CN105180944A - Judgment and compensation method for hull sideslipping speed error - Google Patents

Abstract

The invention discloses a judgment and compensation method for a hull sideslipping speed error. The method comprises the following steps of fixedly arranging an inertia unit of an integrated navigation system on a to-be-tested ship; enabling the ship to do S shaped movement, keeping the rotation speed to be constant in the cornering process, and collecting the data of the integrated navigation system; projecting hull east orientation speed VE and north orientation speed VN recorded during hull cornering on a carrier b system by using a pose transformation matrix, wherein an x axis speed Vx on the carrier system is the hull sideslipping speed, and a y axis speed on the carrier system Vy is a hull fore and aft speed; carrying out difference on a course angle theta3 output by the integrated navigation system to determine the angle size (shown in the description) during cornering, namely a rotation rate r0 during the hull cornering, so as to determine the side slipping coefficient; and when the to-be-tested ship navigates practically, collecting the rotating rate r, transverse acceleration (shown in the description) and a helm angle delta r in real time during cornering, so as to carry out side slipping speed compensation. The method provided by the invention reduces the influence of velocity measurement error of an electromagnetic log on the inertial navigation system, and reduces the maximum error of system output.

Description

A kind of hull slipspeed error judges compensation method

Technical field

The invention belongs to inertial navigation field, particularly relate to a kind of hull slipspeed error and judge compensation method.

Background technology

For the inertial navigation system that electromagnet log is auxiliary, the range rate error of electromagnet log has a strong impact on inertial navigation system performance.Due to the reason such as principle of work and allocation plan of electromagnet log, when causing naval vessel to be turned, electromagnet log cannot measure the slipspeed that naval vessel transversely exists, this part naval vessel slipspeed is equivalent to the range rate error of electromagnet log, thus its auxiliary inertial navigation system transient error can be made to increase.Particularly, when Transfer Alignment is carried out in the sub-inertial navigation in armament systems on warship ship-to-ship, such as speed adds attitude matching and aims at, naval vessel is often needed to do certain motor-driven navigation, now naval vessel slipspeed occurs frequent, the main inertial navigation causing electromagnet log auxiliary is exported transient error and becomes large, thus reduce alignment precision.Such as data display, the inertial navigation systems such as WSN-7A, WSN-7B of USN, for mating the initial alignment of Tomahawk cruise missile, needing to compensate the slipspeed of carrier, but not setting forth compensatory michanism.The domestic report that yet there are no in this respect, needs badly and carries out hull slipspeed error mechanism and compensation method research.

The compensation of slipspeed, according to slipspeed theoretical expression, the demarcation the present invention that finally can be summed up as sideslip coefficient mainly gives slipspeed compensation criterion and sideslip coefficient scaling method.

Summary of the invention

The object of this invention is to provide that a kind of method is simple, reliability is high, hull slipspeed error judges compensation method.

A kind of hull slipspeed error judges compensation method, comprises the following steps,

Step one: be fixedly installed in by the inertance element of integrated navigation system on ship to be measured, ensures inertance element mounting base axis and the dead in line of hull fore and aft;

Step 2: make ship do the motion of S type, keep speed of rotation constant in turning process, gather integrated navigation system data, comprise east orientation speed V _e, north orientation speed V _n, pitching angle theta ₁, roll angle θ ₂and course angle θ ₃;

Step 3: the hull east orientation speed V of record when ship is turned _ewith north orientation speed V _nutilize pose transformation matrix to project to carrier system b to fasten, carrier fastens x-axis speed V _xbe hull slipspeed, carrier fastens y-axis speed V _ybe hull fore and aft speed;

Step 4: by the course angle θ exported integrated navigation system ₃carry out difference can determine its turn time size be the speed of rotation r of hull when turning ₀, and then determine sideslip coefficient wherein L is captain;

Step 5: ship to be measured does real navigation, speed of rotation r, the transverse acceleration of Real-time Collection hull when turning and rudder angle δ _r, carry out slipspeed v=kLr compensation according to three kinds of situations;

Situation one: according to rudder angle δ _rjudge whether that needing to introduce slipspeed compensates, when or δ _rduring=const>0, slipspeed need be compensated;

Situation two: at towed cable maneuvering period, if speed of rotation r and side-slip acceleration meet condition, then compensate slipspeed, t ₀for given threshold time;

Situation three: in towed cable transition period, first determines sliding velocity range [-v _max, v _max], then determine the boundary value [-r of speed of rotation _max, r _max], when speed of rotation is outside border, need to compensate slipspeed.

A kind of hull slipspeed of the present invention error judges that compensation method can also comprise:

1, carry out in slipspeed v=kLr compensation according to three kinds of situations, the priority of three kinds of situations is situation one from high to low successively, situation two, situation three.

2, carrier fastens x-axis speed V _xy-axis speed V is fastened with carrier _yfor:

${\left[\begin{array}{c}{V}_x\\ V_y\end{array}\right]}^b={\left(C_n^b\right)}_{2\×2}\·{\left[\begin{array}{c}{V}_E\\ V_N\end{array}\right]}^n$

$\begin{array}{c}{\left(C_n^b\right)}_{2\×2}=\\ \left[\begin{array}{cc}{\mathrm{cos\θ}}_2{\mathrm{cos\θ}}_3-& {\mathrm{cos\θ}}_2{\mathrm{sin\θ}}_3+\\ {\mathrm{sin\θ}}_2{\mathrm{sin\θ}}_1{\mathrm{sin\θ}}_3& {\mathrm{sin\θ}}_2{\mathrm{sin\θ}}_1{\mathrm{cos\θ}}_3\\ -{\mathrm{cos\θ}}_1{\mathrm{sin\θ}}_3& {\mathrm{cos\θ}}_1{\mathrm{cos\θ}}_3\end{array}\right]\end{array}.$

Beneficial effect:

The present invention judges that when needing to introduce slipspeed compensates accurately, and reasonably introduces slipspeed compensation according to the sideslip coefficient calibrated.Reduce the reason such as principle of work and allocation plan due to electromagnet log, when causing naval vessel to be turned, electromagnet log cannot measure slipspeed that naval vessel transversely exists and impact on inertial navigation system performance.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of slipspeed error compensation criterion of the present invention and method.

Fig. 2 is hull coordinate system schematic diagram of the present invention.

Fig. 3 is hull force analysis schematic diagram of the present invention.

Fig. 4 is that the present invention breaks away prejudgementing criteria analysis schematic diagram.

Hull slipspeed when Fig. 5 is naval vessel of the present invention turning and course angle change (by water currents greatly) schematic diagram.

Hull slipspeed when Fig. 6 is first time period naval vessel of the present invention turning and course angle schematic diagram.

Hull slipspeed when Fig. 7 is the turning of the present invention second time period naval vessel and course angle schematic diagram.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in further details.

The object of the invention is to judge that when should introduce slipspeed compensates, and provide sideslip coefficient scaling method.

The object of the present invention is achieved like this, as shown in Figure 1.

Step one: the permanent gyration model of analysis level, goes out slipspeed expression formula according to hull permanent gyration stress balance equation inference.

Step 2: according to each hydrodynamic parameter characteristic, in conjunction with the international zero dimension system of ship hydrodynamics that international towing basin meeting (ITTC) is recommended, the hydrodynamic force coefficient occurred in slipspeed expression formula step one obtained carries out naturalization.

Step 3: will according to rudder angle δ _rjudge whether that needing to introduce slipspeed compensates, when or δ _rduring=const>0, slipspeed need be compensated.

Step 4: at towed cable maneuvering period, can also use as criterion, wherein exported by inertial navigation system and obtain, get different t according to ship type difference ₀value.

Step 5: in towed cable transition period, first determines acceptable velocity error scope [-v _max, v _max], then determine the boundary value [-r of course angle speed _max, r _max], when course angle speed is outside border, need to compensate slipspeed.

Step 6: be fixedly installed in by the inertance element of integrated navigation system on naval vessel to be measured, ensures that inertance element mounting base axis and hull fore and aft axis strictly overlap.

Step 7: gathering integrated navigation system data, is naval vessel east orientation speed V respectively _e, north orientation speed V _n, and the pitching angle theta on naval vessel ₁, roll angle θ ₂and course angle θ ₃.

Step 8: by for producing enough large slipspeed, making naval vessel do the motion of S type, noting keeping speed of rotation constant in turning process.In the process, V is recorded _e, V _n, θ ₁, θ ₂, θ ₃export.

Step 9: the hull east orientation speed V of record time turning in naval vessel _ewith north orientation speed V _nutilize pose transformation matrix to project to carrier system b to fasten, carrier fastens x-axis speed V _xbe hull slipspeed, carrier fastens y-axis speed V _ybe hull fore and aft speed.

Step 10: by the speed of rotation course angle θ by integrated navigation system export of hull when turning ₃carry out difference can determine its turn time size (namely r), hull fore and aft speed is V _y(namely u), hull slipspeed V _x(namely v), obtained by step 9, all values is substituted into formula the demarcation of sideslip coefficient k can be completed.

This research is by carrying out mathematical modeling to hull slipspeed, research ship body slipspeed is to the Influencing Mechanism of the inertial navigation system transient error of assisting based on electromagnet log, and then design suitable hull slipspeed compensation scheme, reduce electromagnet log range rate error to the impact of inertial navigation system, thus reduce system output maximum error.

The hull east orientation speed V of record time turning in naval vessel _ewith north orientation speed V _nutilize pose transformation matrix to project to carrier system b to fasten, namely

${\left[\begin{array}{c}{V}_x\\ V_y\end{array}\right]}^b={\left(C_n^b\right)}_{2\×2}\·{\left[\begin{array}{c}{V}_E\\ V_N\end{array}\right]}^n$

Wherein

$\begin{array}{c}{\left(C_n^b\right)}_{2\×2}=\\ \left[\begin{array}{cc}{\mathrm{cos\θ}}_2{\mathrm{cos\θ}}_3-& {\mathrm{cos\θ}}_2{\mathrm{sin\θ}}_3+\\ {\mathrm{sin\θ}}_2{\mathrm{sin\θ}}_1{\mathrm{sin\θ}}_3& {\mathrm{sin\θ}}_2{\mathrm{sin\θ}}_1{\mathrm{cos\θ}}_3\\ -{\mathrm{cos\θ}}_1{\mathrm{sin\θ}}_3& {\mathrm{cos\θ}}_1{\mathrm{cos\θ}}_3\end{array}\right]\end{array}.$

Carrier fastens x-axis speed V _xbe hull slipspeed, carrier fastens y-axis speed V _ybe hull fore and aft speed.

During research hull horizontal plane motion, first suppose fixed coordinate system initial point E and coordinate plane E ζ η all in ship center of gravity G institute in the horizontal plane.Therefore, the coordinate of center of gravity G under fixed coordinate system can usually be expressed as (ξ _g, η _g, 0), see Fig. 2.

The permanent gyration of surface level is three-degree-of-freedom motion, has 3 independently state variables in its equation of motion.Setting up equation of motion state variable used has multiple combination available.In order to obtain slipspeed expression formula, one group of state below this selects: u, v, r.

External force suffered by boats and ships, comprises rudder power and propeller thrust, is all water acting force strictly speaking.General always propeller thrust and moment and other water acting force are separated, propeller thrust is designated as X _t, and other water effect force and moments all are all combined and are designated as X, Y, N.In given waters, under hydrostatic condition, to given boats and ships object, the function that X, Y, N will be quantity of state and relevant controlling amount, and be a function determined.Because propeller thrust is not included within X, Y, N, the controlled quentity controlled variable considered is needed to only have rudder angle δ here _r, for the state variable selected, X, Y, N can be expressed as

X＝X(u,v,r,δ _r)(1)

Y＝Y(u,v,r,δ _r)(2)

N＝N(u,v,r,δ _r)(3)

Fig. 3 be hull do permanent revolution time stress balance graph of a relation.Wherein used coordinate system is hull coordinate system, and x-axis positive dirction represents the direction of bow.U represents tangential velocity.X, Y, N are the force and moments suffered by ship.

When permanent revolution, tangential velocity U is constant, therefore each power is zero in tangential making a concerted effort.Each power should equal centripetal force F with joint efforts in circumference radial direction _l.Because angle of revolution speed remains unchanged, therefore total torque is zero.

The form of centripetal force formula is

$F_L=m\sqrt{u^2+v^2}r---\left(4\right)$

Hull coordinate system uranium to dynamic balance relation be

X+X _T+mvr＝0(5)

Y-mur＝0(6)

N＝0(7)

Recover the permanent gyration general equation of surface level that hydrodynamic functional form obtains another form

X(u,v,r,δ _r)+mvr+X _T＝0(8)

Y(u,v,r,δ _r)-mur＝0(9)

N(u,v,r,δ _r)＝0(10)

The permanent gyration general equation of surface level remains nonlinear equation, them at a u=U ₀, v=r=δ _r=0 generate Taylor series, reservation zero degree and once item, obtain linear-apporximation formula:

$Y\≈Y_0+Y_{v}v+Y_{r}r+Y_{{\mathrm{\δ}}_r}{\mathrm{\δ}}_r---\left(11\right)$

$N\≈N_0+N_{v}v+N_{r}r+N_{{\mathrm{\δ}}_r}{\mathrm{\δ}}_r---\left(12\right)$

In formula $Y_0=Y(0,0,0){|}_{u=U_0}$

$N_0=N(0,0,0){|}_{u=U_0}$

$\begin{array}{c}{Y}_v=\frac{\∂Y}{\∂v}{|}_{u=U_0,v=r={\mathrm{\δ}}_r=0}\\ N_v=\frac{\∂N}{\∂v}{|}_{u=U_0,v=r={\mathrm{\δ}}_r=0}\\ .\\ .\\ .\\ N_v=\frac{\∂N}{\∂{\mathrm{\δ}}_r}{|}_{u=U_0,v=r={\mathrm{\δ}}_r=0}\end{array}$

Be called hydrodynamic force coefficient.

Suppose that outboard shape keeps symmetrical to vertical middle face, know Y thus ₀=N ₀=0.Notice this point, linear for function approximate expression (11) formula (12) is substituted into gyration general equation and namely obtains the permanent gyration linear equation of surface level

$Y_{v}v+(Y_r-m{U}_0)r=-Y_{{\mathrm{\δ}}_r}{\mathrm{\δ}}_r---\left(13\right)$

$N_{v}v+N_{r}r=-N_{{\mathrm{\δ}}_r}{\mathrm{\δ}}_r$

Thus, can obtain

$v=\frac{{\mathrm{mU}}_0{N}_{{\mathrm{\δ}}_r}-Y_r{N}_{{\mathrm{\δ}}_r}+N_r{Y}_{{\mathrm{\δ}}_r}}{Y_v{N}_{{\mathrm{\δ}}_r}-N_v{\mathrm{\γ}}_{{\mathrm{\δ}}_r}}\·r---\left(14\right)$

According to each hydrodynamic parameter characteristic, in conjunction with the international zero dimension system of ship hydrodynamics that international towing basin meeting (ITTC) is recommended, numerical value naturalization is carried out to the hydrodynamic force coefficient related in formula (14), derivation hull side slip velocity topology degree expression formula, adopt the zero dimension system that ship hydrodynamics is international herein, this system regulation is using water-mass density (ρ), the ship speed of a ship or plane (U), captain (L) three amount as fundamental quantity.The actual use value of fundamental quantity and operating point value, using the base value as himself zero dimension, are also the foundations determining every other value simultaneously.Dimensionless quantity adds that an apostrophe represents without exception in the upper right corner of this physical quantity symbol, being numerically equal to this physical quantity has because sub-value is divided by the base value of this physical quantity, carries out zero dimension naturalization to the hydrodynamic force coefficient related in formula.

The zero dimension equation of each hydrodynamic force coefficient substitutes in formula (14), obtains through abbreviation

$v=\frac{m^{\′}{N}_{{\mathrm{\δ}}_r}^{\′}-Y_r^{\′}{N}_{{\mathrm{\δ}}_r}^{\′}+N_r^{\′}{Y}_{{\mathrm{\δ}}_r}^{\′}}{Y_v^{\′}{N}_{{\mathrm{\δ}}_r}^{\′}-N_v^{\′}{Y}_{{\mathrm{\δ}}_r}^{\′}}\·L\·r---\left(15\right)$

Because zero dimension coefficient is without unit quantity, make zero dimension system be constant definite value for same ship type k, be referred to as sideslip coefficient, then (15) are expressed as

v＝k·L·r(16)

V=kLr is slipspeed expression formula.

Investigate the turning motion of boats and ships from ship motion angle, its process can be divided into three phases, i.e. maneuvering period, transition period and permanent cycle stage, as shown in Figure 4.Can find out, in maneuvering period amount to be compensated, laterally (sideslip) speed v value is less, and transverse acceleration rotating angular acceleration and rudder angle δ _rsignificantly continue to increase, so now judging whether slipspeed compensates is ideal phase, in addition, can consider to judge in the early stage of transition period according to actual conditions.Three kinds of sideslip compensation judgment methods can be produced thus:

1. rudder angle δ _rthe most direct criterion, when or δ _rduring=const>0, slipspeed need be compensated, but in practical work process, the simple inertial navigation system that relies on is difficult to obtain δ _ror

2. at maneuvering period, can also use as criterion, wherein exported by inertial navigation system and obtain, get different t according to ship type difference ₀value;

3. in transition period, acceptable velocity error scope [-v is first determined _max, v _max], the boundary value [-r of course angle speed can be determined by formula (16) _max, r _max], when course angle speed is outside border, need to compensate slipspeed.

The priority of above-mentioned three kinds of methods be 1. > 2. > is 3..

The key of sideslip coefficient scaling method is the precise speed information that dynamically can provide carrier, and obtaining correct velocity information has two kinds of modes usually, and one is that DGPS etc. defends and leads velocity information, and two is high-precision integrated navigation system velocity informations.Main elaboration utilizes high-precision integrated navigation system to carry out sideslip coefficient scaling method herein.

Utilize high-precision integrated navigation system to provide reference velocity information, when calculating hull slipspeed, need the mounting base axis of integrated navigation system and hull fore and aft axis strictly to overlap, to reduce the measuring error that alignment error is introduced.Utilize integrated navigation system to provide high-precision reference velocity information, hull sideslip coefficient demarcating steps is as follows:

1. the inertance element of integrated navigation system is fixedly installed on naval vessel to be measured, ensures that inertance element mounting base axis and hull fore and aft axis strictly overlap;

2. gathering integrated navigation system data, is naval vessel east orientation speed V respectively _e, north orientation speed V _n, and the pitching angle theta on naval vessel ₁, roll angle θ ₂and course angle θ ₃;

3. for producing enough large slipspeed, making naval vessel do the motion of S type, noting keeping speed of rotation constant in turning process.In the process, V is recorded _e, V _n, θ ₁, θ ₂, θ ₃export;

The hull east orientation speed V of record when 4. being turned in naval vessel _ewith north orientation speed V _nutilize pose transformation matrix to project to carrier system b to fasten, namely

${\left[\begin{array}{c}{V}_x\\ V_y\end{array}\right]}^h={\left(C_n^b\right)}_{2\×2}\·{\left[\begin{array}{c}{V}_E\\ V_N\end{array}\right]}^n---\left(17\right)$

Wherein

$\begin{array}{c}{\left(C_n^b\right)}_{2\×2}=\\ \left[\begin{array}{cc}{\mathrm{cos\θ}}_2{\mathrm{cos\θ}}_3-& {\mathrm{cos\θ}}_2{\mathrm{sin\θ}}_3+\\ {\mathrm{sin\θ}}_2{\mathrm{sin\θ}}_1{\mathrm{sin\θ}}_3& {\mathrm{sin\θ}}_2{\mathrm{sin\θ}}_1{\mathrm{cos\θ}}_3\\ -{\mathrm{cos\θ}}_1{\mathrm{sin\θ}}_3& {\mathrm{cos\θ}}_1{\mathrm{cos\θ}}_3\end{array}\right]\end{array}.$

Carrier fastens x-axis speed V _xbe hull slipspeed, carrier fastens y-axis speed V _ybe hull fore and aft speed;

5. from formula (16), hull sideslip coefficient formulas is

$k=\frac{v}{Lr}---\left(18\right)$

Wherein, ship length L is known, the course angle θ of the speed of rotation of hull when turning by exporting integrated navigation system ₃carry out difference can determine its turn time size (namely r), hull fore and aft speed is V _y(namely u), hull slipspeed V _x(namely v), 4. obtained by step, all values is substituted into formula (18), the demarcation of sideslip coefficient k can be completed.

For checking hull slipspeed model, trying with a river data choosing two periods of hulls turning periods in test figure, measuring equipment adopts PHINS+GPS combined inertial nevigation system (range rate error 0.1m/s).

It should be noted that this Test Data Collecting is tested in Song Hua River, because hull is not navigate by water in hydrostatic field, so current must have an impact to result of calculation.In order to reduce the impact that current calculate hull slipspeed as far as possible, the condition choosing the period of turning is: ensure in hull turning process, and slipspeed is constant value as far as possible.Its reason is: when turning in existence is compared with the region of mass-flow, and think constant because flow direction is approximate, and angle, ship's head constantly changes, the slipspeed causing current to make ship raw is cyclical variation; On the other hand, even can ignore if current are less, then the skid force produced when it is turned to naval vessel is also approximate can be ignored, and so measuring the hull turning slip speed obtained should be comparatively steady, does not have larger fluctuation.As shown in Figure 5, in hull turning process, the hull slipspeed calculated changes to 2m/s from-2m/s, can think what this was mainly caused by current, so this segment data is not suitable as calculate hull sideslip coefficient.

Fig. 6 is the data of the one period of ship turning period satisfied condition, and gets slipspeed average V _x=-1m/s, in period 31.872h ~ 31.905h, rotational speed when course angle variation range is 73.2 ° ~ 240 ° turnings is r=0.0245rad/s, and captain is 8m, all data are substituted in model formation (18), tries to achieve sideslip coefficient k=5.11.

For verifying the hull sideslip coefficient calculated further above, choose data time turning in the second segment naval vessel satisfied condition, as shown in Figure 7.In period 27.717h ~ 27.725h, course angle variation range is 208.7 ° ~ 251.9 °, the rotational speed that approximate treatment obtains when hull is turned is r=0.026rad/s, captain is 8m, all data are substituted in the computing formula (18) of model, calculate hull slipspeed V _x=1.06m/s, with measurement result approximately equal.

It should be noted that due to this test condition do not meet hull sideslip coefficient demarcate experimental enviroment requirement, current, wind and mounting base etc. all can bring impact to result of calculation, so this calibration coefficient can only as rough reference value.

Claims (3)

1. hull slipspeed error judges a compensation method, it is characterized in that: comprise the following steps,

Step one: be fixedly installed in by the inertance element of integrated navigation system on ship to be measured, ensures inertance element mounting base axis and the dead in line of hull fore and aft;

Step 2: make ship do the motion of S type, keep speed of rotation constant in turning process, gather integrated navigation system data, comprise east orientation speed V _e, north orientation speed V _n, pitching angle theta ₁, roll angle θ ₂and course angle θ ₃;

Step 3: the hull east orientation speed V of record when ship is turned _ewith north orientation speed V _nutilize pose transformation matrix to project to carrier system b to fasten, carrier fastens x-axis speed V _xbe hull slipspeed, carrier fastens y-axis speed V _ybe hull fore and aft speed;

Step 4: by the course angle θ exported integrated navigation system ₃carry out difference can determine its turn time size be the speed of rotation r of hull when turning ₀, and then determine sideslip coefficient wherein L is captain;

Step 5: ship to be measured does real navigation, speed of rotation r, the transverse acceleration of Real-time Collection hull when turning and rudder angle δ _r, carry out slipspeed v=kLr compensation according to three kinds of situations;

Situation one: according to rudder angle δ _rjudge whether that needing to introduce slipspeed compensates, when or δ _rduring=const>0, slipspeed need be compensated;

Situation two: at towed cable maneuvering period, if speed of rotation r and side-slip acceleration meet condition, then compensate slipspeed, t ₀for given threshold time;

Situation three: in towed cable transition period, first determines slipspeed scope [-v _max, v _max], then determine the boundary value [-r of speed of rotation _max, r _max], when speed of rotation is outside border, need to compensate slipspeed.

2. a kind of hull slipspeed error according to claim 1 judges compensation method, it is characterized in that: described carries out in slipspeed v=kLr compensation according to three kinds of situations, the priority of three kinds of situations is situation one from high to low successively, situation two, situation three.

3. a kind of hull slipspeed error according to claim 1 judges compensation method, it is characterized in that: described carrier fastens x-axis speed V _xy-axis speed V is fastened with carrier _yfor:

${\left[\begin{array}{c}{V}_x\\ V_y\end{array}\right]}^b={\left(C_n^b\right)}_{2\×2}\·{\left[\begin{array}{c}{V}_E\\ V_N\end{array}\right]}^n$

$\begin{array}{c}{\left(C_n^b\right)}_{2\×2}=\\ \left[\begin{array}{cc}{\mathrm{cos\θ}}_2{\mathrm{cos\θ}}_3-& {\mathrm{cos\θ}}_2{\mathrm{sin\θ}}_3+\\ {\mathrm{sin\θ}}_2{\mathrm{sin\θ}}_1{\mathrm{sin\θ}}_3& {\mathrm{sin\θ}}_2{\mathrm{sin\θ}}_1{\mathrm{cos\θ}}_3\\ -{\mathrm{cos\θ}}_1{\mathrm{sin\θ}}_3& {\mathrm{cos\θ}}_1{\mathrm{cos\θ}}_3\end{array}\right]\end{array}.$