Optimization method for avoiding motion isolation decoupling failure of rotary inertial navigation carrier

Abstract

The invention relates to an optimization method for avoiding motion isolation decoupling failure of a rotary inertial navigation carrier, which comprises the following steps: in four-axis rotary inertial navigation, defining an outer frame, a middle frame, an inner frame, a table body frame, rotation angles corresponding to the four frames and coordinate systems corresponding to the four frames; determining values detected by three gyroscopes of which the rotation angular velocities of the four frames are mapped to the X direction, the Y direction and the Z direction; determining angular velocities sensed by the gyro assembly, wherein the angular velocities comprise an earth rotation angular velocity, a carrier motion angular velocity and a frame rotation angular velocity; decoupling calculation is carried out on the angular velocity of the carrier, and decoupling model formulas under four states that four frames do not participate in motion isolation are obtained; and according to the decoupling model formula, obtaining a basic condition of possible motion decoupling failure and a combination condition of decoupling failure, and determining to adopt the corresponding decoupling model formula to realize motion isolation. The problems that single-axis and double-axis rotary inertial navigation cannot completely isolate carrier motion and three-axis rotary inertial navigation fails in decoupling in carrier motion isolation are effectively solved.

Description

Technical Field

[0001] The present invention relates to the field of inertial navigation technology, and in particular to an optimization method for avoiding the failure of motion isolation and decoupling of a rotating inertial navigation carrier. Background Art

[0002] The rotating inertial navigation system can realize the modulation of the constant error of the inertial device through system-level rotation self-compensation, so that the error can be integrated and offset within a cycle, thereby improving the navigation and positioning accuracy. However, when the carrier has angular motion, the rotation modulation angular motion may be superimposed on the carrier motion, making it impossible to integrate and offset the constant error of the inertial device within the modulation period, affecting the modulation effect. Therefore, when the carrier has angular motion, it should be isolated. At present, the rotating inertial navigation is mainly divided into single-axis, dual-axis, and three-axis rotating inertial navigation systems according to the number of rotating axes. For example, Zhang Lundong et al. studied the influence of the carrier angular motion on the modulation for the single-axis rotating inertial navigation system, and analyzed the heading motion compensation method. (Zhang Lundong, Lian Junxiang, Wu Meiping, Hu Xiaoping. Research on the heading isolation method of the carrier of the single-axis rotating inertial navigation system [J]. Chinese Journal of Scientific Instrumentation, 2012, 33(6): 1247-1252.) Ren Yuan et al. and Deng Zhihong et al. proposed a carrier motion isolation modulation method based on angle control for the dual-axis rotating inertial navigation system, which achieved angular motion isolation in the x and z directions and improved the navigation accuracy of the system. (Ren Yuan, Zhu Ting, Wang Lifen, et al. A method for isolating and modulating the angular motion of the carrier of a dual-axis rotating inertial navigation system [P]. CN 111964696 A, 2020.11; Deng Zhihong, Cai Shanbo, Fu Mengyin, et al. A method for isolating and modulating the angular motion of the carrier of a dual-axis rotating inertial navigation system [P]. CN 105588562A, 2016.05.)

[0003] However, the single-axis or dual-axis rotational inertial navigation systems used in the above studies can only achieve partial isolation of the carrier motion. If motion isolation in three directions is to be achieved, theoretically at least three axes are required. Hu Xiaomao et al. analyzed the three-axis frame and designed a stabilization loop based on angular velocity feedback to achieve the isolation of the carrier angular motion. However, this solution will fail to decouple when the middle frame rotates 90 or 270 degrees, and cannot effectively isolate the carrier motion. (Hu Xiaomao, Yu Hao, Yin Luan. Fiber optic gyro inertial navigation stabilization platform and rotation modulation method [J], 2018, 5(2): 23-28.) Summary of the invention

[0004] The purpose of the present invention is to provide an optimization method to avoid the failure of isolation and decoupling of rotating inertial navigation carrier motion in order to solve the problem that single-axis, dual-axis and three-axis rotating inertial navigation systems cannot achieve complete isolation of carrier motion and decoupling failure of carrier motion.

[0005] The above-mentioned object of the present invention is achieved by the following technical solutions:

[0006] An optimization method for avoiding the failure of the motion isolation and decoupling of a rotating inertial navigation carrier, characterized in that it comprises the following steps:

[0007] Step 1: In the four-axis rotational inertial navigation, define the outer frame, middle frame, inner frame, platform frame, the rotation angles corresponding to the four frames, and the coordinate systems corresponding to the four frames;

[0008] Step 2: In the four-axis rotational inertial navigation, determine the rotational angular velocities of the four frames and map them to the values ​​detected by the three gyroscopes in the X, Y, and Z directions;

[0009] Step 3: In the rotating inertial navigation system, determine the angular velocity that the gyro assembly is sensitive to, including the angular velocity of the earth's rotation, the angular velocity of the carrier's motion, and the angular velocity of the frame's rotation;

[0010] Step 4, decoupling calculation is performed on the carrier angular velocity to obtain decoupling model formulas under four states: the outer frame does not participate in motion isolation, the middle frame does not participate in motion isolation, the inner frame does not participate in motion isolation, and the platform frame does not participate in motion isolation;

[0011] Step 5: Based on the decoupling model formulas under the four states, the basic situations in which motion decoupling failure may occur and the combination situations in which motion decoupling failure may be obtained are obtained, and the corresponding decoupling model formulas are determined to achieve motion isolation.

[0012] Moreover, in step 1, the outer frame is defined as frame 1, the outer frame rotation angle is α, and the corresponding coordinate system is x1y1z1; the middle frame is defined as frame 2, the middle frame rotation angle is β, and the corresponding coordinate system is x2y2z2; the inner frame is defined as frame 3, the inner frame rotation angle is γ, and the corresponding coordinate system is x3y3z3; the platform frame is defined as frame 4, the platform frame rotation angle is θ, and the corresponding coordinate system is x4y4z4; the local geographic coordinate system is x n y n z n .

[0013] Moreover, in step 2, the rotational angular velocities of the four frames are mapped to the values ​​ω detected by the three gyroscopes in the X, Y, and Z directions. px ,ω py ,ω pz for:

[0014]

[0015] In the formula, α is the rotation angle of the outer frame; β is the rotation angle of the middle frame; γ is the rotation angle of the inner frame; θ is the rotation angle of the platform frame; is the angular velocity of the outer frame rotation; is the angular velocity of the middle frame rotation; is the inner frame rotation angular velocity; is the angular velocity of the table frame rotation.

[0016] Furthermore, in step 3,

[0017]

[0018] In the formula, b represents the carrier coordinate system, n represents the navigation coordinate system, e represents the earth coordinate system, p represents the gyro body coordinate system, i represents the inertial coordinate system, and k represents a certain moment. represents the projection of the angular velocity of the p system relative to the b system at time k on the p system, It represents the projection of the angular velocity of the b system relative to the n system at time k in the p system, that is, the carrier angular velocity, It represents the projection of the angular velocity of system n relative to system e at time k in system p, It represents the projection of the angular velocity of the e system relative to the i system at time k in the p system, It represents the projection of the angular velocity of the p system relative to the i system in the p system, that is, the angular velocity sensed by the gyro component;

[0019] and Solve according to the following formula:

[0020]

[0021] In the formula, is the transformation matrix of system n relative to system p at time k, is the projection of the angular velocity of the e system relative to the i system at time k in the e system, that is, the angular velocity of the earth's rotation, which is a constant value. L(k) is the latitude of the carrier at time k, and V E (k) is the eastward velocity of the carrier at time k, V N (k) is the northward velocity of the carrier at time k, and R is the radius of the earth;

[0022] Then, the angular velocity of the carrier at time k, that is, the projection of the angular velocity of the b system relative to the n system in the p system Then the separation can be performed, namely:

[0023]

[0024] Furthermore, in step 4, if the outer frame does not participate in motion isolation, then:

[0025]

[0026] If the middle frame does not participate in motion isolation, then:

[0027]

[0028] If the inner frame does not participate in motion isolation, then:

[0029]

[0030] If the table frame does not participate in motion isolation, then:

[0031]

[0032] Moreover, in step 5, the basic situation of motion decoupling failure is:

[0033] (5.1) When γ approaches 90 or 270 degrees, that is, cosγ approaches 0 and sinγ approaches 1;

[0034] Formula (6), (7) or (8) should be used for motion isolation;

[0035] (5.2) When γ approaches 0 degrees, that is, sinγ approaches 0 and cosγ approaches 1;

[0036] Formula (5), (7) or (8) should be used for motion isolation;

[0037] (5.3) When β approaches 0 degrees, that is, sinβ approaches 0 and cosβ approaches 1;

[0038] Formula (5), (6) or (8) should be used for motion isolation;

[0039] (5.4) When β approaches 90 or 270 degrees, that is, cosβ approaches 0 and sinβ approaches 1;

[0040] Formula (5), (6) or (7) should be used for motion isolation;

[0041] The combination of motion decoupling failure is:

[0042] (5.5) When γ approaches 90 or 270 degrees, that is, cosγ approaches 0, sinγ approaches 1, and β approaches 0 degrees, that is, sinβ approaches 0, cosβ approaches 1;

[0043] Use formula (6) or (8) for motion isolation;

[0044] (5.6) When γ approaches 90 or 270 degrees, that is, cosγ approaches 0, sinγ approaches 1, and β approaches 90 or 270 degrees, that is, cosβ approaches 0, sinβ approaches 1;

[0045] Use formula (6) or (7) for motion isolation;

[0046] (5.7) When γ approaches 0 degrees, that is, sinγ approaches 0, cosγ approaches 1, and β approaches 0 degrees, that is, sinβ approaches 0, cosβ approaches 1;

[0047] Use formula (5) or (8) for motion isolation;

[0048] (5.8)γ approaches 0 degrees, that is, sinγ approaches 0, cosγ approaches 1, and β approaches 90 or 270 degrees, that is, cosβ approaches 0, sinβ approaches 1;

[0049] Formula (5) or (7) should be used for motion isolation.

[0050] The present invention has the advantages and positive effects:

[0051] The present invention constructs an optimization method to avoid the failure of the isolation and decoupling of the motion of the rotating inertial navigation carrier. The four-axis rotating inertial navigation is used to realize the isolation of the carrier motion. After calculating and separating the angular velocity of the carrier, the three frame motors in the four-axis rotating inertial navigation are applied with corresponding angular velocities to achieve the purpose of carrier isolation. This method can solve the problem that the single-axis or dual-axis rotating inertial navigation system cannot achieve the complete isolation of the carrier motion, and at the same time can avoid the decoupling failure caused by the frame locking in the three-axis rotating inertial navigation. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a schematic diagram of the four-axis rotation inertial navigation structure of the present invention;

[0053] Figure 2 It is a schematic diagram of the structure when the table frame of the present invention is locked and the middle frame is rotated 90 or 270 degrees. DETAILED DESCRIPTION

[0054] The structure of the present invention is further described below with reference to the accompanying drawings and by way of examples. It should be noted that the present examples are descriptive rather than restrictive.

[0055] An optimization method to avoid the failure of the rotational inertial navigation carrier motion isolation decoupling, see Figure 1-2 The invention point is: using four-axis rotation inertial navigation to achieve carrier motion isolation, after calculating the carrier angular velocity, applying corresponding angular velocities to three of the frame motors, while achieving the isolation purpose, it can also avoid decoupling failure caused by frame locking.

[0056] In a four-axis frame, if three of the frames are used, there will be a loss of freedom in some cases, but there are various ways to convert them, and appropriate adjustments can be made to avoid related problems. In general, a three-axis rotational inertial navigation system is required to isolate the angular motion of the carrier, but decoupling failure will occur, resulting in the inability to isolate the angular motion of the carrier. Therefore, a four-axis rotational inertial navigation system in the form of a four-frame can be used to avoid this problem.

[0057] Step 1. In the four-axis rotation inertial navigation, define the outer frame as frame 1, the outer frame rotation angle is α, and the corresponding coordinate system is x1y1z1; the middle frame is frame 2, the middle frame rotation angle is β, and the corresponding coordinate system is x2y2z2; the inner frame is frame 3, the inner frame rotation angle is γ, and the corresponding coordinate system is x3y3z3; the platform frame is frame 4, the platform frame rotation angle is θ, and the corresponding coordinate system is x4y4z4; the local geographic coordinate system is x n y n z n .

[0058] Step 2: In the four-axis rotational inertial navigation, the rotational angular velocity of the four frames is mapped to the values ​​ω detected by the three gyroscopes in the X, Y, and Z directions. px ,ω py ,ω pz for:

[0059]

[0060] In the formula, α is the rotation angle of the outer frame; β is the rotation angle of the middle frame; γ is the rotation angle of the inner frame; θ is the rotation angle of the platform frame; is the angular velocity of the outer frame rotation; is the angular velocity of the middle frame rotation; is the inner frame rotation angular velocity; is the angular velocity of the table frame rotation.

[0061] Step 3: In the rotating inertial navigation system, the angular velocity sensed by the gyro assembly is relative to the inertial system, including the angular velocity of the earth's rotation, the angular velocity of the carrier's motion, and the angular velocity of the frame's rotation, as follows:

[0062]

[0063] In the formula, b represents the carrier coordinate system, n represents the navigation coordinate system, e represents the earth coordinate system, p represents the gyro body coordinate system, i represents the inertial coordinate system, and k represents a certain moment. represents the projection of the angular velocity of the p system relative to the b system at time k on the p system, It represents the projection of the angular velocity of the b system relative to the n system at time k in the p system, that is, the carrier angular velocity, It represents the projection of the angular velocity of system n relative to system e at time k in system p, It represents the projection of the angular velocity of the e system relative to the i system at time k in the p system, It represents the projection of the angular velocity of the p system relative to the i system under the p system, that is, the angular velocity sensed by the gyroscope component.

[0064] and It can be solved as follows:

[0065]

[0066] In the formula is the transformation matrix of system n relative to system p at time k, is the projection of the angular velocity of the e system relative to the i system at time k in the e system, that is, the angular velocity of the earth's rotation, which is a constant value. L(k) is the latitude of the carrier at time k, and V E (k) is the eastward velocity of the carrier at time k, V N (k) is the northward velocity of the carrier at time k, and R is the radius of the earth.

[0067] Then, the angular velocity of the carrier at time k, that is, the projection of the angular velocity of the b system relative to the n system in the p system Then the separation can be performed, namely:

[0068]

[0069] The carrier angular velocity obtained at this moment is Then it can be isolated.

[0070] Step 4: To isolate the carrier motion, after calculating the carrier angular velocity, it is necessary to apply the corresponding angular velocity to each frame motor to achieve the purpose of isolation. At this time, it is necessary to decouple the carrier angular velocity. For the carrier angular velocity in three directions, theoretically three axes need to be isolated, but only using a three-axis rotating frame may cause the frame to lock and decoupling to fail.

[0071] For the angular motion isolation in the four-axis rotary inertial navigation system, this situation can be effectively avoided by optimizing the motion mode of each axis. The specific motion decoupling situations at the initial position are classified as follows:

[0072] α is the outer frame rotation angle; β is the middle frame rotation angle; γ is the inner frame rotation angle; θ is the platform frame rotation angle

[0073] If the outer frame does not participate in motion isolation, then:

[0074]

[0075] If the middle frame does not participate in motion isolation, then:

[0076]

[0077] If the inner frame does not participate in motion isolation, then:

[0078]

[0079] If the table frame does not participate in motion isolation, then:

[0080]

[0081] In the formula, are the projections of the carrier angular motion in the x, y, and z directions of the stage coordinate system, α is the rotation angle of the outer frame; β is the rotation angle of the middle frame; γ is the rotation angle of the inner frame; θ is the rotation angle of the stage frame, is the angular velocity of the outer frame rotation; is the angular velocity of the middle frame rotation; is the inner frame rotation angular velocity; is the angular velocity of the table frame rotation.

[0082] It can be seen from the above formula that when a certain axis is locked and does not participate in isolation, the four-axis rotational inertial navigation system degenerates into a three-axis rotational inertial navigation system. Taking the platform frame locked and not participating in motion isolation as an example, when only the outer frame, middle frame and inner frame are used for motion isolation, when the rotation angle β of the middle frame is close to 90 or 270 degrees, cosβ≈0, the frame decoupling will result in infinite angular velocity, so that when the outer frame, middle frame and inner frame are used, the isolation angular motion cannot be applied, as shown below:

[0083]

[0084] The four-axis rotation method can effectively avoid this situation. That is, when the rotation angle β of the middle frame is close to 90 or 270 degrees, the situation represented by formula (9) can be achieved by motion isolation through the outer frame, the middle frame and the platform frame, that is, formula (7). At this time, cosβ≈0, then sinβ≈1, then formula (7) can be converted to:

[0085]

[0086] According to formula (10), when the outer frame, the middle frame and the inner frame are used for motion isolation, effective motion isolation can be achieved by timely conversion of the outer frame, the middle frame and the platform frame, thereby avoiding the situation where decoupling cannot be achieved.

[0087] Step 5. It can be seen from formulas (5)-(8) that the denominator terms in the decoupling formula include cosγ, sinγ, sinβ, and cosβ, that is, the basic situation in which motion decoupling failure may occur is:

[0088] (5.1) When γ approaches 90 or 270 degrees, that is, cosγ approaches 0 and sinγ approaches 1;

[0089] At this time, if formula (5) is used for motion isolation, the decoupling between the middle frame and the platform frame will fail. To avoid this, formula (6), (7) or (8) should be used for motion isolation.

[0090] (5.2) When γ approaches 0 degrees, that is, sinγ approaches 0 and cosγ approaches 1;

[0091] At this time, if formula (6) is used for motion isolation, the decoupling between the outer frame and the platform frame will fail. To avoid this situation, formula (5), (7) or (8) should be used for motion isolation.

[0092] (5.3) When β approaches 0 degrees, that is, sinβ approaches 0 and cosβ approaches 1;

[0093] At this time, if formula (7) is used for motion isolation, the decoupling between the outer frame and the platform frame will fail. To avoid this situation, formula (5), (6) or (8) should be used for motion isolation.

[0094] (5.4)β approaches 90 or 270 degrees, that is, cosβ approaches 0 and sinβ approaches 1.

[0095] At this time, if formula (8) is used for motion isolation, the decoupling between the outer frame and the inner frame will fail. To avoid this, formula (5), (6) or (7) should be used for motion isolation.

[0096] Based on the above basic situations, it can be concluded that the combination of motion decoupling failure is:

[0097] (5.5) When γ approaches 90 or 270 degrees, that is, cosγ approaches 0, sinγ approaches 1, and β approaches 0 degrees, that is, sinβ approaches 0, cosβ approaches 1;

[0098] At this time, if formula (5) or (7) is used for motion isolation, the decoupling of the outer frame, the middle frame and the platform frame will fail. To avoid this situation, formula (6) or (8) should be used for motion isolation.

[0099] (5.6)γ approaches 90 or 270 degrees, that is, cosγ tends to 0, sinγ tends to 1, and β approaches 90 or 270 degrees, that is, cosβ tends to 0, sinβ tends to 1.

[0100] At this time, if formula (5) or (8) is used for motion isolation, the decoupling of the outer frame, inner frame, middle frame and platform frame will fail. To avoid this situation, formula (6) or (7) should be used for motion isolation.

[0101] (5.7)γ approaches 0 degrees, that is, sinγ approaches 0, cosγ approaches 1, and β approaches 0 degrees, that is, sinβ approaches 0, cosβ approaches 1.

[0102] At this time, if formula (6) or (7) is used for motion isolation, the decoupling between the outer frame and the platform frame will fail. To avoid this situation, formula (5) or (8) should be used for motion isolation.

[0103] (5.8)γ approaches 0 degrees, that is, sinγ approaches 0, cosγ approaches 1, and β approaches 90 or 270 degrees, that is, cosβ approaches 0, sinβ approaches 1.

[0104] At this time, if formula (6) or (8) is used for motion isolation, the decoupling of the outer frame, inner frame and platform frame will fail. To avoid this situation, formula (5) or (7) should be used for motion isolation.

[0105] The present invention can effectively solve the problem that single-axis and dual-axis rotary inertial navigation systems cannot completely isolate the carrier motion, and the problem that three-axis rotary inertial navigation systems fail to decouple in the carrier motion isolation.

Claims

1. An optimization method to avoid the failure of the motion isolation and decoupling of a rotating inertial navigation carrier, characterized in that: The steps include: Step 1: In the four-axis rotational inertial navigation, define the outer frame, middle frame, inner frame, platform frame, the rotation angles corresponding to the four frames, and the coordinate systems corresponding to the four frames; Step 2: In the four-axis rotational inertial navigation, determine the rotational angular velocities of the four frames and map them to the values ​​detected by the three gyroscopes in the X, Y, and Z directions; Step 3: In the rotating inertial navigation system, determine the angular velocity that the gyro assembly is sensitive to, including the angular velocity of the earth's rotation, the angular velocity of the carrier's motion, and the angular velocity of the frame's rotation; Step 4, decoupling calculation is performed on the carrier angular velocity to obtain decoupling model formulas under four states: the outer frame does not participate in motion isolation, the middle frame does not participate in motion isolation, the inner frame does not participate in motion isolation, and the platform frame does not participate in motion isolation; Step 5: Based on the decoupling model formulas under the four states, the basic situations in which motion decoupling failure may occur and the combination situations in which motion decoupling failure may be obtained are obtained, and the corresponding decoupling model formulas are determined to achieve motion isolation.

2. The optimization method for avoiding the failure of the rotational inertial navigation carrier motion isolation and decoupling according to claim 1 is characterized in that: In step 1, define the outer frame as frame 1, the outer frame rotation angle is α, and the corresponding coordinate system is x1y1z1; the middle frame is frame 2, the middle frame rotation angle is β, and the corresponding coordinate system is x2y2z2; the inner frame is frame 3, the inner frame rotation angle is γ, and the corresponding coordinate system is x3y3z3; the platform frame is frame 4, the platform frame rotation angle is θ, and the corresponding coordinate system is x4y4z4; the local geographic coordinate system is x n y n z n .

3. The optimization method for avoiding the failure of the rotational inertial navigation carrier motion isolation and decoupling according to claim 2 is characterized in that: In step 2, the rotational angular velocities of the four frames are mapped to the values ​​ω detected by the three gyroscopes in the X, Y, and Z directions. px ,ω py ,ω pz for: In the formula, α is the rotation angle of the outer frame; β is the rotation angle of the middle frame; γ is the rotation angle of the inner frame; θ is the rotation angle of the platform frame; is the angular velocity of the outer frame rotation; is the angular velocity of the middle frame rotation; is the inner frame rotation angular velocity; is the angular velocity of the table frame rotation.

4. The optimization method for avoiding the failure of the rotational inertial navigation carrier motion isolation and decoupling according to claim 3 is characterized in that: In step 3, In the formula, b represents the carrier coordinate system, n represents the navigation coordinate system, e represents the earth coordinate system, p represents the gyro body coordinate system, i represents the inertial coordinate system, and k represents a certain moment. represents the projection of the angular velocity of the p system relative to the b system at time k on the p system, It represents the projection of the angular velocity of the b system relative to the n system at time k in the p system, that is, the carrier angular velocity, It represents the projection of the angular velocity of system n relative to system e at time k in system p, It represents the projection of the angular velocity of the e system relative to the i system at time k in the p system, It represents the projection of the angular velocity of the p system relative to the i system in the p system, that is, the angular velocity sensed by the gyro component; and Solve according to the following formula: In the formula, is the transformation matrix of system n relative to system p at time k, is the projection of the angular velocity of the e system relative to the i system at time k in the e system, that is, the angular velocity of the earth's rotation, which is a constant value. L(k) is the latitude of the carrier at time k, and V E (k) is the eastward velocity of the carrier at time k, V N (k) is the northward velocity of the carrier at time k, and R is the radius of the earth; Then, the angular velocity of the carrier at time k, that is, the projection of the angular velocity of the b system relative to the n system in the p system Then the separation can be performed, namely:

5. The optimization method for avoiding the failure of the rotational inertial navigation carrier motion isolation and decoupling according to claim 4 is characterized in that: In step 4, if the outer frame does not participate in motion isolation, then: If the middle frame does not participate in motion isolation, then: If the inner frame does not participate in motion isolation, then: If the table frame does not participate in motion isolation, then:

6. The optimization method for avoiding the failure of the rotation inertial navigation carrier motion isolation and decoupling according to claim 5 is characterized in that: In step 5, the basic situation of motion decoupling failure is: (5.1) When γ approaches 90 or 270 degrees, that is, cosγ approaches 0 and sinγ approaches 1; Formula (6), (7) or (8) should be used for motion isolation; (5.2) When γ approaches 0 degrees, that is, sinγ approaches 0 and cosγ approaches 1; Formula (5), (7) or (8) should be used for motion isolation; (5.3) When β approaches 0 degrees, that is, sinβ approaches 0 and cosβ approaches 1; Formula (5), (6) or (8) should be used for motion isolation; (5.4) When β approaches 90 or 270 degrees, that is, cosβ approaches 0 and sinβ approaches 1; Formula (5), (6) or (7) should be used for motion isolation; The combination of motion decoupling failure is: (5.5) When γ approaches 90 or 270 degrees, that is, cosγ approaches 0, sinγ approaches 1, and β approaches 0 degrees, that is, sinβ approaches 0, cosβ approaches 1; Use formula (6) or (8) for motion isolation; (5.6) When γ approaches 90 or 270 degrees, that is, cosγ approaches 0, sinγ approaches 1, and β approaches 90 or 270 degrees, that is, cosβ approaches 0, sinβ approaches 1; Use formula (6) or (7) for motion isolation; (5.7) When γ approaches 0 degrees, that is, sinγ approaches 0, cosγ approaches 1, and β approaches 0 degrees, that is, sinβ approaches 0, cosβ approaches 1; Use formula (5) or (8) for motion isolation; (5.8)γ approaches 0 degrees, that is, sinγ approaches 0, cosγ approaches 1, and β approaches 90 or 270 degrees, that is, cosβ approaches 0, sinβ approaches 1; Use formula (5) or (7) for motion isolation.

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