Equivalent kinematics modeling and control method for segment end points of rope-driven segmented linkage mechanical arm

Abstract

The invention relates to a rope-driven segmented linkage mechanical arm segment end point equivalent kinematics modeling and control method. The method comprises the steps that a PYYP configuration linkage segment is equivalent to an RPPP configuration equivalent linkage segment; the RPPP configuration equivalent linkage section is equivalent to a space arc and a straight line section, section space parameters are set and extracted, and the section space parameters comprise a section face angle and an equivalent central angle; driving space-section space kinematics and section space-task space kinematics are obtained; solving inverse kinematics of section space-task space of the rope-driven segmented linkage mechanical arm; performing section space state equivalence on the linkage section under the non-equal-angle linkage; and the pose error of the tail end of the segmented linkage manipulator is decomposed into segment space state parameters of all segments, and closed-loop control is conducted in segment spaces, so that high-precision control over the tail end pose of the rope-driven segmented linkage manipulator is achieved.

Description

Technical Field

[0001] This invention relates to the field of cable-driven segmented linkage robotic arms, and more particularly to a method for equivalent kinematic modeling and control of the endpoints of cable-driven segmented linkage robotic arms. Background Technology

[0002] Traditional articulated robots typically consist of rigid arms linked together, with motors positioned at the joints. They usually possess 6 to 7 degrees of freedom, meeting the needs of some structured applications in industry and agriculture, such as welding, grasping, and transferring. However, in complex obstacle environments and narrow spaces, articulated robots, due to their large structural size and discrete joint arrangement, are often hindered by size limitations during operation, or are prone to collisions with surrounding structures, making it difficult to effectively utilize their degrees of freedom and thus challenging to complete complex tasks in these environments.

[0003] Compared to traditional articulated robots, rope-driven flexible robotic arms offer superior flexibility and environmental adaptability, making them more suitable for operation in such environments. As a type of rope-driven flexible robotic arm, the rope-driven segmented linkage robotic arm further leverages the advantages of rope-driven flexible robotic arms. It connects several joints and arms together to form an arm segment. Passive ropes constrain the motion angle of each joint within the arm segment, enabling each arm segment to exhibit equiangular motion. This extends the arm segment length that the drive motor can control, avoiding the problems of increased drive motor numbers and bulky drive systems associated with increasing arm length to expand workspace, while still retaining the flexible movement capabilities of rope-driven flexible robotic arms.

[0004] However, rope-constrained linkage structures are affected by friction, linear elastic deformation, and hysteresis caused by the ropes, leading to non-uniform angle linkages within the linkage segments. This means that the angles of motion in the same direction within a single linkage segment are no longer the same. While multi-stage linkage mechanisms, created through composite combinations, can reduce joint angle motion errors caused by non-uniform angle linkages to some extent, they cannot eliminate the non-uniform angle linkage issue entirely. Furthermore, non-uniform angle motions within linkage segments cause pose errors at the segment endpoints, resulting in decreased pose accuracy at the end of the segmented linkage manipulator and hindering its application in long-distance precision operations.

[0005] Due to factors such as rope deformation and transmission link friction, the constant-angle bending characteristics of the linkage segment in rope-driven segmented linkage robotic arms are difficult to meet. Traditional kinematic solutions, trajectory planning, and control methods based on the assumption of constant-angle linkage have large errors, making it impossible to complete precise operation tasks. Summary of the Invention

[0006] This invention provides a method for modeling and controlling the equivalent kinematics of the endpoints of a rope-driven segmented linkage robotic arm, aiming to solve at least one of the technical problems existing in the prior art.

[0007] A method for equivalent kinematic modeling and control of the endpoints of a cable-driven segmented linkage robotic arm, the method being applied to a cable-driven segmented linkage robotic arm, and the method comprising the following steps:

[0008] S100, the PYYP configuration (or YPPY configuration) linkage segment based on the rope-driven linkage robot arm has planar motion properties and equal angle motion properties, and the PYYP configuration linkage segment is equivalent to the RPPP configuration equivalent linkage segment;

[0009] S200. The equivalent linkage segment of the RPPP configuration is equivalent to a spatial circular arc and a straight line segment. The segment space parameters are set and extracted, including the segment face angle. and equivalent central angle θ seg,i ;

[0010] S300. Based on the segment space parameters, the segment space state description yields the multi-layer state space relationship of the rope-driven segmented linkage manipulator, and the kinematics of the drive space-segment space and the kinematics of the segment space-task space are obtained.

[0011] S400, based on the improved Forward and Backward Reaching Inverse Kinematics for Continuum Robots (FABRIKc) method, solves the inverse kinematics of the segment space-task space of the rope-driven segmented linkage robot arm;

[0012] S500. With the goal of minimizing the positional error of the equivalent preceding and following linkage segments, the linkage segments under non-equiangular linkage are subjected to segment space state equivalence. An approximate curve of the ideal workspace is obtained using a fitting method, and the actual segment endpoint position P is then determined. seg,a,i The position P of the projection point is obtained by projecting the fitted workspace curve. seg,p,i Then, the position P of the projection point on the ideal motion plane. seg,p,i Transformed into two state variables, the segment angle, in segment space and equivalent central angle θ seg,i ;

[0013] S600, based on the whole arm decomposition motion control block diagram of the rope-driven segmented linkage manipulator, decomposes the position and posture error of the end of the segmented linkage manipulator into the segment space state parameters of each segment, and performs closed-loop control in the segment space to achieve high-precision control of the end position and posture of the rope-driven segmented linkage manipulator.

[0014] The present invention also proposes a rope-driven segmented linkage robotic arm, characterized in that the rope-driven segmented linkage robotic arm comprises:

[0015] A drive box, the drive box including a motor and a transmission mechanism, the transmission mechanism including at least a rope puller, a linear guide rail and a lead screw;

[0016] The linkage manipulator includes multiple linkage arm segments connected in series. Each linkage arm segment has two degrees of freedom, and an end effector is provided on the linkage arm segment at the end.

[0017] The beneficial effects of this invention are:

[0018] This invention discloses a method for equivalent kinematic modeling and decomposition motion control of the end-points of a rope-driven segmented linkage manipulator. A multi-space equivalent kinematic model of the rope-driven segmented linkage manipulator based on the spatial state description of the arm segments is established, overcoming the problem of large errors in models built under the assumption of equal-angle linkage in previous methods, thus improving the accuracy of manipulator modeling. A method for controlling the end-point positions of the rope-driven segmented linkage manipulator based on joint measurement within the linkage segment and end-point solution is proposed, improving the accuracy of end-point position control under non-equal-angle linkage. Furthermore, a method for decomposing the motion of the entire arm of the rope-driven segmented linkage manipulator based on the equivalent end-points is proposed, decomposing the motion of the manipulator's end-point in Cartesian space into segment space and drive space, improving the end-point pose control accuracy of manipulators with non-equal-angle linkage. Attached Figure Description

[0019] Figure 1 This is a flowchart of the equivalent kinematic modeling and decomposition motion control method for the endpoints of a rope-driven segmented linkage robotic arm in an embodiment of the present invention.

[0020] Figure 2 This is a schematic diagram of the structure of the rope-driven segmented linkage manipulator in an embodiment of the present invention.

[0021] Figure 3 This is a schematic diagram of the small figure-eight linkage structure in the same linkage direction in an embodiment of the present invention.

[0022] Figure 4 This is a schematic diagram of the large figure-eight linkage structure in the same linkage direction in an embodiment of the present invention.

[0023] Figure 5 This is a simplified diagram of the PYYP configuration linkage segment in an embodiment of the present invention.

[0024] Figure 6 This is a schematic diagram of the PYYP configuration linkage segment in an embodiment of the present invention.

[0025] Figure 7 This is a schematic diagram of the linkage segment of the RPPP configuration in an embodiment of the present invention.

[0026] Figure 8 This is a schematic diagram of the linkage segment in the space in an embodiment of the present invention.

[0027] Figure 9 This is a schematic diagram of the linkage segment after the arc is equivalent in an embodiment of the present invention.

[0028] Figure 10 This is a schematic diagram of the multi-level state-space kinematic relationship of the rope-driven segmented linkage robotic arm in an embodiment of the present invention.

[0029] Figure 11 This is a schematic diagram of the PY configuration joint mechanism in an embodiment of the present invention.

[0030] Figure 12 This is a schematic diagram of the RP configuration joint mechanism in an embodiment of the present invention.

[0031] Figure 13 This is a schematic diagram of the equivalent arc of the linkage segment of the rope-driven segmented linkage robotic arm in an embodiment of the present invention.

[0032] Figure 14 This is a schematic diagram of the inverse kinematics solution process for the segment space-task space of the rope-driven segmented linkage robotic arm in an embodiment of the present invention.

[0033] Figure 15 This is a schematic diagram showing the comparison between the expected workspace and the actual workspace of a single linkage segment in an embodiment of the present invention (joint angles in the P and Y directions -10° to 10°).

[0034] Figure 16 This is a schematic diagram illustrating the influence of the actual workspace and the ideal workspace on the segment endpoint error in an embodiment of the present invention (using the subtraction of projected components).

[0035] Figure 17 The fitted workspace curve in this embodiment of the invention is located at projection point P. P A schematic diagram of the slope relationship in the vicinity.

[0036] Figure 18 This is a block diagram showing the control of the endpoint position of the rope-driven segmented linkage robotic arm in an embodiment of the present invention.

[0037] Figure 19 This is a schematic diagram of the segment space controller in the segment endpoint position control of an embodiment of the present invention.

[0038] Figure 20 This is a schematic diagram of the whole arm decomposition motion control block diagram of the rope-driven segmented linkage robotic arm in an embodiment of the present invention.

[0039] Figure 21 This is a schematic diagram showing the accumulation of terminal errors of each segment at the end of the rope-driven segmented robotic arm in an embodiment of the present invention.

[0040] Figure 22 This is a schematic diagram illustrating the influence of the rate of change of the motion plane rotation speed of the i-th linkage segment with respect to time on the end velocity in an embodiment of the present invention.

[0041] Figure 23 This is a schematic diagram illustrating the effect of the rate of change of the joint angle motion speed of the i-th linkage segment with respect to time on the end velocity in an embodiment of the present invention. Detailed Implementation

[0042] The following will provide a clear and complete description of the concept, specific structure, and technical effects of the present invention in conjunction with the embodiments and accompanying drawings, so as to fully understand the purpose, solution, and effects of the present invention. It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other.

[0043] It should be noted that, unless otherwise specified, when a feature is referred to as "fixed" or "connected" to another feature, it can be directly fixed or connected to the other feature, or indirectly fixed or connected to the other feature. Furthermore, the descriptions of "upper," "lower," "left," "right," "top," and "bottom" used in this invention are only relative to the relative positional relationships of the various components of the invention in the accompanying drawings.

[0044] Furthermore, unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art. The terminology used in this specification is for the purpose of describing particular embodiments only and not for limiting the invention. The term "and / or" as used herein includes any combination of one or more of the associated listed items.

[0045] It should be understood that although the terms first, second, third, etc., may be used in this disclosure to describe various elements, these elements should not be limited to these terms. These terms are only used to distinguish elements of the same type from one another. For example, without departing from the scope of this disclosure, a first element may also be referred to as a second element, and similarly, a second element may also be referred to as a first element.

[0046] Reference Figures 1 to 23 In some embodiments, a method for describing the equivalent state of a linkage segment and kinematic modeling is provided. This method is applied to a cable-driven segmented linkage robotic arm and includes the following steps:

[0047] S100, the PYYP configuration (or YPPY configuration) linkage segment based on the rope-driven linkage robot arm has planar motion properties and equal angle motion properties, and the PYYP configuration linkage segment is equivalent to the RPPP configuration equivalent linkage segment;

[0048] S200. The equivalent linkage segment of the RPPP configuration is equivalent to a spatial circular arc and a straight line segment. The segment space parameters are set and extracted, including the segment face angle. and equivalent central angle θ seg,i ;

[0049] S300. Based on the segment space parameters, the segment space state description yields the multi-layer state space relationship of the rope-driven segmented linkage manipulator, and the kinematics of the drive space-segment space and the kinematics of the segment space-task space are obtained.

[0050] S400, based on the improved Forward and Backward Reaching Inverse Kinematics for Continuum Robots (FABRIKc) method, solves the inverse kinematics of the segment space-task space of the rope-driven segmented linkage robot arm;

[0051] S500. With the goal of minimizing the positional error of the equivalent preceding and following linkage segments, the linkage segments under non-equiangular linkage are subjected to segment space state equivalence. An approximate curve of the ideal workspace is obtained using a fitting method, and the actual segment endpoint position P is then determined. seg,a,i The position P of the projection point is obtained by projecting the fitted workspace curve. seg,p,i Then, the position P of the projection point on the ideal motion plane. seg,p,i Transformed into two state variables, the segment angle, in segment space and equivalent central angle θ seg,i ;

[0052] S600, based on the whole arm decomposition motion control block diagram of the rope-driven segmented linkage manipulator, decomposes the position and posture error of the end of the segmented linkage manipulator into the segment space state parameters of each segment, and performs closed-loop control in the segment space to achieve high-precision control of the end position and posture of the rope-driven segmented linkage manipulator.

[0053] The beneficial effects of this invention are:

[0054] This invention discloses a method for equivalent kinematic modeling and decomposition motion control of the end-points of a rope-driven segmented linkage manipulator. A multi-space equivalent kinematic model of the rope-driven segmented linkage manipulator based on the spatial state description of the arm segments is established, overcoming the problem of large errors in models built under the assumption of equal-angle linkage in previous methods, thus improving the accuracy of manipulator modeling. A method for controlling the end-point positions of the rope-driven segmented linkage manipulator based on joint measurement within the linkage segment and end-point solution is proposed, improving the accuracy of end-point position control under non-equal-angle linkage. Furthermore, a method for decomposing the motion of the entire arm of the rope-driven segmented linkage manipulator based on the equivalent end-points is proposed, decomposing the motion of the manipulator's end-point in Cartesian space into segment space and drive space, improving the end-point pose control accuracy of manipulators with non-equal-angle linkage.

[0055] It should be noted that the equivalent kinematic modeling and decomposition motion control method for the endpoints of the rope-driven segmented linkage robotic arm described in this invention is based on the equivalent kinematic modeling and decomposition motion control method of application number CN201711471782.7 and invention title "A Two-Degree-of-Freedom Linked Joint Segment and Flexible Robotic Arm". This invention models and decomposes the motion control of the above-mentioned hardware.

[0056] This invention provides a method for equivalent kinematic modeling and decomposed motion control of the endpoints of a rope-driven segmented linkage manipulator. Based on the planar isoangular motion characteristics of the linkage segments, a multi-space equivalent kinematic model of the rope-driven segmented linkage manipulator based on segment spatial state description is established. This overcomes the problem of large errors in models built under the assumption of isoangular linkage, improving the accuracy of manipulator modeling. For non-isoangular linkage characteristics, a method for controlling the endpoint positions of the rope-driven segmented linkage manipulator based on joint measurement and equivalent endpoint solution is proposed, improving the accuracy of endpoint position control under non-isoangular linkage. Addressing the problem that end-effector pose error cannot be eliminated solely through the endpoint position control of a single linkage segment, a method for decomposed motion control of the entire rope-driven segmented linkage manipulator is proposed, improving the end-effector pose control accuracy of non-isoangular linkage manipulators. The aforementioned method for equivalent kinematic modeling and decomposed motion control of the endpoints of the rope-driven segmented linkage manipulator solves the problem of end-effector operation accuracy being affected by non-isoangular linkage conditions in rope-driven segmented linkage manipulators.

[0057] This example applies to a rope-driven segmented robotic arm. (Refer to...) Figure 2 Its basic structure consists of two parts: a drive box and a linkage manipulator. The drive box is the power source for the manipulator and integrates transmission structures such as a motor, a rope puller, a linear guide, and a lead screw. The manipulator is composed of multiple linkage arm segments connected in series, and each arm segment has two degrees of freedom. The end effector of the manipulator can be equipped with a corresponding type of end effector according to the needs of the task.

[0058] The rope-driven segmented linkage robotic arm employs a hybrid active-passive drive system. This means that the linear drive module directly changes the length of the drive rope to move the endpoints of the linkage segment, while the passive linkage ropes form a linkage mechanism that drives the joints within the linkage segment to achieve equiangular movements. The combined action of the active and passive ropes achieves motion control of the segmented linkage robotic arm. For active drive, the rope-driven segmented linkage robotic arm moves the linkage segment by changing the lengths of the three active drive ropes corresponding to that segment. For passive linkage, a typical linkage structure for the rope-driven segmented linkage robotic arm is described below. Figure 3 , Figure 4 Its linkage structure includes a small figure-eight linkage structure ( Figure 3 ) and the large figure-eight linkage structure ( Figure 4 ). Reference Figure 3Taking the linkage rope on one side of the small figure-eight linkage as an example to illustrate its arrangement, the linkage rope is first fixed to point P on the cross axis i. s,1 At that point, around the line-crossing disk O s,i With the line-passing disk O s,i+1 Then fix it to point P on the cross axis i+1. s,2 The arrangement of the connecting ropes on the other side is similar. (Refer to...) Figure 4 The figure-eight linkage is arranged between two non-adjacent booms and two cross-axis hinges forming two unidirectional rotation axes. Taking the linkage rope on one side of the figure-eight linkage as an example, its arrangement is explained. First, the linkage rope is fixed to point P of boom i-2. l,1 At that point, it then passes around the line-passing disk O on the cross axis i-1. l,i-1 At point P l,2 Guided to point P via rope sleeve l,3 Finally, it passes through the line-passing disk O on the cross axis i. l,i Point P, fixed to rod i l,4 The arrangement of the connecting ropes on the other side is similar.

[0059] Furthermore, in step S100, the PYYP configuration (or YPPY configuration) linkage segment of the cable-driven robotic arm possesses planar motion properties, specifically:

[0060] Reference Figure 5 Let L be the j-th link of the i-th linkage segment. i,j Let U be the j-th cross axis of the i-th linkage segment. i,j L i,1 First rotate around the y-axis of the world coordinate system by -q Y,i Then rotate q around the z-axis of the world coordinate system. P,i Then L i,1 Compared to L i-1,m Location i-1,m L i,1 For equation (1):

[0061]

[0062] Where Rot(·,·) is the rotation transformation matrix, the first parameter of Rot(·,·) represents the rotation axis, and the second parameter of Rot(·,·) represents the rotation angle; i-1,m L i,1 For L i,1 In L i-1,m The position in the local coordinate system; c P,i / s P,i For q P,i The cosine value / sine value, i.e., c P,i =cosq P,i / sP,i =sinq P,i c Y,i / s Y,i For q Y,i The cosine value / sine value, i.e., c Y,i =cosq Y,i / s Y,i =sinq Y,i l represents the length of a single boom.

[0064] Similarly, L i,2 Compared to L i,1 Location i,1 L i,2 For equation (2):

[0065]

[0066] i,1 L i,2 For L i,2 In L i,1 The position in the local coordinate system;

[0067] L i,2 The coordinates from L i,1 Transform to L in the local coordinate system i-1,m In the local coordinate system, we obtain equation (3):

[0068]

[0069] in, i-1,m L i,2 For L i,2 In L i-1,m The position in the local coordinate system; c 2Y,i For 2q Y,i The cosine value, i.e., c 2Y,i =cos(2q) Y,i );

[0070] L i-1,m L i,1 and L i,2 The product of the vectors corresponding to the three rods is given by equation (4):

[0071]

[0072] Because of L i-1,m L i,1 and L i,2 The vectors corresponding to the three rods satisfy the condition that the mixed product is zero, therefore, L i-1,m L i,1 and L i,2Located in the same plane, the PYYP configuration (or YPPY configuration) linkage segment of the cable-driven robotic arm has planar motion properties.

[0073] Furthermore, in step S100, the PYYP configuration (or YPPY configuration) linkage segment of the cable-driven robotic arm possesses equal-angle motion properties, specifically:

[0074] The choice of coordinate system is independent of the angle between the two vectors, therefore in L i-1,m In the local coordinate system, L i-1,m and L i,1 The included angle between them is given by equation (5);

[0075]

[0076] In L i,1 In the local coordinate system, L i,1 and L i,2 The included angle between them is given by equation (6):

[0077]

[0078] The results of the two equations above are equal, that is, L i-1,m and L i,1 The included angle between and L i,1 and L i,2 The included angles between them are equal; similarly, L i,j With L i,j+1 The included angle and L i,j+1 With L i,j+2 The included angles are equal.

[0079] Furthermore, in step S100,

[0080] Reference Figure 6 and Figure 7 The PYYP configuration linkage segment is equivalent to the RPPP configuration linkage segment. In the RPPP configuration, R indicates that the first revolute axis is the Roll axis, which is the x-axis in the world coordinate system. The following three Ps indicate that the two adjacent arms of the equivalent linkage segment are connected by a revolute joint, the direction of the rotation axis is the Pitch axis, and the angle of motion around the Pitch axis is equal.

[0081] The RPPP configuration equivalent linkage segment has the same number of booms, active degrees of freedom, and workspace as the PYYP configuration linkage segment, specifically:

[0082] L′ i,1 First at L′ i-1,m Rotate about the x-axis in the local coordinate system Then in U′ i,1 Rotate θ around the y-axis in the local coordinate system e,iGet L′ i,1 Relative to L′ i-1,m Location i-1,m L′ i,1 It is expressed as equation (7):

[0083]

[0084] in, for The cosine value / sine value, i.e. c θ,i / s θ,i For θ e,i The cosine value / sine value, i.e., c θ,i =cosθ e,i / s θ,i =sinθ e,i ;

[0085] L′ i,2 In L′ i,1 Rotate θ around the y-axis in the local coordinate system e,i It is expressed as equation (8):

[0086]

[0087] Then i,1 L′ i,2 Convert to L′ i-1,m In the local coordinate system, it is expressed as equation (9):

[0088]

[0089] Calculate L′ i-1,m L′ i,1 and L′ i,2 The vector hybrid product corresponding to the three rods is expressed as equation (10):

[0090]

[0091] Get L′ i-1,m L′ i,1 and L′ i,2 The vectors corresponding to the three rods satisfy the condition that the mixed product is zero, therefore L′ i-1,m L′ i,1 and L′ i,2 The three rods are in the same plane, meaning that the equivalent RPPP configuration linkage segment also has planar motion properties; at the same time, the angles of motion of each arm rod around the pitch axis in the equivalent RPPP configuration linkage segment are equal, so it also has equal angle motion properties.

[0092] Therefore, the kinematic characteristics of the linkage segment in the RPPP configuration are consistent with those in the PYYP configuration, so the ideal kinematics of the linkage segment in the RPPP configuration and the PYYP configuration are equivalent.

[0093] Furthermore, in step S200, referring to Figure 8 and Figure 9 The equivalent linkage segment of the RPPP configuration is equivalent to a spatial arc inscribed within all the booms of the equivalent linkage segment of the RPPP configuration and two straight segments tangent to the beginning and end of the spatial arc and with a length of half the boom length.

[0094] The segment angle The angle between the plane containing the equivalent circular arc corresponding to the equivalent linkage segment of the RPPP configuration and the z-axis of the local coordinate system of the equivalent linkage segment of the RPPP configuration; the equivalent central angle θ seg,i The central angle of the equivalent circular arc corresponding to the equivalent linkage segment of the RPPP configuration.

[0095] Furthermore, in step S300,

[0096] Reference Figure 10 The multi-layer state space relationship includes drive space, rope space, segment space and task space. The drive space and rope space are mapped one-to-one, which is summarized as the overall analysis of the rope drive space. When the rope length is determined, the overall configuration of the corresponding linkage segment, that is, the segment space parameters, can also be determined.

[0097] Similarly, once the segment space parameters of the linkage segment are determined, the length of the drive rope can be calculated through the configuration of the linkage segment; finally, multiple linkage segments are connected in series to form a complete rope-driven segmented linkage robotic arm, thus mapping from the segment space to the Cartesian space.

[0098] Furthermore, in step S300,

[0099] For the kinematics of the drive space-segment space, the rope length corresponding to the same rope hole position at each cross-axis joint in the same linkage segment is equal due to symmetry. Therefore, the sum of the rope lengths of each drive rope at the joints in a single linkage segment is equal to the number of joints multiplied by the rope length of the drive rope at the first joint.

[0100] Reference Figure 11 and Figure 12 The PY joint is equivalent to an RP configuration joint. In the PY configuration joint, the homogeneous transformation matrix from the local coordinate system {i,j-1} to the local coordinate system {i,j} is... i,j-1 T PY,i,j For equation (11):

[0101]

[0102] Where h is the distance from the center of the cross axis to the plane where the wiring disk is located; Trans(·,·,·) is the translation transformation matrix, and the three parameters represent the components of the translation distance along the x-axis, y-axis and z-axis, respectively;

[0103] In RP configuration joints, the homogeneous transformation matrix from local coordinate system {i,j-1} to local coordinate system {i,j} is... i,j-1 T RP,i,j For equation (12):

[0104]

[0105] Among them, s θ,i / c θ,i For θ e,i The sine / cosine value, θ e,i =θ seg,i / m i m i The number of cross axes in the i-th segment; for The sine / cosine value;

[0106] When the PY joint is equivalent to an RP configuration joint, their homogeneous transformation matrices are the same. Therefore, we can obtain the q when the PY joint is equivalent to an RP configuration joint. P,i q Y,i and θ seg,i , The relationship is given by equation (13):

[0107]

[0108] In an RP-configuration joint, since the same rope passes through the same hole in each local coordinate system, when the PY joint is equivalent to an RP-configuration joint, the rod L... i,j-1 The position A of the rope hole in the local coordinate system {i,j-1} PY-RP,i,k For equation (14):

[0109] i,j-1 A PY-RP,i,k =[0 r cable cosβ i,k r cable sinβ i,k ] T (14)

[0110] Where, r cable The radius of the rope hole position;

[0111] L rod i,j The position B of the rope hole in the local coordinate system {i,j} PY-RP,i,k For equation (15):

[0112] i,j B PY-RP,i,k =[0 r cable cosβ i,k r cable sinβ i,k ] T (15)

[0113] The position of the rope hole in the local coordinate system {i,j} i,j B PY-RP,i,k Rewritten as homogeneous coordinates For equation (16):

[0114]

[0115] And through homogeneous transformation matrix i,j-1 T RP,i,j The position of the rope hole in the local coordinate system {i,j-1} is obtained. i,j-1 B PY-RP,,i,k homogeneous coordinates For equation (17):

[0116]

[0117] Where h is the rod L i,j-1 or rod L i,j Length between the center point of the wiring disk and the center point of the cross axis; For β i,k The sine / cosine value, i.e.

[0118] This allows us to obtain the position of the rope hole in the local coordinate system {i,j-1} of the PY configuration joint. i,j-1 B PY-RP,i,k For equation (18):

[0119]

[0120] Therefore, when the PY configuration joint of the i-th segment is equivalent to the RP configuration joint, the rope length l corresponding to the k-th rope hole position at the joint is... PY-RP,i,k For equation (19):

[0121]

[0122] For the k-th rope fixed to the wiring disk of the last arm of the i-th linkage segment, it passes through all joints of the i-th linkage segment, as well as all joints and all arms of the preceding i-1 linkage segments. Therefore, the total rope length l cable,i,k For equation (20):

[0123]

[0124] The above equation is the inverse kinematics of the driving space-segment space, m u is the number of cross-axis joints in the u-th linkage segment; l is the distance between the center points of two adjacent cross-axis joints.

[0125] Furthermore, refer to Figure 13 In step S300, for the driving space-segment kinematics, the segmented linkage manipulator obtained by connecting multiple linkage segments in series is replaced by the first and last straight lines and several interconnected circular arcs in the middle after being equivalent to circular arcs.

[0126] The length of the first straight segment is given by equation (21):

[0127]

[0128] Among them, P base Position of the segmented linkage manipulator base; P tgc,0 This represents the position of the tangent point between the first straight line and the arc after the arc is equivalent.

[0129] The length of the straight line at the end is given by equation (22):

[0130]

[0131] Among them, P tgc,N For the equivalent arc, P is the location of the point where the tail line is tangent to the arc. e This refers to the end position of the segmented linkage manipulator;

[0132] The equivalent arc length s of the i-th segment in the middle i For equation (23):

[0133]

[0134] To facilitate the solution of forward kinematics, the intersection of two tangents at the same endpoint of the arc is denoted as the virtual joint point P of that arc segment. jv,i Therefore, the arc is equivalent to two points intersecting at the virtual joint P. jv,i Virtual joint rod and Its direction is the same as the two tangents, and its length is the length l from the endpoint to the virtual joint. v,i For equation (24):

[0135]

[0136] After a linkage segment containing an even number of sub-joints is equivalent to a circular arc, the local coordinate system of the i-th circular arc is {i-1}. c homogeneous transformation matrix to its endpoints i-1,c T even,i,c For equation (25):

[0137]

[0138] Homogeneous transformation matrix of a single linkage segment 0 T seg,1 For equation (26):

[0139] 0 T seg,1 =Trans(x,l / 2) 0,c T even,1,c Trans(x,l / 2) (26)

[0140] Homogeneous transformation matrix of the entire linkage segment 0 T seg,3 For equation (27):

[0141]

[0142] Equation (27) is the forward kinematics of the segment space-task space of the rope-driven segmented linkage robotic arm.

[0143] Furthermore, refer to Figure 14 In step S400, the inverse kinematics of the rope-driven segmented linkage robot arm segment space-task space is solved based on the improved forward and backward inverse kinematics method. The solution process includes the following steps:

[0144] Starting from a given target end position and attitude, initialize the counter and calculate the current end error;

[0145] If the error is within the acceptable range, try to analyze the inverse kinematics solution;

[0146] If a solution is found, the process ends; otherwise, it continues with the iteration process. In each iteration, a backward iteration is performed first, starting from the base, calculating the direction and equivalent bending angle of the next segment segment by segment, determining whether the maximum allowable value is exceeded, and updating the virtual rod length and direction accordingly, until the end.

[0147] Then, a forward iteration is performed, starting from the end and updating the direction and bending angle of each segment towards the base. The actual or maximum value is used for updating based on whether the angle exceeds the limit, and the position of the segment is recalculated until it returns to the base.

[0148] After completing one round of updates, recalculate the terminal error and determine whether the conditions are met or the maximum number of iterations is exceeded. Otherwise, continue to the next round of iterations until a solution that meets the error tolerance is obtained or the iteration terminates.

[0149] Furthermore, the forward reach iteration process involves calculating the position P of the tangent point of the arc sequentially from the last segment to the first segment. tgc,i , pointing to t tgc,i and central angle θseg,i For the last segment, the end position P of the linkage segment e,d and pointing to t e,d Given the given quantities, and the end segment containing a rod of length l / 2 tangent to the last arc, we first calculate the position P of the point where the last arc is tangent to the l / 2 rod length. tgc,m ;

[0150] For the remaining segments, the position P of the tangent point between the (i+1)th and the ith arc is... tgc,i It is calculated from the (i+1)th segment, as shown in equation (28):

[0151]

[0152] By P tgc,i The virtual joint position P of the i-th arc is obtained by combining the tangency condition. v,i For equation (29):

[0153] P v,i =P tgc,i -l v,i t tgc,i (29)

[0154] The direction t of the tangent point between the i-th arc and the (i-1)-th arc can be calculated from the virtual joint positions of the i-th arc and the (i-1)-th arc. tgc,i-1 For example, equation (30);

[0155]

[0156] However, for the first segment, the orientation of the base remains unchanged, so no calculation is needed. Instead, calculate the central angle θ of the i-th segment. seg,i For equation (31):

[0157]

[0158] When the central angle θ of the i-th segment seg,i Less than or equal to θ seg,MAX Then, update the virtual rod length l of the i-th segment according to formula (24). v,i And the newly calculated rod length l v,i Substituting into equations (29), (30), and (31), update the virtual joint position P of the i-th arc. v,i The direction of the (i-1)th arc tangent point t tgc,i-1 and the central angle θ of the i-th segment seg,i Finally, the position P of the tangent point between the i-th arc and the (i-1)-th arc is obtained. tgc,i-1 For equation (32):

[0159] P tgc,i-1 =P v,i-l v,i t tgc,i-1 (32)

[0160] When the central angle θ of the i-th segment seg,i Greater than θ seg,MAX At that time, the rope-driven segmented linkage robotic arm cannot move to this configuration, therefore it is necessary to restrict the solution angle to θ. seg,MAX Calculate the direction t of the tangent point between the i-th arc and the (i-1)-th arc. tgc,i-1 For equation (33):

[0161]

[0162] in, 0 R v,i For forward iteration, the virtual joint pose of the i-th segment is as shown in equation (34); The x-axis of the local coordinate system of the i-th virtual joint during forward iteration;

[0163]

[0164] Then update the virtual rod length l of the i-th segment according to equation (24). v,i And the newly calculated rod length l v,i Substituting into equations (29), (30), and (31), update the virtual joint position P of the i-th arc. v,i The direction of the (i-1)th arc tangent point t tgc,i-1 and the central angle θ of the i-th segment seg,i Finally, the position P of the (i-1)th arc tangent point is obtained according to equation (32). tgc,i-1 ;

[0165] The position P of the tangent point between the first segment and the arc of half the rod length tgc,0 The base position P can be calculated. base As in equation (35):

[0166]

[0167] At this point, the base position may not be zero, so it is necessary to move the base position back to its original position through backward iteration.

[0168] Furthermore, the backward iteration process involves calculating the position P of the tangent point of the arc from the first segment to the last segment in the forward iteration. tgc,i , pointing to t tgc,i and central angle θ seg,i For the first segment, the base position P of the linkage segment base and pointing to t baseGiven that the base also has a rod of length l / 2 tangent to the first arc, we first calculate the position P of the point where the first arc tangent to the l / 2 rod. tgc,0 For the remaining i segments, the position P of the point where the (i-1)th arc is tangent to the i-th arc is... tgc,i-1 It is calculated from the (i-1)th segment, as shown in equation (36):

[0169]

[0170] By P tgc,i-1 The virtual joint position P of the i-th arc is obtained from the tangency point condition. v,i As in equation (37):

[0171] P v,i =P tgc,i-1 +l v,i t tgc,i-1 (37)

[0172] The direction t of the tangent point between the i-th and (i+1)-th arcs can be calculated from the virtual joint positions of the i-th and (i+1)-th arcs. tgc,i As in equation (38):

[0173]

[0174] For the last segment, the direction of the end point remains unchanged, so no calculation is needed; calculate the central angle θ of the i-th segment. seg,i As in equation (31);

[0175] When the central angle θ of the i-th segment seg,i Less than or equal to θ seg,MAX Then, update the virtual rod length l of the i-th segment according to formula (24). v,i And the newly calculated rod length l v,i Substituting into equations (37), (38), and (31), update the virtual joint position P of the i-th arc. v,i The direction of the i-th arc tangent point t tgc,i and the central angle θ of the i-th segment seg,i Finally, the position P of the tangent point between the i-th arc and the (i+1)-th arc is obtained. tgc,i As in equation (39):

[0176] P tgc,i =P v,i +l v,i t tgc ,i (39)

[0177] When the central angle θ of the i-th segment seg,i Greater than θ seg,MAX At that time, with θ seg,MAXCalculate the direction t of the tangent point between the i-th arc and the (i+1)-th arc. tgc,i As in equation (40):

[0178]

[0179] in, 0 R′ v,i For backward iteration, the virtual joint pose of the i-th segment is as shown in equation (41):

[0180]

[0181] Rotate θ around the z′ axis of the local coordinate system of the i-th virtual joint during backward iteration. seg,MAX The rotation matrix; Let x′ be the local coordinate system of the i-th virtual joint during backward iteration;

[0182] The position P of the last segment tangent to the arc with half the length of the rod tgc,N The end position P can be calculated. e,cal As shown in equation (42):

[0183]

[0184] After one forward and one backward iteration, it becomes as shown in equation (43):

[0185]

[0186] Calculate the end position error e p and end pointing error e t ;

[0187] After repeated forward and backward iterations, if the end-effector pose error of the entire arm meets the iteration parameter requirement e... p,MAX and e t,MAX This gives the central angle θ of each segment. seg,i , the direction vector t of the tangent point of the circular arc seg,i and position P seg,i .

[0188] Furthermore, in step S400, the result of the inverse solution needs to be transformed into a segment space, where the central angle θ of each segment is... seg,i The angle of the segment can be directly calculated during the inverse solution, while... To change the x-direction vector t of the i-th segment endpoint... seg,i The results are obtained by transforming the coordinates of each segment into the local coordinate system, as shown in equation (44):

[0189]

[0190] in, i tseg,i,y / i t seg,i,z for i t seg,i The components in the y / z directions, and i t i+1 =( 0 R i ) T 0 t i+1 , 0 R i Let be the attitude matrix of the i-th endpoint;

[0191] After calculating the segment space parameters of the i-th segment, the pose of the i-th segment is obtained. 0 T i Calculate the segment space parameters of the (i+1)th segment. and θ seg,i This process continues until all segment spatial parameters of the rope-driven segmented linkage robotic arm are calculated, thus obtaining the inverse kinematics of the segment space-task space of the rope-driven segmented linkage robotic arm.

[0192] Furthermore, in step S500,

[0193] Reference Figure 15 The workspace of a single linkage segment is a curved surface. However, in the actual movement of the linkage segment, due to the existence of unequal angle linkage, the expected workspace and the actual workspace surface do not overlap. In order to make the linkage segment under unequal angle linkage also apply the segment space state description and make the equivalent motion configuration closest to the actual motion configuration, the linkage segment under unequal angle linkage is equivalently segmented with the goal of minimizing the error of the segment endpoint position of the linkage segment before and after equivalence.

[0194] Reference Figure 16 First, set the actual segment endpoint position P. seg,a,i Project onto the ideal workspace, and then use the projection point position P seg,p,i and the expected segment endpoint position P seg,d,i The component of the difference is taken as Δy seg,i and △z seg,i After segment closed-loop control, the projected point position coincides with the desired segment endpoint position, i.e., the actual segment endpoint position P after segment closed-loop control. seg,c,i With respect to the desired segment endpoint position P seg,d,i Closest distance;

[0195] The approximate curve of the ideal workspace is obtained using the fitting method, and the actual segment endpoint position P is used. seg,a,i The position P of the projection point is obtained by projecting the fitted workspace curve. seg,p,i Then, the position P of the projection point on the ideal motion plane. seg,p,iTransformed into two state variables, the segment angle, in segment space and central angle θ seg,i ;

[0196] In cases of non-uniform angle linkage, the actual end position is projected onto the desired workspace to obtain the projection point position, and the segment space parameters corresponding to the projection point position are further calculated. and θ e,i As the equivalent segment spatial parameters of the linkage segment under non-equiangular linkage, the equivalent segment spatial kinematic model parameters are used to replace the joint angles of each cross-axis joint in the P and Y directions in the actual linkage segment, in order to characterize the overall motion of each rope-driven linkage segment, without considering the specific linkage characteristics of the linkage segment or the linkage error of each joint.

[0197] The workspace of a single linkage segment is a non-circular curve, and its parametric equation can be written as shown in equation (45):

[0198]

[0199] An approximate curve of the ideal workspace is obtained using a fitting method, limiting the range of motion of a single joint within a linkage segment to |θ. seg,i |≤90°, discretize its workspace into 1000 points (u j ,x j ), j = 1, 2, ..., 1000, using n fit By fitting a polynomial of degree to its workspace, and by scaling the workspace points to the center during fitting and neglecting small values ​​of the fitting coefficients, an approximate workspace function x(u) in the ideal motion plane can be obtained, as shown in equation (46):

[0200]

[0201] Where, k i For u i The fitting coefficient, u mean / u std x represents the mean / standard deviation of the u-axis of the workspace points after discretization of the workspace. mean / x std The x-axis value is the average / standard deviation of the workspace points after discretization.

[0202] Reference Figure 17 Using a geometric method, the actual segment endpoints are projected onto the fitted workspace curve to obtain the projection point position. The fitted workspace curve is located at the projection point P. P Calculate the coordinates of the projection point based on the slope relationship with the surrounding area.

[0203] The position P of the projection point on the ideal motion plane seg,p,iTransformed into two state variables, the segment angle, in segment space and central angle θ seg,i The position P of the endpoint of the linkage segment after the arc is equivalent seg,i The length projected onto the u-axis in the plane of motion is as shown in equation (50):

[0204]

[0205] Among them, R i Let R be the equivalent radius of the i-th segment. i =l / 2tan(θ) seg,i / 2m i );

[0206] Calculate the y-components of the segment endpoints on the y-axis and z-axis. seg,p,i and z seg,p,i With segmental angle and central angle θ seg,i As in equation (51):

[0207]

[0208] The segment angle is obtained by inverse solution of equation (51). As in equation (52):

[0209]

[0210] In equation (50), u(θ seg,i The expression contains several trigonometric nonlinear terms, making it difficult to use the segment endpoints P. seg,i The central angle θ is obtained by inversely solving the length projected onto the u-axis within the plane of motion. seg,i Therefore, the universal trigonometric formula is used to convert it into an algebraic equation, and then Newton's iteration method is used to obtain an approximate solution to the equation, thereby obtaining the central angle. Taking a segment with 4 joints as an example, u4(θ) seg,i The expression is as shown in equation (53):

[0211]

[0212] Substitute θ using the universal trigonometric function formula. seg,i That is, t = tan(θ) seg,i / 8), as in equation (54):

[0213]

[0214] The solution t of equation (54) can be obtained using Newton's iteration method. * The central angle is obtained. As in equation (55):

[0215]

[0216] Extend equation (55) to a segment m i In the case of a single joint, the universal trigonometric function formula can also be used to substitute u. seg,p,i (θ seg,i θ in ) seg,i That is, t = tan(θ) seg,i / 2m i The solution t of the equation can be obtained using Newton's iteration method. * Furthermore, the central angle can be obtained. As in equation (56):

[0217]

[0218] The segment endpoint position control method based on the segment endpoint equivalence method of the linkage segment ensures the segment endpoint position accuracy to the greatest extent and at the same time ensures the segment endpoint attitude accuracy as much as possible, so that the overall actual configuration of the single linkage segment is close to the desired configuration.

[0219] Furthermore, in step S500, refer to Figure 18 Step S500 includes the following steps:

[0220] Starting with segment endpoint trajectory planning, the desired segment endpoint positions P are obtained. seg,d,i , and the actual position P at the end of the current segment seg,p,i The position error ΔP is obtained by comparison;

[0221] The attitude error was calculated using Cartesian differential kinematics. and angular error △θ seg,i The input is sent to the segment space control law module, and the desired attitude parameters are obtained by combining the Cartesian forward kinematics results. θ seg,d,i The control law output is used to solve for the desired rope drive length l. cable,i,k ;

[0222] The rope drive kinematics is converted into the motion target of each motor, and the motor closed-loop control module drives the rope to pull the specific mechanism, ultimately controlling the joint movement of each drive segment;

[0223] The joint positions of each segment are fed back through the encoder, and the current actual end position P of the segment is obtained through forward kinematics calculation. seg,a,i Used to update the equivalent position P of the segment endpoints in the closed-loop feedback loop. seg,p,i .

[0224] Furthermore, in step S600, the control law for the corresponding segment endpoint position control is derived, and the total differential of equation (51) is obtained to obtain equation (57):

[0225]

[0226] Where A is u(θ) seg,i ) for θ seg,i The derivative of J is given in equation (58). Φ-P The Jacobian matrix represents the velocity from the segment space parameter velocity to the velocity at the segment endpoint position of a single linked segment;

[0227]

[0228] Multiply both sides of equation (57) by the inverse of the Jacobian matrix of the velocity at the end point of a single linked segment from the velocity of the segment space parameter. Equation (59) is obtained. Substituting the segment endpoint error ΔP into equation (59) yields equation (59). seg,i Δy in the y and z directions seg,i , △z seg,i The segment angle error can be obtained. and the error of the central angle △θ seg,i ,

[0229]

[0230] Positional error ΔP at the end of the segment seg,i The specific form is as shown in equation (60).

[0231]

[0232] Substituting equations (49) and (60) into equation (57), the segment endpoint position error is converted into segment surface angle error. and the error of the central angle △θ seg,i .

[0233] Furthermore, in step S500, a segment space controller is provided in the segment endpoint position control, and its input signal is provided by... and △θ seg,i The system consists of a PID controller module, where each signal is processed by its own PID controller module, which includes proportional, integral, and derivative components. The processed signals are then combined by an adder into a single comprehensive control signal output.

[0234] Furthermore, refer to Figure 19 The segment space controller combines PID control and feedforward control. By superimposing the two control methods, it improves the dynamic response of the system and reduces steady-state error. The specific control laws are Equations (61) and (62):

[0235]

[0236] in, For the corresponding PID parameters, K θ,P Kθ,I K θ,D For the corresponding θ seg,pid,i PID parameters;

[0237] The output of the segment space controller obtained from equations (61) and (62) is converted into the length of the three ropes of the linkage segment through the segment space-rope space kinematics, and then input into the motor closed-loop controller; while the motor assembly of the rope-driven segment linkage robot arm uses a motor controller, so the specific control law of the motor closed-loop controller will not be described in detail.

[0238] Furthermore, step S600 includes the following steps:

[0239] The desired end-effector pose X is obtained by trajectory planning in the task space according to task requirements. e,d Then, the expected values ​​of the segment space parameters are obtained through the improved FABRIKc inverse kinematics. θ seg,d,i As a feedforward quantity for kinematic control;

[0240] The value q is measured using the joint encoder within the linkage section. a,i Calculate the current actual end-effector pose X e,a , and the desired end pose X e,d The end-effector pose error ΔX was obtained after comparison. e Then, the pseudo-inverse of the Jacobian matrix from the segment space state parameter velocity to the terminal velocity is obtained. Decompose it into segment space to obtain segment angle error and the error of the central angle △θ seg,i ,

[0241] Combining the feedforward results calculated earlier θ seg,d,i The kinematic relationship between the segment space and the rope-driven space is transformed into the motion of each rope. The motion of the rope-driven segmented linkage robotic arm is further driven by the closed-loop control of the motor, thus realizing closed-loop control in the segment space of the rope-driven segmented linkage robotic arm.

[0242] Furthermore, in step S600,

[0243] The pose error of the segmented linkage manipulator is decomposed into the segment space state parameters of each segment, and closed-loop control is performed in the segment space to achieve high-precision control of the end-effector pose of the rope-driven segmented linkage manipulator.

[0244] Reference Figure 20 First, trajectory planning is performed in the task space according to task requirements to obtain the desired end-effector pose X. e,d Then, the expected values ​​of the segment space parameters are obtained through the improved FABRIKc inverse kinematics. θ seg,d,iAs a feedforward for kinematic control; then the joint encoder value q in the linkage segment is measured. a,i Calculate the current actual end-effector pose X e,a , and the desired end pose X e,d The end-effector pose error ΔX was obtained after comparison. e Then, the pseudo-inverse of the Jacobian matrix from the segment space state parameter velocity to the terminal velocity is obtained. Decompose it into segment space to obtain segment angle error and the error of the central angle △θ seg,i Combining the feedforward results calculated earlier θ seg,d,i ,

[0245] By converting the kinematic relationship between the segment space and the rope-driven space into the motion of each rope, and further driving the rope-driven segmented linkage robotic arm to move through closed-loop motor control, closed-loop control is achieved in the segment space of the rope-driven segmented linkage robotic arm, thereby realizing the decomposed motion control of the entire end effector of the rope-driven segmented linkage robotic arm.

[0246] Furthermore, in step S600,

[0247] By acquiring the angle values ​​q of each joint within the linkage segment in real time P,a,i and q Y,a,i And calculate the end-effector pose error ΔX in real time. e Then, based on segment space-task space differential kinematics, the end-effector pose error ΔX is calculated. e Decomposed into segment angle error and the error of the central angle △θ seg,i The end-effector pose error ΔX is eliminated by using a whole-arm decomposition motion control method. e To achieve high-precision control of the end-effector posture of the rope-driven segmented linkage robotic arm;

[0248] Among them, reference Figure 21 The endpoint position of the ideal segment i is P. seg,d,i The actual endpoint position of the segment is P. seg,a,i Its error △P i As in equation (63):

[0249] △P i =P seg,d,i -P seg ,a,i (63)

[0250] After being accumulated to the end of the boom, the positional error e at the end of the boom p As in equation (64):

[0251]

[0252] The equivalent end-effector posture of the rope-driven segmented linkage robotic arm can be denoted as R using a rotation matrix. e,d The actual end-effector pose is denoted as R. e,a Then the position error e at the end of the entire arm p and attitude error e o As in equations (64) and (65):

[0253]

[0254] Where, n e,d / o e,d / a e,d For R e,d The first / second / third column, n e,a / o e,a / a e,a For R e,a The first / second / third column;

[0255] The encoder is used to obtain the angle values ​​q of each joint in the linkage segment in real time. P,a,i and q Y,a,i The actual position P of the segment terminal is obtained through the DH method. seg,a,i and the actual posture R of the end effector of the entire arm e,a It can also be used to set the current desired segment space state parameters. and θ seg,d,i The positions P of the endpoints of the linked segments are calculated using segment-task-space forward kinematics. seg,d,i Equivalent posture R of the entire arm end effector e,d Therefore, the pose error e of the end effector can be obtained in real time. p and e o ;

[0256] Based on segment-task-space differential kinematics, the end-effector pose error ΔX can be calculated. e Decomposed into segment angle error and the error of the central angle △θ seg,i As in equation (66):

[0257]

[0258] in, Let be the rate of change of the segment space state parameter of the linkage segment with respect to time, as defined in equation (67).

[0259]

[0260] The rate of change of the segment space state parameters of the linked segment with respect to time. and linear velocity v at the end e and angular velocity ω eThe pseudo-inverse of the Jacobian matrix.

[0261] Furthermore, in step S600,

[0262] Reference Figure 22 For the i-th linkage segment, the segment angular velocity For the end linear velocity and angular velocity The effect is shown in equation (68):

[0263]

[0264] in, For the angular velocity of the segment Caused terminal linear velocity / terminal angular velocity; x i-1 r is the x-axis of the local coordinate system {i-1}; i-1,N It is the vector from the end point of the (i-1)th linkage segment to the end position of the rope-driven segmented linkage robot arm;

[0265] For the i-th linkage segment, the central angular velocity For the terminal linear velocity v e and angular velocity ω e The effect needs to be obtained through differentiation;

[0266] Reference Figure 23 The following derives the coordinate system of the motion plane of the i-th linkage segment. Inner central angular velocity For the terminal linear velocity v e and angular velocity ω e Due to the influence of the angular velocity of the center of the i-th linkage segment The final linear velocity v is generated only in the motion plane of the i-th linkage segment. e,θ,i Therefore, the end position P of the rope-driven segmented linkage robotic arm is... e P is obtained by projecting it onto the motion plane of the i-th linkage segment. e,xou,i P e,xou,i linear velocity at point P e The linear velocity is the same at point P; e,xou,i In the local coordinate system The coordinates in the equation are as shown in equation (69):

[0267]

[0268] in, Let be the projection length of the vector from the end point of the i-th linkage segment to the end position of the cable-driven segmented linkage robot arm onto the motion plane of the i-th linkage segment, specifically in the form of equation (70):

[0269]

[0270] in, for The x / z direction components, Specifically as formula (71):

[0271]

[0272] Differentiating equation (69) yields P e,xou,i linear velocity at As in equation (72):

[0273]

[0274] From equation (72), we can know the angular velocity of the center of the i-th linkage segment. The resulting terminal linear velocity v e,θ,i As in equation (73):

[0275]

[0276] Wherein, the coefficient K x K u The expression is as shown in equation (74):

[0277]

[0278] The central angular velocity of the i-th linkage segment The resulting terminal angular velocity ω e,θ,i We obtain, as shown in equation (75):

[0279]

[0280] in, Local coordinate system The v-axis;

[0281] Combining equations (68), (73), and (75), and rewriting them in matrix form, we obtain the segment angular velocity of the i-th linkage segment. and the angular velocity of the center With the terminal linear velocity v e and angular velocity ω e The relationship is as shown in equation (76):

[0282]

[0283] in, J θ,i Let i be the angular velocity of the i-th linkage segment. angular velocity of the center linear velocity v at the end e and angular velocity ω eThe generalized transmission ratio, i.e., the column of the velocity Jacobian matrix from the i-th segment to the end, is specifically expressed as in equation (77):

[0284]

[0285] Extending equation (77) to all linked segments yields the segmental angular velocities of the linked segments. angular velocity of the center linear velocity v at the end e and angular velocity ω e Jacobian matrix J Φ-E As shown in equation (78):

[0286]

[0287] The end pose error ΔX e Substituting equation (78) into the equation, the end-effector pose error ΔX can be calculated. e The error is decomposed into segmental spatial state parameter error ΔΦ, and then the proposed whole-arm decomposed motion control method of the rope-driven segmented linkage robotic arm is used to eliminate the end-effector pose error ΔX. e As shown in equation (79):

[0288]

[0289] in, For J Φ-E The false rebellion.

[0290] Furthermore, in step S600, in order to accelerate the end pose convergence speed of the rope-driven segmented linkage robot arm and avoid the instability of the pseudo-inverse solution of the decomposed motion controller, the virtual joint method, damped least squares method, singular value truncation method and Jacobi matrix block method were used to improve the solution of the whole arm decomposed motion control.

[0291] When the rope-driven segmented linkage robotic arm has strict requirements for the desired end-effector orientation but not strict requirements for the desired posture, adding a virtual roll joint at the end-effector can accelerate its convergence speed to the desired end-effector position and orientation.

[0292] After adding a virtual joint at the end, the Jacobian matrix J Φ-E A column needs to be added to transform it into an extended Jacobian matrix J. Φ-E,EXT As in equation (80):

[0293] J Φ-E,EXT =[J Φ-E J v (80)

[0294] Among them, J v The virtual joint motion velocity to the end-effector linear velocity v e and angular velocity ω eThe generalized transmission ratio is given by equation (81):

[0295]

[0296] Where, x N Let {N} be the x-axis of the local coordinate system;

[0297] At this point, the end-effector pose error ΔX e The formula for decomposing into segment space is shown in equation (82):

[0298]

[0299] Where, △Φ EXT The extended segment space parameter error is specifically in the form of equation (83):

[0300]

[0301] in,

[0302] Furthermore, in actual control, the Jacobian matrix J Φ-E Or extended Jacobian matrix J Φ-E,EXT In certain configurations of the cable-driven segmented linkage manipulator, the matrix becomes ill-conditioned. At this time, the pseudo-inverse solution of the linear equation system obtained by solving equation (79) or equation (82) will be numerically unstable. Therefore, it is necessary to deal with the singularity problem of the Jacobian matrix to avoid the cable-driven segmented linkage manipulator from generating violent shaking during the end pose closed-loop control.

[0303] To overcome the above problems, we can calculate the Jacobian matrix J. Φ-E Or extended Jacobian matrix J Φ-E,EXT The pseudo-reverse time is introduced by adding a damping factor λ, i.e., using damped least squares to solve the linear equation system, to expand the Jacobian matrix J. Φ-E,EXT For example, after adding the damping factor λ, the extended Jacobian matrix J Φ-E,EXT pseudo-reversal As in equation (84):

[0304]

[0305] Where I7 is the identity matrix with a rank of 7;

[0306] The damping factor λ is equivalent to adding a constraint to the solution of a system of linear equations using the least squares method, limiting the size of the solution and thus avoiding computational instability. By adjusting the size of the damping factor λ, accuracy and stability can be balanced. A larger damping factor λ will make the solution smoother, but may deviate from the true solution; a smaller damping factor λ is closer to the minimum norm solution. The selection of the damping factor λ usually requires experience or adjustment; here, the extended Jacobian matrix J is chosen.Φ-E,EXT The largest element is used as the damping factor;

[0307] Furthermore, singular value decomposition can be used to address the instability of pseudo-inverse solutions of ill-conditioned matrices, thereby extending the Jacobian matrix J. Φ-E,EXT For example, when some singular values ​​in the matrix are small, the error is easily amplified when calculating the pseudo-inverse, leading to instability in the pseudo-inverse solution. Therefore, J Φ-E,EXT After singular value decomposition, smaller singular values ​​need to be truncated before calculating the pseudoinverse, thus extending the Jacobian matrix J. Φ-E,EXT Perform singular value decomposition, as shown in equation (85):

[0308]

[0309] Among them, U Φ-E,EXT Σ Φ-E,EXT and V Φ-E,EXT For J Φ-E,EXT The three matrices obtained from singular value decomposition;

[0310] For matrix Σ Φ-E,EXT It is necessary to truncate the smaller singular values ​​and construct its pseudo-inverse matrix. As in equation (86):

[0311]

[0312] in, For matrix The element in the i-th row and j-th column has the specific value as shown in equation (87):

[0313]

[0314] Where τ is the singular value truncation threshold, τ = 10 -3 ;

[0315] Construct the pseudo-inverse matrix with truncated singular values ​​using equation (86). As in equation (88):

[0316]

[0317] Furthermore, substituting equation (88) into equation (82) yields a more stable pseudo-inverse solution;

[0318] When the end position of the cable-driven segmented linkage robotic arm changes within a small range of space, its end posture also changes relatively little. Therefore, based on this characteristic of the cable-driven segmented linkage robotic arm, positional accuracy can be prioritized while slightly sacrificing posture accuracy. This also solves the problem of pseudo-inverse instability caused by ill-conditioned matrices, as shown in equation (89):

[0319]

[0320] Extended Jacobian matrix J Φ-E,EXT It can be divided into two matrices J, upper and lower. Φ-v,EXT and J Φ-ω,EXT J Φ-v,EXT and J Φ-ω,EXT Each of them has a small condition number and is not an ill-conditioned matrix. Therefore, using J Φ-v,EXT The end position error e can be separated. p Decomposing into segment space yields a stable pseudo-inverse solution △Φ′, as shown in equation (90):

[0321]

[0322] in, For J Φ-v,EXT The pseudo-inverse matrix;

[0323] Since Equation (90) can only guarantee the end position accuracy of the rope-driven segmented linkage robot arm, but cannot guarantee the end attitude accuracy, it can only be used as a singularity avoidance method after both the damped least squares solution corresponding to Equation (84) and the singular value decomposition method corresponding to Equation (88) have failed, making the movement of the rope-driven segmented linkage robot arm more stable and safe.

[0324] Furthermore, the present invention also proposes a rope-driven segmented linkage robotic arm, the rope-driven segmented linkage robotic arm comprising:

[0325] A drive box, the drive box including a motor and a transmission mechanism, the transmission mechanism including at least a rope puller, a linear guide rail and a lead screw;

[0326] The linkage manipulator includes multiple linkage arm segments connected in series. Each linkage arm segment has two degrees of freedom, and an end effector is provided on the linkage arm segment at the end.

[0327] Furthermore, the linkage manipulator of the rope-driven segmented linkage robotic arm adopts a hybrid active-passive drive, that is, the linear drive module directly changes the length of the drive rope to realize the movement of the segment endpoints of the linkage segment, and the passive linkage rope forms a linkage mechanism to drive the joints in the linkage segment to realize equal-angle movement. Through the combined action of the active rope and the passive rope, the motion control of the segmented linkage manipulator is realized.

[0328] In terms of active drive, the rope-driven segmented linkage robotic arm realizes the movement of the linkage segment by changing the length of the three active drive ropes of the corresponding linkage segment; one end of the drive rope is fixed on the wiring disk at the end of the linkage segment, and the other end passes through the rope-passing hole designed on the wiring disk of each arm in the linkage segment and is fixed on the rope puller of the linear drive module in the drive box.

[0329] In terms of passive linkage, the linkage structure is an 8-shaped linkage structure, which includes a small 8-shaped linkage structure and a large 8-shaped linkage structure. Both the small 8-shaped linkage structure and the large 8-shaped linkage structure make the movement angle of two adjacent cross axes the same in the linkage direction.

[0330] The above description is merely a preferred embodiment of the present invention. The present invention is not limited to the above-described embodiments. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this disclosure, as long as they achieve the technical effects of the present invention by the same means, should be included within the scope of protection of this disclosure. Within the scope of protection of the present invention, the technical solutions and / or implementation methods can have various modifications and variations.

Claims

1. A method for describing the equivalent state and kinematic modeling of a linkage segment, wherein the method is applied to a cable-driven segmented linkage robotic arm, characterized in that, The method includes the following steps: S100, the PYYP configuration (or YPPY configuration) linkage segment based on the rope-driven linkage robot arm has planar motion properties and equal angle motion properties, and the PYYP configuration linkage segment is equivalent to the RPPP configuration equivalent linkage segment; S200. The equivalent linkage segment of the RPPP configuration is equivalent to a spatial circular arc and a straight line segment. The segment space parameters are set and extracted, including the segment face angle. and equivalent central angle θ seg,i ; S300. Based on the segment space parameters, the segment space state description yields the multi-layer state space relationship of the rope-driven segmented linkage manipulator, and the kinematics of the drive space-segment space and the kinematics of the segment space-task space are obtained. S400. Solve the inverse kinematics of the segment space-task space of the rope-driven segmented linkage robot arm based on the improved forward and backward inverse kinematics method. S500. With the goal of minimizing the positional error of the equivalent preceding and following linkage segments, the linkage segments under non-equiangular linkage are subjected to segment space state equivalence. An approximate curve of the ideal workspace is obtained using a fitting method, and the actual segment endpoint position P is then determined. seg,a,i The position P of the projection point is obtained by projecting the fitted workspace curve. seg,p,i Then, the position P of the projection point on the ideal motion plane. seg,p,i Transformed into two state variables, the segment angle, in segment space and equivalent central angle θ seg,i ; S600, based on the whole arm decomposition motion control block diagram of the rope-driven segmented linkage manipulator, decomposes the position and posture error of the end of the segmented linkage manipulator into the segment space state parameters of each segment, and performs closed-loop control in the segment space to achieve high-precision control of the end position and posture of the rope-driven segmented linkage manipulator.

2. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S100, the PYYP configuration (or YPPY configuration) linkage segment of the cable-driven robotic arm possesses planar motion properties, specifically: Let L be the j-th link of the i-th linkage segment. i,j Let U be the j-th cross axis of the i-th linkage segment. i,j L i,1 First rotate around the y-axis of the world coordinate system by -q Y,i Then rotate q around the z-axis of the world coordinate system. P,i Then L i,1 Compared to L i-1,m Location i-1,m L i,1 For equation (1): Where Rot(·,·) is the rotation transformation matrix, the first parameter of Rot(·,·) represents the rotation axis, and the second parameter of Rot(·,·) represents the rotation angle; i-1,m L i,1 For L i,1 In L i-1,m The position in the local coordinate system; c P,i / s P,i For q P,i The cosine value / sine value, i.e., c P,i =cosq P,i / s P,i =sinq P,i ;c Y,i / s Y,i For q Y,i The cosine value / sine value, i.e., c Y,i =cosq Y,i / s Y,i =sinq Y,i ; l represents the length of a single boom; Similarly, L i,2 Compared to L i,1 Location i,1 L i,2 For equation (2): i,1 L i,2 For L i,2 In L i,1 The position in the local coordinate system; L i,2 The coordinates from L i,1 Transform to L in the local coordinate system i-1,m In the local coordinate system, we obtain equation (3): in, i-1,m L i,2 For L i,2 In L i-1,m The position in the local coordinate system; c 2Y,i For 2q Y,i The cosine value, i.e., c 2Y,i =cos(2q) Y,i ); L i-1,m L i,1 and L i,2 The product of the vectors corresponding to the three rods is given by equation (4): Because of L i-1,m L i,1 and L i,2 The vectors corresponding to the three rods satisfy the condition that the mixed product is zero, therefore, L i-1,m L i,1 and L i,2 Located in the same plane, the PYYP configuration (or YPPY configuration) linkage segment of the cable-driven robotic arm has planar motion properties.

3. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S100, the PYYP configuration (or YPPY configuration) linkage segment of the cable-driven robotic arm possesses equal-angle motion properties, specifically: The choice of coordinate system is independent of the angle between the two vectors, therefore in L i-1,m In the local coordinate system, L i-1,m and L i,1 The included angle between them is given by equation (5); In L i,1 In the local coordinate system, L i,1 and L i,2 The included angle between them is given by equation (6): The results of the two equations above are equal, that is, L i-1,m and L i,1 The included angle between and L i,1 and L i,2 The included angles between them are equal; similarly, L i,j With L i,j+1 The included angle and L i,j+1 With L i,j+2 The included angles are equal.

4. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S100, The PYYP configuration linkage segment is equivalent to the RPPP configuration linkage segment. In the RPPP configuration, R indicates that the first revolute axis is the Roll axis, which is the x-axis in the world coordinate system. The following three Ps indicate that the two adjacent arms of the equivalent linkage segment are connected by a revolute joint, the direction of the rotation axis is the Pitch axis, and the angle of motion around the Pitch axis is equal. The RPPP configuration equivalent linkage segment has the same number of booms, active degrees of freedom, and workspace as the PYYP configuration linkage segment, specifically: L′ i,1 First at L′ i-1,m Rotate about the x-axis in the local coordinate system Then in U′ i,1 Rotate θ around the y-axis in the local coordinate system e,i Get L′ i,1 Relative to L′ i-1,m Location i-1,m L′ i,1 It is expressed as equation (7): in, for The cosine value / sine value, i.e. c θ,i / s θ,i For θ e,i The cosine value / sine value, i.e., c θ,i =cosθ e,i / s θ,i =sinθ e,i ; L′ i,2 At L′ i,1 Rotate θ around the y-axis in the local coordinate system e,i It is expressed as equation (8): Then i,1 L′ i,2 Convert to L′ i-1,m In the local coordinate system, it is expressed as equation (9): Calculate L′ i-1,m L′ i,1 and L′ i,2 The vector hybrid product corresponding to the three rods is expressed as equation (10): Get L′ i-1,m L′ i,1 and L′ i,2 The vectors corresponding to the three rods satisfy the condition that the mixed product is zero, therefore L′ i-1,m L′ i,1 and L′ i,2 The three rods are in the same plane, meaning that the equivalent RPPP configuration linkage segment also has planar motion properties; at the same time, the angles of motion of each arm rod around the pitch axis in the equivalent RPPP configuration linkage segment are equal, so it also has equal angle motion properties. Therefore, the kinematic characteristics of the linkage segment in the RPPP configuration are consistent with those in the PYYP configuration, so the ideal kinematics of the linkage segment in the RPPP configuration and the PYYP configuration are equivalent.

5. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S200 The equivalent linkage segment of the RPPP configuration is equivalent to a spatial arc inscribed within all the booms of the equivalent linkage segment of the RPPP configuration and two straight segments tangent to the beginning and end of the spatial arc, with a length of half the boom length. The segment angle The angle between the plane containing the equivalent circular arc corresponding to the equivalent linkage segment of the RPPP configuration and the z-axis of the local coordinate system of the equivalent linkage segment of the RPPP configuration; the equivalent central angle θ seg,i The central angle of the equivalent circular arc corresponding to the equivalent linkage segment of the RPPP configuration.

6. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S300, The multi-layer state space relationship includes drive space, rope space, segment space and task space. The drive space and rope space are mapped one-to-one, which is summarized as the overall analysis of the rope drive space. When the rope length is determined, the overall configuration of the corresponding linkage segment, that is, the segment space parameters, can also be determined. Similarly, once the segment space parameters of the linkage segment are determined, the length of the drive rope can be calculated through the configuration of the linkage segment; finally, multiple linkage segments are connected in series to form a complete rope-driven segmented linkage robotic arm, thus mapping from the segment space to the Cartesian space.

7. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S300, For the kinematics of the drive space-segment space, the rope length corresponding to the same rope hole position at each cross-axis joint in the same linkage segment is equal due to symmetry. Therefore, the sum of the rope lengths of each drive rope at the joints in a single linkage segment is equal to the number of joints multiplied by the rope length of the drive rope at the first joint. The PY joint can be equivalent to an RP configuration joint. In the PY configuration joint, the homogeneous transformation matrix from the local coordinate system {i,j-1} to the local coordinate system {i,j} is... i,j-1 T PY,i,j For equation (11): Where h is the distance from the center of the cross axis to the plane where the wiring disk is located; Trans(·,·,·) is the translation transformation matrix, and the three parameters represent the components of the translation distance along the x-axis, y-axis and z-axis, respectively; In RP configuration joints, the homogeneous transformation matrix from local coordinate system {i,j-1} to local coordinate system {i,j} is... i,j- 1 T RP,i,j For equation (12): Among them, s θ,i / c θ,i For θ e,i The sine / cosine value, θ e,i =θ seg,i / m i m i The number of cross axes in the i-th segment; for The sine / cosine value; When the PY joint is equivalent to an RP configuration joint, their homogeneous transformation matrices are the same. Therefore, we can obtain the q when the PY joint is equivalent to an RP configuration joint. P,i q Y,i and θ seg,i , The relationship is given by equation (13): In an RP-configuration joint, since the same rope passes through the same hole in each local coordinate system, when the PY joint is equivalent to an RP-configuration joint, the rod L... i,j-1 The position A of the rope hole in the local coordinate system {i,j-1} PY-RP,i,k For equation (14): i,j-1 A PY-RP,i,k =[0 r cable cosβ i,k r cable sinβ i,k ] T (14) Where, r cable The radius of the rope hole position; L rod i,j The position B of the rope hole in the local coordinate system {i,j} PY-RP,i,k For equation (15): i,j B PY-RP,i,k =[0 r cable cosβ i,k r cable sinβ i,k ] T (15) The position of the rope hole in the local coordinate system {i,j} i,j B PY-RP,i,k Rewritten as homogeneous coordinates For equation (16): And through homogeneous transformation matrix i,j-1 T RP,i,j The position of the rope hole in the local coordinate system {i,j-1} is obtained. i,j-1 B PY-RP,,i,k homogeneous coordinates For equation (17): Where h is the rod L i,j-1 or rod L i,j Length between the center point of the wiring disk and the center point of the cross axis; s βi,k / c βi,k For β i,k The sine / cosine value, i.e., s βi,k =sinβ i,k / c βi,k =cosβ i,k ; This allows us to obtain the position of the rope hole in the local coordinate system {i,j-1} of the PY configuration joint. i,j-1 B PY-RP,i,k For equation (18): Therefore, when the PY configuration joint of the i-th segment is equivalent to the RP configuration joint, the rope length l corresponding to the k-th rope hole position at the joint is... PY-RP,i,k For equation (19): For the k-th rope fixed to the wiring disk of the last arm of the i-th linkage segment, it passes through all joints of the i-th linkage segment, as well as all joints and all arms of the preceding i-1 linkage segments. Therefore, the total rope length l cable,i,k For equation (20): The above equation is the inverse kinematics of the driving space-segment space, m u is the number of cross-axis joints in the u-th linkage segment; l is the distance between the center points of two adjacent cross-axis joints.

8. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S300, for the driving space-segment space kinematics, the segmented linkage manipulator obtained by connecting multiple linkage segments in series is replaced by the first and last straight lines and several interconnected circular arcs in the middle after being equivalent to circular arcs. The length of the first straight segment is given by equation (21): Among them, P base Position of the segmented linkage manipulator base; P tgc,0 This represents the position of the tangent point between the first straight line and the arc after the arc is equivalent. The length of the straight line at the end is given by equation (22): Among them, P tgc,N For the equivalent arc, P is the location of the point where the tail line is tangent to the arc. e This refers to the end position of the segmented linkage manipulator; The equivalent arc length s of the i-th segment in the middle i For equation (23): To facilitate the solution of forward kinematics, the intersection of two tangents at the same endpoint of the arc is denoted as the virtual joint point P of that arc segment. jv,i Therefore, the arc is equivalent to two points intersecting at the virtual joint P. jv,i Virtual joint rod and Its direction is the same as the two tangents, and its length is the length l from the endpoint to the virtual joint. v,i For equation (24): After a linkage segment containing an even number of sub-joints is equivalent to a circular arc, the local coordinate system of the i-th circular arc is {i-1}. c homogeneous transformation matrix to its endpoints i-1,c T even,i,c For equation (25): Homogeneous transformation matrix of a single linkage segment 0 T seg,1 For equation (26): 0 T seg,1 =Trans(x,l / 2) 0,c T even,1,c Trans(x,l / 2) (26) Homogeneous transformation matrix of the entire linkage segment 0 T seg,3 For equation (27): Equation (27) is the forward kinematics of the segment space-task space of the rope-driven segmented linkage robotic arm.

9. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S400, the inverse kinematics of the rope-driven segmented linkage robot arm segment space-task space is solved based on the improved forward and backward inverse kinematics method. The solution process includes the following steps: Starting from a given target end position and attitude, initialize the counter and calculate the current end error; If the error is within the acceptable range, try to analyze the inverse kinematics solution; If a solution is found, the process ends; otherwise, it continues with the iteration process. In each iteration, a backward iteration is performed first, starting from the base, calculating the direction and equivalent bending angle of the next segment segment by segment, determining whether the maximum allowable value is exceeded, and updating the virtual rod length and direction accordingly, until the end. Then, a forward iteration is performed, that is, starting from the end, the direction and bending angle of each segment are updated towards the base segment by segment. The actual or maximum value is used to update based on whether the angle exceeds the limit, and the position of the segment is recalculated until it returns to the base. After completing one round of updates, recalculate the terminal error and determine whether the conditions are met or the maximum number of iterations is exceeded. Otherwise, continue to the next round of iterations until a solution that meets the error tolerance is obtained or the iteration terminates.

10. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 9, characterized in that, The forward reaching iteration process involves calculating the position P of the tangent point of the arc sequentially from the last segment to the first segment. tgc,i , pointing to t tgc,i and central angle θ seg,i For the last segment, the end position P of the linkage segment e,d and pointing to t e,d Given the given quantities, and the end segment containing a rod of length l / 2 tangent to the last arc, we first calculate the position P of the point where the last arc is tangent to the l / 2 rod length. tgc,m ; For the remaining segments, the position P of the tangent point between the (i+1)th and the ith arc is... tgc,i It is calculated from the (i+1)th segment, as shown in equation (28): By P tgc,i The virtual joint position P of the i-th arc is obtained by combining the tangency condition. v,i For equation (29): P v,i =P tgc,i -l v,i t tgc,i (29) The direction t of the tangent point between the i-th arc and the (i-1)-th arc can be calculated from the virtual joint positions of the i-th arc and the (i-1)-th arc. tgc,i-1 For example, equation (30); However, for the first segment, the orientation of the base remains unchanged, so no calculation is needed. Instead, calculate the central angle θ of the i-th segment. seg,i For equation (31): When the central angle θ of the i-th segment seg,i Less than or equal to θ seg,MAX Then, update the virtual rod length l of the i-th segment according to formula (24). v,i And the newly calculated rod length l v,i Substituting into equations (29), (30), and (31), update the virtual joint position P of the i-th arc. v,i The direction of the (i-1)th arc tangent point t tgc,i-1 and the central angle θ of the i-th segment seg,i Finally, the position P of the tangent point between the i-th arc and the (i-1)-th arc is obtained. tgc,i-1 For equation (32): P tgc,i-1 =P v,i -l v,i t tgc,i-1 (32) When the central angle θ of the i-th segment seg,i Greater than θ seg,MAX At that time, the rope-driven segmented linkage robotic arm cannot move to this configuration, therefore it is necessary to restrict the solution angle to θ. seg,MAX Calculate the direction t of the tangent point between the i-th arc and the (i-1)-th arc. tgc,i-1 For equation (33): in, 0 R v,i For forward iteration, the virtual joint pose of the i-th segment is as shown in equation (34); The x-axis of the local coordinate system of the i-th virtual joint during forward iteration; Then update the virtual rod length l of the i-th segment according to equation (24). v,i And the newly calculated rod length l v,i Substituting into equations (29), (30), and (31), update the virtual joint position P of the i-th arc. v,i The direction of the (i-1)th arc tangent point t tgc,i-1 and the central angle θ of the i-th segment seg,i Finally, the position P of the (i-1)th arc tangent point is obtained according to equation (32). tgc,i-1 ; The position P of the tangent point between the first segment and the arc of half the rod length tgc,0 The base position P can be calculated. base As in equation (35): At this point, the base position may not be zero, so it is necessary to move the base position back to its original position through backward iteration.

11. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 9, characterized in that, The backward iteration process involves calculating the position P of the tangent point of the arc from the first segment to the last segment in the forward iteration. tgc,i , pointing to t tgc,i and central angle θ seg,i For the first segment, the base position P of the linkage segment base and pointing to t base Given that the base also has a rod of length l / 2 tangent to the first arc, we first calculate the position P of the point where the first arc tangent to the l / 2 rod. tgc,0 For the remaining i segments, the position P of the point where the (i-1)th arc is tangent to the i-th arc is... tgc,i-1 It is calculated from the (i-1)th segment, as shown in equation (36): By P tgc,i-1 The virtual joint position P of the i-th arc is obtained from the tangency point condition. v,i As in equation (37): P v,i =P tgc,i-1 +l v,i t tgc,i-1 (37) The direction t of the tangent point between the i-th and (i+1)-th arcs can be calculated from the virtual joint positions of the i-th and (i+1)-th arcs. tgc,i As in equation (38): For the last segment, the direction of the end point remains unchanged, so no calculation is needed; calculate the central angle θ of the i-th segment. seg,i As in equation (31); When the central angle θ of the i-th segment seg,i Less than or equal to θ seg,MAX Then, update the virtual rod length l of the i-th segment according to formula (24). v,i And the newly calculated rod length l v,i Substituting into equations (37), (38), and (31), update the virtual joint position P of the i-th arc. v,i The direction of the i-th arc tangent point t tgc,i and the central angle θ of the i-th segment seg,i Finally, the position P of the tangent point between the i-th arc and the (i+1)-th arc is obtained. tgc,i As in equation (39): P tgc,i =P v,i +l v,i t tgc , i (39) When the central angle θ of the i-th segment seg,i Greater than θ seg,MAX At that time, with θ seg,MAX Calculate the direction t of the tangent point between the i-th arc and the (i+1)-th arc. tgc,i As in equation (40): in, 0 R′ v,i For backward iteration, the virtual joint pose of the i-th segment is as shown in equation (41): Rotate θ around the z′ axis of the local coordinate system of the i-th virtual joint during backward iteration. seg,MAX The rotation matrix; Let x′ be the local coordinate system of the i-th virtual joint during backward iteration; The position P of the last segment tangent to the arc with half the length of the rod tgc,N The end position P can be calculated. e,cal As shown in equation (42): After one forward and one backward iteration, it becomes as shown in equation (43): Calculate the end position error e p and end pointing error e t ; After repeated forward and backward iterations, if the end-effector pose error of the entire arm meets the iteration parameter requirement e... p,MAX and e t,MAX This gives the central angle θ of each segment. seg,i , the direction vector t of the tangent point of the circular arc seg,i and position P seg,i .

12. The method for describing the equivalent state of a linkage segment and modeling its kinematics according to claim 11, characterized in that, In step S400, the result of the inverse solution still needs to be transformed into a segment space, and the central angle θ of each segment needs to be calculated. seg,i The angle of the segment can be directly calculated during the inverse solution, while... To change the x-direction vector t of the i-th segment endpoint... seg,i The results are obtained by transforming the coordinates of each segment into the local coordinate system, as shown in equation (44): in, i t seg,i,y / i t seg,i,z for i t seg,i The components in the y / z directions, and i t i+1 =( 0 R i ) T0 t i+1 , 0 R i Let be the attitude matrix of the i-th endpoint; After calculating the segment space parameters of the i-th segment, the pose of the i-th segment is obtained. 0 T i Calculate the segment space parameters of the (i+1)th segment. and θ seg,i This process continues until all segment spatial parameters of the rope-driven segmented linkage robotic arm are calculated, thus obtaining the inverse kinematics of the segment space-task space of the rope-driven segmented linkage robotic arm.

13. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S500, The workspace of a single linkage segment is a curved surface. However, in the actual movement of the linkage segment, due to the existence of unequal angle linkage, the expected workspace and the actual workspace surface do not overlap. In order to make the linkage segment under unequal angle linkage also apply the segment space state description and make the equivalent motion configuration closest to the actual motion configuration, the linkage segment under unequal angle linkage is equivalently segmented with the goal of minimizing the error of the segment endpoint position of the linkage segment before and after equivalence. First, determine the actual segment endpoint position P. seg,a,i Project onto the ideal workspace, and then use the projection point position P seg,p,i and the expected segment endpoint position P seg,d,i The component of the difference is taken as Δy seg,i and △z seg,i After segment closed-loop control, the projected point position coincides with the desired segment endpoint position, i.e., the actual segment endpoint position P after segment closed-loop control. seg,c,i With respect to the desired segment endpoint position P seg,d,i Closest distance; The approximate curve of the ideal workspace is obtained using the fitting method, and the actual segment endpoint position P is used. seg,a,i The position P of the projection point is obtained by projecting the fitted workspace curve. seg,p,i Then, the position P of the projection point on the ideal motion plane. seg,p,i Transformed into two state variables, the segment angle, in segment space and central angle θ seg,i ; In cases of non-uniform angle linkage, the actual end position is projected onto the desired workspace to obtain the projection point position, and the segment space parameters corresponding to the projection point position are further calculated. and θ e,i As the equivalent segment spatial parameters of the linkage segment under non-equiangular linkage, the equivalent segment spatial kinematic model parameters are used to replace the joint angles of each cross-axis joint in the P and Y directions in the actual linkage segment, in order to characterize the overall motion of each rope-driven linkage segment, without considering the specific linkage characteristics of the linkage segment or the linkage error of each joint. The workspace of a single linkage segment is a non-circular curve, and its parametric equation can be written as shown in equation (45): An approximate curve of the ideal workspace is obtained using a fitting method, limiting the range of motion of a single joint within a linkage segment to |θ. seg,i |≤90°, discretize its workspace into 1000 points (u j ,x j ), j = 1, 2, ..., 1000, using n fit By fitting a polynomial of degree to its workspace, and by scaling the workspace points to the center during fitting and neglecting small values ​​of the fitting coefficients, an approximate workspace function x(u) in the ideal motion plane can be obtained, as shown in equation (46): Where, k i For u i The fitting coefficient, u mean / u std x represents the mean / standard deviation of the u-axis of the workspace points after discretization of the workspace. mean / x std The x-axis value is the average / standard deviation of the workspace points after discretization. Using a geometric method, the actual segment endpoints are projected onto the fitted workspace curve to obtain the projection point position. The fitted workspace curve is located at the projection point P. P Calculate the coordinates of the projection point based on the slope relationship with the surrounding area. The position P of the projection point on the ideal motion plane seg,p,i Transformed into two state variables, the segment angle, in segment space and central angle θ seg,i The position P of the endpoint of the linkage segment after the arc is equivalent seg,i The length projected onto the u-axis in the plane of motion is as shown in equation (50): Among them, R i Let R be the equivalent radius of the i-th segment. i =l / 2tan(θ) seg,i / 2m i ); Calculate the y-components of the segment endpoints on the y-axis and z-axis. seg,p,i and z seg,p,i With segmental angle and central angle θ seg,i As in equation (51): The segment angle is obtained by inverse solution of equation (51). As in equation (52): In equation (50), u(θ seg,i The expression contains several trigonometric nonlinear terms, making it difficult to use the segment endpoints P. seg,i The central angle θ is obtained by inversely solving the length projected onto the u-axis within the plane of motion. seg,i Therefore, the universal trigonometric formula is used to convert it into an algebraic equation, and then Newton's iteration method is used to obtain an approximate solution to the equation, thereby obtaining the central angle. Taking a segment with 4 joints as an example, u4(θ) seg,i The expression is as shown in equation (53): Substitute θ using the universal trigonometric function formula. seg,i That is, t = tan(θ) seg,i / 8), as in equation (54): The solution t of equation (54) can be obtained using Newton's iteration method. * The central angle is obtained. As in equation (55): Extend equation (55) to a segment m i In the case of a single joint, the universal trigonometric function formula can also be used to substitute u. seg,p,i (θ seg,i θ in ) seg,i That is, t = tan(θ) seg,i / 2m i The solution t of the equation can be obtained using Newton's iteration method. * Furthermore, the central angle can be obtained. As in equation (56): The segment endpoint position control method based on the segment endpoint equivalence method of the linkage segment ensures the segment endpoint position accuracy to the greatest extent and at the same time ensures the segment endpoint attitude accuracy as much as possible, so that the overall actual configuration of the single linkage segment is close to the desired configuration.

14. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, Step S500 includes the following steps: Starting with segment endpoint trajectory planning, the desired segment endpoint positions P are obtained. seg,d,i , and the actual position P at the end of the current segment seg,p,i The position error ΔP is obtained by comparison; The attitude error was calculated using Cartesian differential kinematics. and angular error △θ seg,i The input is sent to the segment space control law module, and the desired attitude parameters are obtained by combining the Cartesian forward kinematics results. θ seg,d,i The control law output is used to solve for the desired rope drive length l. cable,i,k ; The rope drive kinematics is converted into the motion target of each motor, and the motor closed-loop control module drives the rope to pull the specific mechanism, ultimately controlling the joint movement of each drive segment; The joint positions of each segment are fed back through the encoder, and the current actual end position P of the segment is obtained through forward kinematics calculation. seg,a,i Used to update the equivalent position P of the segment endpoints in the closed-loop feedback loop. seg,p,i .

15. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S600, the control law for the corresponding segment endpoint position control is derived. The total differential of equation (51) is obtained, yielding equation (57): Where A is u(θ) seg,i ) for θ seg,i The derivative of J is given in equation (58). Φ-P The Jacobian matrix represents the velocity from the segment space parameter velocity to the velocity at the segment endpoint position of a single linked segment; Multiply both sides of equation (57) by the inverse of the Jacobian matrix of the velocity at the end point of a single linked segment from the velocity of the segment space parameter. Equation (59) is obtained. Substituting the segment endpoint error ΔP into equation (59) yields equation (59). seg,i The components Δy in the y and z directions seg,i , △z seg,i The segment angle error can be obtained. and the error of the central angle △θ seg,i , Positional error ΔP at the end of the segment seg,i The specific form is as shown in equation (60). Substituting equations (49) and (60) into equation (57), the segment endpoint position error is converted into segment surface angle error. and the error of the central angle △θ seg,i .

16. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S500, a segment space controller is provided in the segment endpoint position control, and its input signal is provided by... and △θ seg,i The system consists of a PID controller module, where each signal is processed by its own PID controller module, which includes proportional, integral, and derivative components. The processed signals are then combined by an adder into a single comprehensive control signal output.

17. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 16, characterized in that, The segment space controller combines PID control and feedforward control. By superimposing the two control methods, it improves the dynamic response of the system and reduces steady-state error. The specific control laws are Equations (61) and (62): in, For the corresponding PID parameters, K θ,P K θ,I K θ,D For the corresponding θ seg,pid,i PID parameters; The output of the segment space controller obtained from equations (61) and (62) is converted into the length of the three ropes of the linkage segment through the segment space-rope space kinematics, and then input into the motor closed-loop controller; while the motor assembly of the rope-driven segment linkage robot arm uses a motor controller, so the specific control law of the motor closed-loop controller will not be described in detail.

18. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, Step S600 includes the following steps: The desired end-effector pose X is obtained by trajectory planning in the task space according to task requirements. e，d Then, the expected values ​​of the segment space parameters are obtained through the improved FABRIKc inverse kinematics. θ seg,d,i As a feedforward quantity for kinematic control; The value q is measured using the joint encoder within the linkage section. a,i Calculate the current actual end-effector pose X e,a , and the desired end pose X e,d The end-effector pose error ΔX was obtained after comparison. e Then, the pseudo-inverse of the Jacobian matrix from the segment space state parameter velocity to the terminal velocity is obtained. Decompose it into segment space to obtain segment angle error and the error of the central angle △θ seg,i , Combining the feedforward results calculated earlier θ seg,d,i The kinematic relationship between the segment space and the rope-driven space is transformed into the motion of each rope. The motion of the rope-driven segmented linkage robotic arm is further driven by the closed-loop control of the motor, thus realizing closed-loop control in the segment space of the rope-driven segmented linkage robotic arm.

19. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 18, characterized in that, In step S600, The pose error of the segmented linkage manipulator is decomposed into the segment space state parameters of each segment, and closed-loop control is performed in the segment space to achieve high-precision control of the end-effector pose of the rope-driven segmented linkage manipulator. First, trajectory planning is performed in the task space according to task requirements to obtain the desired end-effector pose X. e,d Then, the expected values ​​of the segment space parameters are obtained through the improved FABRIKc inverse kinematics. θ seg,d,i As a feedforward for kinematic control; then the joint encoder value q in the linkage segment is measured. a,i Calculate the current actual end-effector pose X e,a , and the desired end pose X e,d The end-effector pose error ΔX was obtained after comparison. e Then, the pseudo-inverse of the Jacobian matrix from the segment space state parameter velocity to the terminal velocity is obtained. Decompose it into segment space to obtain segment angle error and the error of the central angle △θ seg,i Combining the feedforward results calculated earlier θ seg,d,i The kinematic relationship between the segment space and the rope-driven space is transformed into the motion of each rope. The motion of the rope-driven segmented linkage robot arm is further driven by the closed-loop control of the motor, thereby realizing closed-loop control in the segment space of the rope-driven segmented linkage robot arm and thus realizing the decomposed motion control of the end effector of the rope-driven segmented linkage robot arm.

20. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S600, By acquiring the angle values ​​q of each joint within the linkage segment in real time P,a,i and q Y,a,i And calculate the end-effector pose error ΔX in real time. e Then, based on segment space-task space differential kinematics, the end-effector pose error ΔX is calculated. e Decomposed into segment angle error and the error of the central angle △θ seg,i The end-effector pose error ΔX is eliminated by using a whole-arm decomposition motion control method. e To achieve high-precision control of the end-effector posture of the rope-driven segmented linkage robotic arm; Wherein, the endpoint position of the ideal segment i is P seg,d,i The actual endpoint position of the segment is P. seg,a,i Its error △P i As in equation (63): △P i =P seg,d,i -P seg,a,i (63) After being accumulated to the end of the boom, the positional error e at the end of the boom p As in equation (64): The equivalent end-effector posture of the rope-driven segmented linkage robotic arm can be denoted as R using a rotation matrix. e,d The actual end-effector pose is denoted as R. e,a Then the position error e at the end of the entire arm p and attitude error e o As in equations (64) and (65): Where, n e,d / o e,d / a e,d For R e,d The first / second / third column, n e,a / o e,a / a e,a For R e,a The first / second / third column; The encoder is used to obtain the angle values ​​q of each joint in the linkage segment in real time. P,a,i and q Y,a,i The actual position P of the segment terminal is obtained through the DH method. seg,a,i and the actual posture R of the end effector of the entire arm e,a It can also be used to set the current desired segment space state parameters. and θ seg,d,i The positions P of the endpoints of the linked segments are calculated using segment-task-space forward kinematics. seg,d,i Equivalent posture R of the entire arm end effector e,d Therefore, the pose error e of the end effector can be obtained in real time. p and e o ; Based on segment-space-task-space differential kinematics, the end-effector pose error ΔX can be calculated. e Decomposed into segment angle error and the error of the central angle △θ seg,i As shown in equation (66): in, Let be the rate of change of the segment space state parameter of the linkage segment with respect to time, as defined in equation (67). The rate of change of the segment space state parameters of the linked segment with respect to time. and linear velocity v at the end e and angular velocity ω e The pseudo-inverse of the Jacobian matrix.

21. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S600, For the i-th linkage segment, the segment angular velocity For the end linear velocity and angular velocity The effect is shown in equation (68): in, For the angular velocity of the segment Caused terminal linear velocity / terminal angular velocity; x i-1 Let r be the x-axis of the local coordinate system {i-1}; i-1,N It is the vector from the end point of the (i-1)th linkage segment to the end position of the rope-driven segmented linkage robot arm; For the i-th linkage segment, the central angular velocity For the terminal linear velocity v e and angular velocity ω e The effect needs to be obtained through differentiation; The following derives the coordinate system of the motion plane of the i-th linkage segment. Inner central angular velocity For the terminal linear velocity v e and angular velocity ω e Due to the influence of the angular velocity of the center of the i-th linkage segment The final linear velocity v is generated only in the motion plane of the i-th linkage segment. e,θ,i Therefore, the end position P of the rope-driven segmented linkage robotic arm is... e P is obtained by projecting it onto the motion plane of the i-th linkage segment. e,xou,i P e,xou,i linear velocity at point P e The linear velocity is the same at point P; e,xou,i In the local coordinate system The coordinates in the equation are as shown in equation (69): in, Let be the projection length of the vector from the end point of the i-th linkage segment to the end position of the cable-driven segmented linkage robot arm onto the motion plane of the i-th linkage segment, specifically in the form of equation (70): in, for The x / z direction components, Specifically, it is as formula (71): Differentiating equation (69) yields P e,xou,i linear velocity at As in equation (72): From equation (72), we can know the angular velocity of the center of the i-th linkage segment. The resulting terminal linear velocity v e,θ,i As in equation (73): Wherein, the coefficient K x K u The expression is as shown in equation (74): The central angular velocity of the i-th linkage segment The resulting terminal angular velocity ω e,θ,i We obtain, as shown in equation (75): in, Local coordinate system The v-axis; Combining equations (68), (73), and (75), and rewriting them in matrix form, we obtain the segment angular velocity of the i-th linkage segment. and the angular velocity of the center With the terminal linear velocity v e and angular velocity ω e The relationship is as shown in equation (76): in, J θ,i Let i be the angular velocity of the i-th linkage segment. angular velocity of the center linear velocity v at the end e and angular velocity ω e The generalized transmission ratio, i.e., the column of the velocity Jacobian matrix from the i-th segment to the end, is specifically expressed as in equation (77): Extending equation (77) to all linked segments yields the segmental angular velocities of the linked segments. angular velocity of the center linear velocity v at the end e and angular velocity ω e Jacobian matrix J Φ-E As in equation (78): The end pose error ΔX e Substituting equation (78) into the equation, the end-effector pose error ΔX can be calculated. e The error is decomposed into segmental spatial state parameter error ΔΦ, and then the proposed whole-arm decomposed motion control method of the rope-driven segmented linkage robotic arm is used to eliminate the end-effector pose error ΔX. e As shown in equation (79): in, For J Φ-E The false rebellion.

22. The method for describing the equivalent state and kinematic modeling of the linkage segment according to claim 1, characterized in that, In step S600, in order to accelerate the end pose convergence speed of the rope-driven segmented linkage robot arm and avoid the instability of the pseudo-inverse solution of the decomposed motion controller, the solution of the whole arm decomposed motion control was improved by using the virtual joint method, damped least squares method, singular value truncation method and Jacobi matrix block method. When the rope-driven segmented linkage robotic arm has strict requirements for the desired end-effector orientation but not strict requirements for the desired posture, adding a virtual roll joint at the end-effector can accelerate its convergence speed to the desired end-effector position and orientation. After adding a virtual joint at the end, the Jacobian matrix J Φ-E A column needs to be added to transform it into an extended Jacobian matrix J. Φ-E,EXT As in equation (80): J Φ-E,EXT =[J Φ-E J v ] (80) Among them, J v The virtual joint motion velocity to the end-effector linear velocity v e and angular velocity ω e The generalized transmission ratio is given by equation (81): Where, x N Let {N} be the x-axis of the local coordinate system; At this point, the end-effector pose error ΔX e The formula for decomposing into segment space is shown in equation (82): Where, △Φ EXT The extended segment space parameter error is specifically in the form of equation (83): in, 23. The method for describing the equivalent state of a linkage segment and modeling its kinematics according to claim 22, characterized in that, In practical control, the Jacobian matrix J Φ-E Or extended Jacobian matrix J Φ-E,EXT In certain configurations of the cable-driven segmented linkage manipulator, the matrix becomes ill-conditioned. At this time, the pseudo-inverse solution of the linear equation system obtained by solving equation (79) or equation (82) will be numerically unstable. Therefore, it is necessary to deal with the singularity problem of the Jacobian matrix to avoid the cable-driven segmented linkage manipulator from generating violent shaking during the end pose closed-loop control. To overcome the above problems, we can calculate the Jacobian matrix J. Φ-E Or extended Jacobian matrix J Φ-E,EXT The pseudo-reverse time is introduced by adding a damping factor λ, i.e., using damped least squares to solve the linear equation system, to expand the Jacobian matrix J. Φ-E,EXT For example, after adding the damping factor λ, the extended Jacobian matrix J Φ-E,EXT pseudo-reversal As in equation (84): Where I7 is the identity matrix with a rank of 7; The damping factor λ is equivalent to adding a constraint to the solution of a system of linear equations using the least squares method, limiting the size of the solution and thus avoiding computational instability. By adjusting the size of the damping factor λ, accuracy and stability can be balanced. A larger damping factor λ will make the solution smoother, but may deviate from the true solution; a smaller damping factor λ is closer to the minimum norm solution. The selection of the damping factor λ usually requires experience or adjustment; here, the extended Jacobian matrix J is chosen. Φ-E,EXT The largest element is used as the damping factor; Furthermore, singular value decomposition can be used to address the instability of pseudo-inverse solutions of ill-conditioned matrices, thereby extending the Jacobian matrix J. Φ-E,EXT For example, when some singular values ​​in the matrix are small, the error is easily amplified when calculating the pseudo-inverse, leading to instability in the pseudo-inverse solution. Therefore, J Φ-E,EXT After singular value decomposition, smaller singular values ​​need to be truncated before calculating the pseudoinverse, thus extending the Jacobian matrix J. Φ-E,EXT Perform singular value decomposition, as shown in equation (85): Among them, U Φ-E,EXT , Σ Φ-E,EXT and V Φ-E,EXT For J Φ-E,EXT The three matrices obtained from singular value decomposition; For matrix Σ Φ-E,EXT It is necessary to truncate the smaller singular values ​​and construct its pseudo-inverse matrix. As in equation (86): in, For matrix The element in the i-th row and j-th column has the specific value as shown in equation (87): Where τ is the singular value truncation threshold, τ = 10 -3 ; Construct the pseudo-inverse matrix with truncated singular values ​​using equation (86). As in equation (88): Furthermore, substituting equation (88) into equation (82) yields a more stable pseudo-inverse solution; When the end position of the cable-driven segmented linkage robotic arm changes within a small range of space, its end posture also changes relatively little. Therefore, based on this characteristic of the cable-driven segmented linkage robotic arm, positional accuracy can be prioritized while slightly sacrificing posture accuracy. This also solves the problem of pseudo-inverse instability caused by ill-conditioned matrices, as shown in equation (89): Extended Jacobian matrix J Φ-E,EXT It can be divided into two matrices J, upper and lower. Φ-v,EXT and J Φ-ω,EXT J Φ-v,EXT and J Φ-ω,EXT Each of them has a small condition number and is not an ill-conditioned matrix. Therefore, using J Φ-v,EXT The end position error e can be separated. p Decomposing into segment space yields a stable pseudo-inverse solution △Φ′, as shown in equation (90): in, For J Φ-v,EXT The pseudo-inverse matrix; Since Equation (90) can only guarantee the end position accuracy of the rope-driven segmented linkage robot arm, but cannot guarantee the end attitude accuracy, it can only be used as a singularity avoidance method after both the damped least squares solution corresponding to Equation (84) and the singular value decomposition method corresponding to Equation (88) have failed, making the movement of the rope-driven segmented linkage robot arm more stable and safe.

24. A rope-driven segmented linkage robotic arm, characterized in that, The cable-driven segmented linkage robotic arm includes: A drive box, the drive box including a motor and a transmission mechanism, the transmission mechanism including at least a rope puller, a linear guide rail and a lead screw; The linkage manipulator includes multiple linkage arm segments connected in series. Each linkage arm segment has two degrees of freedom, and an end effector is provided on the linkage arm segment at the end.

25. The rope-driven segmented linkage robotic arm according to claim 24, characterized in that, The linkage arm of the rope-driven segmented linkage robotic arm adopts a hybrid active-passive drive, that is, the linear drive module directly changes the length of the drive rope to realize the movement of the segment endpoints of the linkage segment, and the passive linkage rope forms a linkage mechanism to drive the joints in the linkage segment to achieve equal-angle movement. Through the combined action of the active rope and the passive rope, the motion control of the segmented linkage robotic arm is realized. In terms of active drive, the rope-driven segmented linkage robotic arm realizes the movement of the linkage segment by changing the length of the three active drive ropes of the corresponding linkage segment; one end of the drive rope is fixed on the wiring disk at the end of the linkage segment, and the other end passes through the rope-passing hole designed on the wiring disk of each arm in the linkage segment and is fixed on the rope puller of the linear drive module in the drive box. In terms of passive linkage, the linkage structure is an 8-shaped linkage structure, which includes a small 8-shaped linkage structure and a large 8-shaped linkage structure. Both the small 8-shaped linkage structure and the large 8-shaped linkage structure make the movement angle of two adjacent cross axes the same in the linkage direction.

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