The control of robotic systems in complex, shared collaborative workspaces presents significant challenges in achieving robust performance and safety when learning from experienced or simulated data is employed in the pipeline. A primary bottleneck is the reliance on coordinate-dependent models, which leads to profound data inefficiency by failing to generalize physical interactions across different frames of reference. This forces learning algorithms to rediscover fundamental physical principles in every new orientation, artificially inflating the complexity of the learning task. This paper introduces a novel framework that synergizes a coordinate-free, unreduced multibody dynamics and kinematics model based on tensor mechanics with a Data-Assisted Control (DAC) architecture. A non-recursive, closed-form Newton-Euler model in an augmented matrix form is derived that is optimized for tensor-based control design. This structure enables a principled decomposition of the system into a structurally certain, physically grounded part and an uncertain, empirical, and interaction-focused part, mediated by a virtual port variable. Then, a complete, end-to-end tensor-invariant pipeline for modeling, control, and learning is proposed. The coordinate-free control laws for the structurally certain part provide a stable and abstract command interface, proven via Lyapunov analysis. Eventually, the model and closed-loop system are validated through simulations. This work provides a naturally ideal input for data-efficient, frame-invariant learning algorithms, such as equivariant learning, designed to learn the uncertain interaction. The synergy directly addresses the data-inefficiency problem, increases explainability and interpretability, and paves the way for more robust and generalizable robotic control in interactive environments.