CN107756400B - 6R robot inverse kinematics geometric solving method based on momentum theory - Google Patents

6R robot inverse kinematics geometric solving method based on momentum theory Download PDF

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CN107756400B
CN107756400B CN201710953599.4A CN201710953599A CN107756400B CN 107756400 B CN107756400 B CN 107756400B CN 201710953599 A CN201710953599 A CN 201710953599A CN 107756400 B CN107756400 B CN 107756400B
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刘志峰
许静静
赵永胜
蔡力钢
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Abstract

The invention discloses a 6R robot inverse kinematics geometric solving method based on a momentum theory, and belongs to the field of robot kinematics inverse solution method research. Establishing a base coordinate system and a tool coordinate system, determining kinematic parameters of the 6R robot through the base coordinate system and the tool coordinate system, and establishing a positive kinematic model. And (3) decomposing the inverse solution motion of the first three joints of the 6R robot and establishing a six-membered quadratic equation set. Initial position q is solved based on positive kinematics model of momentums1Corresponding target position qe1. The method combines geometric description with a momentum theory, has more definite geometric meaning, solves an algebraic equation set by simplifying an inverse solution algorithm, effectively improves the calculation efficiency, and can provide a new inverse solution processing method for the real-time control of the robot motion.

Description

6R robot inverse kinematics geometric solving method based on momentum theory
Technical Field
The invention belongs to the field of robot kinematics inverse solution method research, and particularly relates to a 6R robot inverse kinematics geometric solution method based on a momentum theory.
Background
Kinematic analysis is the basis for realizing motion control, and mainly establishes a mapping model of joint variables and terminal poses. The inverse solution problem, namely the problem of solving joint variables by knowing the end pose, is one of research hotspots in the field of robots, and the solution efficiency directly influences the real-time performance of robot motion control. At present, robot kinematics inverse solution modeling is mainly based on a D-H method and a rotation theory, researchers compare the two methods and find that the application of the latter has the following advantages: the establishment of a local coordinate system can be avoided, a calculation model is simplified, and the singularity generated by local parameters is overcome; the geometric meaning is clear, and the condition and the number of the generated multiple solutions can be conveniently determined. At present, the Paden-Kahan subproblem based on the momentum description is used for solving problems reversely by a wide robot, but the method is not suitable for a six-degree-of-freedom robot with any configuration, so most scholars extend and provide some new subproblem models based on three basic subproblems, and if Tan improves a subproblem II, a mathematical model for a subproblem of 'rotating motion around two non-intersecting axes' is established; chen describes a sub-problem of "rotational motion about three axes, two of which are parallel and out of plane with the third axis", for the inverse solution of which detailed momentum descriptions are made. Based on the subproblem models, the subproblem models are widely applied to the inverse solution problem of the six-degree-of-freedom series robot, and research shows that the inverse kinematics problem of the 6R robot has a closed solution when the 6R robot meets the Piper criterion. The method is characterized in that the inverse kinematics solution of the 6R robot is realized by the Sariyldi and the like based on three sub-problems; the Lushizeng et al reduces the dependency on the subproblems by introducing a Wu method into the rotation method, and converts the inverse solution problem of the 6R robot into the solution problem of a six-element eight-order equation set. In order to further simplify the inverse solution model of the 6R robot, the inverse solution problem of the 6R robot is converted into a six-element quadratic equation set and a solution of the three-element quadratic equation set through a geometric description method, so that the geometric meaning of the robot is clearer, and the solution process is simpler.
Disclosure of Invention
The invention aims to provide a method for solving the inverse kinematics geometry of a 6R robot based on a momentum theory. The method is mainly characterized in that an inverse kinematics model with clear geometric meaning and simple solving process is provided for the inverse kinematics problem of the 6R robot kinematics by combining a geometric description method and a momentum theory.
In order to achieve the purpose, the technical means adopted by the invention is a 6R robot inverse kinematics geometric solving method based on a momentum theory, and the method is realized by the following steps:
and S1, establishing a base coordinate system and a tool coordinate system, determining kinematic parameters of the 6R robot through the base coordinate system and the tool coordinate system, and establishing a positive kinematic model.
S2, as shown in fig. 1, describes the inverse solution motion of the first three joints of the 6R robot: from point qeTo point c2＝[x2 y2z2]A rotational movement around the joint 1; point c2To point c1＝[x1 y1 z1]A rotational movement around the joint 2; point c1To point qsWinding switchThe rotational movement of the segments 3. Wherein q iss，qeInitial and target positions of the robot tip, respectively, are established by x according to the geometric relationship description shown in fig. 11,y1,z1,x2,y2And z2Is a six-membered quadratic system of variables.
S3, solving the equation set by adopting a 'SOLVE' function in MATLAB to obtain c1And c2Is calculated from the coordinate vector of (1).
S4, knowing the inverse solution motion starting and ending positions of the first three joints, and respectively establishing the first three joint angle variables theta of the 6R robot based on the Pasen-Kahan subproblem 11，θ2And theta3And (4) explicitly solving the model and solving.
S5, taking one point on the axis of the joint 6 as the initial position q in the inverse solution movement of the last three jointss1And solving the initial position q based on the momentum positive kinematics models1Corresponding target position qe1。
S6, fig. 2, describes the inverse kinematics of joints 4 and 5 in a decomposition: from point qe1To point c3＝[x3y3z3]The rotational movement of (a); from point c3To point qs1The rotational movement of (a). According to FIG. 2, the x is established according to the geometric relationship3,y3And z3Is a ternary quadratic equation system of variables and is solved by adopting a 'SOLVE' function to obtain c3Is calculated from the coordinate vector of (1).
S7, the inverse solution motion starting and ending positions of the joint 4 and the joint 5 are known, and joint angle variables theta of the joint 4 and the joint 5 are respectively established based on the Pasen-Kahan subproblem 14And theta5And (5) explicitly solving the model and solving.
S8, taking any point q not on the axis of the joint 6s2Solving q based on a momentum positive kinematics models2Corresponding target position qe2Then, the position of the start and end of the inverse kinematics of the joint 6 is known. Solving theta based on subproblem 1 in the same way6。
The method is characterized in that the kinematics inverse solution problem of the 6R robot is converted into the solution of a six-membered quadratic equation set and a three-membered quadratic equation set based on a geometric description method, the inverse solution model is simplified, the geometric significance is more definite, and a certain method support is provided for the real-time motion control of the 6R robot. The method combines geometric description with a momentum theory, has more definite geometric meaning, solves an algebraic equation set by simplifying an inverse solution algorithm, effectively improves the calculation efficiency, and can provide a new inverse solution processing method for the real-time control of the robot motion.
Drawings
FIG. 1 is a geometric description of the inverse kinematics of joints 1, 2 and 3;
FIG. 2 is a geometric description of the inverse kinematics of joints 4 and 5;
figure 3 a 6R robot parameter coordinate system.
Detailed Description
The present invention is described in detail below with reference to the accompanying fig. 1-3.
S1 determining kinematic parameters of 6R robot and establishing positive kinematic model
As shown in fig. 3, the position vectors and rotation vectors of the joints of the 6R robot in the initial state are known as follows:
Figure BDA0001433432300000048
Figure BDA0001433432300000049
wherein r isiI is more than or equal to 1 and less than or equal to 6, and represents the position vector of the i joint in the base coordinate system, omegaiRepresenting the rotation vector of the i-joint.
Based on the rotation theory, the positive kinematics model of the 6R robot is expressed as,
Figure BDA0001433432300000041
wherein g isst(θ),gst(0) Respectively showing the initial pose and the target pose of the tail end of the robot,
Figure BDA0001433432300000042
representing the form of the exponential product of the i-joint rotational motion,
Figure BDA0001433432300000043
in the formula [ theta ]iIs the ith joint angular displacement;
Figure BDA0001433432300000044
is the i joint rotation vector omegaiBy ωi＝[ω1 ω2 ω3]Is defined as
Figure BDA0001433432300000045
Then
Figure BDA0001433432300000046
νiIs the rotational linear velocity of the i-joint motion, vi＝-ωi×ri。
Given the pose of the object(s),
Figure BDA0001433432300000047
s2 six-membered quadratic equation set established for the first three joints
The rotational motion of the first three joints is described based on the momentum theory as,
Figure BDA0001433432300000051
q 'in the formula'sAnd q'eRespectively representing the initial position and the target position of the 6R robot tip in the rotational movement. As can be seen from S1, in this case,
q′s＝[xs ys zs 1]＝[0 744 940 1]
q′e＝[xe ye ze 1]＝[-936.6611 631.7859 570.0752 1]
let the movement pass through point c1And c2The following set of relationships is established according to the geometric description shown in figure 1,
Figure BDA0001433432300000052
wherein q is1，q2And q is3At any point on the axis of rotation of joint 1, joint 2 and joint 3, respectively, and q is taken for simplifying the model1＝[0 0 0]，q2＝[0 150 250]And q is3＝[0 150 800]Then, equation (7) is expressed as the following equation system,
Figure BDA0001433432300000053
solving an equation set (8) by adopting a 'SOLVE' function in MATLAB to obtain x1,y1,z1,x2,y2And z2The solution of (1).
S3 calculating angular displacement theta of joint1，θ2And theta3
Obtain process point coordinates c1＝[x1 y1 z1]And c2＝[x2 y2 z2]Hereinafter, the rotational movements about the joint 1, the joint 2 and the joint 3 will be described respectively as follows,
Figure BDA0001433432300000054
based on the Pasen-Kahan subproblem 1, the display expression of the angular displacement of the joint is obtained as follows,
Figure BDA0001433432300000061
Figure BDA0001433432300000062
Figure BDA0001433432300000063
step (4) establishing a ternary quadratic equation set for the joint 4 and the joint 5
As can be seen from the formula (3),
Figure BDA0001433432300000064
wherein
Figure BDA0001433432300000065
Taking a point q on the axis of the joint 6s1＝[xs1 ys1 zs1]＝[0 744 0]The rotational movement thereof about the joints 4 and 5 is described below,
Figure BDA0001433432300000066
wherein q'e1＝g1q′s1＝[xe1 ye1 ze1 1]。
Let the coordinate of the passing point of motion be c3＝[x3 y3 z3]The following set of relationships is established according to the geometric description shown in figure 2,
Figure BDA0001433432300000067
wherein q is4＝[0 744 940]. Equation (14) is expressed as the following equation,
Figure BDA0001433432300000068
solving the equation set (15) by using a 'SOLVE' function in MATLAB to obtain x3,y3And z3The solution of (1).
Step (5) calculating the angular displacement theta of the joint4And theta5
Obtain process point coordinates c3＝[x3 y3 z3]Hereinafter, the rotational movements about the joints 4 and 5 will be described respectively as follows,
Figure BDA0001433432300000071
obtaining angular displacement theta of joint based on problem 1 of Paden-Kahan son4And theta5The expression of (a) is as follows,
Figure BDA0001433432300000072
Figure BDA0001433432300000073
step (6) calculating the angular displacement theta of the joint6
Take a point q not on the axis of the joint 6s2＝[0 750 940]The rotational movement of which about the joint 6 is described as,
Figure BDA0001433432300000074
wherein
Figure BDA0001433432300000075
Obtaining the angular displacement theta of the joint based on the problem 1 of the Paden-Kahan in the same way6The expression of (a) is as follows,
Figure BDA0001433432300000076
eight sets of inverse kinematics solutions obtained by the above solution are shown in table 1.
TABLE 1 eight kinematic inverse solutions
Figure BDA0001433432300000077

Claims (2)

1. A6R robot inverse kinematics geometric solving method based on a momentum theory is characterized by comprising the following steps: the method comprises the following implementation processes:
s1, establishing a base coordinate system and a tool coordinate system, determining kinematic parameters of the 6R robot through the base coordinate system and the tool coordinate system, and establishing a positive kinematic model;
s2, decomposing and describing inverse solution motions of the first three joints of the 6R robot: from point qeTo point c2＝[x2 y2 z2]A rotational movement around the joint 1; point c2To point c1＝[x1 y1 z1]A rotational movement around the joint 2; point c1To point qsA rotational movement about the joint 3; wherein q iss，qeRespectively establishing x-based initial and target positions of the tail ends of the front three joints of the robot according to the description of the geometric relationship1,y1,z1,x2,y2And z2Is a six-membered quadratic system of variables;
s3, solving the equation set by adopting a 'SOLVE' function in MATLAB to obtain c1And c2The coordinate vector of (2);
s4, knowing the inverse solution motion starting and ending positions of the first three joints, and respectively establishing the first three joint angle variables theta of the 6R robot based on the Pasen-Kahan subproblem 11，θ2And theta3The explicit solution model of (2) and solving;
s5, taking one point on the axis of the joint 6 as the initial position q in the inverse solution movement of the last three jointss1And solving the initial position q based on the momentum positive kinematics models1Corresponding target position qe1；
S6, describing the inverse solution motion decomposition of joints 4 and 5: from point qe1To point c3＝[x3 y3 z3]The rotational movement of (a); from point c3To point qs1The rotational movement of (a); establishing the relation x according to the geometric relation3,y3And z3As a ternary quadratic equation of a variableGroup, and solving by using a 'SOLVE' function to obtain c3The coordinate vector of (2);
s7, the inverse solution motion starting and ending positions of the joint 4 and the joint 5 are known, and joint angle variables theta of the joint 4 and the joint 5 are respectively established based on the Pasen-Kahan subproblem 14And theta5Explicitly solving the model and solving;
s8, taking any point q not on the axis of the joint 6s2Solving q based on a momentum positive kinematics models2Corresponding target position qe2If so, the starting and ending positions of the inverse solution movement of the joint 6 are known; solving theta based on subproblem 1 in the same way6。
2. The inverse kinematics geometric solution method of the 6R robot based on the momentum theory as claimed in claim 1, wherein: s1 determining kinematic parameters of 6R robot and establishing positive kinematic model
The position vector and the rotation vector of each joint in the initial state of the 6R robot are known as follows:
Figure FDA0002729732180000021
Figure FDA0002729732180000022
wherein r isiI is more than or equal to 1 and less than or equal to 6, and represents the position vector of the i joint in the base coordinate system, omegaiA rotation vector representing the i-joint;
based on the rotation theory, the positive kinematics model of the 6R robot is expressed as,
Figure FDA0002729732180000023
wherein g isst(0),gst(theta) respectively representing an initial pose and a target pose of the robot end,
Figure FDA0002729732180000024
representing the form of the exponential product of the i-joint rotational motion,
Figure FDA0002729732180000025
in the formula [ theta ]iIs the ith joint angular displacement;
Figure FDA0002729732180000026
is the i joint rotation vector omegaiBy ωi＝[ω1ω2 ω3]Is defined as
Figure FDA0002729732180000027
Then
Figure FDA0002729732180000028
νiIs the rotational linear velocity of the i-joint motion, vi＝-ωi×ri；
Given the pose of the object(s),
Figure FDA0002729732180000031
s2 six-membered quadratic equation set established for the first three joints
The rotational motion of the first three joints is described based on the momentum theory as,
Figure FDA0002729732180000032
q 'in the formula'sAnd q'eRespectively representing the initial position and the target position of the 6R robot tail end in the rotary motion; as can be seen from S1, in this case,
q′s＝[xs ys zs 1]＝[0 744 940 1]
q′e＝[xe ye ze 1]＝[-936.6611 631.7859 570.0752 1]
let the movement pass through point c1And c2The following set of relationships is established from the geometric description,
Figure FDA0002729732180000033
wherein q is1，q2And q is3At any point on the axis of rotation of joint 1, joint 2 and joint 3, respectively, and q is taken for simplifying the model1＝[0 0 0]，q2＝[0 150 250]And q is3＝[0 150 800]Then, equation (7) is expressed as the following equation system,
Figure FDA0002729732180000034
solving an equation set (8) by adopting a 'SOLVE' function in MATLAB to obtain x1,y1,z1,x2,y2And z2The solution of (1);
s3 calculating angular displacement theta of joint1，θ2And theta3
Obtain process point coordinates c1＝[x1 y1 z1]And c2＝[x2 y2 z2]Hereinafter, the rotational movements about the joint 1, the joint 2 and the joint 3 will be described respectively as follows,
Figure FDA0002729732180000041
based on the Pasen-Kahan subproblem 1, the display expression of the angular displacement of the joint is obtained as follows,
Figure FDA0002729732180000042
Figure FDA0002729732180000043
Figure FDA0002729732180000044
s4 sets up a system of ternary quadratic equations for joint 4 and joint 5
As can be seen from the formula (3),
Figure FDA0002729732180000045
wherein
Figure FDA0002729732180000046
Taking a point q on the axis of the joint 6s1＝[xs1 ys1 zs1]＝[0 744 0]The rotational movement thereof about the joints 4 and 5 is described below,
Figure FDA0002729732180000047
wherein q'e1＝g1q′s1＝[xe1 ye1 ze1 1]；
Let the coordinate of the passing point of motion be c3＝[x3 y3 z3]The following set of relationships is established from the geometric description,
Figure FDA0002729732180000048
wherein q is4＝[0 744 940](ii) a Equation (14) is expressed as the following equation,
Figure FDA0002729732180000049
solving the equation set (15) by using a 'SOLVE' function in MATLAB to obtain x3,y3And z3The solution of (1);
s5 calculating angular displacement theta of joint4And theta5
Obtain process point coordinates c3＝[x3 y3 z3]Hereinafter, the rotational movements about the joints 4 and 5 will be described respectively as follows,
Figure FDA0002729732180000051
obtaining angular displacement theta of joint based on problem 1 of Paden-Kahan son4And theta5The expression of (a) is as follows,
Figure FDA0002729732180000052
Figure FDA0002729732180000053
s6 calculating angular displacement theta of joint6
Take a point q not on the axis of the joint 6s2＝[0 750 940]The rotational movement of which about the joint 6 is described as,
Figure FDA0002729732180000054
wherein
Figure FDA0002729732180000055
Obtaining the angular displacement theta of the joint based on the problem 1 of the Paden-Kahan in the same way6The expression of (a) is as follows,
Figure FDA0002729732180000056
eight sets of inverse kinematics solutions are obtained by the above solution.
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