Notes_04_05 page 1 of 10 Two-dimensional position, velocity and acceleration solutions Kinematically driven motion 1.0 Initialize 1.1 constants 1.2 global locations of fixed points r1 P 1.3 local blueprint locations s i ' P 2.0 Initial estimates 2.1 global locations of origins ri 2.2 orientation angles i r2 2 2.3 assemble into q r3 3 3.0 Explicit time loop 4.0 Position solution 4.1 rip new values for ri and i from qk 4.2 form attitude matrices Ai Bi f̂ i ĝi 4.3 compute global locations for all points ri P 4.4 evaluate constraints KINEMATIC DRIVER 4.5 assemble position constraints KINEMATIC DRIVER Notes_04_05 4.6 evaluate partial derivatives ri KINEMATIC ri DRIVER i KINEMATIC 4.7 assemble Jacobian r 2 KINEMATIC 2 KINEMATIC q 2 DRIVER r 2 DRIVER page 2 of 10 i DRIVER r 3 KINEMATIC 3 KINEMATIC 3 DRIVER r 3 DRIVER 4.8 Newton-Raphson update qk 1 qk q 1 4.9 check convergence - repeat 4.1 through 4.9 if needed 5.0 Velocity solution 5.1 evaluate KINEMATIC 0 DRIVER 5.2 assemble KINEMATIC DRIVER 5.3 compute q q 1 6.0 Acceleration solution 6.1 evaluate KINEMATIC r2 2 q r3 3 for DRIVER 6.2 assemble KINEMATIC DRIVER 6.3 compute q q 1 for r2 2 q r3 3 Notes_04_05 page 3 of 10 Four Bar Example G3 13.151 B C 3 4 2 2 A = G2 AB = 30 cm BC = 60 cm CD = 45 cm AD = 90 cm BG3 = 23 cm DG4 = 24 cm 2 = 65 3 = 13.151 4 = - 65.173 G4 65.173 D D 1 65 1 B2 x2 y2 C4 C3 y3 2 x3 G2 G3 G3 G4 4 3 y4 G4 B3 x4 A2 D4 blueprint information s 2 ' A 0 0 s 2 ' B AB 0 CG 4 0 s 4 'C s 4 ' D DG 4 0 adjust r2 check r2 A r1A driver 2 2 _ START 2 t 2 BG 3 0 s 3 ' B r3 3 r4 r1 A 0 0 s 3 'C CG 3 0 r1 D AD 0 4 r3 B r2 B r4 C r3 C r4 D r1 D Notes_04_05 adjust x 2 y 2 r2 2 2 q 2 x 3 r q q 3 3 y 3 q 3 4 r 3 4 x 4 4 y 4 4 desire REV _ A REV _ B KINEMATIC REV _ C DRIVER REV _ D DRIVER 2 constraints page 4 of 10 r2 A r1 A r3 B r2 B r4 C r3 C r4 D r1 D 2 _ START 0 2 t r2 A 2 s 2 ' A r1 A r A s ' B r A s ' B 3 3 2 2 2 3 C C r4 A 4 s 4 ' r3 A 3 s 3 ' 0 r4 A 4 s 4 ' D r1 D t 2 2 _ START 2 Newton-Raphson position solution qk 1 qk q 1 k Notes_04_05 q JOINT 0 REV _ A 0 2 x1 0 2 x1 REV _ B KINEMATIC q q REV _ C 0 2 x1 DRIVER 0 REV _ D 2 x1 velocity solution q DRIVER page 5 of 10 DRIVER t 2 0 q q 1 8x1 2 acceleration solution q q REV rj P ri P 02x1 REV j 2 A j s j ' P i 2 A i s i 'P DRIVER tt 2 REV _ A 2 A 2 s 2 ' A 2 B 2 B REV _ B 3 A 3 s 3 ' 2 A 2 s 2 ' KINEMATIC 2 2 C C q q REV _ C 4 A 4 s 4 ' 3 A 3 s 3 ' DRIVER 2 4 A 4 s 4 ' D REV _ D DRIVER 0 q q 1 Notes_04_05 page 6 of 10 r2 A 2 s 2 ' A r1 A r A s ' B r A s ' B 3 3 2 2 2 3 C C r4 A 4 s 4 ' r3 A 3 s 3 ' 0 r4 A 4 s 4 ' D r1 D t 2 2 _ START 2 q I 2 I 2 0 2 x 2 0 2x 2 01x 2 B 2 s 2 ' A 0 2 x 2 0 2 x1 0 2 x 2 B 2 s 2 ' B I 2 B 3 s 3 ' B 0 2 x 2 0 2 x1 I 2 B 3 s 3 ' C I 2 0 2 x1 0 2 x 2 0 2 x1 I 2 1 01x 2 0 01x 2 0 2 x1 0 2 x1 B 4 s 4 'C D B 4 s 4 ' 0 Notes_04_05 page 7 of 10 using blueprint information q 0 0 1 0 1 0 1 0 BG 2 S G 2 0 1 BG 2 C G 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 BG 3S G 3 0 0 0 1 BG 3 C G 3 0 0 1 0 CG 3S G 3 1 0 0 1 CG 3 C G 3 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 CG 4 S G 4 CG 4 C G 4 DG 4 S G 4 DG 4 C G 4 0 0 evaluating terms q 0 0 0 0 0 1 0 1 0 0 0 0 1 0 27.189 1 0 5.233 0 1 12.678 0 1 22.397 0 0 0 1 0 8.418 0 0 1 36.030 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 19.059 8.817 21.782 10.077 0 0 AB = 30 cm BC = 60 cm CD = 45 cm AD = 90 cm BG3 = 23 cm DG4 = 24 cm 2 = 65 3 = 13.151 4 = - 65.173 Notes_04_05 page 8 of 10 transpose T q 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 27.189 12.678 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 1 0 0 0 5.233 22.397 8.418 36.030 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 19.059 8.817 21.782 10.077 0 0 0 0 0 0 from Notes_06_01 matrix static force analysis 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 F12x 0 y 0 1 0 0 0 0 0 F12 0 27.189 12.678 0 0 0 0 1 F23x 0 y 1 0 1 0 0 0 0 F23 102.589 0 1 0 1 0 0 0 F34x 109.433 5.233 22.396 8.418 36.030 0 0 0 F34x 0 0 0 1 0 1 0 0 F14x 157.973 0 0 0 1 0 1 0 F14y 122.656 0 0 19.059 8.817 21.782 10.077 0 T12 0 0 0 0 0 0 Notes_04_05 page 9 of 10 Four bar E r2 2 r q 3 3 r4 4 3 REV _ A REV _ B REV _ C REV _ D DRIVER C B 4 2 A D DRIVER 2 2 _ START 2 t DRIVER 4 4 _ CENTER 4 sin 2ft DRIVER 3 c 0 c 2 t c 2 t 2 c 3 t 3 DRIVER 4 3 cons tan t DRIVER x 3 x 3 _ START vx 3 t E E E General slider crank r2 2 r q 3 3 r4 4 REV _ A REV _ B REV _ C D C Q P r4 r4 parallel r1 r1 r4 C r1 P parallel r1 Q r1 P DRIVER 4 C 3 B 2 A In-line slider crank REV _ A REV _ B REV _ C y4 4 DRIVER B 2 A Q 3 4 C P C 4 D Notes_04_05 page 10 of 10 Inverted slider crank r2 2 r q 3 3 r4 4 REV _ A REV _ B REV _ C Q B D C r3 r3 parallel r4 r4 r3 B r4 C parallel r4 D r4 C DRIVER C D B 3 4 2 B A Q 3 Two link manipulator – joint interpolated motion r2 q 2 r3 3 REV _ A REV _ B 2 2 _ START 2 t START t 2 2 _ END 2 _ START / t C 3 2 3 END START / t A B 2 Two link manipulator – straight line interpolated motion r2 q 2 r3 3 REV _ A REV _ B x C 3 x C 3 _ START v C 3X y C 3 y C 3 _ START v C 3Y v C3X x C3 _ END x C3 _ START / t v C3X y C3 _ END y C3 _ START / t t t C 3 A 2 B
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