CN102234048B - Method for correcting speed curve of elevator - Google Patents

Abstract

The invention relates to a method for correcting a speed curve of an elevator. The method is used for correcting a speed curve comprising a deceleration second half travel; and in the deceleration second half travel, the speed is decreased progressively to zero at positive constant jerk. The method comprises the following steps of: detecting the actual position of an elevator car to obtain an actual residual distance; comparing an estimated residual distance with the actual residual distance to obtain a residual distance error value; providing a speed correction curve used for correcting the deceleration second half travel under the condition that the residual distance error value is not zero; and superposing the speed curve and the speed correction curve to obtain the corrected speed curved.

Description

Elevator speed curve modification method

Technical field

The present invention relates to a kind of modification method of elevator speed curve.

Background technology

In order to take the traveling comfort that the passenger takes elevator into account, the velocity curve of elevator is level and smooth continuous curve in all processes, and accelerating curve is curve continuous and that do not have the offset existence.When input lift car dock floor, generate the control elevator accordingly according to the lift car preset movement distance and move the run curve of usefulness, run curve comprise the position to the curve of time, speed to the curve of time, acceleration/accel to the curve of time, the acceleration curve to the time.

As shown in Figure 1, velocity curve has comprised at time durations T usually ₁Interior speed is with the acceleration α of positive constant ₁From zero increase to speed V ₁Acceleration before half trip (be the non-uniform acceleration stroke, acceleration/accel can increase in time); At time durations T ₂Interior speed is with the acceleration/accel a of positive constant ₂From speed V ₁Be incremented to speed V ₂Decide accelerating travel (be the constant acceleration stroke, acceleration/accel is definite value); At time durations T ₃Interior speed is with the acceleration α of negative constant ₃From speed V ₂Be incremented to speed V ₄Acceleration after half trip (be the non-uniform acceleration stroke, acceleration/accel can reduce in time); At time durations T ₄In remain on speed V ₄Constant speed stroke (be the constant speed stroke, acceleration/accel equals zero); At time durations T ₅Interior speed is with the acceleration α of negative constant ₅From speed V ₄Be decremented to speed V ₅Deceleration before half trip (be the non-uniform acceleration stroke, acceleration/accel can reduce in time); At time durations T ₆Interior speed is with the acceleration/accel a of negative constant ₆From speed V ₅Be decremented to speed V ₆Decide retarding travel (be the constant acceleration stroke, acceleration/accel is definite value); At time durations T ₇Interior speed is with the acceleration α of positive constant ₇From speed V ₆Be decremented to half trip (be the non-uniform acceleration stroke, acceleration/accel can increase in time) after zero the deceleration.Can omit above-mentioned constant speed stroke according to circumstances according to the preset movement distance of car, and/or omit and above-mentionedly decide accelerating travel and decide retarding travel.

TaiWan, China patent announcement I295662 communique (calling patent documentation 1 in the following text) discloses the control setup that a kind of companion ladder is used turn-around machine, and it can adjust the run curve of companion ladder according to the car carrying capacity.The Fig. 2 that is disclosed in patent documentation 1 namely is this type of elevator speed curve.

Yet, in the operational process of elevator, hang lift car cable may because of inertia with respect to cable slide around the drive pulley of hanging, or factors such as velocity deviation that produce because of the speed of response of frequency converter cause the actual miles of relative movement of lift car error to be arranged with estimating to move.

For with lift car accurate and smooth-going rest in predetermined dock floor, be necessary to revise the elevator speed curve according to this error.

Summary of the invention

Therefore, the object of the present invention is to provide a kind of velocity curve modification method, it can be with easy mode correction elevator speed curve.

Another object of the present invention is to provide a kind of velocity curve modification method, it can be with smooth-going and continuous mode correction elevator speed curve, thereby makes the predetermined dock floor that rests in that lift car can be accurate and smooth-going.

According to elevator speed curve modification method of the present invention, it is used for revising the velocity curve that comprises the back half trip that slows down, and in this slowed down the back half trip, speed was decremented to zero with the acceleration of positive constant, and this method comprises the following step:

Detect the actual position of lift car, to obtain actual residual distance;

The residual distance of this reality is deducted the residual distance of estimating of this velocity curve, to obtain residual distance error value;

Be worth under the non-vanishing situation in this residual distance error, provide one for the speed correction curve of revising this deceleration back half trip, this speed correction curve comprises the 1st section, the 2nd section, the 3rd section and the 4th section at least; And

With this velocity curve and this speed correction curve superposition, to obtain to revise the back velocity curve;

Wherein this residual distance error value greater than zero situation under,

In the 1st section, the speed correction with the acceleration correction of positive constant from zero increase;

In the 2nd section, this speed correction is incremented to speed correction maxim with the acceleration correction of negative constant;

In the 3rd section, this speed correction is successively decreased from this speed correction maxim with the acceleration correction of negative constant;

In the 4th section, this speed correction is decremented to zero with the acceleration correction of positive constant;

Wherein be worth under the minus situation in this residual distance error,

In the 1st section, this speed correction is successively decreased from zero with the acceleration correction of negative constant;

In the 2nd section, this speed correction is decremented to speed correction minimum value with the acceleration correction of positive constant;

In the 3rd section, this speed correction increases progressively from this speed correction minimum value with the acceleration correction of positive constant;

In the 4th section, this speed correction is incremented to zero with the acceleration correction of negative constant.

The persond having ordinary knowledge in the technical field of the present invention after reading this manual, it will be appreciated that other purpose of the present invention and advantage.

Description of drawings

Fig. 1 shows known elevator speed curve;

Predetermined velocity curve and corresponding accelerating curve in the half trip after the deceleration that the speed that is presented at Fig. 2 reduces with the acceleration of positive constant;

Fig. 3 shows the preceding velocity curve of correction and revises the back velocity curve that wherein residual distance error value is less than zero; And

Fig. 4 shows the preceding velocity curve of correction and revises the back velocity curve that wherein residual distance error value is greater than zero.

The specific embodiment

Hereinafter with reference to description of drawings according to preferred embodiment of the present invention.Although accompanying drawing and preferred embodiment show many details of the present invention, the details shown in the present invention is not limited to.Accompanying drawing only for the usefulness of explanation, is not drawn to scale.

Fig. 2 is presented at after the deceleration that speed successively decreases with the acceleration of positive constant predetermined velocity curve and corresponding accelerating curve in the half trip.This back half trip that slows down is equivalent to most end stroke shown in Figure 1.

When the elevator run curve enters when slowing down the back half trip by deciding retarding travel, decide the acceleration/accel a of retarding travel ₆, the back half trip that slows down rate of onset V ₆And estimate residual apart from S ₇For known.

To be speed be reduced as zero stroke with the acceleration of positive constant to the back half trip owing to slow down, and therefore, the back half trip that slows down can satisfy following formula:

a＝αt(1)

$V=\frac{1}{2}{\mathrm{\αt}}^2---\left(2\right)$

$S=\frac{1}{6}{\mathrm{\αt}}^3---\left(3\right)$

Wherein α is acceleration, and t is the time, and V is speed, and S is distance.

Therefore, estimate residual apart from S ₇Can represent with following formula:

$S_7=\frac{1}{6}{\mathrm{\α}}_7{T}_7^3---\left(4\right)$

α wherein ₇Be the acceleration of the back half trip that slows down, T ₇Time durations (being the travel time of this trip) for the back half trip that slows down.

The rate of onset V of half trip after slowing down ₆Can represent with following formula:

$V_6=\frac{1}{2}{\mathrm{\α}}_7{T}_7^2---\left(5\right)$

After formula (4) and formula (5) rearrangement, obtain:

$S_7=\frac{1}{3}{V}_6{T}_7---\left(6\right)$

Try to achieve time durations T by formula (6) ₇For:

$T_7=\frac{3S_7}{V_6}---\left(7\right)$

According to acceleration/accel a ₆T during in time ₇Calculate acceleration α ₇, acceleration α ₇Can try to achieve via following formula:

${\mathrm{\α}}_7=\frac{a_6}{T_7}=\frac{2V_6^3}{9S_7^2}---\left(8\right)$

Because half trip after deceleration, velocity curve v (t) can represent with following formula:

$v\left(t\right)=\∫\mathrm{adt}$

$=\∫\∫\mathrm{\αdtdt}$

$=\frac{1}{2}{\mathrm{\&alpha;}}_7{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000102600200011.png,HSA00000102600200021.png"\; state="\{\{state\}\}">t</figure-callout>}}^2$

$=\frac{1}{2}\frac{2V_6^3}{S_7^2}{(T_7-t)}^2---\left(9\right)$

According to the embodiment of the invention, can near each dock floor, shield be set.When lift car during near dock floor, can pass through shield, obtain the actual distance between lift car and the dock floor by this, namely actual residual apart from S _R

By relatively actual more residual apart from S _RWith estimate residual apart from S ₇, obtain residual distance error value Δ S ₇, residual distance error value Δ S wherein ₇Can represent with following formula:

ΔS ₇＝S _R-S ₇(10)

There is being residual distance error value to exist under the situation, if velocity curve v (t) is not revised, then can't lift car be being rested in predetermined dock floor according to predetermined velocity curve v (t) accurate and smooth-goingly.

At residual distance error value Δ S ₇Under zero situation, represent the actual residual apart from S of lift car _RResidual apart from S greater than estimating ₇, answer the erection rate curve, make estimate residual apart from S ₇Prolong, make the residual distance of estimating of revised velocity curve realistic residual apart from S _R

At residual distance error value Δ S ₇Under the minus situation, represent the actual residual apart from S of lift car _RResidual apart from S less than estimating ₇, answer the erection rate curve, make estimate residual apart from S ₇Shorten, make the residual distance of estimating of revised velocity curve realistic residual apart from S _R

Fig. 3 shows the preceding velocity curve v (t) of correction and revises back velocity curve v (t) ', wherein residual distance error value Δ S ₇Greater than zero.

Fig. 4 shows the preceding velocity curve v (t) of correction and revises back velocity curve v (t) ', wherein residual distance error value Δ S ₇Less than zero.

To be example with situation shown in Figure 3 below, illustrate by velocity curve v (t) to the correction of revising back velocity curve v (t) '.

For velocity curve v (t) is modified to revised velocity curve v (t) ' smooth-goingly, be provided for the speed correction curve Δ v (t) of erection rate curve v (t).Similarly, speed correction curve Δ v (t) is level and smooth continuous curve in all processes, and acceleration correction curve (be the subdifferential of speed correction curve Δ v (t), (t) represent with Δ v ') is curve continuous and that do not have the offset existence.Speed correction curve Δ v (t) is made up of the 1st section, the 2nd section, the 3rd section and the 4th section, and each section has equal time durations, and the time durations of each section is T ₇/ 4.

In the 1st section, the speed correction with the acceleration correction (α represents with Δ) of positive constant from zero increase to 1/2nd of speed correction limit Δ v.

In the 2nd section, the speed correction is incremented to speed correction limit Δ v with the acceleration correction of negative constant (representing with-Δ α) by 1/2nd of speed correction limit Δ v.

In the 3rd section, the speed correction is decremented to 1/2nd of speed correction limit Δ v with the acceleration correction of negative constant (representing with-Δ α) from speed correction limit Δ v.

In the 4th section, the speed correction is decremented to zero with the acceleration correction (α represents with Δ) of positive constant by 1/2nd of speed correction limit Δ v.

" speed correction " refers to that speed correction curve Δ v (t) puts corresponding value sometime.In this example, speed correction curve Δ v (t) is at T ₇/ 4 time point corresponds to Δ v/2, at T ₇/ 2 time point corresponds to Δ v, at 3T ₇/ 4 time point corresponds to Δ v/2.

" acceleration correction " refers to the difference of the acceleration of the acceleration of velocity curve v (t) and revised velocity curve v (t) ', and represents with Δ α.Be that acceleration correction amount α is the second differential (Δ v " (t)) of speed correction curve Δ v (t), acceleration correction amount α is constant in this example.

Preferably select the acceleration correction amount α of each section, make the absolute value of the acceleration correction in the 1st to 4 section | Δ α | equate.So, can significantly simplify calculating, and residual distance error value Δ S ₇Can represent with following formula:

$\mathrm{\&Delta;}{S}_7=\frac{1}{2}\mathrm{\&Delta;v}{T}_7---\left(11\right)$

Trying to achieve speed correction limit Δ v by formula (11) is:

$\mathrm{\&Delta;v}=\frac{2\mathrm{\&Delta;}{S}_7}{T_7}---\left(12\right)$

At residual distance error value Δ S ₇Under zero situation, speed correction limit Δ v is the maxim of speed correction curve Δ v (t).

According to formula (2), the speed correction when the 1st section finishes

Δ v can represent with following formula:

$\frac{1}{2}\mathrm{\&Delta;v}=\frac{1}{2}\mathrm{\&Delta;\&alpha;}{\left(\frac{1}{4}{T}_7\right)}^2---\left(13\right)$

By formula (13), the acceleration correction amount α that calculates at the 1st section:

$\mathrm{\&Delta;\&alpha;}=\frac{16\mathrm{\&Delta;v}}{T_7^2}---\left(14\right)$

Therefore, the acceleration correction of positive constant can be expressed as Δ α in the 1st section, the acceleration correction of negative constant can be expressed as-Δ α in the 2nd section, the acceleration correction of negative constant can be expressed as-Δ α in the 3rd section, and the acceleration correction of positive constant can be expressed as Δ α in the 4th section.

Afterwards, with velocity curve v (t) and speed correction curve Δ v (t) superposition, to obtain revising back velocity curve v (t) ', revise back velocity curve v (t) ' and can represent with following formula:

v(t)′＝v(t)+Δv(t)(15)

Therefore, can try to achieve correction back velocity curve v (t) ' is:

During the 1st section, the time Situation under,

$v{\left(t\right)}^{\&prime;}=v\left(t\right)+\frac{1}{2}\mathrm{\&Delta;\&alpha;}{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000102600200011.png,HSA00000102600200021.png"\; state="\{\{state\}\}">t</figure-callout>}}^2$

$=\frac{1}{2}{\mathrm{\&alpha;}}_7{(T_7-t)}^2+\frac{1}{2}\mathrm{\&Delta;\&alpha;}{t}^2---\left(16\right)$

During the 2nd section, the time Situation under,

$v{\left(t\right)}^{\&prime;}=v\left(t\right)+[\mathrm{\&Delta;v}-\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(\frac{T_7}{2}-t)}^2]$

$=\frac{1}{2}{\mathrm{\&alpha;}}_7{(T_7-t)}^2+[\mathrm{\&Delta;v}-\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(\frac{T_7}{2}-t)}^2]---\left(17\right)$

During the 3rd section, the time Situation under,

$v{\left(t\right)}^{\&prime;}=v\left(t\right)+[\mathrm{\&Delta;v}-\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(t-\frac{T_7}{2})}^2]$

$=\frac{1}{2}{\mathrm{\&alpha;}}_7{(T_7-t)}^2+[\mathrm{\&Delta;v}-\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(\frac{T_7}{2}-t)}^2]---\left(18\right)$

During the 4th section, the time

Situation under,

$v{\left(t\right)}^{\&prime;}=v\left(t\right)+\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(\frac{T_7}{2}-t)}^2$

$=\frac{1}{2}{\mathrm{\&alpha;}}_7{(T_7-t)}^2+\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(\frac{T_7}{2}-t)}^2---\left(19\right)$

At residual distance error value Δ S ₇Under the minus situation, i.e. situation shown in Figure 4, speed correction curve Δ v (t) is made up of the 1st section, the 2nd section, the 3rd section and the 4th section, and each section has equal time durations, and the time durations of each section is T ₇/ 4.

In the 1st section, the speed correction is decremented to 1/2nd of speed correction limit Δ v with the acceleration correction (α represents with Δ) of negative constant from zero.

In the 2nd section, the speed correction is decremented to speed correction limit Δ v with the acceleration correction of positive constant (representing with-Δ α) by 1/2nd of speed correction limit Δ v.

In the 3rd section, the speed correction is incremented to 1/2nd of speed correction limit Δ v with the acceleration correction of positive constant (representing with-Δ α) from speed correction limit Δ v.

In the 4th section, the speed correction is incremented to zero with the acceleration correction (α represents with Δ) of negative constant by 1/2nd of speed correction limit Δ v.

At residual distance error value Δ S ₇Under the minus situation, speed correction limit Δ v is the minimum value of speed correction curve Δ v (t).

Though in the above-described embodiments, the speed correction curve comprises the 1st section to the 4 sections, but the speed correction curve also can further comprise between acceleration correction amount between the 1st section and the 2nd section be the section of constant and/or between the 3rd section and the 4th section the acceleration correction amount be the section of constant.The speed correction curve can comprise further also that the speed correction is the section of constant between the 2nd section and the 3rd section

Though above-described embodiment to the velocity curve correction once will be appreciated that, also can be before lift car stops, in different time points to the velocity curve correction repeatedly.

Preferably after deceleration half trip during in revise predetermined velocity curve, the time point of namely revising predetermined speed can be slow down the back half trip initial point or enter arbitrary time point after the back half trip that slows down.If the arbitrary time point after entering the back half trip that slows down is just to predetermined velocity curve correction, then employed time durations T when the computation speed correction curve ₇Mean the time point of predetermined velocity curve correction to the speed of predetermined velocity curve is reduced as time durations between zero the time point.

Determine the principle of elevator speed curve to can be applicable to produce the speed correction curve equally.

Though the present invention describes demonstration with reference to preferred embodiment, only will be appreciated that in not breaking away from spirit of the present invention and category for the persond having ordinary knowledge in the technical field of the present invention, still many variations and modification must be arranged.Therefore, the present invention is not restricted to disclosed embodiment, and the equalization that does not namely depart from the present patent application changes and modification, should still belong to covering scope of the present invention.

Claims (9)

1. elevator speed curve modification method, it is used for revising the velocity curve that comprises the back half trip that slows down, and in this slowed down back half trip, speed was reduced as zero with fixing acceleration, and this method comprises the following step:

Detect the actual position of lift car, to obtain actual residual distance;

The residual distance of this reality is cut the residual distance of estimating of this velocity curve, to obtain residual distance error value;

Be worth under the non-vanishing situation in this residual distance error, provide one for the speed correction curve of revising this deceleration back half trip, this speed correction curve comprises the 1st section, the 2nd section, the 3rd section and the 4th section at least; And

With this velocity curve and this speed correction curve superposition, to obtain to revise the back velocity curve;

Wherein this residual distance error value greater than zero situation under,

In the 1st section, the speed correction with the acceleration correction of positive constant from zero increase;

In the 2nd section, this speed correction is incremented to speed correction maxim with the acceleration correction of negative constant;

In the 3rd section, this speed correction is successively decreased from this speed correction maxim with the acceleration correction of negative constant;

In the 4th section, this speed correction is decremented to zero with the acceleration correction of positive constant;

Wherein be worth under the minus situation in this residual distance error,

In the 1st section, this speed correction is successively decreased from zero with the acceleration correction of negative constant;

In the 2nd section, this speed correction is decremented to speed correction minimum value with the acceleration correction of positive constant;

In the 3rd section, this speed correction increases progressively from this speed correction minimum value with the acceleration correction of positive constant;

In the 4th section, this speed correction is incremented to zero with the acceleration correction of negative constant.

2. the method for claim 1, wherein this residual distance error value greater than zero situation under,

In the 1st section, this speed correction with the acceleration correction of positive constant from zero increase to this speed correction peaked 1/2nd;

In the 2nd section, this speed correction is incremented to speed correction maxim with the acceleration correction of negative constant from this speed correction peaked 1/2nd;

In the 3rd section, this speed correction is decremented to this speed correction peaked 1/2nd with the acceleration correction of negative constant from this speed correction maxim;

In the 4th section, this speed correction is decremented to zero with the acceleration correction of positive constant from this speed correction peaked 1/2nd.

3. the method for claim 1 wherein is worth under the minus situation in this residual distance error,

In the 1st section, this speed correction is decremented to 1/2nd of this speed correction minimum value with the acceleration correction of negative constant from zero;

In the 2nd section, this speed correction is decremented to this speed correction minimum value with the acceleration correction of positive constant from 1/2nd of this speed correction minimum value;

In the 3rd section, this speed correction is incremented to 1/2nd of this speed correction minimum value with the acceleration correction of positive constant from this speed correction minimum value;

In the 4th section, this speed correction is incremented to zero with the acceleration correction of negative constant from 1/2nd of this speed correction minimum value.

4. as each described method in the claim 1 to 3, wherein each of the 1st section, the 2nd section, the 3rd section and the 4th section has equal time durations.

5. as each described method in the claim 1 to 3, wherein each the absolute value of acceleration correction of the 1st section, the 2nd section, the 3rd section and the 4th section equates.

6. as each method in the claim 1 to 3, wherein this speed correction maxim or this speed correction minimum value equal:

7. the method for claim 1, wherein this speed correction curve more comprise between acceleration correction amount between the 1st section and the 2nd section be the section of constant and/or between the 3rd section and the 4th section the acceleration correction amount be the section of constant.

8. as claim 1 or 7 described methods, wherein this speed correction curve comprises more that the speed correction is the section of constant between the 2nd section and the 3rd section.

9. elevator speed curve modification method, it is used for revising a velocity curve v (t), and this velocity curve v (t) is included in time durations T at least ₇In, with the acceleration α of speed with constant ₇Be reduced as half trip after zero the deceleration, this method comprises the following step:

Detect the actual position of lift car, actual residual apart from S to obtain _R

Relatively this velocity curve estimate residual apart from S ₇And this reality is residual apart from S _R, to obtain residual distance error value Δ S ₇, Δ S wherein ₇=S _R-S ₇

At this residual distance error value Δ S ₇Under the non-vanishing situation, provide one for the speed correction curve Δ v (t) that revises this deceleration back half trip, this speed correction curve Δ v (t) is made up of the 1st section, the 2nd section, the 3rd section and the 4th section, and each section has equal time durations T ₇/ 4;

Calculate the speed correction limit Δ v of this speed correction curve Δ v (t), wherein $\mathrm{\&Delta;v}=\frac{{2\mathrm{\&Delta;S}}_7}{T_7};$

Calculate the acceleration correction amount α of this speed correction curve Δ v (t) in the 1st section and the 4th section, wherein The acceleration correction of this speed correction curve Δ v (t) in the 2nd section and the 3rd section then is-Δ α;

With this velocity curve v (t) and this speed correction curve Δ v (t) superposition, to obtain to revise back velocity curve v (t) ', v (t) '=v (t)+Δ v (t) wherein, wherein

During the 1st section, the time

Situation under,

$v{\left(t\right)}^{\&prime;}=v\left(t\right)+\frac{1}{2}{\mathrm{\&Delta;\&alpha;t}}^2=\frac{1}{2}{\mathrm{\&alpha;}}_7{(T_7-t)}^2+\frac{1}{2}{\mathrm{\&Delta;\&alpha;t}}^2$

During the 2nd section, the time

Situation under,

$v{\left(t\right)}^{\&prime;}=v\left(t\right)+[\mathrm{\&Delta;v}-\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(\frac{T_7}{2}-t)}^2]=\frac{1}{2}{\mathrm{\&alpha;}}_7{(T_7-t)}^2+[\mathrm{\&Delta;v}-\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(\frac{T_7}{2}-t)}^2]$

During the 3rd section, the time

Situation under,

$v{\left(t\right)}^{\&prime;}=v\left(t\right)+[\mathrm{\&Delta;v}-\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(t-\frac{T_7}{2})}^2]=\frac{1}{2}{\mathrm{\&alpha;}}_7{(T_7-t)}^2+[\mathrm{\&Delta;v}-\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(\frac{T_7}{2}-t)}^2]$

During the 4th section, the time

Situation under,

$v{\left(t\right)}^{\&prime;}=v\left(t\right)+\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(\frac{T_7}{4}-t)}^2=\frac{1}{2}{\mathrm{\&alpha;}}_7{(T_4-t)}^2+\frac{1}{2}\mathrm{\&Delta;\&alpha;}{(\frac{T_7}{4}-t)}^2.$

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