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This paper delves into the investigation of flatness-based active damping control for hybrid vehicle transmissions. The main objective is to improve the current in-production controller performances without the need for additional sensors or observers. The primary goals include improving torque setpoint tracking, enhancing robustness margins, and ensuring zero steady-state torque correction. The investigation proceeds in several steps: Initially, both the general differential flatness property and the identification of flat outputs in linear dynamical systems are revisited. Subsequently, the bond graph formalism is employed to deduce straightforwardly the dynamical equations of the system. Next, a new flat output of the vehicle transmission is identified and utilized to formulate the trajectory tracking controller to align with the required control objectives and to fulfill the system constraints.