2.1. Structure of the DHCR

The prototype and the structure of the DHCR are shown in Fig. 1(a) and (b) respectively. The DHCR consists of an active hemisphere and a passive one. The active hemisphere includes the active hemispherical shell and the radially magnetized NdFeB magnet ring, while the passive hemisphere includes the passive hemispherical shell, the video camera module, the battery, and the radio frequency transmitting module. The active hemispherical shell and the passive one can be fabricated by additive manufacturing. The passive hemisphere is connected with the active hemisphere through the ceramic bearing, thus the active hemisphere can rotate around the passive hemisphere. The main structural parameters are shown in Table I.

Figure 1. The prototype and the structure of the DHCR.

2.2. Three-phase current superposition control

Based on the orthogonal superposition theorem of three alternating magnetic components [Reference Zhang, Wang, Du, Sun and Wang19], the SURMV is superimposed by separately applying tri-phase sine currents to three-axis orthogonal square Helmholtz coils and B are derived as [Reference Zhang, Yu, Yang, Huang and Chen25]

(1) \begin{equation} \boldsymbol{{B}}=\left(\begin{array}{l} B_{x}\\[5pt] B_{y}\\[5pt] B_{z} \end{array}\right)=\left(\begin{array}{l} B_{0}\sqrt{1-\cos ^{2}\theta \cos ^{2}\varphi }\cdot \sin \left(\omega t+\phi _{x}\pm \pi \right)\\[5pt] B_{0}\sqrt{1-\sin ^{2}\theta \cos ^{2}\varphi }\cdot \sin \left(\omega t-\phi _{y}\right)\\[5pt] B_{0}\cos \varphi \sin \left(\omega t\right) \end{array}\right) \end{equation}

where tan $\phi$ _x = tanθ/sin $\varphi$ , tan $\phi$ _y = cotθ/sin $\varphi$ ; B ₀ and $\omega$ are the magnetic flux density and angular velocity of the rotating magnetic vector, respectively; θ and $\varphi$ are the yaw angle and pitch angle of the normal vector n _B of the rotating magnetic vector B , respectively, as shown in Fig. 2.

For the three-axis orthogonal square Helmholtz coils, the relationship of the rotating magnetic vector B and the control input current I can be expressed as follows:

(2) \begin{equation} \boldsymbol{{B}}=\boldsymbol{{KI}} \end{equation}

where K ∈R ^{3 × 3} is the mapping matrix in tesla per ampere from the current input to the magnetic vector

(3) \begin{equation} \boldsymbol{{K}}=\left(\begin{array}{c@{\quad}c@{\quad}c} K_{x} & 0 & 0\\[5pt] 0 & K_{y} & 0\\[5pt] 0 & 0 & K_{z} \end{array}\right) \end{equation}

where K _i (i = x,y,z) are the structural parameters of each coil, respectively [Reference Alamgir, Fang, Gu and Han21, Reference Zhang, Chi and Su26].

Considering the different structural parameters of the three-axis square Helmholtz coils, it is necessary to compensate the control input currents for achieve the uniform magnetic vector. The modified basic triphase sin current equations are derived as

(4) \begin{equation} \boldsymbol{{I}}=\boldsymbol{{K}}^{-\mathrm{1}}\boldsymbol{{B}}=\left(\begin{array}{l} I_{x}\\[5pt] I_{y}\\[5pt] I_{z} \end{array}\right)=\left(\begin{array}{l} K_{\mathrm{z}}/K_{x}I_{0}\sqrt{1-\cos ^{2}\theta \cos ^{2}\varphi }\cdot \sin \left(\omega t+\phi _{x}\pm \pi \right)\\[5pt] K_{\mathrm{z}}/K_{y}I_{0}\sqrt{1-\sin ^{2}\theta \cos ^{2}\varphi }\cdot \sin \left(\omega t-\phi _{y}\right)\\[5pt] I_{0}\cos \varphi \sin \left(\omega t\right) \end{array}\right) \end{equation}

where K _i (i = x,y,z) is the structural parameter of each coil, respectively. The magnetic flux density B ₀ can be obtained from the control input current I ₀, namely, B ₀ = K _z I ₀.

Table I. The structure parameters of the DHCR.

Figure 2. The SURMV generated by the three-axis orthogonal square Helmholtz coils.

Once the desired control parameters (I ₀, $\omega$ , θ and $\varphi$ ) are input into the control system, the coils will generate a sufficient rotating magnetic vector B for posture alignment.

2.3. Manipulation based on tracking effect

As shown in Fig 3, the magnetic torque T is coupled between the radially magnetized magnet ring and the rotating magnetic vector B , which drives the DHCR realizing three basic motions-rotation and tilting motions and 2-D rolling. Before the active hemisphere touches the intestine, the DHCR is in the passive mode for rotation or tilting, while the active hemisphere touches the intestine, the DHCR is in the active mode for rolling locomotion. Tracking effect can navigate the DHCR to convert from the passive mode to the active mode for the rolling locomotion and vice versa. For medical application scenario inside the stomach, as shown in Fig. 3, the detailed operation process is as follows:

Figure 3. The overall medical application scenario of the DHCR and the schemes of the DHCR motions.

Step 1: Posture adjustment in the passive mode can be used for panoramic observation to find the position of the lesion area briefly. First, the doctor applies a vertical upward normal vector n _B0 to make the DHCR in a stable vertical upward with the axis n ₀. Then, the doctor can use a joystick to control the tilting motion with angle $\varphi$ , and the rotation motion around vertical z ₀-axis with angle θ, for surrounding observation ( n ₁ , n ₂ , n ₃ and n ₄). Based on the real-time video, the doctor can determine the position of the lesion area briefly by the image location.

Step 2: Rolling locomotion in the active mode can be used to approach the lesion area. In this case, the doctor attempts to apply the perpendicular normal vector n _B5 with the propagation direction normal vector, and the DHCR realizes rolling locomotion in x ₀ oy ₀ plane along the propagation direction. Thus, the active hemisphere can drive the DHCR to reach the lesion area.

Step 3: Posture adjustment can be used for close observation and further medical treatments. Similar to step 1, the axis of the DHCR is navigated to the orientation n ₆ , n ₇ , n _8, and n ₉. As a single-use product [Reference Van de Bruaene, De Looze and Hindryckx27], the DHCR is automatically excreted from the GI tract after its duty is completed.

Precise control of the posture of the DHCR is the key requirement for achieving 3D maneuvers and for performing dynamic tasks in the future, such as biopsy and localized drug delivery. For the actuation of the DHCR, the strength, frequency, and direction of the SURMV have a great influence on the passive mode. Therefore, by changing the parameters of the SURMV, precise navigation and fine-turning of the motion can be realized to meet the needs of practical applications. Mechanical resistance from its surrounding environment also plays an important role in magnetic navigation. Thus, to navigate the robotic axis n accurately and efficiently, the dynamic stability characteristic of the tracking effect in the passive mode has to be investigated firstly, and the dynamic posture modeling is derived next.

3.1. Establishment of coordinate systems

In passive mode, the active hemisphere at the top is driven to rotate idling, while the passive hemisphere at the bottom dents into the GI tract surface adhered with a layer of oil film by the gravity of the DHCR. Thus, the passive hemisphere slides inside the dent of the GI tract surface, and the DHCR could be regarded as a fix center deflection during the posture adjustment. As shown in Fig. 4, the O-x ₂ y ₂ z ₂ rotating magnetic vector coordinate system and the O-x ₁ y ₁ z ₁ Résal coordinate system are established.

Figure 4. The establishment of the coordinate systems.

The O-x ₂ y ₂ z ₂ rotating magnetic vector coordinate system is transformed from the O-x ₀ y ₀ z ₀ coordinate system by two rotations, respectively, one is around the z ₀-axis by θ (yaw angle) and the other is around the y ₂-axis $\varphi$ (pitch angle). In this coordinate system, the normal vector (objective orientation) n _B is set as the x ₂-axis, and the rotating magnetic vector B just rotates in the y ₂ Oz ₂ plane. The transformation matrix from the O-x ₀ y ₀ z ₀ coordinate system to the O-x ₂ y ₂ z ₂ coordinate system is

(5) \begin{equation} \boldsymbol{{A}}_{2}=\left(\begin{array}{c@{\quad}c@{\quad}c} \cos \theta \cos \varphi & \sin \theta \cos \varphi & \sin \varphi \\[5pt] -\sin \theta & \cos \theta & 0\\[5pt] -\cos \theta \sin \varphi & -\sin \theta \sin \varphi & \cos \varphi \end{array}\right) \end{equation}

In a similar way, the O-x ₁ y ₁ z ₁ Résal coordinate system attached to the DHCR is transformed from the O-x ₀ y ₀ z ₀ coordinate system through rotating $\alpha$ (pitch angle) around z ₀-axis and $\beta$ (yaw angle) around y ₁-axis, and the robotic axis n is the Ox ₁ axis. The transformation matrix from the O-x ₀ y ₀ z ₀ coordinate system to the Résal coordinate system is

(6) \begin{equation} \boldsymbol{{A}}=\left(\begin{array}{c@{\quad}c@{\quad}c} \cos \alpha \cos \beta & \sin \alpha \cos \beta & \sin \beta \\[5pt] -\sin \alpha & \cos \alpha & 0\\[5pt] -\cos \alpha \sin \beta & -\sin \alpha \sin \beta & \cos \beta \end{array}\right) \end{equation}

where the angles $\alpha$ and $\beta$ can describe the posture of the robotic axis n in the O-x ₀ y ₀ z ₀ rotating magnetic vector coordinate system.

The transformation relation from n _ox1 to n _ox0 can be obtained as

(7) \begin{equation} \boldsymbol{{n}}_{ox0}=\boldsymbol{{A}}^{-1}\boldsymbol{{n}}_{ox1} \end{equation}

where the robotic axis n in the O-x ₀ y ₀ z ₀ coordinate system can be expressed as n _ox0 (cos $\alpha$ cos $\beta$ , sin $\alpha$ cos $\beta$ , sin $\beta$ ) and in the O-x ₁ y ₁ z ₁ coordinate system is expressed as n _ox1 (1,0,0).

During posture adjustment, the changing law of the angles $\alpha$ and $\beta$ can reflect the motion of the robotic axis, so as to determine the stability of tracking effect.

3.2. Torque analysis

The external torques acting on the DHCR in posture adjustment include the coupling magnetic torque and the viscoelastic resistance torque between the DHCR and the GI tract.

3.2.1 The coupling magnetic torque

The establishment of the rotating magnetic vector coordinate system and the Résal coordinate system also significantly simplifies the coupling magnetic torque model. The rotating magnetic vector B in the O-x ₂ y ₂ z ₂ coordinate system is expressed as

(8) \begin{equation} \boldsymbol{{B}}_{2}=\left(\begin{array}{c@{\quad}c@{\quad}c} 0 & B_{0}\cos \omega t & B_{0}\sin \omega t \end{array}\right)^{\mathrm{T}} \end{equation}

where B ₀ is the magnetic flux density of the magnetic field.

The magnetic dipole moment m of the permanent magnet can be expressed in the Résal coordinate system as

(9) \begin{equation} \boldsymbol{{m}}=\left(\begin{array}{c@{\quad}c@{\quad}c} 0 & m_{0}\cos \left(\omega t-\delta \right) & m_{0}\sin \left(\omega t-\delta \right) \end{array}\right)^{\mathrm{T}} \end{equation}

where δ is the slip angle between the magnetic dipole moment and the rotating magnetic vector; m ₀ is the modulus of the magnetic dipole moment.

The magnetic torque T can be expressed in the Résal coordinate system as

(10) \begin{equation} \begin{array}{c} \boldsymbol{{T}}=\left(\begin{array}{c@{\quad}c@{\quad}c} T_{x1} & T_{y1} & T_{z1} \end{array}\right)^{\mathrm{T}}=\boldsymbol{{m}}\times \left(\boldsymbol{{A}}^{-1}\boldsymbol{{A}}_{2}\boldsymbol{{B}}_{2}\right)\\[5pt] =m_{0}B_{0}\times \left(\begin{array}{c} \cos \left(\omega t-\delta \right)\left(D_{5}\cos \omega t+D_{6}\sin \omega t\right)-\sin \left(\omega t-\delta \right)\left(D_{3}\cos \omega t+D_{4}\sin \omega t\right)\\[5pt] \sin \left(\omega t-\delta \right)\left(D_{1}\cos \omega t+D_{2}\sin \omega t\right)\\[5pt] -\cos \left(\omega t-\delta \right)\left(D_{1}\cos \omega t+D_{2}\sin \omega t\right) \end{array}\right) \end{array} \end{equation}

where the specific forms of D ₁, D ₂, D ₃, D ₄, D _5, and D ₆ are following as

\begin{equation*} \begin{array}{l} D_{1}=-\mathit{\cos }\mathit{\alpha }\cos \beta \sin \theta +\mathit{\sin }\mathit{\alpha }\cos \beta \cos \theta, \\[5pt] D_{2}=\mathit{\cos }\mathit{\alpha }\mathit{\cos }\mathit{\beta }\cos \theta \sin \varphi +\mathit{\sin }\mathit{\alpha }\mathit{\cos }\mathit{\beta }\sin \theta \sin \varphi -\sin \beta \cos \varphi,\\[5pt] D_{3}=\sin \alpha \sin \theta +\cos \alpha \cos \theta, D_{4}=-\mathit{\sin }\mathit{\alpha }\cos \theta \sin \varphi +\mathit{\cos }\mathit{\alpha }\sin \theta \sin \varphi,\\[5pt] D_{5}=-\mathit{\cos }\mathit{\alpha }\sin \beta \sin \theta -\mathit{\sin }\mathit{\alpha }\sin \beta \cos \theta, \\[5pt] D_{6}=-\mathit{\cos }\mathit{\alpha }\mathit{\sin }\mathit{\beta }\cos \theta \sin \varphi -\mathit{\sin }\mathit{\alpha }\mathit{\sin }\mathit{\beta }\sin \theta \sin \varphi +\cos \beta \cos \varphi. \end{array} \end{equation*}

3.2.2 Dynamic resistance torque

During the rotating process of the DHCR for posture adjustment, the deformed GI tract surface and the digestive fluid exert a viscoelastic damping effect on the DHCR [Reference Flom and Bueche28, Reference Johnson29]. According to the transformation process in Fig. 4, the passive hemisphere of the DHCR rotates around the z ₀-axis and the y ₁-axis. The viscoelastic damping effect on the DHCR can be expressed as

(11) \begin{equation} \left\{\begin{array}{l} M_{fy1}=-k\dot{\beta }\\[5pt] M_{fz2}=-k\dot{\alpha } \end{array}\right. \end{equation}

where $\dot{\alpha }$ and $\dot{\beta }$ are the angular velocity around the z ₀-axis and the y ₁-axis, respectively, and k is the damping coefficient between the DHCR and the contact surface. And the dynamic resistance torque M _f in the Résal coordinate system can be obtained as

(12) \begin{equation} \boldsymbol{{M}}_{f}=\left(\begin{array}{l} M_{fx1}\\[5pt] M_{fy1}\\[5pt] M_{fz1} \end{array}\right)=\left(\begin{array}{l} -k\dot{\alpha }\sin \beta \\[5pt] -k\dot{\beta }\\[5pt] -k\dot{\alpha }\cos \beta \end{array}\right) \end{equation}

3.2.3 Total External Torque

According to Eqs. (11) and (12), the total external torque of DHCR can be written as

(13) \begin{equation} \left\{\begin{array}{l} M_{x1}=T_{x1}-k\dot{\alpha }\sin \beta \\[5pt] M_{y1}=T_{y1}-k\dot{\beta }\\[5pt] M_{z1}=T_{z1}-k\dot{\alpha }\cos \beta \end{array}\right. \end{equation}

When the DHCR reaches the equilibrium state toward the target orientation, without any pitch and yaw motion, the active hemisphere only rotates synchronously with the magnetic vector with the angular velocity $\omega$ , thus, M _x1 = 0.

3.3. Dynamic equations

The DHCR has a multirigid structure with complex internal forces, which need not be considered in the Lagrange equation. Thus, the dynamic movement of the DHCR can be described by the Lagrange equation as

(14) \begin{equation} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_{i}}\right)-\frac{\partial L}{\partial q_{i}}=Q_{i} \end{equation}

where q _i ( $\alpha$ and $\beta$ ) are the generalized coordinate, Q _i is the generalized torque in the generalized coordinate system, and L = T _L -V _L is the Lagrangian, where T _L and V _L are the kinetic and potential energies, respectively.

The angular velocity $\boldsymbol{\omega}$ _a of the active hemisphere and the angular velocity $\boldsymbol\omega$ _p of the passive hemisphere in the Résal coordinate system can be expressed as

(15) \begin{equation} \boldsymbol{\omega }_{a}=\left(\begin{array}{l} \omega _{ax1}\\[5pt] \omega _{ay1}\\[5pt] \omega _{az1} \end{array}\right)=\left(\begin{array}{c@{\quad}c@{\quad}c} \cos \beta & 0 & -\sin \beta \\[5pt] 0 & 1 & 0\\[5pt] \sin \beta & 0 & \cos \beta \end{array}\right)\cdot \left(\begin{array}{l} 0\\[5pt] 0\\[5pt] \dot{\alpha } \end{array}\right)+\left(\begin{array}{l} \omega \\[5pt] \dot{\beta }\\[5pt] 0 \end{array}\right)=\left(\begin{array}{c} \omega -\dot{\alpha }\sin \beta \\[5pt] \dot{\beta }\\[5pt] \dot{\alpha }\cos \beta \end{array}\right) \end{equation}

(16) \begin{equation} \boldsymbol{\omega }_{p}=\left(\begin{array}{l} \omega _{px1}\\[5pt] \omega _{py1}\\[5pt] \omega _{pz1} \end{array}\right)=\left(\begin{array}{c@{\quad}c@{\quad}c} \cos \beta & 0 & -\sin \beta \\[5pt] 0 & 1 & 0\\[5pt] \sin \beta & 0 & \cos \beta \end{array}\right)\cdot \left(\begin{array}{l} 0\\[5pt] 0\\[5pt] \dot{\alpha } \end{array}\right)+\left(\begin{array}{l} 0\\[5pt] \dot{\beta }\\[5pt] 0 \end{array}\right)=\left(\begin{array}{c} -\dot{\alpha }\sin \beta \\[5pt] \dot{\beta }\\[5pt] \dot{\alpha }\cos \beta \end{array}\right) \end{equation}

The kinetic energy T _L of DHCR in the passive mode is

(17) \begin{align} T_{L}&=\frac{1}{2}J_{1}\omega _{ax1}^{2}+\frac{1}{2}J_{2}\omega _{px1}^{2}+\frac{1}{2}J_{e}\left(\omega _{py1}^{2}+\omega _{pz1}^{2}\right) \nonumber \\[5pt] & =\frac{1}{2}J_{1}\left(\omega -\dot{\alpha }\sin \beta \right)^{2}+\frac{1}{2}J_{2}\left(-\dot{\alpha }\sin \beta \right)^{2}+\frac{1}{2}J_{e}\left(\dot{\alpha }\cos \beta \right)^{2}+\frac{1}{2}J_{e}\dot{\beta }^{2} \end{align}

where J ₁ and J ₂ are the polar moments of inertia of the active and passive hemisphere, respectively, J _e is the equatorial moment of inertia of the whole DHCR.

Assuming that the center of the DHCR locates in the zero potential plane, thus, the potential energy V _L of DHCR is

(18) \begin{equation} V_{L}=-{mgl} \, \text{sin} \beta \end{equation}

where l is the distance from the center of gravity to the center of DHCR.

Substituting (17) and (18) into (14), the posture dynamic modeling of the DHCR in passive mode can be obtained:

(19) \begin{equation} \left\{\begin{array}{l} \ddot{\alpha }\left[\left(J_{1}+J_{2}\right)\sin ^{2}\beta +J_{e}\cos ^{2}\beta \right]+2\left(J_{1}+J_{2}-J_{e}\right)\dot{\alpha }\dot{\beta }\sin \beta \cos \beta -J_{1}\omega \dot{\beta }\cos \beta =Q_{1}\\[5pt] J_{e}\ddot{\beta }-\left(J_{1}+J_{2}-J_{e}\right)\dot{\alpha }^{2}\sin \beta \cos \beta +J_{1}\omega \dot{\alpha }\cos \beta +{mgl} \, \text{cos} \beta =Q_{2} \end{array}\right. \end{equation}

where the generalized torque Q ₁ = M _z1cos $\beta$ and Q ₂ = M _y1.

For the dynamic model (19) also can be formulated in a similar manner as the form:

(20) \begin{equation} \boldsymbol{{M}}\!\left(\boldsymbol{{q}}\right)\ddot{\boldsymbol{{q}}}+\boldsymbol{{C}}\!\left(\boldsymbol{{q}},\dot{\boldsymbol{{q}}}\right)\dot{\boldsymbol{{q}}}+\boldsymbol{{G}}\!\left(\boldsymbol{{q}}\right)+\boldsymbol{{M}}_{f}\!\left(\dot{\boldsymbol{{q}}}\right)=\boldsymbol{{T}} \end{equation}

where T is the torque input, $\boldsymbol{{M}}_{f}\!\left(\dot{\boldsymbol{{q}}}\right)$ is the resistance torque, $\boldsymbol{{M}}\!\left(\boldsymbol{{q}}\right)$ and $\boldsymbol{{C}}\!\left(\boldsymbol{{q}},\dot{\boldsymbol{{q}}}\right)$ are the inertia matrix and centrifugal force and Coriolis force matrix, and $\boldsymbol{{G}}\!\left(\boldsymbol{{q}}\right)$ is the gravity item, respectively.

4.1. Critical stability control line and stable domain

In complex GI tract environment, the control parameters (B ₀ and $\omega$ ) need to be adjusted in the stability domain to achieve precise control. Then, the critical values B _c and $\omega$ _c are selected when the system is periodic oscillation, and the critical stability control lines with different k are shown in Fig. 6(a).

In Fig. 6(a), the domain is divided into two parts by the critical line. In the upper region, the system phase diagram in Fig 6(b) with angles $\alpha$ and $\beta$ as state variables is divergent, which indicates that the system is unstable. On the contrary, the DHCR can keep the posture stable in the lower region, and the system phase diagram is asymptotic stable at objective orientation. When the selected parameters are near the critical line, the system phase diagram is periodic oscillation. The lower area can be called the stable domain.

4.2. The influence of parameters on stability

4.2.1 Magnetic flux density

From Fig. 6(a), taking B ₀ as the single variable, when B ₀ is less than B _c, the system can finally stabilize at a predetermined orientation; when B ₀ is greater than B _c, the system is unstable. It shows that with the increase of B ₀, the system changes from a stable state to an unstable state. The simulation results with different magnetic flux density B ₀ in stable domain are shown in Fig. 7(a). The smaller B ₀ can decrease the vibration amplitude, and the time to achieve a stable state. Thus, proper reduction of B ₀ is beneficial for the system to reach a stable state. It is noteworthy that when B ₀ is too smaller, the effect of the resistance torque is enhanced, not conducive to subsequent posture adjustment, thus, this situation should be avoided.

Figure 6. (a) The critical stability control lines with different k; (b) the system phase diagram with the angle $\alpha$ and $\beta$ .

Figure 7. (a) Response curves of the posture angle $\alpha$ with different values of B ₀ and (b) the response curves of the posture angle $\alpha$ with different values of $\omega$ .

6.1. Experimental set-up

In order to verify the theoretical analysis results, the experimental set-up is built, as shown in Fig. 10. The platform consists of the tri-axis orthogonal square Helmholtz coils, the control system of the rotating magnetic vector, the host computer, and the prototype of the DHCR. During posture alignment control, first, when the control parameters (B ₀, $\omega$ , q _d ) are input to the host computer, the control system can input three-phase alternating current to the tri-axis orthogonal square Helmholtz coils to generate the SURMV. Then, the corresponding tracking torque can be generated to rotate the DHCR; thus, the axis of DHCR can keep tracking the normal orientation of the SURMV. The GI tract image can be acquired and displayed on the computer, and observation can be performed based on the image. Finally, we repeat the above steps for the next scan point observation until the GI screening is completed.

Figure 10. The (a) experimental set-up and (b) system architecture of the DHCR system.

6.2. Motion and dynamic characteristics test

In motion and stability test, as shown in Fig. 11, the DHCR is placed in a simulated GI tract environment, in which the tilting motion is tested in z ₀ oy ₀ plane, the rotation motion is tested around the z ₀-axes, and the rolling locomotion is tested along the y ₀-axes. We find that three motions of the DHCR can be realized by the normal vector n _B navigation based on tracking effect.

Figure 11. Motion test of the DHCR.

To verify the posture stability domains by tracking effect, the orientation measuring device of the DHCR axis is developed. The schematic diagram of the device is shown in Fig. 12. The camera module is replaced with a laser diode. The coordinates of the bright trajectory are converted into the amplitude of the oscillation to estimate the stability. The coordinate paper is placed at l = 80 mm above the sphere center of the DHCR. The bright spot of the laser diode on the coordinate paper can reflect the bright trajectory of the axis in real time, and the trajectory can be recorded by the video measuring device. Setting the initial orientation of the DHCR is (0^o,80^o), and the objective orientation of the rotating magnetic vector is (0^o,90^o), which is convenient to measure accurately with the smaller magnetic field error and to read the posture of the DHCR.

Figure 12. The (a) schematic diagram and (b) orientation measuring device.

By observing the trajectory of the DHCR axis, it can be found that there are three kinds of responses under different parameters: asymptotic stable response, periodic oscillation response, and divergent response, as shown in Fig. 13, consistent with the Fig. 6(b).

Figure 13. The snapshot of the trajectory of the DHCR axis.

The prerequisite for obtaining accurate posture alignment is to ensure the stability of dynamic tracking effect. Thus, to obtain the stability domains of control parameters, the critical stability values are measured with different magnetic flux densities B ₀, angular velocities $\omega$ , and the damping coefficients k of the DHCR. Figure 14 is a comparison diagram of the measured values and calculated values of the critical stability control lines. Because the effect of GI tract environment on the DHCR is simplified in the theoretical model, the calculated values are smaller than the measured values, while the differences between two values are smaller, indicating that the numerical results of the critical control line are close to the experimental results. We also notice that when k increases, the stability domain of magnetic flux density and angular velocity is significantly expanded, and the difference between the calculated values and the measured values decreases, indicating that the larger k is conducive to stability.

Figure 14. Critical stability control lines with different k.

Then, the experiments are carried out to verify the influence of control parameters (magnetic flux density B ₀ and angular velocity $\omega$ ) in stability domain on stability. The amplitude of the oscillation of the DHCR can be used to evaluate the stability of tracking effect. Figure 15 shows the amplitude of the oscillation of the DHCR when B ₀ is from 6 to 10mT (when $\omega$ = 20π rad/s), and $\omega$ is from 12 to 20 rad/s (when B ₀ = 6.4mT). From the Fig. 15, it can be found that the amplitude of the oscillation is directly proportional to magnetic flux density and inversely proportional to the angular velocity. The experiment results are consistent with the theoretical calculation in Fig. 6. In general, in the case of the asymptotic stable response, we can obtain:

• The magnetic flux density B ₀ is smaller, and its stability is better.

• The larger the angular velocity $\omega$ (larger than the critical value $\omega$ _c ) is beneficial to its stability.

• The larger damping coefficient k is beneficial to its stability.

Figure 15. The amplitude of the oscillation of the DHCR versus the magnetic flux density B ₀ and angular velocity $\omega$ .

6.3. Posture alignment performance test

To verify validity and efficiency of the dynamic tracking effect, the posture alignment performance tests are carried out. The sliding mode control algorithm is embedded into the control system, and the corresponding tracking torque can be generated to rotate the DHCR after the control parameters are input. Then the yaw angle and pitch angle errors of the DHCR are measured. The results are compared with open loop control, as shown in Table III. The results show that both control methods can achieve posture alignment with small errors, less than the ALICE system [Reference Lee, Choi, Go, Jeong, Young Ko, Park and Park16], and the sliding mode control method further improves the posture alignment performance.

Table III. Posture tracking error verification results.

Figure 16. The curves of the (a) the yaw angle and pitch angle and (b) trajectory of the DHCR’s axis.

Figure 16 shows the curves of the yaw angle and pitch angle and trajectory of the DHCR’s axis when the target point is (70^o, 80^o). Once the target orientation parameters are determined, the DHCR can respond quickly under the uniform SURMV and eventually stabilized near the objective orientation. Thus, it can be seen that the axis of the robot will track the normal vector of the rotating magnetic vector efficiently and precisely.