The design of a robot system for assembling numerous parts, including screws, washers, nuts, and other irregular parts, is important in manufacturing. Although a robot can flexibly manipulate various parts with the aid of vision and force sensors, some challenges remain, for example, precise orientation and precise insertion when uncertainty exists in the positioning and manipulation of irregular objects. In this work, we present a method to design the orienting strategy for positioning the irregular parts based on the constraints formed by the fixture, thus the irregular parts such as the motor can be precisely localized only with the robot squeezing actions. We then develop a peg-in-hole insertion strategy using the force constraints formed by the environment, where fewer robot actions are required to insert the peg, thus enabling efficient and precise peg-in-hole insertions. Finally, we developed a robot assembly system with three robots from Universal Robots to perform all the assembly tasks, including picking, holding, placing, and inserting. Experiments on the assembly of a belt drive unit reveal the efficiency of the proposed method.