CN106844951B - Method and system for solving inverse kinematics of super-redundant robot based on segmented geometric method - Google Patents

Method and system for solving inverse kinematics of super-redundant robot based on segmented geometric method Download PDF

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CN106844951B
CN106844951B CN201710040121.2A CN201710040121A CN106844951B CN 106844951 B CN106844951 B CN 106844951B CN 201710040121 A CN201710040121 A CN 201710040121A CN 106844951 B CN106844951 B CN 106844951B
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elbow
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shoulder
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CN106844951A (en
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徐文福
牟宗高
梁斌
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Shenzhen Graduate School Harbin Institute of Technology
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
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    • B25J9/1605—Simulation of manipulator lay-out, design, modelling of manipulator

Abstract

The invention discloses a method for solving the inverse kinematics of a super-redundant robot based on a piecewise geometric method, which comprises the following steps: the system segments the mechanical arm of the n-degree-of-freedom robot into a shoulder, an elbow and a wrist; the system solves and determines a head end node S of the shoulder according to the wrist tail end node T0Shoulder and elbow intersection E0And the intersection point W of the elbow and the wrist0The location of each node; the system solves the angles of all joints of the shoulder; the system solves each joint angle of the elbow; the system solves the angles of the joints of the wrist. The invention also provides a system for solving the inverse kinematics of the super-redundant robot based on the piecewise geometry method, which starts with the solution of the positions of all the selected key nodes based on the target position and determines the reasonable configuration of the super-redundant mechanical arm in a 3D space according to the expected tail end position and the posture of the super-redundant mechanical arm.

Description

Method and system for solving inverse kinematics of super-redundant robot based on segmented geometric method
Technical Field
The invention relates to the field of inverse kinematics of robots, in particular to a method and a system for solving inverse kinematics of a super-redundant mechanical arm formed by serially connecting universal joints based on a piecewise geometric method.
Background
Compared with the traditional mechanical arm, the super-redundant mechanical arm has more degrees of freedom, and shows extremely strong flexibility in a complex and multi-obstacle environment, so that the super-redundant mechanical arm is widely applied to tasks of overhauling, maintaining, rescuing and the like of large equipment in the nuclear power field and the aerospace field. Motion planning is a key point and difficulty in the research of the hyper-redundancy robot mechanics, however, inverse kinematics solution is a necessary link of the motion planning. Generally, inverse kinematics solution methods for redundant robotic arms can be divided into three categories: (1) algebraic method (2) iterative method (3) geometric method. Due to the existence of the super-redundant degree of freedom, the super-redundant mechanical arm can realize some additional tasks such as: optimizing joint velocity, avoiding obstacles, avoiding singular configurations, avoiding joint limits, minimizing drive joint moments.
Takegaki attributes the inverse kinematics problem of the super-redundant mechanical arm to the problem of determining the ridge line configuration changing along with time, and provides a new method for determining the optimal configuration of the mechanical arm based on a kinematics formula. Chirikjian proposes the concept of a ridge line method aiming at the kinematics planning of the super-redundancy mechanical arm, lays the theoretical basis of the kinematics of the super-redundancy robot, adopts the ridge line method to describe the macroscopic configuration of the super-redundancy mechanical arm, and obtains a plurality of important conclusions. Mochiyama carries out deep research on the control problem of the super-redundant rigid connecting rod mechanical arm formed by two-degree-of-freedom joints by adopting a space curve method. Some researchers have applied the mode function method to the motion planning of planar super-redundant robots. One variable in the curvature function defined by the planar ridge controls the direction of the ridge at the root, and the other variables are used to achieve obstacle avoidance. Meanwhile, a simple plane ridge line fitting method is introduced, namely, the tangent line of the ridge line is used as the direction of the rod piece, and the method effectively reduces the position error of the end effector. The artificial neural network method is widely applied to inverse solution of an industrial mechanical arm, but the size of a required training set can be changed significantly along with the change of the number of degrees of freedom, and the requirement of real-time performance can not be met due to the extension of the learning process.
In order to reduce the problem of high calculation amount caused by a large number of degrees of freedom of the super-redundant mechanical arm, the technology needs to be improved.
Disclosure of Invention
In order to solve the technical problems, the invention aims to provide a method and a system for solving the inverse kinematics of a super-redundant robot based on a segmented geometry method.
The technical scheme adopted by the invention is as follows:
the invention provides a method for solving the inverse kinematics of a super-redundant robot based on a segmental geometry method, which comprises the following steps:
the system represents a robot arm segment of an n-degree-of-freedom robot as comprising a shoulder, an elbow, a wrist, where n > 8;
the system is based on the wrist end node T and the position I of the grabbing targetThen, the head end node S of the shoulder is determined by solving0Shoulder and elbow intersection E0And the intersection point W of the elbow and the wrist0The location of each node;
the system solves the angles of all joints of the shoulder according to the selected arm type angle;
the system solves each joint angle of the elbow;
the system utilizes the intersection point W of the elbow and the wrist0And solving the angle of each joint of the wrist by the wrist tail end node T.
As a refinement of this solution, the shoulder comprises 4 degrees of freedom and the wrist comprises 2 degrees of freedom.
As an improvement of the technical scheme, the system utilizes the spatial circular arc parameters to solve the angles of all the joints of the elbow.
As an improvement of the technical scheme, the method also comprises the following steps: the system is based on the cross node E of the shoulder and the elbow0And the intersection point W of the elbow and the wrist0The space circular arc parameters are obtained α,
Figure BDA0001213126940000031
the value of phi.
As an improvement of the technical scheme, the system of the step solves and determines a head end node S of the shoulder according to the consistency of the positions of the wrist tail end node T and the grabbing target0Shoulder and elbow intersection E0And the intersection point W of the elbow and the wrist0The location of each node, wherein:
the cross joint W of the elbow and the wrist0Can be expressed as a position of (a) or (b),
Figure BDA0001213126940000032
wherein [ p ] isx,py,pz]TIs a coordinate system ∑nThe origin position of (a); d is in the coordinate
The desired direction vector under {0 }; leIndicating the length of the endmost link of the wrist.
Further, the intersection node E of the shoulder and the elbow0The position of (d) can be expressed as:
Figure BDA0001213126940000033
wherein the shoulder includes 4 degrees of freedom, then S2And E0Coincidence, said l represents non-terminal
The length of the end connecting rod, and n is the total freedom degree of the robot.
Further, each node E of the elbow0,E1,…,En/2-3Can be expressed as:
Figure BDA0001213126940000041
wherein j is 1, …, n/2-3.
Further, the system calculates the intersection node E of the shoulder and the elbow0And selecting an arm type angle psi to obtain the angle of each joint of the shoulder.
On the other hand, the invention also provides a system for solving the inverse kinematics of the super-redundant robot based on the segmented geometric method, which comprises the following steps:
a first module for performing a step system representing a robot arm segment of an n-degree-of-freedom robot as including a shoulder, an elbow, a wrist, wherein n > is 8;
a second module, which is used for solving and determining a head end node S of the shoulder according to the consistency of the wrist tail end node T and the position of the grabbing target by the system for executing the steps0Shoulder and elbow intersection E0And the intersection point W of the elbow and the wrist0The location of each node;
the third module is used for executing the step system to solve the angle of each joint of the shoulder according to the selected arm type angle;
the fourth module is used for executing the step system to solve each joint angle of the elbow;
a fifth module for performing the step system utilizing an intersection of the elbow and wristFork node W0And solving the angle of each joint of the wrist by the wrist tail end node T.
The invention has the beneficial effects that: according to the method and the system for solving the inverse kinematics of the super-redundant robot based on the segmented geometry method, the robot of the multi-degree-of-freedom mechanical arm is divided into three parts by geometric segmentation, based on the target position, the reasonable configuration of the super-redundant mechanical arm is determined in a 3D space according to the expected tail end position and the attitude of the super-redundant mechanical arm based on the solution of the positions of all selected key nodes, and the method and the system have the advantages of high calculation efficiency, convenience, rapidness, unique solution and high accuracy.
Drawings
The following further describes embodiments of the present invention with reference to the accompanying drawings:
FIG. 1 is a schematic diagram of a D-H coordinate system according to an embodiment of the present invention;
FIG. 2 is a schematic view of a segment rule of a super redundant manipulator according to a second embodiment of the present invention;
FIG. 3 is a schematic illustration of the piecewise geometry rule cost effective of the third embodiment of the present invention;
FIG. 4 is a fourth embodiment of the present invention relative to a reference plane S0E0E1Schematic arm angle diagram of (1).
Detailed Description
It should be noted that the embodiments and features of the embodiments in the present application may be combined with each other without conflict.
The invention provides a super-redundancy mechanical arm inverse kinematics solving method based on a segmental geometric method, aiming at solving the problem of inverse kinematics solving of a super-redundancy mechanical arm. The method can determine the reasonable configuration of the super-redundant manipulator in the 3D space according to the expected tail end position and the expected attitude of the super-redundant manipulator. The method can obtain the required configuration by adjusting a small number of geometric parameters according to specific task requirements. All joint angles are solved in an analytic solution mode, the calculation efficiency is high, the method is suitable for real-time online control, and meanwhile, the method is also suitable for various super-redundancy configurations.
The invention provides a method for solving the inverse kinematics of a super-redundant robot based on a segmental geometry method, which comprises the following steps:
the system represents a mechanical arm segment of an n-degree-of-freedom robot as comprising a shoulder, an elbow and a wrist, wherein n is 2a, and a is an integer greater than or equal to 4; because this type of arm is the universal joint constitution, a universal joint comprises two degrees of freedom.
The system solves and determines a head end node S of the shoulder according to the consistency of the positions of the wrist tail end node T and the grabbing target0Shoulder and elbow intersection E0And the intersection point W of the elbow and the wrist0The location of each node;
the system solves the angles of all joints of the shoulder according to the selected arm type angle;
the system solves each joint angle of the elbow;
the system utilizes the intersection point W of the elbow and the wrist0And solving the angle of each joint of the wrist by the wrist tail end node T.
As a refinement of this solution, the shoulder comprises 4 degrees of freedom and the wrist comprises 2 degrees of freedom.
As an improvement of the technical scheme, the system utilizes the spatial circular arc parameters to solve the angles of all the joints of the elbow.
As an improvement of the technical scheme, the method also comprises the following steps: the system is based on the cross node E of the shoulder and the elbow0And the intersection point W of the elbow and the wrist0The space circular arc parameters are obtained α,
Figure BDA0001213126940000061
the value of phi.
As an improvement of the technical scheme, the system of the step solves and determines a head end node S of the shoulder according to the consistency of the positions of the wrist tail end node T and the grabbing target0Shoulder and elbow intersection E0And the intersection point W of the elbow and the wrist0The location of each node, wherein:
the cross joint W of the elbow and the wrist0Can be expressed as a position of (a) or (b),
Figure BDA0001213126940000062
wherein [ p ] isx,py,pz]TIs a coordinate system ∑nThe origin position of (a); d is the desired direction vector under the coordinate system {0 }; leIndicating the length of the endmost link of the wrist.
Further, the intersection node E of the shoulder and the elbow0The position of (d) can be expressed as:
Figure BDA0001213126940000071
wherein the shoulder includes 4 degrees of freedom, then S2And E0And (4) overlapping, wherein l represents the length of the non-tail end connecting rod, and n is the total freedom degree of the robot.
Further, each node E of the elbow0,E1,…,En/2-3Can be expressed as:
Figure BDA0001213126940000072
wherein j is 1, …, n/2-3.
Further, the system calculates the intersection node E of the shoulder and the elbow0And selecting an arm type angle psi to obtain the angle of each joint of the shoulder.
On the other hand, the invention also provides a system for solving the inverse kinematics of the super-redundant robot based on the segmented geometric method, which comprises the following steps:
a first module for performing a step system representing a robot arm segment of an n-degree-of-freedom robot as including a shoulder, an elbow, a wrist, wherein n > is 8;
a second module, which is used for solving and determining a head end node S of the shoulder according to the consistency of the wrist tail end node T and the position of the grabbing target by the system for executing the steps0Shoulder and elbow intersection E0And the intersection point W of the elbow and the wrist0The location of each node;
the third module is used for executing the step system to solve the angle of each joint of the shoulder according to the selected arm type angle;
the fourth module is used for executing the step system to solve each joint angle of the elbow;
a fifth module for performing the step system using the intersection node W of the elbow and wrist0And solving the angle of each joint of the wrist by the wrist tail end node T.
The algorithm overall principle mainly comprises the following steps:
referring to fig. 1, a schematic diagram of a D-H coordinate system according to an embodiment of the invention is shown. The D-H coordinate system of the super-redundancy mechanical arm is similar to that of a human arm, and the super-redundancy mechanical arm is divided into three sections: shoulder (shoulder S), elbow (elbow E), wrist (wrist W), and key nodes are denoted by the symbol S0，E0，W0And T, refer to FIG. 2.
Referring to fig. 3, a schematic diagram of a third embodiment of the present invention is shown for cost-effective segment geometry legislation. Wherein the shoulder comprises the first four joints theta1To theta4Is configured to locate an elbow initial point E0While one degree of freedom is redundant, the arm angle parameter ψ is used to express this redundancy.
Referring to FIG. 4, a fourth embodiment of the present invention is shown with respect to a reference plane S0E0E1Schematic arm angle diagram of (1). When the desired elbow initial point E0It is known that all 4 joints can be solved at a specific arm angle. Theta5To thetan-2The elbow joint is formed, and similar to the arm of a human, the elbow joint can enable the tool at the tail end of the mechanical arm to move in a large range in space, and meanwhile, the elbow joint also plays an important role in matching a specific task configuration. In order to solve the elbow multi-joint angles, the scheme adopts a fitting method based on a space circular arc, so that all even-numbered joint motion angles are equal, and all odd-numbered joint motion angles are equal. The processing mode can effectively avoid the internal singularity of the joint and the overrun movement of the joint. The wrist joint is formed by the last universal joint (theta)n-1To thetan) For determining the pointing direction of the end effector.
The following notation is defined for ease of discussion:
n is the number of degrees of freedom of the super-redundant mechanical arm.
θiAn ith joint angle, i ═ 1,2.. n;
Suthe position of the shoulder gimbal can be expressed as
Figure BDA0001213126940000091
EvElbow gimbal position, can be expressed as
Figure BDA0001213126940000092
W0Wrist gimbal position, which can be expressed as
Figure BDA0001213126940000093
T-tool coordinate center position, which can be expressed as T (x)T,yT,zT)；
O0Coordinate system {0} origin position;
OEorigin position of coordinate system { E }, and S2(E0) The posture of the coordinate system { E } is the same as the posture of the coordinate system {0 }; wherein S2And E0Representing the same position, En/2-3And W0Representing the same location.
First, a key point T, W is determined0,S2(E0),S0The value of (c). Key point T, S0The values of (d) are the same as the tool tip desired position and the base position, respectively, and can be considered known. The orientation of the end effector relative to the coordinate system {0} is used to solve for the point W0,S2(E0) From the locations of the wrist and elbow points, α for the spatial arc parameters can then be solved,
Figure BDA0001213126940000094
the value of phi.
Secondly, selecting an arm type angle to solve the shoulder joint angle;
thirdly, solving the elbow joint angle by using the spatial circular arc parameters;
and fourthly, solving the wrist joint angle by utilizing the wrist position and the effector position.
The key point solving steps are as follows:
assume that a given end effector position and pose is:
Figure BDA0001213126940000101
wherein [ n ]x,ny,nz]T,[ox,oy,oz]TAnd [ a ]x,ay,az]TIs a unit vector representing the x-axis, y-axis, z-axis of the end effector. [ p ]x,py,pz]TIs a coordinate system ∑nThe origin position of (a);0Rnis a rotation matrix of 3 × 3.
S0Is the origin of the global coordinate system, and can be considered as zero in all coordinate values;
solving the wrist point: the desired direction vector is set to D in the coordinate system {0}, which can be determined by a given rotation matrix0RnFind, therefore, the wrist point can be defined as:
Figure BDA0001213126940000102
wherein leIndicating the length of the endmost link of the wrist.
And (3) solving a shoulder end point: in the coordinate system {0}, the coordinates of the shoulder end point (elbow start point) can be expressed as
Figure BDA0001213126940000103
l represents the non-end link length; except for the end rod leAnd the other connecting rods are all equal in length. Solving the space arc parameters:
the fitting configuration of the elbow is a space circular arc, and a node falling on the circular arcAre respectively represented as E0,E1,…,En/2-3(W0). Correspondingly, the connecting rod forming the elbow is the chord length of the space circular arc, and the first chord length is E0E1Its vector in space can be represented as:
Figure BDA0001213126940000111
this spatial arc, such as α,
Figure BDA0001213126940000112
φ, defined as follows:
α∈[0,2π]vector (1)
Figure BDA0001213126940000113
And plane YE-ZEThe included angle between the two;
Figure BDA0001213126940000114
vector quantity
Figure BDA0001213126940000115
In the plane YE-ZEProjection on and the YE-an angle between axis;
phi epsilon < -pi, pi > is the included angle between adjacent connecting rods.
Based on the spatial arc parameters α,
Figure BDA0001213126940000116
phi, elbow node E0,E1,…,En/2-3Can be expressed as:
Figure BDA0001213126940000117
wherein j is 1, …, n/2-3.
According to the above En/2-3(W0) And E0(S2) Has determined, the spatial arc parameters α,
Figure BDA0001213126940000118
phi can be obtained by substituting the boundary value j of n/2-3 into (4) in equation (4).
Shoulder joint solution
Shoulder joint node S2The position relative to the coordinate system 0 depends on the first four joint angles, i.e. theta1To theta4. This in effect constitutes a 4 degree of freedom robot arm, since S2Has already found out the position of theta1To theta4Can be obtained in the same way. Obviously, one degree of freedom is redundant for spatial 3-coordinate positioning, and in order to utilize the redundant degree of freedom, the scheme defines a parameter, namely the arm type angle psi. Referring to FIG. 4, S based on arm angle ψ1Is defined as S1ψWhen S is1ψAfter the position of (2) is determined, θ1,θ2And theta3,θ4Can be based on S1ψAnd S2The spatial position of (2) is obtained.
Solving for S1ψIn space of
The reference plane is defined as S0E0E1，S1The possible spatial positions form a spatial circle, the center of which is defined as S, as shown in FIG. 41cWhile falling on the plane S0E0E1And two points on the spatial circle are defined as S1aAnd
Figure BDA0001213126940000121
the arm angle is defined as the slave plane S0S1aS2To the plane S0S1ψS2Angle of (S)1ψSatisfies the following conditions:
a)S1ψin the plane S1ψS1cS1The above step (1);
b) point S1ψDistance to coordinate system {0} is l;
c) the arm angle psi is defined as S1cS1aAnd S1cS1ψThe included angle of (a).
Solving for theta1To thetan：
Figure BDA0001213126940000122
Figure BDA0001213126940000123
Figure BDA0001213126940000124
Figure BDA0001213126940000125
Figure BDA0001213126940000126
Figure BDA0001213126940000127
Figure BDA0001213126940000131
Figure BDA0001213126940000132
Figure BDA0001213126940000133
Figure BDA0001213126940000134
Figure BDA0001213126940000135
θn-1＝arctan2(ynr/lecn,xnr/lecn) (16)
According to the method and the system for solving the inverse kinematics of the super-redundant robot based on the segmented geometry method, the robot of the multi-degree-of-freedom mechanical arm is divided into three parts by geometric segmentation, based on the target position, the reasonable configuration of the super-redundant mechanical arm is determined in a 3D space according to the expected tail end position and the attitude of the super-redundant mechanical arm based on the solution of the positions of all selected key nodes, and the method and the system have the advantages of high calculation efficiency, convenience, rapidness, unique solution and high accuracy.
While the preferred embodiments of the present invention have been illustrated and described, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (9)

1. A method for solving the inverse kinematics of a super-redundant robot based on a piecewise geometric method is characterized by comprising the following steps:
the system represents a robot arm segment of an n-degree-of-freedom robot as comprising a shoulder, an elbow, a wrist, where n > 8;
the system solves and determines a head end node S of the shoulder according to the consistency of the positions of the wrist tail end node T and the grabbing target0Shoulder and elbow intersection E0And the intersection point W of the elbow and the wrist0The location of each node;
the system solves the angles of all joints of the shoulder according to the selected arm type angle;
the system solves each joint angle of the elbow;
the system utilizes the intersection point W of the elbow and the wrist0And solving each joint angle of the wrist by a wrist tail end node T;
the angles of the joints of the shoulder are calculated as follows:
Figure FDA0002519846670000011
θ1＝arctan2(ys1/lc2,xs1/lc2)；
Figure FDA0002519846670000012
θ3＝arctan2(ys2r/lc4,xs2r/lc4)。
2. the method for solving the inverse kinematics of the hyper-redundant robot based on the piecewise geometry method of claim 1, wherein the shoulder comprises 4 degrees of freedom and the wrist comprises 2 degrees of freedom.
3. The method for solving the inverse kinematics of the hyper-redundant robot based on the piecewise geometry of claim 2, wherein the system uses spatial circular arc parameters to solve for each joint angle of the elbow.
4. The method for solving the inverse kinematics of the hyper-redundant robot based on the piecewise geometric method according to claim 3, further comprising the steps of: the system is based on the cross node E of the shoulder and the elbow0And the intersection point W of the elbow and the wrist0The space circular arc parameters are obtained α,
Figure FDA0002519846670000013
a value of phi;
wherein, α∈ [0, 2 pi ]]Represents a vector
Figure FDA0002519846670000014
The included angle between the plane YE-ZE and the plane,
Figure FDA0002519846670000015
representing a vector
Figure FDA0002519846670000016
In the plane YE-ZEProjection on and the YE-clips between axisAngle phi ∈ [ -pi, pi]And indicates the angle between adjacent links.
5. The method for solving the inverse kinematics of the hyper-redundant robot based on the piecewise geometry method according to claim 4, wherein the step system solves the determination of the head end node S of the shoulder according to the coincidence of the wrist end node T and the position of the grabbing target0Shoulder and elbow intersection E0And the intersection point W of the elbow and the wrist0The location of each node, wherein:
the cross joint W of the elbow and the wrist0Can be expressed as a position of (a) or (b),
Figure FDA0002519846670000021
wherein [ p ] isx,py,pz]T is the origin position of the coordinate system Σ n; d is the desired direction vector under the coordinate system {0 }; le denotes the length of the endmost link of the wrist.
6. The method for solving the inverse kinematics of the hyper-redundant robot based on the piecewise geometry method according to claim 5, wherein the intersection node E of the shoulder and the elbow0The position of (d) can be expressed as:
Figure FDA0002519846670000022
wherein the shoulder includes 4 degrees of freedom, then S2And E0And (4) overlapping, wherein l represents the length of the non-tail end connecting rod, and n is the total freedom degree of the robot.
7. The method for solving the inverse kinematics of the hyper-redundant robot based on the piecewise geometric method according to claim 6, wherein each node E of the elbow0,E1,…,En/2-3Can be expressed as:
Figure FDA0002519846670000023
wherein j is 1, …, n/2-3.
8. The method for solving the inverse kinematics of the hyper-redundant robot based on the piecewise geometry method according to claim 7, wherein the system is configured to calculate the intersection node E of the shoulder and the elbow0And selecting an arm type angle psi to obtain the angle of each joint of the shoulder.
9. A system for solving the inverse kinematics of a super-redundant robot based on a piecewise geometric method is characterized by comprising the following steps:
a first module for performing a step system representing a robot arm segment of an n-degree-of-freedom robot as including a shoulder, an elbow, a wrist, wherein n > is 8;
a second module, which is used for solving and determining a head end node S of the shoulder according to the consistency of the wrist tail end node T and the position of the grabbing target by the system for executing the steps0Shoulder and elbow intersection E0And the intersection point W of the elbow and the wrist0The location of each node;
the third module is used for executing the step system to solve the angle of each joint of the shoulder according to the selected arm type angle;
the fourth module is used for executing the step system to solve each joint angle of the elbow;
a fifth module for performing the step system using the intersection node W of the elbow and wrist0And solving each joint angle of the wrist by a wrist tail end node T;
the angles of the joints of the shoulder are calculated as follows:
Figure FDA0002519846670000031
θ1＝arctan2(ys1/lc2,xs1/lc2)；
Figure FDA0002519846670000032
θ3＝arctan2(ys2r/lc4,xs2r/lc4)。
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