Adding class to nested UL in menu
You might get into the sub menu element with the ['#below'] object. not tested.

&lt;?php

   /**
    * Implements theme_menu_link__[MENU NAME].
    */
   function YOURTHEME_menu_link__main_menu(&amp;$variables) {       


        $element = $variables['element'];
        $sub_menu = '';         

        $element['#attributes']['class'][] = 'dropdown mm-menu-fw';        
        $element['#localized_options']['attributes']['class'][] = 'url-link';

        if ($element['#below']) 
        {
         $sub_menu = $element['#below']['#attributes']['class'][]= 'NEED-TO-ADD-CLASS';
         drupal_render($sub_menu);
        }

       $output = l($element['#title'], $element['#href'], $element['#localized_options']);

       return '&lt;li' . drupal_attributes($element['#attributes']) . '&gt;' . $output . $sub_menu . "&lt;/li&gt;\n";

 }
?&gt;