3D 视觉

We place the attention token on the group: a token is an element $g_i$ of a matrix Lie group $G$ -- a bare transformation, with no feature payload and no external action $ρ(g)$ carrying it. To our knowledge this is the first attention construction whose tokens are bare matrix Lie group elements: their score is the closed-form algebra norm of the relative pose rather than a learned kernel, and it reaches the affine full-frame groups that every irrep- or surjective-exp-based method must exclude. We call it Lie-Algebra Attention. Once tokens are group elements, the rest follows with none of the usual representation-theoretic machinery. The relative geometry of a pair is canonical, $g_i^{-1} g_j$, so the pairwise invariant $w_{ij} = \log(g_i^{-1} g_j)$ is intrinsic rather than designed; equivariance under the diagonal $G$-action is tautological, and the cocycle condition holds automatically. The attention score is the negative squared algebra norm, $s_{ij} = -\|\log(g_i^{-1} g_j)\|_λ^2/τ$: the canonical proximity kernel under a block-weighted Frobenius inner product, with no irreducible representations, spherical harmonics, Clebsch-Gordan products, or learned kernel. The construction applies to any matrix Lie group on a chosen logarithm chart containing the relative poses, including the non-compact non-abelian affine groups with scale and shear that no vector-token attention method reaches: neither the irrep tradition nor surjective-exp methods. Three sequence-completion experiments, on SE(2), SO(3), and Aff(2), bear this out: the closed-form score matches a learned MLP kernel on the same invariant and outperforms it on SE(2), using 50 to 80x fewer score parameters, while a vector-token baseline breaks invariance by five to twelve orders of magnitude.

Robot learning has advanced rapidly in learning control, but learning the physical body of a robot remains much more difficult because jointly searching over design and control creates a very large combinatorial problem. Here, we present a data-driven framework for generating robot hands from human demonstrations. Instead of learning a complex controller together with each candidate design, we generate robot hand designs using the same simple control policy used after fabrication: matching fingertip positions through inverse kinematics. Using more than 4 million frames of human fingertip motion from everyday manipulation, our algorithm optimizes tree-structured robot hands to reproduce desired target motions. The framework produced both a 6-degree-of-freedom (DoF) general-purpose hand and lower-DoF task-specific hands with spatial four-bar mimic joints. To accelerate the search over designs, we trained a reinforcement-learning (RL) actor to propose good hand designs and joint angles, reducing search time from hours to minutes. We fabricated the mechanisms directly as one-piece articulated structures with print-in-place joints. In real-world experiments, the 6-DoF hand achieved highly accurate teleoperated fingertip tracking better than available commercial robot hands, whereas the specialized 3-DoF hands reproduced structured human and synthetic trajectories with reduced mechanical complexity. These results showed that large-scale human motion data can be used not only to train robot controllers but also as a reference for optimizing and generating the physical embodiment of robots.

We study a novel type of mechanical structures, composed of spatial four-bar linkages, that are bistable, that is, they allow for two distinct configurations. These structures have an interpretation as quad nets in the Study quadric which we use to prove existence of assemblies with an unbounded number of links and joints. We propose a purely geometric construction of such objects, starting from infinitesimally flexible quad nets in Euclidean space and applying Whiteley de-averaging. This point of view situates the problem within the broader framework of discrete differential geometry and enables the construction of bistable structures from well-known classes of quad nets, such as discrete minimal surfaces. In contrast to many other construction methods for bistable structures, our approach does not rely on numerical optimization and it allows for simple control of relevant geometric parameters such as axis positions and snap angles.