Rapid and accurate initial alignment of the low-cost MEMS IMU chip dedicated for tilted RTK receiver

Abstract

The new global navigation satellite system real-time kinematic (RTK) receivers with tilt compensation extend the applicability of high-precision RTK positioning to previously restrictive environments since the pole is no longer required to be maintained vertically. The rapid and accurate initial alignment of the built-in low-cost inertial measurement unit (IMU) is a critical technical issue and currently remains a challenge for the tilted RTK. We address this problem by proposing a method specific to the determination of the IMU initial heading for the tilted RTK. In this approach, the IMU heading is calculated from the angle between the inertial navigation system (INS)- and RTK-indicated position increment vectors in the horizontal plane based on the fact that the INS- and RTK-indicated trajectories in the short term are similar in shape. Instead of conventional INS mechanization, the INS-indicated trajectory is determined using the gyro-derived attitude and pole length by treating the pole movement as the motion of a rigid body with a fixed point to enhance the trajectory accuracy. The experiment results indicate that the initial heading is accurately determined to 1.15° at a 98.2% confidence level within only 2–3 s when using a dollar-level IMU chip without any additional sensors, e.g., a magnetometer. The proposed algorithm satisfactorily meets the requirements of the tilted RTK application, and its accuracy and convergence speed are higher than those of the current methods reported in publications and product sheets.

Appendix

In this appendix, we address the question of how accurate the attitude angles should be for the tilt compensation when 2 cm compensation accuracy is required in the tilted RTK surveying. We will illustrate that the roll and pitch angles can be initialized with sufficient accuracy quickly even for low-cost MEMS IMU. Then, the derivation of the initial heading accuracy requirement is sketched.

Roll and Pitch Angle Initialization

When the INS remains stationary with respect to the earth, the roll and pitch angles, as defined in the methodology section, can be determined through the accelerometer leveling process in a few seconds, even for a low-cost MEMS IMU. The resulting roll and pitch errors may be expressed as follows for the particular situation in which the body frame is nominally aligned with the local geographic frame (Titterton and Weston 2004):

$$\delta \theta = -\frac{\delta f_x}{g}$$

(22)

where ${{\delta }}\phi$ and ${{\delta }}\theta$ denote the errors in the initial roll and pitch angles, respectively. ${{\delta f}}_{{x}}$ and ${{\delta f}}_{{y}}$ are the accelerometer measurement error in x-axis and y-axis, respectively. g is the magnitude of the local gravitational acceleration.

Assume the typical MEMS accelerometers are accurate to 1000 mGal, then a 0.06° roll and pitch accuracy can be achieved, which is sufficiently accurate for the RTK tilt compensation.

Heading Accuracy Requirement Analysis

Here, we analyze the heading accuracy requirement for the tilt compensation. Recall Eq. (1), the tilt compensation from GNSS antenna phase center to the pole tip in the n-frame is given by:

$$\varvec{l}^{{n}} { = }{\mathbf{C}}_{{b}}^{{n}} \varvec{l}^{{b}}$$

(23)

where $\varvec{l}^{{n}}$ is the lever-arm vector resolved in the n-frame, i.e., the tilt compensation component; The lever-arm vector in the b-frame is $\varvec{l}^{{b}} = \left[ {\begin{array}{*{20}c} 0 & 0 & {{l}} \\ \end{array} } \right]^{T}$, where l denotes the length between the GNSS antenna phase center and the pole tip. The direction cosine matrix in terms of Euler angles, as defined in the methodology section, is given by Savage (2007):

$${\mathbf{C}}_{{b}}^{{n}} = \left[ {\begin{array}{*{20}c} {{{c}}\theta {{c}}\psi } & {{{s}}\phi {{s}}\theta {{c}}\psi - {{c}}\phi {{s}}\psi } & {{{c}}\phi {{s}}\theta {{c}}\psi + {{s}}\phi {{s}}\psi } \\ {{{c}}\theta {{s}}\psi } & {{{s}}\phi {{s}}\theta {{s}}\psi + {{c}}\phi {{c}}\psi } & {{{c}}\phi {{s}}\theta {{s}}\psi - {{s}}\phi {{c}}\psi } \\ { - {{s}}\theta } & {{{s}}\phi {{c}}\theta } & {{{c}}\phi {{c}}\theta } \\ \end{array} } \right]$$

(24)

where c and s refer to the cosine and sine operators, respectively. The computed direction cosine matrix which contains errors due to the errors in roll, pitch and heading Euler angles, is written as follows (Titterton and Weston 2004):

$${\hat{\mathbf{C}}}_{{b}}^{{n}} = \left[ {{\mathbf{I}} - {\varPsi }} \right]{\mathbf{C}}_{{b}}^{{n}}$$

(25)

where I is the 3-by-3 identity matrix and ${\varPsi }$ is given by:

$${\varPsi } = \left[ {\begin{array}{*{20}c} 0 & { - {{\delta }}\psi } & {{{\delta }}\theta } \\ {{{\delta }}\psi } & 0 & { - {{\delta }}\phi } \\ { - {{\delta }}\theta } & {{{\delta }}\phi } & 0 \\ \end{array} } \right]$$

(26)

where the elements, ${{\delta }}\phi$, ${{\delta }}\theta$, and ${{\delta }}\psi$ correspond to the errors in roll, pitch and heading Euler angles, respectively. The tilt compensation error, denoted by ${{\delta }}\varvec{l}^{{n}}$, due to the errors in the computed attitude matrix is given by:

$${{\delta }}\varvec{l}^{{n}} = {\hat{\mathbf{C}}}_{{b}}^{{n}} \varvec{l}^{{b}} - {\mathbf{C}}_{{b}}^{{n}} \varvec{l}^{{b}}$$

(27)

Substituting (25) into (27) yields

$${{\delta }}\varvec{l}^{{n}} = - {\varPsi }{\mathbf{C}}_{{b}}^{{n}} \varvec{l}^{{b}}$$

(28)

Substituting (24) and (26) into (28) yields the three elements of ${{\delta }}\varvec{l}^{{n}}$ as:

$$\left\{ {\begin{array}{*{20}l} {{{\delta l}}_{{N}} = {{\delta }}\psi \cdot {{l}}\left( {{{c}}\phi {{s}}\theta {{s}}\psi - {{s}}\phi {{c}}\psi } \right) - {{\delta }}\theta \cdot {{l}}\left( {{{c}}\phi {{c}}\theta } \right)} \hfill \\ {{{\delta l}}_{{E}} = - {{\delta }}\psi \cdot {{l}}\left( {{{c}}\phi {{s}}\theta {{c}}\psi + {{s}}\phi {{s}}\psi } \right) + {{\delta }}\phi \cdot {{l}}\left( {{{c}}\phi {{c}}\theta } \right)} \hfill \\ {{{\delta l}}_{{D}} = - {{\delta }}\phi \cdot {{l}}\left( {{{c}}\phi {{s}}\theta {{s}}\psi - {{s}}\phi {{c}}\psi } \right) + {{\delta }}\theta \cdot {{l}}\left( {{{c}}\phi {{s}}\theta {{c}}\psi + {{s}}\phi {{s}}\psi } \right)} \hfill \\ \end{array} } \right.$$

(29)

Equation (29) shows that the heading angle error, i.e., ${{\delta }}\psi$, only influences the tilt compensation in the horizontal direction. The roll and pitch angle errors, i.e., ${{\delta }}\phi$ and ${{\delta }}\theta$, have an effect on both the horizontal and vertical directions. Even for MEMS IMU, the roll and pitch angles can be computed accurately corresponding to the previous analysis. The components of the tilt compensation errors induced by ${{\delta }}\psi$, are given by:

$$\left\{ {\begin{array}{*{20}l} {{{\delta l}}_{{{{\delta }}\psi ,{{N}}}} = {{\delta }}\psi \cdot {{l}}\left( {{{c}}\phi {{s}}\theta {{s}}\psi - {{s}}\phi {{c}}\psi } \right)} \hfill \\ {{{\delta l}}_{{{{\delta }}\psi ,{{E}}}} = - {{\delta }}\psi \cdot {{l}}\left( {{{c}}\phi {{s}}\theta {{c}}\psi + {{s}}\phi {{s}}\psi } \right)} \hfill \\ \end{array} } \right.$$

(30)

where ${{\delta l}}_{{{{\delta }}\psi ,{{N}}}}$ and ${{\delta l}}_{{{{\delta }}\psi ,{{E}}}}$ are the north and east component of the δψ-induced tilt compensation errors in the horizontal direction. Thus, the magnitude of the combined δψ-induced tilt compensation errors in the horizontal direction is computed as follows:

$${{\delta l}}_{{{{\delta }}\psi ,{{H}}}} = \sqrt {{{\delta l}}_{{{{\delta }}\psi ,{{N}}}}^{2} + {{\delta l}}_{{{{\delta }}\psi ,{{E}}}}^{2} } = {{\delta }}\psi \cdot {{l}}\sqrt {1 - {{c}}^{2} \phi {{c}}^{2} \theta }$$

(31)

Substituting (24) into (23), the third element of the vector $\varvec{l}^{{n}}$, i.e., the vertical component of the tilt compensation, is obtained as:

$${{l}}_{{D}}^{{n}} = {{l}}\cos \phi \cos \theta$$

(32)

According to (Luo et al. 2018), the tilt compensation relates to the tilt angle (denoted by α), which is defined as the angle between the local zenith and the pole, as depicted in Fig. 1. We can readily write the vertical component of the tilt compensation with respect to the tilt angle α, as:

$${{l}}_{{D}}^{{n}} = {{l}}\cos \alpha$$

(33)

Comparing (32) and (33) yields

$$\cos \alpha = \cos \phi \cos \theta$$

(34)

Substituting (34) into (31) yields

$$\begin{array}{*{20}c} {{{\delta l}}_{{{{\delta }}\psi ,{{H}}}} = {{\delta }}\psi \cdot {{l}}\sin \alpha } & {\left( {0^\circ \le \alpha < 90^\circ } \right)} \\ \end{array}$$

(35)

The above equation demonstrates that tilt compensation errors are introduced as a result of the heading angle error, i.e., δψ, the length of the offset between GNSS antenna phase center to the pole tip, and the tilt angles. It should be noted that in (31) and (35), we denote ${{\delta l}}_{{{{\delta }}\psi ,{{H}}}}$ as the magnitude of the horizontal compensation error by ignoring the sign of the quantity. Figure 11 is a plot of the threshold of the heading angle error versus the tilt angles when 2 cm tilt compensation accuracy should be assured for a 2-m lever arm. For example, for a 2 m pole, a 1.15° accuracy is required when the pole is tilted 30° and 2.0° accuracy is required when the pole is tilted 17°.

Fig. 11

Heading angle accuracy requirement versus tilt angle by assuming the pole length l = 2.0 m

Impacts of the Gyro Bias and Random Noise on the Heading Alignment

The random noise in the gyro angular rate measurements is integrated to produce an attitude random walk error. The standard deviation of a random walk process is proportional to the square root of the integration time (Groves 2013). It is common for manufacturers to specify the random noise using the angular random walk (ARW) or power spectral density (PSD). The ARW for the random noise of low-cost gyro chips used in tilted RKT receivers is typically 0.3°/ $\sqrt h$, please refer to the specifications of ICM-20602 listed in Table 1 in the experimental description. This means that after 1-hour integration, the standard deviation of the attitude error is 0.3°. As the proposed heading alignment can be completed in 2 s, the attitude drift produced by random noise will be $0.3 \cdot \sqrt {2/3600} \approx 0.01$ °. Therefore, the impact of the random noise on the gyro-derived attitude is negligibly small.

The fixed gyro bias may comprise the run-to-run variation plus the residual fixed bias remaining after sensor calibration. A fixed gyro bias produces an approximately linear attitude drift within a short time, as follows:

$${{\delta }}\beta = {{b}}_{{g}} {{t}}$$

(36)

where ${{\delta }}\beta$ is the attitude drift, and ${{b}}_{{g}}$ is the fixed gyro bias. The low-cost gyro chips employed in tilted RTK receivers may exhibit large run-to-run gyro biases up to 1.0°/s. In practice, this large fixed gyro bias could be initially estimated by averaging the gyro measurements when the IMU remains stationary. Please note that the earth rotation rate, i.e., the actual input of the gyros when stationary which is about 15°/h, is ignored here and therefore is left as an error source of such gyro bias estimation. Another error source is the impact of the gyro noise to the averaging value of the gyro measurement. Averaging the stationary gyro measurements is a process to filtering out the random noise. As long as the random noise could be averaged out to a sufficient level, the run-to-to bias could be estimated to a certain extent. The standard deviation of the average gyro angular rate measurement varies inversely proportionate to the square root of the averaging time. If the tilted RTK receiver remains stationary for 3 s during roll and pitch initialization and the ARW is 0.3°/ $\sqrt h$, as mentioned above, the standard deviation of the average gyro measurement error is $0.3/\sqrt {3/3600 } \approx 10.4 ^\circ /{{h}}$. Considering this residual impact of the random noise plus the ignored earth rotation rate (15°/h), the run-to-run gyro bias could be estimated to the level of $\sqrt {10.4^{2} + 15.0^{2} }$ = 18.2°/h. After compensating the estimated bias to the gyro measurement, the residual run-to-run bias is reduced to a magnitude of approximately 18.2°/h. Therefore, the attitude drift induced by the gyro bias does not exceed 0.01°, please refer to (36), when the alignment is accomplished within 2 s, which is negligibly small.