JP3397647B2 - Propulsion control device - Google Patents

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a setting input section capable of inputting a terrain condition in a propulsion section from a propulsion start position to a propulsion target position when propelling a propelling body in the ground along a propulsion planning line. Relates to a propulsion control device.

[0002]

2. Description of the Related Art A propulsion device used in a conventional propulsion method is provided at a propulsion body to be pushed into the ground and at a propulsion start position (hereinafter simply referred to as "start position") for pushing the propulsion body into the ground. It includes a propulsion device, a propulsion control device that controls the propulsion work, a position detection device that is provided at a propulsion arrival target position (hereinafter, simply referred to as “target position”) for detecting the current position of the propulsion body, and the like. At the time of the propulsion work, the worker first determines the propulsion plan line in consideration of the horizontal distance and height from the start position to the target position, the minimum radius of curvature that can be propelled by the propulsion body, and the like. And during the promotion work,
While comparing the current position of the propelling body shown on the display unit of the propulsion control device with the propulsion planning line, the worker needs to appropriately correct the propulsion direction of the propulsion body.

[0003]

The propulsion planning line can be determined relatively smoothly when there is no obstacle or the like on the way from the start position to the target position. However, there are often obstacles in the middle of the propulsion route, for example, a gutter is provided beside the road that is the starting position. In such a case, the operator is required to determine whether or not the obstacle can be avoided, in addition to the determination of the propulsion planning line. For this reason, the determination of the promotion plan line becomes complicated, and there is still room for improvement in order to carry out the promotion work efficiently. An object of the present invention is to eliminate such drawbacks of the conventional technique and provide a propulsion control device capable of quickly determining whether or not propulsion is possible when an obstacle exists in the propulsion section.

[0004]

[Means for Solving the Problems]

(Structure 1) As described in claim 1, the propulsion control device of the present invention is arranged such that the coordinate values of the propulsion start position, the coordinate values of the propulsion arrival target position, and the propulsion start position and the propulsion arrival target position are between Whether the propulsion body can reach the propulsion arrival target position based on the respective coordinate values and the minimum propulsion curvature of the propulsion body, having coordinate input means for inputting the coordinate values of the obstacle located It is characterized in that it has a judgment and notification means for judging whether or not to notify. (Operation / Effect) In this configuration, in addition to inputting the coordinate value of the propulsion start position and the coordinate value of the propulsion arrival target position,
Since the coordinate values of obstacles located between the propulsion start position and the propulsion arrival target position can be input, a propulsion planning line for reaching the propulsion target position can be set and the obstacle can be set. It is possible to quickly determine whether or not an object can be avoided. Therefore, according to the propulsion control device of the present invention, the propulsion work can be made simple and efficient.

(Structure 2) In the propulsion control device of the present invention, as described in claim 2, the judgment notifying means is based on the respective coordinate values and the minimum propulsion curvature of the propulsion body.
When a propulsion planning line consisting of a straight propulsion section extending in the horizontal direction from the propulsion start position and a bending section having a minimum propulsion curvature is obtained, and the coordinate values of the obstacle are located above the propulsion planning line. In addition, it can be configured to determine that the propulsion body can reach the propulsion arrival target position. (Action / Effect) The obstacle corresponds to, for example, a gutter beside a road. In most cases, such an obstacle extends from the ground side to a predetermined depth in the ground. Therefore, if a promotion planning line that avoids obstacles on the lower side is obtained as in this configuration, it can be determined that the promotion along the promotion planning line can be reliably performed. Further, when the propulsion section is viewed from the side and the area surrounded by the propulsion planning line and the ground surface is regarded as an area where an obstacle can exist, the propulsion planning line formed by this configuration is formed. Region will have the largest area. That is, in the present configuration, the linear propulsion section extending in the horizontal direction and the area above the bending section formed with the minimum propulsion curvature after that correspond to the above-mentioned area. This is because the area becomes maximum. This is clear from the fact that, if the bending section is configured by a propulsion planning line having a larger curvature, the propulsion planning line is displaced upward as compared with the propulsion planning line having the smallest propulsion curvature. In other words, as a general rule, it is impossible to carry out propulsion through the area below the propulsion planning line obtained by this configuration, and according to this configuration, it is possible to obtain a boundary condition as to whether or not propulsion is possible while avoiding obstacles. it can. As a result, the worker can also try to set another propulsion planning line in which the curvature of the bending section is set large.

[0006]

BEST MODE FOR CARRYING OUT THE INVENTION Embodiments of the present invention will be described below with reference to the drawings.

(Outline) FIG. 1 shows an example of the structure of a propulsion apparatus S according to the present invention. The propulsion control device of the present invention, for example, in order to pull a gas pipe, a water pipe, etc. buried underground in the road 1 from a work pit 2 formed in the road 1 to a predetermined position on a residential land 3 adjacent to the road 1. Is a control device provided in the propulsion device S for pushing the propelling body 4 into the ground from the work pit 2 and forming the propulsion hole 5 reaching the predetermined position of the residential land 3.

(Propulsion device) The propulsion device S is the propulsion device 6
And a propulsion unit 4, and a propulsion control device 7 for controlling the propulsion unit 6 and the like. The propulsion device 6 is installed and used in the work pit 2 and the like, and includes a pressing mechanism 8 including a hydraulic jack 8a for pressing the propulsion body 4 into the soil. The propulsion control device 7 is
Although not shown in the drawings, it has portability, and a strap attachment portion is provided so that an operator can hang it from the shoulder. The propulsion body 4 to be pushed into the soil is connected to the tip of the propulsion device 6. The propulsion body 4 has a propulsion head 4a attached to its tip and the propulsion head 4a.
And a plurality of propulsion pipes 4b sequentially inserted and connected between the propulsion device 6 and the propulsion device 6. Inside the propulsion head 4a, a pressure receiving surface portion 9 rotatable about an axis substantially parallel to the axis Z of the propulsion head 4a and an axis substantially parallel to the axis Z of the propulsion head 4a are provided. The transmitter coil 10 is included. The target position P ₂ is is provided with a pair of receiving coils, the transmitting coil 10 to identify the approximate position of the propellant 4 by detecting the voltage induced by magnetic field originated.

(Terrain condition / operating condition) The propulsion control device 7 of the present invention is a propulsion planning line L in a section where actual propulsion is performed.
To calculate. As a result, the propulsion control device 7 determines whether or not the terrain allows propulsion work, and displays the position of the propulsion head 4a during the propulsion work. For this purpose, prior to the propulsion work, the operator inputs the terrain condition A from the installation position of the propulsion device 6, that is, the starting position P ₁ to the target position P ₂ and the operating condition B of the propulsion device S. Figure 1
Shows an example of the terrain of the propulsion section forming the propulsion hole 5.

In this embodiment, a gutter 11 which is an obstacle 11 adjacent to the road 1 from a work pit 2 provided on the road 1
The case of avoiding a and promoting to the residential land 3 having a predetermined height is shown. As the terrain condition A, a numerical value that can specify the coordinate value of the start position P ₁ or the target position P ₂ is input. Specifically, as shown in FIG. 1, a road depth 12 from the surface of the road 1 to the propulsion device 6 and a gutter depth 1 from the surface of the road 1 to the lower end of the gutter 11a.
3, height of residential land from the surface of road 1 to the surface of residential land 1
4, the amount of moat 15 provided in advance at the target position P ₂ to reach the propulsion body 4, the road distance 1 from the propulsion device 6 to the residential land boundary 1
6. Enter the residential land distance 17 from the residential land boundary to the target position P ₂ . On the other hand, as the operating condition B, for example, the propulsion unit 4
Enter the propulsion minimum radius of curvature R of. Here, the propulsion minimum radius of curvature R refers to the minimum radius when the propulsion direction is changed within the vertical plane when the propulsion body 4 propels. As shown in FIGS. 1 and 2, the propelling body 4 includes a propulsion head 4a at the most tip and a plurality of propulsion pipes 4b connected to the rear of the propulsion head 4a. These propulsion pipes 4b are connected to each other using a pin 18 provided in the horizontal direction. As a result,
The propelling pipes 4b are prevented from bending horizontally in accordance with the propulsion, and can be bent only within the vertical plane.
However, even when the propulsion pipes 4b are bent in the vertical plane, the stopper mechanism 19 provided at the connecting portion between the propulsion pipes 4b prevents the propulsion pipes 4b from bending at a predetermined angle or more. Therefore, the propulsion minimum radius of curvature R is
The curvature of the propulsion path appears when the stopper mechanism 19 is operating between the propulsion pipes 4b. With this configuration, excessive bending of the propulsion path can be prevented, and the thrust of the propulsion body 4 can be ensured. However, the propulsion minimum radius of curvature R is uniquely determined by the propulsion pipe 4b used. Therefore, it is usually treated as a fixed value, and the minimum radius of curvature R of the propulsion is not input at each propulsion work.

The terrain condition A and the operating condition B are shown in FIG.
As shown in, the setting is performed using the setting input unit 20 which is the coordinate input means provided in the propulsion control device 7. The setting input section 20
Is composed of, for example, a keyboard, and specific numerical values of the above-mentioned topographical condition A can be sequentially input. The input is performed according to an instruction of a display unit 21 provided beside the input unit 20 which is, for example, a liquid crystal display. The display unit 21
Is the first screen G1 for inputting the terrain condition A etc.
Then, it is possible to switch to the second screen G2 that displays the position of the propelling body 4 when actually propelling. Topographic condition A
When inputting, etc., the screen is switched to the first screen G1.
The display unit 21 is provided with a cover body 21a that can be opened and closed. The cover body 21a has a function of protecting the display unit 21 when the propulsion control device 7 is stored and transported, and is a shade when the display unit 21 is in use.
Has the function of ensuring visibility. As shown in Figure 2,
On the first screen G1, the approximate topography that has been input in advance or the side-view topography of the propulsion section set in the preceding propulsion work is displayed. Further, the position where each numerical value of the terrain condition A should be input is displayed. These input positions are configured to sequentially blink, and the worker inputs a predetermined condition according to the movement of the blinking positions.

(Determination of Propulsion Possible) In principle, the propulsion device S according to the present invention starts propulsion in the horizontal direction from the start position P ₁ . Then, when the predetermined position is reached, the propulsion direction is bent vertically upward, and the minimum propulsion radius of curvature R
It is intended to reach the target position P ₂ performs propulsion in. After that, the point where the propulsion direction is bent is the bending point Pc.
Called. In the propulsion control device 7 of the present invention, when the obstacle 11 exists between the start position P ₁ and the target position P ₂ , propulsion work can be performed based on the input terrain condition A and operating condition B. Determine whether or not. The result of the judgment is notified to the worker by the judgment notifying means H.

Whether or not the propulsion work is possible is performed by geometrical calculation based on the input operating condition B. An example thereof is shown in FIG. The horizontal distance X from the start position P ₁ to the target position P ₂ , the target height h, the minimum radius of curvature R of the propelling body 4 and the position of the obstacle 11 are used for the determination. In this case, the horizontal distance X is a distance obtained by adding the road distance 16 and the residential land distance 17. Further, the target height h is the height obtained by subtracting the digging amount 15 from the height obtained by adding the residential land height 14 to the road depth 12. The minimum radius of curvature R for propulsion is uniquely determined by the propelling body 4 used, and in this case, it is 2.8 m. In the case of this embodiment, the obstacle 11 is the gutter 11a, and the gutter depth 13 is the position thereof.

Based on the above conditions, a bending point Pc at which the straight-ahead propulsion initially started in the horizontal direction is bent upward is calculated.
The method will be described with reference to FIG. First, the propulsion head 4a from the bending point Pc to the target position P ₂
The deflection angle θ _{2 of} is calculated. Center Q of the minimum thrust curvature
Is located in a direction perpendicular to the straight line portion of the propulsion planning line L. In other words, it is above the propulsion planning line L at a right angle direction from the bending point Pc on the propulsion planning line L. Assuming a bending point Pc and a center of curvature Q as shown in FIG. 4, the deflection angle θ ₂
Is the corner PcQP ₂ . Further, QPc = QP ₂ = R. In addition, the point where a perpendicular line is drawn from P ₂ to QPc is set as point C. On the other hand, a point D perpendicular to the target position P _{2 with} respect to the extension of the initial promotion plan line P ₁ Pc is defined as D.
In FIG. 4, P ₁ D = horizontal distance X and P ₂ D =
The target height h. Since QC = QPc-CPc, Rcos θ ₂ = R−h cos θ ₂ = (R−h) / R, and the deflection angle θ ₂ can be obtained. When the deflection angle θ ₂ is obtained, the horizontal distance Xc of the bending section after the bending point Pc is Xc = PcD = CP ₂ = R · sin θ ₂ . As a result, the horizontal distance Xs of the section where straight ahead is performed
Can be obtained by Xs = X−Xc = X−R · sin θ ₂ . That is, in this case, from the starting position P ₁ , the above-mentioned Xs is driven straight ahead in the horizontal direction, and the bending point P
After passing c, the pressure receiving surface portion 9 of the propulsion head 4a is rotated downward, the propulsion head 4a is deflected upward, and bending propulsion is performed with the propulsion minimum curvature radius R.

As described above, the propulsion control device 7 calculates the propulsion planning line L. However, depending on the terrain condition A,
In some cases, propulsion work cannot be performed. Therefore, following the calculation of the promotion planning line L, the validity of the promotion planning line L is judged by the judgment notifying means H. The judgment notifying means H
First determines whether the obtained Xs is a positive value. X
When s is a negative value, the bending point Pc is the starting position P ₁
Since it is located on the front side of the propulsion direction, the judgment notifying means H displays, for example, "the start-up distance of the propulsion body is insufficient". In this case, the propulsion start direction must be set upward from the beginning, and installation of the propulsion device S becomes complicated, which lowers the efficiency of propulsion work. On the other hand, if the value of the horizontal distance Xs in the section where straight ahead is to be performed is positive, no notification is given and the next judgment is made.

The propulsion planning line L obtained above does not consider the obstacle 11 in the middle of the propulsion planning line L. Therefore,
In order to carry out the promotion without any obstacle, it is necessary to judge whether or not the obstacle 11 can be avoided. That is, the judgment notifying means H compares the propulsion planning line L with the position of the lower end of the side groove 11a, and the position of the lower end of the side groove 11a is located below the propulsion planning line L. When it is determined, it is determined that the vehicle cannot be promoted, and the display unit 21 displays, for example, "hit an obstacle". In this case, it is necessary to take measures such as further digging down and setting the start position P ₁ . On the other hand, when it is determined that the lower end portion of the side groove 11a is located above the propulsion planning line L, no notification is given and the display unit 21 is displayed on the second screen G.
The instruction to switch to 2 is displayed.

It should be noted that the propulsion planning line L obtained above is the planning line for propulsing deepest in the ground to avoid the obstacle 11 existing in the propulsion section, and can be said to be the safest planning line. When the promotion section is viewed from the side, the promotion plan line L
When the region surrounded by the ground surface and the ground surface is regarded as a region in which the obstacle 11 can exist, the above-mentioned region formed by the propulsion planning line L obtained as described above has the maximum area, in other words, The propulsion planning line L is the planning line farthest from the obstacle 11. This is clear from the fact that if the curvature of the bending section is set to be larger, the propulsion planning line L is displaced further upward. On the contrary, as a general rule, the propulsion work passing through the region below the propulsion planning line L obtained as described above becomes impossible. When the propulsion planning line L is obtained as described above, it is possible to obtain the boundary condition as to whether or not propulsion is possible while avoiding the obstacle 11. As a result, the worker can also try to set another propulsion planning line L in which the curvature of the bending section is separately set to be large.

Further, the judgment notifying means H also judges the applicable range of the propulsion device S. That is, when the horizontal distance X exceeds 10 m, and when the target height h
If the distance exceeds 3 m, the accuracy with which the propelling body 4 reaches the target position P ₂ may be reduced. This is because if the horizontal distance X is too long, the frictional force applied to the propulsion body 4 and the propulsion pipe 4b by the earth pressure becomes excessive and it becomes difficult to push in the propulsion body 4. Also, the target height h is 3 m
If it exceeds the minimum propulsion radius of curvature R of the propulsion body 4 beyond the range, an area where the propulsion body 4 recedes and rises as the propulsion body 4 is deflected upward, and not only an unnatural propulsion planning line L but also a propulsion planning line L However, there may be a problem such as once crossing a residential land boundary. Therefore, even when the horizontal distance X and the target height h exceed the above range,
The display unit 21 is configured to display that fact.

(Propulsion work) Topographic condition A and operating condition B
After inputting, the display of the display unit 21 is switched to the second screen G2 when the propulsion work is possible. An example of the second screen G2 is shown in FIG. To switch to the second screen G2,
This is performed using the button of the setting input unit 20. 2nd screen G
2 shows a schematic view relating to a side view and a plan view of the propulsion section, and also includes the propulsion planning line L and the propulsion head 4a.
The current position of is displayed on each schematic. Also,
The attitude of the propulsion head 4a is also displayed on the side display. That is, the inclination angle of the axis Z of the propulsion head 4a with respect to the horizontal direction and the orientation of the pressure receiving surface portion 9 at the tip of the propulsion head 4a with respect to the axis Z of the propulsion head 4a are displayed.

The propulsion of the propulsion head 4a is performed by pushing the propulsion pipes 4b sequentially connected to the rear of the propulsion head 4a into the ground by the propulsion device 6.
The pushing of the propulsion head 4a is performed by, for example, alternately repeating a pushing operation and a retracting operation. The push-in / pull-out operation of the propulsion unit 4 is performed by using the driving operation unit 22 of the propulsion control device 7.

<Propulsion Direction Control of Propulsion Head in Vertical Direction> The position of the propulsion head 4a shown in the side view display of the second screen G2 is obtained from the attitude of the propulsion head 4a and the propulsion distance of the propulsion head 4a. The attitude of the propulsion head 4a refers to the inclination angle of the axis Z of the propulsion head 4a with respect to the horizontal direction. Since the propulsion head 4a is provided with the inclinometer 23 as shown in FIG. 2, the inclination angle of the propulsion head 4a can always be known. The direction control of the propulsion head 4a is performed by appropriately rotating the pressure receiving surface portion 9 provided at the tip of the propulsion head 4a. In other words, the pressure receiving surface portion 9 is the same as that shown in FIG.
As shown in FIG. 3, the thrust head 4a is processed obliquely with respect to the axis Z, and the advancing direction of the thrust head 4a is bent by the lateral component of earth pressure acting on the pressure receiving surface portion 9. Therefore, when advancing the propulsion head 4a in a specific direction, the pressure receiving surface portion 9 is provided on the side opposite to the direction.
You should turn. On the other hand, the propulsion distance of the propulsion head 4a is
For example, it is determined by integrating the strokes of pushing the propulsion pipe 4b with the hydraulic jack 8a or by counting the number of connected propulsion pipes 4b. The propulsion control device 7 performs integration / counting of these. In this way, the current position of the propulsion head 4a can be almost specified by cumulatively evaluating the change in the propulsion distance and the posture of the propulsion head 4a during the propulsion distance while considering the initial propulsion start posture of the propulsion head 4a. it can.

<Control of the propulsion direction of the propulsion head in the horizontal direction> As described above, since the propulsion tubes 4b are connected by the horizontal pin 18, the structure is such that it is difficult to deflect in the horizontal direction. However, in reality, since the propulsion body 4 is bent, the propulsion head 4a is displaced in the horizontal direction as the propulsion is performed. Therefore, in order to control the propulsion direction of the propulsion head 4a in the horizontal direction, as shown in FIG. 6, the transmitting coil 10 for generating a magnetic field is provided in the propulsion head 4a, and a pair of receiving coils for detecting the magnetic field. 24 is provided at the target position P ₂ . In the transmitter coil 10, the axis of the transmitter coil 10 has a propulsion head 4
It is provided so as to be parallel to the axis Z of a. A current that has been subjected to a predetermined modulation is applied to the oscillator coil 10,
This produces a magnetic field. On the other hand, the pair of receiving coils 24 are provided at symmetrical positions at the target position P ₂ with a vertical plane including the propulsion planning line L in between. As a general rule, the axes of the two receiving coils 24 are parallel to each other, and they are also installed parallel to the propulsion planning line L in plan view. The pair of receiving coils 24 includes the transmitting coil 1
Electromagnetic induction occurs due to the magnetic field generated by 0, and a voltage is generated in each receiving coil 24. By comparing these voltage values, the position of the propulsion head 4a is specified. In other words, when the propulsion head 4a is propelling correctly along the propulsion planning line L, the number of magnetic flux lines passing through both receiving coils 24 becomes equal, and the voltage value generated in the pair of receiving coils 24 becomes equal. Are equal. However, when the propulsion head 4a is displaced toward the one receiving coil 24 side with respect to the propulsion planning line L, the number of magnetic flux lines passing through both receiving coils 24 is different, and each of them is different. There is a difference in the voltage value generated in the receiving coil 24. In the receiving coil 24 on the side where the propulsion head 4a is displaced, more lines of magnetic force pass, so that the receiving coil 24 on the side exhibits a high voltage value. The position of the propulsion head 4a with respect to the target position P ₂ is specified by comparing and evaluating the magnitude of the difference between the voltage values. If the propulsion head 4
When a is displaced to the left toward the target position P ₂ , the position of the propulsion head 4a is plotted on the left side with respect to the propulsion planning line L in the plane display of the second screen G2. The distance between the position of the plot and the propulsion planning line L is
The longer the displacement of the propulsion head 4a with respect to the target position P _{2, the} longer the display.

(Effect) With the propulsion control device 7 of the present invention, in addition to inputting the coordinate value of the propulsion start position P _{1 and} the coordinate value of the propulsion arrival target position P ₂ , the propulsion start position P ₁ Since the coordinate value of the obstacle 11 located between the propulsion arrival target position P ₂ can be input, the propulsion planning line L for making the propulsion body 4 reach the propulsion arrival target position P ₂ can be set. Therefore, it is possible to quickly determine whether the obstacle 11 can be avoided. Therefore, according to the propulsion control device 7 of the present invention, the propulsion work can be made simple and efficient.

It should be noted that in the description of the above claims, reference numerals are given to refer to the drawings for convenience of comparison with the drawings, but the present invention is not limited to the configuration of the accompanying drawings by the entry. Absent.

[Brief description of drawings]

FIG. 1 is a side view showing a configuration of a propulsion device and an outline of a propulsion section.

FIG. 2 is an explanatory view showing the structure of a propelling body.

FIG. 3 is an external view of the propulsion control device and an example of the first screen shown on the display unit.

FIG. 4 is an explanatory diagram for obtaining a propulsion planning line by geometric calculation.

FIG. 5 is an external view of the propulsion control device and an example of the second screen shown on the display unit.

FIG. 6 is an explanatory diagram related to horizontal control of a propulsion head.

[Explanation of symbols]

4 Propulsion body 11 Obstacle 20 Setting input section A Topographical condition H Judgment and notification means L Propulsion planning line P ₁ Propulsion start position P ₂ Propulsion target position

─────────────────────────────────────────────────── ─── Continuation of the front page (72) Inventor Takashi Togawa 1-1-1, Hama, Amagasaki-shi, Hyogo Prefecture Kubota Technology Development Laboratory (72) Inventor Yukishige Yamada 1-1-1 Hamama, Amagasaki-shi Hyogo Stock (56) Reference JP-A-8-303182 (JP, A) JP-A-11-62478 (JP, A) JP-A-8-303181 (JP, A) JP-B-2- 59932 (JP, B2) (58) Fields surveyed (Int.Cl. ⁷ , DB name) E21D 9/06 311

Claims (2)

(57) [Claims] 1. A propulsion start position (P ₁ ) when propelling a propulsion body (4) along a propulsion planning line (L) in the ground.
A propulsion control device having a setting input section (20) capable of inputting a terrain condition (A) in a propulsion section from the vehicle to the propulsion arrival target position (P ₂ ), the coordinate value of the propulsion start position (P ₁ ) and , inputs the coordinate values of the propulsion position to be reached (P _2), the coordinate values of the obstacle located between these propulsion start position (P ₁₎ and the propulsion position to be reached (P ₂₎ (11) Whether or not the propulsion body (4) can reach the propulsion arrival target position (P ₂ ) based on the respective coordinate values and the minimum propulsion curvature of the propulsion body (4), having coordinate input means. A propulsion control device having a judgment notifying means (H) for judging and notifying whether or not. 2. The judgment notifying means (H) horizontally extends from the propulsion start position (P ₁ ) based on the respective coordinate values and the minimum propulsion curvature of the propulsion body (4). When a propulsion planning line (L) including a linear propulsion section and a bending section having the minimum propulsion curvature is obtained, and the coordinate value of the obstacle (11) is located above the propulsion planning line (L). The propulsion control device according to claim 1, wherein the propulsion body (4) is configured to determine that the propulsion target position (P ₂ ) can be reached.