International Journal of Advanced Mechatronics and Robotics (IJAMR) Vol. 3, No. 1, January-June 011; pp. 9-0; International Science Press, ISSN: 0975-6108 A Simulation Study on Joint Velocities and End Effector Deflection of a Flexible Two Degree Freedom Composite Robotic Arm A. Purushotham 1 & J. Anjaneyulu 1 Professor, Mechanical Engineering, Sreenidhi Institute of Science & Technology, (An Autonomous Institution) Hyderabad-501 301, India E-mail: apurushotham@rediffmail.com Assistant Professor, Mechanical Engineering, Vasavi College of Engineering, Hyderabad- 501 031, India E-mail: anjaneyulu_jalleda@yahoo.co.in ABSTRACT The dynamic response of joint velocities and end effector deflection of a two degree-of-freedom planar composite robot arm is presented in this paper. For the purpose of analysis, the material of links of the arm of the robot are assumed to be fabricated from either aluminum or laminated composite materials. The numerical solution to obtain the joint velocities and end effector deflections of the robot arm is sought in MATLAB Code. The simulation studies examine the interaction between the flexible and the rigid body motions of the robot arm. After numerical investigation, it is found that there is a advantages of using composites in the structural design of robotic manipulators as composite arms improves in the accuracy end effector path. Keywords: Flexibility, End Effector, Advanced Composites, Dynamic Modeling. NOTATION [m] : Inertia matrix of manipulators of size x l 1 l I 1 I g d θ : Rotating Link length : Sliding link length : Mass moment inertia of rotating length about at C.G : Mass moment inertia of sliding link about its C.G : Acceleration due to gravity : Linear displacement of sliding link : Angular displacement of rotating link q : Generalized coordinates variables θ d θd ω = v = θ d t t T
10 International Journal of Advanced Mechatronics and Robotics D h G (a,b) X : Inertia matrix of manipulator : Centrifugal and coriolis component matrix : Gravity matrix : Linearization point : State space variable vector T : Joint torque at the joint 1 F : Displacement force at Joint (x g,y g ) : Coordinates of CGs in inertial frame A ρ E M K : Amplitude vector : Density of link : Young s modulus : Mass matrix : Stiffness matrix 1. INTRODUCTION Many robotic applications require fast and precise robot arms. Generally fast maneuverability is achieved by lightweight robotic manipulators. However, lighter structures fabricated from conventional metals, tend to be weaker and would undergo large deformations; thus deteriorating the accuracy of the path of end effector of many industrial robots. The accuracy of end effector path can be significantly improved by modifying their existing structural design with the focus being on maximizing the stiffness-to-weight ratio of the arms. The higher stiffness to weight ratio is achieved by either optimizing the geometric dimensions of the structure [1, ], or by implementing lighter and stiffer materials such as composites [, 3]. A quick and relatively inexpensive way of evaluating such modifications is simulation studies. Therefore, An dynamic model is derived which include all the coupling terms between the rigid and the flexible motions of the manipulator and couples the rigid body motion of the structure with its linear elastic deformations. In literature, three methods, the assumed modes method [4-6], the finite element method [7-15] and the lumped mass method [16 and 17], are commonly available to get the approximate solution of the dynamic model of the flexible robotic manipulators. Most studies related to dynamic modeling of flexible robot arms consider open-kinematic chains with revolute joints. Very few papers address the modeling problem of robotic manipulators with prismatic joints [5, 6, 14, 15]. Chalhoub and Ulsoy [5] used the assumed modes method to discretize the elastic deformation of a flexible link connected to a prismatic joint. The effect of the variation of the length of the flexible beam on the admissible functions was considered in a quasi-static fashion. Wang and Wei [6] derived expressions for admissible functions that are dependent on spatial coordinate and time. Song and Haug
A Simulation Study on Joint Velocities and End Effector Deflection of a Flexible 11 [14] and Pan et al., [15] developed a general formulation to model flexible mechanisms with revolute and prismatic joints. The derivation is based on the finite element method. The compatibility conditions associated with revolute and prismatic joints are imposed by kinematic constraints. The equations governing the rigid and flexible motions of open- kinematic chains are coupled and highly nonlinear. The nonlinearity is large and cannot be eliminated as these terms influence the dynamic response large extent when bodies treated highly flexible and high velocities. Therefore, the equations of motions have to be solved with these terms. In this paper, the solution of the dynamic model for a planar, two-link, flexible robotic manipulator is done numerically using Gear s Method [14] with the help MATLAB software. The dynamic equations developed for this flexible manipulator is similar to the dynamic models developed by Azhdari A et.al [15]. The paper has been organized in three sections. The description of manipulator structure is given in the section- and Numerical results are presented in the section-3. In the last section, the work done in this study is summarized and the main conclusions are highlighted.. DESCRIPTION OF DYNAMIC MODELDEVELOPED.1 Description of Manipulator The manipulator shown in Figure. 1 consist of two links. Link 1 rotates about vertical member while link slides in link one. The links which are modeled solid circular cylinders beams Figure 1: Geometry of the Manipulator
w h e r e, m(q) 1 International Journal of Advanced Mechatronics and Robotics considered as rigid bodies studies. These circular cylinders beams made hallow and considered as flexible links. The flexible links are modeled as beams with annular cross-sections. The dynamic analysis is carried out with solid and hallowlinks assumed to be made up of Aluminum and hallow links assumed to be made up of Composites. The material properties and geometric dimensions of the system are given in Table I. The purpose of simulation studies is to examine the effect of the elastic deformation when arms are manufactured with light structure links of hallow cross section made up of composites and to demonstrate the advantages of incorporating composites in the structural design of robotic manipulators. Table I Geometric and Material Properties of Links of Robotic Arm Geometric Property Material Property Outer diameter of link1 (cm) Mass density of Aluminum (Kg/m 3 ).643*10 3 Inner diameter of link1 (cm) 1 E of Aluminum (N/m ) 0.7*10 11 Length of Rotating Link1 (cm) 60-10 G of Aluminum (N/m ) 0.6*10 11 Outer diameter of link (cm) 1 Mass density of Graphite /Epoxy 1.3*10 3 (Kg/m 3 ) Inner diameter of link (cm) 0.5 E of Graphite /Epoxy (N/m ) 1.77*10 11 Length of sliding Link (cm) 50-100 G of Graphite /Epoxy (N/m ) 0.0376*10 11. Dynamic Equation Formulation The Euler-Lagrange equations of motion are generally applied to derive the dynamic equations of a serial manipulator, which possess n degrees of freedom. The dynamic equations of a serial rigid manipulator [15] in terms of joint variables [q] given as: ( ) [ m]( q +,)() h q q + G q = T (1) is the inertia matrix of manipulator, h(q,q ) is a n n matrix related to the centrifugal and Coriolis terms, G(q) Gravity terms and T is the joint generalized force/ torques. The link s mass and the center of mass of each link in D-H frame are required to form the inertia matrix m, Corilisis and centrifugal matrix h and gravity matrix G and they are given by m1l 1 + I1 + md + I 0 θ md 0 0 ω + ( )( v) 0 m 0 ω + m d d 0 v () m1lcos 1 + md g θ T1 + = m g sin θ T Based on the geometry details, mass properties and relative positions of links, The matrices D, h and G are evaluated. The D, h, and G matrices are separately evaluated for ()
A Simulation Study on Joint Velocities and End Effector Deflection of a Flexible 13 the manipulator structure when links are solids with aluminum, hallow with aluminum and hallow with composites. Following linearization scheme adopted for centrifugal and coriolis components of dynamic equation at a critical point of configuration (a, b) f (,)( d,)( f )( a b ) f θ = + θ a + f a b θ d The state space variable is introduced as given below: x1 θ x ϖ X = = x 3 d x4 v Now the dynamic equation converted in to state space as shown below: X = AX + BU Y = CX + DU [A] is square matrix of size of 4 by 4, [C] is output matrix and input matrix is given by B 0 1 m 0 1 m 11 = and D is null matrix The input torques has been defined as given below: T U = F This equation is solved using MATLAB code to obtain joint variables: Joint angles and joint velocities. Then the linear displacement, velocities and accelerations are obtained by using following positional relations: x = l cosθ g1 1 g1 1 g g.3 Finite Element Formulation y = l sin θ x = d cos( θ =,) x d θ y = d sin( θ =,) y d θ The two links are divided in to j number of element with the idealization of frame elements. The number of nodes made is k. g g (3) (4) (5) (6) (7) (8)
14 International Journal of Advanced Mechatronics and Robotics Then the finite element equation is given by: Ma + Ka = F (t) (9) The solution of finite element equation can be sought with a = A sin ωt (10) Where A is nodal displacements vector which is given by A = a1 a ak M is Global mass matrix; K is Global stiffness matrix and F Global load vector. The each element Mass Matrix is given by (11) M Each Element stiffness matrix is given by j ρal ρal 0 0 0 0 3 6 13ρAl 11ρAl 9 ρal 13ρAl 0 35 10 70 40 3 3 ρal 13ρAl ρal 0 105 40 140 = ρal 0 0 3 13 ρal ρal 5 10 3 ρal 105 (1) K j EA EA 0 0 0 0 l l 1 EI 6EI 1EI 6EI 3 0 3 l l l l 6 EI EI 0 0 l l = EA 0 0 l 1EI 6EI 3 l l 4EI l The natural frequencies i of overall manipulator can be obtained with the following equation: K i (13) ω M = 0, i = 1 to k (14)
A Simulation Study on Joint Velocities and End Effector Deflection of a Flexible 15 The amplitude equation for each natural frequencies i is given by A i. i K ωi M A = 0 (15) From the amplitude equation, the Amplitude vectors are for each frequency obtained as The Amplitude vectors are normalized with following relations: i i A i T i AN = where Si = A [ M ] A S (16) Then the normalized vector of all amplitude can be written as A N = The global load vector is written as i 1 k AN, AN,, A N (17) F1 x F 1y 0 F x F = F y 0 F kx Note that each term of Force vector is to be expressed as (18) F = F 0 sin ϖt (19) The inertia forces at nodes corresponding to COG are obtained with accelerations which are found from dynamic equation and are given by F x (t) = F y (t) = m x m x 1 1g g m y m y 1 1g g (0) (1) Some F the nodal forces may be taken as Zero based on the nodal boundary conditions.
16 International Journal of Advanced Mechatronics and Robotics The normalized Force vector f is obtained as: f = A N F () The generalized matrix P is obtained as f ( m,1) P = ( ω ) ϖ m, m = 1 to k (3) The final solution, i.e. nodal amplitude vector is calculated as: A = A N P (4) 3. NUMERICAL RESULTS AND DISCUSSION In the numerical study, three models have been developed. The manipulator made up of links with circular solid cylinder with Aluminum is termed as mode-l, manipulator made up of links with hallow circular cylinder with Aluminum is termed as mode- and manipulator made up of links with hallow circular cylinder with composites is termed as model-3. The effect of the flexibility of the links on the rigid body behavior of the robot arm is illustrated by comparing the joint velocities and End effector deflection obtained from these three different models of the robotic manipulator. The saw-tooth profile of force and torque, is applied at the joints of the manipulator in all cases. The joint 1 is a rotary joint and a torque shown in Figure is assumed. At joint, a force (shown in figure ) similar variation of joint 1 is assumed to be applied in the model. The variation of applied in put indicate that input at joints is gradually increases and then suddenly brought to zero. This type of input generally comes across in many manipulators. Figure : Applied Joints Inputs The response obtained from the first model, whose both links are assumed to be rigid (rigid-rigid), is given in Figures 3a and 3b. The results of the second model, which includes the flexibility of both links (flexible-flexible), are shown in Figures 4a and 4b. Acomparison
A Simulation Study on Joint Velocities and End Effector Deflection of a Flexible 17 of the simulation results shows that the effect of the flexibility in links causing vibratory motion and it is more pronounced in joint1 rather than joint. Figure 3: Joint Velocities When Both Manipulators are Rigid Figure 4: Joint Velocities When Both Manipulators are Flexible The accuracy of the end effector deflections predicted by models is also assessed. This is performed by comparing the results obtained from the rigid-rigid and the flexibleflexible models of the robot arm when subjected to the saw tooth torque and force. The results are illustrated in Figures 5a-b. The flexible -flexible model exhibits higher frequency and higher amplitude than the rigid-rigid model at steady state. However The deflection variation continued in rigig-rigid model. This indicates that the energy stored in a flexible manipulator would be dissipated quickly.
18 International Journal of Advanced Mechatronics and Robotics (a) Rigid-Rigid Figure 5: Deflection of End Effector (b) Flexible-Flexible The superiority of advanced composites over conventional metals is demonstrated by comparing the response of an hallow aluminum arms to the response of a hallow graphite/ epoxy arms. The properties of graphite/epoxy composites of laminations type [ 45 / 45 ] and type [+40 /40 ] have been used in evaluation of mass and inertia matrices for the dynamic models. A valid comparison between the performance of manipulators which are made of aluminum materials and graphite/epoxy materials is done by prescribing the deflections of end effector of the robotic arm arm. This is because graphite/epoxy is lighter than aluminum and the application of a set of saw-tooth torque and force to the robot arm would generate higher speed, and consequently, higher inertial forces in the composite arm than in the aluminum one. Figure 6 shows that the composite arm exhibits less vibration than the arm made of aluminum.the [+40 /40 ] laminated arm has the lowest end effector (a) Composite of type [ 45 / 45 ] (b) Composite of type [ 40 / 40 ] Figure 6: Deflection of End Effector
A Simulation Study on Joint Velocities and End Effector Deflection of a Flexible 19 deflection than [ 45 /+45 ] laminated composite. (See Figure 6a and 6b). A slight change of the angle of fiber orientation in the graphite/epoxy, from 45 to 40 demonstrates less end effector deflections.changing the orientation of lamina changes the transformed stiffness coefficients. Thus transverse deflection of [ 40 /+40 ] laminate is smaller than the one corresponding to [ 45 /+45 ]4 laminate. 4. SUMMARY AND CONCLUSIONS In this work, the results of dynamic model of a planar, revolute-prismatic robot arm are presented. Both links of the robotic manipulator are considered to be rigid, flexible and made up of composites. All coupling terms between the rigid and the flexible motions of the robot arm are included in the model. The joint velocities and deflections of the manipulator are simulated by using MATLAB code. As the governing equations of motion which are in the form a set of stiff, coupled and highly nonlinear, Gear s method, which is best suited to handle stiff differential equations, is employed to obtain the numerical solutions. The effect of the flexibility of links on motion of the manipulator is significant and cannot be ignored in the context of accurate path track of end effector. In addition, the inclusion of the flexibility in the links of the arm leads to an overestimated end effector deflections at steady state. The use of advanced composite materials in the fabrication of lightweight and highspeed robotic manipulators leads to improved end effector positional accuracy and to lower torque/ force requirements at the joints. The selection of the angle of fiber orientation and material properties in the design of a robot arm plays an important role in the reduction of the end effector deflections. ACKNOWLDGMENTS Authors also thank Dr. T.Ch.SivaReddy, Head Mechanical Engg. Sreenidhi Institute of Science and Technology (An Autonomous Institution) Hyderabad, A.P. India-501301 for his encouragement and help in carrying out this research work. REFERENCES [1] Imam, I. and Sandor, G. N., High Speed Mechanism Design: A General Analytical Approach, ASME Journal of Engineering for Industry, 97(), 1975, 609-68. [] Liao, D. X., Sung, C. K., and Thompson, B. S., The Design of Flexible Robotic Manipulators with Optimal Arm Geometries Fabricated from Composite Laminates with Optimal Material Properties, The International Journal of Robotics Research, 6(3), 1987, 14-137. [3] Thompson, B. S. and Gandhi, M. V., The Finite Element Analysis of Mechanism Components made from Fiberreinforced Composite Materials, ASME Paper 80-Det-63. [4] Book, W., J., Maizza-Neto, O., and Whitney D. E., Feedback Control of Two Beams, Two Joint Systems with Distributed Flexibility, ASME Journal of Dynamic Systems, Measurement and Control 97(4), 1975, 44-431.
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