**Mechanization Equations**

**Derivation of position, velocity, and orientation update equations.**

The core of inertial navigation involves integrating the data from accelerometers and gyroscopes to update the position, velocity, and orientation of an object over time. The derivation of these update equations is based on fundamental principles of mechanics:

**Position Update**: The position of the object is updated by integrating its velocity over time. Mathematically, this can be expressed as:

P(t)=P(t0)+∫t0tV(τ)dτ\mathbf{P}(t) = \mathbf{P}(t_0) + \int_{t_0}^{t} \mathbf{V}(\tau) d\tauP(t)=P(t0)+∫t0tV(τ)dτ

where P(t)\mathbf{P}(t)P(t) is the position at time ttt, and V(τ)\mathbf{V}(\tau)V(τ) is the velocity at time τ\tauτ.

**Velocity Update**: The velocity is updated by integrating the acceleration measurements from the accelerometers:

V(t)=V(t0)+∫t0tA(τ)dτ\mathbf{V}(t) = \mathbf{V}(t_0) + \int_{t_0}^{t} \mathbf{A}(\tau) d\tauV(t)=V(t0)+∫t0tA(τ)dτ

where A(τ)\mathbf{A}(\tau)A(τ) is the acceleration at time τ\tauτ.

**Orientation Update**: The orientation (attitude) is updated by integrating the angular velocity measurements from the gyroscopes. This involves quaternion or rotation matrix representations to handle the non-linear nature of rotational kinematics:

R(t)=R(t0)⋅Q(t)\mathbf{R}(t) = \mathbf{R}(t_0) \cdot \mathbf{Q}(t)R(t)=R(t0)⋅Q(t)

where R(t)\mathbf{R}(t)R(t) is the rotation matrix at time ttt, and Q(t)\mathbf{Q}(t)Q(t) is the quaternion representing the integrated angular velocity.

These equations are solved continuously to provide real-time updates of the object’s navigational state.