CN114347005B - Rope traction parallel robot continuous reconstruction planning method capable of avoiding obstacles - Google Patents

Rope traction parallel robot continuous reconstruction planning method capable of avoiding obstacles Download PDF

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CN114347005B
CN114347005B CN202210268127.6A CN202210268127A CN114347005B CN 114347005 B CN114347005 B CN 114347005B CN 202210268127 A CN202210268127 A CN 202210268127A CN 114347005 B CN114347005 B CN 114347005B
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rope
point
parallel robot
space
guide pulley
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CN114347005A (en
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尚伟伟
王晓燕
张彬
张飞
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University of Science and Technology of China USTC
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University of Science and Technology of China USTC
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/003—Programme-controlled manipulators having parallel kinematics
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/10—Programme-controlled manipulators characterised by positioning means for manipulator elements
    • B25J9/104—Programme-controlled manipulators characterised by positioning means for manipulator elements with cables, chains or ribbons

Abstract

The invention discloses a continuous reconstruction planning method for a rope traction parallel robot capable of avoiding obstacles, which comprises the following steps: 1) introducing a virtual rope leading-out point according to the position relation of the rope and the guide pulley, constructing a robot kinematic model and considering a closed chain structure of the robot to obtain a closed chain constraint equation; 2) according to the manifold generated by the closed-chain constraint equation, obtaining an orthogonal substrate for establishing a cutting space by adopting a mode of locally approximating the manifold by the cutting space; 3) performing collision-free reconstruction path search in the manifold cutting space; 4) and solving the actual rope length and the position of the actual rope leading-out point according to the reconstruction path to obtain the configuration change of the robot, thereby realizing collision-free continuous reconstruction planning of the robot. According to the invention, the obstacle avoidance capability of the rope traction parallel robot is improved by performing continuous reconstruction planning on the rope traction parallel robot.

Description

Rope traction parallel robot continuous reconstruction planning method capable of avoiding obstacles
Technical Field
The invention relates to the field of configuration planning of rope traction parallel robots, in particular to a continuous reconfiguration planning method of a rope traction parallel robot capable of avoiding obstacles.
Background
The rope traction parallel robot has the advantages of high load-weight ratio, large working space, easiness in assembly and the like, but due to the fixed spatial position distribution of the ropes, collision among the ropes, between the ropes and an obstacle and between the movable platform and the obstacle is easy to occur. Therefore, the problem of planning obstacle avoidance paths of the fixed rope traction parallel robot is very complex, and obstacle avoidance movement is difficult to realize in practical application. The reconfigurable rope traction parallel robot can automatically change the self structure, particularly flexibly change the spatial position distribution of each rope by adjusting the position of a rope leading-out point, thereby realizing effective obstacle avoidance in the movement process, but the length of the rope and the position of the rope leading-out point can influence the obstacle avoidance and need to be comprehensively considered. In addition, the movable platform of the reconfigurable rope traction parallel robot is driven by a plurality of ropes simultaneously, the positions of the rope leading-out points are variable, the relations between the ropes and between the ropes and the rope leading-out points are highly coupled, and difficulty is increased for continuous reconfiguration planning of the rope traction parallel robot.
The Chinese patent application CN202110488846.4 discloses a method for a rope-traction parallel robot to bypass obstacles on the same plane through plane rotation reconstruction, but the included angle between the ropes is fixed, and the tail end movable platform and the ropes only move in a plane instead of three-dimensional, so that the obstacle avoidance effect is greatly limited.
In view of the above, the present invention is particularly proposed.
Disclosure of Invention
The invention aims to provide a continuous reconfiguration planning method for a rope-traction parallel robot capable of avoiding an obstacle, which can realize continuous reconfiguration by continuously and simultaneously changing the position of a rope leading-out point and the rope length, improve the obstacle avoiding capability of the rope-traction parallel robot in a complex environment and further solve the technical problems in the prior art.
The purpose of the invention is realized by the following technical scheme:
the embodiment of the invention discloses a rope traction parallel robot continuous reconstruction planning method capable of avoiding obstacles, which comprises the following steps:
step 1, introducing a virtual rope leading-out point according to the position relation between a rope of the rope traction parallel robot and a guide pulley, constructing a kinematic model of the rope traction parallel robot, and obtaining a closed chain constraint equation according to a closed chain structure of the rope traction parallel robot;
step 2, according to the manifold generated by the closed-chain constraint equation obtained in the step 1, locally approximating the manifold by a cutting space to obtain an orthogonal base for establishing the manifold cutting space, and establishing the manifold cutting space in the joint space of the rope traction parallel robot through the orthogonal base;
step 3, in the manifold cutting space obtained in the step 2, utilizing a fast expansion random tree algorithm to perform collision-free reconstruction path search between the initial reconstruction path point and the target reconstruction path point, performing force feasible detection and collision detection on the configuration formed by the expansion points obtained by search, and finding out a force feasible and collision-free reconstruction path;
and 4, solving an actual rope leading-out point position and an actual rope length according to the reconstruction path obtained in the step 3, and obtaining a series of collision-free changing configurations of the rope-traction parallel robot according to the actual rope leading-out point position and the actual rope length, namely completing collision-free continuous reconstruction planning of the rope-traction parallel robot.
Compared with the prior art, the rope traction parallel robot continuous reconstruction planning method capable of avoiding the obstacle has the beneficial effects that:
by introducing virtual rope leading-out points, not only is the kinematics modeling of the robot simplified, but also the difficulty of the guide pulley model in calculating the rope length is simplified. The position of the movable platform can be obtained through analysis by a kinematic model, the traditional Levenberg-Marquardt iterative solution mode is replaced, and the solution efficiency of the position of the movable platform is improved. Considering the closed chain structure of the rope traction parallel robot, the joint space is reduced into a low-dimensional manifold by establishing a constraint equation, a reconstruction path meeting collision-free and force-feasible constraints is searched in a manifold cutting space, and the rope length and the rope leading-out point position are adjusted at the same time, so that the obstacle avoidance capability of the rope traction parallel robot is improved.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required to be used in the description of the embodiments are briefly introduced below, and it is obvious that the drawings in the description below are only some embodiments of the present invention, and it is obvious for those skilled in the art that other drawings can be obtained according to the drawings without creative efforts.
Fig. 1 is a flowchart of a method for continuous reconfiguration planning of a rope-traction parallel robot capable of avoiding obstacles according to an embodiment of the present invention.
Fig. 2 is a flowchart of performing collision-free reconstructed path search on manifold cut space according to an embodiment of the present invention.
Fig. 3 is a schematic structural diagram of a rope-traction parallel robot capable of avoiding obstacles according to an embodiment of the present invention.
Fig. 4 and 5 are schematic diagrams of kinematic parameters of a rope-traction parallel robot capable of avoiding obstacles according to an embodiment of the present invention.
Fig. 6 is a schematic diagram of performing collision-free reconstruction path search on manifold cut space according to an embodiment of the present invention.
Fig. 7 is a schematic diagram of on-slice spatial sampling according to an embodiment of the present invention.
Fig. 8 is a schematic diagram of a guide pulley and a rope of a rope traction parallel robot with a suspension configuration capable of avoiding obstacles according to an embodiment of the present invention.
Fig. 9 is a schematic diagram of a guide pulley and a rope of a rope traction parallel robot with an unhindered configuration capable of avoiding obstacles according to an embodiment of the invention.
The part names corresponding to the respective marks in the figure are: 11-a fixed frame; 12-rope take-off, 121-guide pulley, 122-slider; 13-a rope; 14-moving the platform; 15-a motor driving the lead screw; 16-a motor driving the drum; 17-a reel; 18-vertical lead screw.
Detailed Description
The technical scheme in the embodiment of the invention is clearly and completely described below by combining the specific content of the invention; it is to be understood that the described embodiments are merely exemplary of the invention, and are not intended to limit the invention to the particular forms disclosed. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the present invention without making any creative effort, shall fall within the protection scope of the present invention.
The terms that may be used herein are first described as follows:
the term "and/or" means that either or both can be achieved, for example, X and/or Y means that both cases include "X" or "Y" as well as three cases including "X and Y".
The terms "comprising," "including," "containing," "having," or other similar terms of meaning should be construed as non-exclusive inclusions. For example: including a feature (e.g., material, component, ingredient, carrier, formulation, material, dimension, part, component, mechanism, device, process, procedure, method, reaction condition, processing condition, parameter, algorithm, signal, data, product, or article of manufacture), is to be construed as including not only the particular feature explicitly listed but also other features not explicitly listed as such which are known in the art.
The term "consisting of … …" is meant to exclude any technical feature elements not explicitly listed. If used in a claim, the term shall render the claim closed except for the inclusion of the technical features that are expressly listed except for the conventional impurities associated therewith. If the term occurs in only one clause of the claims, it is defined only as specifically listed in that clause, and elements recited in other clauses are not excluded from the overall claims.
Unless expressly stated or limited otherwise, the terms "mounted," "connected," and "secured," etc., are to be construed broadly, as for example: can be fixedly connected, can also be detachably connected or integrally connected; can be mechanically or electrically connected; they may be connected directly or indirectly through intervening media, or they may be interconnected between two elements. The specific meaning of the above terms herein can be understood by those of ordinary skill in the art as appropriate.
When concentrations, temperatures, pressures, dimensions, or other parameters are expressed as ranges of values, the ranges of values should be understood to specifically disclose all ranges formed by any pair of upper values, lower values, or preferred values within the range, regardless of whether the ranges are explicitly recited; for example, if a numerical range of "2 ~ 8" is recited, then the numerical range should be interpreted to include ranges of "2 ~ 7", "2 ~ 6", "5 ~ 7", "3 ~ 4 and 6 ~ 7", "3 ~ 5 and 7", "2 and 5 ~ 7", and the like. Unless otherwise indicated, the numerical ranges recited herein include both the endpoints thereof and all integers and fractions within the numerical range.
The terms "central," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," "counterclockwise," and the like are used in an orientation or positional relationship that is indicated based on the orientation or positional relationship shown in the drawings for ease of description and simplicity of description only, and are not intended to imply or imply that the referenced device or element must have a particular orientation, be constructed and operated in a particular orientation, and are therefore not to be considered limiting herein.
The method for planning the continuous reconstruction of the rope-traction parallel robot capable of avoiding the obstacle provided by the invention is described in detail below. Details which are not described in detail in the embodiments of the invention belong to the prior art which is known to the person skilled in the art. The examples of the present invention, in which specific conditions are not specified, were carried out according to the conventional conditions in the art or conditions suggested by the manufacturer. The reagents and instruments used in the examples of the present invention are not specified by manufacturers, and are conventional products commercially available.
As shown in fig. 1, an embodiment of the present invention provides a method for planning continuous reconfiguration of a rope-drawn parallel robot capable of avoiding an obstacle, where continuous reconfiguration is implemented by continuously and simultaneously changing a position of a rope leading-out point and a rope length, so as to improve an obstacle avoidance capability of the rope-drawn parallel robot in a complex environment, and the method includes:
step 1, introducing a virtual rope leading-out point according to the position relation between a rope of the rope traction parallel robot and a guide pulley, constructing a kinematic model of the rope traction parallel robot, and obtaining a closed chain constraint equation according to a closed chain structure of the rope traction parallel robot;
step 2, according to the manifold generated by the closed-chain constraint equation obtained in the step 1, locally approximating the manifold by a cutting space to obtain an orthogonal base for establishing the manifold cutting space, and establishing the manifold cutting space in the joint space of the rope traction parallel robot through the orthogonal base;
step 3, in the manifold cutting space obtained in the step 2, performing collision-free reconstruction path search between the initial reconstruction path point and the target reconstruction path point by using a fast expansion random tree algorithm, performing force feasible detection and collision detection on a configuration formed by the expansion points obtained by search, and finding out a force feasible and collision-free reconstruction path;
and 4, solving the position and the length of the actual rope leading-out point according to the reconstruction path obtained in the step 3, and obtaining a series of collision-free changing configurations of the rope-traction parallel robot according to the position and the length of the actual rope leading-out point, namely completing the collision-free continuous reconstruction planning of the rope-traction parallel robot.
In the above method, the continuously reconfigured planned rope-pulling parallel robot comprises (see fig. 3, 4 and 5):
a fixed frame,
Figure 315319DEST_PATH_IMAGE001
A rope leading-out device,
Figure 294777DEST_PATH_IMAGE001
A vertical screw rod,
Figure 871251DEST_PATH_IMAGE002
A motor,
Figure 63198DEST_PATH_IMAGE001
A winding drum,
Figure 674308DEST_PATH_IMAGE001
A rope and a movable platform; wherein, the first and the second end of the pipe are connected with each other,
each rope leading-out device consists of a sliding block and a guide pulley, and the guide pulley is connected and arranged on the sliding block and can synchronously move along with the sliding block;
the inner part of the fixed frame is arranged in a surrounding distribution way
Figure 792044DEST_PATH_IMAGE001
Each vertical screw is provided with a sliding block of a rope leading-out device, one end of each vertical screw is connected with a motor, and the vertical screws can rotate under the driving of the motors and drive the sliding blocks to move up and down;
a winding drum is correspondingly arranged on the ground in the fixed frame below each vertical screw rod, a rope is wound on each winding drum, each winding drum is connected with a motor, and the winding drums can rotate under the driving of the motors to wind and unwind the connected ropes;
the other end of each rope sequentially rounds a guide pulley of a rope leading-out device above the rope and then is connected with the movable platform, and each rope suspends the movable platform in the fixed frame;
the position and the rope length of a rope leading-out point of the rope traction parallel robot can be continuously changed at the same time, and continuous reconstruction is realized.
In the method, in the rope traction parallel robot, the rope is divided into three sections by the guide pulley, wherein the three sections are respectively a rope before entering the guide pulley, a rope wound on the guide pulley and a rope connected with the movable platform after leaving the guide pulley.
In step 1 of the method, a virtual rope leading-out point is introduced according to the position relation between the rope of the rope traction parallel robot and the guide pulley, a kinematic model of the rope traction parallel robot is constructed, and a closed chain constraint equation is obtained according to the closed chain structure of the rope traction parallel robot, wherein the closed chain constraint equation comprises the following steps: taking the ground center of the fixed frame of the rope traction parallel robot as the global coordinate system of the rope traction parallel robot
Figure 121394DEST_PATH_IMAGE003
Origin of (2)
Figure 117032DEST_PATH_IMAGE004
；
The rope is used for drawing the position vector of the movable platform of the parallel robot
Figure 582648DEST_PATH_IMAGE005
It is shown that,
Figure 139794DEST_PATH_IMAGE006
、
Figure 690861DEST_PATH_IMAGE007
、
Figure 224610DEST_PATH_IMAGE008
respectively moving platform in the global coordinate system of the rope-towed parallel robot
Figure 544733DEST_PATH_IMAGE009
A shaft,
Figure 36894DEST_PATH_IMAGE010
Shaft and
Figure 573793DEST_PATH_IMAGE011
coordinates on an axis; by using
Figure 911233DEST_PATH_IMAGE012
A virtual rope exit point is shown which represents,
Figure 85862DEST_PATH_IMAGE013
，
Figure 217766DEST_PATH_IMAGE014
the number of the rope leading-out points is the number of the ropes, the virtual rope leading-out point is the intersection point of the extension lines of the two ropes, one is the extension line of the rope before entering the guide pulley, the other is the extension line of the rope after leaving the guide pulley and connected with the movable platform, and the virtual rope leading-out point is
Figure 9005DEST_PATH_IMAGE012
A position vector of
Figure 386022DEST_PATH_IMAGE015
In which
Figure 680737DEST_PATH_IMAGE016
And
Figure 249122DEST_PATH_IMAGE017
is a constant that is known to be a constant,
Figure 996498DEST_PATH_IMAGE018
is a variable; by using
Figure 675741DEST_PATH_IMAGE019
Representing virtual rope exit points
Figure 824962DEST_PATH_IMAGE012
And moving platform
Figure 351800DEST_PATH_IMAGE020
The length vector of the virtual rope between the two ropes, the size of the length vector of the virtual rope is as follows:
Figure 320893DEST_PATH_IMAGE021
（1）；
four parameters can be used for each kinematic branching equation
Figure 538248DEST_PATH_IMAGE022
And
Figure 604293DEST_PATH_IMAGE023
it is shown that there is, among others,
Figure 281524DEST_PATH_IMAGE024
for virtual rope-leading-out points
Figure 3492DEST_PATH_IMAGE025
In that
Figure 24538DEST_PATH_IMAGE026
Coordinates in the axial direction;
Figure 882773DEST_PATH_IMAGE027
as a virtual rope length vector
Figure 963861DEST_PATH_IMAGE028
The die of (2);
Figure 671661DEST_PATH_IMAGE029
as a virtual rope length vector
Figure 496397DEST_PATH_IMAGE030
In that
Figure 474718DEST_PATH_IMAGE031
Projection of the axial plane and
Figure 461128DEST_PATH_IMAGE032
the included angle is in the positive direction of the axis;
Figure 157689DEST_PATH_IMAGE033
as a virtual rope length vector
Figure 22002DEST_PATH_IMAGE034
And
Figure 120408DEST_PATH_IMAGE035
the included angle of the negative shaft direction is shared by the rope traction parallel robot
Figure 543299DEST_PATH_IMAGE036
A moving branch chain, the rope pulls the position vector of the moving platform of the parallel robot
Figure 461576DEST_PATH_IMAGE037
One of the branch equations is expressed as:
Figure 628115DEST_PATH_IMAGE038
（2）；
subtracting the residual branch chain equation from a branch chain equation in the rope traction parallel robot to obtain
Figure 315449DEST_PATH_IMAGE039
The closed-chain constraint equation is:
Figure 407776DEST_PATH_IMAGE040
（3）；
in the formula (3), the reaction mixture is,
Figure 813350DEST_PATH_IMAGE041
，
Figure 783580DEST_PATH_IMAGE042
is the number of the equations of the kinematic branched chain,
Figure 856578DEST_PATH_IMAGE043
，
Figure 621272DEST_PATH_IMAGE044
and
Figure 750027DEST_PATH_IMAGE045
are all serial numbers of the branch chain equation,
Figure 258369DEST_PATH_IMAGE046
is shown as
Figure 185873DEST_PATH_IMAGE047
Strip and the first
Figure 121468DEST_PATH_IMAGE048
The bars are not the same branch equation.
In step 2 of the above method, the rope pulls the joint space of the parallel robot
Figure 501634DEST_PATH_IMAGE049
For coordinates in
Figure 46623DEST_PATH_IMAGE050
It is shown that the process of the present invention,
Figure 828634DEST_PATH_IMAGE051
wherein, in the process,
Figure 935130DEST_PATH_IMAGE052
，
Figure 802592DEST_PATH_IMAGE053
；
according to the closed chain constraint equation of the rope traction parallel robot, the rope is connectedJoint space of cable traction parallel robot
Figure 652736DEST_PATH_IMAGE049
Manifold with reduced dimension to low dimension
Figure 790719DEST_PATH_IMAGE054
；
Using a cutting space to align said manifold
Figure 68117DEST_PATH_IMAGE054
Local approximation of the small neighborhood is performed to derive an orthogonal basis for establishing manifold tangent space
Figure 157295DEST_PATH_IMAGE055
；
If it is
Figure 811131DEST_PATH_IMAGE055
Is located at
Figure 302155DEST_PATH_IMAGE056
The orthogonal basis of the point-tangent space then satisfies the following relationship:
Figure 750454DEST_PATH_IMAGE057
（4）；
in the above-mentioned formula (4),
Figure 91043DEST_PATH_IMAGE058
to restrain
Figure 282990DEST_PATH_IMAGE059
A Jacobian matrix of (d);
Figure 628520DEST_PATH_IMAGE060
is a zero matrix;
Figure 513300DEST_PATH_IMAGE061
is a unit matrix;
the orthogonal substrate
Figure 577071DEST_PATH_IMAGE055
Is a Jacobian matrix
Figure 74173DEST_PATH_IMAGE062
Through a null space of
Figure 274211DEST_PATH_IMAGE063
The Jacobian matrix is calculated by decomposition
Figure 329891DEST_PATH_IMAGE064
In the null space, i.e. established at
Figure 880958DEST_PATH_IMAGE056
The stream of points cuts the space.
In step 3 of the above method, a collision-free reconstruction path search is performed between the initial reconstruction path point and the target reconstruction path point by using a fast-expanding random tree algorithm in the following manner, and a configuration formed by the expansion points obtained by the search is subjected to force feasible detection and collision detection, so as to find out a force feasible and collision-free reconstruction path (see fig. 2), including:
step 31, selecting a cutting space: randomly selecting a cutting space from a cutting space set by using a fast expanding random tree algorithm
Figure 414708DEST_PATH_IMAGE065
Determining the cutting space
Figure 469252DEST_PATH_IMAGE065
Root node of
Figure 194369DEST_PATH_IMAGE066
And orthogonal substrates
Figure 967153DEST_PATH_IMAGE067
If the cutting space is selected for the first time, only one initial reconstruction path point is selected from the cutting space set
Figure 304593DEST_PATH_IMAGE068
For the root node's tangent space, there is only one initial reconstructed path point in the random tree
Figure 744802DEST_PATH_IMAGE068
；
Step 32, cutting the space to sample: in the cutting space
Figure 876706DEST_PATH_IMAGE065
The coordinate of the sampling point in the cutting space is
Figure 903830DEST_PATH_IMAGE069
The coordinates of which in the joint space of the rope-towed parallel robot are
Figure 513803DEST_PATH_IMAGE070
The conversion relationship is as follows:
Figure 808518DEST_PATH_IMAGE071
（5）；
step 33, expanding in the cutting space: search the same space
Figure 111323DEST_PATH_IMAGE065
Median coordinate
Figure 124279DEST_PATH_IMAGE072
The Euclidean distance closest point has the coordinate of the closest point in the joint space of the rope traction parallel robot
Figure 803522DEST_PATH_IMAGE073
The coordinates in the tangential space are
Figure 451278DEST_PATH_IMAGE074
In the cutting space
Figure 190564DEST_PATH_IMAGE065
Secondary coordinates of formula (6) in middle press
Figure 425237DEST_PATH_IMAGE075
To the coordinate
Figure 642591DEST_PATH_IMAGE069
By a fixed step length
Figure 911899DEST_PATH_IMAGE076
：
Figure 57971DEST_PATH_IMAGE077
（6）；
To obtain a cutting space
Figure 779940DEST_PATH_IMAGE065
The expansion point in the tangent space
Figure 800985DEST_PATH_IMAGE065
The center coordinate is
Figure 924799DEST_PATH_IMAGE078
Will coordinate
Figure 5888DEST_PATH_IMAGE078
Substituting the expansion point into the formula (5) to obtain the coordinate of the expansion point in the joint space of the rope traction parallel robot
Figure 713687DEST_PATH_IMAGE079
；
Step 34, extension point detection: to the coordinate is
Figure 272845DEST_PATH_IMAGE079
Is subjected to force feasibility detection and collision detection if the coordinates are
Figure 251165DEST_PATH_IMAGE079
Determining that the rope-traction parallel robot collides or does not meet the force feasible condition if the configuration of the expansion point fails to be detected, and giving up the coordinates as
Figure 503155DEST_PATH_IMAGE079
Go to step 31 to restart the search; if the coordinates are
Figure 465294DEST_PATH_IMAGE079
Detecting the extension point of the object to be detected, and converting the coordinates of the object to coordinates
Figure 329607DEST_PATH_IMAGE079
Equation of constraint
Figure 162434DEST_PATH_IMAGE059
To calculate the coordinates as
Figure 585325DEST_PATH_IMAGE079
Is located at a distance from the manifold
Figure 769182DEST_PATH_IMAGE080
Error of (2)
Figure 670142DEST_PATH_IMAGE081
：
Figure 623054DEST_PATH_IMAGE082
（7）；
Detecting the error
Figure 738819DEST_PATH_IMAGE083
Whether the 2 norm of (2) exceeds a threshold value
Figure 144393DEST_PATH_IMAGE084
If not, the coordinate is
Figure 114623DEST_PATH_IMAGE079
Adding the extension point of (2) into the random tree; otherwise, the coordinate is defined as
Figure 656463DEST_PATH_IMAGE079
Is mapped to the manifold by Newton's iteration
Figure 421157DEST_PATH_IMAGE085
Generating a mapping point, replacing the coordinates with the mapping point
Figure 815491DEST_PATH_IMAGE079
Adding the expansion point into a random tree, establishing a new cutting space by using the mapping point, and adding the new cutting space into a cutting space set;
step 35, detecting whether the target point is reached: if the coordinates are
Figure 792674DEST_PATH_IMAGE079
The extension point and the target reconstruction path point
Figure 720179DEST_PATH_IMAGE086
Between Euclidean distance is less than fixed step length for expansion
Figure 655774DEST_PATH_IMAGE087
Then will be
Figure 35940DEST_PATH_IMAGE079
And with
Figure 816814DEST_PATH_IMAGE086
Connecting to obtain a path in a cutting space, otherwise, returning to the step 31 for searching until the path is found successfully;
step 36, path point mapping: in the mapping process, Newton iteration method is adopted to map all points of the path in the tangent space found in the step 35 to the manifold
Figure 97360DEST_PATH_IMAGE085
As a force feasible and collision-free reconstruction path, the end condition of the iteration is
Figure 203857DEST_PATH_IMAGE083
Is less than 2 norm
Figure 71318DEST_PATH_IMAGE088
，
Figure 655884DEST_PATH_IMAGE089
。
In the feasible detection of the force in the step 34, the balance equation of the rope traction parallel robot dynamic platform is established as follows:
Figure 292401DEST_PATH_IMAGE090
（8）；
in the above-mentioned formula (8),
Figure 569799DEST_PATH_IMAGE091
a structural matrix corresponding to the rope traction parallel robot;
Figure 160442DEST_PATH_IMAGE092
a rope tension vector for the rope-towed parallel robot;
Figure 548698DEST_PATH_IMAGE093
the rope pulls the movable platform of the parallel robot to bear the resultant external force;
Figure 39723DEST_PATH_IMAGE094
and
Figure 488021DEST_PATH_IMAGE095
respectively a minimum rope tension vector and a maximum rope tension vector;
equating said formula (8) to said formula (9):
Figure 64496DEST_PATH_IMAGE096
（9）；
in the above formula (9), matrix
Figure 256443DEST_PATH_IMAGE097
Sum vector
Figure 366088DEST_PATH_IMAGE098
According to said structure by moving the hyperplane methodMatrix of
Figure 250868DEST_PATH_IMAGE099
Maximum rope tension vector
Figure 314639DEST_PATH_IMAGE095
Minimum rope tension vector
Figure 44697DEST_PATH_IMAGE094
Obtaining; determining coordinates of an extension point in joint space
Figure 510314DEST_PATH_IMAGE079
Whether the formed configuration satisfies the formula (9), if so, confirming that the passing force is feasible, and keeping the coordinate as
Figure 67459DEST_PATH_IMAGE079
The extension point of (a); otherwise, confirming that the failure force can be detected, and discarding the coordinate as
Figure 87368DEST_PATH_IMAGE079
The extension point of (a).
In the collision detection in the step 34, the rope is modeled as a straight line segment, the obstacle is modeled as a convex polyhedron, a collision detection algorithm is adopted to respectively detect whether the rope collides with the obstacle, the rope collides with the rope, and the movable platform collides with the obstacle, and if no collision exists, the coordinates are kept as
Figure 152276DEST_PATH_IMAGE079
The extension point of (a); if a collision occurs, the discarded coordinates are
Figure 472399DEST_PATH_IMAGE079
The extension point of (a).
In the method, the collision detection algorithm adopts any one of a separation axis theorem algorithm and a Gilbert-Johnson-Keerthi algorithm.
In step 4 of the above method, the actual rope leading-out point position is solved according to the reconstruction path in the following manner, including:
setting the rope at the entry tangent point
Figure 167822DEST_PATH_IMAGE100
Enters a guide pulley at the exit tangent point
Figure 206185DEST_PATH_IMAGE101
Out of the guide pulley and into the tangent point
Figure 42161DEST_PATH_IMAGE100
The actual rope drawing point of the parallel robot drawn by the rope is calculated by the following equation (10)
Figure 951211DEST_PATH_IMAGE102
The positions of (A) are:
Figure 348694DEST_PATH_IMAGE103
（10）；
in the formula (10), the reaction mixture is,
Figure 608774DEST_PATH_IMAGE104
for actual rope draw-off points
Figure 749906DEST_PATH_IMAGE100
The position vector of (a), wherein,
Figure 779042DEST_PATH_IMAGE105
and
Figure 583312DEST_PATH_IMAGE106
for virtual rope-leading-out points
Figure 596267DEST_PATH_IMAGE107
In that
Figure 275510DEST_PATH_IMAGE108
And
Figure 424732DEST_PATH_IMAGE109
the known coordinates of the direction of the light beam,
Figure 632859DEST_PATH_IMAGE110
to reconstruct the pathpoint values in the path,
Figure 366067DEST_PATH_IMAGE111
is the radius of the guide pulley;
solving the actual rope length according to the reconstruction path in the following way, including:
reconstructing path routing parameters
Figure 849001DEST_PATH_IMAGE112
And
Figure 852729DEST_PATH_IMAGE113
the composition is that the formula (2) is used to calculate the position vector of the movable platform
Figure 497337DEST_PATH_IMAGE114
(ii) a Using the position vector of the actual rope exit point
Figure 219305DEST_PATH_IMAGE115
Calculating the center of the guide pulley
Figure 240351DEST_PATH_IMAGE116
A position vector of (a); using position vectors of moving platforms
Figure 600050DEST_PATH_IMAGE117
And the center of the guide pulley
Figure 681139DEST_PATH_IMAGE116
The position vector of (2) to solve the moving platform
Figure 624824DEST_PATH_IMAGE117
To the center of the guide pulley
Figure 449560DEST_PATH_IMAGE116
Is a distance of
Figure 162301DEST_PATH_IMAGE118
；
Calculating the point of tangency of the rope leaving the guide pulley by means of the following equation (11)
Figure 414291DEST_PATH_IMAGE119
And moving platform
Figure 609387DEST_PATH_IMAGE120
Length of rope in between
Figure 972235DEST_PATH_IMAGE121
The method comprises the following steps:
Figure 539483DEST_PATH_IMAGE122
（11）；
in the above-mentioned formula (11),
Figure 962374DEST_PATH_IMAGE123
，
Figure 880651DEST_PATH_IMAGE124
and
Figure 47190DEST_PATH_IMAGE125
are respectively a distance
Figure 235989DEST_PATH_IMAGE118
In that
Figure 829781DEST_PATH_IMAGE126
Distance components in three directions of the axis;
if the rope pulls the parallel robot in a suspended configuration, the angle of wrap of the rope on the guide pulley is
Figure 235354DEST_PATH_IMAGE127
Equal to:
Figure 940005DEST_PATH_IMAGE128
（12）；
if the rope-traction parallel robot is in a non-suspended configuration, the winding angle of the rope on the guide pulley
Figure 481845DEST_PATH_IMAGE127
Equal to:
Figure 246539DEST_PATH_IMAGE129
（13）；
the actual length of the rope
Figure 637944DEST_PATH_IMAGE130
The movable platform is connected for the length of the rope wound on the guide pulley and after leaving the guide pulley
Figure 146285DEST_PATH_IMAGE120
The actual rope length is calculated by the following formula (14)
Figure 808211DEST_PATH_IMAGE130
The method comprises the following steps:
Figure 743806DEST_PATH_IMAGE131
（14）。
in conclusion, the method provided by the embodiment of the invention not only simplifies the kinematic modeling of the robot by introducing the virtual rope leading-out point, but also simplifies the difficulty of the rope length calculation of the guide pulley model. The position of the movable platform can be obtained through analysis by a kinematic model, the traditional Levenberg-Marquardt iterative solution mode is replaced, and the solution efficiency of the position of the movable platform is improved. Considering the closed chain structure of the rope traction parallel robot, the joint space is reduced into a low-dimensional manifold by establishing a constraint equation, a reconstruction path meeting collision-free and force-feasible constraints is searched in a manifold cutting space, and the rope length and the rope leading-out point position are adjusted at the same time, so that the obstacle avoidance capability of the rope traction parallel robot is improved.
In order to more clearly show the technical solutions and the technical effects provided by the present invention, the method for continuous reconfiguration and planning of a rope-traction parallel robot capable of avoiding obstacles provided by the embodiment of the present invention is described in detail with specific embodiments below.
Example 1
The embodiment provides a continuous reconfiguration planning method for a rope traction parallel robot capable of avoiding obstacles, which is carried out according to the following steps (see fig. 1):
step 1, introducing a virtual rope leading-out point by combining the position relation between a rope and a guide pulley of the rope traction parallel robot, constructing a robot kinematics model, and obtaining a closed chain constraint equation according to a closed chain structure of the robot: taking the ground center of the fixed frame of the rope traction parallel robot as the global coordinate system of the rope traction parallel robot
Figure 858392DEST_PATH_IMAGE003
Of (2)
Figure 671890DEST_PATH_IMAGE004
；
By using
Figure 453901DEST_PATH_IMAGE012
（
Figure 560397DEST_PATH_IMAGE013
) A virtual rope leading-out point is shown, which is the intersection point of two rope extension lines, one is the extension line of the rope before entering the pulley, and the other is the extension line of the rope connected with the movable platform after leaving the guide pulley
Figure 896701DEST_PATH_IMAGE012
A position vector of
Figure 746845DEST_PATH_IMAGE015
In which
Figure 117783DEST_PATH_IMAGE016
And
Figure 159295DEST_PATH_IMAGE017
is a constant that is known to be a constant,
Figure 514053DEST_PATH_IMAGE018
is a variable; by using
Figure 636730DEST_PATH_IMAGE019
Representing virtual rope exit points
Figure 393334DEST_PATH_IMAGE012
And moving platform
Figure 841632DEST_PATH_IMAGE020
The length vector of the virtual rope in between,
Figure 418107DEST_PATH_IMAGE027
as a virtual rope length vector
Figure 111519DEST_PATH_IMAGE028
The die of (a) is used,
Figure 457050DEST_PATH_IMAGE029
is recorded as a virtual rope length vector
Figure 76250DEST_PATH_IMAGE030
In that
Figure 140021DEST_PATH_IMAGE031
Projection of a plane and
Figure 135659DEST_PATH_IMAGE132
the included angle is formed in the positive direction,
Figure 123248DEST_PATH_IMAGE133
as a virtual rope length vector
Figure 913349DEST_PATH_IMAGE134
And
Figure 198837DEST_PATH_IMAGE035
clip with negative axisAngle (see fig. 4); for the moving platform position vector of the robot
Figure 998166DEST_PATH_IMAGE037
Showing that said rope-drawn parallel robots have in common
Figure 52710DEST_PATH_IMAGE036
The moving branch chains are connected to the same moving platform, and one branch chain equation is used for subtracting the residual branch chain equation to obtain the moving branch chain
Figure 13712DEST_PATH_IMAGE039
A constraint equation;
step 2, according to the manifold generated by the closed-chain constraint equation, obtaining an orthogonal substrate for establishing a cutting space by adopting the local approximate manifold of the cutting space: considering a constraint equation to reduce the dimensions of a joint space into a low-dimensional manifold, because the manifold is difficult to be globally expressed by independent variables, using a tangent space to carry out local approximation of a small neighborhood on the manifold, deducing a Jacobian matrix of the constraint equation, and obtaining a substrate for establishing the tangent space through a null space of the Jacobian matrix;
and 3, performing collision-free reconstruction path search in the manifold cutting space: it is known that
Figure 553540DEST_PATH_IMAGE068
And
Figure 890981DEST_PATH_IMAGE135
to use the parameter
Figure 65610DEST_PATH_IMAGE022
And
Figure 197514DEST_PATH_IMAGE023
（
Figure 457594DEST_PATH_IMAGE136
) The represented initial reconstruction path point and the target reconstruction path point are fused with a fast expansion random tree algorithm to expand a random tree in a tangent space and find a force feasible and collision-free weightConstructing a path;
step 4, solving the positions of the actual rope length and the actual rope leading-out point: and solving the actual rope length and the actual rope leading-out point position according to the reconstruction path to obtain a series of change configurations of the robot, so as to realize collision-free continuous reconstruction planning of the robot.
In step 1, as shown in fig. 3, 4 and 5, the rope-traction parallel robot mainly includes: a fixed frame,
Figure 598726DEST_PATH_IMAGE001
A rope leading-out device,
Figure 860817DEST_PATH_IMAGE001
A lead screw,
Figure 429202DEST_PATH_IMAGE001
A rope is arranged,
Figure 176578DEST_PATH_IMAGE002
A motor,
Figure 590242DEST_PATH_IMAGE001
A winding drum and a movable platform; the rope leading-out device mainly comprises a sliding block and a rope guide pulley;
Figure 739464DEST_PATH_IMAGE001
the motor drives the lead screw to rotate, so that the sliding block is driven to move up and down, and the position of the guide pulley arranged on the sliding block is changed along with the lead screw, so that the robot is reconstructed; in addition, the
Figure 478749DEST_PATH_IMAGE001
The motor is connected with the winding drum and used for winding and unwinding the rope. The rope from the reel end is connected with the movable platform after passing through the guide pulley
Figure 214886DEST_PATH_IMAGE020
Global coordinate system
Figure 432241DEST_PATH_IMAGE003
Of (2)
Figure 435969DEST_PATH_IMAGE004
The fixed frame is positioned on the ground; by using
Figure 346156DEST_PATH_IMAGE012
（
Figure 802546DEST_PATH_IMAGE013
) A virtual rope leading-out point which is the intersection point of two rope extension lines, one is the rope extension line before entering the pulley, the other is the rope extension line connected with the movable platform after leaving the guide pulley, and the virtual rope leading-out point
Figure 823591DEST_PATH_IMAGE012
Is a position vector of
Figure 445940DEST_PATH_IMAGE015
，
Figure 995870DEST_PATH_IMAGE012
Is stippled to
Figure 205135DEST_PATH_IMAGE006
、
Figure 29871DEST_PATH_IMAGE007
The position of the direction is fixed, and the direction is fixed,
Figure 8192DEST_PATH_IMAGE008
the change of the direction position represents the change of the robot configuration; for the moving platform position vector of the robot
Figure 994602DEST_PATH_IMAGE005
Indicating that a virtual cord outlet point is to be connected
Figure 927048DEST_PATH_IMAGE012
And moving platform
Figure 289896DEST_PATH_IMAGE020
The virtual rope length vector between is represented as
Figure 388303DEST_PATH_IMAGE019
The size is expressed as:
Figure 811194DEST_PATH_IMAGE021
（1）；
will vector
Figure 463892DEST_PATH_IMAGE028
And
Figure 630431DEST_PATH_IMAGE035
the included angle of the negative axis is recorded as
Figure 81879DEST_PATH_IMAGE033
，
Figure 675671DEST_PATH_IMAGE137
Will be
Figure 815665DEST_PATH_IMAGE138
Is recorded as a vector
Figure 520316DEST_PATH_IMAGE139
In that
Figure 327735DEST_PATH_IMAGE140
Projection of plane and
Figure 593894DEST_PATH_IMAGE141
the positive direction angle (see figure 5),
Figure 221184DEST_PATH_IMAGE142
said rope-drawn parallel robots having
Figure 729526DEST_PATH_IMAGE143
Four parameters can be used for each moving branch chain and each moving branch chain equation
Figure 391451DEST_PATH_IMAGE144
And
Figure 327046DEST_PATH_IMAGE145
the position of the movable platform can be expressed by one of the branch chain equations as follows:
Figure 441633DEST_PATH_IMAGE146
（2）；
the robot moving branched chain is connected to the same moving platform, and the equation of one branched chain is subtracted by the equation of the rest branched chain to obtain
Figure 252201DEST_PATH_IMAGE147
The constraint equation:
Figure 768633DEST_PATH_IMAGE148
（3）；
in the above-mentioned formula (3),
Figure 609550DEST_PATH_IMAGE149
，
Figure 477012DEST_PATH_IMAGE150
is the number of the equations of the kinematic branched chain,
Figure 327156DEST_PATH_IMAGE151
，
Figure 963674DEST_PATH_IMAGE152
and
Figure 476957DEST_PATH_IMAGE153
are all serial numbers of the branch chain equation,
Figure 831715DEST_PATH_IMAGE154
is shown as
Figure 219971DEST_PATH_IMAGE152
Strips andfirst, the
Figure 710995DEST_PATH_IMAGE153
The bars are not the same branch equation.
In the step 2, the robot has the whole joint space
Figure 424873DEST_PATH_IMAGE155
Four parameters can be used
Figure 1348DEST_PATH_IMAGE156
And
Figure 426251DEST_PATH_IMAGE157
（
Figure 37361DEST_PATH_IMAGE149
) Representing; the movable platform is driven by a plurality of guide pulleys and ropes simultaneously, and parameters
Figure 656561DEST_PATH_IMAGE156
And
Figure 454753DEST_PATH_IMAGE157
are highly coupled, and closed chain constraints exist between them, which space the joints
Figure 450390DEST_PATH_IMAGE155
Reducing the dimension to a low-dimensional manifold
Figure 916007DEST_PATH_IMAGE158
(ii) a To satisfy the closed-chain constraint between the four parameters, a manifold is required
Figure 941994DEST_PATH_IMAGE158
Upsampling; considering that the manifold is difficult to be globally represented by an independent variable, a series of tangential spaces are adopted to carry out local approximation of a tiny neighborhood on the manifold;
suppose that
Figure 493061DEST_PATH_IMAGE159
Is located at
Figure 26810DEST_PATH_IMAGE160
An orthogonal basis of the spot-cut space that satisfies:
Figure 346933DEST_PATH_IMAGE161
（4）
in the above-mentioned formula (4),
Figure 573515DEST_PATH_IMAGE162
to restrain
Figure 110414DEST_PATH_IMAGE163
The jacobian matrix of (a) is,
Figure 182275DEST_PATH_IMAGE164
is a matrix of the unit, and is,
Figure 622483DEST_PATH_IMAGE165
is a zero matrix, orthogonal basis
Figure 754387DEST_PATH_IMAGE166
Is a Jacobian matrix
Figure 14468DEST_PATH_IMAGE167
Through which it can pass
Figure 890020DEST_PATH_IMAGE168
Is calculated by decomposition at
Figure 686200DEST_PATH_IMAGE050
A tangent space of points is established;
in the step 3, as shown in fig. 6, the performing of the collision-free reconstruction path search in the manifold cutting space mainly includes (see the flow of fig. 2):
step 31, selecting a cutting space: randomly selecting a cutting space from a cutting space set by using a fast expanding random tree algorithm
Figure 989005DEST_PATH_IMAGE169
Determining the cutting space
Figure 736381DEST_PATH_IMAGE169
Root node of
Figure 415624DEST_PATH_IMAGE170
And orthogonal substrates
Figure 564846DEST_PATH_IMAGE171
If the tangent space is selected for the first time, only one initial reconstruction path point is selected from the tangent space set
Figure 304132DEST_PATH_IMAGE172
For the root node's tangent space, there is only one initial reconstructed path point in the random tree
Figure 37339DEST_PATH_IMAGE172
；
Step 32, cutting the space to sample: as shown in fig. 7, in the cutting space
Figure 989115DEST_PATH_IMAGE169
The coordinate of the sampling point in the cutting space is
Figure 258422DEST_PATH_IMAGE069
The coordinates in the joint space of the rope traction parallel robot are
Figure 168609DEST_PATH_IMAGE173
The conversion relationship is as follows:
Figure 890577DEST_PATH_IMAGE071
（5）；
step 33, expanding in the cutting space: search the same space
Figure 413088DEST_PATH_IMAGE065
Neutral separation
Figure 5743DEST_PATH_IMAGE174
The Euclidean distance closest point has the coordinate of the closest point in the joint space
Figure 352411DEST_PATH_IMAGE073
The coordinates in the tangential space are
Figure 296096DEST_PATH_IMAGE075
(ii) a In the cutting space
Figure 120833DEST_PATH_IMAGE065
From
Figure 99153DEST_PATH_IMAGE075
To the direction of
Figure 420036DEST_PATH_IMAGE069
By a fixed step length
Figure 851018DEST_PATH_IMAGE076
：
Figure 213866DEST_PATH_IMAGE077
（6）；
Obtaining an extension point in the cutting space, the coordinate of which in the cutting space is
Figure 46693DEST_PATH_IMAGE078
Will be
Figure 204004DEST_PATH_IMAGE078
Substituting the expansion point into the formula (5) to obtain the coordinates of the expansion point in the joint space
Figure 387861DEST_PATH_IMAGE079
；
Step 34, extension point detection: to pair
Figure 55865DEST_PATH_IMAGE079
The formed configuration is subjected to force feasibility detection and collision detection if
Figure 477619DEST_PATH_IMAGE079
Fail detection, determination
Figure 71411DEST_PATH_IMAGE079
Abandoning the parallel robot when collision occurs or the rope traction does not meet the force feasible condition
Figure 476985DEST_PATH_IMAGE079
Go to step 31 to restart the search; if it is not
Figure 447215DEST_PATH_IMAGE079
By detecting, then calculating
Figure 989055DEST_PATH_IMAGE079
From the manifold
Figure 252284DEST_PATH_IMAGE080
Error of (2)
Figure 145153DEST_PATH_IMAGE081
(see FIG. 7):
Figure 387916DEST_PATH_IMAGE082
（7）
detecting the error
Figure 49841DEST_PATH_IMAGE081
Whether the 2 norm of (2) exceeds a threshold value
Figure 985436DEST_PATH_IMAGE084
If not, will
Figure 100023DEST_PATH_IMAGE079
Adding the obtained mixture into a random tree; otherwise, it will
Figure 647941DEST_PATH_IMAGE079
Mapping to the manifold by Newton's iteration method
Figure 429952DEST_PATH_IMAGE080
Generating a mapping point, and replacing the mapping point with the mapping point
Figure 270869DEST_PATH_IMAGE079
Adding the new space into a random tree, establishing a new cutting space by using the mapping points, and adding the new cutting space into a cutting space set;
step 35, detecting whether the target point is reached: if it is not
Figure 138331DEST_PATH_IMAGE079
Reconstructing path points with a target
Figure 988476DEST_PATH_IMAGE135
The Euclidean distance between them is less than fixed step length for expansion
Figure 359414DEST_PATH_IMAGE076
Then will be
Figure 135347DEST_PATH_IMAGE079
And
Figure 224526DEST_PATH_IMAGE175
connecting to obtain a path in a cutting space, otherwise, returning to the step 31 for searching until the path is found successfully;
step 36, path point mapping: because the reconstructed path point is partially positioned in the manifold cutting space, all points of the obtained path in the cutting space are mapped to the manifold in the mapping process by adopting a Newton iteration method
Figure 612782DEST_PATH_IMAGE176
The end condition of the iteration is
Figure 103806DEST_PATH_IMAGE083
Is less than 2 norm
Figure 552105DEST_PATH_IMAGE089
。
In the force feasible detection of step 34, establishing a balance equation of the rope traction parallel robot moving platform as follows:
Figure 394159DEST_PATH_IMAGE090
（8）；
in the above-mentioned formula (8),
Figure 87570DEST_PATH_IMAGE177
a structural matrix corresponding to the parallel robot is pulled by the rope;
Figure 167522DEST_PATH_IMAGE092
is the rope tension vector;
Figure 52301DEST_PATH_IMAGE093
the movable platform is subjected to external force;
Figure 116072DEST_PATH_IMAGE094
and
Figure 846131DEST_PATH_IMAGE095
respectively a minimum rope tension vector and a maximum rope tension vector;
the above formula (8) is equivalent to the following formula:
Figure 46168DEST_PATH_IMAGE096
（9）；
in the above formula (9), the matrix
Figure 600384DEST_PATH_IMAGE097
Sum vector
Figure 151451DEST_PATH_IMAGE098
According to the structural matrix by moving the hyperplane method
Figure 685200DEST_PATH_IMAGE099
Maximum rope tension vector
Figure 739744DEST_PATH_IMAGE095
Minimum rope tension vector
Figure 966326DEST_PATH_IMAGE094
Obtaining;
judgment of
Figure 240575DEST_PATH_IMAGE079
Whether the formed configuration is satisfied (9), and if so, determining that the force is available for detection; otherwise, abandon
Figure 578015DEST_PATH_IMAGE079
。
In the collision detection of the step 34, the rope is modeled as a straight line segment and the obstacle is modeled as a convex polyhedron by neglecting the looseness and deformation of the rope, whether the rope collides with the obstacle, the rope collides with the rope and the movable platform collides with the obstacle is respectively detected by adopting a separation axis theorem algorithm or a Gilbert-Johnson-Keerthi algorithm, and if no collision occurs, the collision is retained
Figure 18224DEST_PATH_IMAGE079
(ii) a If a collision occurs, discarding
Figure 884549DEST_PATH_IMAGE079
。
In said step 4, the rope is at the entry point, as shown in fig. 8 and 9
Figure 410208DEST_PATH_IMAGE100
Point into a guide pulley at the point of exit tangent
Figure 285760DEST_PATH_IMAGE101
The point is moved away from the pulley and,
Figure 813431DEST_PATH_IMAGE100
the position of the point is recorded as the position of the actual rope exit point, and the position vector is
Figure 116237DEST_PATH_IMAGE178
；
Figure 129192DEST_PATH_IMAGE179
Is a movable platform
Figure 808435DEST_PATH_IMAGE180
And
Figure 957657DEST_PATH_IMAGE181
the length of the rope between the points is,
Figure 431363DEST_PATH_IMAGE182
is the winding angle of the rope on the guide pulley,
Figure 167500DEST_PATH_IMAGE183
in order to guide the radius of the pulley,
Figure 384855DEST_PATH_IMAGE184
is the center of the guide pulley;
the position of an actual rope leading-out point of the rope traction parallel robot is calculated by the following formula (10)
Figure 654162DEST_PATH_IMAGE100
Comprises the following steps:
Figure 298770DEST_PATH_IMAGE103
（10）；
in the formula (10), the meaning of each parameter is:
Figure 755159DEST_PATH_IMAGE100
for the rope to enter the entry tangent point of the guide pulley, the
Figure 776205DEST_PATH_IMAGE100
The position vector of the point is
Figure 398554DEST_PATH_IMAGE104
Wherein
Figure 214063DEST_PATH_IMAGE105
And
Figure 423328DEST_PATH_IMAGE106
for virtual rope draw-off points
Figure 982485DEST_PATH_IMAGE107
In that
Figure 960805DEST_PATH_IMAGE108
And
Figure 947216DEST_PATH_IMAGE109
the known coordinates of the direction of the light,
Figure 145241DEST_PATH_IMAGE185
in order to reconstruct the values in the path points,
Figure 508090DEST_PATH_IMAGE111
is the guide pulley radius;
using the position vector of the actual rope exit point
Figure 340916DEST_PATH_IMAGE115
Calculating the center of the guide pulley
Figure 498228DEST_PATH_IMAGE184
A position vector of (a);
reconstructing path routing parameters
Figure 682085DEST_PATH_IMAGE112
And
Figure 81580DEST_PATH_IMAGE113
the position vector of the movable platform can be obtained by using the formula (2)
Figure 34492DEST_PATH_IMAGE114
Thereby moving the platform
Figure 628285DEST_PATH_IMAGE117
To the centre of the rope guide pulley
Figure 33858DEST_PATH_IMAGE116
Of (2) is
Figure 4088DEST_PATH_IMAGE118
Is calculated. By using
Figure 811507DEST_PATH_IMAGE186
And
Figure 812087DEST_PATH_IMAGE187
expressed as a distance
Figure 704956DEST_PATH_IMAGE118
In that
Figure 947719DEST_PATH_IMAGE188
Distance components in three directions of the axis, so
Figure 875224DEST_PATH_IMAGE189
Equal to:
Figure 545239DEST_PATH_IMAGE122
（11）
in the above-mentioned formula (11),
Figure 689519DEST_PATH_IMAGE123
；
if the rope-pulling parallel robot is in a suspended configuration, as shown in fig. 8, the winding angle of the rope on the guide pulley
Figure 735973DEST_PATH_IMAGE127
Equal to:
Figure 252405DEST_PATH_IMAGE190
（12）；
if the rope-pulling parallel robot is in an unsuspended configuration, as shown in fig. 9, the winding angle of the rope on the guide pulley
Figure 358901DEST_PATH_IMAGE127
Equal to:
Figure 226363DEST_PATH_IMAGE191
（13）；
the rope is divided into three sections by the guide pulley, namely the rope before entering the pulley, the rope wound on the guide pulley and the rope between the movable platform and the movable platform after leaving the guide pulley. The sum of the length of the rope wound on the guide pulley and the length of the rope connected with the movable platform after leaving the guide pulley is the actual length of the rope, and the actual length of the rope is calculated
Figure 810928DEST_PATH_IMAGE127
And
Figure 948911DEST_PATH_IMAGE192
then solving the actual rope length through the following formula:
Figure 226308DEST_PATH_IMAGE193
（14）；
the actual rope exit point position and the actual rope length are thus available.
In summary, compared with the prior art, the continuous reconstruction planning method of the embodiment of the present invention has at least the following beneficial effects:
(1) by introducing the virtual rope leading-out point, the influence of the guide pulley model on the kinematic modeling of the robot is simplified, the difficulty of the guide pulley model on the calculation of the length of the rope is also simplified, and the kinematic model of the robot is convenient to derive.
(2) The position of the movable platform can be obtained through analysis by a kinematic model, the traditional Levenberg-Marquardt iterative solution mode is replaced, and the solution efficiency of the position of the movable platform is improved.
(3) Considering the coupling between the moving branched chains of the rope-traction parallel robot capable of avoiding obstacles, reducing the dimensions of joint space into low-dimensional manifold by establishing a constraint equation, and performing local approximation of a small neighborhood on the manifold by adopting a series of tangential spaces to ensure that the local part of the manifold has an expression, thereby solving the problem that the manifold is difficult to be expressed globally by independent variables.
(4) By searching a reconstruction path meeting collision-free and force-feasible constraints in the manifold cutting space, the rope traction parallel robot can simultaneously adjust the rope length and the rope leading-out point position, and the obstacle avoidance capability of the rope traction parallel robot is improved.
Those of ordinary skill in the art will understand that: all or part of the processes of the methods for implementing the embodiments may be implemented by a program, which may be stored in a computer-readable storage medium, and when executed, may include the processes of the embodiments of the methods as described above. The storage medium may be a magnetic disk, an optical disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), or the like.
The above description is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are also within the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims. The information disclosed in this background section is only for enhancement of understanding of the general background of the invention and should not be taken as an acknowledgement or any form of suggestion that this information forms the prior art already known to a person skilled in the art.

Claims (7)

1. A rope traction parallel robot continuous reconstruction planning method capable of avoiding obstacles is characterized by comprising the following steps:
step 1, introducing a virtual rope leading-out point according to the position relation between a rope of the rope traction parallel robot and a guide pulley, constructing a kinematic model of the rope traction parallel robot, and obtaining a closed chain constraint equation according to a closed chain structure of the rope traction parallel robot;
the rope-drawn parallel robot of the continuous reconstruction plan includes: a fixed frame,
Figure 291586DEST_PATH_IMAGE001
A rope leading-out device,
Figure 865656DEST_PATH_IMAGE001
A vertical screw rod,
Figure 438720DEST_PATH_IMAGE002
A motor,
Figure 781845DEST_PATH_IMAGE001
A winding drum,
Figure 782162DEST_PATH_IMAGE001
A rope and a movable platform; wherein the content of the first and second substances,
each rope leading-out device consists of a sliding block and a guide pulley, and the guide pulley is connected and arranged on the sliding block and can synchronously move along with the sliding block;
the inner part of the fixed frame is arranged in a surrounding distribution way
Figure 328681DEST_PATH_IMAGE001
Each vertical screw is provided with a sliding block of a rope leading-out device, one end of each vertical screw is connected with a motor, and the vertical screws can rotate under the driving of the motors and drive the sliding blocks to move up and down;
a winding drum is correspondingly arranged on the ground in the fixed frame below each vertical screw rod, a rope is wound on each winding drum, each winding drum is connected with a motor, and the winding drums can rotate under the driving of the motors to wind and unwind the connected ropes;
the other end of each rope sequentially rounds a guide pulley of a rope leading-out device above the rope and then is connected with the movable platform, and each rope suspends the movable platform in the fixed frame;
the position of a rope leading-out point of the rope traction parallel robot and the rope length can be continuously changed at the same time, so that continuous reconstruction is realized;
in the rope traction parallel robot, the rope is divided into three sections by a guide pulley, namely a rope before entering the guide pulley, a rope wound on the guide pulley and a rope connected with a movable platform after leaving the guide pulley;
in the step 1, a virtual rope leading-out point is introduced according to the position relation between the rope of the rope traction parallel robot and the guide pulley, a kinematic model of the rope traction parallel robot is constructed, and a closed chain constraint equation is obtained according to the closed chain structure of the rope traction parallel robot, wherein the closed chain constraint equation comprises the following steps: taking the ground center of the fixed frame of the rope traction parallel robot as the global coordinate system of the rope traction parallel robot
Figure 689124DEST_PATH_IMAGE003
Origin of (2)
Figure 903068DEST_PATH_IMAGE004
；
The rope is used for drawing the position vector of the movable platform of the parallel robot
Figure 326483DEST_PATH_IMAGE005
It is shown that,
Figure 625878DEST_PATH_IMAGE006
、
Figure 540744DEST_PATH_IMAGE007
、
Figure 858462DEST_PATH_IMAGE008
respectively moving platform in the global coordinate system of the rope-towed parallel robot
Figure 200581DEST_PATH_IMAGE009
A shaft,
Figure 987272DEST_PATH_IMAGE010
Shaft and
Figure 955097DEST_PATH_IMAGE011
coordinates on an axis; by using
Figure 143633DEST_PATH_IMAGE012
A virtual rope exit point is shown which represents,
Figure 640342DEST_PATH_IMAGE013
，
Figure 179908DEST_PATH_IMAGE014
the number of the rope leading-out points is the number of the ropes, the virtual rope leading-out point is the intersection point of the extension lines of the two ropes, one is the extension line of the rope before entering the guide pulley, the other is the extension line of the rope after leaving the guide pulley and connected with the movable platform, and the virtual rope leading-out point is
Figure 436577DEST_PATH_IMAGE012
A position vector of
Figure 725957DEST_PATH_IMAGE015
Wherein
Figure 144300DEST_PATH_IMAGE016
And
Figure 171162DEST_PATH_IMAGE017
in order to be a known constant, the constant,
Figure 215210DEST_PATH_IMAGE018
is a variable; by using
Figure 847180DEST_PATH_IMAGE019
Representing virtual rope exit points
Figure 233162DEST_PATH_IMAGE012
And moving platform
Figure 465429DEST_PATH_IMAGE020
The length vector of the virtual rope in between, the size of the length vector of the virtual rope is:
Figure 63900DEST_PATH_IMAGE021
（1）；
four parameters can be used for each kinematic branching equation
Figure 330802DEST_PATH_IMAGE022
And
Figure 356527DEST_PATH_IMAGE023
it is shown that there is, among others,
Figure 826823DEST_PATH_IMAGE024
for virtual rope-leading-out points
Figure 481183DEST_PATH_IMAGE025
In that
Figure 353324DEST_PATH_IMAGE026
Coordinates in the axial direction;
Figure 815529DEST_PATH_IMAGE027
as a virtual rope length vector
Figure 22388DEST_PATH_IMAGE028
The mold of (4);
Figure 962662DEST_PATH_IMAGE029
as a virtual rope length vector
Figure 689310DEST_PATH_IMAGE030
In that
Figure 571684DEST_PATH_IMAGE031
Projection of the axial plane and
Figure 282151DEST_PATH_IMAGE032
the included angle is in the positive direction of the axis;
Figure 744226DEST_PATH_IMAGE033
as a virtual rope length vector
Figure 856538DEST_PATH_IMAGE034
And
Figure 394967DEST_PATH_IMAGE035
the included angle of the negative shaft direction is shared by the rope traction parallel robot
Figure 409426DEST_PATH_IMAGE036
The moving branch chain, the rope pulls the position vector of the moving platform of the parallel robot
Figure 691503DEST_PATH_IMAGE037
One of the branch equations is expressed as:
Figure 392743DEST_PATH_IMAGE038
（2）；
subtracting the residual branch chain equation from a branch chain equation in the rope traction parallel robot to obtain
Figure 882499DEST_PATH_IMAGE039
A closed-chain constraint equation, which is:
Figure 770820DEST_PATH_IMAGE040
（3）；
in the formula (3), the reaction mixture is,
Figure 122167DEST_PATH_IMAGE041
，
Figure 927181DEST_PATH_IMAGE042
is the number of the equations of the kinematic branched chain,
Figure 72991DEST_PATH_IMAGE043
，
Figure 714188DEST_PATH_IMAGE044
and
Figure 587335DEST_PATH_IMAGE045
are all serial numbers of the branched chain equation,
Figure 263167DEST_PATH_IMAGE046
denotes the first
Figure 845458DEST_PATH_IMAGE047
Strip and the first
Figure 226149DEST_PATH_IMAGE048
The bars are not the same branch equation;
step 2, according to the manifold generated by the closed-chain constraint equation obtained in the step 1, locally approximating the manifold by adopting a cutting space to obtain an orthogonal base for establishing the manifold cutting space, and establishing the manifold cutting space in the joint space of the rope traction parallel robot through the orthogonal base;
step 3, in the manifold cutting space obtained in the step 2, performing collision-free reconstruction path search between the initial reconstruction path point and the target reconstruction path point by using a fast expansion random tree algorithm, performing force feasible detection and collision detection on a configuration formed by the expansion points obtained by search, and finding out a force feasible and collision-free reconstruction path;
and 4, solving an actual rope leading-out point position and an actual rope length according to the reconstruction path obtained in the step 3, and obtaining a series of collision-free changing configurations of the rope-traction parallel robot according to the actual rope leading-out point position and the actual rope length, namely completing collision-free continuous reconstruction planning of the rope-traction parallel robot.
2. The obstacle-avoidance rope-drawn parallel robot continuous reconstruction planning method according to claim 1, wherein in the step 2, the rope-drawn parallel robot joint space
Figure 653719DEST_PATH_IMAGE049
For coordinates of (1)
Figure 167746DEST_PATH_IMAGE050
It is shown that,
Figure 655359DEST_PATH_IMAGE051
wherein, in the process,
Figure 271148DEST_PATH_IMAGE052
，
Figure 17256DEST_PATH_IMAGE053
；
according to the closed chain constraint equation of the rope traction parallel robot, the rope is pulled into the joint space of the parallel robot
Figure 402101DEST_PATH_IMAGE054
Manifold with reduced dimension to low dimension
Figure 795036DEST_PATH_IMAGE055
；
Using a cutting space to align said manifold
Figure 147389DEST_PATH_IMAGE055
Local approximation of the tiny neighborhood is performed to derive an orthogonal basis for establishing manifold-cut space
Figure 182341DEST_PATH_IMAGE056
；
If it is
Figure 687272DEST_PATH_IMAGE056
Is located at
Figure 497446DEST_PATH_IMAGE057
The orthogonal basis of the point-tangent space then satisfies the following relationship:
Figure 87828DEST_PATH_IMAGE058
（4）；
in the above-mentioned formula (4),
Figure 926471DEST_PATH_IMAGE059
to restrain
Figure 269596DEST_PATH_IMAGE060
A Jacobian matrix of (d);
Figure 4334DEST_PATH_IMAGE061
is a zero matrix;
Figure 65700DEST_PATH_IMAGE062
is a unit matrix;
the orthogonal substrate
Figure 708034DEST_PATH_IMAGE056
Is a Jacobi matrix
Figure 921977DEST_PATH_IMAGE063
Through a null space of
Figure 342463DEST_PATH_IMAGE064
The Jacobian matrix is calculated by decomposition
Figure 907437DEST_PATH_IMAGE065
Of (2)In space, i.e. built in
Figure 822303DEST_PATH_IMAGE057
The manifold of points cuts the space.
3. The method for planning continuous reconstruction of rope-drawn parallel robots capable of avoiding obstacles according to claim 2, wherein in the step 3, a fast-expanding random tree algorithm is used to search for collision-free reconstruction paths between an initial reconstruction path point and a target reconstruction path point, and a configuration formed by expansion points obtained by the search is subjected to force feasible detection and collision detection to find out a reconstruction path with feasible force and no collision, and the method comprises the following steps:
step 31, selecting a cutting space: randomly selecting one cutting space from the cutting space set by using fast expanding random tree algorithm
Figure 142950DEST_PATH_IMAGE066
Determining the cutting space
Figure 485070DEST_PATH_IMAGE066
Root node of
Figure 537340DEST_PATH_IMAGE067
And orthogonal substrates
Figure 239585DEST_PATH_IMAGE068
If the cutting space is selected for the first time, only one initial reconstruction path point is selected from the cutting space set
Figure 428121DEST_PATH_IMAGE069
For the root node's tangent space, there is only one initial reconstructed path point in the random tree
Figure 941142DEST_PATH_IMAGE069
；
Step 32, cutting space sampling: in the cutting space
Figure 729976DEST_PATH_IMAGE066
The coordinates of the sampling point in the tangent space are
Figure 986645DEST_PATH_IMAGE070
The coordinates of which in the joint space of the rope-towed parallel robot are
Figure 29687DEST_PATH_IMAGE071
The conversion relationship is as follows:
Figure 228456DEST_PATH_IMAGE072
（5）；
step 33, expanding in the cutting space: search the same space
Figure 724159DEST_PATH_IMAGE066
Median coordinate
Figure 784519DEST_PATH_IMAGE071
The Euclidean distance closest point has the coordinate of the closest point in the joint space of the rope traction parallel robot
Figure 928406DEST_PATH_IMAGE073
The coordinates in the tangential space are
Figure 48809DEST_PATH_IMAGE074
In the cutting space
Figure 297387DEST_PATH_IMAGE066
Secondary coordinates of formula (6) in middle press
Figure 145127DEST_PATH_IMAGE075
To the coordinate
Figure 162761DEST_PATH_IMAGE070
By a fixed step length
Figure 454065DEST_PATH_IMAGE076
：
Figure 173628DEST_PATH_IMAGE077
（6）；
To obtain a cutting space
Figure 841370DEST_PATH_IMAGE066
The extension point in (1) is in the tangent space
Figure 447932DEST_PATH_IMAGE066
The middle coordinate is
Figure 159405DEST_PATH_IMAGE078
Will coordinate
Figure 116997DEST_PATH_IMAGE078
Substituting the expansion point into the formula (5) to obtain the coordinate of the expansion point in the joint space of the rope traction parallel robot
Figure 57271DEST_PATH_IMAGE079
；
Step 34, extension point detection: to the coordinate is
Figure 36116DEST_PATH_IMAGE079
Is subjected to a force feasibility detection and a collision detection if the coordinates are
Figure 934801DEST_PATH_IMAGE079
Determining that the rope-traction parallel robot collides or does not meet the force feasible condition if the configuration of the expansion point fails to be detected, and giving up the coordinates as
Figure 114110DEST_PATH_IMAGE079
The extension point of (2), turnRestarting the search at step 31; if the coordinates are
Figure 107343DEST_PATH_IMAGE079
Detecting the extended point of the image data to obtain coordinates
Figure 954076DEST_PATH_IMAGE079
Equation of constraint
Figure 758084DEST_PATH_IMAGE080
The calculated coordinates are
Figure 939535DEST_PATH_IMAGE079
Is located at a distance from the manifold
Figure 487191DEST_PATH_IMAGE055
Error of (2)
Figure 188431DEST_PATH_IMAGE081
：
Figure 147029DEST_PATH_IMAGE082
（7）；
Detecting the error
Figure 832088DEST_PATH_IMAGE081
Whether the 2 norm of (a) exceeds a threshold
Figure 917856DEST_PATH_IMAGE083
If not, the coordinate is
Figure 719940DEST_PATH_IMAGE079
Adding the extension point of (2) to the random tree; otherwise, the coordinate is defined as
Figure 865750DEST_PATH_IMAGE079
Is mapped by Newton's iteration methodTo the manifold
Figure 772527DEST_PATH_IMAGE055
Generating a mapping point, replacing the coordinates with the mapping point as
Figure 645674DEST_PATH_IMAGE079
Adding the expansion point into a random tree, establishing a new cutting space by using the mapping point, and adding the new cutting space into a cutting space set;
step 35, detecting whether the target point is reached: if the coordinates are
Figure 321505DEST_PATH_IMAGE079
The extension point and the target reconstruction path point
Figure 621906DEST_PATH_IMAGE084
Between Euclidean distance is less than fixed step length for expansion
Figure 15978DEST_PATH_IMAGE085
Then will be
Figure 443548DEST_PATH_IMAGE079
And with
Figure 223154DEST_PATH_IMAGE084
Connecting to obtain a path in a cutting space, otherwise, returning to the step 31 for searching until the path is found successfully;
step 36, path point mapping: adopting Newton iteration method to map all points of the path in the tangent space found in the step 35 to the manifold
Figure 710767DEST_PATH_IMAGE055
As a force feasible and collision-free reconstruction path, the end condition of the iteration is
Figure 592136DEST_PATH_IMAGE081
Is less than 2 norm
Figure 810015DEST_PATH_IMAGE086
，
Figure 194860DEST_PATH_IMAGE087
。
4. The method for planning continuous reconstruction of rope-drawn parallel robot capable of avoiding obstacles according to claim 3, wherein in the force feasibility detection of step 34, the balance equation for establishing the moving platform of the rope-drawn parallel robot is as follows:
Figure 853374DEST_PATH_IMAGE088
（8）；
in the formula (8), the reaction mixture is,
Figure 471306DEST_PATH_IMAGE089
a structural matrix corresponding to the parallel robot is pulled by the rope;
Figure 240679DEST_PATH_IMAGE090
a rope tension vector for the rope-towed parallel robot;
Figure 480031DEST_PATH_IMAGE091
the rope pulls the movable platform of the parallel robot to bear the resultant external force;
Figure 558714DEST_PATH_IMAGE092
and
Figure 414675DEST_PATH_IMAGE093
respectively a minimum rope tension vector and a maximum rope tension vector;
equating said formula (8) to said formula (9):
Figure 722159DEST_PATH_IMAGE094
（9）；
in the above formula (9), matrix
Figure 330864DEST_PATH_IMAGE095
Sum vector
Figure 331181DEST_PATH_IMAGE096
According to said structural matrix by moving the hyperplane method
Figure 408858DEST_PATH_IMAGE097
Maximum rope tension vector
Figure 55388DEST_PATH_IMAGE093
Minimum rope tension vector
Figure 269332DEST_PATH_IMAGE092
Obtaining; determining coordinates of an extension point in joint space
Figure 706130DEST_PATH_IMAGE079
Whether the formed configuration satisfies the formula (9) or not, and if so, confirming that the force is available for detection, and keeping the coordinate as
Figure 254792DEST_PATH_IMAGE079
The extension point of (a); otherwise, it is determined that the failed force can be detected, and the discarded coordinate is
Figure 904079DEST_PATH_IMAGE079
The extension point of (a).
5. The method for planning continuous reconstruction of rope-drawn parallel robot capable of avoiding obstacles according to claim 4, wherein in the collision detection of step 34, the rope is modeled as a straight line segment, the obstacle is modeled as a convex polyhedron, and a collision detection algorithm is adopted to detect the rope and the obstacle, the rope and the rope, the moving platform and the moving platform respectivelyWhether the obstacle collides or not, if not, the coordinate is kept as
Figure 238108DEST_PATH_IMAGE079
The extension point of (a); if a collision occurs, the discarded coordinates are
Figure 95075DEST_PATH_IMAGE079
The extension point of (a).
6. The method for continuous reconstruction and planning of rope-drawn parallel robots capable of avoiding obstacles as claimed in claim 5, wherein the collision detection algorithm adopts any one of a split-axis theorem algorithm and a Gilbert-Johnson-Keerthi algorithm.
7. The method for planning continuous reconstruction of rope-drawn parallel robots capable of avoiding obstacles according to claim 1, wherein in the step 4, the actual rope leading-out point position is solved according to the reconstruction path in the following manner, including:
setting the rope at the entry tangent point
Figure 881765DEST_PATH_IMAGE098
Enters a guide pulley at the exit tangent point
Figure 334743DEST_PATH_IMAGE099
Out of the guide pulley and into the tangent point
Figure 772546DEST_PATH_IMAGE100
The actual rope drawing point of the parallel robot drawn by the rope is calculated by the following equation (10)
Figure 551147DEST_PATH_IMAGE100
The positions of (A) are:
Figure 825133DEST_PATH_IMAGE101
（10）；
in the formula (10), the reaction mixture is,
Figure 333999DEST_PATH_IMAGE102
for actual rope draw-off points
Figure 377042DEST_PATH_IMAGE100
The position vector of (a), wherein,
Figure 326543DEST_PATH_IMAGE103
and
Figure 71514DEST_PATH_IMAGE104
for virtual rope draw-off points
Figure 131874DEST_PATH_IMAGE105
In that
Figure 295002DEST_PATH_IMAGE106
And
Figure 399093DEST_PATH_IMAGE107
the known coordinates of the direction of the light,
Figure 647672DEST_PATH_IMAGE108
to reconstruct the pathpoint values in the path,
Figure 495411DEST_PATH_IMAGE109
is the radius of the guide pulley;
solving the actual rope length according to the reconstruction path in the following way, including:
reconstructing path routing parameters
Figure 513046DEST_PATH_IMAGE110
And
Figure 804350DEST_PATH_IMAGE111
composition using the formula (2) Determining a motion platform position vector
Figure 520983DEST_PATH_IMAGE112
(ii) a Using the position vector of the actual rope exit point
Figure 923146DEST_PATH_IMAGE113
Calculating the center of the guide pulley
Figure 795287DEST_PATH_IMAGE114
A position vector of (a); using position vectors of moving platforms
Figure 506760DEST_PATH_IMAGE115
And the center of the guide pulley
Figure 933193DEST_PATH_IMAGE114
The position vector of (2) to solve the moving platform
Figure 404626DEST_PATH_IMAGE115
To the center of the guide pulley
Figure 380541DEST_PATH_IMAGE114
Is a distance of
Figure 748068DEST_PATH_IMAGE116
；
Calculating the point of tangency of the rope leaving the guide pulley by means of the following equation (11)
Figure 192956DEST_PATH_IMAGE117
And moving platform
Figure 186189DEST_PATH_IMAGE118
Length of rope in between
Figure 32922DEST_PATH_IMAGE119
The method comprises the following steps:
Figure 836930DEST_PATH_IMAGE120
（11）；
in the above-mentioned formula (11),
Figure 755732DEST_PATH_IMAGE121
，
Figure 303388DEST_PATH_IMAGE122
and
Figure 270207DEST_PATH_IMAGE123
are respectively a distance
Figure 228804DEST_PATH_IMAGE124
In that
Figure 648284DEST_PATH_IMAGE125
Distance components in three directions of the axis;
if the rope pulls the parallel robot into a suspended configuration, the angle of wrap of the rope on the guide pulley is
Figure 983320DEST_PATH_IMAGE126
Equal to:
Figure 804645DEST_PATH_IMAGE127
（12）；
if the rope-traction parallel robot is in an unsuspended configuration, the winding angle of the rope on the guide pulley
Figure 684877DEST_PATH_IMAGE126
Equal to:
Figure 575341DEST_PATH_IMAGE128
（13）；
the actual length of the rope
Figure 464800DEST_PATH_IMAGE129
The movable platform is connected for the length of the rope wound on the guide pulley and after leaving the guide pulley
Figure 140632DEST_PATH_IMAGE118
The actual rope length is calculated by the following formula (14)
Figure 703681DEST_PATH_IMAGE129
The method comprises the following steps:
Figure 832174DEST_PATH_IMAGE130
（14）。
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