CN110744547B - Continuous body mechanical arm inverse kinematics modeling method based on segmented constant curvature - Google Patents

Continuous body mechanical arm inverse kinematics modeling method based on segmented constant curvature Download PDF

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CN110744547B
CN110744547B CN201911089006.XA CN201911089006A CN110744547B CN 110744547 B CN110744547 B CN 110744547B CN 201911089006 A CN201911089006 A CN 201911089006A CN 110744547 B CN110744547 B CN 110744547B
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inverse kinematics
constant curvature
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杜付鑫
卢佳佳
张涛
王得晨
岳明君
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Shandong University
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    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
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Abstract

The invention discloses a continuous body mechanical arm inverse kinematics modeling method based on a sectional constant curvature, which solves the problems that the traditional inverse kinematics model aiming at a sectional constant curvature line-driven continuous body mechanical arm can not obtain an analytic solution and the tail end gesture of the mechanical arm is not considered, and has the technical scheme that: the method comprises the following steps: establishing a geometric model of a mechanical arm joint based on a segmental constant curvature hypothesis to obtain a mapping relation between the bending direction and the bending angle of the mechanical arm and the length of a driving wire; extracting a mechanical arm bone trunk curve model, and establishing a mapping relation from the tail end position of the mechanical arm to the bending direction and the bending angle of the mechanical arm; and obtaining constraint conditions of the pose of the tail end of the mechanical arm based on the quaternion, establishing an inverse kinematics model containing the position and the posture of the tail end of the mechanical arm, and obtaining an analytic solution.

Description

Continuous body mechanical arm inverse kinematics modeling method based on segmented constant curvature
Technical Field
The invention relates to the technical field of mechanical arm control, in particular to a continuous body mechanical arm inverse kinematics modeling method based on a segmented constant curvature.
Background
The multi-redundancy characteristic of the flexible mechanical arm enables the flexible mechanical arm to have better flexibility than the traditional discrete mechanical arm, so that the flexible mechanical arm is widely applied to the related fields of industry, medical treatment and the like, and the flexible mechanical arm can be generally divided into two types: soft bodies and continuous bodies. The soft mechanical arm mostly utilizes a bionic technology to imitate living things in nature from behavior or function, such as octopus tentacles, elephants noses, inchworms and the like, is generally made of elastic materials such as silica gel, rubber and the like, and has infinite freedom degree theoretically. Continuum robots typically have several identical or similar units in tandem or have slots cut into the sidewall of a tubular material, which have a limited number of degrees of freedom compared to soft robots, but are better in load bearing and handling than soft robots. The multi-redundancy characteristic of the continuum mechanical arm enables the continuum mechanical arm to obtain excellent flexibility compared with the traditional discrete mechanical arm, but simultaneously, the forward and inverse kinematics solution of the continuum mechanical arm is difficult, and further development of the continuum mechanical arm in related fields is restricted.
The kinematics model aims to establish a mutual mapping relation between the length of the driving wire and the terminal pose of the continuum mechanical arm. As shown in fig. 1, a driving space, a form space and a working space are established, the driving space includes a driving wire length L, the form space includes a manipulator bending angle Θ and a bending direction Φ, and the working space includes a manipulator tip position and a manipulator attitude Q. The inverse kinematics mapping establishment process is equally divided into two steps, and the mapping establishment sequence is morphological space-driving space and working space-morphological space.
The inventor finds that many scholars adopt different methods to deeply research the kinematics problem of the continuum mechanical arm, for example, a modal method abstracts the macroscopic geometric characteristics of the mechanical arm into a backbone curve with constant curvature change according to the tail end position of the mechanical arm, and solves the inverse kinematics by establishing a shape function; if a forward-inverse kinematics model between the driving force and the tail end position of the mechanical arm is established on the basis of a cantilever beam theory aiming at the notch type continuum mechanical arm; if an ellipse integral is used for providing an instantaneous positive kinematics analysis formula of the multi-backbone continuum mechanical arm, a mapping relation from a morphological space to a working space is established; such as a method for solving the inverse kinematics of a continuum manipulator based on a jacobian matrix, and the like, these conventional methods mainly have the following problems: firstly, only the positive kinematics problem is solved, and the inverse kinematics problem is not considered; secondly, although the forward and inverse kinematics problem is solved, an inverse kinematics analytic solution is not deduced; thirdly, the kinematic model has low calculation efficiency and does not meet the real-time control requirement of the mechanical arm; and fourthly, the tail end posture problem of the mechanical arm is not considered.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a continuous body mechanical arm inverse kinematics modeling method based on a sectional constant curvature, which is characterized in that a mechanical arm inverse kinematics solving geometric model is established by extracting a mechanical arm skeleton curve based on a sectional constant curvature hypothesis; by introducing quaternion, constraint conditions of the terminal pose of the continuum mechanical arm are given; an inverse kinematics model including the terminal position and attitude is established, and an analytical solution is derived.
The invention adopts the following technical scheme:
a continuous body mechanical arm inverse kinematics modeling method based on a segmented constant curvature comprises the following steps:
establishing a geometric model of a mechanical arm joint based on a segmental constant curvature hypothesis to obtain a mapping relation between a bending direction and a bending angle of a mechanical arm and the length of a driving wire;
extracting a backbone curve model of the mechanical arm, and establishing a mapping relation from the tail end position of the mechanical arm to the bending direction and the bending angle of the mechanical arm;
and (3) obtaining a constraint condition of the pose of the tail end of the mechanical arm based on the quaternion, establishing an inverse kinematics model containing the position and the pose of the tail end of the mechanical arm, and obtaining an analytic solution.
Further, the continuum mechanical arm controls the bending motion of the continuum mechanical arm through two groups of driving wires, and the lengths of the two groups of driving wires are expressed as follows:
Figure GDA0002774885680000031
Figure GDA0002774885680000032
Figure GDA0002774885680000033
Figure GDA0002774885680000034
wherein the content of the first and second substances,
Figure GDA0002774885680000035
d represents the diameter of the circumference of the driving wire, N represents the number of the joints of the mechanical arm, and P1、P3And P2、P4The positions of two groups of driving wires are respectively expressed, theta represents the bending angle of the mechanical arm, h represents the length of the driving wire between the two units when the joint does not deflect, and C is a constant.
Further, C ═ NH0+HB+HEWherein H isBIndicating the height of the robot arm base, HEIndicating the height of the end unit of the robot arm, H0To representThe height of the cell.
Further, in the above-mentioned case,
Figure GDA0002774885680000036
wherein HbAnd dbThe chamfer size of the cell is indicated, H the height of the middle cell and D the maximum diameter of the cell.
Further, the end pose Q of the arm is (x, y, z, α, β, γ)TWherein x, y and z represent three-dimensional coordinates of a Cartesian coordinate system of the terminal position, and alpha, beta and gamma represent Euler angles of the terminal attitude.
Further, in the above-mentioned case,
Figure GDA0002774885680000037
wherein, C0＝HE-HB-H。
Further, a constraint condition is obtained according to the quaternion property:
Figure GDA0002774885680000041
Figure GDA0002774885680000042
Figure GDA0002774885680000043
wherein, alpha, beta and gamma represent the tail end postures of the mechanical arm.
Further, the length of the driving wire can be obtained by giving the end pose Q satisfying the constraint condition.
Compared with the prior art, the invention has the beneficial effects that:
(1) the method solves the problems that the traditional inverse kinematics model aiming at the segmented constant-curvature linear drive continuum mechanical arm cannot obtain an analytic solution and the tail end posture of the mechanical arm is not considered, provides a new modeling thought for the continuum mechanical arm, and can realize low-delay control on the position or posture of the continuum mechanical arm by using the method;
(2) the method is based on the assumption of sectional constant curvature, and a mechanical arm inverse kinematics solving geometric model is established by extracting a mechanical arm skeleton curve; by introducing quaternion, constraint conditions of the terminal pose of the continuum mechanical arm are given; an inverse kinematics model including the terminal position and attitude is established, and an analytical solution is derived.
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The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate embodiments of the application and, together with the description, serve to explain the application and are not intended to limit the application.
FIG. 1 is a schematic diagram of forward and inverse kinematics mapping;
FIG. 2 is a schematic diagram of a continuum robot structure according to a first embodiment of the invention;
FIG. 3 is a side view of a continuum robot arm in accordance with a first embodiment of the invention;
FIG. 4 is a schematic illustration of the articulation of a continuum robot arm, in accordance with an embodiment of the present invention;
FIG. 5 is a schematic view of a joint of a continuum robot arm in the absence of deflection according to a first embodiment of the present invention;
FIG. 6 is a schematic view of a joint of the continuum robot arm at an angle θ according to the first embodiment of the present invention;
FIG. 7 is a schematic cross-sectional view of a continuum robot according to a first embodiment of the invention;
FIGS. 8-9 are schematic diagrams of backbone curve models of a continuum robot arm according to an embodiment of the invention;
fig. 10 is a schematic view of a backbone curve model of a mechanical arm after quaternion introduction according to a first embodiment of the invention;
the surgical instrument comprises a base 1, a base 2, a joint 3, a middle unit 4, a rubber tube 5, a driving wire 6, a surgical instrument 7 and a tail end unit.
Detailed Description
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the disclosure. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present application. As used herein, the singular forms "a", "an", and/or "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof;
as introduced in the background art, the prior art has the defects that the traditional inverse kinematics model aiming at the segmented constant-curvature linear drive continuum mechanical arm cannot obtain an analytic solution and does not consider the tail end posture of the mechanical arm, and the invention provides the segmented constant-curvature based inverse kinematics modeling method for the continuum mechanical arm to solve the technical problems.
The first embodiment is as follows:
the present invention will be described in detail with reference to fig. 2 to 10, and specifically, the structure is as follows:
the continuum robot arm structure is shown in FIG. 2, with three degrees of freedom. The continuous mechanical arm main body is connected with a surgical instrument 6 and comprises a plurality of middle units 3, joints 2, a base 1, a tail end unit 7 and a rubber tube 4, the rubber tube 4 is used as a framework to sequentially connect all the units in series, each two units form one joint through spherical surface contact, deflection of the joints is controlled by retraction and extension changes of a driving wire 5, and deflection superposition causes overall bending motion of the mechanical arm. The four driving wires 5 are uniformly distributed on the same circumference, the two driving wires 5 which are centrosymmetric can control the mechanical arm to do forward and reverse bending motion in a plane by winding and unwinding, and the four driving wires 5 can realize the bending motion of the mechanical arm in any direction in an XY plane by winding and unwinding simultaneously.
In the continuous body mechanical arm inverse kinematics modeling method based on the segmented constant curvature, firstly, a geometric model of a mechanical arm joint is established, and a mapping relation between a mechanical arm bending direction and a bending angle to a driving wire length is obtained; then, establishing a mapping relation from the tail end position of the mechanical arm to the bending direction and the bending angle of the mechanical arm by extracting a mechanical arm bone trunk curve model; by introducing quaternion, providing a constraint condition of the terminal pose of the continuum mechanical arm; and finally, establishing an inverse kinematics model containing the terminal position and the terminal attitude, and deducing an analytic solution.
In particular, the method comprises the following steps of,
first, morphology space-drive space:
the continuum mechanical arm joint is formed by two adjacent units in spherical contact, as shown in FIG. 5, the parameter D represents the maximum diameter of the unit, D represents the diameter of the circumference where the driving wire is located, H represents the height of the middle unit, H0Represents the length of the driving wire in the cell, HbAnd dbThe chamfer size of the unit is shown, h represents the length of the drive wire between the two units when the joint is not deflected, h0The distance between the driving wires of the two units when the joint deflects is shown, and the following principle is known:
Figure GDA0002774885680000061
Figure GDA0002774885680000062
as shown in fig. 6, when the joint is deflected by an angle θ in the plane of the drive wire, it can be deduced from the geometrical relationship:
Figure GDA0002774885680000071
Figure GDA0002774885680000072
wherein h islAnd hrThe length of the two driving wires which are centrosymmetric between the two units.
Assuming that the continuum mechanical arm has N joints, according to the principle of piecewise constant curvature, the deflection angle of each joint is equal and is θ, then:
Ll＝Nhl+C
Lr＝Nhr+C (4)
wherein C is NH0+HB+HEWhich is always constant, LlAnd LrIs the length of a group of driving wires in the mechanical arm HBIndicating the height of the robot arm base, HEIndicating the robot arm end unit height.
Order to
Figure GDA0002774885680000073
The following can be derived from equations (3) and (4):
Figure GDA0002774885680000074
the deflection of the joint results in a bending motion of the mechanical arm, with the arm bending angle Θ being expressed as:
Θ＝Nθ (6)
there is a maximum value for the arm bending angle, which is determined by the unit own parameters of the joint:
Figure GDA0002774885680000075
the continuum manipulator controls its bending motion by two sets of drive wires, P as shown in FIG. 71、P3And P2、P4Respectively representing the positions of two groups of driving wires, assuming that the mechanical arm bends along the direction with the included angle phi with the X axis, the phi belongs to [0,2 pi ], the deflection angle of each joint is theta, and P is1 E、P2 E、P3 E、P4 EAre respectively P1、P2、P3、P4At the equivalent point on the imaginary axis X', the lengths of the two sets of drive wires in the robot arm are expressed as:
Figure GDA0002774885680000081
Figure GDA0002774885680000082
Figure GDA0002774885680000083
Figure GDA0002774885680000084
wherein
Figure GDA0002774885680000085
When Φ is 0, in particular, the robot arm is bent in the positive direction along the X axis,
Figure GDA0002774885680000086
at this time
Figure GDA0002774885680000087
The size is the same and varies with the variation of theta.
Two, working space-form space
Pose of end of arm can be Q ═ x, y, z, alpha, beta, gamma)TWhere X, Y, Z are the three-dimensional coordinates of a cartesian coordinate system characterizing the tip position, and α, β, γ are the RPY (rotation order X-Y-Z) euler angles characterizing the tip pose.
As shown in fig. 8 and 9, when
Figure GDA0002774885680000088
According to the triangle geometrical relationship, the following can be obtained:
Figure GDA0002774885680000089
Figure GDA00027748856800000810
Figure GDA00027748856800000811
Figure GDA0002774885680000091
substituting equation (12) into equation (13) and eliminating the redundant solution yields
Figure GDA0002774885680000092
Wherein, C0＝HE-HB-H。
When in use
Figure GDA0002774885680000093
Equations (10), (11) and (14) can still be obtained based on the above method, and will not be described herein.
The continuum robot arm has three degrees of freedom, and therefore there must be a constraint that limits the end pose Q. As shown in fig. 10, the transformation process of the coordinate systems { B } to { E } can be understood as a rotation transformation R (Θ, R) around the unit vector R (-sin Φ, cos Φ,0) and a translation transformation along l. R (theta, R) can be represented by a unit quaternion q ═ eta + uiB+vjB+wkBIs uniquely represented and has
Figure GDA0002774885680000094
Wherein iB、jB、kBThree basis vectors of the coordinate system B. By substituting equations (11) and (14) into equation (15), an expression in which q is expressed by the end position coordinates x, y, z can be obtained:
Figure GDA0002774885680000095
the quaternion property can indicate that the alpha, beta and gamma representing the tail end attitude of the mechanical arm have the following relationship with the quaternion q:
Figure GDA0002774885680000101
Figure GDA0002774885680000102
Figure GDA0002774885680000103
Figure GDA0002774885680000104
Figure GDA0002774885680000105
Figure GDA0002774885680000106
Figure GDA0002774885680000107
the constraint conditions are obtained by combining the formulas (17) and (18):
Figure GDA0002774885680000108
Figure GDA0002774885680000109
Figure GDA00027748856800001010
the combination of the formulas (10), (15) and (18) can obtain another expression form of the constraint
x＝f(α，β，γ)
y＝g(α，β，γ)
z＝h(α，β，γ) (20)
And (3) completing establishment of inverse kinematics mapping of the continuum mechanical arm, obtaining inverse kinematics analytic solutions of formulas (8), (10), (14), (19) and (20), and solving the length of the driving wire by giving a terminal pose Q meeting the constraint condition (19) or (20).
The above description is only a preferred embodiment of the present application and is not intended to limit the present application, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, improvement and the like made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims (7)

1. A continuous body mechanical arm inverse kinematics modeling method based on a segmented constant curvature is characterized by comprising the following steps:
establishing a geometric model of a mechanical arm joint based on a segmental constant curvature hypothesis to obtain a mapping relation between a bending direction and a bending angle of a mechanical arm and the length of a driving wire;
the continuum mechanical arm controls the bending motion of the continuum mechanical arm through two groups of driving wires, wherein the lengths of the two groups of driving wires are expressed as follows:
Figure FDA0002809413220000011
Figure FDA0002809413220000012
Figure FDA0002809413220000013
Figure FDA0002809413220000014
wherein the content of the first and second substances,
Figure FDA0002809413220000015
Φ∈[0,2π)
d represents the diameter of the circumference of the driving wire, N represents the number of the joints of the mechanical arm, and P1、P3And P2、P4Respectively representing the positions of two groups of driving wires, theta represents the bending angle of the mechanical arm, h represents the length of the driving wire between the two units when the joint does not deflect, and C is a constant;
extracting a backbone curve model of the mechanical arm, and establishing a mapping relation from the tail end position of the mechanical arm to the bending direction and the bending angle of the mechanical arm;
and (3) obtaining a constraint condition of the pose of the tail end of the mechanical arm based on the quaternion, establishing an inverse kinematics model containing the position and the pose of the tail end of the mechanical arm, and obtaining an analytic solution.
2. The method for modeling the inverse kinematics of the continuum manipulator based on piecewise constant curvature according to claim 1, wherein C ═ NH0+HB+HEWherein HBIndicating the height of the robot arm base, HEIndicating the height of the end unit of the robot arm, H0Indicating the drive wire length within the cell.
3. The modeling method for the inverse kinematics of the continuum manipulator according to claim 2,
Figure FDA0002809413220000021
wherein HbAnd dbThe chamfer size of the cell is indicated, H the height of the middle cell and D the maximum diameter of the cell.
4. The modeling method for the inverse kinematics of the continuum manipulator based on piecewise constant curvature of claim 1, wherein the pose Q of the end of the manipulator is (x, y, z, α, β, γ)TWherein x, y and z represent three-dimensional coordinates of a Cartesian coordinate system of the terminal position, and alpha, beta and gamma represent Euler angles of the terminal attitude.
5. The method for modeling the inverse kinematics of a continuum manipulator according to claim 4,
Figure FDA0002809413220000022
wherein, C0＝HE-HB-H, wherein H represents the height of the intermediate unit, HBIndicating the height of the robot arm base, HEIndicating the robot arm end unit height.
6. The method for modeling the inverse kinematics of the continuum manipulator based on the piecewise constant curvature according to claim 1, wherein the constraint condition is obtained according to quaternion properties:
Figure FDA0002809413220000023
Figure FDA0002809413220000024
Figure FDA0002809413220000025
wherein, alpha, beta and gamma are RPY (revolution number) for representing the terminal postureThe sequence X-Y-Z Euler angles, X, Y, Z being the three-dimensional coordinates of a Cartesian coordinate system characterizing the position of the tip, C0＝HE-HB-H, wherein H represents the height of the intermediate unit, HBIndicating the height of the robot arm base, HEIndicating the robot arm end unit height.
7. The method for modeling the inverse kinematics of the continuum manipulator based on the piecewise constant curvature of claim 6, wherein the length of the driving wire can be obtained by giving an end pose Q satisfying constraint conditions.
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