Programmatically update Menu Item in Component in joomla - php

Question

I'm creating a component in Joomla, where among other things, can create and update menus, and I have successfully used this answer to create a new menu item.
https://joomla.stackexchange.com/questions/5104/programmatically-add-menu-item-in-component?newreg=1e7c576205354c0795a55607bc3e2508
$menuTable = JTableNested::getInstance('Menu');
// which menu you want to add to -
$menutype = 'thisismymenusname';
// this is heading menu item but what data you have and require will vary per case - just look at an appropriate row in yr menu table
$menuData = array(
'menutype' => $menutype,
'title' => $table->alias,
'alias' => $table->alias,
'path' => $table->alias,
'type' => 'heading',
'component_id' => 0,
'language' => '*',
'published' => 1,
);
// this item is at the root so the parent id needs to be 1
$parent_id = 1;
$menuTable->setLocation($parent_id, 'last-child');
// save is the shortcut method for bind, check and store
if (!$menuTable->save($menuData))
{
$this->setError($menuTable->getError());
return false;
}
But I'm unable to find the correct way to update a menu item, like the title and alias. But it is unsuccessful. I have tried using this with no success.
$menu_title = $_POST['title'];
$menu_alias = JFilterOutput::stringURLSafe($menu_title);
$menuTable = JTableNested::getInstance('Menu');
// this is heading menu item but what data you have and require will vary per case - just look at an appropriate row in yr menu table
$menuData = array(
'title' => $menu_title,
'alias' => $menu_alias,
);
// this item is at the root so the parent id needs to be 1
$parent_id = ($_POST['parent'] != "" ? $_POST['parent'] : '1');
$menuTable->setLocation($parent_id, 'last-child');
// save is the shortcut method for bind, check and store
if (!$menuTable->set($_POST['menuid'],$menuData)) {
$this->setError($menuTable->getError());
exit;
}
Can anyone help?