On the Dynamic Modelling of a Novel Serial-parallel Robot

Nội dung text: On the Dynamic Modelling of a Novel Serial-parallel Robot

1. Tuyển tập Hội nghị khoa học toàn quốc lần thứ nhất về Động lực học và Điều khiển Đà Nẵng, ngày 19-20/7/2019, tr. 206-214, DOI 10.15625/vap.2019000280 On the Dynamic Modelling of a Novel Serial-parallel Robot Chu Anh MY* Vu Minh HOAN *Department of Special Robotics and Mechatronics, Le Quy Don Technical University 236 Hoang Quoc Viet, Hanoi, Vietnam Department of Machinery Mechanics, Le Quy Don Technical University 236 Hoang Quoc Viet, Hanoi, Vietnam E-mail: mychuanh@yahoo.com Abstract This paper addresses the kinematic and dynamic modelling of a novel serial-parallel robot arm which has been designed and implemented for hot press forging industry in Vietnam. The robot is capable of replacing workers for transferring hot and heavy billets between a heating furnace and a hydraulic press machine of a forging station. To meet all functional requirements of the forging process automation, the robot was designed with a particular architecture of five DOFs, which combines a serial-link robot arm and some local closed chain mechanisms. All the active joints of the robot are actuated by hydraulic cylinders. The control system of the robot was designed mainly based on the industrial PLC units, which also (a) plays a role as a central control unit controlling all the components of the entire forging station. Since some local closed mechanisms are added to the main kinematic chain of the robot, the number of the generalized coordinates of the mechanical system is increased, and the mathematical modelling for the robot is more complex that requires a formulation of constraint equations with respect to the local closed chains. By using the Lagrangian formulation with Lagrangian Multipliers, the dynamic equations are first derived with respect to all extended generalized coordinates. Then a compact form of the dynamic equations is yielded by cancelling the Multipliers. Since the obtained dynamic equations are expressed in terms of independent generalized coordinates which are selected according to active joint variables of the (b) arm, the equations could be best suitable for control law design and implementation. The simulation of the forward and inverse Fig. 1. a) 3D design of the robot, b) a robot picture kinematics and dynamics of the arm demonstrates the motion behavior of the robot system. . To begin a forging press cycle, workers grip a raw Keywords: Manufacturing Robot, Novel robot design, Robot workpiece from a workpiece collecting area, move and control, Serial-parallel robot, Robot kinematics, Robot place it onto the heating furnace. After the temperature of dynamics, Local closed mechanism. the workpiece in the furnace reaches to a required value (11000C, for example) the workers grip again and 1. Introduction transfer the heated workpiece into the die mounted on the press machine. After that the machine presses the Forging press is the process of shaping a hot workpiece to obtain the shaped part as required. The use workpiece that is placed in a die by applying hydraulic of this workpiece transferring method increases the pressure. This type of forging is usually done on a downtime and causes the worker fatigue and danger since forging press machine which applies gradual pressure on the workers handle hot and heavy workpieces directly. To the forging die. In general, a conventional workshop of support workers transferring workpieces, assistive hydraulic hot forging press is mainly comprised of a robotic systems are needed for such the manufacturing heating furnace and a forging press machine. workshops. Thereby, a cost-effective robotic system which can pickup-transfer-release heavy workpieces with
2. Chu A My & Vu M Hoan persistent result and high efficiency was designed and theory was used in [9], and the matrix approach was implemented (see Fig. 1). The use of the robot in employed in [11] for the dynamics and control of the manufacturing is also to improve safety by handling hot parallel robots. A general solution to the problem of workpieces in the unsafe environment instead of the dynamic modelling and analysis of parallel robots was operators, and finally the manual operation is replaced presented in [12]. For the issue of control law design and with a more precise method using the designed robotic development, [14] investigated a model-based technique, system. [15,16,17] addressed the robust control algorithms, In recent years, most of industrial manipulators whereas the sliding mode control was designed in [18]. commonly used in industries are usually actuated by The earlier foundation for control law designed for the electric motors, such as the welding robots, the assembly parallel robots one can find in [19], in which the robots, etc. The use of electric motors actuating active computed torque technique was used as for those of the robot joints possesses several advantages: easy to control, common serial manipulators. The control law design for high positioning accuracy, and high flexibility. However, a 6-DOF hexapod robot was presented in [20], and a if a manipulator is designed to operate in a large technique to improve the control quality for parallel workspace with high loading capability, the use of robots was investigated in [21]. electric motors for the design could lead to a very heavy In recent years, there have been several attempts to architecture of the robot. Counterweights could be added investigate the dynamic modelling and analysis of the to balance to shaking forces. In that case, hydraulic hybrid serial – parallel robots [22-28]. Ibrahim & Khalil cylinders driving robot joints is often used for the design. [22] studied recursive solutions for obtaining the inverse The presence of hydraulic cylinders increases the and direct dynamic models of hybrid robots that are stiffness of robot structure so that the robot is capable of constructed by serially connected non-redundant parallel handling heavy parts in a larger operational space. modules. Pisla et al. [23] addressed geometric and Moreover, the counterweights could be avoided since the kinematic models of a surgical hybrid robot used for cylinders actuating revolute joints play a role of auxiliary camera and active instruments positioning. Tanev [24] links appended to the main structure. Though there will investigated a kinematic analysis of a hybrid be advantages when using hydraulic cylinders for robot parallel–serial robot manipulator which consists of two designs, the presence of cylinders in a robot architecture serially connected parallel mechanisms. Each mechanism involves complex procedures for the mathematical has three degrees of freedom. Xu et al. [25] presented a modelling, analysis and control. The addition of cylinders particular hybrid manipulator for computer controlled to the conventional serial kinematic chain of robot arm ultra-precision freeform polishing. This hybrid architecture could constitute local closed kinematic manipulator is composed of a 3-DOF parallel module, a chains within the entire robot mechanism. Issues 2-DOF serial module and a turntable providing a related to the hybrid serial – parallel feature of the robot redundant DOF. Zeng & Fang [26] studied a method of structure, the geometry, the mass and the inertia of structural synthesis and analysis of serial–parallel hybrid cylinders must be taken into account. mechanisms with spatial multi-loop kinematic chains In the literature, numerous works have been carried based on the displacement group theory. Zeng & Ehmann out to investigate several aspects related to the dynamic [27] presented a design method for parallel hybrid-loop modeling and analysis of serial manipulators and parallel manipulators, which is based on the constrained motion robots [1-21]. The fundamentals of kinematic and properties of related spatial over constrained linkages and dynamic modelling and analysis of serial manipulators general parallel mechanisms. Zhang & Gao [28] can be found in [1,2], where Denavit-Hartenberg investigated a method for performance optimization of a approach was mostly used for the kinematic modelling 5-DOF compliant hybrid parallel robot and D’Alembert-Lagrange Formulation for the dynamic micromanipulator. modelling. As for more complex robotic systems, there Though there has been a number of researches has been a number of researches dealing with different dealing with the dynamic modelling and analysis of issues related to the kinematics and the dynamics as well. different robot architectures, little attention has been paid The research presented in [3] studied the kinematic and to robots actuated by hydraulic actuators. Issues related dynamic modelling for closed chain manipulators, [4] to the mathematical modelling of manipulators which addressed algorithms for the dynamic analysis of serial have local closed kinematic chains constituted by adding robots having a large number of joints, [5] investigated hydraulic cylinders seem to be overlooked. the inverse kinematics and dynamics of the redundant In this paper, the dynamic modelling and analysis of robots, [6] studied the dynamics of mobile serial such the forging robot arm which has four local closed manipulators. In parallel with researches concerning with chains appended to a conventional arm mechanism are the serial robot dynamics, a massive number of addressed. First, the kinematic equations and constraints researches related to the dynamics and control of parallel equations are derived, by using the notation of Denavit – robots has been addressed as well such as publications Hartenberg. Second, Lagrange’s formulation is used to [7-21]. The researches [8,10,13] investigated methods for establish differential equations of motion, with the use of the inverse and forward dynamic modelling and analysis Lagrangian multipliers. Finally, based on the formulated of the 3-PRS type parallel manipulators. The Screw DAEs, a technique of Lagrangian multipliers cancellation
3. On the dynamic modelling of a novel serial-parallel robot is employed to obtain ODEs describing the dynamics of variable 𝜃2), and the second one EG drives the joint F the entire robot system. Finally, the dynamic analysis is (joint variable 𝜃5). Due to the use of the cylinders, four carried out with the help of the symbolic and numeric closed kinematic chains are constituted: ABCD, EFGH, computing Maple and Matlab softwares. As a result, BPQFB and FQSMF. The robot have four DOFs and the proposed investigation could be used further for the four active joints. The first active joint is a revolute joint purposes of dynamic simulation and control law design (joint 1) with joint variable 𝜃1. The second and the third for a class of hydraulic architecture robots. active joints are prismatic joints (with the joint variables d4 and d7 ) because of the relative motion of the two parts of the hydraulic cylinders. The last active joint is a 2. Kinematics of the robot revolute joint (joint 9) with joint variable 𝜃9.Therefore, among eight joint variables qqq,,,,,dd qq,, Let’s consider a robot in Fig 2. Different from the 14792 35 qqq,, , q and q (see Fig. 1), only three joint conventional robot arm architecture, the considering 6 8 10 11 12 manipulator has two hydraulic cylinders. The first variables q1 , d4 , d7 andq9 are independent. cylinder AC is to drive the revolute joint B (joint Fig. 2. The mechanical model of the robot In Fig. 2, O0x0y0z0 is defined as a reference frame, i 𝜃𝑖 𝑑𝑖 𝑎𝑖 𝛼𝑖 and OExEyEzE as an end effector frame, of which the 1 𝜃1 l1 0 p /2 origin locates at point E. Other local coordinate systems 2 𝜃 0 l 0 attached to links are defined by using Denavit – 2 2 5 𝜃5 0 l 0 Hartenberg (DH) notations. ll110111222122,,,,,,, lllll 5 llllllll23,,, 3 5 51 , 52 ,,, 6 8 9 are geometric parameters of 8’ 𝜃8 0 l8 0 links. The main mechanism of the arm includes three 8 p /2 0 0 p /2 links: link 1, link 2, link 5, link 8, and link 9. The first 9 𝜃9 l9 0 0 local closed chain (ABCDA) involves link 1, link 2, link Table 2. The DH parameters related to the local closed 3 and link 4. The second local closed chain (KFGHK) chain ABCDA relates to link 2, link 5, link 6 and link 7. The third local closed chain (BPQFB) involves link 1, link 2, link 10 i 𝜃𝑖 𝑑𝑖 𝑎𝑖 𝛼𝑖 and link 11. The fourth local closed chain (FQSMF) 3’ 𝜃3 0 l3 0 relates to link 5, link 8, link 11 and link 12. Based on all 3 p /2 0 0 p /2 the coordinates in Fig. 1 which are defined with respect 4 0 d4 0 -p /2 to the DH convention, the parameters of the robot links 1’ 0 0 0 -p /2 are presented in the following tables. 1 0 l11 0 p /2 Table 1. The DH parameters related to the main mechanism 2 𝜃2 0 l22 0
4. Chu A My & Vu M Hoan Table 3. The DH parameters related the local closed chain Table 5. The DH parameters related the local closed chain KFGHK FQSMF i 𝜃𝑖 𝑑𝑖 𝑎𝑖 𝛼𝑖 i 𝜃𝑖 𝑑𝑖 𝑎𝑖 𝛼𝑖 2 𝜃2 0 l2 0 5 𝜃5 0 l5 0 11 𝜃11 0 l 0 8 𝜃80 0 l 0 12 l12 1’ 0 0 0 -p /2 11 𝜃11 0 12 p /2 l l 1 0 12 0 p /2 12 𝜃12 0 5 0 l 10 𝜃10 0 12 0 Based on the parameters in Tables 1,2,3,4 and 5, the Table 4. The DH parameters related the local closed chain transformation matrices describing the motion of any BPQFB two successive coordinate systems can be derived i 𝜃𝑖 𝑑𝑖 𝑎𝑖 𝛼𝑖 accordingly. For the main mechanism, the vector of the end working point R is computed as follows: 2 𝜃2 0 l2 0 E 11 0 0 𝜃11 l12 1’ 0 0 0 -p /2 l 1 0 12 0 p /2 l 10 𝜃10 0 12 0 ộcos(cos(qqqllll ++ ) cos q + cos( qqq +++ ) cos( qqq ++ )ự ờ 15 2 5 2 2 8 2 5 8 9 2 5 8ỳ R=sin(cos(ờ qqqllll ++ ) cos q + cos( qqq +++ ) cos( qqq ++ )ỳ (1) E ờ 152 5 2 2 8 2 58 9 2 58ỳ ờ ỳ ờởll15 ++++++ sin(qq 2 5) l 228sin q lsin( qqq 2 5 8) +++l9258sin(qqq ) ỷỳ R ộdlcosqq+ cos ự The matrix E describes the kinematics of motion of ờ 4333ỳ one active joint and three passive joints. In other words, 2 ờ ỳ C is calculated as C =+dl4333sinqq sin . Therefore, R characterizes the kinematic relationship of the end ờ ỳ E ờ 0 ỳ link of the arm with the use of only one independent ở ỷ joint variable q1 and other two dependent joint two independent constraint equations involved in this variables qq25, and q8 . closed chain are determined as In order to determine the kinematic relationship with ùỡ fl=-+cosqq dl cos = 0 ù 1222( 43) 3 all independent joint variables, the constraints raised due ớ (4) ù fl=-+=sinqq dl sin + l 0 to the local closed chains need to be taken into account. ợù 2222() 43 311 ùỡ 222 Constraint equations ù (ld3++ 4) ( l 11) -( l 22 ) ùsin()q = ù 3 ù 2()ldl3411+ Consider the first local closed chain (ABCDA). In ớù (5) order to write constraint equations regarding of this ù 2 22 ù ()ld34+ l 1122 l closed feature of the mechanism, a virtual cut at the joint ùsin()q2 = ù 2ll C is made, so that transformation matrices written with ợù 11 22 respect to two routines ABC and ADC, respectively, are Similarly, the other two constraint equations involved in yielded as the second local closed chain KFGHK are obtained as ộựcosqq- sin 0l cos q 22222 ỡ =+ =qqq ờỳù fl3515217cos ld cos 66 l cos 6 0 ờỳqq q+ ớ (6) sin2222211 cos 0ll sin ù H = ờỳ (2) ợù fl4515= =sinqqq d 7 sin 66 l sin 6 0 ABC ờỳ0010 ờỳỡ 222 ờỳù ()()()ld+- l - l ờỳ0001 ù 6 7 51 21 ởỷùcos()q5 = ù 2 ll ộự sinqq cos 0dl cos qq + cos ù ()()51 21 ờỳ334333 ớ (7) ờỳù 22 2 cosqq334333-+ sin 0dl sin qq sin ù ()()()ld+- l + l H = ờỳ ù 6 7 51 21 ADC ờỳ(3) ùcos()q6 = ờỳ0010 ù 2()(lld+ ) ờỳợù 21 6 7 ởỷờỳ0001 The other four constraint equations involved in the In the first routine, the position C1 of the point C second local closed chain BPQFB and FQSMF are obtained as ộựl cosq ùỡ fl=++-=cosqq l cos q l cos q 0 ờỳ22 2 ù 5122112() 22 10 1 ờỳ ớ is C =+ll22sin q 2 11 , and in the second one, the point ù (8) ờỳ ù f6122112=++ llllsin()qq sin q 22 sin q 1012= ờỳ ợù ởỷ0
5. On the dynamic modelling of a novel serial-parallel robot ỡqq= T ù 10 2  ộ ự ù E = xEEE()ty () tz () t can be numerically computed ớù (9) ở ỷ ù p ùqq11=- 2 ợù 2 and shown in the following Figs. (3,4), and Fig. 5, ùỡ respectively. ù fl7125=++-+-=cos()qq 805 l cos q 5 l 5 cos ( q 11121211 q ) l cos q 0 ớù ù ù fl8125=++-+-=sin()qq 805 l sin q 5 l 5 sin ( q 11121211 q ) l sin q 0 ợù (10) 0.2 vxE 0.15 vyE ùỡqqq=- vzE ớù 12 5 11 (11) 0.1 ùqqq80=- 11 5 ợù 0.05 p 0 Based on Eqs. (9,11), and qq880=-, we have 2 -0.05 -0.1 qqq825=- - (12) Forward kinematics simulation -0.15 -0.2 -0.25 0 5 10 15 times [s] T Fig. 5 E = ộxty tz tự ởờ EEE() () ()ỷỳ 3. Dynamics of the robot Let’s denote T s = [qqqqqqqqqq147923568101112dd ] is the vector of redundant generalized coordinates, T q = [qq1479dd ] is the vector of independent generalized coordinates, T T Fig. 3. E = ộựx ty tz t z = [qqq qqq q q] is the vector of ởỷờỳEEE() () () 235 68101112 dependent generalized coordinates, T f = [ f12345678fffffff] is the vector of constraint equations, ảf Φ = is the Jacobian matrix of constraint equations, s ảs T λ = [lllllll1234567 l 8] is the vector of Lagrangian Multipliers, and T τ = [MFFM147 90 0 0 0 0000] is the vector of applied torque/forces. By using the Lagrangian formulation, the system of equations of motion including constraint equations for the robot can be written as T MssCsssgs(),,,,tttt+++= (  )  () Φλτs () (13) Fig. 4. A trajectory of E in the workspace fs( ,t) = 0 (14) The global mass matrix can be computed as [1] Based on Eqs. (1,5,7,12), with the inputs given as 12 Ms(),tm=+ JJTTT J AIAJ ồ()iTii T R i ii i R i (15) p i=1 q1 ()tt=´ , dt4 ( )=+0.1 0.01 ´ t and 18 where mi and Ii are the mass and the inertia of body i , respectively; J and J are translational and Ti Ri rotational Jacobian matrices of body , accordingly. dt7 ( )=+0.1 0.01 ´ t, the position of the end effector i T The angular velocity of the links are calculated as E = ộựx ty tz t and the velocity ởỷEEE() () () follows:
6. Chu A My & Vu M Hoan ộự0 ộổửl ự ờỳ ờỗld+-4 ữcosqq cos ỳ ờỳ ờốứỗ 342 ữ 1 3ỳ ωωω1811== =ờỳ0 ờ ỳ ờỳ ờổửỳ q l4 ởỷờỳ1 r =+-ờỗld ữsinqq cos ỳ C4ờỗ 3 4ữ 1 3 ỳ ờốứ2 ỳ ờ ỳ ộ qq sin ự ờ ổửl ỳ ờ 2 1 ỳ ờ ỗld+-4 ữsin q + lỳ ờ ỳ ốứỗ 342 ữ 310 ωω==ờ-cosq q ỳ ởờ ỷỳ 210ờ 2 1ỳ ờ q ỳ ở 1 ỷ ộ 1 ự ờ llcosqqq cos()++ cos qq cos ỳ ờ 2 51 25212ỳ ờ ỳ ộ qq sin ự ờ 1 ỳ ờ 31ỳ r =++ờ llsinqqq cos() sin qq cos ỳ ờ ỳ C5ờ 2 5 1 2 5 2 1 2 ỳ ωω==ờ-cosqq ỳ ờ ỳ 34 31 ờ 1 ỳ ờ ỳ qq++ q + ờ q ỳ ờ lll525221sin() sin ỳ ở 1 ỷ ởờ 2 ỷỳ ộự ờỳsin qq12()+ q 5 ộ 1 ự ờỳ ờ ll61cosqqq cos() 262312++ cos qq cos ỳ ờỳ ờ 2 ỳ ωω5 ==-12cosqq 1() 2 + q 5 ờ ỳ ờỳ ờ 1 ỳ ờỳ =++qqq qq ờỳ rC6ờ ll 6sin 1 cos() 2 6 23 sin 1 cos 2 ỳ q1 ờ 2 ỳ ởỷờỳ ờ ỳ ờ 1 ỳ lllsinqq++ sin q + ộự ờ 6262321() ỳ ờỳsin qq12()+ q 6 ởờ 2 ỷỳ ờỳ ờỳ ωω67==-ờỳcosqq 126() + q ộổửl ự ờỳ ờỗld+-7 ữcosqqq cos() + + l cos qq cos ỳ ờỳq ờốứỗ 672 ữ 1 26231 2ỳ ởỷờỳ1 ờ ỳ ờổửl ỳ r =+-ờỗld7 ữsincosqqq ++ l sincos qqỳ C7ờỗ 6 7ữ 1() 2 6 23 1 2 ỳ ộựqq cos ờốứ2 ỳ ờỳ91 ờ ỳ ờỳ ờ ổử ỳ ω = qq sin l7 ữ 991ờỳ ờ ỗld+-ữsin()qq + + l sin q + l ỳ ờỳ ờ ốứỗ 672 ữ 262321ỳ ờỳq ở ỷ ởỷ1 ộ ổử1 ự ờcosqqqqỗ ll+++ cos l cos ữỳ ờ 185ỗỳỗ () 25 2 2ữ The vectors of the mass center of the links are ờ ốứ2 ỳ determined as follows: ờ ổử1 ỳ r =+++ờ sinqqqqỗ ll cos l cos ữỳ C81852522ờ ỗ () ữỳ ộ 0 ự ờ ốứỗ2 ỳ ờ ỳ ờ ỳ r = ờ 0 ỳ lllsinqq++ sin q + C1 ờ ỳ ờ 525221() ỳ ờ ỳ ờ ỳ ởl1 /2ỷ ởờ ỷỳ ộ ổử1 ự ộự1 ờ qqqqỗ ++ + + ữỳ cos1985ỗ lll cos() 25 l 2 cos 2ữ ờỳl212cosqq cos ờ ốứỗỳ2 ữ ờỳ2 ờ ỳ ờỳ ờ ỳ ờỳ1 ờ ổử1 ỳ r =++++sinqqqqỗ lll cos() l cos ữ rC2= ờỳl 2sinqq 1 cos 2 C9ờ 1ỗ 985 2 5 2 2ữỳ ờỳ2 ờ ốứ2 ỳ ờỳ ờ ỳ ờỳ1 ờ lll525221sin()qq++ sin q + ỳ ờỳll221sin q + ờ ỳ ởỷờỳ2 ởờ ỷỳ ộự1 ộ 1 ự ờỳl cosqq cos ờ l212cosqq cos ỳ ờỳ2 313 ờ 2 ỳ ờỳ ờ ỳ ờỳ1 ờ 1 ỳ r = l sinqq cos rC10= ờ l 2sinqq 1 cos 2 ỳ C3ờỳ 3 1 3 ờ 2 ỳ ờỳ2 ờ ỳ ờỳ ờ 1 ỳ ờỳ1 lllsin q ++ ờỳll3310sin q + ờ 22121ỳ ởỷờỳ2 ởờ 2 ỷỳ
7. On the dynamic modelling of a novel serial-parallel robot ộự ờỳ ộ ảảffự ờỳl cosqq cos ΦΦΦ==ộựờ ỳ (18) ờỳ212 sqzởỷờỳờảảq z ỳ r = ờỳl sinqq cos ở ỷ C11ờỳ 2 1 2 ờỳ1 Let’s consider the following expression ờỳlll22sin q ++ 121 ởỷờỳ2 TTộ -1 T ự REΦΦ=-ờ , qz()ỳ (19) ở ỷ ộựổử1 Hence ờỳcosqqqqỗ ll cos()++ cos ữ ờỗ15ốứ2 25 2 2ữ ỳ ờỳRΦTT= 0 (20) ờỳổử1 s r =++ờỳsinqqqqỗ ll cos cos ữ C12ờỳ 1ỗ 5() 2 5 2 2 ữ ờỳốứỗ2 On the other hand ờỳ ờỳ1 s=Rq ờỳllll5sin()qq 2++ 5 2 sin q 2121 ++ (21) ởỷờỳ2 s=Rq+Rq   (22) The matrix of Coriolis and Centrifugal terms is calculated as [1] Substituting (22, 23, 24) into (15) yields ảảMs(),,tt1 ổ Ms() ỗ Mq++= Cq  Gq τ , (23) Css(),,t =Ä-Ä() Enn sỗ () s E q ảảss2 ốỗ (16) where T Based on the calculation of the total potential energy of MRMsR= (),t , the entire system, the terms related to the gravity effect is calculated as CRMsRCssR=+T ()(),,,tt ( ) , GR= T g()s,t , ộựảP T gs,t = ờỳ T () (17) and . ởỷờỳảs τRτq = ()t In order to analyze the dynamic behavior of the robot Eq (23) is expressed in term of independent generalized system, the DAEs (13,14) need to be transformed in a T way that the Multipliers are cancelled. coordinates, q = [qq1479dd ] . Note that all the ảf formulations above are implemented and demonstrated Rewrite Φ = in the following form: s ảs in Maple environment. The following section shows the numeric solutions of the forward and inverse dynamic issues. Forward dynamics simulation The dynamical parameters of the robot system are given in Tab. 6. Table 6. The parameters of the robot Center of gravity Inertia I Link Mass (kg) xC yC zC Ixx Iyy Izz Ixy Iyz Izx 1 0 lC1 0 m1 = 81,3 I1x I1y I1z 0 0 0 2 lC2 0 0 m2 =50,6 I2x I2y I2z 0 0 0 3 0 0 lC3 m3 = 25,6 I3x I3y I3z 0 0 0 4 0 lC4 0 m4 =5,70 I4x I4y I4z 0 0 0 5 lC5 0 0 m5 =58,4 I5x I5y I5z 0 0 0
8. Chu A My & Vu M Hoan 6 0 0 lC6 m6 = 25,6 I6x I6y I6z 0 0 0 7 0 lC7 0 m7 =5,70 I7x I7y I7z 0 0 0 8 0 0 lC8 m8 =37,7 I8x I8y I8z 0 0 0 9 0 0 lC9 m9 = 73,1 I9x I9y I9z 0 0 0 10 l 0 0 m = 7,20 I I I 0 0 0 C10 10 10x 10y 10z 11 l 0 0 m =12,9 I I I 0 0 0 C11 11 11x 11y 11z 12 l 0 0 m = 6,50 I I I 0 0 0 C12 12 12x 12y 12z l = 0.5m; l = 0.2 m; l = 0.2 m;l = 0.6 m; l = 0.2 m; l = 0.4 m; l = 0.4 m; l = 0.2 m; l = 1 10 11 2 20 21 22 23 3 0.4 m ; l = 0.4 m; l = 0.4 m; l = 0.3 m; l = 0.1 m; l = 0.4 m; l = 0.4 m. l = 0.2 m; l = 0.2 m; 4 5 51 52 6 7 8 9 We assume that ll= /2 = 0.25 m; ll= /2 = 0.3 m; ll= /2 = 0.2 m; ll= /2 = 0.2 m; C11 C22 C33 C44 ll= /2 = 0.5 m; ll= /2 = 0.2m; l = l /2 = 0.2 m; ll= /2 = 0.1 m; ll= /2 = 0.1 m; C55 C66 C7 7 C88 C99 22pp ll= /2 = 0.3 m; ll= /2=0.1m; ll= /2 = 0.1 m; -ÊÊq ; 0.1Ê<dl; 0.1Ê<dl. C10 2 C11 11 C12 5 331 44 77 We assume that each link of the robot is considered as a slender rod rotating about one end. The moments of inertia of the links are calculated as 2 2 2 2 2 2 Iml111x = /3; I1y = 0 ; Iml111z = /3; I2x =0 ; Iml222y = /3; Iml222z = /3; Iml333x = /3; Iml333y = /3; 2 2 2 2 I3z = 0 ; Iml444x = /3; I4 y = 0 ; Iml444z = /3; I5x =0; Iml555y = /3; Iml555z = /3; 2 2 2 2 2 2 Iml666x = /3; Iml666y = /3; I6z = 0 ; Iml777x = /3; I7 y = 0 ; Iml777z = /3; Iml888x = /3; Iml888y = /3; 2 2 2 2 I8z = 0 ; Iml999x = /3 ; Iml999y = /3 ; I9z = 0 ; I10 x = 0 ; Iml10y = 10 2 /3 ; Iml10z = 10 2 /3 I11x =0 ; 2 2 2 2 Iml11y = 11 12 /3; Iml11z = 11 12 /3; I12 x = 0 ; Iml12y = 12 5 /3; Iml12z = 12 5 /3 The applied torque/forces are given as M1 (tt)=´10 sin( 2 ) Ft4 ( )=´+1000( 5 2 t) Ft=´+500 5 2 t 7 ( ) ( ) M tt=´0.5 sin 2 9 ( ) ( ) Teta1 [rad]; d4 [m]; d7 [m];Teta9 [rad] [rad]; d4 [m]; d7 [m];Teta9 Teta1 Fig. 6 shows the time evolution of q1 ()t , d4 (t) , d7 (t) and q9 ()t . The shape of the curves reflect well the evolution of the input Fig. 6. The time evolution of q , d , d and q forces/torques. 1 4 7 9
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