The 39-year-old rocker and 27-year-old actress have been married for nearly six years. They announced their separation in August.
According to papers filed Friday in Superior Court, Robinson is seeking shared custody of the couple's son, Ryder Russell Robinson, who will turn 3 in January.
Hudson, daughter of Goldie Hawn, was nominated for a supporting-actress Oscar for her role as rock groupie Penny Lane in 2000's "Almost Famous." Other screen credits include "You, Me and Dupree" and "How to Lose a Guy in 10 Days."
A call to Hudson representative Brad Cafarelli for comment after business hours Friday was not immediately returned.
Robinson is the lead singer of the Black Crowes, a rock group that released its first CD, "Shake Your Money Maker," in 1990. The band's hits include "Hard to Handle" and "She Talks to Angels."