The mission dubbed Ax-1 is to last around 10 days, said Axiom Space president and CEO Michael Suffredini.
During an initial uncrewed test flight in December 2019, the Starliner capsule failed to dock at the ISS and returned to Earth prematurely.
Soon, space "tourists" -- civilians who are not professional astronauts -- will take their places on Dragon.
It left for the ISS on 2 November 2019 in the Cygnus capsule and returned on 14 January 2021 aboard SpaceX's Dragon capsule.
The four astronauts on Crew-1 are NASA astronauts Victor Glover, Michael Hopkins, Shannon Walker and JAXA astronaut Soichi Noguchi.
When completed by late 2022, Tianhe is expected to weigh about 66 tons, a fraction the size of the International Space Station.
The current population includes six Americans, two Russians, two Japanese and one French.
French astronaut Thomas Pesquet, Americans Shane Kimbrough and Megan McArthur and Japan's Akihiko Hoshide will be launching today.
"The first core module of the new Russian orbital station is in the works," Rogozin said in a statement.
This will be the first crew flight using a recycled Falcon and Dragon as both were designed for reuse.
This was the second missions for Kate Rubins, and Sergei Ryzhikov Ryzhikov but first mission for Sergei Kud-Sverchkov.
CNES will also be supplying fireproof carry bags made in France to shield equipment from shocks and radiation.
He was added to the next Soyuz crew, in order to keep a constant US presence on the ISS, as operations would be hampered if no Americans were on board.
More than 3,000 experiments have been carried out in this microgravity laboratory, which flies at an average of 400 kilometres above Earth, at 28,000 km/h.
he Starliner's first crewed flight is currently scheduled for September.
The eight existing solar panels on the ISS are 12 to 20 years old which is past their design lifetime and deteriorating.
The ship is carrying apples, tomatoes, nuts, smoked salmon, Parmesan and cheddar cheeses, caramels and coconut strips along with science experiments.
The agency also said the “time has come” to put disabled people into space as part of an initiative called the Parastronaut Feasibility Project.
The spacecraft is carrying 1,400 kg of research and crew supplies, freshwater, nitrogen gas and propellant for ISS' Zvezda service module propulsion system.
The mission will be the first time that two international partner crew members are flying together, NASA said.