Duchess does it when she wants a belly rub or when she doesn't want to be picked up. If I call her and she doesn't feeling like coming, she will very slowly walk over with her head down and about 2ft from me she will stop, drop and roll all in slow motion.
She does this if she thinks I want to pick her up.
I did have a rescue yorkie who was submissive when I first brought her home and she would roll on her back and start shaking. It broke my heart. She was used for breeding and was neglected
.
But within a few months, lots of love and patience. she went from being shy and submissive to a diva. Rest in Peace Brandi