Magento: Get all configurable products' children from a category

gistfile1.php

<?php
require_once 'app/Mage.php';
Mage::app(); 

$categoryId = 86;

$products = Mage::getModel('catalog/product')->getCollection()
        ->addAttributeToSelect('*')
        ->addAttributeToFilter('type_id', 'configurable')
        ->joinField('category_id', 'catalog/category_product', 'category_id', 'product_id = entity_id', null, 'left')
         ->addAttributeToFilter('category_id', array(
             array('finset' => $categoryId)
           ));

$ids = array();
$skus = array();

foreach($products as $product) {
    $childProducts = Mage::getModel('catalog/product_type_configurable')
                    ->getUsedProducts(null, $product);   
    
    foreach($childProducts as $child) {
        $ids[] = $child->getId();
        $skus[] = $child->getSku();
    }
}

$ids = implode(', ', $ids);
$skus = implode(', ',$skus);

Zend_Debug::dump($ids);
Zend_Debug::dump($skus);