Toward An Analytic Theory of Intrinsic Robustness for Dexterous Grasping

Abstract

Conventional approaches to grasp planning require perfect knowledge of an object's pose and geometry. Uncertainties in these quantities induce uncertainties in the quality of planned grasps, which can lead to failure. Classically, grasp robustness refers to the ability to resist external disturbances after grasping an object. In contrast, this work studies robustness to intrinsic sources of uncertainty like object pose or geometry affecting grasp planning before execution. To do so, we develop a novel analytic theory of grasping that reasons about this intrinsic robustness by characterizing the effect of friction cone uncertainty on a grasp's force closure status. We apply this result in two ways. First, we analyze the theoretical guarantees on intrinsic robustness of two grasp metrics in the literature, the classical Ferrari-Canny metric and more recent min-weight metric. We validate these results with hardware trials that compare grasps synthesized with and without robustness guarantees, showing a clear improvement in success rates. Second, we use our theory to develop a novel analytic notion of probabilistic force closure, which we show can generate unique, uncertainty-aware grasps in simulation.

Figures (6)

• Figure 1: This work develops an analytic theory of uncertainty-aware grasping, which enables an analysis of the uncertainty tolerance of classical grasp metrics, and the development of new, probabilistic approaches. Above we plot a grasp obtained by optimizing our proposed probabilistic metric, PONG, on a toy example with varying surface normal uncertainty. While there exist many robust grasps for the true geometry, the grasp synthesized using our probabilistic metric places the fingertips in the minimum-uncertainty region.
• Figure 2: (a) A set of nominal basis wrenches and their hull $\mathcal{C}_{\bar{w}}$ with a Ferrari-Canny metric of $\varepsilon$. (b) The value of $\varepsilon$ is sensitive to some perturbations (e.g., $w_2$) but not others (e.g., $w_1$). We derive guarantees on the size of allowable perturbations such that $0\in\mathcal{C}_w$. (c) The guarantees from Theorem \ref{['thm:containment']}. The shaded regions $\mathcal{S}_i$ indicate areas where $w_i$ may lie, independent of other basis wrenches, while still guaranteeing $0\in\mathcal{C}_w$. Note how the $\mathcal{S}_i$ are anisotropic. (d) The Ferrari-Canny metric $\varepsilon$ also provides (strictly worse) guarantees as a result. Note this cartoon shows a planar wrench space, which is actually $\mathbb{R}^6$.
• Figure 3: Left. Hardware setup. We synthesized grasps on eight objects for an Allegro hand mounted on a Franka Research 3. From left to right: 3D-printed part, goblet, box, mug, cube, bottle, conditioner, and apple. We employed an eye-in-hand setup with a Zed camera to capture images used to train a NeRF prior to grasping. Only monocular RGB was used. Right. Representative FRoGGeR-synthesized grasps.
• Figure 4: Hardware results. FRoGGeR-synthesized grasps are of higher quality and have a much higher success rate (34/40) than the baseline, which only ensures force closure and does not optimize for grasp quality, resulting in many low-quality grasps and a lower success rate (21/40). This plot also confirms the validity of Theorem \ref{['thm:min_weight_bound']}, as a linear lower bound can be seen.
• Figure 5: More examples of grasps on a synthetically-uncertain spherical manipuland. The surface normal tangent covariance at each point on the sphere is (proportional to) an isotropic monotonically-increasing function of the real part of the (2,4)-spherical harmonic. The synthesized grasps tend to avoid placing the fingertips on the red uncertain regions while remaining kinematically feasible. They were synthesized in 6.47 and 4.80 seconds respectively.

Theorems & Definitions (17)

• Theorem 1
• proof
• Corollary 1
• proof
• Definition 1
• Theorem 2
• Lemma 1
• proof
• proof
• Lemma 2