def connected_children_names(obj, bone_name):
    """ Returns a list of bone names (in order) of the bones that form a single
        connected chain starting with the given bone as a parent.
        If there is a connected branch, the list stops there.
    """
    bone = obj.data.bones[bone_name]
    names = []
    while True:
        connects = 0
        con_name = ''
        for child in bone.children:
            if child.use_connect:
                connects += 1
                con_name = child.name
        if connects == 1:
            names += [con_name]
            bone = obj.data.bones[con_name]
        else:
            break
    return names
