Mxsmanic wrote
How do autopilots make coordinated turns even when they cannot control
the rudder?

Swept wing and some straight wing aircraft have independent Yaw Damper(s)
that control the rudder(s). Their primary function is to control (prevent)
dutch roll. They operate with the autopilot on or off.

From Wikipedia, the free encyclopedia

A yaw damper is a device used on many aircraft (usually jets and
turboprops) to damp (reduce) the rolling and yawing oscillations due to
Dutch roll mode[1]. It involves yaw rate sensors and a processor that
provides a signal to an actuator connected to the rudder. The use of the
yaw damper helps to provide a better ride for passengers, and on some
aircraft is a required piece of equipment to ensure that the aircraft
stability remains within certification values.

The term is often mis-pronounced as a "yaw dampener".



Bob Moore