Flexible Joint

A joint with noticeable elasticity between the motor and the link, so the motor's angle doesn't equal the link's actual angle.

A flexible joint is one where non-negligible elasticity exists between the motor and the link it drives. The most common source is a harmonic drive: its flexspline achieves near-zero backlash by deforming under load, which also introduces torsional elasticity. Spong's classic 1987 model treats such a joint as a torsional spring connecting the motor side and the link side, τ = K(θ − q): θ is the motor's angle referred to the output side, q is the link's actual angle, K is the joint stiffness, and τ is the torque transmitted across the spring. This adds an extra set of states per joint and raises the order of the dynamics, causing two problems: the motor encoder no longer reads the link's true angle, leaving a position offset at the end effector, and the low-frequency vibration mode this creates makes high-gain control prone to oscillating. Some designs introduce flexibility on purpose, such as series elastic actuators, which use the spring's deflection to measure force and improve safety during collisions.

ExampleWhen a lightweight robot arm carrying a payload stops suddenly, the end effector often sways back and forth briefly before settling — the flexibility introduced by the harmonic drive is one cause.

Also called

Elastic Joint, Joint Flexibility

Related

Strain Wave Gear (Harmonic Drive) · Series Elastic Actuator (SEA) · Stiffness · Compliance · Actuator Modeling (Actuator Network) · Joint Torque Sensor

Sources

Spong: Modeling and Control of Elastic Joint Robots (J. Dyn. Sys., Meas., Control 1987)
Lynch & Park, Modern Robotics（8.9.5 Joint and Link Flexibility）