We propose a visual-inertial odometry (VIO) framework that fuses inertial measurement unit (IMU) data in the body frame based on kinematic relations. The IMU measurements are directly used as the parameters of kinematic constraints rather than integration. The VIO problem is formulated as an ordinary differential equation (ODE) constrained optimization and solved through a discretize-then-optimization manner. The kinematic ODE constraints are approximated in a least-square formulation and then transformed as error terms, where the IMU noise is handled during the optimization rather than accumulated during the integration. A keyframe-based VIO system is introduced, where the prediction, tracking, bundle adjustment, and IMU initialization are uniformly implemented based on the proposed kinematic constraints. Experiments show that the proposed method achieves competitive results on real-world data compared to the state-of-the-art methods.