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.
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 Binh1, *, Do Dang Khoa2, Phan Dang Phong3, Do Sanh2 Viet Tri University of Industry, Tien Cat Ward, Viet Tri City, Phu Tho Province Hanoi University of Science and Technology, No.1 Dai Co Viet Str., Ha Noi National Research Institute of Mechanical Engineering, No Pham Van Dong Str., Ha Noi * Email: vubinhchc@gmail.com Received: 15 March 2018; Accepted for publication: 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 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] DYNAMICAL MODEL Let consider a scleronomous holonomic system, whose position is defined by the Lagrange coordinates q j ( j 1, n) and the generalized forces noted as Qj ( j 1, n) , respectively From now Program motion of unloading manipulators 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]: Aq = Q(0) + Q(1) + Q(2) where: A= aij i , j 1, n (1) is a (nxn) regular matrix called the inertia matrix; q is a (nx1) matrix of T generalized acelerations, q q1 q2 qn ;, 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]: g (t , q1, q2 , , qn ) ; (2) 1, r There exist two approaches to solve the stated problem as follows: Method 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 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: Aq = Q(0) + Q(1) + Q(2) + U (3) where U is the (nx1) matrix of the form: U U1 U U n T (4) Its components are defined by means of following equations [6]: GA 1U G(*) (5) where: G g j ;g n G (0) ( i, j g ;G* qj j g qi q j qi q j G (0) n j g qj t GA-1 (Q (0) + Q (1) - Q (2) ) g ) t 1, n ( 1, r ; j 1, n) (6) 1, r and Q(1) ,Q(2) 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: Aq = Q(0) + Q(1) + Q(2) + U +R (7) 663 Vu Duc Binh, Do Dang Khoa, Phan Dang Phong, Do Sanh for the given program being the first integral of the system, it is neccessary to realize the condition DR = where D is (kxn) matrix whose elements are the coefficients to express all of the generalized acceleration qi (i 1, n) in terms of the independent generalized accelerations q (( 1, k n r ) Therefore, we have: D(Aq - Q (0) - Q (1) - Q (2) - U) = g (t , q j , q j ) (8) It is noted that the control forces U j are the forces acting on the system of interest In the particular case, they may be components of the force Q(0) 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 J 2, 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: y x 2(l1 l2 ) (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 The Investigated unloading manipulator 664 Program motion of unloading manipulators where OK 2(l1 l2 ); OL l1 l2 The considered manipulator of interest is of degrees of freedom Let choose q 1, 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 q 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: DAq = D(Q0 + Q(1) + Q(2) 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: U M1 M2 F T In order to calculate the above matrices, we use the method of transmission matrix [7,8] For this aim, let us introduce the symbols: dq1 dq2 ; q5 ; q6 dt dt and develop the matrices: q4 t1 cos q1 sin q1 t11 sin q1 cos q1 r1 c1 ; r2 sin q1 cos q1 ; dq3 ; q7 dt t2 cos q2 sin q2 cos q1 sin q1 ; t21 0 sin q2 cos q2 c2 ;r3 c3 ; r d q1 ; q8 dt sin q2 l1 cos q2 ; t3 cos q2 sin q2 ; t31 0 l1 ; P1 m1g ; P2 d q2 ; q9 dt d q3 dt (11) q3 ; 0 0 0 0 0 m2 g ; P3 m3 g ; P 0 mg (12) The coordinates of the mass centers C1, C2, C3 and of the load are defined by following formulas: r01 = t1r1 ; Potential energy r02 = t1t r2 ; r30 = t1t t3r3 ; r0 = t1t 2t 3r (13) can be written as: r01T P1 + r02T P2 + r03T P3 + r0T P [m1c1 (m2 m3 m)l1 ]g sin q2 [m2c2 m3c3 ml3 (m3 m)q3 ]g sin(q1 q2 ); (14) Therefore, generalized forces Q(0) and the matrices of control forces can be expressed as: 665 Vu Duc Binh, Do Dang Khoa, Phan Dang Phong, Do Sanh q1 Q (0) q2 q3 q q M1 M2 F ;U (15) q where , , 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: x l1 cos q1 (l3 q3 )cos(q1 q2 ) ; y l1 sin q1 (l3 q3 )sin(q1 (16) q2 ) into the expressions (9), we obtain now: f l1 sin q1 (l3 q3 )sin(q1 q2 ) 2(l1 cos q1 (l3 q3 )cos(q1 (17) q2 )) 2(l1 l2 ) and we have: df dt f1 [(cos(q1 q2 ) 2sin(q1 + q2 )(l3 q3 ) (2sinq1 + cosq1 )l1 ]q4 (18) [(l3 q3 )[cos(q1 + q2 )+ 2sin(q1+ q2)]q5 + [sin(q1+ q2) - 2cos(q1+ q2)]q6 Hence, matrix D is given by: D13 ; D13 D23 D d31 / d33 ; D23 f ; d32 q1 d32 / d33 ; d31 f ; d33 q2 f q3 (19) where: d31 [(c os(q1 + q2 ) + 2sin(q1 + q2 ))(l3 + q3 ) + (cosq1 + 2sinq1 )l1 ]; d32 = [cos(q1 + q2 ) + 2sin(q1 + q2 ))(l3 + q3 )]; d33 = [ sin(q1 + q2 ) - cos(q1 + q2 )] (20) The Q(1) , Q(1) 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]: m1r1T t11T t11r1 a11 m2 r2T t2T t11T t11t2 r2 m3 (l12 c32 q32 J1 + J + J 2c3 q3 T T m2 r2T t21 t1 t11t2 r2 a12 m3 (c32 q32 c3 q3 l1 cos q2 (c3 T T T 2 21 1 21 a22 m r t t tt r m(l q q3 ) m(l12 2l1 cos q2 (c3 T T m3 r3T t3T t21 t1 t11t2t3 r3 T T T m3 r T3 t31 t2 t1 t11t2t3 r3 a13 m3 r3T t3T t2T t11T t11t2 t3 r3 T T T T 3 21 1 21 3 m3 r t t t t t t r 2l3 q3 ) J T T T T 3 31 1 21 3 a23 m r t t t t t t r T T T 666 a33 m3 r3T t31 t2 t1 t1t2t31r3 T T T T 31 1 31 mr t t t t t t r c12 q32 J1 q32 2q3l3 J2 J3 J2 J3 m2 (l12 2(l1 cos q2 (l3 T T mr3T t3T t21 t1 t1t21t3 r q3l3 l1 cos q2 (l3 q3 )) J ml1 sin q2 (m m3 ) m2 c22 m3 (c32 c22 ) q3 )) m2 (c22 c2l1 cos q2 ) (m m3 )l2 sin q2 J3 T T T mr3T t31 t2 t1 t1t21t3 r l32 T T mr T t3T t21 t1 t11t2t3 r q3 )) m(l32 T T T mr3T t31 t2 t1 t11t2t3 r3 mr T t2T t11T t11t2 r q32 2c3 q3 ) J3 a13 T T T m3 r T3 t31 t2 t1 t11t2t3 r3 a22 m2 r2 t21t1 t1t21r2 T T T mr3T t31 t2 t1 t11t2t3 r3 Program motion of unloadingT manipulators T T T T T T 3 m(l q T T mr3T t3T t21 t1 t1t21t3 r m3 r t3 t21t1 t1t21t3 r3 2l3 q3 ) J (m m3 )l2 sin q2 m2 c22 m3 (c32 q32 2c3 q3 ) J3 a23 T T T m3 r3T t31 t2 t1 t1t21t3 r3 T T T mr3T t31 t2 t1 t1t21t3 r ml1 sin q2 a33 T T T m3 r3T t31 t2 t1 t1t2t31r3 T T T mr T t31 t2 t1 t1t2t31r (m m3 ) (21) Q(1) , Q(*) - (3x1) matrices can be written as: Q(1) (1) Q1(1) Q2(1) Q3(1) T ; (22) Q2(1) Q 0.5qT D1 Aq; Q(2) D1Aq1* D2 Aq*2 Q3(1) 0.5qT D2 Aq; 0.5qT D3 Aq D3 Aq*3 ; where q,q* are (3x1) matrices and Di A(i 1, n) is a (3x3) matrix: q q4 q5 q6 T a11 q1 a12 q1 a13 q1 D1A ; q1* q42 a12 q1 a22 q1 a23 q1 q5 q4 a13 q1 a23 q1 a33 q1 q6 q4 ; q2* a11 q2 a12 q2 a13 q2 ; D2 A q52 q4 q5 a12 q2 a22 q2 a23 q2 q6 q5 ; q3* a13 q2 a23 ; D3 A q2 a33 q2 q4 q6 a11 q3 a12 q3 a13 q3 q5 q6 q62 a12 q3 a22 q3 a23 q3 a13 q3 a23 ; (23) q3 a33 q3 M1 M2 F (24) The equations of motion for robotic arm are of the form: D31 D32 a11 a12 a13 a12 a22 a23 a13 a23 a33 q7 q8 q9 Q1(0) Q2(0) Q3(0) D31 D32 Q1(1) Q2(1) Q3(1) Q1(*) Q2(*) Q3(*) which can be written as: (a11 D31 )q7 (a12 D31a23 )q8 (a13 D31 )q9 Q1(0) Q1(1) Q1* D31 (Q3(0) Q3(1) Q3* ) M1 D31F (a12 D32 )q7 (a22 D32 a23 )q8 (a 23 D32 )q9 Q2(0) Q2(1) Q2* D32 (Q3(0) Q3(1) Q3* ) M D32 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: f1 d31q4 d32 q5 d33q3 (26) The problem is solved by the system of equations (25) and (27) now f2 df1 dt ( d31q7 d32 q4 q1 d32 q8 d32 q5 q2 d33q9 ( d31 q4 q1 d32 q6 )q5 q3 ( d31 q5 q2 d33 q4 q1 d31 q6 )q4 ) q3 d33 q5 q2 d33 q6 )q6 q3 (27) To solve these equations, it is possible to use software Here the Maple software is used Results of numerical simulation 667 Vu Duc Binh, Do Dang Khoa, Phan Dang Phong, Do Sanh Numerical simulation of robotic arm is performed with the following parameters: l1 = m, l3 = 0.5 m, m1 = kg, m2 = kg, m3 = kg, m = kg, = rad, = rad, = 0.1 rad, c1 = 0.5 m, c2 = 0.25 m, c3 = 0.25 m, J1 = 0.02 kgm2, J2 = 0.01 kgm2, J3 = 0.05 kgm2, g = 10 m/s2, l0 = 0.01 m, M1 = 25 Nm, M2 = 0.1 Nm, F = 0.05 N The initial conditions are: q1(0) = rad, q2(0) = rad, q3(0) = m, q4(0) = 0.03 rad/s, q5(0) = -0.01 rad/s, q6(0) = m/s, Figure Graph of rotation angle q1 and q2 and the displacement q3 of the plunger Figure Graph of angular velocities q4, q5 and velocity q6 668 Program motion of unloading manipulators Figure Graph of orbital motion 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 Galiulin F R - Constructing Systems of Controlled Motion Controlled motion, Publisher “Nauka” (in Russian), 1971 Do Sanh - On the principle of Compatibility and the Equations of Motion of a Constrained Mechanical System, ZAMM, pp 210-212, (1980) Do Sanh - On the Problem of First Integrals of Mechanical Systems, Problems of Nonlinear Vibration, Nr 20, pp 55-70, 1980, Warsaw Erughin N P - Construting a Set of Differential Equations Having Given Trajectory, Applied Mathematics and Mechanics (PMM), No 6, 1952 (in Russian) Do Sanh, Dinh Van Phong, Do Dang Khoa, Tran Duc - A Method for Solving the Motion of Constrained Systems, Proceedings of the 16th Asian Pacific Vibration Conference APVC 2015, Ha Noi, Viet Nam, 2015, pp 532-537, (APVC 2015) Do Sanh - Motion of Constrained Mechanical Systems, Thesis of Science Doctor Hanoi University of Science and Technology, 1984 (in Vietnamese) Do Sanh, Analytical Mechanics, Publ Bachkhoa, Ha Noi, 2008 (in Vietnamese) Do Sanh, Do Dang Khoa, Analytical Dynamics, Publ Bachkhoa, 2017 (in Vietnamese) 669 ... Figure The Investigated unloading manipulator 664 Program motion of unloading manipulators where OK 2(l1 l2 ); OL l1 l2 The considered manipulator of interest is of degrees of freedom Let choose... Graph of rotation angle q1 and q2 and the displacement q3 of the plunger Figure Graph of angular velocities q4, q5 and velocity q6 668 Program motion of unloading manipulators Figure Graph of orbital... acting on the system of interest In the particular case, they may be components of the force Q(0) MOTION INVESTIGATION OF AN UNLOADING MANIPULATOR Consider the motion of three-link unloading manipulator: