Sensors & Transducers 21 by IFSA http://www.sensorsportal.com Research on Kinematics for Inhibition Fluttering of Picking Robot Arm Zhang Zhiyong, * Tang Jinglei, Huang lvwen, Li Heqing College of information Engineering, Northwest A&F University, Yangling, Shaanxi, 7121, China E-mail: zzy@nwsuaf.edu.cn, tangjinglei@nwsuaf.edu.cn Received: 18 September 21 /Accepted: 22 November 21 /Published: December 21 Abstract: To solve basic problems on restraining fluttering of picking robot arm, this paper used the theory on forward kinematics of picking arm through joint movement, mapping, transformation and matrix operations, obtained the redundant problems equation affecting picking robot arm motion. Simulation results showed each joint displacement curve of the robot arm realized the smooth transition, which is differential coefficient and continuous, thus verified the correctness of mechanical arm kinematics, provided theoretical basis on effectively restraining joints fluttering problems. Copyright 21 IFSA. Keywords: Homogeneous coordinates, Posture, Matrix transformation, Forward kinematics. 1. Introduction Picking robot arm manipulator consists of the upper arm, forearm, picking arm and shoulder joint, elbow and wrist joints, which is called -DOF. All the arm joints are open rod chain connecting type structure driven by corresponding joint motor. Picking operations directly in contact with the fruit picking arm mounted on the wrist on wheeled mobile platform, its mechanics body structure diagram is shown as Fig. 1. Picking robot arm constitute -DOF RRR rotation structure. The first freedom degree pedestal is fixed on the mobile trolley to support the rotating shoulder joint, the second and third freedom degrees are the axis of rotation, lift picking robot gripper to move in any direction on space, and according to the picking control instruction, send hand to stay in picking apple's position, so as to complete picking ripe fruits. At present, in order to achieve the automatic picking, such as spraying, transplanting and other agricultural operations process, have developed with a finger, attractor, scissors and other styles of picking gripper [1]. Picking objects are studied in this paper mainly apple fruit given size priority to 75-85 mm, the actual picking methods of cut stem near the fruit trees side according to apple's physical properties, which is equipped with rotary cutter inside of the clamping device, driven by the servo motor fixed on the back of picking arm to rotate cutting fruit stalk. The paper aimed at picking arm motion analysis, whose essence is proposing the each joint movement rule of, the joint is composed of series of joint connection space-open-chain mechanism. In order to restrain fluttering and improve stability control, its kinematic and dynamic location including picking arm posture analysis must be analyzed to draw a picking arm forward and inverse kinematics equations, and solve the problem of redundant movement [2]. 198 Article number P_174
2. Picking Arm Homogeneous Coordinate Transformation Matrix Homogeneous coordinate transformation matrix (HCTM) is used to describe the relationship between two rigid body space position of 4 4 matrix in the robot kinematics, this paper apply this concept to describe the spatial geometric relationships compositing of picking arm system between ideal posture and the actual position []. HCTM of picking arm comprehensive express on axis translation around the axis of rotation. In order to apply the same matrix to rotation and translation, HCTM is introduced therefore. adjacent joints depends on two rotation and translation indicated by four parameters [4]. General rotation and transformation of picking point will simplify dynamics equation. That is helpful to improve moving arm state control, to open up new ways to perfect the kinematics and dynamics analysis. Any space picking point A shown as Fig. 2 rotate around unit vector k bypassing the original point and turning angle A. k x, k y, k z standing for vector k are respectively coordinate component fixed reference axis system X, Y, Z meeting the 2 2 2 conditions of k k k 1. X Y Z Fig. 2. Rotation transformations bypass origin point for picking point. Fig. 1. Mechanical noumenon architecture of applepicking-arm. 2.1. Picking Point General Rotation Transformation This can be verified that the arbitrary unit vector k rotated around original point turning the rotor angle q, the rotation operator can be written as following. Formula (1) so-called general homogeneous transformation is given, including various special expression of homogeneous transformation rotating around X-axis, Y-axis and Z-axis. Usually, aim to analysis on picking robot arm kinematics, the coordinate transformation between kxkxvers c kyk Xvers kzs kzkxvers kys kxkyvers kzs kykyvers c kzkyvers kxs Rot( k, ), (1) kxkyvers kys kykzvers kxs kzkzvers c 1 where vers 1 cos, s sin, c cos. On the other hand, if the rotation operator is given, according to the Formula (2), the equivalent rotation vector k and equivalent angle q is calculated as following nx ox ax ny oy ay R nz oz az 1 (2) 1 sin 2 tan nx oy az 1 oz ay k X 2sin ax nz ky 2sin ny ox k Z 2sin o a a n n o 2 2 2 Z Y X Z Y X o a a n n o 2 2 2 Z Y X Z Y X, () 199
where if the rotor angle q stands between o and 18 o, the sign in formula approaches "+"; if q is very small, it is difficult to determine the revolving shaft; if q stands nearly o or 18 o, the revolving shaft is unsure completely. The above-mentioned rotation operator not only applied to the rotation of the picking point transformation, but also extended to vector, generalized rotation coordinate system, objects, etc. point G 2 ; if the arm doesn't move, the picking wrist rotate 9 o in anticlockwise around Z 1 -axis, the picking gripper can reach point G, the expression in matrix on picking arm coordinate system {G 2 } and {G } can be calculated. 2.2. Picking Operator Left and Right Multiplication Rules If the relative fixed coordinate system is transformed, picking operator is left multiplied; if the relative moving coordinate system is transformed, the picking operator is right multiplied. As for a picking points U in the given coordinates, the position vector U=[7,, 2, 1] T shown as Fig., the point rotate 9 o in anticlockwise around Z-axis, then rotate 9 o around Y-axis, the rotation transformation point W can be obtained. Fig. 4. Rotation motion for picking robot forearm and wrist. In view of the arm rotating around a fixed axle can be looked as transformation relative to fixed coordinate system, therefore Fig.. Two rotation transformations for picking point. Therefore W Rot(Y,9 )Rot(Z,9 ) U 1-1 7 1-1 1 1 2 1 1 1-1 7 2-1 = 7 1 2 1 1 1 (4) Fig. 4 shows the picking wrist joint with one rotational freedom degree. The starting posture matrix can be calculated by experimental simulation 1 2 1 6 G1-1 2 1 (5) If the forearm rotate 9 o in anticlockwise around Z -axis, the picking gripper can reach -1 1 G2 Rot(Z,9 ) G1 1 1 1 2-1 -6 1 6 1 2-1 2-1 2 1 1 (6) Moreover, the picking arm rotate around the wrist joints, which is rotation transform relative to the moving coordinate system 1 2 1 6 G G1Rot(Z,9 ) -1 2 1-1 2 1 2 1-1 6 1-1 2 1 1. Homogeneous Transformation of Picking Arm Translation.1. Translation in the Space Rectangular Coordinate System (7) Based on general translation movement of picking translation theory, the picking point can be regarded as rotation and translation revolving around 2
the basis point in the coordinate system, which can be attributed to the same matrix equation, this method is convenient to calculate [5-7], therefore, as for relative complex problem on picking kinematics, especially for the multiple-joints trajectory system research, it is very meaningful to utilize homogeneous transformation method. As shown in Fig. 5, if any picking point A in the space coordinates ( XA, YA, ZA)translates to point A ', the coordinates ' ' ' would be turned to ( X, Y, Z ). Thus A A A ' X X A A X ' Y Y A A Y ' Z Z A A Z (8).2. Translation Transformation Between the Picking Coordinate System and the Target Coordinate System The homogeneous transformation formula (1) of picking point similarly apply to the objects, coordination system etc., above-mentioned left and right multiplication operator also apply to the translation of homogeneous transformation. Fig. 5 stands for three kind of situations between picking arm and coordination system in transformation of translation as following 1) The moving coordinate system {A} translate (-1, 2, 2) to approach {A } relative to the fixed axis X, Y, Z ; 2) The moving coordinate system {A} translates (2, 2, -1) to approach {A } relative to the coordinate system axis X, Y, Z; ) After the picking apples Q translate (2, 6, ) to approach Q relative to a fixed coordinate system X, Y, Z. The matrix belonging to moving coordinates A, Q and two homogeneous coordinate transformation translational operators can be both calculated in Matlab 1 1 1 1 A -1 1 1 Fig. 5. Translation transformation for picking points. Or can be written as the following X A 1 XX A Y A 1 Y Y A Z A 1 ZZ A 1 1 1 Also can be abbreviated ' A Trans( X, Y, Z)A, (9) where Trans( X, Y, Z) stands for picking point translation operator of coordinate homogeneous transformation, X, Y, Z stands translation operator along the axis X, Y, Z respectively, which can be expressed as following 1 X 1 Y Trans( X, Y, Z) 1 Z 1 (1) 1 1 1 1 1 1 1 1 2 2 2 2 Q 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 Trans( X, Y, Z ) 1 2 (11) 1 The matrix expression belonging to coordinate system {A }, {A } and object Q Can be calculated. Because of coordinate system {A } is converted to the moving {A} by translation transformation along a fixed coordinate system, the matrix expression of {A } shown as following according to the operator left multiplying. 1 1 1 2 A' Trans( 1,2,2) A 1 2 1-1 1-1 -1 1-1 -1 1-1 1 1 (12) Similarly, coordinate system {A } is converted to the moving {A} by translation transformation 21
along coordinate system. The matrix expression of {A } shown as following according to the operator right multiplying. -1 1-1 1 A'' ATrans ( 1,2,2) -1 1 1 1 1-1 -1 1 2 2 1 2-1 -1 1 1 (1) Moreover, the picking apples Q translate the homogeneous coordinate relative to fixed coordinates by Matlab, transformation operator can be shown as following 1 2 1 6 ATrans( X, Y, Z) 1 1 1 2 1 6 Q ' Trans(2,6,) Q 1 (14) 1-1 1-1 -1 1-1 7-1 1-1 1 1 1 Coordinate system {A }, {A } and the picking apple objects is shown as Fig. 6 through translation transform. 4. Picking Arm Forward Kinematics Analysis on forward kinematics of picking arm is solved posture by picking arm joint variables, also known as forward problem. According to Section 1.1 and 1.2, as for studying on joints variable of picking arm, first of all, build joint coordinate system, then calculated the homogeneous coordinate transformation matrix A i (i=1, 2, ) used to describe relative translation and rotation of the arm coordinate system. Where A 1 describe posture on upper arm versus shoulder joint coordinate system, A 2 describe posture on fore arm versus the upper arm, A describe posture on picking arm coordinate system versus the forearm posture. Now for the question of -DOF arm QUAD in this paper, picking arm coordinate system, namely arm coordinate System, relative to the upper arm the coordinate system is be represent by the homogeneous transformation matrix 2 T. 2 T A A A (15) 2 1 The homogeneous transformation matrix T of picking gripper versus the robot body coordinate system T AA A, (16) 1 2 where T can be written as T. Recent studies [8-1] have revealed the D-H method is used to describe the posture transformation relationship between adjacent connection rod matrix, which is general primary method on robot manipulator kinematics research, and based on that author regards picking arm QUAD as system verification objects, how to establish gradually the kinematic equation of the arm based on the D-H method, picking arm structure schematic shown as Fig. 6 is describe as following. 1) Establishing the D-H coordinate system To establish picking arms coordinate system according to the d-h, Z -axis rotate along the joint axis 1, Z i -axis along the joint i+1, supposing all X i axis parallel to the base coordinate system X in Fig. 7, Y i -axis is defined by the right hand coordinate system. 2) Determining the D-H parameters and joints variables of picking arm Experiment is given as shown in Table 1 QUAD picking arm D-H parameters and joint variables. Table 1. D-H parameters for QUAD picking-arm. Fig. 6. Translation transformations between picking motion coordinate and picked objects. CR V α a d Cos α Sin α No. 1 1-9 o 1 No. 2 2 9 o d2 1 No. d o d 1 22
*Note that CR stands for Connecting Rod, V stands Variable angle. ) Solving the posture matrix A i between arms According to Table 1, the D-H parameters and the homogeneous transformation matrix formula, A 1, A 2, A can be obtained as following. ox cos 1(cossin 1) oy sin 1( cos2 sin) cos 1) oz sin1cos2 ax cos1sin 2 ay sin1cos az sincos 2 PX dcos1sind2sin1 PY dsin1sin2 d2cos1 PZ dcos2 5. The Experimental Simulation and Analysis Fig. 7. Diagram of picking-arm for QUAD. cos1 sin1 sin1 cos1 A1 1 1 cos2 sin2 sin2 cos2 A2 1 d2 1 1 1 A 1 d 1 4) Solving the picking arm kinematic equation nx ox ax PX ny oy ay P Y T AA 1 2A, (17) nz oz az PZ 1 where T stands for posture of picking gripper in the base coordinate system. nx cos 1(cos2 sin 2) sin ny sin 1(coscos 1) nz sin1cos2 To verify the established kinematics equation, 8 different picking arm states were measured experimentally, the angle and the corresponding arm space d (Table 1) was substituted respectively into forward kinematics equations (17), the corresponding parameter of picking arm posture shown in Table 2 can be obtained, omit the posture parameters n, o, a involving picking point location in the table is reflected as the following. All parameters of the picking points can obtained using the VC visual programming according to equations (17), there into, eight picking point position parameter in Table 2 is shown as following 1 1 1 1 65 T (:,:,1) 1 1 1 88 1.4 1..29 1.8 1.84 1.84 T (:,:, 2).54.54.54 475 1.67.17. 1.7. 92 T (:,:,) 1.. 1. 475 1.45.62.68 9.9 1.27.54 492 T (:,:,8).15 1.8.99 871 1 According to the above established forward kinematics model, position error program in the direction of the x, y, z axis in Matlab is designed, picking arm simulation diagram is shown as Fig. 8, as can be seen from the Table 2, the position error distribute within [.29,.18] mm range, the relative error is small, picking arm kinematics model is proved to be correct. 2
No. Table 2. The parameter of forward kinematics for QUAD picking-arm. Joint Angle/rad 1 2 Terminal Position/mm Px Py Pz 1 65 88 2.4-1.4.28 65 475 1..6-1.1-1 92 47 4-1.2.4 -. 9 297 88 5 2. 1.4.7-1281 444 476 6 1.97-1..9-1274 -92 474 7 1.67.25 1.9 15-228 88 8 1.2-2.7.28-9 492-871 6. Conclusions This paper aimed at resolving -joint-variables and picking arm posture solution problem, used the homogeneous coordinate and the target homogeneous matrix representation, got the represent method of the arm posture, furthermore, used D-H method for forward and inverse kinematics solution and simulation, and conducted picking arm statics calculation on the basis of Jacobean matrix analysis. Simulation experiments showed that the picking robot arm joint displacement curve smoothing, which proved that the established picking arm inverse kinematics accuracy. The kinematics solution for picking arm trajectory planning and for subsequent picking model provided theory basis on stability control. Acknowledgements This work is funded by National Nature Science Foundation of China (No. 11175) and Technology Innovation Foundation of Northwest A&F University (No. QN2151 and No. QN21169). Fig. 8. Trajectory errors of picking-arm for QUAD. The picking gripper is driven from the initial state (, 65, 88) to the target point (9, 492, 871) in the linear motion, each joint angular displacement curve is shown as Fig. 9 within 2 seconds, each joint displacement curve has meet the smooth transition, derivative and continuously, thereby the inverse kinematics of robotic arm is verified the validity providing theoretical evidence on avoiding effectively the joints fluttering. Fig. 9. Diagram of joints angle displacement curves for picking-arm. References [1]. Bradley D. A., Seward D. W., The development, control and operation of an autonomous robotic excavator, Journal of Intelligent & Robotic Systems, 21, 1, 1998, pp. 7-97. [2]. Farahmand F., Pourazad M., Moussavi Z., An intelligent assistive apple picking robotic manipulator, in Proceedings of the IEEE Conf Eng Med Biol Soc., 5,, 25, pp. 528-51. []. Moreno-Valenzuela J., Orozco-Manríquez E., A new approach to motion control of torque-constrained manipulators by using time-scaling of reference trajectories, Journal of Mechanical Science and Technology, 2, 12, 29, pp. 221-225. [4]. Thuilot B., C. Cariou, P. Martinet and M. Berducat, Automatic Guidance of a Farm Tractor Relying on a Single CP-DGPS, Autonomous Robots, 1, 1, 22, pp. 5-71. [5]. Sakai S., Iida M., Osuka K., Umeda M., Design and control of a heavy material handling manipulator for agricultural robots, Autonomous Robots, 25,, 28, pp. 189-198. [6]. Kuffner J., Nishiwaki K., Kagami S., Inaba M., Inoue H., Motion Planning for Humanoid Robots, Dario P., Chatila R., Robotics Research, Springer Berlin/Heidelberg, 5, 2, 25, pp. 65-74. [7]. Hao Y. X., Agrawal S., Formation Planning and Control of UGVs with Trailers, Autonomous Robots, 19,, 25, pp. 257-268. [8]. Lozanop T., Asimple motion planning algorithm for general robot manipulators, IEEE of Robotics and Automation,,, 25, pp. 224-28. [9]. Penne R., Smet E., Klosiewicz P., A Short Note on Point Singularities for Robot Manipulators, Journal of Intelligent & Robotic Systems, 62, 2, 211, pp. 25-212. 24
[1]. Tzafestas Spyros G., Handbook of Industrial Robotics, Shimon Y. Nof (ed.), Journal of Intelligent & Robotic Systems, 28,, 2, pp. 295-. [11]. Pedamallu C., Ozdamar L., Solving kinematics problems by efficient interval partitioning, Optimization and Engineering, 5,, 211, pp. 459-568. [12]. Van Henten E. J., Hemming J., van Tuijl B. A. J., Kornet J. G., Meuleman J., Bontsema J. and van Os E. A., An Autonomous Robot for Harvesting Cucumbers in Greenhouses, Autonomous Robots, 1,, 22, pp. 241-258. [1]. Harper C., Virk G., Towards the Development of International Safety Standards for Human Robot Interaction, International Journal of Social Robotics, 2,, 25, pp. 229-24. [14]. Wang Z. H. G., Zhang L. B., Bao G. J., Qian S. M., Yang Q. H., Design and control of integrated pneumatic dexterous robot finger, Journal of Central South University of Technology, 18, 4, 211, pp. 115-1122. [15]. Yahya H. Zweiri, Identification schemes for unmanned excavator arm parameters, International Journal of Automation and Computing, 5, 2, 28, pp. 185-192. [16]. Mysorewala M., Popa D., Lewis F., Multi-Scale Adaptive Sampling with Mobile Agents for Mapping of Forest Fires, Journal of Intelligent & Robotic Systems, 54, 4, 25, pp. 55-42. 21 Copyright, International Frequency Sensor Association (IFSA). All rights reserved. (http://www.sensorsportal.com) 25