The main task of navigation is to measure and use the basic information of the vehicle such as real-time position, velocity, orientation and distance. It also guides the vehicle from a position to a predetermined position with the help of navigation sensors.
Inertial navigation is an autonomous navigation method. It is also one of the most important and main modern navigation methods. At present, almost all advanced integrated navigation methods are based on inertial navigation.
Suppose a car moves in a straight line at a uniform speed on the road, and the distance S it travels depends on the speed v and the travel time t, that is,
\( S=v\cdot t \)
If the car moves in a linear motion with variable speed, the speed is V (t), and the initial position of the car is set to \( \mathbf{X}_0 \), the instantaneous position of the car at time t is
\( \mathbf{X}(t)=\mathbf{X}_0+\underset{}{\overset{}{\int}}v(t)\,{\mathrm{d}}t \)
Another example is an aircraft taxiing along a runway for takeoff. It will start moving from a stationary state with a certain acceleration \( \mathbf{a}_0 \) due to the thrust of the engine. As time goes by, its speed increases until it leaves the ground and flies to the destination at a certain speed. Obviously, the speed of flight will depend on the magnitude of the acceleration and the time. That is, the speed is the integral of the acceleration, which can be expressed as,
\( v\left({t}_k^{}\right)=v\left({t}_0^{}\right)+\int_{t_0}^{t_k}a\left(t\right)dt \)
In the above formula, \( v\left({t}_0^{}\right) \) is the velocity vector of the vehicle at the initial time (Here is zero).
The instantaneous position of the aircraft depends on the magnitude of the velocity and the flight time. This means the position is equal to the integral of the velocity, which can be written as,
\( r\left({t}_k^{}\right)=r\left({t}_0^{}\right)+\int_{t_0}^{t_k}v\left(t\right)dt \)
In the above formula, \( r\left({t}_0^{}\right) \) is the position vector of the aircraft at the initial time.
Inertial Navigation System (INS) uses such a physical method to realize navigation and positioning. It uses an accelerometer to measure the acceleration of the vehicle. It uses a gyro-stabilized platform to simulate the horizontal plane, and establishes a spatial rectangular coordinate system. The three coordinate axes point to the east(E), to the north(N) and Up(U), which is usually called the East-North-Up Coordinate System.
During the movement of the vehicle, the gyro is used to make the platform always track the local horizontal plane. The three axes always point to the east, north and up orientations. An east accelerometer, a north accelerometer, and a vertical accelerometer are installed in the orientations of these three axes. The eastward accelerometer measures the acceleration ae of the vehicle along the east-west orientation. The northward accelerometer measures the acceleration an of the vehicle along the north-south orientation. The vertical accelerometer measures the acceleration au along the up orientation. By integrating the acceleration components in these three orientations, the velocity components of the vehicle along these three orientations can be obtained as，
\( \left. \begin{array}{llllllllll} {v}_e\left({t}_k\right)={}v_e\left({t}_0\right){}+\int_{t_0}^{t_k}a_edt{}\\ {v}_n\left({t}_k\right)={}v_n\left({t}_0\right){}+\int_{t_0}^{t_k}a_ndt{}\\ {v}_u\left({t}_k\right)={}v_u\left({t}_0\right){}+\int_{t_0}^{t_k}a_udt{} \end{array}\right\} \)
Usually, the position of the vehicle on the Earth is expressed in longitude, latitude, and elevation. After the velocity integration, we obtain the following formula,
\( \left. \begin{array}{llllllllll} \lambda={}\lambda_0{}+\int_{t_0}^{t_k}\dot\lambda dt{} \\ L={}L_0+\int_{t_0}^{t_k}\dot L dt{}\\ h={}h_0+\int_{t_0}^{t_k}\dot h dt{}\\ \end{array}\right\} \)
In the above formula, \( \lambda_0 \), \( L_0 \) and \( h_0 \) are the initial position of the vehicle. \( \dot\lambda \), \( \dot L \) and \( \dot h \) are the time change rates in longitude, latitude, and elevation, respectively. The following formula is obtained by calculating the motion velocity,
\( \left. \begin{array}{llllllllll} {{\dot{\lambda}=\frac{v_e}{(N+h)\cos L}}} \\ {{\dot{L}=\frac{v_{n}}{M+h}}} \\ {{\dot{h}=v_{u}}} \end{array}\right\} \)
Then we can get the instantaneous position of the vehicle,
\( \left. \begin{array}{llllllllll} \lambda={}\lambda_0{}+\int_{t_0}^{t_k}\frac{v_e}{(N+h)\cos L} dt{} \\ L={}L_0+\int_{t_0}^{t_k}\frac{v_{n}}{M+h} dt{}\\ h={}h_0+\int_{t_0}^{t_k}v_{u} dt{}\\ \end{array}\right\} \)
In the above formula, M and N represent the curvature radius of meridian and equator of the earth respectively. If the earth is approximately regarded as a sphere with radius R, then M=N=R.
Because the initial positions (\( \lambda_0 \), \( L_0 \) and \( h_0 \) ) must be known in advance and input to the inertial navigation system, so inertial navigation belongs to relative positioning.
- Accelerometer. It is used to measure the motion acceleration of the vehicle. Usually there should be 3 accelerometers on the 3 coordinate axes.
- Gyrostabilized platform. It provides an accurate coordinate reference for the accelerometer to keep the accelerometer always measuring the acceleration along the three axial directions.
- Navigation computer. It is used to complete the navigation. According to the measured acceleration signal, the vehicle’s velocity, position and other navigation parameters are calculated. At the same time, in order to ensure that the platform is always horizontal and pointing north when it moves with the aircraft and the earth rotates, the command signal to correct the platform position should be calculated constantly, and the toxic acceleration should be calculated and compensated.
- Display terminal. It can be used to output and display navigation parameters. There are also other necessary control operations, such as input of initial data.
- Power supply and accessories.
According to the installation ways of inertial measurement unit (IMU)on the vehicle, the inertial navigation system can be divided into platform inertial navigation system and strap-down inertial navigation system.
Platform inertial navigation system has an inertial navigation unit, that is inertial measurement unit / IMU. The IMU is installed on the inertial platform. According to the different coordinate system simulated by the platform, the platform inertial navigation system is divided into space-stable inertial navigation system and local horizontal plane inertial navigation system.
The platform of space-stable inertial navigation system is relatively stable in inertial space and is used to simulate a certain inertial coordinate system. The separation of gravitational accelerations and the compensation of other unwanted accelerations are done entirely by computer. This kind of system is mostly used to control the launch vehicle and some spacecraft.
The platform of this system simulates a certain local horizontal coordinate system. This ensures that the reference plane composed of the sensitive axes of the two horizontal accelerometers always tracks the local horizontal plane. Such systems are mostly used for aircraft that move near the surface, such as airplanes and cruise missiles.
The platform of the platform inertial navigation system can isolate the angular vibration of the vehicle. It also can provide a better working environment for the inertial measurement unit. Because the navigation coordinate system is established directly on the platform, it needs small amount of calculation to extract useful signals, but its structure is complex and its size is large.
In the strapdown inertial navigation system, the inertial measurement unit / IMU is installed directly on the vehicle. There is no solid platform, and the sensitive axis of the IMU is arranged in the three axis direction of the vehicle coordinate system. It replaces the physical platform with a “mathematical platform” stored in the computer.
In the motion, the gyro measures the angular velocity measurement of the vehicle with respect to the inertial reference frame. The gyro also calculates the coordinate transformation matrix from the vehicle coordinate system to the navigation (calculation) coordinate system. Through this matrix, the acceleration information measured by the accelerometer is transformed to the navigation (calculation) coordinate system. Then the navigation calculation is carried out to obtain the required navigation parameters.
Since there is no physical platform, compared with the platform inertial navigation system, the strapdown inertial navigation system has simple structure, small size and easy maintenance. However, the inertial measurement unit is directly mounted on the vehicle. This will have poor working conditions and reduce the accuracy of the IMU. Since the acceleration component output by the three accelerometers is along the axial direction of the vehicle coordinate system, it must be converted into the acceleration component of the navigation coordinate system by the computer (This transformation acts as a “mathematical platform”), so the calculation is much larger.
From the above analysis, it can be seen that inertial navigation uses inertial measurement unit / IMU to measure the linear motion and angular motion of the vehicle in the inertial space. The navigation computer calculates the navigation parameters such as position, velocity and attitude of the vehicle
The inertial navigation system has the following outstanding advantages,
- Autonomy, without any navigation station, the navigation function can be completed by the inertial navigation system itself.
- Concealment, it does not send any signal outside the vehicle, and no one can search or find its working information.
- Anti-interference, not affected by the external electromagnetic environment, no one can implement electromagnetic interference and control it.
- Global, no geographical restrictions, with global navigation capability.
- Continuity, it can provide navigation information continuously and in real time.
- Completeness, it can provide not only the position information of the vehicle but also the attitude, speed and time information.
The basic principle of inertial navigation determines that it must use accelerometers to measure the acceleration of the vehicle, and then the position of the vehicle can be obtained after two integral operations. In this way, a constant error in the accelerometer measurement will cause a position error that grows quadratically with time. Therefore, when it is used alone, there is a disadvantage that the navigation error accumulates with time. It is usually necessary to use other navigation systems to correct its error periodically.
