An Intraoperative Method for Measuring Acetabular Cup Inclination and Anteversion in Total Hip Arthroplasty Using an Inertial Measurement Unit

The rotation matrix is a fundamental tool for describing the rotation of a rigid body. Rotation about a single axis of the coordinate system is called elementary rotation, and any rotation can be decomposed into a combination of elementary rotations about three axes [20]. The elementary rotation by angle α about the x-axis is represented by a rotation matrix R_x(α) as shown in (4). (4) Rxα=1000cosα−sinα0sinαcosα {{\boldsymbol {R}}_x}\left( \alpha \right) = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & {\cos \alpha } & { - \sin \alpha } \cr 0 & {\sin \alpha } & {\cos \alpha } \cr } } \right]

Similarly, rotation by angle β about the y-axis and rotation by angle γ about the z-axis are the other two elementary rotations and can also be represented by R_y(β) and R_z(γ) as shown in (5) and (6), respectively. (5) Ryβ=cosβ0sinβ010−sinβ0cosβ {{\boldsymbol {R}}_y}\left( \beta \right) = \left[ {\matrix{ {\cos \beta } & 0 & {\sin \beta } \cr 0 & 1 & 0 \cr { - \sin \beta } & 0 & {\cos \beta } \cr } } \right] (6) Rzγ=cosγ−sinγ0sinγcosγ0001 {{\boldsymbol {R}}_z}\left( \gamma \right) = \left[ {\matrix{ {\cos \gamma } & { - \sin \gamma } & 0 \cr {\sin \gamma } & {\cos \gamma } & 0 \cr 0 & 0 & 1 \cr } } \right]

There are two different ways to define the composition of multiple elementary rotations, namely extrinsic rotations and intrinsic rotations [21]. Extrinsic rotations refer to rotations that are applied to a space-fixed coordinate system. When composing extrinsic rotations, the order of rotations is crucial. For example, applying extrinsic rotations in the order of (R₀,R₁,⋯,R_n) results in a net rotation R_e as shown in (7). (7) Re=RnRn−1⋯R1R0 {{\boldsymbol {R}}_{\rm{e}}} = {{\boldsymbol {R}}_{\rm{n}}}{{\boldsymbol {R}}_{{\rm{n}} - 1}}\; \cdots {{\boldsymbol {R}}_1}{{\boldsymbol {R}}_0}

In contrast, intrinsic rotations involve applying each rotation about the axes of the current coordinate system (also known as a body-fixed coordinate system), which is dynamically updated after each rotation. When combining multiple intrinsic-type elementary rotations into a net rotation, the order of matrix multiplication is the opposite of that for extrinsic rotations. For example, the net rotation R_i from a series of intrinsic rotations in the order of (R₀,R₁,⋯,R_n) can be expressed as (8), where the matrix multiplication order follows the sequence of rotations. (8) Ri=R0R1⋯Rn {{\boldsymbol {R}}_{\rm{i}}} = {{\boldsymbol {R}}_0}{{\boldsymbol {R}}_1} \cdots {{\boldsymbol {R}}_{\rm{n}}}

In 1776, Leonhard Euler proposed Euler angles, which describe the relative orientation of coordinate systems by decomposing the rotational transformation into elementary rotations about three axes [22]. In this paper, the Euler angles, known as Tait-Bryan angles, are used, which were proposed by Peter Guthrie Tait and George H. Bryan in the 19th century. These Euler angles, denoted as (α,β,γ), decompose the rotation of a rigid body into three sequential elementary rotations, as shown in (9), (9) Rzyx=RzγRyβRxα==cβcγsαsβcγ−cαsγcαsβcγ+sαsγcβsγsαsβsγ+cαcγcαsβcγ−sαsγ−sβsαcβcαcβ \matrix{ {{{\boldsymbol {R}}_{zyx}}} \hfill & { = {{\boldsymbol {R}}_z}\left( \gamma \right){{\boldsymbol {R}}_y}\left( \beta \right){{\boldsymbol {R}}_x}\left( \alpha \right) = } \hfill \cr {} \hfill & { = \left[ {\matrix{ {{{\rm{c}}_\beta }{{\rm{c}}_\gamma }} & {{{\rm{s}}_\alpha }{{\rm{s}}_\beta }{{\rm{c}}_\gamma } - {{\rm{c}}_\alpha }{{\rm{s}}_\gamma }} & {{{\rm{c}}_\alpha }{{\rm{s}}_\beta }{{\rm{c}}_\gamma } + {{\rm{s}}_\alpha }{{\rm{s}}_\gamma }} \cr {{{\rm{c}}_\beta }{{\rm{s}}_\gamma }} & {{{\rm{s}}_\alpha }{{\rm{s}}_\beta }{{\rm{s}}_\gamma } + {{\rm{c}}_\alpha }{{\rm{c}}_\gamma }} & {{{\rm{c}}_\alpha }{{\rm{s}}_\beta }{{\rm{c}}_\gamma } - {{\rm{s}}_\alpha }{{\rm{s}}_\gamma }} \cr { - {{\rm{s}}_\beta }} & {{{\rm{s}}_\alpha }{{\rm{c}}_\beta }} & {{{\rm{c}}_\alpha }{{\rm{c}}_\beta }} \cr } } \right]} \hfill \cr } where R_x(α), R_y(β), and R_z(γ) are the elementary rotations, as shown in (4)–(6), c_α is short for cos α and s_α is short for sin α, and R_zyx is the net rotation matrix defined by three elementary intrinsic rotations in the axis order of z-y-x. The net rotation is decomposed as: first, rotation R_z(γ) by angle γ about its own z-axis (also called yaw); second, rotation R_y(β) by angle β about its updated y-axis (called pitch); and finally, rotation R_x(α) by angle α about its rotated x-axis (called roll) [20].

Based on this equation, the inverse transformation from rotation matrices to Euler angles (solution for β in the range (−π/2,π/2)) can also be derived, as shown in (10), (10) α=Atan2r32,r33β=Atan2−r31,r322+r332γ=Atan2r21,r11 \left\{ {\matrix{ {\alpha = {\rm{A}}\tan 2\left( {{r_{32}},{r_{33}}} \right)} \hfill \cr {\beta = {\rm{A}}\tan 2\left( { - {r_{31}},\sqrt {r_{32}^2 + r_{33}^2} } \right)} \hfill \cr {\gamma = {\rm{A}}\tan 2\left( {{r_{21}},{r_{11}}} \right)} \hfill \cr } } \right. where r_ij represents the element of the matrix R_zyx in the i-th row and j-th column, and Atan2 is the four-quadrant inverse tangent function designed to resolve ambiguities in angle calculation by considering the signs of its parameters.

In addition to describing rotational movement, a rotation matrix also represents the coordinate transformation relationship before and after rotation. In the IMU-based solution, Euler angles are the output of the IMU sensor and are converted to the rotation matrix, denoted as RIW {\boldsymbol {R}}_{\rm{I}}^{\rm{W}} , which represents the instantaneous orientation of {I} with respect to {W}. Assuming p^W denotes the coordinates of a fixed vector p in {W}, RI1W {\boldsymbol {R}}_{{\rm{I}}1}^{\rm{W}} and RI2W {\boldsymbol {R}}_{{\rm{I}}2}^{\rm{W}} are the rotation matrices obtained when the intrinsic-type IMU is in two different attitudes, and p^I1 and p^I2 denote the coordinates of p in {I} of the first and second attitude, respectively, the transformation of these coordinates can be expressed as (11) according to (8), (11) pW=RI1WpI1=RI1WRI2I1pI2=RI2WpI2 {{\boldsymbol {p}}^{\rm{W}}} = {\boldsymbol {R}}_{I1}^{\rm{W}}{{\boldsymbol {p}}^{{\rm{I}}1}} = {\boldsymbol {R}}_{{\rm{I}}1}^{\rm{W}}{\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1}{{\boldsymbol {p}}^{{\rm{I}}2}} = {\boldsymbol {R}}_{{\rm{I}}2}^{\rm{W}}{{\boldsymbol {p}}^{{\rm{I}}2}} where RI2I1 {\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1} represents the relative rotational transformation of the IMU's second attitude with respect to the first attitude. It can be calculated as (12), (12) RI2I1=RI1W−1RI2W=RI1WTRI2W {\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1} = {\boldsymbol {R}}_{{\rm{I}}1}^{{{\rm{W}}^{ - 1}}}{\boldsymbol {R}}_{{\rm{I}}2}^{\rm{W}} = {\boldsymbol {R}}_{{\rm{I}}1}^{{{\rm{W}}^{\rm{T}}}}{\boldsymbol {R}}_{{\rm{I}}2}^{\rm{W}} where RI1WT {\boldsymbol {R}}_{{\rm{I}}1}^{{{\rm{W}}^{\rm{T}}}} is the transpose of RI1W {\boldsymbol {R}}_{{\rm{I}}1}^{\rm{W}} , and RI1WT=RI1W−1 {\boldsymbol {R}}_{{\rm{I}}1}^{{{\rm{W}}^{\rm{T}}}} = {\rm{R}}_{{\rm{I}}1}^{{{\rm{W}}^{ - 1}}} since it is a rotation matrix.

Assuming the patient and the operating table remain relatively immobile throughout the surgery, the orientation of the operating table can be used to represent the patient's body orientation. For ease of discussion, take the left lateral decubitus position shown in Fig. 2(b) as an example. Since the operating table is horizontally placed, the x_B-axis is considered parallel to the z_W-axis, meaning that: (13) eBxW=eWzW=0,0,1T {\boldsymbol {e}}_{{\rm{B}}x}^{\rm{W}} = {\boldsymbol {e}}_{{\rm{W}}z}^{\rm{W}} = {\left( {0,0,1} \right)^{\rm{T}}} where eBxW {\boldsymbol {e}}_{{\rm{B}}x}^{\rm{W}} and eWzW {\boldsymbol {e}}_{{\rm{W}}z}^{\rm{W}} are the standard basis vectors of the x_B-axis and the z_W-axis in {W}, respectively.

Thus, the transformation relationship between {B} and {W} can be determined if the orientation of the y_B-axis or the z_B-axis is acquired. During registration, an IMU is used to measure the coordinates of the unit vector e_Bz in {W} along the z_B-axis, which is parallel to the long axis of the operating table. The IMU is mounted on the operating table in any orientation, and the operating table is rotated about its long axis by a small angle (about 10°). By recording the Euler angles from IMU before and after rotation, the axis of this rotation — i.e., the z_B-axis — can be obtained. Suppose (α₁,β₁,γ₁) and (α₂,β₂,γ₂) are the Euler angles from the IMU before and after rotation, respectively; the relative rotational transformation can be derived from (9) and (12) as shown in (14). (14) RI2I1=RxTα1RyTβ1RzTγ1Rzγ2Ryβ2Rxα2 {\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1} = {\boldsymbol {R}}_x^{\rm{T}}\left( {{\alpha _1}} \right){\boldsymbol {R}}_y^{\rm{T}}\left( {{\beta _1}} \right){\boldsymbol {R}}_z^{\rm{T}}\left( {{\gamma _1}} \right){{\boldsymbol {R}}_z}\left( {{\gamma _2}} \right){{\boldsymbol {R}}_y}\left( {{\beta _2}} \right){{\boldsymbol {R}}_x}\left( {{\alpha _2}} \right)

Convert the rotational matrix RI2I1 {\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1} in (14) into the axis/angle representation, i.e., parameterize this rotation as a unit vector u = (u_x,u_y,u_z)^T along the rotation axis and a counterclockwise rotation angle φ about this axis, which are expressed as (15) and (16), respectively, (15) cosφ=12trRI2I1−1sinφ=1−cos2φ \left\{ {\matrix{ {\cos \varphi = {1 \over 2}\left[ {{\rm{tr}}\left( {{\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1}} \right) - 1} \right]} \hfill \cr {\sin \varphi = \sqrt {1 - {{\cos }^2}\varphi } } \hfill \cr } } \right. (16) ux=12sinφRI2I13,2−RI2I12,3uy=12sinφRI2I11,3−RI2I13,1uz=12sinφRI2I12,1−RI2I11,2 \left\{ {\matrix{ {{u_x} = {1 \over {2\sin \varphi }}\left[ {{\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1}\left( {3,2} \right) - {\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1}\left( {2,3} \right)} \right]} \hfill \cr {{u_y} = {1 \over {2\sin \varphi }}\left[ {{\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1}\left( {1,3} \right) - {\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1}\left( {3,1} \right)} \right]} \hfill \cr {{u_z} = {1 \over {2\sin \varphi }}\left[ {{\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1}\left( {2,1} \right) - {\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1}\left( {1,2} \right)} \right]} \hfill \cr } } \right. where trRI2I1 {\rm{tr}}\left( {{\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1}} \right) denotes the trace of RI2I1 {\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1} , while RI2I1(i,j) {\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1}({\rm{i}},{\rm{j}}) represents the element in its i-th row and j-th column. Since RI2I1 {\boldsymbol {R}}_{{\rm{I}}2}^{{\rm{I}}1} is an intrinsic-type relative rotation matrix, the coordinates of u (i.e., e_Bz) in (16) are actually in the system {I} before rotation, and can be transformed from {I} to {W} using the transformation matrix RI1W {\boldsymbol {R}}_{{\rm{I}}1}^{\rm{W}} , as shown in (17). (17) eBzW=uW=RI1WeBzI1=RI1Wu {\boldsymbol {e}}_{{\rm{B}}z}^{\rm{W}} = {{\boldsymbol {u}}^{\rm{W}}} = {\boldsymbol {R}}_{{\rm{I}}1}^{\rm{W}}{\boldsymbol {e}}_{{\rm{B}}z}^{{\rm{I}}1} = {\boldsymbol {R}}_{{\rm{I}}1}^{\rm{W}}{\boldsymbol {u}}

At this point, the orientations of the x_B-axis and the z_B-axis are both obtained, so the coordinates of the unit vector along the y_B-axis in {W} are derived from (13) and (17), as shown in (18). (18) eByW=eBzW×eBxW {\boldsymbol {e}}_{{\rm{B}}y}^{\rm{W}} = {\boldsymbol {e}}_{{\rm{B}}z}^{\rm{W}} \times {\boldsymbol {e}}_{{\rm{B}}x}^{\rm{W}}

According to the definition of the rotation matrix, the transformation matrix RBW {\boldsymbol {R}}_{\rm{B}}^{\rm{W}} of the body system {B} relative to the world system {W} can be finally expressed as shown in (19), where eBxB {\boldsymbol {e}}_{{\rm{B}}x}^{\rm{B}} , eByB {\boldsymbol {e}}_{{\rm{B}}y}^{\rm{B}} , and eBzB {\boldsymbol {e}}_{{\rm{B}}z}^{\rm{B}} are the coordinates of the orthonormal basis vectors of {B} in its own system, meaning that the matrix eBxBeByBeBzB \left[ {\matrix{ {{\boldsymbol {e}}_{{\rm{B}}x}^{\rm{B}}} & {{\boldsymbol {e}}_{{\rm{B}}y}^{\rm{B}}} & {{\boldsymbol {e}}_{{\rm{B}}z}^{\rm{B}}} \cr } } \right] is an identity matrix. (19) RBW=RBWeBxBeByBeBzB=eBxWeByWeBzW {\boldsymbol {R}}_{\rm{B}}^{\rm{W}} = {\boldsymbol {R}}_{\rm{B}}^{\rm{W}}\left[ {\matrix{ {{\boldsymbol {e}}_{{\rm{B}}x}^{\rm{B}}} & {{\boldsymbol {e}}_{{\rm{B}}y}^{\rm{B}}} & {{\boldsymbol {e}}_{{\rm{B}}z}^{\rm{B}}} \cr } } \right] = \left[ {\matrix{ {{\boldsymbol {e}}_{{\rm{B}}x}^{\rm{W}}} & {{\boldsymbol {e}}_{{\rm{B}}y}^{\rm{W}}} & {{\boldsymbol {e}}_{{\rm{B}}z}^{\rm{W}}} \cr } } \right]

With the transformation relationship RBW {\boldsymbol {R}}_{\rm{B}}^{\rm{W}} from {B} to {W} established, the registration phase is complete. Compared to other registration methods, this one does not require inserting any markers into the body, avoiding additional damage to the bones. Meanwhile, it is also independent of bony landmarks. However, this registration relies on the assumption of relative immobility between the patient and the operating table, so proper patient positioning and leveling of the operating table are critical to the success of the registration phase.

During this phase, an IMU is mounted on the acetabular cup axis with a specific orientation to measure the cup's inclination and anteversion. Mount the IMU on the acetabular cup impactor such that the x-o-y plane of the IMU coordinate system {I} is perpendicular to the acetabular cup axis. In this case, the axis of the acetabular cup can be represented by the z-axis of {I}. Let n denote the axial vector along the cup axis as shown in (1), so n = e_Iz, and its coordinates in {I} are as shown in (20). (20) nI=eIzI=(0,0,1)T {{\boldsymbol {n}}^{\rm{I}}} = {\boldsymbol {e}}_{{\rm{Iz}}}^{\rm{I}} = {(0,0,1)^{\rm{T}}}

During the THA procedure, the IMU mounted on the cup axis can measure and output the Euler angles (α_c,β_c,γ_c), so the real-time orientation of {I} relative to {W}, denoted as RIcW {\boldsymbol {R}}_{{\rm{Ic}}}^{\rm{W}} , can be derived using (21), which in turn allows the coordinates of n in {W} to be computed as shown in (22). (21) RIcW=RzγcRyβcRxαc {\boldsymbol {R}}_{{\rm{Ic}}}^{\rm{W}} = {{\boldsymbol {R}}_z}\left( {{\gamma _c}} \right){{\boldsymbol {R}}_y}\left( {{\beta _c}} \right){{\boldsymbol {R}}_x}\left( {{\alpha _c}} \right) (22) nW=RIcWnI=RIcWeIzI {{\boldsymbol {n}}^{\rm{W}}} = {\boldsymbol {R}}_{{\rm{Ic}}}^{\rm{W}}{{\boldsymbol {n}}^{\rm{I}}} = {\boldsymbol {R}}_{{\rm{Ic}}}^{\rm{W}}{\boldsymbol {e}}_{{\rm{Iz}}}^{\rm{I}}

Since the transformation matrix of {B} relative to {W} has been established during the registration phase, as given by (19), combining it with (22) yields the coordinates of n in {B}, as shown in (23). (23) nB=RWBnW=RBW−1nW=RBWTRIcWnI {{\boldsymbol {n}}^{\rm{B}}} = {\boldsymbol {R}}_{\rm{W}}^{\rm{B}}{{\boldsymbol {n}}^{\rm{W}}} = {\boldsymbol {R}}_{\rm{B}}^{{{\rm{W}}^{ - 1}}}{{\boldsymbol {n}}^{\rm{W}}} = {\boldsymbol {R}}_{\rm{B}}^{{{\rm{W}}^{\rm{T}}}}{\boldsymbol {R}}_{{\rm{Ic}}}^{\rm{W}}{{\boldsymbol {n}}^{\rm{I}}}

The coordinates n_B = (n_xB,n_yB,n_zB)^T of the vector n in {B} have been obtained at this point. As previously analyzed in (2) and (3) in the problem definition section, the inclination θ_i and anteversion θ_a of the acetabular cup can finally be computed using (3), and the measurement phase is finished.

This experiment aimed to test the fundamental functionality of the IMU used in this paper, i.e., its performance in estimating its own rotational angles. The IMU was mounted on the 3-axis tilt table at an arbitrary initial orientation. The tilt table was then directed to perform a series of rotations: it was first rotated about the z-axis from 0° to 20° and then in the reverse direction to −20°, with the measured values (true values) from the tilt table and the estimated values from the IMU recorded at every 5° intervals; subsequently, the identical rotation procedure and data recording were conducted for the y-axis and x-axis, respectively. The results and the errors are shown in Fig. 5.

Fig. 5.

Angle estimation accuracy of the IMU for single-axis rotation. The true values (blue line) and estimated values (orange line) are referenced to the left vertical axis, while their absolute errors (green line) are referenced to the right vertical axis.

The results demonstrated that, irrespective of the rotation axis, the estimated angles were in good agreement with the true ones, and the absolute errors stayed within ±1°. Importantly, the magnitude of error showed a clear trend of increasing with the rotation angle. The error is attributed to three primary factors: IMU sensor intrinsic noise, the data fusion latency, and mechanical jitter during rotation. RMSE and mean absolute percentage error (MAPE) were acquired as shown in (24) and (25), and were listed in Table 2 to evaluate the angle estimation accuracy.

(24) RMSE=1n∑i=1nEsti−Truei2 {\rm{RMSE}} = \sqrt {{1 \over n}\sum\nolimits_{i = 1}^n {{{\left( {{\rm{Es}}{{\rm{t}}_i} - {\rm{Tru}}{{\rm{e}}_i}} \right)}^2}} }

(25) MAPE=1n∑i=1nEsti−TrueiTruei {\rm{MAPE}} = {1 \over n}\sum\nolimits_{i = 1}^n {\left| {{{{\rm{Es}}{{\rm{t}}_i} - {\rm{Tru}}{{\rm{e}}_i}} \over {{\rm{Tru}}{{\rm{e}}_i}}}} \right|}

In the registration phase, establishing the transformation between coordinate systems {B} and {W} depends on accurately acquiring the orientation of the z-axis of {B} by rotating the operating table. Therefore, the goal of this experiment was to test how well the rotation axis coordinates can be acquired. In the experiment, the IMU was attached to the turntable of the 3-axis tilt table. The tilt table was then rotated about its y-axis to simulate the operating table’s rotation about its long axis during the registration phase, and our method estimated the coordinates of the unit vector along this rotation axis. An algorithm computed and recorded the rotation axis vector coordinates for this movement after the IMU sent the Euler angle data (before and after rotation) to the PC via Bluetooth.

Fig. 6.

Coordinates of rotation axis unit vector u^W in {W} acquired during the registration phase using (17), recorded at four positions.

The tilt table was positioned at four different locations on the test tabletop for the experiment. In multiple measurements at Position 1, the tilt table rotated by a fixed 10° about its y-axis, but the IMU was secured to the tilt table in varying poses each time to test whether the IMU pose affects the rotation axis vector acquisition. For Positions 2–4, the IMU remained in a consistent pose, while the tilt table’s rotation angles about the y-axis varied from 3° to 15°, aiming to investigate the effect of rotation angle. The rotation axis vector coordinates obtained at different tilt table positions, IMU orientations, and rotation angles are plotted in a scatter plot (Fig. 6).

Since the tilt table sits on the test table, its y-axis remains almost horizontal. For the estimated rotation axis vector coordinates uW=(uxW,uyW,uzW)T {{\boldsymbol {u}}^{\rm{W}}} = {(u_x^{\rm{W}},u_y^{\rm{W}},u_z^{\rm{W}})^{\rm{T}}} , the values of uzW u_z^{\rm{W}} should be close to zero, with only uxW u_x^{\rm{W}} and uyW u_y^{\rm{W}} changing across different positions. Among all 20 test results (part of the data are listed in Table 3), only 4 showed minor anomalies where uzW>0.1 \left| {u_z^{\rm{W}}} \right| > 0.1 (likely caused by measurement jitter or noise), while the remaining 16 had uzW \left| {u_z^{\rm{W}}} \right| within 0.04. Although there are 3 anomalies in the 5 tests at Position 1, the estimated vector directions remained largely consistent for each position. This indicates that fixing the IMU on the operating table at any angle can achieve registration, though accuracy may improve when the IMU is in specific poses. The well-performing Pose A was selected for testing at Positions 2-4, and the results were better than those at Position 1, revealing that uzW \left| {u_z^{\rm{W}}} \right| tends to increase when the rotation angle is very small. This suggests the rotation angle of the operating table during the registration phase should not be too small.

Table 3.

Tilt table position	IMU pose	Rotation angle	Estimated axis vector uxW,uyW,uzW \left( {u_x^{\rm{W}},u_y^{\rm{W}},u_z^{\rm{W}}} \right)
Position 1	Pose A	10°	(−0.586, 0.810, −0.024)
	Pose B		(−0.565, 0.819, 0.103)
	Pose C		(−0.579, 0.808, −0.111)
	Pose D		(−0.601, 0.799, −0.022)
	Pose E		(−0.553, 0.812, −0.186)
Position 2	Pose A	3°	( 0.993, 0.119, 0.020)
		6°	( 0.993, 0.120, 0.006)
		9°	( 0.993, 0.119, 0.006)
		12°	( 0.993, 0.117, −0.004)
		15°	( 0.994, 0.114, 0.007)

To verify the accuracy of our method, the 3-axis tilt table was used to simulate the human body coordinate system, while the threaded rod screwed into the center served as the acetabular cup axis, representing the implantation angle of the cup. Through geometric derivation, the cup's θ_i and θ_a were converted into the tilt table’s rotation angles about its x-axis and y-axis, respectively.

The experiment began after the registration was done. The IMU was fixed on the top circular turntable, facing up to ensure its z_I-axis was parallel to the cup axis. Then the tilt table was rotated about its x-axis and y-axis to move the cup axis to a series of preset orientations (changing from (25°, 5°) to (40°, 20°) at 5° intervals). Meanwhile, the PC received real-time Euler angle data from the IMU and immediately estimated the current θ_i and θ_a. The experimental results were saved as 2D vectors (θ_i,θ_a) and plotted on a 2D coordinate graph, as shown in Fig. 7.

Fig. 7.

Comparison of estimated (θ_i,θ_a) values vs. true values. A larger magnitude of the error vector indicates greater error.

The measurement accuracy was evaluated by the RMSE of both angles and the magnitude of the error vector, i.e., θi,θaEst.−θi,θaTrue \left| {{{\left( {\left( {{\theta _{\rm{i}}},{\theta _{\rm{a}}}} \right)} \right)}_{{\rm{Est}}{\rm{.}}}} - {{\left( {\left( {{\theta _{\rm{i}}},{\theta _{\rm{a}}}} \right)} \right)}_{{\rm{True}}}}} \right| . As shown in Table 4, the results verified that the proposed method achieved high measurement accuracy, with the RMSE for both θ_i and θ_a below 0.3°, and the error vector magnitude generally below 0.714° across all test positions. The result is far smaller than the ±10° tolerance range specified in the safe zone. Thus, using the method proposed in this paper can ensure, with high probability, that the acetabular cup is implanted within the safe zone.

Table 4.

RMSE	θi,θaEst.−θi,θaTrue \left| {{{\left( {\left( {{\theta _{\rm{i}}},{\theta _{\rm{a}}}} \right)} \right)}_{{\rm{Est}}{\rm{.}}}} - {{\left( {\left( {{\theta _{\rm{i}}},{\theta _{\rm{a}}}} \right)} \right)}_{{\rm{True}}}}} \right|
θi	θa	Min.	Mean	Max.
0.278°	0.296°	0.073°	0.373°	0.714°

This experiment is divided into two parts. In the proposed method, when mounting the IMU onto the acetabular cup impactor, it is only required that its z_I-axis is parallel to the acetabular cup axis. The first part of this experiment was to verify this claim. We selected two specific acetabular cup orientations — (θ_i,θ_a) = (25°,10°) and (35°,20°) — for testing. During the tests, the IMU’s pose was changed by rotating the circular turntable at the top of the tilt table, with the direction of the IMU z_I-axis unchanged. The estimated results and the error vectors were recorded to analyze the influence of the IMU’s pose, as shown in Fig. 8. The results showed that the average magnitude of the error vectors was 0.987°, which increased compared to that before the IMU pose change (possibly due to jitter generated when rotating the turntable), but still remained within the tolerance range, indicating the method remains effective.

Fig. 8.

Comparison of error vectors in different IMU poses.

Due to the IMU drift error, which accumulates over time, the method proposed in this paper may become less effective over long-term use. Therefore, the second part of this experiment was designed to test the impact of the IMU drift. Similar to Experiment 3, the IMU measured θ_i and θ_a at a series of preset acetabular cup angles, with its pose unchanged. The difference in this experiment was that the procedure was conducted again after 10 and 20 minutes. The influence of drift was analyzed by comparing the error vectors of the estimated values at different times (as shown in Fig. 9).

Fig. 9.

Comparison of error vectors at different times.

The RMSE of measurements and the average magnitude of error vectors at different times are listed in Table 5. Obviously, the IMU drift affected measurement accuracy, as all error results increased progressively with the extension of the test duration. However, even when the test was conducted after 20 minutes, the average error increased approximately 3.5 times, and the absolute errors remained within ±3°, largely meeting the accuracy requirements for acetabular cup orientation measurement. Considering that the acetabular cup implantation procedure in surgery takes approximately 30 minutes, it is recommended that surgeons perform secondary registration midway through the operation to recalibrate the IMU’s angle measurements.

This work shows advantages across multiple aspects, including high prosthesis compatibility, non-invasiveness, and no radiation exposure when compared with other existing technologies, as presented in Table 6. In particular, its angular measurement error is significantly lower than that of other technologies. However, it must be noted that this excellent angular measurement accuracy is associated with the experimental environment. This work was tested on a precise 3-axis tilt table, which eliminated errors caused by patient position shifts and human soft tissue compression. Therefore, the previously mentioned experiments primarily verified the theoretical correctness and feasibility of the proposed method under ideal conditions. Meanwhile, it also indicates that this work needs to be further validated through in vitro tests and clinical trials in the future to examine its effectiveness in complex real surgical scenarios.