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Controlling Hybrid Machine Tools concerning Error Compensation of Chain Elements

Controlling Hybrid Machine Tools concerning Error Compensation of Chain Elements Hindawi Journal of Robotics Volume 2022, Article ID 4366888, 9 pages https://doi.org/10.1155/2022/4366888 Research Article Controlling Hybrid Machine Tools concerning Error Compensation of Chain Elements 1 1 1 Khanh Duong-Quoc , Thuy Le-Thi-Thu , Long Pham-Thanh , and Ngoc Nong-Minh Division of Mechatronics, Faculty of Mechanical Engineering, ai Nguyen University of Technology, ai Nguyen 24100, Vietnam ai Nguyen University, ai Nguyen 24100, Vietnam Correspondence should be addressed to uy Le-i-u; hanthuyngoc@tnut.edu.vn Received 13 December 2021; Accepted 10 February 2022; Published 7 April 2022 Academic Editor: Yaoyao Wang Copyright © 2022 Khanh Duong-Quoc et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper introduces a methodology for controlling parallel robots in case they are used as a kind of specialized fixture to expand the technological capabilities of machines. e parallel robot is mounted on the workbench to extend the number of degrees of freedom. However, there are always measurable kinematic errors of the workbench which will be eliminated by the robot’s motion. e actual working motion of the robot is then still performed by its active joints. erefore, the displacement of each movable joint is now decided by two sources, one is due to the error compensation motion of the workbench, the other is the required work movement. According to the superposition principle, these two motions are combined into a single displacement characteristic curve to control the robot. e base exchange technique to determine the error compensation motion of the workbench, the technique of solving the inverse kinematics problem by the generalized reduced gradient (GRG) method, and the principle of joint motion combination are then introduced in detail in the paper. Finally, an example with the hexapod is presented. e obtained results, which use the robot itself to generate error-compensated movements of the workbench by means of the base exchange technique, will open up the possibility of intervening in hybrid machine systems to ensure the desired forming accuracy without no hardware intervention required. mainly used in machine tools [2]. Zhu et al. [3] considered 1. Introduction the 3-TPS hybrid machine tool (In there, the parallel has three legs, each leg has three joints: T-Twisting joint, Since their first appearance for more than two decades, P-Prismatic joint, and S- Spherical joint). e research has parallel robots have been expected to replace conventional solved the problem of selecting motor parameters and Computer Numerical Controlled (CNC) machines with mechanism design. Zhang et al. [4] presented a novel parallel serial kinematics. However, there are many unresolved manipulator with one translational and two rotational de- problems existed, especially in machining applications that grees of freedom (DOF). is 5 DOF hybrid kinematic require precision, rigidity, dexterity, and large workspaces machine tool is used in the aerospace field for large het- [1]. erogeneous complex structural component machining. In To improve the problem of limited working space while this research, a three-degree-of-freedom 2PRU-PRPS par- preserving the characteristics such as high rigidity, high allel manipulator is proposed to increase stiffness (P denotes precision, and high speed of parallel robots, many scholars the Prismatic active joint, R denotes the Rotational joint, and have researched, designed, and applied a combination of U denotes the Universal joint). Song et al. [5] performed chain mechanisms (such as CNC machines) and parallel kinematic modal characteristic analysis on the 4PRR-P robots, called hybrid mechanisms. Hybrid mechanisms are 2 Journal of Robotics is a spherical joint. Tao et al. [31] analyzed the kinematics hybrid mechanism. In the studies of hybrid machine tools, 5- axis hybrid milling machines are mentioned most, especially and workspace for a 4 DOF palletizing robot hybrid structure with a base, a waist weld frame, and a parallel in terms of kinematics, dynamics, and control [6–12]. Antonov [13] presented a forward and inverse kinematics machine arm drive. Harib et al. [1] studied the mobility of solution for a five-degree-of-freedom parallel-serial ma- three classes of mechanisms (serial, parallel, and hybrid nipulator. Similarly, the problem of kinematic analysis for kinematic structures) and focused on the mobility according hybrid machine tools was performed by Kang et al. [14], to their application in parallel and hybrid structures to Harib et al. [9], etc. reduce the number of passive joints. Zhang et al. [4] pre- In addition, some studies on hybrid structures used in sented a novel parallel manipulator with one translational degree of freedom and two degrees of rotation (PRU) that other fields are also presented. Zhou [15] investigated a robot system for subretical insertion integrated with intraoperative can be used to form a 5 DOF composite kinematic machine tool for machining large heterogeneous complex structural optical coherence tomography. is is a micromanipulator hybrid system of two parallel and one serial structure. components in the aerospace field. Many scholars have studied the hybrid structure and Meanwhile, a serial-parallel hybrid worm-like robot with 9 DOF is proposed to move on typical terrains such as the hexapod robots. Some focus is on exploiting, optimizing, stair, the slope, the gap, and the narrow passage [16]. and effectively using the features of these structures. Al- In hybrid mechanisms, the parallel part has outstanding though the parallel mechanism has high accuracy, the cu- properties that cannot be obtained by chain structures such mulative error leads to the inaccuracy of the end-effector as rigidity, light structure, high precision, and high speed when combined with the serial structure. To ensure accuracy [17]. Hexapods or Stewart platforms are typical parallel while expanding the working space, improving the tech- nological capabilities of the hexapod, a solution when the structures with 6 legs and are used in many different ap- plications [18]. Many studies on hexapods can be found as in hexapod is mounted on a CNC workbench will be presented in this paper. [19–21]. e error on the end-effector of a small hexapod milling machine is corrected using a coordinate measuring machine [19]. e hexapod is used as a coordinate mea- 2. Principle of End-Effector Error suring machine in [20]. Hexapods are used to measure forces Compensation by the Base Exchange Method in space when machining in [21]. Recently, 6-leg parallel configurations are being applied is paper presents research on a hybrid machine tool, the to machines to improve the rigidity, stability, and accuracy combination structure of CNC machine table and parallel [22]. e end-effector is used as the main-stock or work- robot hexapod 6SPS. bench of the machine tools [23–25]. ere are 4 reference systems in Figure 1 as follows: Ni et al. [26] developed an integrated geometric error O : the frame of reference attached to the machine; modelling method under the unified coordinate system framework for a general five-axis hybrid machine tool O : the frame of reference attached to the cutting tool; (including 3-PRS parallel spindle head and X–Y serial O : the frame of reference attached to the fixed plate workbench) based on small perturbation theory and a vector center of the parallel robot; chain method. Antonov et al. [13] presented inverse and O : the frame of reference attached to the center of the forward kinematics and workspace analysis of a novel 5- moving plate of the parallel robot; DOF (3T2R) parallel-serial (hybrid) manipulator, in which a parallel kinematic series is generated in the form of a tripod, Since the robot is attached to the workbench, O and O 0 M and a serial part with two DOFs (2T motion pattern) based can be converted to each other by the following axis transfer on two carriages displaced in perpendicular directions relationship: (3T2R). Zhou et al. [27] proposed an intelligent algorithm based on the extreme learning machine (ELM) and se- O .T � O . (1) 0 1 M quential mutation genetic algorithm (SGA) to determine the e cutting tool trajectory described in the O frame of inverse kinematic solutions of a robot controller with six 0 reference is as (2): degrees of freedom. Huang et al. [28] introduced the con- figuration and calibration method of the 3-P(4R) S-XY O (2) f (xyz) � 0. hybrid machine tool. Based on the simplification of the parallel 3-P(4R)S to 3-PRS, error models and error kine- e kinematic equation of the robot in the O frame of matics are derived to integrate all possible manufacturing reference is as (3): and assembly errors in the calibration process. Guo et al. [29] f xyz, q . . . q � 0. (3) proposed the design and kinematic analysis of a 5 DOF 2 1 6 hybrid manipulator with five-face machining capability. e desired trajectory of the workbench is as (4) With the same opinion, Zhang et al. [30] solved the inverse kinematics problem by the closed-loop vector method for a f (xyz) � 0. (4) hybrid mechanism with a fixed base, a movable platform and four variable length legs (2RPU and 2SPR), where R is a e actual trajectory of the table, including its kinematic rotary joint, P is a prismatic joint, U is a universal joint, and S error, is (5) Journal of Robotics 3 O O M M δ (δx,δy,δz) � f (xyz) − f (xyz) . (6) pi 3 pi 4 pi If the hexapod robot stays still and the workbench moves, this error will be copied to the O frame of reference according to the rule (6). In order for the table not to transmit this error from O to O , assuming the point O is Or 0 1 1 O1 fixed at coordinates f (xyz) , it is necessary to control the 1 pi moving joints of the hexapod mechanism so that the point O O M M O moves from f (xyz) to f (xyz) . In other words, a 4 pi 3 pi movement −δ (δx, δy,δz) of the robot mechanism gen- pi erated at point P propagating in the direction from O to O i 1 0 will eliminate the transmission error +δ (δx,δy,δz) in the pi O0 direction from O to O . To calculate this compensatory 0 1 OM motion of the robot, the base conversion and quantitative problem (7) allows to determine the motion of each joint: −1 O O (7) 􏽨f xyz, q . . . q 􏼁 􏽩 � −δ (δx, δy, δz) . 2 1 6 pi In which, the left side of (7) is the robot kinematics equation that changes from the O frame of reference to the O frame, the right side of (7) implies that the point O Figure 1: Equipment layout and the frame of reference. 1 0 O O M M moves from f (xyz) to f (xyz) . However, since the 4 pi 3 pi left side of (7) is written in the O frame of reference, its right O 1 side −δ (δx, δy, δz) has become as follows: (5) f (xyz) � 0. pi O O 1 0 (8) δ (δx, δy,δz) � δ (δx, δy, δz) .T , pi pi 2 is means that the hybrid machine tool will combine the movement of the workbench and the 6DOF hexapod where T is the axis transfer matrix between O and O in the 2 0 1 robot as a fixture to expand the technological possibilities for home position of the hexapod robot. orbital completion (2). However, instead of the ideal tra- jectory (4), due to the position error, the workbench moves 3. Motion of Intermediate Joints of the Robot along the trajectory (5), which then affects the workpiece as the path (2) is no longer correct. e hexapod robot itself has To expand the workspace of the parallel robot, the hexapod is a very high rigidity, the errors of the fixture can be neglected, placed on the workbench of the 3D CNC machine tool. is while to achieve the desired accuracy in shaping, the errors combination will bring great benefits as mentioned above. of the workbench need to be compensated. ere are two However, as a serial mechanism, the workbench has con- ways to do it as follows: siderable errors, especially after a period of work. In this paper, we propose a method in which the hexapod will (i) e first option is to correct the workbench itself so eliminate errors caused by the workbench at the same time that the actual trajectory (5) returns to the ideal one (simultaneously) with its own operation. e basis of this (4), this option needs to interfere with the machine activity is shown as follows: tool’s hardware and cannot eliminate the errors of the workbench thoroughly and permanently. erefore, it is not the subject of this paper. 3.1. Using a Hexapod Robot as a Motion Mechanism. e (ii) e second option is to accept the error of the differential mechanism is to aggregate two sources of motion workbench, and its actual trajectory is (5). Since the with different properties into one, it is widely used in gear hexapod has extra degrees of freedom, they can be processing. In cars, the differential mechanism is used with used to create compensatory motions to eliminate the opposite function, i.e., dividing torque and revolutions to errors in the workbench. Simultaneously, the robot both output shafts. Likewise, here, the hexapod plays a also combines with the workbench to create the similar role as a differential mechanism when merges the movements according to (2) to complete its task. movement from two component movements. us, the robot’s motion now has two functions, ere are two problems and a three-step process to shaping through the movement of the workbench implement the control model as shown in Figure 2 as and eliminating the errors created by the workbench. follows: is process requires no hardware intervention as (i) Problem 1: determine the characteristics of each the first option and it allows long-term precision joint when the robot performs the error-compen- mastery. is is the objective of this paper. sating motion of the floor according to (7). is To eliminate the kinematic errors of the workbench, first process is repeated for all keypoints on the trajectory th it is necessary to evaluate the error at the i point of the to build a group of compensatory properties which is trajectory as follows: the solution of (7) and is symbolled as follows: 4 Journal of Robotics e workbench’s errors e parallel robot e synthesize trajectories e operating trajectories Workspace Joint space Figure 2: Description of input/output data relationships when controlling the robot considering the error of the floor. compen 4. An Example with Hexapod q � q , . . . q . (9) 1 6 e illustrated computational model is a hybrid structure: a (ii) Problem 2: determine the motion characteristics of parallel hexapod robot is placed on a CNC machine each joint when the robot executes the trajectory of workbench. e error of the workbench will be compensated the cutting tool according to (2). e kinematic by the hexapod at the same time as the orbital movement model of this problem is as follows: required by the robot. e trajectory to be taken is a circle O O (10) with a diameter of 300 mm. e operation process un- f xyz, q . . . q 􏼁 � f (xyz) , 2 1 6 1 dergoes these two movements: the workbench moves in a feed circle of diameter 150 mm, the center of the moving plate of (11) q � q , . . . q 􏼁 . 1 6 the tool-mounted robot also moves with a circle of diameter 150 mm. As mentioned above, the parallel robot has very us, the total displacement of each joint on the hexapod high accuracy, so the error caused by the robot can be ig- robot along the toolpath while compensating the error of the nored. At this time, the error caused by the workbench floor, will be (12): moving with the orbit of 150 mm will be compensated by the movement of the hexapod itself. compen feed (12) q � q + q . e accuracy of the end-effector is assured by the fol- lowing steps. ese data are a direct output to control the active joints to achieve the desired trajectory (4) instead of without compensating this trajectory (5). 4.1. Kinematic Model of the Hexapod Robot with SPS Con- e three-step control process mentioned above is as figuration and Hybrid Structure. e linear drive leg in the follows: hexapod structure makes the vector loop form through one leg and it takes the form of a tetragonal loop instead of a Step 1: set up and solve equation (7); pentagonal loop (Figure 3). Step 2: set and solve equation (10); Because b and p cross each other in space, it requires a i i Step 3: determine the kinematic properties according to rotation matrix R to turn them coplanar. e equation of RPY equation (12). the closed loop on the quadrilateral is written as follows: → → (13) l � − b + t + R .p , i � 1, . . . , 6. i i RPY i 3.2. Methods for Preparing Kinematic Data for Robots. To prepare the kinematic data of the robot, it is necessary to e instantaneous value and direction cosine must be perform the three-step process mentioned at the end of preknown to describe the detailed coordinates of the node Section 3.1, of which the most important is solving equations Bi. Notice that in this case, the rotation around the AiBi axis (7) and (10). ese two equations have the same nonlinear does not create a new kinematic solution, so the joint S is and transcendent nature with the same solving method. In modeled by two variables u and v equivalent to the U this paper, we will apply the GRG method to solve these (cardan) joint, consider the diagram of Figure 4: th equations, this problem is mentioned in the documents erefore, for the detailed equation of the i leg is [20, 32, 33] and not repeated here. written as follows: l c(u).c(v) x p cβ.cc sα.sβ.cc − cα.sc cα.sβ.cc + sα.sc x i Ai x Bi ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ l c(u).s(v) ⎥ � −⎢ y ⎥ + ⎢ p ⎥ + ⎢ cβ.sc sα.sβ.sc + cα.cc cα.sβ.sc − sα.cc ⎥.⎢ y ⎥. (14) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ i ⎥ ⎢ Ai ⎥ ⎢ y ⎥ ⎢ ⎥ ⎢ Bi ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ l s(u) z p −sβ sα.cβ cα.cβ z i Ai z Bi Journal of Robotics 5 → → w −1 → − P � b − l − R ∗ a , RPY i Px x l .Cu .Cv x B u 4 1 Bi i i i Ai B 1 ⎥ (15) ⎡ ⎢ ⎤ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ 4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −⎢ Py ⎥ � ⎢ y ⎥ − ⎢ l .Cu .Sv ⎥ − 􏼂R 􏼃 ∗ ⎢ y ⎥, ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Bi ⎥ ⎢ i i i ⎥ RPY ⎢ Ai ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Pz z l .Su z Bi i i Ai where the elements of the inverse matrix RRPY are as follows: a11 � (cos(β+c)+cos(β-c))/2, a12 � (sin(β+c)-sin(β- c))/2, a13 � -sin(β), a21 � (-cos(α+β+c)+cos(α-β+c)-cos(α+ 1 A A 4 i β-c)+cos(α-β-c)-2∗ sin(α+c)+2∗ sin(α-c))/4, a22 � (2∗ cos(α+ c)+2∗ cos(α-c)-sin(α+β+c)+sin(α-β+c)+ sin(α+β-c)-sin(α- th Figure 3: Development of the i branch of a driven parallel robot β-c))/4, a23 � (sin(α+β)+sin(α-β))/2, a31 � (-2∗ cos(α+c)+2∗ type P. cos(α-c)+sin(α+β+c)-sin(α-β+c)+ sin(α+β-c)-sin(α-β-c))/4, a32 � (-cos(α+β+c)+cos(α-β+c)+cos(α+β-c)-cos(α-β-c)-2∗ sin (α+c)-2∗ sin(α-c))/4, a33 � (cos(α+β)+cos(α-β))/2. th erefore, the system of kinematic equations for the i leg of the hexapod robot applying the base exchange method is as follows: (i) Px �x - l .C .C - ((cos(β+c)+cos(β-c))/2).x - Bi i ui vi Ai ((sin(β+c)-sin(β-c))/2).y - (-sin(β)).z Ai Ai (ii) Py � y - l .C .S - ((-cos(α+β+c)+cos(α-β+c)- Bi i ui vi cos(α+β-c)+cos(α-β-c)-2∗sin(α+c)+2∗sin(α-c))/ 4).x - ((2∗cos(α+c)+2∗cos(α-c)-sin(α+β+c)+ Ai sin(α-β+c)+sin(α+β-c)-sin(α-β-c))/4).y - ((sin Ai (α+β)+sin(α-β))/2).z Ai Figure 4: Schematic illustration of calculating the coordinates of (iii) Pz � z -l .S - ((-2∗cos(α+c)+2∗cos(α-c)+sin(α+ Bi i ui the endpoint AiBi through its direction cosine. β+c)-sin(α-β+c)+sin(α+β-c)-sin(α-β-c))/4).x - Ai ((-cos(α+β+c)+cos(α-β+c)+cos(α+β-c)-cos(α-β-c)- Retrieve for 6 legs to get full kinematics equations. 2∗sin(α+c)-2∗sin(α-c))/4). y - ((cos(α+β)+cos(α- Ai β))/2). z Ai 4.2. Kinematic Error of Workbench. e workbench makes e optimal function is developed to solve the the motion according to the desired trajectory, which is a problem according to the GRG numerical method as circle of diameter 150 mm. In real motion, there is an error at follows: every single point on the trajectory. In this model, the author (Px + xBi - li.Cui.Cvi - ((cos(ßβ+c)+cos(β-c))/2).xAi - uses the errors of the workbench in [34]. Totally 64 points ((sin(ßβ+c)-sin(β-c))/2).yAi - (-sin(β)).zAi)^2 � 0 are surveyed on the orbital circle. e desired and actual trajectory of the workbench is (Py + yBi - li.Cui.Svi - ((-cos(aα+ßβ+c)+cos(α- β+c)-cos(aα+β-c)+cos(α-β-c)-2∗sin(aα+c)+2∗sin(α- shown in Figure 5: c))/4).xAi - ((2 cos(aα+c)+2∗cos(α-c)-sin(aα+ßβ+ c)+sin(α-β+c)+sin(aα+β-c)-sin(α-β-c))/4).yAi - ((sin 4.3. Error Compensation Motion. e error compensation (aα+β)+sin(α-β))/2).zAi)^2 � 0. (Pz + zBi -li.Sui - ((-2∗ values, after being taken from the inverse of the coordinates cos(aα+c)+2∗cos(α-c)+sin(aα+ßβ+c)-sin(α-β+c)+sin of the errors, will be used as input for the hexapod dis- (aα+β-c)-sin(α-β-c))/4).xAi - ((-cos(aα+ßβ+c)+cos(α- placement kinematics problem. GRG numerical method β+c)+cos(aα+β-c)-cos(α-β-c)-2 sin(aα+c)-2∗sin(α-c))/ with optimal function is applied to solve this problem. e 4). yAi - ((cos(aα+β)+cos(α-β))/2). zAi) ^2 � 0 displacements of the joint’s variables and the auxiliary pa- rameters at the error compensation points are determined: Similar process is needed to do for the 6 legs of the th From the kinematic diagram of the leg i hexapod to solve the kinematic problem. when trans- ferring axis is as shown in Figure 6, the kinematic vector e inverse kinematic problem solved by equation of the robot when transferring axis is as follows: numerical method GRG is applied to determine joint 6 Journal of Robotics Figure 5: Simulation of the desired and actual trajectory of the workbench-detailed view on circle segment; see difference between both trajectories (red line: desired trajectory; black line: actual trajectory with error). th Figure 6: Kinematic diagram of the leg i when transferring axis. 0.1 0.05 13579 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 -0.05 -0.1 Δl1 Δl4 Δl2 Δl5 Δl3 Δl6 Figure 7: Graph of 6 control joint variables of 6 robot legs used to compensate for the workbench error. variables and sub-parameters at 64 survey points-64 (Px - xAi + Cβ.Cc.xBi + (Sα.Sβ.Cc - Cα.Sc).yBi + coordinates of error compensation points. e values of 6 (Cα.Sβ.Cc + Sα.Sc).zBi - li.Cui.Cvi)2 � 0 active joint variables at 64 survey points are shown in (Py - yAi + Cβ.Sc.xBi + (Sα.Sβ.Sc + Cα.Cc).yBi + Figure 7. (Cα.Sβ.Sc - Sα.Cc).zBi - li.Cui.Svi)2 � 0 (Pz - zAi - Sβ.xBi + Sα.Cβ. yBi + Cα.Cβ.zBi - li.Sui)2 � 0 4.4. Feed Movement. e hexapod robot performs a e GRG numerical method is applied to solve these optimal movement trending a circular tool with a diameter of functions. Solving the inverse kinematic problem at 64 survey 150 mm. Use equation (15) for 6 legs and change them to the points will give 64 sets of values of joint variables and optimal form: mm Journal of Robotics 7 13579 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 l1 (mm) l4 (mm) l2 (mm) l5 (mm) l3 (mm) l6 (mm) Figure 8: Control joint variables when performing the feed movement. 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 L1 (mm) L1 (mm) L1 (mm) L1 (mm) L1 (mm) L1 (mm) Figure 9: Control joint variables for the robot to perform two functions simultaneously. 00.200 00.180 00.160 00.140 00.120 00.100 00.080 00.060 00.040 00.020 00.000 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 Error before compensation Error aer compensation Figure 10: Pre- and post-compensated orbital errors at 64 survey points. Error (mm) mm 8 Journal of Robotics [3] L. Da Zhu, J. S. Shi, G. Q. Cai, and W. S. Wang, “Kinematic/ corresponding sub-parameters. In which, the value sets of 6 dynamic simulation based on ADAMS for 3-TPS hybrid control joint variables are illustrated as in the following graph machine tool,” Dongbei Daxue Xuebao/Journal of North- (Figure 8): eastern University, vol. 28, no. 10, pp. 1473–1476, 2007. [4] H. Zhang, H. Fang, Y. Fang, and B. Jiang, “Workspace analysis 4.5. Synthetic Motion. e synthetic motion in the of a hybrid kinematic machine tool with high rotational hexapod control is the value of the active joint variable applications,” Mathematical Problems in Engineering, when participating in two processes: error compensation vol. 2018, pp. 1–12, 2018. [5] W. Song, Y. Xu, Y. Lv, Y. Liu, and Z. Liang, “Kinematics for the CNC workbench and the main feed movement of analysis of a 4PRR-P hybrid machining mechanism,” MATEC 150 mm. Web of Conferences, vol. 232, pp. 03054–03056, 2018. e data set for the synthetic motion processes is shown [6] S. Son, T. Kim, S. E. Sarma, and A. 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Controlling Hybrid Machine Tools concerning Error Compensation of Chain Elements

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Copyright © 2022 Khanh Duong-Quoc et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Journal of Robotics Volume 2022, Article ID 4366888, 9 pages https://doi.org/10.1155/2022/4366888 Research Article Controlling Hybrid Machine Tools concerning Error Compensation of Chain Elements 1 1 1 Khanh Duong-Quoc , Thuy Le-Thi-Thu , Long Pham-Thanh , and Ngoc Nong-Minh Division of Mechatronics, Faculty of Mechanical Engineering, ai Nguyen University of Technology, ai Nguyen 24100, Vietnam ai Nguyen University, ai Nguyen 24100, Vietnam Correspondence should be addressed to uy Le-i-u; hanthuyngoc@tnut.edu.vn Received 13 December 2021; Accepted 10 February 2022; Published 7 April 2022 Academic Editor: Yaoyao Wang Copyright © 2022 Khanh Duong-Quoc et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper introduces a methodology for controlling parallel robots in case they are used as a kind of specialized fixture to expand the technological capabilities of machines. e parallel robot is mounted on the workbench to extend the number of degrees of freedom. However, there are always measurable kinematic errors of the workbench which will be eliminated by the robot’s motion. e actual working motion of the robot is then still performed by its active joints. erefore, the displacement of each movable joint is now decided by two sources, one is due to the error compensation motion of the workbench, the other is the required work movement. According to the superposition principle, these two motions are combined into a single displacement characteristic curve to control the robot. e base exchange technique to determine the error compensation motion of the workbench, the technique of solving the inverse kinematics problem by the generalized reduced gradient (GRG) method, and the principle of joint motion combination are then introduced in detail in the paper. Finally, an example with the hexapod is presented. e obtained results, which use the robot itself to generate error-compensated movements of the workbench by means of the base exchange technique, will open up the possibility of intervening in hybrid machine systems to ensure the desired forming accuracy without no hardware intervention required. mainly used in machine tools [2]. Zhu et al. [3] considered 1. Introduction the 3-TPS hybrid machine tool (In there, the parallel has three legs, each leg has three joints: T-Twisting joint, Since their first appearance for more than two decades, P-Prismatic joint, and S- Spherical joint). e research has parallel robots have been expected to replace conventional solved the problem of selecting motor parameters and Computer Numerical Controlled (CNC) machines with mechanism design. Zhang et al. [4] presented a novel parallel serial kinematics. However, there are many unresolved manipulator with one translational and two rotational de- problems existed, especially in machining applications that grees of freedom (DOF). is 5 DOF hybrid kinematic require precision, rigidity, dexterity, and large workspaces machine tool is used in the aerospace field for large het- [1]. erogeneous complex structural component machining. In To improve the problem of limited working space while this research, a three-degree-of-freedom 2PRU-PRPS par- preserving the characteristics such as high rigidity, high allel manipulator is proposed to increase stiffness (P denotes precision, and high speed of parallel robots, many scholars the Prismatic active joint, R denotes the Rotational joint, and have researched, designed, and applied a combination of U denotes the Universal joint). Song et al. [5] performed chain mechanisms (such as CNC machines) and parallel kinematic modal characteristic analysis on the 4PRR-P robots, called hybrid mechanisms. Hybrid mechanisms are 2 Journal of Robotics is a spherical joint. Tao et al. [31] analyzed the kinematics hybrid mechanism. In the studies of hybrid machine tools, 5- axis hybrid milling machines are mentioned most, especially and workspace for a 4 DOF palletizing robot hybrid structure with a base, a waist weld frame, and a parallel in terms of kinematics, dynamics, and control [6–12]. Antonov [13] presented a forward and inverse kinematics machine arm drive. Harib et al. [1] studied the mobility of solution for a five-degree-of-freedom parallel-serial ma- three classes of mechanisms (serial, parallel, and hybrid nipulator. Similarly, the problem of kinematic analysis for kinematic structures) and focused on the mobility according hybrid machine tools was performed by Kang et al. [14], to their application in parallel and hybrid structures to Harib et al. [9], etc. reduce the number of passive joints. Zhang et al. [4] pre- In addition, some studies on hybrid structures used in sented a novel parallel manipulator with one translational degree of freedom and two degrees of rotation (PRU) that other fields are also presented. Zhou [15] investigated a robot system for subretical insertion integrated with intraoperative can be used to form a 5 DOF composite kinematic machine tool for machining large heterogeneous complex structural optical coherence tomography. is is a micromanipulator hybrid system of two parallel and one serial structure. components in the aerospace field. Many scholars have studied the hybrid structure and Meanwhile, a serial-parallel hybrid worm-like robot with 9 DOF is proposed to move on typical terrains such as the hexapod robots. Some focus is on exploiting, optimizing, stair, the slope, the gap, and the narrow passage [16]. and effectively using the features of these structures. Al- In hybrid mechanisms, the parallel part has outstanding though the parallel mechanism has high accuracy, the cu- properties that cannot be obtained by chain structures such mulative error leads to the inaccuracy of the end-effector as rigidity, light structure, high precision, and high speed when combined with the serial structure. To ensure accuracy [17]. Hexapods or Stewart platforms are typical parallel while expanding the working space, improving the tech- nological capabilities of the hexapod, a solution when the structures with 6 legs and are used in many different ap- plications [18]. Many studies on hexapods can be found as in hexapod is mounted on a CNC workbench will be presented in this paper. [19–21]. e error on the end-effector of a small hexapod milling machine is corrected using a coordinate measuring machine [19]. e hexapod is used as a coordinate mea- 2. Principle of End-Effector Error suring machine in [20]. Hexapods are used to measure forces Compensation by the Base Exchange Method in space when machining in [21]. Recently, 6-leg parallel configurations are being applied is paper presents research on a hybrid machine tool, the to machines to improve the rigidity, stability, and accuracy combination structure of CNC machine table and parallel [22]. e end-effector is used as the main-stock or work- robot hexapod 6SPS. bench of the machine tools [23–25]. ere are 4 reference systems in Figure 1 as follows: Ni et al. [26] developed an integrated geometric error O : the frame of reference attached to the machine; modelling method under the unified coordinate system framework for a general five-axis hybrid machine tool O : the frame of reference attached to the cutting tool; (including 3-PRS parallel spindle head and X–Y serial O : the frame of reference attached to the fixed plate workbench) based on small perturbation theory and a vector center of the parallel robot; chain method. Antonov et al. [13] presented inverse and O : the frame of reference attached to the center of the forward kinematics and workspace analysis of a novel 5- moving plate of the parallel robot; DOF (3T2R) parallel-serial (hybrid) manipulator, in which a parallel kinematic series is generated in the form of a tripod, Since the robot is attached to the workbench, O and O 0 M and a serial part with two DOFs (2T motion pattern) based can be converted to each other by the following axis transfer on two carriages displaced in perpendicular directions relationship: (3T2R). Zhou et al. [27] proposed an intelligent algorithm based on the extreme learning machine (ELM) and se- O .T � O . (1) 0 1 M quential mutation genetic algorithm (SGA) to determine the e cutting tool trajectory described in the O frame of inverse kinematic solutions of a robot controller with six 0 reference is as (2): degrees of freedom. Huang et al. [28] introduced the con- figuration and calibration method of the 3-P(4R) S-XY O (2) f (xyz) � 0. hybrid machine tool. Based on the simplification of the parallel 3-P(4R)S to 3-PRS, error models and error kine- e kinematic equation of the robot in the O frame of matics are derived to integrate all possible manufacturing reference is as (3): and assembly errors in the calibration process. Guo et al. [29] f xyz, q . . . q � 0. (3) proposed the design and kinematic analysis of a 5 DOF 2 1 6 hybrid manipulator with five-face machining capability. e desired trajectory of the workbench is as (4) With the same opinion, Zhang et al. [30] solved the inverse kinematics problem by the closed-loop vector method for a f (xyz) � 0. (4) hybrid mechanism with a fixed base, a movable platform and four variable length legs (2RPU and 2SPR), where R is a e actual trajectory of the table, including its kinematic rotary joint, P is a prismatic joint, U is a universal joint, and S error, is (5) Journal of Robotics 3 O O M M δ (δx,δy,δz) � f (xyz) − f (xyz) . (6) pi 3 pi 4 pi If the hexapod robot stays still and the workbench moves, this error will be copied to the O frame of reference according to the rule (6). In order for the table not to transmit this error from O to O , assuming the point O is Or 0 1 1 O1 fixed at coordinates f (xyz) , it is necessary to control the 1 pi moving joints of the hexapod mechanism so that the point O O M M O moves from f (xyz) to f (xyz) . In other words, a 4 pi 3 pi movement −δ (δx, δy,δz) of the robot mechanism gen- pi erated at point P propagating in the direction from O to O i 1 0 will eliminate the transmission error +δ (δx,δy,δz) in the pi O0 direction from O to O . To calculate this compensatory 0 1 OM motion of the robot, the base conversion and quantitative problem (7) allows to determine the motion of each joint: −1 O O (7) 􏽨f xyz, q . . . q 􏼁 􏽩 � −δ (δx, δy, δz) . 2 1 6 pi In which, the left side of (7) is the robot kinematics equation that changes from the O frame of reference to the O frame, the right side of (7) implies that the point O Figure 1: Equipment layout and the frame of reference. 1 0 O O M M moves from f (xyz) to f (xyz) . However, since the 4 pi 3 pi left side of (7) is written in the O frame of reference, its right O 1 side −δ (δx, δy, δz) has become as follows: (5) f (xyz) � 0. pi O O 1 0 (8) δ (δx, δy,δz) � δ (δx, δy, δz) .T , pi pi 2 is means that the hybrid machine tool will combine the movement of the workbench and the 6DOF hexapod where T is the axis transfer matrix between O and O in the 2 0 1 robot as a fixture to expand the technological possibilities for home position of the hexapod robot. orbital completion (2). However, instead of the ideal tra- jectory (4), due to the position error, the workbench moves 3. Motion of Intermediate Joints of the Robot along the trajectory (5), which then affects the workpiece as the path (2) is no longer correct. e hexapod robot itself has To expand the workspace of the parallel robot, the hexapod is a very high rigidity, the errors of the fixture can be neglected, placed on the workbench of the 3D CNC machine tool. is while to achieve the desired accuracy in shaping, the errors combination will bring great benefits as mentioned above. of the workbench need to be compensated. ere are two However, as a serial mechanism, the workbench has con- ways to do it as follows: siderable errors, especially after a period of work. In this paper, we propose a method in which the hexapod will (i) e first option is to correct the workbench itself so eliminate errors caused by the workbench at the same time that the actual trajectory (5) returns to the ideal one (simultaneously) with its own operation. e basis of this (4), this option needs to interfere with the machine activity is shown as follows: tool’s hardware and cannot eliminate the errors of the workbench thoroughly and permanently. erefore, it is not the subject of this paper. 3.1. Using a Hexapod Robot as a Motion Mechanism. e (ii) e second option is to accept the error of the differential mechanism is to aggregate two sources of motion workbench, and its actual trajectory is (5). Since the with different properties into one, it is widely used in gear hexapod has extra degrees of freedom, they can be processing. In cars, the differential mechanism is used with used to create compensatory motions to eliminate the opposite function, i.e., dividing torque and revolutions to errors in the workbench. Simultaneously, the robot both output shafts. Likewise, here, the hexapod plays a also combines with the workbench to create the similar role as a differential mechanism when merges the movements according to (2) to complete its task. movement from two component movements. us, the robot’s motion now has two functions, ere are two problems and a three-step process to shaping through the movement of the workbench implement the control model as shown in Figure 2 as and eliminating the errors created by the workbench. follows: is process requires no hardware intervention as (i) Problem 1: determine the characteristics of each the first option and it allows long-term precision joint when the robot performs the error-compen- mastery. is is the objective of this paper. sating motion of the floor according to (7). is To eliminate the kinematic errors of the workbench, first process is repeated for all keypoints on the trajectory th it is necessary to evaluate the error at the i point of the to build a group of compensatory properties which is trajectory as follows: the solution of (7) and is symbolled as follows: 4 Journal of Robotics e workbench’s errors e parallel robot e synthesize trajectories e operating trajectories Workspace Joint space Figure 2: Description of input/output data relationships when controlling the robot considering the error of the floor. compen 4. An Example with Hexapod q � q , . . . q . (9) 1 6 e illustrated computational model is a hybrid structure: a (ii) Problem 2: determine the motion characteristics of parallel hexapod robot is placed on a CNC machine each joint when the robot executes the trajectory of workbench. e error of the workbench will be compensated the cutting tool according to (2). e kinematic by the hexapod at the same time as the orbital movement model of this problem is as follows: required by the robot. e trajectory to be taken is a circle O O (10) with a diameter of 300 mm. e operation process un- f xyz, q . . . q 􏼁 � f (xyz) , 2 1 6 1 dergoes these two movements: the workbench moves in a feed circle of diameter 150 mm, the center of the moving plate of (11) q � q , . . . q 􏼁 . 1 6 the tool-mounted robot also moves with a circle of diameter 150 mm. As mentioned above, the parallel robot has very us, the total displacement of each joint on the hexapod high accuracy, so the error caused by the robot can be ig- robot along the toolpath while compensating the error of the nored. At this time, the error caused by the workbench floor, will be (12): moving with the orbit of 150 mm will be compensated by the movement of the hexapod itself. compen feed (12) q � q + q . e accuracy of the end-effector is assured by the fol- lowing steps. ese data are a direct output to control the active joints to achieve the desired trajectory (4) instead of without compensating this trajectory (5). 4.1. Kinematic Model of the Hexapod Robot with SPS Con- e three-step control process mentioned above is as figuration and Hybrid Structure. e linear drive leg in the follows: hexapod structure makes the vector loop form through one leg and it takes the form of a tetragonal loop instead of a Step 1: set up and solve equation (7); pentagonal loop (Figure 3). Step 2: set and solve equation (10); Because b and p cross each other in space, it requires a i i Step 3: determine the kinematic properties according to rotation matrix R to turn them coplanar. e equation of RPY equation (12). the closed loop on the quadrilateral is written as follows: → → (13) l � − b + t + R .p , i � 1, . . . , 6. i i RPY i 3.2. Methods for Preparing Kinematic Data for Robots. To prepare the kinematic data of the robot, it is necessary to e instantaneous value and direction cosine must be perform the three-step process mentioned at the end of preknown to describe the detailed coordinates of the node Section 3.1, of which the most important is solving equations Bi. Notice that in this case, the rotation around the AiBi axis (7) and (10). ese two equations have the same nonlinear does not create a new kinematic solution, so the joint S is and transcendent nature with the same solving method. In modeled by two variables u and v equivalent to the U this paper, we will apply the GRG method to solve these (cardan) joint, consider the diagram of Figure 4: th equations, this problem is mentioned in the documents erefore, for the detailed equation of the i leg is [20, 32, 33] and not repeated here. written as follows: l c(u).c(v) x p cβ.cc sα.sβ.cc − cα.sc cα.sβ.cc + sα.sc x i Ai x Bi ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ l c(u).s(v) ⎥ � −⎢ y ⎥ + ⎢ p ⎥ + ⎢ cβ.sc sα.sβ.sc + cα.cc cα.sβ.sc − sα.cc ⎥.⎢ y ⎥. (14) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ i ⎥ ⎢ Ai ⎥ ⎢ y ⎥ ⎢ ⎥ ⎢ Bi ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ l s(u) z p −sβ sα.cβ cα.cβ z i Ai z Bi Journal of Robotics 5 → → w −1 → − P � b − l − R ∗ a , RPY i Px x l .Cu .Cv x B u 4 1 Bi i i i Ai B 1 ⎥ (15) ⎡ ⎢ ⎤ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ 4 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −⎢ Py ⎥ � ⎢ y ⎥ − ⎢ l .Cu .Sv ⎥ − 􏼂R 􏼃 ∗ ⎢ y ⎥, ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Bi ⎥ ⎢ i i i ⎥ RPY ⎢ Ai ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Pz z l .Su z Bi i i Ai where the elements of the inverse matrix RRPY are as follows: a11 � (cos(β+c)+cos(β-c))/2, a12 � (sin(β+c)-sin(β- c))/2, a13 � -sin(β), a21 � (-cos(α+β+c)+cos(α-β+c)-cos(α+ 1 A A 4 i β-c)+cos(α-β-c)-2∗ sin(α+c)+2∗ sin(α-c))/4, a22 � (2∗ cos(α+ c)+2∗ cos(α-c)-sin(α+β+c)+sin(α-β+c)+ sin(α+β-c)-sin(α- th Figure 3: Development of the i branch of a driven parallel robot β-c))/4, a23 � (sin(α+β)+sin(α-β))/2, a31 � (-2∗ cos(α+c)+2∗ type P. cos(α-c)+sin(α+β+c)-sin(α-β+c)+ sin(α+β-c)-sin(α-β-c))/4, a32 � (-cos(α+β+c)+cos(α-β+c)+cos(α+β-c)-cos(α-β-c)-2∗ sin (α+c)-2∗ sin(α-c))/4, a33 � (cos(α+β)+cos(α-β))/2. th erefore, the system of kinematic equations for the i leg of the hexapod robot applying the base exchange method is as follows: (i) Px �x - l .C .C - ((cos(β+c)+cos(β-c))/2).x - Bi i ui vi Ai ((sin(β+c)-sin(β-c))/2).y - (-sin(β)).z Ai Ai (ii) Py � y - l .C .S - ((-cos(α+β+c)+cos(α-β+c)- Bi i ui vi cos(α+β-c)+cos(α-β-c)-2∗sin(α+c)+2∗sin(α-c))/ 4).x - ((2∗cos(α+c)+2∗cos(α-c)-sin(α+β+c)+ Ai sin(α-β+c)+sin(α+β-c)-sin(α-β-c))/4).y - ((sin Ai (α+β)+sin(α-β))/2).z Ai Figure 4: Schematic illustration of calculating the coordinates of (iii) Pz � z -l .S - ((-2∗cos(α+c)+2∗cos(α-c)+sin(α+ Bi i ui the endpoint AiBi through its direction cosine. β+c)-sin(α-β+c)+sin(α+β-c)-sin(α-β-c))/4).x - Ai ((-cos(α+β+c)+cos(α-β+c)+cos(α+β-c)-cos(α-β-c)- Retrieve for 6 legs to get full kinematics equations. 2∗sin(α+c)-2∗sin(α-c))/4). y - ((cos(α+β)+cos(α- Ai β))/2). z Ai 4.2. Kinematic Error of Workbench. e workbench makes e optimal function is developed to solve the the motion according to the desired trajectory, which is a problem according to the GRG numerical method as circle of diameter 150 mm. In real motion, there is an error at follows: every single point on the trajectory. In this model, the author (Px + xBi - li.Cui.Cvi - ((cos(ßβ+c)+cos(β-c))/2).xAi - uses the errors of the workbench in [34]. Totally 64 points ((sin(ßβ+c)-sin(β-c))/2).yAi - (-sin(β)).zAi)^2 � 0 are surveyed on the orbital circle. e desired and actual trajectory of the workbench is (Py + yBi - li.Cui.Svi - ((-cos(aα+ßβ+c)+cos(α- β+c)-cos(aα+β-c)+cos(α-β-c)-2∗sin(aα+c)+2∗sin(α- shown in Figure 5: c))/4).xAi - ((2 cos(aα+c)+2∗cos(α-c)-sin(aα+ßβ+ c)+sin(α-β+c)+sin(aα+β-c)-sin(α-β-c))/4).yAi - ((sin 4.3. Error Compensation Motion. e error compensation (aα+β)+sin(α-β))/2).zAi)^2 � 0. (Pz + zBi -li.Sui - ((-2∗ values, after being taken from the inverse of the coordinates cos(aα+c)+2∗cos(α-c)+sin(aα+ßβ+c)-sin(α-β+c)+sin of the errors, will be used as input for the hexapod dis- (aα+β-c)-sin(α-β-c))/4).xAi - ((-cos(aα+ßβ+c)+cos(α- placement kinematics problem. GRG numerical method β+c)+cos(aα+β-c)-cos(α-β-c)-2 sin(aα+c)-2∗sin(α-c))/ with optimal function is applied to solve this problem. e 4). yAi - ((cos(aα+β)+cos(α-β))/2). zAi) ^2 � 0 displacements of the joint’s variables and the auxiliary pa- rameters at the error compensation points are determined: Similar process is needed to do for the 6 legs of the th From the kinematic diagram of the leg i hexapod to solve the kinematic problem. when trans- ferring axis is as shown in Figure 6, the kinematic vector e inverse kinematic problem solved by equation of the robot when transferring axis is as follows: numerical method GRG is applied to determine joint 6 Journal of Robotics Figure 5: Simulation of the desired and actual trajectory of the workbench-detailed view on circle segment; see difference between both trajectories (red line: desired trajectory; black line: actual trajectory with error). th Figure 6: Kinematic diagram of the leg i when transferring axis. 0.1 0.05 13579 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 -0.05 -0.1 Δl1 Δl4 Δl2 Δl5 Δl3 Δl6 Figure 7: Graph of 6 control joint variables of 6 robot legs used to compensate for the workbench error. variables and sub-parameters at 64 survey points-64 (Px - xAi + Cβ.Cc.xBi + (Sα.Sβ.Cc - Cα.Sc).yBi + coordinates of error compensation points. e values of 6 (Cα.Sβ.Cc + Sα.Sc).zBi - li.Cui.Cvi)2 � 0 active joint variables at 64 survey points are shown in (Py - yAi + Cβ.Sc.xBi + (Sα.Sβ.Sc + Cα.Cc).yBi + Figure 7. (Cα.Sβ.Sc - Sα.Cc).zBi - li.Cui.Svi)2 � 0 (Pz - zAi - Sβ.xBi + Sα.Cβ. yBi + Cα.Cβ.zBi - li.Sui)2 � 0 4.4. Feed Movement. e hexapod robot performs a e GRG numerical method is applied to solve these optimal movement trending a circular tool with a diameter of functions. Solving the inverse kinematic problem at 64 survey 150 mm. Use equation (15) for 6 legs and change them to the points will give 64 sets of values of joint variables and optimal form: mm Journal of Robotics 7 13579 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 l1 (mm) l4 (mm) l2 (mm) l5 (mm) l3 (mm) l6 (mm) Figure 8: Control joint variables when performing the feed movement. 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 L1 (mm) L1 (mm) L1 (mm) L1 (mm) L1 (mm) L1 (mm) Figure 9: Control joint variables for the robot to perform two functions simultaneously. 00.200 00.180 00.160 00.140 00.120 00.100 00.080 00.060 00.040 00.020 00.000 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 Error before compensation Error aer compensation Figure 10: Pre- and post-compensated orbital errors at 64 survey points. Error (mm) mm 8 Journal of Robotics [3] L. Da Zhu, J. S. Shi, G. Q. Cai, and W. S. Wang, “Kinematic/ corresponding sub-parameters. 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Journal of RoboticsHindawi Publishing Corporation

Published: Apr 7, 2022

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