Position and attitude determination by integrated GPS/SINS/TS for feed support system of FAST

To ensure high-precision position and attitude measurement, the integration measurement system uses highprecision measurement devices such as SINS, GPS receiver and TS. The error distribution of each device is shown in Tables 1 to 3.

The process of the measurement system is shown in Figure 3. The integration measurement system starts, TS initializes to find the measurement targets and receives the real-time data from the weather station. GPS receivers search for the available satellite signals and establish a communication link with the local reference stations. SINS conducts initial alignment to determine the initial attitude of the measured object; i.e., yaw, pitch and roll. Then the SINS updates the attitude, speed and position, and finally Kalman filter is used for information integration to obtain the high-accurate measurement results. Because the feed support system moves at a low speed, the yaw error calculated by SINS is large, and a multiple GPS antennas scheme is adopted to determine the yaw of the feed cabin. Both the A-B axis and the Stewart manipulator are rigid bodies, consequently the yaw of the Stewart manipulator can be derived by extrapolating attitude of the GPS/SINS integration system. When TS cannot use, the position and attitude of the Stewart manipulator are calculated with the feed cabin's measurement information and the moving value of A-B axis and six bars in the feed cabin. The A-B axis and six bars are controlled by the optical encoder multiturn absolute, this encoder's attitude accuracy is less than 0.4'', its position accuracy is less than 1 mm. This precision can meet the requirement, such as attitude accuracy is 0.1° and position accuracy is 3 mm, for measuring the Stewart manipulator.

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The initial alignment of SINS is a process of determining the reference navigation coordinate frame. SINS is just a startup. The orientation of the feed-cabin body coordinate frame (b-frame), relative to each axis of the reference navigation coordinate frame (n-frame), is completely unknown or not accurate enough. Therefore, it cannot enter the navigation state immediately. The coarse alignment method can determine the angle between b-frame and n'-frame with two vectors such as g^b and ${\omega }_{ie}^{b}$. When the feed support system is stationary or moving at a constant speed, the gyros measure the similar earth rotation rate, the accelerometers measure the local gravitational acceleration (Gu et al. 2008).

The attitude matrix ${C}_{b}^{{n}^{^{\prime} }}$, which relates the body frame (b-frame) to the computational navigation frame (n'-frame), could be calculated by the following equation:

$$\begin{eqnarray}&&{C}_{b}^{{n}^{^{\prime} }}={\left[\begin{array}{c}{({g}^{n})}^{T}\\ {({\omega }_{ie}^{n})}^{T}\\ {({g}^{n}\times {\omega }_{ie}^{n})}^{T}\end{array}\right]}^{-1}\left[\begin{array}{c}{({g}^{b})}^{T}\\ {({\omega }_{ie}^{b})}^{T}\\ {({g}^{b}\times {\omega }_{ie}^{b})}^{T}\end{array}\right],\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\mathop{\omega }\limits^{\sim }}_{ib}^{b}\approx {\omega }_{ie}^{b},\\ {\mathop{f}\limits^{\sim }}^{b}\approx -{g}^{b},\end{array}\end{eqnarray} \tag{ 2 }$$

where gⁿ is the gravity vector in n frame, ${\mathop{f}\limits^{\sim }}^{b}$ is specific force vector from accelerometers output. ${\omega }_{ie}^{n}$ is the earth rotation rate resolved in n frame, ${\mathop{\omega }\limits^{\sim }}_{ib}^{b}$ is angular rate vector from gyro output.

After coarse alignment, a fine alignment follows. This calculates the misalignment angle (ϕ) between n'-frame and n-frame. The attitude matrix ${C}_{b}^{{n}^{^{\prime} }}$ is modified by ϕ to get a more accurate attitude matrix ${C}_{b}^{n}$:

$$\begin{eqnarray}&&{C}_{b}^{n}={C}_{{n}^{^{\prime} }}^{n}{C}_{b}^{{n}^{^{\prime} }}=[I+(\phi \times )]{C}_{b}^{{n}^{^{\prime} }}.\end{eqnarray} \tag{ 3 }$$

The updating algorithm of SINS consists of position-updating, velocity-updating and attitude-updating. The attitude -updating is the core, and its precision plays a decisive role in the accuracy of the SINS. The attitude-updating can be calculated as follows:

$$\begin{eqnarray}&&{\mathop{\boldsymbol{C}}\limits^{.}}_{b}^{n}={\boldsymbol{C}}_{b}^{n}({\boldsymbol{\omega}}_{nb}^{b}\times ).\end{eqnarray} \tag{ 4 }$$

The velocity-updating can be described as follows:

$$\begin{eqnarray}&&{\dot{V}}^{n}={C}_{b}^{n}{f}^{b}-(2{\omega }_{ie}^{n}+{\omega }_{en}^{n})\times {V}^{n}+{g}^{n}.\end{eqnarray} \tag{ 5 }$$

The position-updating can be calculated as follows:

$$\begin{eqnarray}&&{\dot{r}}^{n}={V}^{n}-{\omega }_{en}^{n}\times {r}^{n},\end{eqnarray} \tag{ 6 }$$

where superscript n refers to the n-frame, superscript b refers to the b-frame. ${C}_{b}^{n}$ is attitude matrix, $\mathop{{\boldsymbol{C}}_{b}^{n}}\limits^{.}$ is attitude rate matrix. ${\boldsymbol{\omega}}_{nb}^{b}$ is the angular rate vector of b-frame with respect to n-frame projected in b-frame, ${\omega }_{en}^{n}$ is the angular rate vector of n-frame with respect to earth frame (e-frame) projected in n-frame. Vⁿ is velocity vector, ${\dot{V}}^{n}$ is velocity rate vector, rⁿ is position vector, $\mathop{{\boldsymbol{r}}^{n}}\limits^{.}$ is position rate vector.

By simultaneous Equations (4) to (6), the attitude, velocity and position of the carrier can be calculated. However, the SINS has inherent defects such as divergence of navigation accuracy over time and poor long-term stability. To solve this problem, the measurement system aids the GPS and TS devices.

The essence of integration measurement is state estimation. The Kalman filter is a set of mathematical equations that provides an efficient computational means to estimate the state of a process, andminimizes the rootmean squared error of that state (Welch & Bishop 2001). The state vector of the Kalman filter is 15 dimensions in the integration measurement system, which is described as follows:

$$\begin{eqnarray}\begin{array}{lll}{\boldsymbol{X}} & = & [{\phi }_{E},{\phi }_{N},{\phi }_{U},\delta {V}_{E},\delta {V}_{N},\delta {V}_{U}\\ & & {\delta L,\delta \lambda ,\delta h,{\varepsilon }_{x},{\varepsilon }_{y},{\varepsilon }_{z},{\nabla }_{x},{\nabla }_{y},{\nabla }_{z}]}^{T},\end{array}\end{eqnarray} \tag{ 7 }$$

where E, N and U are, respectively, east, north and up. The last six terms are the gyro zero drift error and the accelerometer sensor error, which are repeatability errors of inertial devices that have great impact on the accuracy of the system. The first nine terms are,respectively, attitude, velocity, and position error vectors, which can be derived from Equations (4) to (6).

The derivation of attitude error vectors is as follows:

$$\begin{eqnarray}&&{\omega }_{nb}^{b}={\omega }_{ib}^{b}-{\omega }_{in}^{b}.\end{eqnarray} \tag{ 8 }$$

Substitute Equations (8) and (3) into Equation (4) and use the attitude quaternion method, and obtain the attitude error vector Equation (9):

$$\begin{eqnarray}&&\dot{\phi }=\phi \times {\omega }_{in}^{n}+\delta {\omega }_{in}^{n}-{C}_{b}^{n}([\delta {K}_{G}]+[\delta G]){\omega }_{ib}^{b}-{\varepsilon }^{n}.\end{eqnarray} \tag{ 9 }$$

The derivation of velocity error vectors is as follows:

Equation (5) is the ideal velocity equation, and the actual velocity equation is as follows:

$$\begin{eqnarray}&&{\dot{\mathop{V}\limits^{\sim }}}^{n}={\mathop{C}\limits^{\sim }}_{b}^{n}{\mathop{f}\limits^{\sim }}^{b}-(2{\mathop{\omega }\limits^{\sim }}_{ie}^{n}+{\mathop{\omega }\limits^{\sim }}_{en}^{n})\times {\mathop{V}\limits^{\sim }}^{n}+{\mathop{g}\limits^{\sim }}^{n}.\end{eqnarray} \tag{ 10 }$$

Subtract Equation (10) from Equation (5)

$$\begin{eqnarray}\begin{array}{lll}\delta {\dot{V}}^{n} & = & [(I-\phi \times ){C}_{b}^{n}({f}^{b}+\delta {f}^{b})-{C}_{b}^{n}{f}^{b}]\\ & - & \{[2({\omega }_{ie}^{n}+\delta {\omega }_{ie}^{n})+({\omega }_{en}^{n}+\delta {\omega }_{en}^{n})]\\ & \times & ({V}^{n}+\delta {V}^{n})-(2{\omega }_{ie}^{n}+{\omega }_{en}^{n})\times {V}^{n}\}\\ & + & \delta {g}^{n}.\end{array}\end{eqnarray} \tag{ 11 }$$

Expand Equation (11) and omit the second order small quantities of the error, the velocity error vectors Equation (12):

$$\begin{eqnarray}\begin{array}{lll}\delta {\dot{V}}^{n} & = & -{\phi }^{n}\times {f}^{n}+{C}_{b}^{n}([\delta {K}_{A}]+[\delta A]){f}^{b}\\ & & +\delta {V}^{n}\times (2{\omega }_{ie}^{n}+{\omega }_{en}^{n})\\ & & +{V}^{n}\times (2\delta {\omega }_{ie}^{n}+\delta {\omega }_{en}^{n})+{\nabla }^{n}.\end{array}\end{eqnarray} \tag{ 12 }$$

The derivation of position error vectors is as follows:

$$\begin{eqnarray}&&{\omega }_{en}^{n}=\left[\begin{array}{c}-\displaystyle \frac{{V}_{N}}{{R}_{M}+h}\\ \displaystyle \frac{{V}_{E}}{{R}_{N}+h}\\ \displaystyle \frac{{V}_{E}}{{R}_{N}+h}\tan L\end{array}\right].\end{eqnarray} \tag{ 13 }$$

Substitute the angular rate of b-frame ${\omega }_{en}^{n}$ into Equation (6), and solve the differential equation to get the position vectors, they are as follows:

$$\begin{eqnarray}&&\dot{L}=\displaystyle \frac{1}{{R}_{M}+h}{V}_{N},\,\dot{\lambda }=\displaystyle \frac{\sec L}{{R}_{N}+h}{V}_{E},\,\dot{h}={V}_{U}.\end{eqnarray} \tag{ 14 }$$

Take the derivatives of Equation (14), and get Equations (15)–(17):

$$\begin{eqnarray}&&\delta \dot{L}=\delta {V}_{N}/({R}_{M}+h)-\delta h{V}_{N}/{({R}_{M}+h)}^{2},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}\begin{array}{lll}\delta \dot{\lambda } & = & \delta {V}_{E}/({R}_{N}+h)\sec L\\ & & +\delta L\tan L\sec L{V}_{E}/({R}_{N}+h)\\ & & -\delta h{V}_{E}\sec L/{({R}_{N}+h)}^{2},\end{array}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\delta \dot{h}=\delta {V}_{U},\end{eqnarray} \tag{ 17 }$$

where L is latitude, λ is longitude, h is height. K_G is the gyro scale coefficient error, G is the gyro installation error. K_A is the accelerometer scale coefficient error, A is the accelerometer installation error. R_M and R_N are constants.

In integration measurement system, the measurement error is often used as a state vector. Because the Kalman filter is a linear filter, the prediction equation of the measurement error is linear. Generally, the measurement error is relatively small, its higher order items can be ignored to simplify the prediction equation of the measurement error (Yan 2007). The Kalman filter equation includes state equation and measurement equation, which are as follows:

$$\begin{eqnarray}&&{X}_{k}=F{X}_{k-1}+{W}_{k-1},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{Z}_{k}=H{X}_{k}+{V}_{k},\end{eqnarray} \tag{ 19 }$$

where subscript k − 1 is the last time, k is the current time. F is the state transition matrix, H is the measurement matrix, W is the system noise, V is the measurement noise, Z is a measurement vector, in the integration of GPS and SINS, the form of measurement vector Z_{G – S} is as Equation (20), and in the integration of TS and SINS, the form of measurement vector Z_{T − S} is as Equation (21):

$$\begin{eqnarray}\begin{array}{lll}{Z}_{G-S} & = & [{V}_{GE}-{V}_{SE},{V}_{GN}-{V}_{SN},{V}_{GU}-{V}_{SU},\\ & & {{L}_{G}-{L}_{S},{\lambda }_{G}-{\lambda }_{S},{h}_{G}-{h}_{S}]}^{T},\end{array}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}\begin{array}{lll}{Z}_{T-S} & = & [{V}_{TE}-{V}_{SE},{V}_{TN}-{V}_{SN},{V}_{TU}-{V}_{SU},\\ & & {{L}_{T}-{L}_{S},{\lambda }_{T}-{\lambda }_{S},{h}_{T}-{h}_{S}]}^{T}.\end{array}\end{eqnarray} \tag{ 21 }$$

In practice, due to the strong observability of position and velocity vector, the initial value of state vector and corresponding root mean square error matrix is allowed to be relatively large, and they will converge rapidly with the update of filtering. Secondly, the smaller the variance matrix of system noise is, the lower the utilization rate of the measurement vector is. However, the utilization rate of measurement vector is higher when the variance matrix of measurement noise is smaller, and vice versa (Yan 2019). The Kalman filter automatically adjusts the utilization rate of state information and measurement information according to the size of state noise and measurement noise, which makes the most reasonable estimation of the current state. In the process of filtering, the optimal estimation value of state vector is continuously used to modify the SINS calculation value, to make it close to the real position and attitude value. The feedback of Kalman filter is helpful to keep the measurement error equation linear.

The GPS/SINS measurement scheme is used for the feed cabin. The measurement data of total station can be used as a reference. The total station's attitude measurement accuracy is 0.5'', the position measurement accuracy is 5 mm, which are all superior to the measurement requirements of the feed cabin. At the outermost edge of the feed cabin, there are six optical targets. The total station can track these six targets to work out the real-time position and attitude of the feed cabin, which can be used as a reference value to verify the accuracy of GPS/SINS results.

In the accuracy experiment of GPS/SINS, because the difference between TS data and GPS/SINS integrated data is a very small value compared with the experimental data range, TS data and GPS/SINS integrated data cannot be differentiated in a figure. Therefore, only the integrated GPS/SINS result are shown in Figures 4 and 5, the error results between GPS/SINS data and TS data are shown in Figures 6 and 7.

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From Table 4, the maximum root mean squared error of the angle is 0.095°, the maximum root mean squared error of the position is 5.62 mm. These values are less than 15 mm and 0.1° as the precision for measuring the feed cabin, so the position and attitude accuracy of integrated GPS/SINS can meet the measurement requirements of the feed cabin.

The TS/SINS measurement scheme is used for the Stewart manipulator. According to the design requirements, the position measurement accuracy of the Stewart manipulator is 3 mm. However, in a scale of 206 meters and a height of 140 m, there is no measurement equipment with a higher precision than 3 mm. No specific measuring equipment can be found as a reference. From the perspective of practical application, the main task of the Stewart manipulator is making telescope receivers to accurately accept the radio source signal. Therefore, the astronomical observation trajectory is used as a reference for the measurement accuracy of the Stewart manipulator.

The astronomical observation trajectory (azimuth and zenith angle) in the horizontal coordinate system can be calculated according to the right ascension, declination and observation time of the source 0029+349, as shown in Figure 8. With this value, the planned position trajectory (X-Y-Z) of Stewart manipulator in the FAST coordinate system can be derived, then the planned attitude trajectory (yaw-pitch-roll) of Stewart manipulator can be calculated according to the deformation strategy of the telescope. By comparing the error between the planned trajectory of the Stewart manipulator and the TS/SINS integrated measurement, we can verify accuracy of TS/SINS Results. The planned trajectory of the Stewart manipulator and the integrated TS/SINS result of tracking source are shown in Figures 9 and 10. In Figure 9, the red line is the planned position trajectory of the Stewart manipulator and the blue line is the TS/SINS integrated position. In Figure 10, there is the red line is the planned attitude trajectory of the Stewart manipulator and the blue line is TS/SINS integrated attitude. The error results between the planned trajectory and TS/SINS data are shown in Figures 11 and 12.

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From Table 5, the maximum root mean squared error of the angle is 0.093°, the maximum root mean squared error of the position is 2.76 mm. These values are less than 3 mm and 0.1° as the precision for measuring the Stewart manipulator, so the position and attitude accuracy of integrated TS/SINS can meet the measurement requirements of the Stewart manipulator.

In Table 6, the errors between reference and measuring in changing source are shown. For the feed cabin in the changing source state, the maximum root mean squared error of the angle is 0.094°, the maximum root mean squared error of the position is 14.56 mm. For the Stewart manipulator in the changing source state, the maximum root mean squared error of the angle is 0.093°, the maximum root mean squared error of the position is 2.99 mm. Consequently, the position and attitude accuracy of integrated measurement can meet the requirements of the changing source state.