Displacement detection sensor, control device, and control system

Abstract

A displacement detecting sensor has two encoders installed to first and second links connected in a manner capable of relative movement with respect to a predetermined axis, the two encoders have two scale members different in distance from the axis, two detectors that detect scales of the two scale members, respectively, a positional relationship of the scale members and the detectors changes according to an external force received by a front end of the second link, two scales detected by the two encoders produce a difference, and displacement of the front end of the second link can be detected based on the two scales.

Description

TECHNICAL FIELD

[0001] The present application relates to a mechanical control, and particularly to a displacement detection sensor capable of detecting displacement of a front end of a machine, a control device, and a control system. BACKGROUND

[0002] Industrial robots, machine tools, and the like machines have a plurality of links that are coupled in a manner that can move relative to each other. In applications in which a front end of a machine has a tool or a workpiece to perform machining, the position of the front end of the machine changes due to the influence of the flexure of the links, the twist of the link coupling portions, and the like caused by external forces. Since the displacement of the front end of the machine directly affects the machining accuracy, a machine that has a small displacement caused by external forces is required. In contrast, a technique is known in which the position of an output shaft is controlled by a encoder installed on the output shaft, thereby eliminating the influence of the twist of the coupling portion, backlash, and the like. However, in the case where the position of the output shaft is controlled by the encoder installed on the output shaft, the displacement of the front end of the machine caused by the flexure of the link that is located further forward than the coupling portion cannot be addressed. As a technology related to the present application, for example, the following document is known.

[0003] In Patent Literature 1, a robot arm is disclosed in which a main arm that bears a main load, a sub-arm that has a fixed end fixed to a root side of the main arm and a free end that extends to a front end side of the main arm, and a displacement detector that is installed on the free end side and detects the flexure of the main arm caused by the load are provided, and the positioning of the arm front end is performed using the displacement detector to correct the flexure.

[0004] In Patent Literature 2, a polygonal robot is described in which, in a multi-joint robot, in addition to a rotation angle detector of a motor that drives a swing arm, an angle detector that detects an actual angle formed by the swing arms is directly coupled or attached via a speed-up gear on each joint.

[0005] In Patent Literature 3, an encoder device is described in which, in an encoder device that has two encoders, a first scale installed on a first shaft and a second scale installed on a second shaft are arranged adjacent to each other, and the second scale has a light transmission portion for detecting the first scale.

[0006] PRIOR ART DOCUMENTS

[0007] PATENT LITERATURE

[0008] Patent Literature 1: Japanese Patent Application Laid-Open No. 61-125786

[0009] Patent Literature 2: Japanese Patent Application Laid-Open No. 64-45592

[0010] Patent Literature 3: Japanese Patent Application Laid-Open No. 2016-3947 SUMMARY

[0011] Problem to be solved by the Invention

[0012] The present invention has been made in view of the foregoing problems, and has an object to provide a technique capable of detecting displacement of a front end of a machine with high precision.

[0013] Means for solving the problem

[0014] One embodiment of the present disclosure provides a displacement detection sensor including two encoders attached to first and second links that are connected in a manner that allows relative movement with respect to a predetermined axis, the two encoders including two scale members having different distances from the axis, two detectors that detect scales of the two scale members, respectively, and a position relationship between the scale members and the detectors changes in accordance with an external force received by a front end of the second link, two scales detected by the two encoders differ, and displacement of the front end of the second link can be detected based on the two scales.

[0015] Another embodiment of the present disclosure provides a control device that controls a machine including a plurality of links to which the displacement detection sensor described above is attached, the control device including a movement command generation unit that generates a movement command of the machine based on a correction amount corresponding to displacement of a front end of each of the plurality of links.

[0016] Another embodiment of the present disclosure provides a control system including the displacement detection sensor described above, a machine including a plurality of links to which the displacement detection sensor is attached, and a control device that controls the machine, the control device including a movement command generation unit that generates a movement command of the machine based on a correction amount corresponding to displacement of a front end of each of the plurality of links.

[0017] Effects of the Invention

[0018] According to the embodiments of the present disclosure, displacement of a front end of a machine can be detected with high precision. BRIEF DESCRIPTION OF DRAWINGS

[0019] Figure 1 is a plan view of a machine including the displacement detection sensor of the first embodiment.

[0020] Figure 2 is a perspective view of a machine including the displacement detection sensor of the first embodiment.

[0021] Figure 3 is a side view of a machine including the displacement detection sensor of the first embodiment.

[0022] Figure 4 is Figure 3 is a partial enlarged view of the displacement detection sensor.

[0023] Figure 5 is a geometric diagram showing a calculation example of the displacement of the front end of the link.

[0024] Figure 6 is a block diagram of the control system of the first embodiment.

[0025] Figure 7 is a side view showing the mechanism of the displacement detection sensor of the second embodiment.

[0026] Figure 8 is a geometric diagram showing a calculation example of the displacement of the front end of the link. DETAILED DESCRIPTION

[0027] Embodiments of the present disclosure will be described below in detail with reference to the accompanying drawings. In the drawings, the same or similar components are designated by the same or similar reference numerals. In addition, the embodiments described below do not limit the technical scope of the invention described in the scope of claims and the meanings of terms.

[0028] Figure 1 and Figure 2 is a plan view and a perspective view showing the mechanism 10 of the displacement detection sensor 20 of the first embodiment. The mechanism 10 is, for example, an industrial robot, a machine tool, or the like. The mechanism 10 has a first link 11 and a second link 12 that are linked in a manner capable of relative movement with respect to an axis O. The displacement detection sensor 20 is mounted on the first link 11 and the second link 12. The first link 11 and the second link 12 are, for example, linked in a manner capable of rotation about the axis O. The second link 12 is driven by an actuator. The actuator is, for example, a motor 13 and a speed reducer 14. The actuator can also be composed of a direct drive motor that does not have a speed reducer.

[0029] The displacement detection sensor 20 has two encoders 21, 22 mounted on the first link 11 and the second link 12. The two encoders 21, 22 are, for example, rotary encoders that detect the relative angle of the first link 11 and the second link 12. The two encoders 21, 22 are, for example, optical encoders, but can also be encoders of other types such as magnetic encoders. The two encoders 21, 22 have two scale members 21a, 22a that differ in distance from the axis O to the detection position, and detectors 21b, 22b that detect the scales of the two scale members 21a, 22a. The two scale members 21a, 22a are, for example, two annular members that differ in diameter and are arranged on concentric circles centered on the axis O. On the other hand, the two detectors 21b, 22b are arranged at positions opposite to the two scale members 21a, 22a, respectively.

[0030] The displacement detection sensor 20 can further have a base member 23 that supports the two detectors 21b, 22b. The base member 23 is, for example, a rod-shaped member that has a fixed end 23a fixed to the front end of the second link 12 and a free end 23b disposed at the joint of the first link 11 and the second link 12 without being restrained. The detectors 21b, 22b are attached to the free end 23b of the base member 23, and the scale members 21a, 22a are fixed to the first link 11. Thus, the positional relationship between the scale members 21a, 22a and the detectors 21b, 22b changes depending on the external force received by the front end of the second link 12, and the two scales detected by the two encoders 21, 22 differ.

[0031] Figure 3 is a side view of the mechanism 10 that represents the displacement detection sensor 20 of the first embodiment. In this figure, the second link 12 before receiving an external force is depicted by a double-dotted line, and the second link 12 that is deflected by receiving an external force is depicted by a solid line. As indicated by the arrow, when the front end of the second link 12 receives an external force, the front end of the second link 12 (the fixed end 23a of the base member 23) moves from position A to position B due to the deflection of the second link 12, the twisting of the link joint, and the like. That is, the front end of the second link 12 is displaced by δ. On the other hand, although the fixed end 23a of the base member 23 is fixed to the front end of the second link 12, the free end 23b is not restrained, and thus the base member 23 itself does not deform (i.e., does not deflect). Therefore, the positional relationship between the detectors 21b, 22b and the scale members 21a, 22a changes by the amount of displacement δ of the front end of the second link 12, and the two scales detected by the two encoders 21, 22 differ.

[0032] Figure 4 is Figure 3 is a partial enlarged view of the displacement detection sensor 20. The positional relationship between the detectors 21b, 22b and the scale members 21a, 22a changes depending on the displacement δ of the front end of the second link 12, and the two scales, i.e., the two angles θ1, θ2 detected by the two encoders 21, 22 differ. The two angles θ1, θ2 are the changes in angle before and after the external force is applied. The displacement detection sensor 20 can detect the displacement δ of the front end of the second link 12 based on the two angles θ1, θ2.

[0033] The displacement detection sensor 20 can further have a displacement calculation section (not shown) that calculates the displacement of the front end of the second link 12 based on the two angles θ1, θ2, the two distances r1, r2 from the axis O to the detection positions C, D (or C’, D’) of the two scale members 21a, 22a (refer to Figure 4 ), and the distance L from the front end of the second link 12 (the fixed end 23a of the base member 23) to the detection position C of the scale member 22a (refer to Figure 3), to calculate the displacement δ of the front end of the second link 12. r1, r2 are radii of the scale members 21a, 22a, respectively, and L is a distance from the fixed end 23a of the base member 23 before the external force is applied to the detection position C of the scale member 22a, and thus r1, r2, L are known values. Although not shown, the displacement calculation section is, for example, a computer device provided with a processor that executes a program, a memory, an input / output section, and the like, but can also be constituted by a semiconductor integrated circuit such as an FPGA (field-programmable gate array) or an ASIC (application specific integrated circuit) that does not execute a program.

[0034] Figure 5 is a geometric figure showing a calculation example of the displacement δ of the front end of the link. First, the displacement detection sensor 20 stores r1, r2, L, which are known values, and the scales of the two encoders 21, 22 before the external force is applied in the memory. Then, the displacement detection sensor 20 acquires the scales of the two encoders 21, 22 after the external force is applied to acquire the two angles θ1, θ2.

[0035] In Figure 5 , when a line segment EF parallel to a line segment AC connecting the position A of the front end of the second link before the external force is applied (the position of the fixed end of the base member) and the detection position C of the scale member and passing through the detection position C' of the scale member after the external force is applied is drawn, and furthermore, a perpendicular BE is drawn from the position B of the fixed end of the base member after the external force is applied to the line segment EF, the intersection of the line segment AC and the perpendicular BE is set as G, the length of the line segment EG is set as L2, and the length of the perpendicular BE is set as L3, the displacement δ of the front end of the second link can be represented by the following equation.

[0036] [Equation 1]

[0037] δ = L3 - L2 ··· Equation 1

[0038] In addition, when the angle ∠BC'E of the base member after the external force is applied is set as β, and the length of the side BC' from the position B of the fixed end of the base member to the detection position C' of the scale member is set as L', L3 can be represented by the following equation.

[0039] [Equation 2]

[0040] L3 = L' x sin β ··· Equation 2

[0041] As to L', since the distance between the scale member and the detector changes slightly after the external force is applied, if the amount of the change is set as ΔL, the length L' of the side BC' is expressed as L' = L + ΔL. Here, ΔL is sufficiently small with respect to L, and thus can be approximated as L' = L. Therefore, L3 can also be expressed by the following equation.

[0042] [Equation 3]

[0043] L3 = L x sin β... Equation 3

[0044] On the other hand, L2 is equal to the length of the side C'H which is a perpendicular line drawn from the detection position C' of the scale member to the line segment AO, and thus can be expressed by the following equation according to the angle θ2 and the radius r2 of the scale member.

[0045] [Equation 4]

[0046] L2 = r2 x sin θ2... Equation 4

[0047] If β, which is not a known value, is found according to Equation 3 and Equation 4, the displacement δ of the front end of the second link is found. Here, when ∠D'C'O is defined as α, β can be expressed by the following equation.

[0048] [Equation 5]

[0049] β = α - θ2... Equation 5

[0050] If α, which is not a known value, is found according to Equation 5, the displacement δ of the front end of the second link is found. Here, when attention is paid to ΔOC'D' and the length of the side C'D' is L1, α can be expressed by the following equation according to the law of cosines.

[0051] [Equation 6]

[0052]

[0053] If L1, which is not a known value, is found according to Equation 6, the displacement δ of the front end of the second link is found. Here, further attention is paid to ΔOC'D', the length of the side OD' is r1, the length of the side OC' is r2, and the angle ∠C'OD' sandwiched by these two sides is the difference between the two angles θ1 and θ2, and thus L1 can be expressed by the following equation according to the law of cosines.

[0054] [Equation 7]

[0055]

[0056] Here, since r1, r2, θ1, and θ2 are known, the displacement δ of the front end of the second link is found.

[0057] In addition, in the case where the displacement δ of the front end of the second link is corrected, the second link needs to be rotated from the position B around the axis O to the position A. If the rotation angle is set as the correction amount ε, the displacement calculation section can further calculate the correction amount ε that corrects the displacement δ of the front end of the second link. The correction amount ε can be expressed by the following equation.

[0058] [Equation 8]

[0059]

[0060] Here, the side GO is the sum of the side EC' and the side C'F. If attention is paid to the triangle BC'E, the side EC' can be expressed by the following equation.

[0061] [Equation 9]

[0062] EC' = L x cos β Equation 9

[0063] In addition, when attention is paid to the triangle C'OF, the side C'F can be expressed by the following equation.

[0064] [Equation 10]

[0065] C'F = r2 x cos θ2 Equation 10

[0066] The correction amount ε can be calculated from Equation 9 and Equation 10.

[0067] Figure 6 is a block diagram of a control system 1 of a first embodiment. The control system 1 includes a displacement detection sensor 20, a machine 10 including a plurality of links on which the displacement detection sensor 20 is mounted, and a control device 30 that controls the machine 10. The machine 10 is, for example, a vertical multi-joint robot. The machine 10 includes a motor 13 and the displacement detection sensor 20 at a joint portion of each of the plurality of links. The displacement detection sensor 20 is, for example, a rotary encoder. The displacement detection sensor 20 can detect a displacement δ of a front end of each of the plurality of links. If the displacement δ of the front end of each of the plurality of links is corrected, the displacement of the front end of the machine 10 can be corrected.

[0068] The control device 30 includes a motion command generation section 33 that generates a motion command of the machine 10 in accordance with a teaching program. The motion command of the machine 10 includes, for example, a position command, a speed command, a torque command, and the like of the motor 13. The motion command generation section 33 generates the motion command of the machine 10 in accordance with a correction amount ε corresponding to the displacement δ of the front end of each of the plurality of links. The motion command generation section 33 is, for example, constituted by a controller that controls the motor 13, an amplifier, and the like.

[0069] The control device 30 can also include a memory 31 that stores θ1, θ2, r1, r2, and L, and a displacement calculation section 32 that calculates the displacement δ of the front end of each of the plurality of links based on θ1, θ2, r1, r2, and L. The displacement calculation section 32 is, for example, a computer device including a processor that executes a program, a memory, an input / output section, and the like, but can also be constituted by a semiconductor integrated circuit such as an FPGA (field-programmable gate array), an ASIC (application specific integrated circuit), or the like that does not execute a program. In the case where the displacement calculation section 32 is provided in the control device 30, it is not necessary to provide the displacement calculation section in the displacement detection sensor 20. In this case, the displacement detection sensor 20 can output the two angles θ1, θ2 detected by the two encoders to the control device 30 according to an external force received by the front end of the machine 10.

[0070] The displacement calculation section 32 can further calculate, for each link, a correction amount ε that corrects the displacement δ of the front end of each of the plurality of links. The motion instruction generation section 33 generates a motion instruction of the machine 10 based on these correction amounts ε.

[0071] Note that the structure and operation of the displacement detection sensor 20, the control device 30, and the control system 1 of the first embodiment are one example and can be appropriately changed. For example, the base member 23 can not support the two detectors 21b, 22b, but can support the two scale members 21a, 22a. That is, the scale members 21a, 22a can be attached to the free end 23b of the base member 23, and the detectors 21b, 22b can be fixed to the first link 11. Alternatively, for example, the displacement detection sensor 20 can not include the base member 23, and the machine 10 can include the base member 23.

[0072] Alternatively, the first link 11 and the second link 12 can not be connected in a manner that allows rotation about the axis O, but can be connected in a manner that allows straight movement along the axis. That is, the two encoders 21, 22 can not be rotary encoders, but can be linear encoders.

[0073] Figure 7is a side view of the machine 10 that represents the displacement detection sensor 20 of the second embodiment. Hereinafter, only the structure and operation different from the displacement detection sensor 20 of the first embodiment will be described. The first link 11 and the second link 12 are, for example, joined in a manner that enables straight movement along the axis O. The encoders 21, 22 are, for example, linear encoders that detect the relative positions of the first link 11 and the second link 12. The two scale members 21a, 22a are, for example, two linear members whose distances from the axis O to the detection positions are different, and are disposed on parallel lines of the axis O. On the other hand, the two detectors 21b, 22b are disposed at positions opposite to the two scale members 21a, 22a, respectively. The detectors 21b, 22b are mounted at the free end 23b of the base member 23, and the scale members 21a, 22a are fixed to the first link 11. Thus, the positional relationship of the scale members 21a, 22a and the detectors 21b, 22b changes according to the displacement δ of the front end of the second link 12, and a difference is generated in the two scales detected by the two encoders 21, 22, i.e., in the two positions P1, P2.

[0074] The positional relationship of the detectors 21b, 22b and the scale members 21a, 22a changes according to the displacement δ of the front end of the second link 12, and a difference is generated in the two scales detected by the two encoders 21, 22, i.e., in the two positions P1, P2. The two positions P1, P2 are the positional changes before and after the action of the external force. The displacement detection sensor 20 can detect the displacement δ of the front end of the second link 12 on the basis of the two positions P1, P2.

[0075] The displacement detection sensor 20 can further have a displacement calculation section (not shown) that calculates the displacement δ of the front end of the second link 12 on the basis of the two distances r1, r2 from the axis O to the detection positions C, D of the two scale members 21a, 22a, and the distance L from the fixed end 23a of the base member 23 to the detection position C of the scale member 22a, in addition to the two positions P1, P2. r1, r2 are the distances from the axis O to the detection positions C, D of the scale members 21a, 22a, respectively, and L is the distance from the fixed end 23a of the base member 23 before the action of the external force to the detection position C of the base member 22a, and thus r1, r2, L are known values.

[0076] Figure 8 is a geometric diagram that represents a calculation example of the displacement δ of the front end of the link. First, the displacement detection sensor 20 stores r1, r2, L, which are known values, and the scales of the two encoders 21, 22 before the action of the external force in a memory. Then, the displacement detection sensor 20 acquires the scales of the two encoders 21, 22 after the action of the external force and acquires the two positions P1, P2.

[0077] In Figure 8When a line segment AC parallel to the line segment AC that connects the position A (the position of the fixed end of the base member) before the external force is applied and the detection position C of the scale member and that passes the detection position C' of the scale member after the external force is applied, and a perpendicular line BE drawn from the position B of the fixed end of the base member after the external force is applied to the line segment EF are drawn, and a point of intersection of the line segment AC and the perpendicular line BE is set as G, a length of the line segment EG is set as L2, and a length of the perpendicular line BE is set as L3, a displacement δ of the front end of the second link can be expressed by the following equation.

[0078] [Equation 11]

[0079] δ = L3 - L2 · · Equation 11

[0080] In addition, when an angle ∠BC'E of the base member after the external force is applied is set as β, and a length of the side BC' from the position B of the fixed end of the base member to the detection position C' of the scale member is set as L', L3 can be expressed by the following equation.

[0081] [Equation 12]

[0082] L3 = L' x sin β · · Equation 12

[0083] As for L', since a distance between the scale member and the detector is slightly changed after the external force is applied, if the amount of change is set as ΔL, the length L' of the side BC' is expressed as L' = L + ΔL. Here, ΔL is sufficiently small with respect to L, and thus can be approximated as L' = L. Therefore, L3 can also be expressed by the following equation.

[0084] [Equation 13]

[0085] L3 = L x sin β · · Equation 13

[0086] On the other hand, L2 is equal to a length from the detection position C of the scale member to the detection position C', and thus can be expressed by the following equation.

[0087] [Equation 14]

[0088] L2 = P2 · · Equation 14

[0089] If β, which is not a known value, is found from Equation 13 and Equation 14, a displacement δ of the front end of the second link is found. Here, when ΔC'FD' is focused on, β can be expressed by the following equation.

[0090] [Equation 15]

[0091]

[0092] Here, since r1, r2, P1, P2 are known, the displacement δ of the front end of the second link is found.

[0093] In addition, in the case where the displacement δ of the front end of the second link is corrected, it is necessary to make the second link straightly advance from the position B along the axis O to the position A. If the straight advance position is set as the correction amount ε, the displacement calculation section can further calculate the correction amount ε that corrects the displacement δ of the front end of the second link. The correction amount ε is equal to the displacement δ, and thus can be expressed by the following equation.

[0094] [Equation 16]

[0095] ε = δ ··· Equation 16

[0096] According to the above embodiment, the displacement of the front end of the machine 10 can be detected with high precision. In addition, the displacement of the front end of the machine 10 can be corrected with high precision according to various link mechanisms.

[0097] Further, the program executed by the above-described processor can be provided recorded on a non-transitory recording medium, such as a CD-ROM or the like, which is readable by a computer, or can be provided by being distributed from a server device on a WAN (wide area network) or LAN (local area network) via wire or wirelessly.

[0098] In the present specification, various embodiments are described, but the present application is not limited to the above-described embodiments, and it is to be understood that various modifications can be made within the scope recited in the range of the claimed invention.

[0099] Explanation of Symbols

[0100] 1 control system;

[0101] 10 machine;

[0102] 11 first link;

[0103] 12 second link;

[0104] 13 motor;

[0105] 14 speed reducer;

[0106] 20 displacement detection sensor;

[0107] 21, 22 encoder;

[0108] 21a, 22a scale member;

[0109] 21b, 22b detector;

[0110] 23 base member;

[0111] 23a fixed end;

[0112] 23b free end;

[0113] 30 control device;

[0114] 31 memory;

[0115] 32 displacement calculation section;

[0116] 33 action instruction generation section;

[0117] O axis line;

[0118] δ displacement;

[0119] ε correction amount;

[0120] L distance from the fixed end of the base member to the detection position of the scale member;

[0121] r1, r2 distance from the axis line to the detection position of the scale member;

[0122] θ1, θ2 two angles detected by the two encoders;

[0123] P1, P2 two positions detected by the two encoders.

Claims

1. A displacement detecting sensor characterized by comprising two encoders installed to first and second links connected in a manner capable of relative movement with respect to a predetermined axis, the two encoders comprising two scale members different in distance from the axis to a detection position, two detectors each detecting a scale of the two scale members, a positional relationship of the scale members to the detectors changing in accordance with an external force received by a front end of the second link, a difference being produced in two scales detected by the two encoders, and a displacement of the front end of the second link being able to be detected based on the two scales.

2. The displacement detecting sensor according to claim 1, characterized by further comprising a base member supporting one of the detectors and the scale members, the base member comprising a fixed end fixed to the front end of the second link and a free end disposed at a connection portion of the first and second links without being restrained, the one of the detectors and the scale members being installed at the free end, and the other of the detectors and the scale members being fixed to the first link.

3. The displacement detecting sensor according to claim 2, characterized by further comprising a displacement calculating section calculating the displacement of the front end of the second link based on the two scales, and the two different distances from the axis to the detection positions of the two scale members, and a distance from the front end of the second link to the detection position of the scale member, in addition to the two scales.

4. The displacement detecting sensor according to claim 3, characterized in that the displacement calculating section further calculates a correction amount corresponding to the displacement of the front end of the second link.

5. The displacement detecting sensor according to any one of claims 1 to 4, characterized in that the two encoders are rotary encoders detecting a relative angle of the first and second links connected in a manner capable of rotating about the axis.

6. The displacement detecting sensor according to any one of claims 1 to 4, characterized in that the two encoders are linear encoders detecting a relative position of the first and second links connected in a manner capable of advancing straight along the axis.

7. A control device controlling a machine provided with a plurality of links installed with the displacement detecting sensor according to any one of claims 1 to 6, characterized by comprising an operation command generating section generating an operation command of the machine based on a correction amount corresponding to the displacement of a front end of each of the plurality of links.

8. The control device according to claim 7, characterized in that ​ ​ ​ ​ ​ ​ ​ The control device further includes a displacement calculation unit that calculates the displacement of the front end of each of the plurality of links based on two different distances from the axis to the detected positions of the two scale members and a distance from the front end of the second link to the detected position of the scale member in addition to the two scales.

9. The control device according to claim 8, wherein The displacement calculation unit further calculates, for each of the links, the correction amount corresponding to the displacement of the front end of each of the plurality of links.

10. A control system characterized by, including: The displacement detection sensor according to any one of claims 1 to 6; a machine including the plurality of links on which the displacement detection sensor is mounted; and a control device that controls the machine, The control device includes an operation instruction generation unit that generates an operation instruction of the machine based on a correction amount corresponding to the displacement of the front end of each of the plurality of links.

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