Planar Vector Equations in Engineering*

Int. J. Engng Ed. Vol., No.,. 40±406, 006 0949-149X/91 $3.00+0.00 Printed in Great Britain. # 006 TEMPUS Pulications. Planar Vector Equations in Engineering* H. R. MOHAMMADI DANIALI Deartment of Mechanical Engineering, Faculty of Engineering, Mazandaran University, Baol, Mazandaran, Iran. Email: mohammadi@nit.ac.ir A novel method to solve lanar vector equations in two unknowns is outlined and develoed and its alication is illustrated in the context of kinematic analysis. The rolem is classified in terms of all four ossiilities concerning the cominations of unknown airs of vector magnitudes and directions, which may arise in the course of formulation. Through extensive use of 1 and 1 matrix multilication and the lanar, roer orthogonal oerator, it is elieved that this method offers advantages of simlicity and comutational efficiency as well as roustness, comared to conventional aroaches. Keywords: Planar mechanism; vector equation. INTRODUCTION MANY ENGINEERING rolems are formulated with lanar vector equations. Traditionally, these are often solved manually, with grahical construction done to scale or else as a guide to tedious calculations. The use of comuters led to the develoment of a systematic comutational aroach to solutions which offers certain concetual advantage as well as the ovious imrovement in seed and recision. The most familiar methods used to imlement comutational solutions include:. analysis. comlex algera [1]. Chace's method [] Analytical and comlex algera methods require the solution of simultaneous equations. In the case of nonlinear rolems, these often lead to algeraic maniulation. Chace [] took advantage of the conciseness of vector notation to otain exlicit closed-form solutions to lanar vector equations. However, his method involves converting two-dimensional equations into three-dimensional equations and solving the latter. Therefore, the method is very time-consuming. More recently, an orthogonal oerator for the synthesis and analysis of some lanar rootic maniulators in the lane was introduced [3±7]. A novel way to solve such rolems, ased on the oerator introduced y Angeles [3], will e considered herein. In this method, exlicit closedform solutions to all ossile lanar vector equations are formulated. Moreover, the lanar equations are solved in the lane, not in three dimensions. Therefore, any software imlementation of this method will e very efficient. The further advantage of this method arises when one deals with multi-ody kinematics or dynamics. In * Acceted 4 Octoer 005. these cases the rolems are formulated using matrix comutations. This method conveniently can e assemled in matrix form. CLASSIFICATION OF PLANAR VECTOR EQUATIONS Any lanar vector equation can e solved for two unknowns. Deending on the unknowns, four distinct cases occur. Consider the vector equation: c ˆ a 1 in which c ˆ c^c, a ˆ a^a and ˆ ^, where a, and c are scalars, reresenting the magnitudes of the vectors, and ^a, ^ and ^c are unit vectors, reresenting their directions. Then the rolem can e classified, according to the unknowns, as: A. Magnitude and direction of the same vector are the unknowns. B. The magnitudes of two different vectors are the unknowns. C. The magnitude of one vector and direction of another one are the unknowns. D. The directions of two different vectors are the unknowns. The rolem of solving Equation (1) for the unknowns at hand, although aarently linear and quite straightforward, is, in fact quadratic, excet for cases A and B. Indeed, constraints of the form k^xk 1 ˆ 0 must e added to the quadratic Cases C and D, where x is the vector whose direction is unknown. SOLUTIONS OF PLANAR VECTOR EQUATIONS In this section we resent the solutions to the lanar vector equations y using the oerator E, a 40

Planar Vector Equations in Engineering 403 orthogonal matrix which rotates vectors in a lane through a counter-clockwise angle of 908, i.e., E ˆ 0 1 1 0 A. Magnitude and direction of the same vector are the unknowns, i.e., c ˆ c^c of Equation (1) is sought, which is a trivial case. B. The magnitudes of two different vectors, let us say a and, are the unknowns. Then, we ostmultily oth sides of Equation (1)-transosed y E^a, namely: c T E^a ˆ a T E^a T E^a where E^a is erendicular to a. Therefore, the first term of the right hand side of Equation () vanishes. Thus, it can e solved for as follows: ˆ ct E^a 3 ^ T E^a Similarly, one may ost-multily oth sides of Equation (1)-transosed y E^ and solve the resulting equation for a, namely: a ˆ ct E^ 4 ^a T E^ C. The magnitude of one vector and the direction of another, say a and ^, are the unknowns. Then, we ost-multily oth sides of Equation (1), transosed y E^a, namely: c T E^a ˆ a^a T E^a ^ T E^a 5 As exlained earlier, a T E^a ˆ 0. Then, solving Equation (5) for ^ T E^a leads to: ^ T E^a ˆ ct E^a 6 Let us consider ^a, E^a and ^ as shown in Fig. 1. Therefore, Equation (6) leads to: cos ' ˆ ct E^a 7 Thus, Therefore, we have: sin ' ˆ 1 cos ' ^ ˆ cos 'E^a sin '^a 8 9 Finally, one can easily solve Equation (1) for a as: a^a ˆ c 10 D. The direction of two different vectors, let us say ^a and ^ are the unknowns. Then, define ˆ ^c and ˆ E, as shown in Fig.. Therefore, we have: a^a ˆ u v 11 ^ ˆ u c v 1 One may comute the magnitudes of vectors a and as: u v ˆ a 13 u c v ˆ 14 Solving Equations (13, 14) for u and v leads to: v ˆ a c c u ˆ a v 15 16 Sustituting the values of u and v into Equations (13, 14) yields: ^a ˆ u a E^c v a ^c ^ ˆ u 17 c v E^c ^c 18 APPLICATIONS AND EXAMPLES It is elieved that the foregoing methods for solving lanar vector equations may e emloyed to great advantage when alied to the analysis and synthesis of mechanisms and rootic maniulators. In what follows, some examles are resented in suort of this contention. Examle 1: Direct kinematics of the slider-crank The direct kinematics of the slider-crank mechanism of Fig. 3 is the suject of this examle. Direct-kinematics rolems are those where actuator variales are secified and from these, one seeks to estalish the Cartesian co-ordinates of the end-effector (EE). Here, the actuator variale, angle, is given and the dislacement of slider C along the X axis is sought. The closure vector equation for the dislacement is given as: Fig. 1. Unknown direction of a and magnitude of. r 1 r r 3 ˆ r 4 19

404 H. Daniali Fig.. Two unknown directions. where r 1, r, r 3 and r 4 are the vectors directed from O to A, A to B, B to C and O to C, resectively. Equation (19) can e written as: r 4^r 4 r 3^r 3 ˆ r 1^r 1 r ^r 0 in which fr i g 4 1 and f^r ig 4 1 are, resectively, the magnitudes and the direction vectors of vectors fr i g 4 1. In the foregoing equation r 4 and ^r 3 are unknowns. Therefore, this is a class-c rolem. Thus, the solutions are given y Equations (7±10) through the sustitution of the resective given values as follows: cos ' ˆ r 1 r T E^r 4 r 3 sin ' ˆ 1 cos ' ^r 3 ˆ cos 'E^r 4 sin '^r 4 r 4^r 4 ˆ r 1 r r 3^r 3 1 3 4 Examle : Slider-crank velocity analysis In the mechanism of Fig. 3, given the velocity of lock C, v C, find the angular velocities of links AB and BC. The closure velocity vector equation can e written as: v C ˆ v B v C=B 5 in which v B and v C=B are the velocities of oint B and of oint C with resect to B, resectively. But we have: v B ˆ! Er v C=B ˆ! 3 Er 3 6 7 where! and! 3 are the angular velocities of links AB and BC, resectively. Sustituting the values of v B and v C=B from Equations (6, 7) into Equation (5) leads to: v C ˆ! Er! 3 Er 3 8 In Equation (8) the magnitudes of vectors! Er and! 3 Er 3 are the unknowns. Therefore, this is a class-b rolem whose solution is outlined aove. To find!, we may re-multily oth sides of Equation (8) y r T 3, namely: r T 3 v C ˆ! r T 3 Er 0 Solving the foregoing equation for! leads to: 9! ˆ rt 3 v C r T 3 Er 30 Similarly, re-multilying oth sides of Equation (8) y r T and solving the resulting equation for! 3 yields:! 3 ˆ rt v C r T Er 3 31 Examle 3: Slider-crank acceleration analysis In the mechanism of Fig. 3, if the acceleration a C of lock C is given, find the angular accelerations of links AB and BC. The closure vector equation for acceleration can e written as: a C ˆ a B a C=B 3 where a B and a C=B are the accelerations of oint B Fig. 3. Offset slider crank mechanism.

Planar Vector Equations in Engineering 405 and of oint C with resect to B, resectively, and can e written as: a B ˆ! r Er a C=B ˆ! 3 r 3 3 Er 3 33 34 in which and 3 are the angular accelerations of links AB and BC, resectively. Sustituting the values of a B and a C=B from the foregoing equations into Equation (3), uon simlification, leads to: Er 3 Er 3 ˆ a C! r! 3 r 3 35 In Equation (35) the magnitudes of vectors Er and 3 Er 3 are the unknowns. Therefore, this is a class-b rolem, whose solution is outlined aove. To find, we re-multily oth sides of Equation (35) y r T 3, namely: r T 3 Er ˆ r T 3 a C! rt 3 r! 3 rt 3 r 3 Solving the foregoing equation for leads to 36 ˆ rt 3 a C! rt 3 r! 3 rt 3 r 3 r T 3 Er 37 Similarly, re-multilying oth sides of Equation (35) y r T and solving the resulting equation for 3 yields: 3 ˆ rt a C! rt r! 3 rt r 3 r T Er 3 38 Examle 4: Velocity analysis of a lanar 3-dof, 3-RRR arallel maniulator A lanar 3-dof, arallel maniulator [8], with only revolute joints, is deicted in Fig. 4. Its motors P 1, P and P 3 are fixed to the ase. Finding the maniulator Jacoian matrices is the key issue in isotroic design, singularity analysis and dynamic formulation [3, 5]. Herein, we comute the Jacoian matrices of the maniulator under study. In lanar mechanisms, the relationshi etween the actuated-joint-velocity vector: _ ˆ _ 1 ; _ ; _ 3 Š T and the twist or Cartesian-velocity vector t ˆ! v C Š T, can e written as: J _ Kt ˆ 0 39 where J and K are the two 3 3 Jacoian matrices of the maniulator. Moreover,! is the angular velocity of the EE. The velocity v C of the EE can e written for the ith leg as: where v C ˆ v Ai v Qi =A i v C=Qi 40 v Ai ˆ _ i Ea i 41a v Qi =A i ˆ _Er i 41 v C=Qi ˆ!Es i 41c in which _ i and _ i are the angular velocities of links P i A i and A i Q i, resectively. Moreover, a i, r i and s i are vectors from P i to A i, A i and Q i to C, resectively. Sustituting the values of v Ai, v Qi =A i and v C=Qi from Equations (41a±) into Equation (40) leads to: _ i Ea i _ i Er i!es i v C ˆ 0 4 where _ i, the velocity of an unactuated joint, should e eliminated. This can e done y multilying Equation (4) y r T i, namely: r T i _ i Ea i r T i!es i r T i v C ˆ 0 43 Writing the aove equation for i ˆ 1; ; 3, we otain: J _ Kt ˆ 0 44 The 3 3 Jacoian matrices J and K are given as: r T 1 Ea 3 1 0 0 6 J 0 r T Ea 7 4 0 5 45a 0 0 r T 3 Ea 3 r T 1 Es 3 1 r T 1 6 K r T Es r T 7 4 5 45 r T 3 Es 3 r T 3 Fig. 4. Planar 3-dof, 3-RRR arallel maniulators. Using the E oerator, the Jacoian matrices are comuted ased on invariants of the maniulator. This can find vast alications in velocity, dynamic and singularity analyses as well as in the isotroic designs of rootic maniulators.

406 H. Daniali CONCLUSIONS A novel method to solve all ossile lanar vector equations was outlined. It has een roven that the method is alicale to oth simle mechanisms and comlex rootic maniulators. The simlicity, comutational efficiency, and roustness of the method were more ovious when one dealt with rootic maniulators in which the analysis required matrix comutations. REFERENCES 1. J. E. Shigley and J. Uicker, Theory of Machines and Mechanisms, McGraw-Hill (1995).. M. A. Chace, Vector analysis of linkages, ASME J. Eng. Ind., ser. B., 55(3), 1963,. 89±97. 3. J. Angeles, Fundamentals of Rootic Mechanical Systems, Sringer, New York (1997). 4. D. Chalat and J. Angeles, On the kinetostatic otimization of revolute-couled lanar maniulators, Mechanism and Machine Theory, 37, 00,. 351±374. 5. H. Mohammadi Daniali, P. Zsomor-Murray and J. Angeles, Singularity analysis of a general class of lanar arallel maniulators, Proc. IEEE Int. Conf. Rootics and Automation, 1995, Nagoya, Jaan,. 1547±155. 6. J. Angeles and M. A. Gonzalez-Palacios, The loo_olygon method of analysis of lanar linkages, Proc. Sixth IFToMM Int. Symosium on Linkages and Comuter-Aided Design Methods-Theory and Practice of Mechanisms, Bucharest, 1, 1993,. 11±19. 7. R. Sinatra and J. Angeles, A novel aroach to the teaching of lanar mechanism dynamics, a case study, Int. J. Mechanical Engineering Education, 31(3),. 01±14. 8. C. M. Gosselin and J. Sefrioui, Polynomial solutions for the direct kinematic rolem for the lanar three-degree-of-freedom arallel maniulators, Proc. Fifth Int. Conf. on Advanced Rootics, 1991, Pisa,. 114±119. Hamid Reza Mohammadi Daniali graduated as a Mechanical Engineer at Ferdoosi University in 1965. He received the M.Sc. in 1968 from Tehran University and in 1995, the Ph.D. degree in Mechanical Engineering from McGill University. Since 1995 Daniali is with the Deartment of Mechanical Engineering at Mazandaran University. His research interests focus on theoretical kinematics and arallel maniulators.