The invention discloses a bucket foundation with the perpendicularity convenient to control, an auxiliary assembly and a construction method. The bucket foundation comprises a bucket foundation body which is an inverted bucket body, three positioning sleeves which are evenly distributed in the annular direction and penetrate through a top plate of the bucket body are arranged on the inner wall of the bucket body, and a water in/out valve connected with a water drawing system is arranged on the upper end face of the bucket body. The auxiliary assembly comprises a triangular positioning frame and three positioning steel piles used for vertically driving the auxiliary assembly into the seabed. The three positioning steel piles are movably sleeved with the bucket foundation body, and the bucket body is immersed into a sea to make contact with the seabed and penetrates to the designed depth. The perpendicularity problem in the immersing and penetrating process of the bucket foundation is converted into the perpendicularity problem of the positioning steel piles, as long as the perpendicularity of the positioning steel piles meets requirements, the perpendicularity of the bucket foundation automatically meets requirements, subsequent leveling and correction are not needed, and thus the construction process of the bucket foundation is greatly simplified, the construction difficulty is greatly lowered, the operation time at sea is shortened, and the construction cost is lowered.