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Efficient calculation of the inverse dynamics of robotic mechanisms serves as a basis for a variety of control schemes. Classical algorithms for rigid multibody systems do not apply when more realistic modeling is required. The computational treatment of the emerging non-linear equations augmented by drivetrain dynamics and elasticity imposes various challenges on the algorithmic solutions. This paper presents a method for recursive calculation of the inverse dynamics of real-world robots considering elasticity and gyroscopic effects introduced by the drivetrains. A study of the equations of motion elucidates the demand for analytical derivatives of the dynamics up to a maximum order depending on the number of joints. This problem is solved for the first time by means of higher-order time derivatives of spatial operators proposed by the spatial operator algebra.