2DOF

2DOF, short for two degrees of freedom, is a term used in physics and engineering to describe a system whose configuration can be specified by two independent generalized coordinates. The degrees of freedom correspond to independent directions in which the system can move or be reconfigured. In practice, two degrees of freedom may be realized as two rotational coordinates, two translational coordinates, or a combination thereof. The concept is often contrasted with systems that have more or fewer coordinates needed to specify state.

In mechanical systems, a common 2DOF realization is a planar two-link robotic arm, also called a 2R manipulator, with two rotary joints. A point at the end-effector has Cartesian position (x, y) determined by joint angles q1 and q2 and link lengths L1, L2. For a planar 2R arm, x = L1 cos q1 + L2 cos(q1+q2); y = L1 sin q1 + L2 sin(q1+q2). Other 2DOF realizations include two independently actuated linear actuators guiding a body in the plane. In general, not all DOF need to be actuated; some may be passive and constrained by joints or gears.

Equation of motion for a 2DOF mechanical system can be derived with Lagrangian mechanics, yielding a coupled set of second-order differential equations in the generalized coordinates. In control theory, a 2-DOF controller is a design that uses separate paths for reference tracking and disturbance rejection, potentially providing improved setpoint accuracy and robustness compared with single-DOF feedback controllers.

Applications of 2DOF systems appear in robotics, vehicle suspensions, biomechanics, and vibration isolation. The analysis emphasizes kinematic reach, workspace, singularities, and dynamic coupling, while design considerations balance actuation, control bandwidth, and mechanical complexity.