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# GEOM MECH ROBOTS 1 EML 6281

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This 6 page Class Notes was uploaded by Paige Cruickshank on Friday September 18, 2015. The Class Notes belongs to EML 6281 at University of Florida taught by Carl D. Iii Crane in Fall. Since its upload, it has received 11 views. For similar materials see /class/206707/eml-6281-university-of-florida in Engineering Mechanical at University of Florida.

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Date Created: 09/18/15

Group 4 7R Mechanism Reverse Analysis Problem Statement giVeni 0 12 0 23 0 34 145 156 167 0 71 a12 a23 a34 a45 356 367 a71 SI 32 33 S4 35 36 S7 97 nd 91 92 93 94 95 96 Approach Obtain 4 equations of the form ai x22 bi x2 116 ei x22 fixz g 0 i14 101 where the coefficients ai through g are quadratic in the tanhalf angle of 91 Using Sylvester s method will result in a 16th degree polynomial in X1 1021 Derivation of first pair of equations Write the vector loop equation as R R633 R356 0 102 where R633 s s6 a67a 7 57s7 a7la71 sls1 ana12 szs2 a23a23 103 R356 33s3 a34a34 54s4 a45345 sss5 a56a56 104 Projecting the vector loop equation onto 356 and S6X356 gives R633 356 R356 356 R6 ltS6Xa56gt R3 Sam Evaluating the scalar products of these equations gives H1c6 H2s6 J10 H2c6H1s6 J20 105 106 1017 1018 where H1 and H2 contain the sines and cosines of 01 and 02 and J1 and J2 contain the sines and cosines of 04 and 05 H1 H2 J1 J2 Adding 1017 to X6 times 1018 gives H1 06 S X H2 06X6 6 J2X6 J1 0 Subtracting X6 times 1017 from 1018 gives H1 56 06X6 H2 06 S X J1X6 J2 0 The following trig identities are given 56 06X6 X6 06 S X 1 1015 1016 107 108 1019 1020 1021 1022 Substituting these trig identities into 1019 and 1020 gives H2 J2 X6 H1 J1 0 H1 J1 X6 H2 J2 0 1023 Derivation of the second pair of equations Using equation 102 the following equations can be written gait Rwgtlt83 so 12623 8002 as 023 R339 gtltSg so R356 60023 as 1 3016723 39R63923XS3 39Ss X355 R6723 39S3R6723 39 S6 X355 2 023 R339 gtltSg 86 Xasa R356 60023 86 gtlta56gt These two equations can be factored as H306 H456 J320 H3 6H406 J40 1023 1024 1034 1035 1037 1038 where H3 and H4 contain the sines and cosines of 01 and 02 and J3 and J4 contain the sines and cosines of 04 and 05 H3 H4 J3 J4 10267 10268 10160 10161 Using the trig identities of1021 and 1022 allows 1037 and 1038 to be rewritten as H4J4 X Hs J30 lo44 Hs l Js X6H4J40 lo45 Equations 1017 1018 1044 and 1045 can be factored as M102 N152 01 X6 P102 Q1 52 R1 M1 c4 N1 s4 01 X6 Pf c4 Qi s4 Ri i 14 10277 where Mi MU c1 M13 s1 M13 etc 10278 Mi MU c5 MLZ s5 ML3 etc 10279 All the terms MU through R L3 are known in terms of the constant mechanism parameters and the input angle 07 They are defined in 10280 10281 10282 and 10283 The four equations represented by 10277 may be written in matrix format as follows T1 a T2 b 10284 where MLZ M13 N11 N1 2 N13 011 01 2 01 3 P11 P1 2 P13 Q11 Q11 Q13 R1 R1 2 M22 M23 N21 N2 2 N23 021 022 023 P21 P2 2 P23 Q21 Q22 Q23 R21 R22 M32 M33 N11 N32 N33 031 032 033 P11 P3 2 P33 Q31 Q32 Q33 R3 R32 M42 M43 N41 N42 N43 041 042 04 3 Eu P4 2 P43 Q41 Q42 Q43 RAJ R42 10285 71576 P u T a C102X6 S102X6 C2X6 C152X6 SIS2X6 S2X6 ch6 S1X6 X6 0102 s102 02 0152 S152 52 C1 51 1 1 0286 Mi1 Mi2 M13 N1 Ni2 Ni3 Oi1 Oi2 Oi3 P111 P112 P1 3 Ri1 Rig R13 T2 N2 1 Nivu N123 0121 021 023 P213 P22 P2 3 R2 R21 R23 M33 M31 M33 N3 N31 N33 031 032 033 P3 P31 P3 3 Q3 Q31 Q33 R33 R31 R33 LMVAJ M41 M43 N41 N42 N43 041 042 043 P43 P41 P4 3 Q41 Q42 Q43 Riu Rim R431 1 0287 b C4C5X6 C4S5X6 C4X6 S4C5X6 S4S5X6 S4X6 C5X6 S5X6 X6 C4C5 C4S5 C4 S4C5 5455 S4 C5 S5T 1 0288 Note that all the terms in T1 and T2 are known in terms of the given mechanism parameters and the angle 07 It is still necessary to replace the vector b in terms of the sines and cosines of 01 and 02 and the tanhalf angle X6 1029 Elimination of 04 and 05 to obtain inputoutput equation 16 equations are obtained in terms of the sines and cosines of 01 02 04 and 05 and the tanhalf angle X6 These equations can be written in matriX format as A b c 10404 where A is a 16gtlt16 matrix b and c are length 16 vectors A is defined in 10405 b was defined previously in 10288 and c is defined in 10407 All terms in A are known The vector c contains the sines and cosines of 01 and 02 and the tanhalf angle X6

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