3.1 Calibration of the RGB-D camera and IMU

Before the experiment, the gyroscope and accelerometer in the IMU are calibrated using the IMU Toolkit (Valenti et al., Reference Valenti, Dryanovski and Xiao2015), and the RGB-D camera is calibrated using the checkerboard method (Wiedemeyer, Reference Wiedemeyer2015). The transformation matrix of the camera coordinate system with respect to the navigation coordinate system is ${\boldsymbol C}({{\psi_0},{\phi_0},{\theta_0}} )$, where ${\psi _0}$ is the initial heading angle, ${\phi _0}$ is the initial roll angle and ${\theta _0}$ is the initial pitch angle.

The IMU containing a three-axis magnetometer is mounted on the ArUco plane as shown in Figure 3. The ArUco is shifted from Position 1 to Position k(k ≥ 3), and the magnetic field intensity $({m_x^k,m_y^k,m_z^k} )$ and acceleration $({a_x^k,a_y^k,a_z^k} )$ are recorded from the IMU. ${\theta _k}$ and ${\phi _k}$ can be calculated using Equation (2).

(2)\begin{equation}\left( {\begin{array}{@{}c@{}} {{\theta_k}}\\ {{\phi_k}} \end{array}} \right)\textrm{ = }\left( {\begin{array}{@{}c@{}} {\textrm{asin}\left( {\dfrac{{a_x^k}}{g}} \right)}\\ {\textrm{atan}2({a_y^k,a_z^k} )} \end{array}} \right)\end{equation}

$({M_x^k,M_y^k,M_z^k} )$ is the magnetic field intensity in the navigation coordinate system. The heading angle ${\psi _k}$ of the ArUco is calculated using Equations (3)–(6):

(3)\begin{gather}{({M_x^k,M_y^k,M_z^k} )^\textrm{T}}\textrm{ = }{}_b^n{\boldsymbol C}({{\psi_k},{\phi_k},{\theta_k}} ){({m_x^k,m_y^k,m_z^k} )^\textrm{T}}\end{gather}

(4)\begin{gather}M_x^k\textrm{ = }{m_x}\cos {\theta _k}\textrm{ + }{m_y}\sin {\theta _k}\sin {\phi _k}\textrm{ + }{m_z}\cos {\phi _k}\sin {\theta _k}\end{gather}

(5)\begin{gather}M_y^k\textrm{ = }{m_y}\cos {\phi _k} - {m_z}\sin {\phi _k}\end{gather}

(6)\begin{gather}{\psi _k}\textrm{ = arctan}\frac{{M_y^k}}{{M_x^k}}\end{gather}

The rotation matrix ${\boldsymbol C}_k^A$ of the ArUco relative to the camera coordinate system is determined through the ArUco detection library in OpenCV (Romero-Ramirez et al., Reference Romero-Ramirez, Muñoz-Salinas and Medina-Carnicer2018). ${\psi _0}$,${\phi _0}$ and ${\theta _0}$ are calculated as follows:

(7)\begin{equation}\psi _0^\ast \textrm{ = }\mathop {\textrm{arc}\min }\limits_{{\psi _0}} \sum\limits_k {\log [{{{({{\boldsymbol C}({{\psi_0},{\phi_0},{\theta_0}} ){\boldsymbol C}_k^A} )}^\textrm{T}}{}_b^n{\boldsymbol C}({{\psi_k},{\theta_k},{\phi_k}} )} ]}\end{equation}

Figure 3. Calibration of the initial attitude angle

3.2 Tracking of the wall climbing robot

Only the area occupied by the wall-climbing robot in the field of the RGB-D camera is changed during the localisation, which is the region of interest (ROI). To speed up the proposed algorithm, a target-tracking algorithm based on a combination of deep learning and KCF (Henriques et al., Reference Henriques, Caseiro, Martins and Batista2014) is used to obtain the ROI and preliminary position of the robot.

Figure 4 shows the tracking of the wall-climbing robot . Firstly, Yolo-Tiny (Redmon and Farhadi, Reference Redmon and Farhadi2018) is used to find the initial location of target rectangle. Then, taking the initial location and the current frame as the input, KCF is adopted for real-time tracking in the next frame. The Yolo-Tiny is used to relocate the target rectangle when the response value is less than the threshold.

Figure 4. Tracking module

3.3 3D reconstruction of steel components

During the movement of the wall-climbing robot, there is no change in the pixels of the large component in the images acquired by the RGB-D camera. The 3D reconstruction of the steel component is conducted to obtain the 3D point cloud, which is used in the subsequent localisation to obtain the normal vector of the robot chassis, as shown in Figure 5.

Figure 5. 3D reconstruction module

Figure 6 shows a schematic of the pinhole camera model, which describes the mathematical relationship between a point in 3D space and its projection onto the image plane. The pixel coordinate system UOV is defined with the U and V axes parallel to the X _R and Y _R axes, respectively. The centre point C (c _x, c _y) in the camera coordinate system is (0, 0, f), and f is the focal length. A point A(x, y, z) in space is projected to point a(u, v) on the image plane, and the 3D coordinates of A in the camera coordinate system are obtained as follows:

(8)\begin{equation}{({x,y,z} )^\textrm{T}} = {\left( {\frac{{u - {c_x}}}{{{f_x}}},\frac{{v - {c_y}}}{{{f_y}}},1} \right)^\textrm{T}}d/s\end{equation}

where s is the distance scale of the RGB-D camera, and d is the distance from the point to the camera plane. The parameters ${c_x}$, ${c_y}$, ${f_x}$ and ${f_y}$ are obtained by calibration.

Figure 6. The pinhole camera model

3.4 Location based on normal vector projection

Since the range of the target rectangle (obtained in Section 3.2) exceeds the size of the robot itself, only a rough estimate of the position can be obtained after tracking the robot. To solve this problem, a location method based on normal vector projection is proposed.

As shown in Figure 7, in the field of view of the RGB-D camera, the points B ₁-B ₅ on the large component move along the optical path L₁–L₅ to points A ₁–A ₅ on the robot shell, respectively, due to the occlusion of the wall-climbing robot. The lengths of the lines L₁ and L₂ on the top of the robot are reduced by | A ₁B ₁| and | A ₂B ₂|, respectively, in the direction of the light path, while the lengths of the lines L₃-L₅ on the side of the robot are reduced by | A ₃B ₃|, | A ₄B ₄|, and | A ₅B ₅|, respectively.

Figure 7. Length changes in the depth direction caused by the robot

Let point a(u, v) in the pixel coordinate system correspond to point $A({x,y,z} )$ on the robot. When there is no occlusion caused by the robot, the point a corresponds to point $B({x^{\prime},y^{\prime},z^{\prime}} )$ on the surface of the large component, and the inclined line ${\boldsymbol AB}$ is expressed as follows:

(9)\begin{equation}{\boldsymbol AB}\textrm{ = }\left( {\frac{{u - {c_x}}}{{{f_x}}},\frac{{v - {c_y}}}{{{f_y}}},1} \right)\frac{1}{s}\varDelta d({u,v} )\end{equation}

where $\varDelta d({u,v} )$ is the depth change caused by robot occlusion at the pixel a(u, v).

However, the length of ${\boldsymbol AB}$ is related to (u, v) and cannot be directly used to distinguish between the point clouds on top of and around the robot shell, according to Equation (9).

Let ${\boldsymbol n}({n_x^k,n_y^k,n_z^k} )$ be the normal vector of the robot chassis, which is collinear with the normal vector (as shown in Figure 5) in the contact area. The inclined line ${\boldsymbol AB}$ is projected along the vector ${\boldsymbol n}$ to obtain h(j) in Equation (10). The height of the robot is H. The value of h(j) corresponding to the top of the robot is approximately equal to H, while the side is less than H:

(10)\begin{equation}h(j )= {\boldsymbol AB} \cdot {\boldsymbol n}\end{equation}

In the 3D point set $\{{{k_j}|{h(j )= H} } \}$ of the robot top, the midpoint $P({{p_x},{p_y},{p_z}} )$ is selected as the position of the robot.

(11)\begin{equation}{({{p_x},{p_y},{p_z}} )^\textrm{T}}\textrm{ = }{\{{{k_j}|h(j )= H} \}_{0.5}}\end{equation}

3.5 Attitude estimation based on EKF

After the wall-climbing robot reaches the destination, it also needs to adjust the posture of the mechanical arm according to its own attitude in several industrial tasks such as welding, cleaning and detection. Due to the interference of strong magnetic field, a magnetometer cannot be used to correct the heading angle of the wall-climbing robot. Here, an attitude-estimation method based on EKF is proposed to reduce the error accumulation of the heading angle.

The wall-climbing robot moves at a low speed on the surface of large components with an acceleration close to zero. The three-axis output $({a_x^k,a_y^k,a_z^k} )$ of the accelerometer in IMU is approximately equal to the three components of the gravity vector in the body coordinate system. The pitch angle ${\theta _k}$ and roll angle ${\phi _k}$ can be calculated using Equation (2).

In the EKF, the state ${{\boldsymbol x}_k}$ consists of the cosine and the sine of the heading angle:

(12)\begin{equation}{{\boldsymbol x}_k} = {({\cos {\psi_k}\;\sin {\psi_k}} )^\textrm{T}}\end{equation}

where ${{\boldsymbol x}_k}$ can be obtained by multiplying the first two elements of the first column of ${}_b^n{{\boldsymbol C}_k}$ by the factor ${({\cos {\theta_k}} )^{\textrm{ - }1}}$. ${}_b^n{{\boldsymbol C}_k}$ is updated using Equations (13)–(15).

(13)\begin{gather}d({{}_b^n{{\boldsymbol C}_k}} )/dt = {}_b^n{{\boldsymbol C}_k}{[{\omega \times } ]_k}\end{gather}

(14)\begin{gather}{[}{\omega \times } ]\textrm{ = }\left[ {\begin{array}{@{}ccc@{}} 0& { - ({{\omega_z} - {b_z}} )}& {{\omega_y} - {b_y}}\\ {{\omega_z} - {b_z}}& 0& { - ({{\omega_x} - {b_x}} )}\\ { - ({{\omega_y} - {b_y}} )}& {{\omega_x} - {b_x}}& 0 \end{array}} \right]\end{gather}

(15)\begin{gather}{}_b^n{{\boldsymbol C}_{k + 1}} = {}_b^n{{\boldsymbol C}_k} + [{d({{}_b^n{{\boldsymbol C}_k}} )/dt} ]\cdot {t_s}\end{gather}

where $\omega _x^k$, $\omega _y^k$ and $\omega _z^k$ are the angular velocities of IMU at time k. $b_x^k$, $b_y^k$ and $b_z^k$ are the angular velocity drifts of the IMU at time k.

With the pitch angle ${\theta _k}$ at time k obtained by Equation (2), the state transition function of the EKF is shown in Equation (16).

(16)\begin{equation}{{\boldsymbol x}_{k + 1}} = {{\boldsymbol F}_k} \cdot {{\boldsymbol x}_k}\textrm{ + }{\boldsymbol W}_\psi ^k\end{equation}

(17)\begin{equation}{{\boldsymbol F}_k}(1,1) = \frac{{\cos {\theta _k} + {t_s}\varpi _z^k\sin {\phi _k}\sin {\theta _k} - {t_s}\varpi _y^k\cos {\phi _k}\sin {\theta _k}}}{{\cos {\theta _{k + 1}}}}\end{equation}

\[{{\boldsymbol F}_k}(1,2) = \frac{{ - {t_s}\varpi _z^k\cos {\phi _k} - {t_s}\varpi _y^k\sin {\phi _k}}}{{\cos {\theta _{k + 1}}}}\]

\[{{\boldsymbol F}_k}(2,1) = \frac{{{t_s}\varpi _z^k\cos {\phi _k} + {t_s}\varpi _y^k\sin {\phi _k}}}{{\cos {\theta _{k + 1}}}}\]

\[{{\boldsymbol F}_k}(2,2) = \frac{{\cos {\theta _k} + {t_s}\varpi _z^k\sin {\phi _k}\sin {\theta _k} - {t_s}\varpi _y^k\cos {\phi _k}\sin {\theta _k}}}{{\cos {\theta _{k + 1}}}}\]

(18)\begin{equation}\left\{ {\begin{array}{@{}c} {\varpi_y^k = \omega_y^k - b_y^k}\\ {\varpi_z^k = \omega_z^k - b_z^k} \end{array}} \right.\end{equation}

where t_s is the sampling time of IMU, and ${\boldsymbol W}_\psi ^k$ is the zero mean white Gaussian noise.

The normal vector ${\boldsymbol n}({n_x^k,n_y^k,n_z^k} )$ of the robot chassis is calculated through the 3D point cloud of large components obtained in Section 3.3. The vector ${\left( {\begin{array}{*{20}{c}} 0& 1& 0 \end{array}} \right)^\textrm{T}}$ is determined by converting ${\boldsymbol n}({n_x^k,n_y^k,n_z^k} )$ from the navigation coordinate system to the robot body coordinate system,

(19)\begin{equation}{({0\;1\;0} )^\textrm{T}} = {}_b^n{{\boldsymbol C}^\textrm{T}} \cdot {({n_x^k\;n_y^k\;n_z^k} )^\textrm{T}}\end{equation}

Solving the simultaneous Equations (1) and (19) and eliminating θ_k, we get

(20)\begin{equation} \cos {\phi _k} = \left[{\begin{array}{@{}ccc@{}} { - \sin {\psi_k}}& {\cos {\psi_k}}& 0 \end{array}} \right] \cdot {\boldsymbol n}\end{equation}

According to the above relationship between ${\boldsymbol n}({n_x^k,n_y^k,n_z^k} )$ and the heading angle, the measurement function of the EKF is divided into two cases.

1. (i) $n_y^k \ne 0$ or $n_x^k \ne 0$:

   (21)\begin{align} {}^1{{\boldsymbol z}_k} & =\cos {\phi _k}\nonumber\\ {}^1{{\boldsymbol z}_{k\textrm{|}k - 1}} & = {H_1}{x_{k|k - 1}} + {}^1{v_k}\nonumber\\ {H_1} & = \left({\begin{array}{@{}cc@{}} {n_y^k}& { - n_x^k}\end{array}}\right) \end{align}

where ${}^1{v_k}$ is the measurement noise.

The state noise autocorrelation matrix ${{\boldsymbol Q}_k}$ and covariance matrix ${}^1{{\boldsymbol R}_k}$ are expressed as

(22)\begin{gather}{{\boldsymbol Q}_k}\textrm{ = }t_\textrm{s}^2\left( {\begin{array}{@{}cc@{}} {\sigma_{\psi 1}^2}& 0\\ 0& {\sigma_{\psi 2}^2} \end{array}} \right)\end{gather}

(23)\begin{gather}{}^1{{\boldsymbol R}_k}\textrm{ = }\sigma _\phi ^\textrm{2}\end{gather}

where $\sigma _{\psi 1}^2$and $\sigma _{\psi 2}^2$ are the estimated variances of $\cos {\psi _k}$ and $\sin {\psi _k}$, respectively. $\sigma _\phi ^2$ is the measurement noise variance of $\cos {\phi _k}$.

${\boldsymbol d}$ is the normalised factor of the state ${{\boldsymbol x}_k}$, which is defined as

(24)\begin{equation}{\boldsymbol d}= \left( {\begin{array}{@{}cc@{}} {{{|{\cos {\psi_k}} |}^{ - 1}}}& 0\\ 0& {{{|{\sin {\psi_k}} |}^{ - 1}}} \end{array}} \right)\end{equation}

The update steps of the EKF are:

\[{{\boldsymbol P}_{k|k - 1}} = {{\boldsymbol F}_{k - 1}}{{\boldsymbol P}_{k - 1|k - 1}}{\boldsymbol F}_{k - 1}^\textrm{T} + {{\boldsymbol Q}_{k - 1}}\]

\[{{\boldsymbol K}_k} = {{\boldsymbol P}_{k|k - 1}}{{\boldsymbol H}^\textrm{T}}{({{\boldsymbol H}{{\boldsymbol P}_{k|k - 1}}{{\boldsymbol H}^\textrm{T}} + {}^1{{\boldsymbol R}_k}} )^{ - 1}}\]

\[{x^{\prime}_{k|k}} = {{\boldsymbol x}_{k|k - 1}} + {{\boldsymbol K}_k}({{}^1{{\boldsymbol z}_k} - {}^1{{\boldsymbol z}_{k|k - 1}}} )\]

\[{{\boldsymbol x}_{k|k}} = {\boldsymbol d}{x^{\prime}_{k|k}}\]

\[{{\boldsymbol P^{\prime}}_{k|k}} = ({{\boldsymbol I} - {{\boldsymbol K}_k}{\boldsymbol H}} ){{\boldsymbol P}_{k|k - 1}}{({{\boldsymbol I} - {{\boldsymbol K}_k}{\boldsymbol H}} )^\textrm{T}} + {{\boldsymbol K}_k}{}^1{{\boldsymbol R}_k}{\boldsymbol K}_k^\textrm{T}\]

\[{\boldsymbol J} = \frac{{\partial ({{{\boldsymbol x}_k}{\boldsymbol d}} )}}{{\partial ({{{\boldsymbol x}_k}} )}}\]

(25)\begin{equation}{{\boldsymbol P}_{k|k}} = {\boldsymbol J}{{\boldsymbol P^{\prime}}_{k|k}}{{\boldsymbol J}^T}\end{equation}

1. (ii) $n_y^k = 0$ and $n_x^k = 0$. That is, the wall-climbing robot moves in the horizontal plane. Under magnetic interference conditions, the principal components analysis (PCA) method (Tharwat, Reference Tharwat2016) is adopted to calculate the first principal component ${\boldsymbol \tau }({\tau_x^k,\tau_y^k,\tau_z^k} )$ of the 3D point set $\{{{k_j}|{h(j )= H} } \}$ of the robot top, which is the orientation of the wall-climbing robot, as shown in Figure 7.

The vector ${\boldsymbol \tau }({\tau_x^k,\tau_y^k,\tau_z^k} )$ is parallel to the z _b axis of the robot body coordinate system ox _by _bz _b. The vector ${\left( {\begin{array}{*{20}{c}} 0& 0& 1 \end{array}} \right)^\textrm{T}}$ is determined by converting ${\boldsymbol \tau }({\tau_x^k,\tau_y^k,\tau_z^k} )$ from the navigation coordinate system to the robot body coordinate system.

Replacing ${\left( {\begin{array}{*{20}{c}} 0& 1& 0 \end{array}} \right)^\textrm{T}}$ and ${\boldsymbol n}({n_x^k,n_y^k,n_z^k} )$ in Equation (19) with ${\left( {\begin{array}{*{20}{c}} 0& 0& 1 \end{array}} \right)^\textrm{T}}$ and ${\boldsymbol \tau }({\tau_x^k,\tau_y^k,\tau_z^k} )$, we get

(26)\begin{equation}\left[ {\begin{array}{@{}cc@{}} {\cos {\psi_k}}& {\sin {\psi_k}} \end{array}} \right] = \left[ {\begin{array}{@{}cc@{}} { - \tau_y^k}& {\tau_x^k} \end{array}} \right]\end{equation}

According to Equation (26), the measurement function of the EKF is expressed as

(27)\begin{align} {}^2{{\boldsymbol z}_k} & ={\left({\begin{array}{@{}cc@{}} {\tau_y^k}& {\tau_x^k} \end{array}} \right)^\textrm{T}}\nonumber\\ {}^2{{\boldsymbol z}_{k|k - 1}} & ={H_2}{x_{k|k - 1}} + {}^2{v_k}\nonumber\\ {H_2} & = \left( {\begin{array}{@{}cc@{}} { - 1}& 0\\ 0& 1 \end{array}}\right)\end{align}

where ${}^2{v_k}$ is the measurement noise.

The covariance matrix ${}^2{{\boldsymbol R}_k}$ is shown in Equation (28).

(28)\begin{equation}{}^2{{\boldsymbol R}_k} = \left( {\begin{array}{@{}cc@{}} {\sigma_{\tau y}^2}& 0\\ 0& {\sigma_{\tau x}^2} \end{array}} \right)\end{equation}

where $\sigma _{\tau y}^2$ and $\sigma _{\tau x}^2$ are the estimated variances of $\tau _y^k$ and $\tau _x^k$, respectively.

The EKF update steps in (ii) are the same as that in (i), and ${}^1{z_k}$,${}^1{z_{k|k - 1}}$,${H_1}$,${}^1{{\boldsymbol R}_k}$ and ${}^1{v_k}$ are replaced by ${}^2{z_k}$,${}^2{z_{k|k - 1}}$,${H_2}$,${}^2{{\boldsymbol R}_k}$ and ${}^2{v_k}$ in Equation (25).

4.1 Evaluation of the normal vector estimation ability

As an important intermediate variable of the RGBD-IMU-AL method, the normal vector of the robot chassis directly affects the accuracy of the autonomous localisation. The normal vector error ${e_n}$ is defined in Equation (29), where ${{\boldsymbol n}_g}$ is the real normal vector, obtained by the ArUco marker detection method:

(29)\begin{equation}{e_n}\textrm{ = }{\boldsymbol n} \wedge {{\boldsymbol n}_g}\textrm{ = }\arccos \left( {\frac{{{\boldsymbol n} \cdot {{\boldsymbol n}_g}}}{{|{\boldsymbol n} ||{{{\boldsymbol n}_g}} |}}} \right)\end{equation}

When the wall-climbing robot moves in a room with a horizontal plane, its chassis normal vector is fixed and equal to the normal vector of that plane. Therefore, the focus is on evaluating the normal vector of the robot chassis when the robot moves on the steel component (Figure 8b). As shown in Figure 10, the normal vector error does not exceed 3° in the proposed method. According to Section 3.4, the normal vector error determines the accuracy of positioning. According to Equation (20) in Section 3.5, the heading angle is estimated using the normal vector as an observation vector. Reducing the estimation error of normal vector is a focus of our future work.

Figure 10. Errors of the robot chassis normal vector in different paths: (a) Path 1; (b) Path 2

4.2 Evaluation of the position estimation ability

The MS3D method has higher positioning accuracy than other fixed-vision methods, so its performance is compared with that of the RGBD-IMU-AL method. The absolute trajectory error (ATE) ${e_p}$ is defined in Equation (30), where $({{p_x},{p_y},{p_z}} )$ is obtained in Equation (11) and $({{p_{gx}},{p_{gy}},{p_{gz}}} )$ is the real position of the robot:

(30)\begin{equation}{e_p}\textrm{ = }\sqrt {||{{p_x} - {p_{gx}}} ||_2^2\textrm{ + }||{{p_y} - {p_{gy}}} ||_2^2\textrm{ + }||{{p_z} - {p_{gz}}} ||_2^2}\end{equation}

The absolute trajectories of the wall-climbing robot are illustrated in Figure 9. It is clear that the trajectories obtained by the RGBD-IMU-AL method are consistent with the ground truth.

The maximum ATE values for RGBD-IMU-AL and MS3D method are 0⋅02 m and 0⋅17 m, respectively. It can be seen from the peaks in Figure 11 that the RGBD-IMU-AL method has a much smaller deviation from the real value. When the robot moves on the cylindrical steel component, the mean absolute error of the proposed method and MS3D method are 0⋅01 m and 0.075 m, respectively. When the robot moves in the room, the mean absolute error of the proposed method and MS3D method are 0⋅013 m and 0⋅075 m, respectively.

Figure 11. Comparison between absolute errors of the proposed and MS3D methods in different paths: (a) Path 1; (b) Path 2; (c) Path 3; (d) Path 4

The experiments results prove that the RGBD-IMU-AL method can effectively locate the wall-climbing robot, meeting the positioning requirements of industrial tasks and path planning.

4.3 Evaluation of the attitude estimation ability

The existing fixed-vision methods are not suitable for the attitude estimation of low-texture wall-climbing robots (Figure 8a) in terms of real-time performance and accuracy. Therefore, we use the IMU-based DCM method (Wang and Rajamani, Reference Wang and Rajamani2018) to compare with the RGBD-IMU-AL method under magnetic interference conditions. Since the cited two methods estimate the roll angle and pitch angle through the direction of gravity, as shown in Equation (2), we focus on the performance of heading-angle estimation.

Figure 12 illustrates the attitude errors represented by Euler angles. When the robot moves on the cylindrical steel component, the maximum pitch angle error is 2⋅1°, and the maximum roll angle error is 2⋅4°, as shown in Figures 12a and 12b. When the robot moves in the room, the horizontal ground drives the pitch angle (θ = 0°) and the roll angle (ϕ = −90°) to be fixed, according to the definition of the navigation coordinate system in Figure 1.

Figure 12. Attitude error of the proposed method in different paths. The roll angle error and the pitch angle error are zero in Path 3 and Path 4: (a) Path 1; (b) Path 2; (c) Path 3; (d) Path 4

It is evident that the maximum heading angle error using the DCM method is 4⋅7°, which accumulates over time, while it remains within 3⋅1° using the RGBD-IMU-AL method. The heading-angle error caused by Equation (16) can be reduced by the EKF observation in Equation (21) or Equation (27), which is related to the normal vector of the robot chassis. When the robot moves on the cylindrical steel component, the mean heading error of the proposed method and MS3D method are 0⋅19° and 2⋅25°, respectively. When the robot moves in the room, the mean heading error of the proposed method and MS3D method are 0⋅73° and 2⋅17°, respectively. Therefore, the RGBD-IMU-AL method provides a novel solution for the estimation of heading angle under magnetic interference conditions.