To achieve rapid and flexible vertical profile exploration of deep-sea hybrid structures, a multi-joint autonomous underwater vehicle (MJ-AUV) with orthogonal joints was designed. This paper focuses on the 3-dimensional (3D) modeling and attitude control of the designed vehicle. Considering the situation of gravity and buoyancy imbalance, a 3D model of the MJ-AUV was established according to Newton’s second law and torque balance principle. And then the numerical simulation was carried out to verify the credibility of the model. To solve the problems that the pitch and yaw attitude of the MJ-AUV are coupled and the disturbance is unknown, a linear quadratic regulator (LQR) decoupling control method based on a linear extended state observer (LESO) was proposed. The system was decoupled into pitch and yaw subsystems, treated the internal forces and external disturbances of each subsystem as total disturbances, and estimated the total disturbances with LESO. The control law was divided into two parts. The first part was the total disturbance compensator, while the second part was the linear state feedback controller. The simulation results show that the overshoot of the controlled system in the dynamic process is nearly 0 rad, reaching the design value very smoothly. Moreover, when the controlled system is in a stable state, the control precision is within 0.005%.