ALT Finite-dimensional Koopman models enable efficient linear prediction and control of nonlinear robotic systems. However, models learned purely from trajectory data may violate the energetic structure of the underlying mechanics, producing predictions that exhibit artificial energy growth and diverge under recursive propagation. This work presents a structure-preserving Koopman framework for Euler-Lagrange systems built on generalized-momentum coordinates. The momentum transformation exposes the mechanical actuation as a known, state-independent port, which is preserved explicitly in the lifted dynamics. A structure-constrained neural architecture is developed to jointly learn the lifting functions and a port-Hamiltonian Koopman generator, rendering the learned dynamics passive by construction rather than through penalty terms or post-hoc projection. A Cayley-midpoint discretization further preserves the corresponding storage-dissipation balance exactly in discrete time. These properties

ALT Humanoid robots often execute motion commands through whole-body controllers (WBCs) that track targets while maintaining balance and stability. However, most WBCs are blind to scene geometry, which can lead to collisions from imperfect target motions that are geometrically unsafe due to perception, planning, or teleoperation errors. We propose RECAL, a Robot–Environment Cross-Attention Layer that wraps a blind WBC to trade off target tracking against collision avoidance using external scene geometry. RECAL supports collision-aware tracking of floating-base and end-effector commands, including collision avoidance for held objects. It represents the robot, held objects, and environment as point clouds, using cross-attention between robot/object points and the environment to produce geometry-aware control features. In simulation, RECAL improves collision avoidance while preserving target-tracking performance across frozen-arm and adaptive-arm locomotion, object-carrying, and standing-mani