The agility and dexterity of robotic platforms places increasing demands on understanding and computing their intrinsic and interaction dynamics for design, analysis, control and autonomy. However, a major impediment has been the complexity of these dynamics for moderate to large degree of freedom robots - especially ones subject to time-varying environments and constraints. We describe the Spatial Operator Algebra (SOA) methodology that exploits underlying structure to develop analytical and computational methods to overcome the dynamics complexity. SOA defines a family of mathematical operators that can be used to concisely express and compute at low-cost a large family of key robotics quantities such as Jacobians, the mass matrix, operational space inertias, sensitivities, flexible body and contact dynamics etc. Using the constraint-embedding approach, the SOA methods can be extended from tree to closed-chain mechanisms. We will present an overview of SOA, and the SOA-based DARTS computational architecture and software for robotic platforms. We will highlight recent advancements, and describe applications for robot modeling and analysis, and its use as an embedded modeling layer for multi-arm robot manipulation and mobility. We will also describe the DARTS based Dshell modeling and simulation toolkit which is in use by a number of NASA and other projects including Mars landers, Mars Helicopter, launch vehicles, wheeled and legged platforms and under-actuated robots.