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This paper introduces a formulation for Modelica MultiBody models in which angular coordinates in closed-loop systems are replaced by direction cosine coordinates, and the orthonormality violations of the introduced direction cosine coordinates are controlled by Baumgarte-type correction terms at the derivative level. Thus, the position constraints are transformed into linear systems, while the acceleration-level and velocity-level equations remain linear. In favorable cases, the Jacobian becomes constant and dynamic switching of dummy states at the position level is no longer required. Numerical experiments indicate good stability and modest speed improvements.