CN105180889B - A kind of dynamic rotary attitude measuring and method for drilling well - Google Patents

A kind of dynamic rotary attitude measuring and method for drilling well Download PDF

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CN105180889B
CN105180889B CN201410277772.XA CN201410277772A CN105180889B CN 105180889 B CN105180889 B CN 105180889B CN 201410277772 A CN201410277772 A CN 201410277772A CN 105180889 B CN105180889 B CN 105180889B
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msub
msubsup
accelerometer
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CN105180889A (en
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郭秋芬
李佳
刘庆成
李梅英
李纬燕
胡雄
吕伟
岳步江
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Aerospace Science and Industry Inertia Technology Co Ltd
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Abstract

The present invention relates to a kind of dynamic rotary attitude measuring and method for drilling well.The device includes：Suspension slurry and resistance to compression cylinder, resistance to compression cylinder is interior to be provided with power panel, circuit board, magnetic sensor, accelerometer and gyroscope.This method measures the rotating speed of magnetic field of the earth component, terrestrial gravitation component and whole device respectively by magnetic sensor, accelerometer and gyroscope first, and being physically entered of gathering of each sensor is converted into voltage output；Then the rotating speed obtained according to alignment error coefficient and gyroscope, harmful acceleration item is removed in the output item of accelerometer, realizes error compensation；Then the magnetic field of the earth and terrestrial gravitation data gathered according to accelerometer and magnetic sensor calculates the dynamic rotary posture of drilling well.The present invention can measure the drilling well posture under dynamic rotary state, can effectively improve the operating efficiency of dynamic well system.

Description

A kind of dynamic rotary attitude measuring and method for drilling well
Technical field
The invention belongs to oil drilling technical field, is related to the attitude measurement of drilling well, is particularly carrying out rotary drilling Dynamic rotary attitude measuring and measuring method under Special use environment.
Background technology
Since the 1980s, domestic and international most of oil field subsequently enters development late stage, and new exploratory area block is mostly in Particular surroundings, such as ocean, Tan Hai, desert area, this allows for exploration and development difficulty and cost greatly increases.On the other hand, Old filed is faced with substantial amounts of edge oil-gas reservoir, independent small oil-gas reservoir, Complicated Fault-Block Hydrocarbon Reservoir, ultra-thin oil gas to realize stable yields The development problem of the difficult-to-produce reserves such as Tibetan.The needs of exploration and development situation promote the differentiation and development of well type, extended reach well, surpass The complex structural wells such as oil sheet horizontal well, multi-branched well ratio shared in oilfield prospecting developing is increasing.Rotation is led To drilling technology precisely in order to the high new drilling techniques for adapting to the needs of these complex structural well drillng operations and occurring, are The research emphasis and developing direction of modern steerable drilling.
When carrying out drillng operation using rotary steerable drilling technology, it is not necessary to which the can that frequently makes a trip realizes three-dimensional well TRAJECTORY CONTROL, and there is the advantages of well track is more smooth, and extended distance is bigger, to ensureing well track quality, improve drilling well Speed and efficiency, meet that the needs of boring complex structural well is all significant.But these existing well systems can only be in static state In the case of calculate drilling well posture, it is necessary to when system bore stop after carry out attitude algorithm again, it is impossible in system drilling process in real time Attitude information is provided, so that the operating efficiency of system is than relatively low.
The content of the invention
The present invention provides a kind of device and method that can be measured to drilling well posture under current intelligence, solves drilling well System can only carry out the problem of posture clearing under quiescent conditions, can effectively improve the operating efficiency of dynamic well system.
To achieve the above object, the technical solution adopted by the present invention is as follows：
A kind of dynamic rotary attitude measuring for drilling well, including：
Suspension slurry, for carrying the rotary steering system of drilling rig；
Resistance to compression cylinder, with suspension slurry connection, power panel, circuit board, magnetic sensor, accelerometer and gyro are provided with it Instrument；The magnetic sensor, accelerometer and gyroscope measure magnetic field of the earth component, terrestrial gravitation component and whole device respectively Rotating speed；The circuit board gathers the measurement data of the magnetic sensor, accelerometer and gyroscope in real time, and then calculates dynamic Drilling well posture in state rotation status.
Further, in addition to a mechanical framework, it is fixed in the resistance to compression cylinder, the power panel, circuit board, magnetic pass Sensor, accelerometer and gyroscope are fixed on the mechanical framework.
Further, the mechanical framework is connected by way of screw thread with resistance to compression cylinder, and power panel and circuit board pass through spiral shell Nail is connected with mechanical framework, and magnetic sensor is using in the groove that is fixed in mechanical framework by the way of glue seals, inertial sensor It is connected by screw with mechanical framework fastening.
Further, the suspension slurry is connected by screw thread and connector with the resistance to compression cylinder.
Further, the circuit board is connected by connector with the magnetic sensor, accelerometer and gyroscope, electricity Road plate is provided with A/D conversion chips and single-chip microcomputer, and wherein A/D conversion chips are used for the output electricity for each sensor that will be detected Press and digital quantity is converted into by analog quantity, single-chip microcomputer gathers the data of digital quantity in real time, to carry out the posture of next step and thing The calculating of reason amount.
Further, in each sensor (magnetic sensor, accelerometer and gyroscope), circuit board and power panel and machinery The place of framework installation is provided with some seal grooves, is mounted therein with sealing ring, plays multi-sealed effect, prevent in the course of work Down-hole liquid enters in device.
A kind of dynamic rotary attitude measurement method of drilling well using said apparatus, its step include：
1) by magnetic sensor, accelerometer and gyroscope measure respectively magnetic field of the earth component, terrestrial gravitation component and The rotating speed of whole device, being physically entered of gathering of each sensor is converted into voltage output；
2) rotating speed obtained according to alignment error coefficient and gyroscope, harmful accelerate is removed in the output item of accelerometer Item is spent, realizes error compensation；
3) magnetic field of the earth and terrestrial gravitation data gathered according to accelerometer and magnetic sensor calculates the dynamic of drilling well Rotation attitude.
Further, clocking capability is set in the circuit board, when step 3) first determines whether system time is more than calculating Between point, if greater than calculate time point then carry out Attitude Calculation, otherwise continue time cumulation and data storage.
The dynamic rotary attitude measuring and method of the present invention has the characteristics of real-time, is not required to after system brill stops Attitude algorithm is carried out again, can measure the drilling well posture under dynamic rotary state, posture is provided in real time in system drilling process Information, the operating efficiency of system is effectively raised, played an important roll to dynamic drilling well.
Brief description of the drawings
Fig. 1 is the structure chart of dynamic posture rotary measurement device.
Fig. 2 is the schematic shapes of suspension slurry.
Fig. 3 is the flow chart that attitude measurement is carried out using dynamic posture rotary measurement device.
Wherein：1. suspension slurry；2. resistance to compression cylinder；3. power panel；4. circuit board；5. fluxgate (magnetic sensor)；6. machinery Framework；7. sealing ring；8. accelerometer；9. gyroscope.
Embodiment
Below by specific embodiments and the drawings, the present invention will be further described.
Fig. 1 is the structure chart of the dynamic rotary attitude measuring of the present invention.The device includes：Suspension slurry 1；Resistance to compression cylinder 2；Power panel 3；Circuit board 4；Fluxgate 5 (magnetic sensor)；Mechanical framework 6；Sealing ring 7；Accelerometer 8；Gyroscope 9, its In 8 and 9 be inertial sensor.
Wherein, suspension slurry 1 is located at the both ends of resistance to compression cylinder 2, and Fig. 2 is the concrete shape enlarged drawing of suspension slurry 1.Set on suspension slurry There is projection so that whole device more easily can be connected with rotary steering system others drilling rig, improve installation accuracy And operating efficiency.
Suspension slurry 1 is connected by screw thread and connector with resistance to compression cylinder 2.Resistance to compression cylinder 2 is realized using compression-resistant materials such as stainless steels. Mechanical framework 6 is connected by way of screw thread with resistance to compression cylinder 2.Power panel 3 and circuit board 4 are connected by screw with mechanical framework 6, Magnetic sensor 5 is fixed in the groove in mechanical framework 6 by the way of glue envelope, and inertial sensor 8,9 passes through screw and machine The fastening of tool framework is connected.Electrical connection between all parts is connected by the way of connector.Inertial sensor gyroscope 9 is negative The rotating speed of duty measurement whole system, serves a crucial effect.Novelty of the invention utilizes magnetic sensor and inertia Sensor gyroscope collective effect carries out the attitude measurement of drilling well.Pacify in sensor, circuit board and power panel and mechanical framework The place of dress is designed with seal groove, for installing sealing ring 7, plays multi-sealed effect after being connected with resistance to compression cylinder, prevents Down-hole liquid enters in this device in the course of work.
Power panel 3 is used to power for other parts for needing electricity consumption in device.Circuit board 4 passes through connector and each biography Sensor 5,8,9 is connected, circuit board 4 is provided with A/D conversion chips and single-chip microcomputer, and wherein A/D conversion chips are for that will detect The output voltage of each sensor is converted into digital quantity by analog quantity, and single-chip microcomputer gathers the data of digital quantity in real time, to carry out The calculating of the posture and physical quantity of next step.
In drilling process, dynamic rotary attitude measuring of the invention is mounted in the rotary steering system of drilling rig On, rotary drilling is carried out by motor driven system.In drilling process, the collecting computer (single-chip microcomputer) on circuit board 4 is at a high speed Magnetic and inertial sensor data are gathered, and the voltage analog for sensor being exported by the A/D conversion chips on circuit board 4 turns Change digital quantity into, then calculated using these data, obtain the posture of system, and send the data to ground host machine, reach To the purpose of control system drilling direction.In the case where there is tachometric survey, data high-speed collection, filtering are carried out, according to gyro The rotating speed of sensor collection, acceleration compensation, Zhi Hougen are carried out to magnetic sensor and inertial sensor using least square method According to the physical quantity calculate attitude angle after data are average and need.
Specifically, the flow of dynamic rotary attitude measurement is carried out as shown in figure 3, specifically including as follows using said system Step：
1) gathered data
In the course of work, magnetic sensor, accelerometer and gyroscope measure magnetic field of the earth component, terrestrial gravitation point respectively The rotating speed of amount and whole device, each sensor according to it is sensitive to be physically entered, be converted into voltage output.Circuit board 4 passes through Connector is connected with each sensor 5,8,9, detects the output voltage of each sensor, and core is changed by the A/D in circuit board 4 Piece, voltage analog is converted into digital quantity, the single-chip microcomputer in circuit board 4 gathers the data of digital quantity in real time, to carry out down The calculating of the posture and physical quantity of one step.
2) error compensation
In this device, accelerometer is the sensor of sensitive acceleration.And in instrument rotary course, have harmful add Speed is added in the acceleration of sensitivity and eliminated, it is necessary to which this is harmful into acceleration, can just obtain correctly useful acceleration. The rotating speed obtained according to alignment error coefficient and gyroscope, harmful acceleration item is cut i.e. in the output item of accelerometer Can.
3) judge whether system time is more than and calculate time point
Clocking capability is set in circuit board 4, and system time refers to the time of clocking capability acquisition, and calculating time point refers to setting Time for once calculating of progress, for example 10s can be set to.I.e. in calculating process, once calculated per 10s.If it is System time cumulation is equal to 10s, then carries out next step Attitude Calculation, otherwise continue time cumulation and data storage.
4) attitude algorithm is carried out
A) attitude angle hole angle is calculated
The output expression formula of accelerometer is as follows：
Vi=αi+kiaib+kωiaib
Wherein：ViFor the output voltage of 3 accelerometer x, y, z；αiFor the zero bias of 3 accelerometer x, y, z；kiFor 3 The constant multiplier of accelerometer x, y, z；kωiFor harmful acceleration factor of 3 accelerometer x, y, z, this coefficient is by gyroscope The rotating speed measured obtains；aibFor the input of 3 accelerometer x, y, z.
By above formula, according to output voltage ViObtain the input a of accelerometeribAfterwards, posture corner well is calculated using formula below Oblique angle：
B) drift azimuth is calculated
The output expression formula of magnetic sensor is as follows：
Vmi=αmi+kmiMib
Wherein：VmiFor 3 magnetic sensors X, Y, Z output voltage；αmiFor 3 magnetic sensors X, Y, Z zero bias； kmiFor 3 magnetic sensors X, Y, Z constant multiplier；MibFor 3 magnetic sensors X, Y, Z input.
The calculation formula of drift azimuth is as follows：
Wherein, Mv、Mn, r, s be intermediate variable, then the expression formula of drift azimuth is as follows：
ψ=arctan (S)
The present invention enters Mobile state measurement using magnetic sensor, inertial sensor, and sensor error, attitude error are entered Row compensation, drilling well gradient slow-speed of revolution dynamic accuracy can reach 0.5 °, high speed dynamic precision can reach 1 °, the azimuth slow-speed of revolution is moved State precision can reach 1 °, high speed dynamic precision can reach 2 °, can be good at meeting actual requirement of engineering.
The above embodiments are merely illustrative of the technical solutions of the present invention rather than is limited, the ordinary skill of this area Technical scheme can be modified by personnel or equivalent substitution, without departing from the spirit and scope of the present invention, this The protection domain of invention should be to be defined described in claim.

Claims (9)

  1. A kind of 1. dynamic rotary attitude measuring for drilling well, it is characterised in that including：
    Suspension slurry, for carrying the rotary steering system of drilling rig；
    Resistance to compression cylinder, with suspension slurry connection, power panel, circuit board, magnetic sensor, accelerometer and gyroscope are provided with it；Institute State magnetic sensor, accelerometer and gyroscope and measure turning for magnetic field of the earth component, terrestrial gravitation component and whole device respectively Speed；The circuit board gathers the measurement data of the magnetic sensor, accelerometer and gyroscope in real time, according to alignment error system The rotating speed that number and gyroscope obtain, harmful acceleration item is removed in the output item of accelerometer, realizes error compensation, Jin Ergen The magnetic field of the earth and terrestrial gravitation data that are gathered according to accelerometer and magnetic sensor calculate the drilling well appearance in dynamic rotary state State, the formula for calculating attitude angle hole angle are：
    <mrow> <mi>&amp;theta;</mi> <mo>=</mo> <mi>arctan</mi> <mfrac> <msqrt> <mrow> <msubsup> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </msqrt> <mrow> <mo>|</mo> <msubsup> <mi>a</mi> <mrow> <mi>z</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>|</mo> </mrow> </mfrac> <mo>;</mo> </mrow>
    Wherein, aibFor the input of 3 accelerometer x, y, z；
    Calculate drift azimuth formula be：
    ψ=arctan (S),
    <mrow> <mi>s</mi> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>Y</mi> <mi>b</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>X</mi> <mi>b</mi> </mrow> </msub> <mo>)</mo> <msqrt> <mrow> <msubsup> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>z</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </msqrt> </mrow> <mrow> <msub> <mi>M</mi> <mrow> <mi>Z</mi> <mi>b</mi> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>a</mi> <mrow> <mi>z</mi> <mi>b</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>X</mi> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>Y</mi> <mi>b</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow>
    Wherein, MibFor 3 magnetic sensors X, Y, Z input.
  2. 2. it is used for the dynamic rotary attitude measuring of drilling well as claimed in claim 1, it is characterised in that：Also include a machinery Framework, it is fixed in the resistance to compression cylinder, the power panel, circuit board, magnetic sensor, accelerometer and gyroscope are fixed on this On mechanical framework.
  3. 3. it is used for the dynamic rotary attitude measuring of drilling well as claimed in claim 2, it is characterised in that：The mechanical framework It is connected by way of screw thread with resistance to compression cylinder, power panel and circuit board are connected by screw with mechanical framework, and magnetic sensor is adopted The mode sealed with glue is fixed in the groove in mechanical framework, and inertial sensor is connected by screw with mechanical framework fastening.
  4. 4. it is used for the dynamic rotary attitude measuring of drilling well as claimed in claim 1, it is characterised in that：The suspension slurry is logical Screw thread and connector is crossed to be connected with the resistance to compression cylinder.
  5. 5. it is used for the dynamic rotary attitude measuring of drilling well as claimed in claim 1, it is characterised in that：The circuit board leads to Cross connector with the magnetic sensor, accelerometer and gyroscope to be connected, circuit board is provided with A/D conversion chips and monolithic Machine, wherein A/D conversion chips are used to the output voltage of each sensor detected being converted into digital quantity, monolithic by analog quantity Machine gathers the data of digital quantity in real time, and carries out the calculating of drilling well posture and physical quantity.
  6. 6. it is used for the dynamic rotary attitude measuring of drilling well as claimed in claim 1, it is characterised in that：In the resistance to compression cylinder Provided with some seal grooves, sealing ring is provided with the seal groove.
  7. 7. a kind of dynamic rotary attitude measurement method of drilling well using claim 1 described device, its step include：
    1) magnetic field of the earth component, terrestrial gravitation component and whole are measured by magnetic sensor, accelerometer and gyroscope respectively The rotating speed of device, being physically entered of gathering of each sensor is converted into voltage output；
    2) rotating speed obtained according to alignment error coefficient and gyroscope, harmful acceleration is removed in the output item of accelerometer , realize error compensation；
    3) magnetic field of the earth and terrestrial gravitation data gathered according to accelerometer and magnetic sensor calculates the dynamic rotary of drilling well Posture.
  8. 8. method as claimed in claim 7, it is characterised in that：Clocking capability is set in the circuit board, and step 3) first determines whether Whether system time, which is more than, calculates time point, then carries out Attitude Calculation if greater than time point is calculated, otherwise continues the time Accumulative and data storage.
  9. 9. method as claimed in claim 7 or 8, it is characterised in that step 3) calculate dynamic rotary posture method be：
    A) the output expression formula of accelerometer is：
    Vi=αi+kiaib+kωiaib,
    Wherein：ViFor the output voltage of 3 accelerometer x, y, z；αiFor the zero bias of 3 accelerometer x, y, z；kiFor 3 acceleration The constant multiplier of degree meter x, y, z；kωiFor harmful acceleration factor of 3 accelerometer x, y, z, this coefficient is measured by gyroscope Rotating speed obtain；aibFor the input of 3 accelerometer x, y, z；
    By above formula, according to output voltage ViObtain the input a of accelerometeribAfterwards, attitude angle hole angle is calculated using formula below：
    <mrow> <mi>&amp;theta;</mi> <mo>=</mo> <mi>arctan</mi> <mfrac> <msqrt> <mrow> <msubsup> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </msqrt> <mrow> <mo>|</mo> <msubsup> <mi>a</mi> <mrow> <mi>z</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>|</mo> </mrow> </mfrac> <mo>;</mo> </mrow>
    B) the output expression formula of magnetic sensor is：
    Vmi=αmi+kmiMib,
    Wherein：VmiFor 3 magnetic sensors X, Y, Z output voltage；αmiFor 3 magnetic sensors X, Y, Z zero bias；kmiFor 3 Individual magnetic sensor X, Y, Z constant multiplier；MibFor 3 magnetic sensors X, Y, Z input；Order
    <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>X</mi> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>Y</mi> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mi>z</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>Z</mi> <mi>b</mi> </mrow> </msub> </mrow> <msqrt> <mrow> <msubsup> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>z</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </msqrt> </mfrac> <mo>,</mo> </mrow>
    <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>M</mi> <mrow> <mi>X</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>M</mi> <mrow> <mi>Y</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>M</mi> <mrow> <mi>Z</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>M</mi> <mi>v</mi> <mn>2</mn> </msubsup> </mrow> </msqrt> <mo>,</mo> </mrow>
    <mrow> <mi>r</mi> <mo>=</mo> <mfrac> <mrow> <msub> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>Y</mi> <mi>b</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>X</mi> <mi>b</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <msqrt> <mrow> <msubsup> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </msqrt> </mrow> </mfrac> <mo>,</mo> </mrow>
    <mrow> <mi>s</mi> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>Y</mi> <mi>b</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>X</mi> <mi>b</mi> </mrow> </msub> <mo>)</mo> <msqrt> <mrow> <msubsup> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>z</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </msqrt> </mrow> <mrow> <msub> <mi>M</mi> <mrow> <mi>Z</mi> <mi>b</mi> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>a</mi> <mrow> <mi>z</mi> <mi>b</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow> <mi>x</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>X</mi> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mi>y</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow> <mi>Y</mi> <mi>b</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow>
    Wherein, Mv、Mn, r, s be intermediate variable, then the expression formula of drift azimuth is：ψ=arctan (S).
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