the material itself, it can be enormously computationally complex to model an object's surface compliance. Therefore it is often necessary to generate the stiffness map of an object by direct sampling through palpation or applying a force and measuring the subsequent displacement. Then by densely palpating over every point on an object's surface, a time-consuming process task for a very fine sampling, a stiffness map for an entire object can be generated. Alternatively, it is also possible to only palpate a subset of points from the surface. Then by applying regression, the sampled stiffness data can be used to estimate a function between points on a given surface (known as inputs or predictors) and the stiffness data. Finally, using this function, an object's entire stiffness map can be extrapolated.
Previous work on stiffness mapping has specifically been interested in Gaussian Process Regression (GPR) due to its ability to estimate uncertainty about its predicted stiffness values. This uncertainty estimate helps direct the sampling of stiffness data, guiding where to palpate, and ideally leading to a more accurate estimate of an object's stiffness map with fewer data points. A key component of GPR is its kernel or covariance function which measures the similarity among our regression's predictors. For stiffness mapping, we assume that our predictor's covariance or points on the object's surface are correlated as a function of geodesic distance; points nearby on the surface of an object have similar stiffness values. However, GPR requires a smooth kernel function and thereby a smooth distance function between inputs. Consequently, stiffness mapping through GPR struggles with surfaces with a non-smooth geodesic distance function. Notably, this causes difficulty when the surface of our object is modeled using a discrete representation, for example a mesh, which is a common representation used in many fields, including robotics. Due to the mesh's discrete nature, the resulting geodesic distances between points are non-smooth, leading to an invalid kernel function and thus a poor fit of our data.
The main contribution of this thesis is a new method of GPR for fitting a function between predictors sampled from a surface and an associated surface-dependent scalar distribution. The main focus of this work is using GPR in tandem with an embedding or mapping of our predictors into a higher dimensional Euclidean space. By mapping our predictors into a Euclidean space we can use the smooth, Euclidean distance metric to measure the similarities between data points, thereby ensuring a smooth kernel function. Specifically, this thesis presents a supervised learning approach for constructing embeddings where the embedded surface is constructed solely based on its observed stiffness data.
We evaluate our supervised method for embedding by testing its ability to build a map of an object's stiffness using a series of synthetic and real-world stiffness data sets. Then we compare these results against alternative techniques, such as unsupervised embedding methods, which have been the focus of research to this point. Unsupervised embedding methods construct an embedding as a function of some predetermined cost function instead of the observed data and is applied to the predictors as a preprocessing step before regression occurs. We find that our supervised embedding better predicts our underlying stiffness distributions over existing unsupervised embedding techniques. Although this work focuses on stiffness mapping, it can be applied to fit any data with an underlying function mapping between points sampled from a manifold to some scalar value. We hope to explore other distributions with a similar relationship in our future work, such as mapping an object's local thermal or friction information.