In reference to the patent: US 6,151,713
Is this patent expired?
Has anyone else patented it?
Can I take over the patent and how much would it cost me?
If you look at the "Fee Status", you will see that it is "Lapsed". This means that the patent is no longer enforceable. The reason for this is found at the bottom of the Google Patents page, in the Legal Events section:
Jan 22, 2004 FPAY Fee payment Year of fee payment: 4 May 28, 2008 FPAY Fee payment Year of fee payment: 8 Jul 9, 2012 REMI Maintenance fee reminder mailed Nov 28, 2012 LAPS Lapse for failure to pay maintenance fees Jan 15, 2013 FP Expired due to failure to pay maintenance fee Effective date: 20121128
As you can see, the third maintenance fee was not paid, and the patent lapsed on November 28, 2012.
There are no other patent grants in the patent family (e.g., international filings, continuations), and the same claims cannot be re-patented because this patent would serve as Prior Art, making the invention Public Domain.
If the patent were still enforceable, you could potentially license or purchase the patent from the current assignee. Of course, that is not necessary.
If you really wanted the patent, there is one slim possibility of working with the assignee to get the maintenance fee paid and revive the patent, but it has been so long (over two years since the failure to pay) that it would probably not be approved. Then you would need to get the patent reassigned with the cooperation of the assignee, and arrange mutually acceptable terms for the cost. This whole "what if" scenario would be expensive, time consuming, would likely fail and would only get a few years of coverage before the patent expires completely.