A method for solving inverse kinematics of a redundant spatial manipulator with bias

Abstract

The application discloses a kind of biased redundant space mechanical arm inverse kinematics solving method, and relates to a kind of mechanical arm inverse kinematics solving method.First, the kinematics model of redundant mechanical arm is established in conformal space, according to the end pose information of a path point in continuous path tracking movement and the inverse kinematics solving process corresponding to last path point, redundant parameter is obtained, and then the joint position information under the path point is solved, finally the joint is connected in sequence to obtain arm rod information, and the joint rotation angle is calculated according to the change of arm rod.In the execution of continuous path tracking task, the redundant characteristics can be fully utilized, which is helpful to select appropriate configuration in the numerous solutions of redundant mechanical arm inverse kinematics combined with actual task, and has good rapidity and robustness.

Description

Technical Field

[0001] This invention relates to a method for solving the inverse kinematics of a robotic arm, and more particularly to a method for solving the inverse kinematics of an offset redundant spatial robotic arm, belonging to the technical field of redundant robotic arm inverse kinematics. Background Technology

[0002] Redundant spatial manipulators are manipulators with a joint space dimension greater than the operational space dimension. They are commonly used for on-orbit space servicing. The most typical example is a redundant spatial manipulator with a joint configuration of 3 degrees of freedom for the shoulder joint, 1 degree of freedom for the elbow joint, and 3 degrees of freedom for the wrist joint, with joint offsets. Redundant spatial manipulators have the redundancy characteristic that the joint variables can freely change without affecting the end-effector pose after a given end-effector pose. This characteristic gives them advantages such as obstacle avoidance, singularity avoidance, and the ability to overcome the influence of joint limitations on the manipulator's operating range. Furthermore, redundant spatial manipulators have more options for arm configuration during obstacle avoidance path planning, thereby improving work efficiency and reducing energy consumption.

[0003] However, the redundancy also leads to the following problems in the motion planning of redundant space robotic arms: (1) Space station robotic arms undertake important tasks such as transporting cargo outside the cabin, and fast algorithms are needed to control their motion; (2) There are countless solutions to the inverse kinematics of redundant robotic arms. It is necessary to select a suitable solution in combination with the actual task and the optimization goal during the calculation process. How to quickly obtain a suitable and accurate inverse kinematic solution is a technical problem that technicians in the field urgently need to solve. Summary of the Invention

[0004] To address the shortcomings of the prior art, this invention provides a method for solving the inverse kinematics of an offset redundant spatial manipulator. This method can fully utilize redundancy when performing continuous path tracking tasks, and helps to select a suitable configuration from countless solutions of the inverse kinematics of the redundant manipulator in combination with the actual task. It has good speed and robustness.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: a method for solving the inverse kinematics of an offset redundant spatial manipulator, comprising the following steps:

[0006] Step 1: Establish the conformal geometric model of the robotic arm based on its configuration and parameters, and derive the kinematic equations.

[0007] 1.1 Establishing the conformal geometric model of the robotic arm

[0008] basis vectors of conformal geometric algebra in Euclidean space Introduce two additional basis vectors on top of this. and Establish a world coordinate system The joints and levers of the robotic arm are represented using conformal geometry algebra theory:

[0009] The joints of the robot arm are represented as points, denoted as follows:

[0010]

[0011] where, is a point in Euclidean space, is a point in conformal space;

[0012] The links of the robot arm are represented as straight line segments connecting two adjacent joint points, denoted as follows:

[0013]

[0014] where, and are two adjacent joint points, respectively;

[0015] 1.2 Establishing a conformal geometric algebra kinematics model

[0016] In the conformal space, the rotational motion of a joint is represented by a motion operator:

[0017] The rotational operator is denoted as follows:

[0018]

[0019] where, is the rotational axis direction vector, is the rotation angle;

[0020] The geometric body after rotation is denoted as follows:

[0021]

[0022] where, is the expression before rotation, is the expression after rotation;

[0023] The translation operator is denoted as follows:

[0024]

[0025] where, is the translation direction vector, and its magnitude is the translation distance;

[0026] The geometric body after translation is denoted as follows:

[0027]

[0028] where, is the expression before translation, This is the translated form of expression;

[0029] The motion operator for a joint is represented as follows:

[0030]

[0031] The transformation equations between adjacent joints are expressed as follows:

[0032]

[0033] in, Representing the The position or posture of each joint. Representing the The position or posture of each joint. Indicates from the first Motion operators for each joint;

[0034] Given a set of joint variables ,in Representing the joint space, the kinematic equations for solving the end effector pose are as follows:

[0035]

[0036]

[0037]

[0038] in, This represents the position of joint one. Represents the position of the end effector. This represents the proximity vector of the end effector in the initial state. The proximity vector represents the end effector. This represents the attitude vector of the end effector in its initial state. Represents the attitude vector of the end effector. Indicates the first Motion operators for each joint ;

[0039] Step 2: Select the redundancy parameter vector based on the configuration characteristics of the biased redundant space robot arm.

[0040] A typical offset redundant space robot arm has the characteristic that the axes of three joints are always parallel to each other. The direction vectors of the joint axes of these three joints are chosen as the redundant parameter vectors.

[0041] Step 3: Track the motion of the first continuous path according to the given path. Given the end-position pose information of each path point, determine the joint positions of joint 1, joint 6, and joint 7 corresponding to that path point.

[0042] Firstly, a path planning algorithm is used to generate a series of discrete points, and then the robot arm performs a continuous path tracking motion, sequentially tracking the end effector pose corresponding to each discrete point from the starting point to the ending point. Inverse kinematics needs to be solved at each discrete point. When solving the inverse kinematics of the path point, the known information is the position of the end effector corresponding to the path point obtained in the previous path planning algorithm , the approach vector and the pose vector , the joint node position of joint one is obtained by moving the base joint node along the direction of arm one by the corresponding arm length, and the end effector position is regarded as the joint seven node position, and the joint seven position is moved along the approach vector by the corresponding arm length to obtain the joint six node position.

[0043] Step four: the redundant parameter vector corresponding to the current path point is solved by using the inverse solution result of the previous path point

[0044] The redundant parameter vector determined by the inverse kinematics solution of the first path point is introduced into the inverse kinematics solution process of the first path point, and the redundant parameter vector required for the inverse kinematics solution of the first path point is solved by the conformal geometric method, and the motion energy change between adjacent path points is minimized by selecting the redundant parameter vector, and the redundancy reduction process is completed. The redundant parameter vector determined by the inverse kinematics solution of the first path point is introduced into the inverse kinematics solution process of the first path point, and the redundant parameter vector required for the inverse kinematics solution of the first path point is solved by the conformal geometric method, and the motion energy change between adjacent path points is minimized by selecting the redundant parameter vector, and the redundancy reduction process is completed. Step five: according to the redundant parameter vector obtained in step four, the remaining joint node positions except the information solved in step three are solved by the conformal geometric method

[0045] According to the redundant parameter vector obtained in step four, the remaining joint node positions except the information solved in step three are solved by the conformal geometric method in the conformal space by performing conformal geometric operations on the constraint relationships of planes, spheres and straight lines.

[0046] Step six: the joint nodes obtained in step five are connected in sequence to obtain the arm information, and the rotation angle of each joint is solved according to the change of the arm to obtain the finite group inverse solution of the first path point

[0047] Each arm between the base and the end effector is regarded as a directed straight line segment, and when the joint variable changes, the corresponding arm moves in the rotation plane with the joint axis direction as the normal vector. The joint variable is solved by solving the change of the directed straight line segment represented by the arm, and the finite group solution of the inverse kinematics of the robot arm corresponding to the first path point is obtained.

[0048] Each arm between the base and the end effector is regarded as a directed straight line segment, and when the joint variable changes, the corresponding arm moves in the rotation plane with the joint axis direction as the normal vector. The joint variable is solved by solving the change of the directed straight line segment represented by the arm, and the finite group solution of the inverse kinematics of the robot arm corresponding to the first path point is obtained.

[0049] ​​​Step seven: selecting the unique solution matched with the path tracking task from the finite group inverse solution obtained in step six

[0050] Firstly, the solution not meeting the joint rotation angle restriction is removed from the finite group inverse solution, and then the unique solution matched is selected according to the principle of minimum energy change, so that the sum of the absolute values of the angle changes in the inverse kinematics solution of the adjacent two path points is minimum, thus the inverse kinematics solution corresponding to the first path point in the execution of the continuous path tracking motion is completed, and the inverse kinematics solution of the second path point is obtained by repeating the steps three to seven from the starting point to the ending point.

[0051] Compared with the prior art, the present application introduces the basic theory of conformal geometric algebra into the inverse kinematics research of the bias type redundant spatial manipulator, and has the beneficial effects as follows:

[0052] (1) According to the configuration characteristics of the bias type redundant spatial manipulator, the present application provides a method for solving the inverse kinematics in the execution of the continuous path tracking motion task in the conformal space, compared with the prior art, the present application realizes the synchronous motion of the joints, can fully exert the redundant characteristics of the manipulator while executing the task, has good robustness, and has obvious geometric significance;

[0053] (2) The present application solves the inverse kinematics analytical solution by using the conformal geometric algebra theory, avoids the complex matrix operation and transcendental equation solving, improves the rapidity of the solution, has higher calculation efficiency compared with other numerical methods, and has important application significance. BRIEF DESCRIPTION OF DRAWINGS

[0054] Figure 1 is the flowchart of the present application;

[0055] Figure 2 is the conformal geometric model of the bias type redundant spatial manipulator in the embodiment;

[0056] Figure 3 is the process schematic diagram of selecting the redundant parameter vector in the embodiment;

[0057] Figure 4 is the schematic diagram of the end pose information of a path point in the continuous path tracking motion process and the process of solving the redundant parameter vector in the embodiment;

[0058] Figure 5 is the solving process schematic diagram of the circular ring in the joint node solving in the embodiment;

[0059] ​​Figure 5 Part b is the joint point in the embodiment. Solving for midpoint pairs A schematic diagram of the solution process;

[0060] Figure 6 Part a is the joint point in the embodiment. Solving the middle ring A schematic diagram of the solution process;

[0061] Figure 6 Part b is the joint point in the embodiment. Solving for midpoint pairs A schematic diagram of the solution process;

[0062] Figure 7 Part a is the joint point in the embodiment. Solving the middle ring A schematic diagram of the solution process;

[0063] Figure 7 Part b is the joint point in the embodiment. Solving for midpoint pairs A schematic diagram of the solution process;

[0064] Figure 8 This is a curve showing the change in joint rotation angles when the robotic arm tracks a random path during the verification phase of the embodiment.

[0065] Figure 9 This is a curve showing the change in joint rotation angles when the robotic arm tracks a circular path during the verification phase of the embodiment.

[0066] Figure 10 This is the end effector path obtained by substituting the inverse kinematics solution into the kinematic equations when the robotic arm tracks a circular path during the implementation verification stage. Part a is a three-dimensional schematic diagram, and part b is its projection on the xz plane.

[0067] Figure 11 This is a curve showing the change in joint rotation angles when the robotic arm tracks a rectangular path during the verification phase of the embodiment.

[0068] Figure 12 This is the end effector path obtained by substituting the inverse kinematics solution into the kinematic equations when the robotic arm tracks a rectangular path during the verification phase of the embodiment. Part a is a three-dimensional schematic diagram, and part b is its projection onto the xz plane. Detailed Implementation

[0069] The technical solutions in the present application will be described clearly and completely below in connection with the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative efforts belong to the protection scope of the present application.

[0070] As shown in the figure, a bias type redundant spatial mechanical arm inverse kinematics solving method comprises the following steps: Figure 1

[0071] Step one: establishing a conformal geometric model of the mechanical arm according to the configuration and parameters of the mechanical arm, and deducing kinematics equations

[0072] 1.1 Establishing a conformal geometric model of the mechanical arm

[0073] Conformal geometric algebra introduces two additional basis vectors and and on the basis of the basis vectors of Euclidean space , so as to expand Euclidean space to conformal space, realizing the representation of geometric objects such as points, straight lines, planes and spheres in the conformal space.

[0074] Establishing a world coordinate system When studying the kinematics of the mechanical arm in the conformal space, the joints of the mechanical arm are represented as points, and the arm rods of the mechanical arm are represented as straight line segments obtained by connecting two adjacent joint nodes, and the joints and arm rods of the mechanical arm are represented by using the conformal geometric algebra theory.

[0075] The joints of the mechanical arm are represented as follows:

[0076]

[0077] wherein, is a point in Euclidean space, is the representation form of the point in the conformal space.

[0078] The arm rods of the mechanical arm are represented as follows:

[0079]

[0080] wherein, and are two adjacent joint nodes respectively.

[0081] According to the configuration and parameters of the mechanical arm, the conformal geometric model of the mechanical arm can be constructed;

[0082] 1.2 Establishing a conformal geometric algebra kinematics model

[0083] In the conformal space, the rotational motion of a joint can be represented by a motion operator.

[0084] The rotational operator is represented as follows:

[0085]

[0086] where, is the rotational axis direction vector, is the rotation angle.

[0087] The rotated geometric body is represented as follows:

[0088]

[0089] where, is the expression before rotation, is the expression after rotation.

[0090] The translational operator is represented as follows:

[0091]

[0092] where, is the translational direction vector with a magnitude of the translational distance.

[0093] The translated geometric body is represented as follows:

[0094]

[0095] where, is the expression before translation, is the expression after translation.

[0096] The motion operator of a joint is represented as follows:

[0097]

[0098] The transformation equation between adjacent joints is represented as follows:

[0099]

[0100] where, represents the position or pose of the th joint, represents the position or pose of the th joint, indicates the motion operator from the th joint.

[0101] For a seven-degree-of-freedom robotic arm, given a set of joint variables where, The joint space is represented, and the kinematics equation for solving the end effector pose is expressed as follows:

[0102]

[0103]

[0104]

[0105] wherein, represents the position of the first joint, represents the position of the end effector, represents the approach vector of the end effector in the initial state (no change in the joint angle), represents the approach vector of the end effector, represents the pose vector of the end effector in the initial state (no change in the joint angle), represents the pose vector of the end effector, represents the motion operator of the first joint, .

[0106] This step establishes the kinematics model of the robot arm in the conformal space, which is the theoretical basis for kinematics analysis in the conformal space;

[0107] Step 2: Select the redundancy parameter vector according to the configuration characteristics of the bias type redundant space robot arm

[0108] When solving the inverse kinematics of the robot arm, a null solution will be generated, and the redundancy parameter vector needs to be determined according to the special arm position relationship in the robot arm configuration, and then the appropriate finite group solution is selected from the null solution. The bias type redundant space robot arm has the characteristic that the axes of the three consecutive joints are parallel at all times, and the joint axis direction vectors of the three joints are selected as the redundancy parameter vector;

[0109] Step 3: According to the given end pose information of the first path point in the continuous path tracking motion, the joint positions of the first joint, the sixth joint and the seventh joint corresponding to the path point are solved

[0110] In actual situations, before the robot arm performs a task, a path planning algorithm is first used to generate an end effector motion path composed of a series of discrete points, and then the robot arm performs continuous path tracking motion, sequentially tracking and performing the end effector pose corresponding to each discrete point from the starting point to the ending point. Inverse kinematics needs to be solved at each discrete point, and the known information for solving the inverse kinematics of the path point is the position , approach vector and pose vector of the end effector corresponding to the path point obtained in the previous path planning algorithm​​ Since the base of the robot arm is fixed, moving the joint point of joint one along the direction of arm one by the corresponding arm length can obtain the joint point position of joint one, and the end effector position is considered as the joint point position of joint seven. Moving the position of joint seven along the approaching vector by the corresponding arm length can obtain the joint point position of joint six, which provides a prerequisite for solving the redundant parameter vector;

[0111] Step four: using the inverse solution of the previous path point to solve the redundant parameter vector corresponding to the current path point

[0112] When the inverse kinematics of the first path point is solved, the inverse kinematics of the second path point is solved, and the redundant parameter vector of the inverse kinematics of the first path point is determined. The inverse kinematics of the second path point is solved by introducing the redundant parameter vector into the inverse kinematics solving process of the second path point, and the redundant parameter vector required for solving the inverse kinematics of the second path point is obtained by using the conformal geometric method, so that the motion configuration optimization of the robot arm is completed while solving the redundant parameter vector. By selecting a suitable vector from the set of redundant parameters, the minimum change in motion energy between adjacent path points is achieved, and the redundant process of inverse kinematics solving of the robot arm is completed; When the inverse kinematics of the first path point is solved, the inverse kinematics of the second path point is solved, and the redundant parameter vector of the inverse kinematics of the first path point is determined. The inverse kinematics of the second path point is solved by introducing the redundant parameter vector into the inverse kinematics solving process of the second path point, and the redundant parameter vector required for solving the inverse kinematics of the second path point is obtained by using the conformal geometric method, so that the motion configuration optimization of the robot arm is completed while solving the redundant parameter vector. By selecting a suitable vector from the set of redundant parameters, the minimum change in motion energy between adjacent path points is achieved, and the redundant process of inverse kinematics solving of the robot arm is completed; When the inverse kinematics of the first path point is solved, the inverse kinematics of the second path point is solved, and the redundant parameter vector of the inverse kinematics of the first path point is determined. The inverse kinematics of the second path point is solved by introducing the redundant parameter vector into the inverse kinematics solving process of the second path point, and the redundant parameter vector required for solving the inverse kinematics of the second path point is obtained by using the conformal geometric method, so that the motion configuration optimization of the robot arm is completed while solving the redundant parameter vector. By selecting a suitable vector from the set of redundant parameters, the minimum change in motion energy between adjacent path points is achieved, and the redundant process of inverse kinematics solving of the robot arm is completed; Step five: according to the redundant parameter vector obtained in step four, the remaining joint point positions except the information obtained in step three are solved by using the conformal geometric method

[0113] The offset type redundant space robot arm has the characteristics of mutual perpendicularity of each arm at any time and parallelism of the axes of three consecutive joints at any time. According to this characteristic and the redundant parameter vector obtained in step four, the remaining joint point positions except the information obtained in step three are solved by using the conformal geometric method in the conformal space by performing conformal geometric operations on the constraint relationships of planes, spheres and straight lines. Due to the different types of robot arms, multiple solutions may occur in the solving process of this step;

[0114] Step six: connecting the joint points obtained in step five in sequence to obtain the arm length information, and solving the rotation angles of each joint according to the change of the arm to obtain the finite group inverse solution corresponding to the first path point

[0115] The connection line of adjacent joint positions is a straight line segment of the arm, and each arm between the base and the end effector is regarded as a directed straight line segment. When the joint variable changes, the corresponding arm moves in a rotation plane with the joint axis direction as the normal vector. By solving the change of the directed straight line segment represented by the arm, the joint variable is solved, and the finite group solution of the inverse kinematics of the robot arm corresponding to the first path point is obtained;

[0116] The connection line of adjacent joint positions is a straight line segment of the arm, and each arm between the base and the end effector is regarded as a directed straight line segment. When the joint variable changes, the corresponding arm moves in a rotation plane with the joint axis direction as the normal vector. By solving the change of the directed straight line segment represented by the arm, the joint variable is solved, and the finite group solution of the inverse kinematics of the robot arm corresponding to the first path point is obtained; The connection line of adjacent joint positions is a straight line segment of the arm, and each arm between the base and the end effector is regarded as a directed straight line segment. When the joint variable changes, the corresponding arm moves in a rotation plane with the joint axis direction as the normal vector. By solving the change of the directed straight line segment represented by the arm, the joint variable is solved, and the finite group solution of the inverse kinematics of the robot arm corresponding to the first path point is obtained;​​​

[0117] Step seven: select the unique solution matching the path tracking task from the finite set of inverse solutions obtained in step six

[0118] Due to the different arm types of the manipulator, the same end pose may correspond to multiple sets of inverse solutions after determining the redundant parameters. In the screening process, first, remove the solutions that do not meet the joint rotation angle limit from the finite set of inverse solutions, and then select the unique solution that matches according to the principle of minimum energy change, so that the sum of the absolute values of the angle changes in the inverse kinematics solutions of the adjacent two path points is the smallest. Thus, the inverse kinematics solution corresponding to the first path point in the execution of the continuous path tracking motion is completed, and the inverse kinematics solution corresponding to the first path point in the execution of the continuous path tracking motion is completed. From the starting point to the end point, steps three to seven are repeated in turn, and the inverse kinematics solution of the whole process of the manipulator executing the continuous path tracking task is obtained.

[0119] Embodiment

[0120] In this embodiment, the base position of the bias type redundant spatial manipulator is fixed at the origin of the conformal space , and the given manipulator configuration is combined with Figure 2 , wherein the joint positions of joint one to joint seven are , there is a bias at the elbow joint , there is a bias at the wrist joint , and the parameters of each arm of the manipulator are shown in Table 1:

[0121] Table Parameters of each arm of the bias type redundant spatial manipulator (unit: m)

[0122]

[0123] Step 1: if the joint variables of the given manipulator are , the transformation equation between each joint is represented as follows:

[0124]

[0125] wherein, and satisfy the following relationship:

[0126]

[0127] The position of the end effector , the approach vector , the attitude vector can be represented as follows:

[0128]

[0129] ​Step 2: The redundant space manipulator has the characteristics of fixed base position, each arm link is perpendicular to each other at any time, and the continuous three joints axis are parallel to each other at any time. The continuous three parallel arm links with special position relationship are selected The direction vector is the redundant parameter vector.

[0130] Combining Figure 3 , a ball is made with the center as and the radius as . , a ball is made with the center as and the radius as . The intersection of the two balls is a torus , and the unit vector of the line connecting the joint points and is the direction vector of the continuous three parallel arm links . . .

[0131] Step 3: Use the input parameters of inverse kinematics to solve the joint positions of joint one, joint six and joint seven , and .

[0132] Combining Figure 4 , the position of the end effector at the first path point of the continuous path and radial tracking motion is given as , the approach vector is , and the attitude vector is . Now, the inverse kinematics of the manipulator is solved for this path point.

[0133] Combining the manipulator configuration characteristics shown in Figure 1 , we have

[0134] The joint position at joint seven is represented as follows:

[0135]

[0136] The joint position at joint six is represented as follows:

[0137]

[0138] The joint position at joint one is represented as follows:

[0139]

[0140] ​Step 4: The redundant parameter vector is solved by using the conformal geometry method, and the appropriate configuration is selected in the inverse solution without arrays.

[0141] Combining Figure 4 Fig. 2, according to step 2, the inverse solution of the first path point is solved, and the point selected in the process is named as , the projection of to the plane where the circular ring corresponding to the first path point is located, the connection between the center of the circular ring and the projection point, and the intersection of the line and the circular ring at the point , the unit vector of the vector is the unit vector of the parameter vector corresponding to the first path point in the redundant parameter vector without arrays, that is, the redundant parameter vector to be solved, which is named as . Step 5: According to the redundant parameter vector determined in step 4, the positions of the joint points except the information solved in step 3 are solved by using the conformal geometry method. Combining Fig. 3, a sphere is made with as the center and as the radius, a plane is made through the point

[0142] with as the normal vector, the sphere and the plane intersect at a circular ring

[0143] ; a plane is made through the point Figure 5 with as the normal vector, the plane and the circular ring intersect at the point pair , that is, the point pair where the joint point is located, which is expressed as follows:

[0144]

[0145] Among the point pair , the two points represent the positions of the point in different directions of the joint of the mechanical arm.

[0146] Combining Figure 6 Fig. 4, a sphere is made with as the center and as the radius, a plane is made through the point with the close vector as the normal vector, and the sphere and the plane intersect at a circular ring​​​​​​​​​​​​​​​ the plane of the sphere and the plane intersect at a torus ; the plane passing through the point with normal vector , the plane intersects the torus at the point pair , which is the point pair where the joint is located, is expressed as follows:

[0147]

[0148] where the two points in the point pair represent the positions of the point in different orientations of the joint of the robot arm.

[0149] As shown in Figure 7 , a sphere is made with the center and the radius , a sphere is made with the center and the radius , and the two spheres are compared to the torus ; the plane passing through the point with normal vector , the plane intersects the torus at the point pair , which is the point pair where the joint is located, is expressed as follows:

[0150]

[0151] where the two points in the point pair represent the positions of the point in different orientations of the joint of the robot arm.

[0152] The joint is moved along the parameter vector by the length of the corresponding link to obtain the position of the joint , which is expressed as follows:

[0153]

[0154] When the position of the joint is solved by using this method, 8 groups of solutions are generated, because in the above formula calculation, the point , ,​​​​ is presented by point-to-point, parameter , , The corresponding different mechanical arm posture will produce different joint positions.

[0155] Step 6: The joint points obtained in step 5 are connected in turn to obtain the arm rod information, and the rotation angles of each joint are calculated according to the change of the arm rod.

[0156] Definition and are two vectors, and the binary unit vector represented by the rotation plane is represented as follows:

[0157]

[0158] Among them, is the unit vector of , is the unit vector of .

[0159] The angle between and is represented as follows:

[0160]

[0161] SSRMS configuration mechanical arm joint rotation angle The rotation axes are in turn, and the rotation angles are determined by the rotation reference vectors in turn.

[0162] When each joint does not rotate, the configuration of the mechanical arm in the conformal space is shown in Figure 2 , and the initial rotation reference vectors of joint one to joint seven are in turn. After movement, the rotation angle of joint one can be determined according to the positions of the joint points , .

[0163]

[0164] At this time, the rotation reference vector of joint two is , and the rotation angle of joint two is:

[0165]

[0166] In this way, the rotation angles of each joint can be obtained.

[0167] Step 7: At this time, the first For each path point, there are 8 sets of inverse kinematic solutions. First, it is determined whether each set of solutions satisfies the joint rotation angle constraint. After removing the solutions that do not satisfy the joint rotation angle constraint, the remaining solutions are... The inverse solution is then calculated, and based on the principle of minimizing energy change, the inverse solution in each group is compared to the first group. Changes in the inverse kinematics at each path point:

[0168]

[0169] in, Indicates the first The change in the inverse solution of the system Indicates the first The path point corresponding to the th path point Joints in inverse solution Numerical value No. Joints in the inverse solution of path points The value is chosen such that The smallest The inverse solution of the group is used as the first in continuous path tracking motion. The final inverse kinematic solution for each path point.

[0170] To verify the effectiveness of this technical method, the following is combined with... Figures 8 to 12 The technical solution of the present invention will be further described. Computer simulation is used to perform tracking simulations of multiple continuous paths. During the simulation, given the starting point and the end pose of a continuous path, linear interpolation is performed on the position and quaternion Lerp interpolation is performed on the pose to plan the path between the starting point and the end point, thereby obtaining a continuous path.

[0171] Given a starting position of (-4, 3, -2) and an initial proximity vector (0,1,0) is the initial sliding vector. The endpoint is (-1,0,0), the destination is (-2,4,2), and the endpoint proximity vector is (-1,0,0). The endpoint sliding vector is (2,1,0). Given (-1, 2, 0), the above algorithm is used for continuous path interpolation, and the inverse kinematics is solved for each interpolation point to obtain the joint angle variation curves of joints one to seven. Figure 8 As shown, the number of interpolation points is 1000, and the calculation time is approximately 0.227988ms.

[0172] By drawing a circle parallel to y=1.62 with center (-3.67, 0.17) and radius 0.2m, the curves showing the changes in the rotation angles of each joint are obtained. Drawing lines on a plane Figure 9As shown, the joint angle size obtained by solving is used to solve the end point position by forward kinematics, and the end point path is drawn combined with Figure 10 As shown, the number of interpolation points is 3000, and the calculation time is about 0.772113ms.

[0173] On the plane parallel to Draw a rectangle on the plane, and obtain the joint rotation angle size change curve combined with Figure 11 As shown, the joint angle size obtained by solving is used to solve the end point position by forward kinematics, and the end point path is drawn combined with Figure 12 As shown, the number of interpolation points is 1000, and the calculation time is about 0.264052ms.

[0174] In the present application, according to the redundancy characteristics of the biased redundant spatial manipulator, a method for solving the inverse kinematics problem of the redundant manipulator in the conformal space is given, the inverse kinematics result under the given end pose in the continuous path tracking motion process can be solved by using the special arm rod direction vector parameterization method, compared with the joint angle parameterization method commonly used in the inverse kinematics solving of the redundant manipulator, the redundancy characteristics of the manipulator can be fully utilized while performing the task, so that the joint angles change simultaneously, and the appropriate configuration is found in the inverse kinematics solution, which has obvious geometric significance.

[0175] It can be seen from the above solving process that the method given by the present application can solve the inverse kinematics analytical solution, and the complex matrix operation and transcendental equation solving are avoided by using the conformal geometric algebra theory. Taking the biased redundant spatial manipulator in the present embodiment as an example, the inverse kinematics of the path tracking task containing 1000 interpolation points is solved by using the conformal geometric algebra method and the joint angle parameterization method respectively for 15 times in the Ubuntu 20.04, Intel(R) Core(TM) i3-4160 environment, the average value of the solving time of the conformal geometric algebra method is 0.231695ms, and the variance is 5.12E-05, the average value of the solving time of the joint angle parameterization method is 0.23527ms, and the variance is 0.00028. Through the comparison of the experimental data, it can be found that the conformal geometric algebra method improves the speed of inverse kinematics solving, and enhances the robustness of the solving process.

[0176] It is apparent to a person skilled in the art that the present application is not limited to the details of the above-described exemplary embodiments, but that it can be implemented in other embodiments without departing from the spirit or essential characteristics of the application. Therefore, the embodiments should be considered in all respects as illustrative and not restrictive, the scope of the application being defined by the appended claims rather than by the above description, and all changes coming within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. No reference signs in the claims should be considered as limiting the scope of the claims to the features to which the reference signs are attached.

[0177] Furthermore, it should be understood that although the description is made on embodiments, not every embodiment contains only one independent technical solution, and the description is made in this way only for the sake of clarity, and a person skilled in the art should consider the description as a whole, and the technical solutions in each embodiment can also be combined appropriately to form other embodiments that can be understood by a person skilled in the art.

Claims

1. A method for solving inverse kinematics of a redundant spatial manipulator with bias, comprising the following steps: Step 1: Establishing a conformal geometric model of the manipulator according to its configuration and parameters, and deriving kinematic equations 1.1 Establishing the conformal geometric model of the manipulator Conformal geometric algebra introduces two additional basis vectors e0 and e ∞ , a world coordinate system {e1, e2, e3} is established, and the joints and arm links of the robot arm are represented using conformal geometric algebra theory: The joints of the manipulator are represented as points, and are represented as follows: P = p + 0.5p 2 e ∞ +e0 wherein p is a point in Euclidean space, and P is the representation of point p in the conformal space; The arm of the manipulator is represented as a straight line segment connecting two adjacent joint nodes, and is represented as follows: L * = P1 ^ P2 ^ e ∞ P1 and P2 are two adjacent joint nodes, respectively; 1.2 Establishing the conformal geometric algebra kinematics model In the conformal space, the rotational motion of the joint is represented by a motion operator: The rotation operator is represented as follows: I is the rotation axis direction vector, and θ is the rotation angle; The geometry after rotation is represented as follows: wherein o1 is the expression form before rotation, o rotated is the expression form after rotation; The translation operator is represented as follows: t is the translation direction vector, and its size is the translation distance; The geometry after translation is represented as follows: where o2 is the representation before translation, o translated is the representation after translation. The motion operator of the joint is represented as follows: M = TR The transformation equation between adjacent joints is represented as follows: where o i+1 represents the position or pose of the i+1th joint, o i represents the position or pose of the ith joint, M i denotes the motion operator from the ith joint; Given a set of joint variables q = (q1,... qn) e C, where C represents the joint space, the kinematics equation for the end-effector pose is given by: n ​ where P 1o represents the position of joint one, p e represents the position of the end effector, a n represents the approach vector of the end effector in the initial state, a represents the approach vector of the end effector, o n represents the pose vector of the end effector in the initial state, o represents the pose vector of the end effector, M i (q i ), i = 1, 2,..., n represents the motion operator of the i-th joint, n = 7; Step two: selecting the redundant parameter vector according to the configuration characteristics of the biased redundant spatial manipulator The typical biased redundant spatial manipulator has the characteristic that the axes of the three consecutive joints are parallel at all times, and the joint axis direction vectors of the three joints are selected as the redundant parameter vector; Step three: according to the end pose information of the i th path point in the given continuous path tracking motion, the joint positions of joint one, joint six and joint seven corresponding to the path point are solved Firstly, a path planning algorithm is used to generate a series of discrete points, and then the robot arm performs a continuous path tracking motion, sequentially tracking the end effector pose corresponding to each discrete point from the starting point to the ending point. Inverse kinematics needs to be solved at each discrete point. When solving the inverse kinematics of the path point, the known information is the position p e , the approach vector a and the pose vector o of the end effector corresponding to the path point obtained in the previous path planning algorithm. The joint node position of joint one is obtained by moving the base joint node along the direction of arm one by the corresponding arm length. The end effector position is regarded as the joint seven joint node position. The joint six joint node position is obtained by moving the joint seven position along the approach vector a by the corresponding arm length. Step four: the redundant parameter vector corresponding to the current path point is solved by using the inverse kinematics solving result of the previous path point The redundant parameter vector determined by the inverse kinematics solving of the i-1 th path point is introduced into the inverse kinematics solving process of the i th path point, and the redundant parameter vector required for the inverse kinematics solving of the i th path point is solved by the conformal geometric method, and the adjacent path points are selected to realize the minimum change of the motion energy, and the redundant reduction process is completed; Step five: according to the redundant parameter vector obtained in step four, the remaining joint node positions except the information solved in step three are solved by using the conformal geometric method According to the redundant parameter vector obtained in step four, the remaining joint node positions except the information solved in step three are solved by performing conformal geometric operations on the constraint relationships of planes, spheres and straight lines in the conformal space; Step six: the joint nodes obtained in step five are connected in sequence to obtain the arm information, and the rotation angles of the joints are solved according to the change of the arm, to obtain the finite group inverse solution corresponding to the i th path point Each arm between the base and the end effector is regarded as a directed straight line segment, and when the joint variable changes, the corresponding arm moves in the rotation plane with the joint axis direction as the normal vector, and the joint variable is solved by solving the change of the directed straight line segment represented by the arm, to obtain the finite group solution of the inverse kinematics of the i th path point. Step seven: selecting the unique solution matching the path tracking task from the inverse kinematics solutions of the finite group First, remove the solutions that do not meet the joint rotation angle limit from the inverse kinematics solutions of the finite group, then select the unique solution that matches the path tracking task according to the principle of minimum energy change, so that the sum of the absolute values of the angle changes in the inverse kinematics solutions of the adjacent two path points is minimized, thus completing the inverse kinematics solution corresponding to the i th path point when executing the continuous path tracking motion. Repeat steps three to seven from the starting point to the end point to obtain the inverse kinematics solution of the whole process of the robot arm executing the continuous path tracking task.

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