Planning for quasi-static manipulation tasks via an intrinsic haptic metric: a book insertion case study

TL;DR

This work reframes quasi-static manipulation as planning on an implicit equilibrium manifold defined by the manipulation potential $W({\boldsymbol{z}}, {\boldsymbol{u}})$, enabling contact-aware planning through an intrinsic haptic metric. By representing objects with differentiable proxies (superellipses) and solving for contact points along the equilibrium manifold, the approach automatically evolves contact phases and discovers wedging-in strategies without manual phase design. An adaptive ODE lifts control trajectories onto $\mathcal{M}^{eq}$, while a haptic obstacle constrains motion to a single manifold branch, ensuring stable, differentiable optimization. The framework is validated on a crowded bookshelf insertion task using DMPs and black-box optimization to learn policies that reduce contact forces and robustly handle variations in initial pose and external stiffness, with potential applicability to general rigid-body manipulation.

Abstract

Contact-rich manipulation often requires strategic interactions with objects, such as pushing to accomplish specific tasks. We propose a novel scenario where a robot inserts a book into a crowded shelf by pushing aside neighboring books to create space before slotting the new book into place. Classical planning algorithms fail in this context due to limited space and their tendency to avoid contact. Additionally, they do not handle indirectly manipulable objects or consider force interactions. Our key contributions are: i) reframing quasi-static manipulation as a planning problem on an implicit manifold derived from equilibrium conditions; ii) utilizing an intrinsic haptic metric instead of ad-hoc cost functions; and iii) proposing an adaptive algorithm that simultaneously updates robot states, object positions, contact points, and haptic distances. We evaluate our method on a crowded bookshelf insertion task, and it can be generally applied to rigid body manipulation tasks. We propose proxies to capture contact points and forces, with superellipses to represent objects. This simplified model guarantees differentiability. Our framework autonomously discovers strategic wedging-in policies while our simplified contact model achieves behavior similar to real world scenarios. We also vary the stiffness and initial positions to analyze our framework comprehensively. The video can be found at https://youtu.be/eab8umZ3AQ0.

Figures (10)

• Figure 1: Inserting a book into a crowded shelf filled with books (blue), where the space is insufficient for a new book (orange).
• Figure 2: Adaptive ODE allows to lift control ${\boldsymbol{\mathbf{u}}}(t)$ onto $\mathcal{M}^{eq}$. ${\boldsymbol{\mathbf{u}}}$ is extended by moving along $\dot{{\boldsymbol{\mathbf{u}}}}$ and updating ${\boldsymbol{\mathbf{z}}}$ on the $\mathcal{M}^{eq}$ via Eq. \ref{['eq:adaptive ODE']}. Blue arrow denotes ${\boldsymbol{\mathbf{z}}}$ linear approximation as the variation of ${\boldsymbol{\mathbf{u}}}$, red arrow represents Newton-Raphson ‘infinitesimal’ adjustment. Policy can be terminated by haptic obstacle.
• Figure 3: Behavior of proxy on superellipse.
• Figure 4: A book need to be inserted to a narrow shelf, where the remaining space is not enough for insertion. The manipulated book ${\boldsymbol{\mathbf{z}}}_b$ is controlled with impedance control policy ${\boldsymbol{\mathbf{u}}}$. Contact interaction is captured by proxy, and the resistance among the books on the bookshelf is captured by external stiffness ${\boldsymbol{\mathbf{K}}}_i$.
• Figure 5: our BBO framework performs multiple rollouts with different DMP parameters, ${\boldsymbol{\mathbf{\Theta}}}^r$, calculating the corresponding cost (haptic distance) $\phi^r$ for each rollout. The optimal parameter, $\hat{{\boldsymbol{\mathbf{\Theta}}}}$, is then updated, and this process is repeated until the cost converges.