Impact of changing the position of the tool point on the moving platform on the dynamic performance of a 3RRR planar parallel manipulator

Transcription

1 IOSR Journal of Mechanical an Civil Engineering (IOSR-JMCE) e-issn: 78-84,p-ISSN: 0-4X, Volume, Issue 4 Ver. I (Jul. - Aug. 05), PP Impact of changing the position of the tool point on the moving platform on the ynamic performance of a RRR planar parallel manipulator Roshy Foaa Abo-Shanab Department of Mechanical Engineering, Faculty of Engineering / Kafrelsheikh University, Egypt. Abstract: In this paper, the impact of changing the location of the tool point, on the moving platform, on the ynamics of a planar parallel manipulator is investigate. Lagrange- Alembert formulation is use to evelop the ynamic moel of the present manipulator. To evaluate the ynamic performance of the parallel manipulator, the input efforts an energy consumptions are calculate for the manipulator when the eneffector is positione at ifferent locations on the moving platform an executing given esire trajectories. The manipulator s imensions an parameters are kept the same uring the optimization process an only the position of the tool-point on the moving platform is change. The ynamic performance of the manipulator is then evaluate an optimize. It is shown that locating the en-effector of the manipulator at an optimum position reuces the generalize forces require to rive the manipulator. It also reuces the energy consumption of the manipulator. Keywors - Dynamics, energy consumption, Lagrange- Alembert, optimization, parallel manipulators, trajectory. I. Introuction In recent years, many stuies have focuse on parallel manipulators. Since their en-effector, moving platform, is sustaine by several kinematic chains, parallel manipulators can achieve better structural an ynamic properties with less structural mass. Some of the avantages offere by parallel manipulators, when properly esigne, inclue a high loa-to-weight ratio, high stiffness, an positioning accuracy. However, parallel manipulators are ifficult to esign, since the relationships between esign parameters an the workspace, an behavior of the manipulator throughout the workspace, are not intuitive by any means []. In aition, the performances of parallel manipulators are very sensitive to their imensioning. Therefore, a thorough analysis of the kinematic an ynamic behavior of the parallel manipulators shoul be evelope for optimal esign of these machines []. Parallel manipulators also are more energy efficient than serial manipulators. Li an Gary [] showe that over a range of conitions, the average energy usage of the parallel manipulator was etermine to be % of the serial manipulator s. In this respect, Pellicciari et al. [4] showe a slight improve in the energy consumption on favor of parallel manipulators in pick an place inustrial robots application. In a previous article [5], the author stuie the kinematic behavior of a RRR planar manipulator as the location of the tool point, on the moving platform, changes. It was shown that changing the location of the eneffector on the moving platform greatly affects the kinematics of a parallel manipulator as the area of the workspace changes as well as other performance inices such as the global conitioning inex. It was recommene that the location of the en-effector on the moving platform shoul be consiere while optimizing the performance of a parallel manipulator. The ynamic performance of the parallel manipulator has not been stuie in the article. Dynamical analysis of parallel manipulators is complicate by the existence of the multiple close-loop chains. Several approaches have been propose incluing the Newton-Euler formulation [-8], the Lagrangian formulation [9-], an the Lagrange-D Alembert formulation [-]. For the Newton-Euler formulation, one first carries out a etaile force an torque analysis of each rigi link with some physical knowlege such as Newton s thir law, an then apply Newton s secon law an Euler s equation to each of the rigi links to obtain a set of secon orer orinary ifferential equations in the position an angular representation of each rigi link. Finally together with the kinematic constraints, the set of equations can be simplifie or solve, till the esire form of ynamics equation is obtaine. The Lagrangian approach is a more efficient than Newton- Euler metho as it eliminates the unwante reaction forces an moments at the outset. However, because of the numerous constraints impose by the close loops of parallel manipulator, eriving explicit equations of motion in terms of a set of inepenent generalize coorinates becomes a prohibitive task. Therefore, the Lagrangian equations are written in terms of a set of reunant coorinates. The formulation then requires a set of constraint equations erive from the kinematics of the manipulator. Final equations of motion are erive an arrange in DOI: / Page

2 Impact of changing the position of the tool point on the moving platform on the two sets. One contains Lagrange multipliers as the only unknown, an the other contains the generalize forces contribute by the actuator as the traitional unknowns [7]. Lagrange- Alembert formulation allows eriving the equations of motion without explicitly solving for the instantaneous constraint forces present in the system. This procees by projecting the motion of the system into the feasible irections an ignoring the forces of constraints irections. By oing so, a more concise escription of the ynamics can be obtaine [8]. Yiu et al. [9] reviewe various methos use in eriving the ynamic equations for parallel manipulators. They propose to cut the links instea of cutting joints to obtain the tree system, so that all the joints torques, incluing joint friction, can be incorporate in the ynamic equations. Khan et al. [0] evelope a moular an recursive formulation for the inverse ynamics of parallel architectures base on the concept of ecouple natural orthogonal complement. They applie the evelope metho to erive the ynamic equations of a RRR planar parallel manipulator. They cut the moving platform into three parts to form three open chains, to be able to apply torques at the joints, an to inclue the joint friction. Wu et al. [] investigate an compare the ynamics of the planar -DOF 4-RRR, -RRR an - RRR parallel manipulators. The -RRR parallel manipulator has only two limbs an one of these limbs has two active (actuate) joints whereas the 4-RRR planar manipulator has a reunant actuator. They showe that the sum of the absolute values of riving torques of the -RRR manipulator has the largest range. They conclue that the -RRR parallel manipulator has the worst ynamic performance among the three planar -DOF parallel mechanisms an, in some regions of the workspace, the ynamic performance of the 4-RRR manipulator is better than that of the -RRR one. Ruiz et al. [] stuie the impact of kinematic an actuation reunancy on the energy efficiency of planar parallel kinematic machines. They conclue that optimal energy efficient trajectories are epenent on the manipulator architecture an reunant parallel manipulators are more energy efficient than non-reunant manipulators. Generally, it is consiere that, given a esire trajectory, the robot that has the lowest input efforts an lowest energy consumption along the trajectory has the best performance. The objective of this text is to show the effects of changing the location of the tool point, on the moving platform, on the ynamics an energy efficiency of planar parallel manipulators. To the best of the author s knowlege, none of the previous work has iscusse this problem. The manipulator geometry is presente in Section. In Section, the ynamic moel of the stuie manipulator is evelope using Lagrange- Alembert formulation. Two case stuies are use to evaluate the ynamic performance an energy consumption of the manipulator an iscussion of the simulation results are presente in Section 4. In Section 5, conclusions are escribe. II. Manipulator Geometry The manipulator consiere in this work is a RRR planar parallel mechanism. A schematic iagram of the manipulator is shown in Figure. The manipulator consists of a moving equilateral triangular platform of length h connecte to a fixe equilateral triangular base of length by three limbs. Each limb consists of two links; the first link is connecte to the groun by means of a revolute joint ientifie by the letter B i an is actuate by a rotary actuator. The three actuators, one for each limb, control the three egrees of freeom of the moving platform (x, y, an φ). Two coorinate systems are efine to escribe the motion of the moving platform. The first coorinate system is B attache to the fixe base (with origin O an θ ψ axes x an y) an is calle the reference frame while the secon coorinate system is attache A to the moving frame (with origin O' an axes x' y an y'). In the present work, we change the location of the manipulator en-effector, the y' C x' position of O' in Figure, within the area of the moving triangle C C C to fin the x ψ A O φ b optimal position with respect to the ynamic ψ C performance of the manipulator. The pose of O' A C a the en-effector is expresse relative to the θ θ B reference frame by the position vector Z = B x y φ. The input angles Θ = [θ θ θ ] is represente by the angular positions of the Fig.. A -RRR parallel manipulator revolute actuators measure from the x-axis of the reference coorinate system. The inverse DOI: / Page

3 Impact of changing the position of the tool point on the moving platform on the kinematics an Jacobian analysis of the manipulator were presente by the author in a previous article [] an are presente in the appenices for the convenience of the reaer. α III. Dynamic Analysis A parallel manipulator can be consiere as a mechanical system with configuration q R n subject to a set of holonomic constrains, for the present manipulator, q = [θ θ θ x y φ] T. A constraint is sai to holonomic if it restricts the motion of the system to a smooth hypersurface in the, unconstraine, configuration space Q [8]. Holonomic constraints can be represente locally as algebraic constraints on the configuration space, h i q = 0, i =,.., k () where k is the number of linearly inepenent constraints. Each h i is a mapping from the configuration space Q to R restricts the motion of the system. Let L(q, q ) represents the Lagrangian for the unconstraine system. We assume that the constraints are everywhere smooth an linearly inepenent an that the forces of constraint o no work on the system. The equations of motion are forme by consiering the constraint forces as an aitional force which affects the motion of the system. Hence, the ynamics can be written in vector form as t q = q AT λ + τ () h h q q n () h k h k q n q n where A = h q = The columns of A T form non-normalize bases for the constraint forces an λ R k are calle Lagrange multipliers an gives the relative magnitue of the forces of constraints. τ represents nonconservative an externally applie forces. At a given configuration q R n, the instantaneous set of irections in which the system is allowe to move is given by the null space of the constraint matrix, A(q). Let the vector δq R n, which satisfies A(q) δq = 0, a virtual isplacement. Then δw = τ δq is calle the virtual work ue to a force τ acting along a virtual isplacement δq. D Alembert s principle states that the forces of constraint o no virtual work. Hence, A T (q)λ δq = 0 (4) To eliminate the constraint forces from Equation (), the equations of motion is projecte onto the linear subspace generate by the null space of A(q). t q τ δq = 0 (5) q where δq R n satisfies the constraints A(q) δq = 0 Rearrange the matrix A(q) to take the following form A q = A Θ q A z q where A Θ q = h Θ t q, A z τ δq = q q = h Z, Θ = [θ θ θ ] T, an Z = [x y φ] T. Let τ = τ Θ ; F where τ Θ = τ ; τ ; τ is a column vector of the actuating torques an F = F x ; F y ; τ φ is a column vector of the external forces an torques. Then, Equation (5) can be written as follows: τ t Θ Θ Θ δθ t Θ t Θ t Z Z F δz = 0 τ Θ Θ δθ + F δz = 0 t Z Z τ Θ Θ δθ + F A t Z Z Z(q) A Θ (q)δθ = 0 x' l l α O' α C C Fig.. Definition of the moving plate parameters DOI: / Page y' l C

4 Impact of changing the position of the tool point on the moving platform on the Since δθ is free, then t Θ Θ τ Θ + A Θ (q) T A Z (q) T t Z Then the actuator torques can be written as follows: τ Θ = t Θ Θ + A Θ(q) T A Z (q) T t Z Z F = 0 F () Z Equation () represents the ynamic moel of a general parallel manipulator that is subject to holonomic constraints an calculates irectly the actuator torques without explicitly calculating the Lagrange multipliers. For the present manipulator, ifferent terms of Equation () can be etermine as follows. First, the constraint equations of the manipulator, see Figure, can be written as: h i = x x Bi a cos θ i l i cos α i + φ + y y Bi a sin θ i l i sin α i + φ b = 0 (7) Then, h 0 0 A θ q = 0 h 0 (8) 0 0 h an A Z q = where h x h y h φ h x h y h φ h x h y h φ (9) h ij = h i θ j = 0 h ii = h i θ i = a for i j x x Bi sin θ i y y Bi cos θ i + l i sin α i + φ θ i h ix = h i x = x x Bi a cos θ i l i cos(α i + φ) h iy = h i y = y y Bi a sin θ i l i sin(α i + φ) h iφ = h i φ = l i Since x x Bi sin(α i + φ) y y Bi cos(α i + φ) a sin α i + φ θ i A(q) δq = 0, then θ = A θ q A z q z Let A θz q = A Θ (q) A Z (q) A θz q = h h h h x h y h φ h x h y h φ h x h y h φ (0) h x h y h φ h h h A θz q = h x h h y h h φ h () h x h y h φ h h h Let A θz (q) = t A θz (q), the elements of A θz (q) can be calculate as follows: t t t h ix h ii h iy h ii h iφ h ii = h ix h ii h ix h ii h ii = h iy h ii h iy h ii h ii = h iφ h ii h iφ h ii h ii DOI: / Page

5 Impact of changing the position of the tool point on the moving platform on the where h ix = x + a sin θ i θ i + l i sin(α i + φ) φ h iy = y a cos θ i θ i l i cos(α i + φ) φ h iφ = l i x sin(α i + φ) + x x Bi cos(α i + φ)φ y cos(α i + φ) + y y Bi sin(α i + φ) φ a cos α i + φ θ i φ θ i h ii = a x x Bi cos θ i θ i + x sin θ i + y y Bi sin θ i θ i y cos θ i + l i cos α i + φ θ i φ θ i If the motion of the tool-point of the parallel kinematic machine is efine in the Cartesian coorinates, the time erivatives of the joint variables can be calculate as follows. θ = A θz (q)z θ = A θz (q)z + A θz (q)z () () Next, the Lagrangian function of the manipulator is calculate. The manipulator is planar moves in a horizontal plane, therefore, potential energy is zero an the Lagrangian function is simply equals to the total kinetic energy of the manipulator. The kinetic energy of the manipulator can be ivie into three parts; the kinetic energy of the first link in each limb, the secon link in each limb, an the moving platform. First, the kinetic energy, K ai, of the first link in each limb (Link B i A i ), note that the limbs are ientical : K ai = m av ai + I aθ i (4) v ai = a θ i is the velocity of the center of mass of link B i A i, m a is the mass of link B i A i, I a = m aa is the mass moment of inertia of link B i A i about an axis passing through its center of mass an parallel to z-axis, an a is the length of link B i A i. Then K ai = m aa θ i (5) The kinetic energy of the secon link, K bi, in each limb (Link A i C i ): K bi = m bv bi + I b θ i + ψ i () where v bi is the velocity of the center of mass of link A i C i, m b is the mass of link A i C i, an I b = m bb is the mass moment of inertia of link A i C i about an axis passing through its center of mass an parallel to z-axis. The kinetic energy of the secon link of each limb is erive as a function of x, y, φ, θ, θ, an θ an their time erivatives an ψ, ψ, ψ an their time erivatives are eliminate. K bi = m b x + y + a θ i + l i φ i θ i x sin θ i y cos θ i + l i φ x sin α i + φ y cos α i + φ al i φ θ i cos α i + φ θ i (7) Finally, the kinetic energy of the moving plate K p : K p = m pv p + I pφ where v p is the velocity of the origin of the moving coorinate system that attache to the moving plate an I p is the mass moment of inertia of the moving platform. From (5), () an (7), the total kinetic energy of the manipulator is i= i + L = m b + m p x + m b + m p y + l + l + l m b + I p φ + m a + m b a θ m bφ i= l i sin α i + φ a m b i= sin θ i θ i x + m bφ i= l i cos α i + φ + a m b i= cos θ i θ i y a m bφ i= l i cos α i + φ θ i θ i (8) Taking the erivatives of the Lagrangian function (9) with respect to the six generalize coorinates, we get x y an = 0, (9) = 0, (0) DOI: / Page

6 φ = m b Impact of changing the position of the tool point on the moving platform on the i= l icos α i + φ x + l i sin α i + φ y φ + a m bφ i= l i sin α i + φ θ i θ i () = a m θ i b x cos θ i + y sin θ i θ i a l i m b sin α i + φ θ i θ iφ, i =,, an. () φ = m b t t t t x y φ θ i i= l icos α i + φ x + l i sin α i + φ y φ + a m bφ i= l i sin α i + φ θ i θ i () = m b + m p x + m bφ i= l i sin α i + φ a m b i= sin θ i θ i + m bφ i= l i cos α i + φ a m b i= cos θ i θ i (4) = m b + m p y m b l i= icos α i + φ φ i= l i sin α i + φ φ + a m b i= cos θ i θ i i= i (5) a m b sin θ i θ = l + l + l m b + I p φ + m b i= l isin α i + φ x l i cos α i + φ y a m b i= cos α i + φ θ i θ i + m bφ i= l i cos α i + φ x + l i sin α i + φ y + a m b i= sin α i + φ θ i φ θ i θ i () = m a + m b a θ i a m b x sin θ i y cos θ i + l i cos(α i + φ θ i )φ a m b x θ i cos θ i + y θ isin θ i l i sin α i + φ θ i (φ θ i)φ, i =,, an. (7) Equations (8) to (9) an (9) to (7) are substitute in Equation () to obtain the riving torques. Meanwhile, the energy consumption of the parallel manipulator can be expresse as follows [9]: t E = f i= τ i θ i t t o where t o an t f are the start time an en time, respectively of the motion of the manipulator. (8) IV. Simulation Results The evelope schemes are applie to the present manipulator, shown in Figure. The coorinates of the points of connection of the manipulator with the fixe base are: B 00, 7. mm, B 00, 7. mm an B 0, 4.4 mm. The following numerical values are use for the ifferent manipulator imensions: a = 50 mm, b = 7.5 mm, h = 50 mm. The author showe in a previous article [] that these imensions give the maximum reachable workspace of the manipulator. Inertia moments an masses of ifferent links of the manipulator are taken as follows: m a = kg, m b = 4.5 kg, m p = kg, an I p = 0.0 kg.m. The external forces, F = F x F y τ φ, are assume to be constant uring the motion with the following values: F x = 0 N, F y = 0 N, an τ φ = 0 N.m. To evaluate the ynamic performance of the parallel manipulator, the input efforts an energy consume are calculate for the manipulator when the en-effector is positione at ifferent locations on the moving platform an executes given esire trajectories. Figure shows the ifferent locations of the tool point on the moving platform, these locations are chosen base on the similarity of the platform (equilateral triangle), other locations are expecte to give similar results. C Two trajectories are assigne for the motion of the tool point. The first trajectory is a straight-line path the tool point moves from the initial location at Z = π/ T, where x an y are in millimeters an φ is in raians, to the final position at Z = π/ T with cycloial motion: x = x o l t/t /π sin πt/t (9) Where y an φ are constants uring the path an l = 80 mm, is the total istance travele by the tool point uring the task. The secon trajectory is to move the tool point on a circular path of raius R = 40 mm an a center at (x o = 0 mm, y o = 00 mm, ), starts from the initial location at Z = π/ T. The equations of the motion trajectory are: DOI: / Page 7 C C Fig.. Different locations for the tool point on the moving platform. 4 5

7 x-tool point (mm) Z = x o + R cos πt/t y o + R sin πt/t π/ Impact of changing the position of the tool point on the moving platform on the where T = 4 s, is the total motion time in both cases. For the secon trajectory, the tool point makes a complete circle in 4 secons. The orientation of the moving platform is kept constant uring the motion at φ = π/ an the integration time is chosen to be t = ms. Figures 4 an 5 show the trajectories of the moving platform uring the two cases. A MATLAB program is evelope to calculate the actuator torques for both trajectories consiering the tool point at ifferent locations as shown in Fig.. The program also calculates the energy require to execute the manipulator motion. The results show that the input efforts are the lowest when the tool point is locate at Position. Fig. 5 an Fig. show the require torques to rive the manipulator through a straight-line path while the tool point is at Position 4 an Position, respectively. Fig. 7 shows the sum of torques when the tool point is locate at Position, 4, an 5. Figs. 9 to show similar results when the tool point is moving on a circular path. The two sets of results, moving on a straight-line path an on a circular path, give the same inication about the ynamic performance of the manipulator, Position give the lowest input effort to rive the manipulator through the esire trajectories. The optimization toolbox fminimax of MATLAB, which applies Quasi-Newton algorithm, is use to fin the location of the tool point on the moving platform that optimizes the energy consumption uring the execution of trajectory in the two cases. For both cases, it is foun that Position gives the minimum energy consumption. The results are verifie using a MATLAB program. The energy consumption of the manipulator uring the execution of the tasks when the tool point is positione at ifferent locations is calculate. As seen in Fig., Position gives the lowest energy consumption for both trajectories (0) Fig. 4. x- Coorinate of the tool point uring the first case. V. Conclusion The present work investigates the effects of changing the position of the en-effector, on the moving platform, on the ynamic performance of a -RRR planar parallel manipulator. The ynamic equations of the parallel manipulator were evelope using Lagrange Alembert metho. The ynamic performance of the manipulator was then optimize as the location of the tool point on the moving platform changes. All the imensions an parameters of the manipulator are kept the same uring the optimization process. To the best of the author knowlege, none of the previous research ha aresse this problem. It is shown that precisely locating the suitable tool point position on the moving platform reuces the input efforts an the energy consumption. For the present manipulator, the reuctions of the energy consumption were.4% an 8.7% for the first an secon cases, respectively. DOI: / Page

8 Sum of Torques (N.m) Torque (N.m) Torque (N.m) Impact of changing the position of the tool point on the moving platform on the 0 - T T T Fig.. Driving torque for the straight line path, tool point at position T T T Fig. 7. Driving torque for the straight line path, tool point at position Point Point 4 Point Fig. 8. Sum of the absolute values of the riving torque for the straight line path, tool point at ifferent positions. DOI: / Page

9 Sum of Torques (N.m) Torque (N.m) Torque (N.m) Impact of changing the position of the tool point on the moving platform on the T T T Fig. 9. Driving torque for the circular line path, tool point at position T T T Fig. 0. Driving torque for the circular line path, tool point at position Point Point 4 Point Fig.. Sum of the absolute values of the riving torque for the circular line path, tool point at ifferent positions. DOI: / Page

10 Energy Consumption (J) Impact of changing the position of the tool point on the moving platform on the Circular line path Straight line path Different positions on the moving platform Fig.. Energy consumption uring the straight line path an circular line path, tool point at ifferent positions. References [] C. Gosselin, L. Perrault, an C. Viallancourt, Simulation an computer-aie kinematic esign of three-egree-of-freeom spherical parallel manipulator, Journal of Robotic Systems,, 995, [] J. P. Merlet, Jacobian, manipulability, conition number, an accuracy of parallel robots, ASME J. Mech. Des. 8, 005, [] Y. Li an M.B. Gary, Are Parallel Manipulators More Energy Efficient? Proceeings of 00 IEEE International Symposium on computational Intelligence in Robotics an Automation, Banff, Alberta, Canaa, 00, 4-4. [4] M. Pellicciari, G. Berselli, F. Leali, an A. Vergnano, A metho for reucing the energy consumption of pick an place inustrial robots, Mechatronics, (), 0, -4. [5] R.F. Abo-Shanab, Effect of changing the position of tool point on the moving platform on the kinematics of a RRR Planar Parallel Manipulator, Applie Mechanics an Materials Vols , 04, [] W.Q.D. Do an D.C.H. Yang, Inverse Dynamic analysis an simulation of a platform type of a robot, Journal of Robotic Systems, 5(), 988, [7] P. Guglielmetti an R. Longchamp, A close form inverse ynamic moel of the elta parallel robot, Proceeings of the 994 International Feeration of Automatic Control Conference on Robot Control, 994, [8] K.Y. Tsai an D. Kohli, Moifie Newton-Euler computational scheme for ynamic analysis an simulation of parallel manipulators with application to configuration base on R-L actuators, Proceeings of the 990 ASME Design Engineering Technical Conferences, Vol. 4, Boston, Massachusetts, 990, -7. [9] S. Liu, Z. Zhu, Z. Sun, an G. Cao, Kinematics an ynamic analysis of a three-egree of freeom parallel manipulator, Journal of Central South University, Vol.,04, 0-. [0] Y.Nakamura, an M. Ghooussi, Dynamics Computation of Close-Link Robot Mechanisms with Nonreunant an Reunant Actuators, IEEE Trans. Robot. Automat., vol.5(), pp. 94 0, 989. [] Y.Naramura, an Katsu Yamane, Dynamics Computation of Structure-Varying Kinematic Chains an Its Application to Human Figures, IEEE Trans. Robot. Automat., (), 000, 4 4. [] F.C. Park, J. Choi, an S.R. Ploen, Symbolic Formulation of Close Chain Dynamics in Inepenent Coorinates, Mechanism an Machine Theory, 999, vol. 4, [] A. Coourey an E. Buret, A boy oriente metho for fining a linear form of the ynamic equation of fully parallel robots, Proceeings of 997 IEEE International conference on Robotics an Automation, 997, -8. [4] L.W. Tsai, solving the inverse ynamics of parallel manipulators by the principle of virtual work, Proceeings of 998 ASME Design Engineering Technical Conferences, DETC98/MECH-585. [5] J. Wang an C.M. Gosselin, Dynamic analysis of spatial four-egrees-of-freeom parallel manipulators, Proceeings of 997 ASME Design Engineering Technical Conferences, DETC97/DAC-759. [] H. Cheng, Y. Yiu, an Z. Li, Dynamics an control of reunantly actuate parallel manipulators, IEEE/ASME Transactions on Mechatronics, 8(4), 00, [7] L.W. Tsai, Robot Analysis: the mechanics of serial an parallel manipulators, (John Wiley an Sons, USA 999). [8] R. Murray, Z.X. Li, an S. Sastry, A Mathematical Introuction to Robotic Manipulation, (CRC Press, 994). [9] Y.K. Yiu, H. Cheng, Z.H. Xiong, G.F. Liu, an Z.X. Li, On Dynamics of Parallel Manipulators, Proceeings of 000 IEEE international Conference on Robotics an Automation, 00, [0] W. Khan, V. Krovi, S. Saha, an J. Angeles, Recursive Kinematics an Inverse Dynamics for a R Parallel Manipulator, Journal of Dynamic Systems, Measurements, an Control, Vol. 7, 005, [] J. Wu, J. Wang, an Z. You, A comparison stuy on the ynamics of planar -DOF 4RRR, RRR, an -RRR parallel manipulator, Robotics an Computer-Integrate Manufacturing, vol. 7, 0, [] A.G. Ruiz, J.V.C. Fontes, an M.M. a Silva, The impact of the Kinematics an Actuation Reunancy on the Energy Consumption of Planar Parallel Kinematic Machines, Proceeings of the XVII International Symposium on Dynamic Problems of Mechanics, DIN-05-00, 05. DOI: / Page

11 Impact of changing the position of the tool point on the moving platform on the [] R.F. Abo-Shanab, Optimization of the Workspace of a R Planar Parallel Manipulator, Proceeings of the n International Conference on Mechanical an Electronics Engineering (ICMEE 00), paper number M8, Vol., 00, Appenix A Inverse Kinematics From the geometry of the manipulator, shown in Figures an, a vector loop equation can be written for each limb as OO = OB i + B i A i + A i C i + C i O (A) where i =,,. Expaning (), we get x = x Bi + a cos θ i + b cos θ i + ψ i + l i cos α i + φ y = y Bi + a sin θ i + b sin θ i + ψ i + l i sin α i + φ (A) (A) The efinitions of the angles α, α, an α are shown in Figure. Squaring (A) an (A) an summing the results, we get b = x x Bi a cos θ i l i cos α i + φ + y y Bi a sin θ i l i sin α i + φ. (A4) Now, expaning (A4) an putting the result in the following form: e i sin θ i + e i cos θ i + e i = 0, where e i = a ρ i sin α i + φ + a y Bi y e i = a ρ i cos α i + φ + a x Bi x e i = x + y + a + ρ i b x x Bi y y Bi + x Bi + y Bi + l i cos α i + φ x Bi x + l i sin α i + φ y Bi y. Substitute the following trigonometric ientities in (A5) (A5) (A) (A7) (A8) sin θ i = t i +t, cos θ i = t i i +t, an t i = tan θ i we obtain e i e i t i + e i t i + e i + e i = 0, (A9) then θ i = tan e i ± e i + e i e i e i e i (A0) Three cases coul be foun when solving (0). The first case when the solution gives two ifferent real roots. This means that for each given moving platform location, there are two possible configurations for every limb. The secon case, when it yiels a ouble root, this means that this limb is in a fully stretche out or fole back configuration an is calle the singular configuration. The thir case, when the solution yiels no real roots, the specifie moving platform location is not reachable, i.e., this location is out of the manipulator workspace [7]. Appenix B Jacobian Analysis of the Manipulator In this section, the analytical evelopment of the manipulator s Jacobian matrix is presente. For each limb, ifferentiating (A) an (A), we get: x = a sin θ i θ i b sin θ i + ψ i θ i + ψ i l i sin α i + φ φ, (A) y = a cos θ i θ i + b cos θ i + ψ i θ i + ψ i + l i cos α i + φ φ. (A) Solving (A) an (A) to eliminate ψ i, we get cos θ i + ψ i x + sin θ i + ψ i y l i sin θ i + ψ i α i + φ φ = a sin ψ i θ i. (A) Equation (A) is written in the matrix form as follows: J z Z = J Θ Θ, (A4) DOI: / Page

12 Impact of changing the position of the tool point on the moving platform on the cos β sin β l sin β α + φ where J z = cos β sin β l sin β α + φ, β i = θ i + ψ i, an J Θ = cos β sin β l sin β α + φ a sin ψ a sin ψ a sin ψ In the above expression, J z an J Θ are two separate Jacobian matrices, these matrices can be combine to obtain a single matrix that establishes the inverse transformation between the input an output velocities: Θ = J Z, (A5) where J = J Θ J z corresponing to the inverse Jacobian of a serial manipulator. DOI: / Page