Greenpeace and other conservation groups oppose the operation, saying the companies have shown limited ability to clean up the mess if a major spill were to occur.
Among the six activists onboard the rig is New Zealander Johno Smith.
"This pristine environment needs protecting for future generations and all life that will call it home," Mr Smith says. "But instead Shell's actions are exploiting the melting ice to increase a manmade disaster. Climate change is real and already inflicting pain and suffering on my brothers and sisters in the Pacific."
Shell faced problems in 2012, including the grounding of a drill rig whilst it was crossing the Gulf of Alaska.
In December 2014, Noble Drilling, a sub-contractor of Shell and owner of the other drill rig, called the Noble Discoverer, pleaded guilty to committing eight felonies relating to Shell's past failed drilling attempts.
Greenpeace believe this shows Shell cannot be trusted to conduct operations in the Arctic.
Kelly Op De Weegh, a spokesperson for Shell USA, says the corporation's complaint to the federal court in Anchorage is asking for the Greenpeace protesters to be removed from their vessel. They also seek to curb any similar actions that may occur in the future.
Shell argue that the demonstrations are extremely dangerous, particularly on the high seas.
The complaint also claimed that the illegal actions could delay or prevent Shell from conducting the permitted exploration.
The five other protesters onboard the vessel with New Zealander Johno Smith are Aliyah Field from the USA, Andreas Widlund from Sweden, Miriam Friedrich from Austria, Zoe Buckley Lennox from Australia, and Jens Loewe from Germany.
Smith is hopeful that the demonstration will increase awareness of Shell's activities.
"I believe that shining a light on what Shell is doing will encourage more people to take a strong stand against them and other companies who are seeking to destroy this planet for profit. I'm just one voice out here, but I know I'm not alone, and millions if not billions of voices demanding the right to safe and healthy lives will have a huge chance of changing things."
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