Program motion of unloading manipulators

Vietnam Journal of Science and Technology 56 (5) (2018) 662-670 DOI: 10.15625/2525-2518/56/5/11811 PROGRAM MOTION OF UNLOADING MANIPULATORS Vu Duc Binh 1, * , Do Dang Khoa 2 , Phan Dang Phong 3 , Do Sanh 2 1 Viet Tri University of Industry, Tien Cat Ward, Viet Tri City, Phu Tho Province 2 Hanoi University of Science and Technology, No.1 Dai Co Viet Str., Ha Noi 3 National Research Institute of Mechanical Engineering, No. 4 Pham Van Dong Str., Ha Noi * Email: vubinhchc@gmail.com Received: 15 March 2018; Accepted for publication: 9 June 2018 Abstract. In the paper, the program motion of an unloading manipulator which is treated as a first integral of the considered system, is investigated. Currently, the popular way to solve such problem is the method of Lagrange multipliers only. In the paper, the authors use another approach, the Principle of Compatibility, in which the required program is treated as one of motion equations of the system. In the particular case, the program is considered as one of first integrals of the system. For illustrating the proposed method, the motion of an unloading manipulator of three degrees of freedom is considered. Keywords: first integral, transmission matrix method, unloading manipulator, the principle of compatibility. Classification numbers: 5.3.5; 5.3.6 1. INTRODUCTION Consider a manipulator whose grippers must move a load along a prescribed trajectory. Such problem has been studied in many works in [1-4] and still come into much attention by many researchers. Up to now, the problem is solved by the Lagrange multipliers method only. However, the method owns some inconveniences due to adding more variables, Lagrange multipliers, to equations of motion. It is important that by using this method the opportunity of controlling the manipulator will be taken away. In this paper the proposed method overcome such inconveniences by not using the Lagrange multipliers and therefore the motion of the considered system is described in terms of generalized coordinates only. For solving the stated problem it is used the method proposed in [5], in that work, the method of transmission matrix is applied to derive the system equations of motion and the equation of trajectory of the required program is treated as the first integral [6]. 2. DYNAMICAL MODEL Let consider a scleronomous holonomic system, whose position is defined by the Lagrange coordinates ( 1, )jq j n and the generalized forces noted as Qj ( 1, )j n , respectively. From now Program motion of unloading manipulators 663 on the symbols are used: matrices in bold letters, vectors considered as column matrices, the letter T at upper right corner denotes the matrix transposition. Assume that (nxn) inertia matrix is denoted by A, (nx1) matrix of generalized forces is denoted by Q (0) , the equations of mechanical systems are written in matrix form [5-8]: (2)(0) (1)Aq = Q + Q + Q (1) where: A= , 1,ij i j n a is a (nxn) regular matrix called the inertia matrix;q is a (nx1) matrix of generalized acelerations, 1 2 . . , T nq q qq ; Q (0) -a (nx1) matrix of generalized forces derived from the system’s potential, the active and dissipation forces; Q(1), Q(2)- (nx1) matrices, which are determined by the inertia matrix as described in [5-8]. As known, the motion of the system subjected to the program motion can be written as follows [5-8]: 1, 2( , ,.., ) 0 ; 1,ng t q q q r (2) There exist two approaches to solve the stated problem as follows: Method 1. The required program is treated as ideal constraints and the method of Lagrange multipliers is used. As known, the method is not simple because it requires to calculate the Lagrange multipliers. Method 2. The stated problem is solved by means of the Principle of Compatibility. According to this, the program is considered as part of the motion equations of the system. Thus, it is neccessary to add some forces as the control inputs on the considered system. According to this method the motion equations are written as follows: (0) (1) (2)Aq = Q + Q + Q +U (3) where U is the (nx1) matrix of the form: 1 2 . . T nU U UU (4) Its components are defined by means of following equations [6]: 1 (*) 0GGA U (5) where: * (0) 1, 2 2 2 (0) 2 , 1 1 1, ; ; ( 1, ; 1, ) G ( ) j j n j n n i j i j ji j j r g g g r j n q g g g q q q q q t t -1 (0) (1) (2)G G G GA (Q + Q - Q ) (6) and (1) (2)Q ,Q are (nx1) matrices of inertia forces defined by the inertia matrix A [5-8]. In this paper the following method is proposed By using the method in [6], the required program is treated as a first integral of the system. Originally, the program motion is not described by the system’s equations of motion (1). Therefore, it is necessary to act some control forces on the system. By doing that, the motion equations of the system are written as follows: (0) (1) (2)Aq = Q + Q + Q +U +R (7) Vu Duc Binh, Do Dang Khoa, Phan Dang Phong, Do Sanh 664 for the given program being the first integral of the system, it is neccessary to realize the condition. DR = 0 where D is (kxn) matrix whose elements are the coefficients to express all of the generalized acceleration ( 1, )iq i n in terms of the independent generalized accelerations q . ( ( 1, )k n r .Therefore, we have: (0) (1) (2) ( , , ) 0j jg t q q D(Aq - Q - Q - Q - U) = 0 (8) It is noted that the control forces jU are the forces acting on the system of interest. In the particular case, they may be components of the force (0)Q . 3. MOTION INVESTIGATION OF AN UNLOADING MANIPULATOR Consider the motion of three-link unloading manipulator: link OA has length l1, mass m1 with the mass center at the rotatory joint O. Link AD, a cylinder rotating about the axis A, has mass m2, and the mass center at C2 (AC2=c2). Piston BC has mass m3, and the mass center C3 (BC3=c3). The inertia moment of the link OA about the rotatory axis O denoted by J1. The inertia moments of the links AD and BC about the its mass center denoted by J2, J3, respectively. The links OA and AD are exerted by the couples M1 and M2, respectively. Cylinder B is subjected to the expulsive force F. The friction forces in the articulated joints and the slip joint are neglected. The load is treated a point mass. The required program is of the form: 1 22 2( ) 0y x l l (9) where y and x are the coordinates of the load. It means that the load must be moved along the trajectory described by the equation (9), i.e. the inclined line KL (Fig.1). Figure 1. The Investigated unloading manipulator. Program motion of unloading manipulators 665 where 1 2 1 22( );OK l l OL l l The considered manipulator of interest is of 3 degrees of freedom. Let choose q1, q2, q3 as the generalized coordinates, where q1 is the position angle of the link OA with respect to the fixed axis Ox, q3 is the position angle of the link AD with respect to the link OA, and q3 is the displacement of the piton BC with respect to the cylinder AD (see Fig. 1). The motion equations of the manipulator are rewritten from (8) as: ( )0 (1) (2)DAq = D Q +Q +Q U (10) where A-the (3x1) inertia matrix, Q (0)–the (3x1) matrix of the potential forces and dissipative forces, Q (1) , Q (2)–the (3x1) inertia forces, U-the (3x1) control forces, which are of the form: 1 2 T M M FU In order to calculate the above matrices, we use the method of transmission matrix [7,8]. For this aim, let us introduce the symbols: 2 2 2 1 2 3 1 2 3 4 5 6 7 8 92 2 2 ; ; ; ; ; dq dq dq d q d q d q q q q q q q dt dt dt dt dt dt (11) and develop the matrices: 1 1 2 2 1 3 1 1 1 2 2 2 3 1 1 2 2 11 1 1 21 2 2 31 cos sin 0 cos sin 1 0 sin cos 0 ; sin cos 0 ; 0 1 0 ; 0 0 1 0 0 1 0 0 1 sin cos 0 sin cos 0 0 0 1 cos sin 0 ; cos sin 0 ; 0 0 0 0 0 0 0 0 q q q q l q t q q t q q t q q q q t q q t q q t 1 2 3 1 1 2 3 1 2 2 3 3 0 (12) 0 0 0 0 0 0 0 0 ; 0 ;r 0 ; 0 ; 1 ; ; ; (12) 1 1 1 1 0 0 0 c c c l r r r P m g P m g P m g P mg The coordinates of the mass centers C1, C2, C3 and of the load are defined by following formulas: 1 1 02 1 2 2 30 1 2 3 3 0 1 2 301r = t r ; r = t t r ; r = t t t r ; r = t t t r (13) Potential energy can be written as: 01 1 02 2 03 3 0 1 1 2 3 2 2 2 3 3 3 3 3 1 2[ ( ) ] sin [ ( ) ] sin( ); T T T T 1m c m m m l g q m c m c ml m m q g q q r P + r P + r P + r P (14) Therefore, generalized forces (0)Q and the matrices of control forces can be expressed as: Vu Duc Binh, Do Dang Khoa, Phan Dang Phong, Do Sanh 666 1 4 1 1 (0) 2 5 2 2 3 6 3 ; q q M q M q F q q Q U (15) where 1 2 3, , are viscous resistance coefficients of moments and forces acting on links OA, AD, and BC, respectively. To calculate matrix D, we substitute the expression of y, and x in terms of the generalized coordinates: 1 1 3 3 1 2 1 1 3 3 1 2cos ( )cos( ) ; sin ( )sin( )x l q l q q q y l q l q q q (16) into the expressions (9), we obtain now: 1 1 3 3 1 2 1 1 3 3 1 2 1 2sin ( )sin( ) 2( cos ( )cos( )) 2( ) 0f l q l q q q l q l q q q l l (17) and we have: 1 1 2 3 3 3 3 [( ( ) 2 )( ) ( ] [( )[ ] [ ( ] 1 2 1 1 1 4 1 2 6 df f cos q q sin(q + q l q 2sinq +cosq )l q dt l q cos(q + q )+ 2sin(q1+ q2) q5+ sin q1+ q2) - 2cos(q1+ q2) q (18) Hence, matrix D is given by: 13 13 31 33 23 32 33 31 32 33 23 1 2 3 1 0 ; / ; / ; ; ; 0 1 D f f f D d d D d d d d d D q q q D (19) where: 31 2[(c ( ) ( ))( ) ( ) ] [ ) ( ))( )] [ ( ) ( )] 1 2 1 3 3 1 1 1 32 1 2 1 2 3 3 33 1 2 1 2 d os q + q + 2sin q + q l + q + cosq + 2sinq l ; d = cos(q + q + 2sin q + q l + q ; d = sin q + q - cos q + q (20) The (1) (1),Q Q generalized forces of inertia forces are calculated by means of the inertia matrix A. By means of the method of the transmission matrix, the elements of inertia matrix are follows [6-8]: 2 2 11 1 1 11 11 1 2 2 2 11 11 2 2 3 3 3 2 11 11 2 3 3 2 11 11 2 1 2 3 2 1 2 2 2 2 2 2 2 2 3 1 3 3 3 3 1 2 3 3 1 3 1 3 3 3 1 2 3 3 ( ) ( 2 2 cos ( ) ( 2 2( cos ( )) T T T T T T T T T T T T 1 2 3 a m r t t r m r t t t t r m r t t t t t t r mr t t t t r J J J m l c m l c q c q l q c q m l l c q q l l q l q J + J + J 2 12 2 2 21 1 11 2 2 3 3 3 21 1 11 2 3 3 3 21 1 11 2 3 2 3 2 2 2 1 2 2 2 2 2 3 3 3 3 3 1 2 3 3 3 3 3 3 1 2 3 3 2 3 ( cos ) ( cos ( )) ( cos ( )) T T T T T T T T T T Ta m r t t t t r m r t t t t t t r mr t t t t t t r J J m c c l q m c q c q l q c q m l q q l l q l q J J 13 3 3 31 2 1 11 2 3 3 3 31 2 1 11 2 3 3 3 2 2 2 2 2 22 2 2 21 1 1 21 2 3 3 3 21 1 1 21 3 3 3 3 21 1 1 21 3 2 2 3 3 3 3 3 2 2 3 3 3 3 2 3 23 3 3 31 2 ( ) sin ( 2 ) ( 2 ) T T T T T T T T T T T T T T T T T T T T T T a m r t t t t t t r mr t t t t t t r m m l q a m r t t t t r m r t t t t t t r mr t t t t t t r m c m c q c q m l q l q J J a m r t t 1 1 21 3 3 3 31 2 1 1 21 3 1 2 33 3 3 31 2 1 1 2 31 3 31 2 1 1 2 31 3 sin ( ) T T T T T T T T T T T T T t t t t r mr t t t t t t r ml q a m r t t t t t t r mr t t t t t t r m m Program motion of unloading manipulators 667 13 3 3 31 2 1 11 2 3 3 3 31 2 1 11 2 3 3 3 2 2 2 2 2 22 2 2 21 1 1 21 2 3 3 3 21 1 1 21 3 3 3 3 21 1 1 21 3 2 2 3 3 3 3 3 2 2 3 3 3 3 2 3 23 3 3 31 2 ( ) sin ( 2 ) ( 2 ) T T T T T T T T T T T T T T T T T T T T T T a m r t t t t t t r mr t t t t t t r m m l q a m r t t t t r m r t t t t t t r mr t t t t t t r m c m c q c q m l q l q J J a m r t t 1 1 21 3 3 3 31 2 1 1 21 3 1 2 33 3 3 31 2 1 1 2 31 3 31 2 1 1 2 31 3 sin ( ) T T T T T T T T T T T T T t t t t r mr t t t t t t r ml q a m r t t t t t t r mr t t t t t t r m m (1) (*),Q Q - (3x1) matrices can be written as: (1) (1) (1)(1) 1 2 3 (1) (1) (1) 1 2 3 ; 0.5 ; 0.5 ; 0.5 T T T T Q Q Q Q Q Q1 2 3 Q q D Aq q D Aq q D Aq (22) where *q,q are (3x1) matrices and ( 1, )i niD A is a (3x3) matrix: * 2 * 2 * 2 4 5 6 1 4 5 4 6 4 2 4 5 5 6 5 3 4 6 5 6 6; ; ; T q q q q q q q q q q q q q q q q q qq q q q 11 12 13 11 12 13 11 12 13 1 1 1 2 2 2 3 3 3 12 22 23 12 22 23 1 2 1 1 1 2 2 2 13 23 33 13 23 33 1 1 1 2 2 2 ; ; a a a a a a a a a q q q q q q q q q a a a a a a q q q q q q a a a a a a q q q q q q 3D A 0 D A D A 12 22 23 3 3 3 13 23 33 3 3 3 ; a a a q q q a a a q q q (23) The equations of motion for robotic arm are of the form: (0) (1) (*) 11 12 13 7 11 1 1 31 31 (0) (1) (*) 12 22 23 8 22 2 2 32 32 (0) (1) (*) 13 23 33 9 3 3 3 1 0 1 0 0 1 0 1 a a a q Q Q Q M D D a a a q Q Q Q M D D a a a q Q Q Q F (24) which can be written as: (0) (1) (0) (1)* * 11 31 7 12 31 23 8 13 31 9 1 31 3 1 311 1 3 3 (0) (1) (0) (1)* * 12 32 7 22 32 23 8 23 32 9 2 32 3 2 322 2 3 3 ( ) ( ) ( ) (Q ) ( ) ( ) ( ) (Q ) a D q a D a q a D q Q Q Q D Q Q M D F a D q a D a q a D q Q Q Q D Q Q M D F (25) The system of equations (25) and (18) to solve the problem is a system of differential equations. In the work [5], a solution to the problem is proposed by solving the secondary differential equations when the equations of motion are expressed as: 1 31 4 32 5 33 3 0f d q d q d q (26) The problem is solved by the system of equations (25) and (27) now 12 31 7 32 8 33 9 df f d q d q d q dt 31 31 31 4 5 6 4 1 2 3 ( ) ) d d d q q q q q q q 32 32 32 33 33 334 5 6 5 4 5 6 6 1 2 3 1 2 3 ( ) ( ) d d d d d d q q q q q q q q q q q q q q (27) To solve these equations, it is possible to use software. Here the Maple software is used. Results of numerical simulation (2) * * * 1 1 2 2 3;3Q D Aq D Aq D Aq (21) Vu Duc Binh, Do Dang Khoa, Phan Dang Phong, Do Sanh 668 Numerical simulation of robotic arm is performed with the following parameters: l1 = 1 m, l3 = 0.5 m, m1 = 1 kg, m2 = 1 kg, m3 = 1 kg, m = 5 kg, 1 = 0 rad, 2 = 0 rad, 3 = 0.1 rad, c1 = 0.5 m, c2 = 0.25 m, c3 = 0.25 m, J1 = 0.02 kgm 2 , J2 = 0.01 kgm 2 , J3 = 0.05 kgm 2 , g = 10 m/s 2 , l0 = 0.01 m, M1 = 25 Nm, M2 = 0.1 Nm, F = 0.05 N. The initial conditions are: q1(0) = 0 rad, q2(0) = 0 rad, q3(0) = 0 m, q4(0) = 0.03 rad/s, q5(0) = -0.01 rad/s, q6(0) = 0 m/s, Figure 2. Graph of rotation angle q1 and q2 and the displacement q3 of the plunger. Figure 3. Graph of angular velocities q4, q5 and velocity q6. Program motion of unloading manipulators 669 Figure 4. Graph of orbital motion. 4. CONCLUSIONS In the paper a new method for controlling the manipulator to move a load along a required trajectory is considered. Such problem belongs to type of controlling the program motion. This is one of the most important problems in controlling manipulators. This paper proposes a method for the problem of interest based on of a new point of view that the program motion can be treated as a first integral of the considered system. Acknowledgements. The research was supported by Hanoi University of Science and Technology and National Research Institute of Mechanical Engineering. REFERENCES 1. Galiulin F. R. - Constructing Systems of Controlled Motion Controlled motion, Publisher “Nauka” (in Russian), 1971. 2. Do Sanh - On the principle of Compatibility and the Equations of Motion of a Constrained Mechanical System, ZAMM, pp. 210-212, (1980). 3. Do Sanh - On the Problem of First Integrals of Mechanical Systems, Problems of Nonlinear Vibration, Nr 20, pp. 55-70, 1980, Warsaw. 4. Erughin N. P. - Construting a Set of Differential Equations Having Given Trajectory, Applied Mathematics and Mechanics (PMM), No 6, 1952 (in Russian). 5. Do Sanh, Dinh Van Phong, Do Dang Khoa, Tran Duc - A Method for Solving the Motion of Constrained Systems, Proceedings of the 16 th Asian Pacific Vibration Conference APVC 2015, Ha Noi, Viet Nam, 2015, pp. 532-537, (APVC 2015). 6. Do Sanh - Motion of Constrained Mechanical Systems, Thesis of Science Doctor Hanoi University of Science and Technology, 1984 (in Vietnamese). 7. Do Sanh, Analytical Mechanics, Publ. Bachkhoa, Ha Noi, 2008 (in Vietnamese). 8. Do Sanh, Do Dang Khoa, Analytical Dynamics, Publ. Bachkhoa, 2017 (in Vietnamese).