This 12g shotgun prototype made by engineers from General Dynamics, Kriss USA, and some Chinese companies features a reciprocating counterweight systems to reduce recoil.

The patent (US patent # 5585590) was granted to a gentleman named Frederic D. Ducolon in 1996. It describes a mechanism with a recoiling barrel assembly (barrel and bolt carrier group) and a counterweight assembly attached to the rearward recoiling parts via a system of cables and pulleys. Upon firing, the barrel assembly recoils rearward and forces (via the pulleys and cables) the counterweight to move forward twice the distance of the recoiling parts movement. The whole mechanism then returns back driven by the return springs. The inventor also notes that preferably the counterweight assembly will also have a throw-weight mechanism that will be shifted forward and downward at an angled declining path (58 on the drawing below).

Quote from the patent:

"The present invention provides a recoil counter-vectoring gun with a gun barrel assembly and counterweight assembly independently slidably mounted on a frame. The gun barrel assembly includes a barrel and a receiver having a bolt therein and will longitudinally reciprocate on the frame between a forward position and a rearward position. A means is provided for firing the gun when the gun barrel assembly is substantially at its forwardmost position. An adjustable recoil counter-vectoring mechanism is provided which comprises a counterweight assembly which longitudinally reciprocates on the frame between a forward position and a rearward position independent of the gun barrel assembly, a fixed pulley mounted on the frame, a traveling pulley mounted on the gun barrel assembly, and a cable means. The cable means has a first end fixed relative to the frame. It then passes over the traveling pulley, the fixed pulley, and toward a second end which is attached to the counterweight assembly. When the gun is fired, the gun barrel assembly recoils rearwardly such that rearward movement of the traveling pulley causes the cable means to pull the counterweight assembly forward on the frame a distance substantially equal to twice the recoil distance of the gun barrel assembly.

Another aspect of the present invention includes providing at least one throwweight assembly mounted on the counterweight assembly for movement between a forward and rearward position. The throwweight is shifted forwardly on the counterweight assembly at a predetermined position relative to full forward movement of the counterweight assembly. In preferred form, the movement of the throwweight is at a downward angle relative to sliding movement of the counterweight assembly and may be guided along a flexible cable having opposite ends supported by the counterweight assembly."

https://www.thefirearmblog.com/blog/2018/04/26/experimental-recoil-counter-vectoring-shotgun/