Artio works flawlessly on my SSL e-commerce site with virutuemart.
Except:

if I type in www.my_site_name.com or my_site_name.com or my_site_name in the URL address bar, I get:

Found
The document has moved here.
Additionally, a 400 Bad Request error was encountered while trying to use an ErrorDocument to handle the request.

If I type in my_site_name.com there is no problem.

I am not good with htaccess scripting so I need help for a proper redirect of the non SSL info.

My .htaccess reads:

RewriteEngine On
RewriteCond %{SERVER_PORT} !^443$
RewriteRule ^(.*)$ www.my_site_name.com/$1 [R]

RewriteCond %{REQUEST_URI} (/|\.htm|\.php|\.html|/[^.]*)$ [NC]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule (.*) index.php

The first three lines were recommended by the server, the following are recommended by Artio/Joomla...

Please let me know if there is a way to correct this issue so that users can have a seamless redirect from non-SSL addresses in the address bar.

Well www.my_site_name.com and my_site_name.com are technically not the same URLs and they may be completely different sites. So if you want them to be same, should should choose one to be your \"main\" domain -- also configure Joomla! and VM to use that (with/without https) (so lets assume you will choose www.my_site_name.com) and the request to (my_site_name.com) should be automatically forwarded to www with 301 - Permanently moved header. These rules can be added to .htaccess or VirtualHost settings. For details, check Apache docs.

The second issue is switching beween http and https. The rules you have will redirect all traffic not coming through port 443 (https) to https at www.my_site_name.com. (is that wanted behaviour?) In that case, you should make sure it exists, or change the rule to forward not to www.my_site_name.com but to my_site_name.com.