Multiscale Deformable Objects Manipulation Via Wavelet-Decomposed Boundary Element Method

Robotic deformable object manipulation (DOM) faces critical challenges in industrial and medical applications due to under-actuation, unpredictable deformation, and partial observability. Model-free methods often suffer from unstable Jacobians arising from ill-conditioned observations, while physics-based models typically depend on precise parameters and volumetric meshing, limiting their real-time practicality. We propose a wavelet-boundary element method (BEM) framework that leverages multiscale wavelet descriptors to control 3D deformations directly from efficient feedback modalities, such as contours and curves. By coupling wavelets with BEM, we derive an analytical deformation Jacobian that functions independently of material stiffness (e.g., Young's modulus), relying solely on an online-calibrated Poisson's ratio. This mesh-free formulation significantly enhances real-time performance and robustness against sensor occlusion. Validated in simulation and on the da Vinci Research Kit (dVRK) with phantom and ex vivo animal tissue, our method achieves millimetre-level accuracy. Comparative studies against Fourier-based, model-free, and online finite element method (FEM) approaches demonstrate superior stability and computational efficiency. Notably, our framework achieves convergence speeds significantly faster than online FEM by avoiding volumetric computations, while resolving ill-conditioning through spatial-frequency localization. This work advances deformable object manipulation in unstructured environments, particularly in surgical robotics, where stability under partial observability is essential. Project page: https://junleihu.github.io/projects/dwtbem/.