MMS I, Lecture 11 Course content MM1 Basic geometry and rotations MM2 Rotation parameters and kinematics MM3 Rotational Dynamics MM4 Manipulator Kinematics MM5 Manipulator Dynamics MMS I, Lecture 12 Area of use Roll Pitch Yaw MMS I, Lecture 13 Content off to day Vectors and coordinatsystems Direct cosinus matrices (DCM) Dirivitives in rotating coordinatsystems (Transport theorem) Ortogonal coordinat systems: Transformation T from one CS to another: T: R 3 R 3 Tv Tw = v w (preserve distance) Tv x Tw = v x w (preserve angle) T(v x w ) = v x w MMS I, Lecture 14 Basic Geometry P 33 22 11 O x3x3 x2x2 x1x1 Vectors R 3 OP = ( p 1, p 2, p 3 ) T = (x 1,x 2,x 3 ) T x1x2x3x1x2x3 x = [ 1 2 3 ] For ortogonal coordinat cystems: i i = 1 ; 1 x 2 = 3 i x i = 0 1 x 3 = - 2 2 x 3 = 1 {A} x x = x 1 1 + x 2 2 + x 3 3 x i i x i = x i i=1 3 MMS I, Lecture 15 Kinematics Definition: Description of motion regardless of masses, forces and torques Geometric description over time Start Finish f(s(t)) v(t) a(t) no forces no torques Missing?? MMS I, Lecture 16 Dynamics Definition: Description of motion depending on masses M, inertia I, forces F and torques N Start Finish f(s(t)) v(t) a(t) M F N I (t) Dynamic F N v(t) (t) Kinematics s(t) (t) (t) (t) a(t) (t) MMS I, Lecture 17 Rotation matrix Direct cosine 33 22 11 {A} {U} 33 22 11 1 = C 11 1 + C 12 2 + C 13 3 2 = C 21 1 + C 22 2 + C 23 3 3 = C 31 1 + C 32 2 + C 33 3 1 C 11 C 12 C 13 2 = C 21 C 22 C 23 3 C 31 C 32 C 33 123123 = C AU 123123 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 C AU = r r= r 1 1 + r 2 2 + r 3 3 = r 1 1 + r 2 2 + r 3 3 C AU is the rotationsmatrix fra A U MMS I, Lecture 18 Direct cosine cont. Proporties of C AU : 1. C AU C AU = I 2. C AU = C AU 3. det (C AU C AU ) = det I = det (C AU ) 2 = 1 det (C AU ) = ( i 1 ) 2 + ( i 2 ) 2 + ( i 3 ) 2 = 1 i = (1,2,3,) T 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 T T = C AU C UA = 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 C AU = C UA T MMS I, Lecture 19 Euler angels (3-2-1) 33 1 2 22 1 11 con 3 sin 3 0 sin 3 con C 3 ( 3 ) = con 2 0 sin sin 2 0 con 2 C 2 ( 2 ) = con 1 sin 1 0 sin 1 con 1 C 1 ( 1 ) = C UA = C UV C VW C WA = C 1 ( 1 )C 2 ( 2 )C 3 ( 3 ) {A} {W} {V} {U} MMS I, Lecture 110 Euler angels (3-2-1) cont. c 2 c 3 c 2 s 3 -s 2 s 1 s 2 c 3 c 1 c 3 s 1 s 2 s 3 c 1 s 3 s 1 c 2 c 1 s 2 c 3 + s 1 s 3 c 1 s 2 s 3 s 1 c 3 c 1 c 2 Euler angels (3-1-3) Orbit planes c c - s s c s c +c c s s s -c s -s c c -s s +c c c s c s s -c s c C C C = Euler angels (2-3-1) NASA c 2 c 3 s 3 - s 2 c 3 -c 1 c 2 s 3 + s 1 s 2 c 1 c 3 c 1 s 2 s 3 + s 1 c 2 s 1 c 2 s 3 + c 1 s 2 -s 1 c 2 -s 1 s 2 s 3 + c 1 c 2 Pitch Yaw Roll C 1 C 3 C 2 = C 1 C 2 C 3 = MMS I, Lecture 111 Vector differentiation Angular velocity: 11 22 x P O = d dt = x i = i i = 1,2,3 dii dt 11 11 33 22 22 11 {A} {U} = 1 1 + 2 2 + 3 3 U Something rotten! MMS I, Lecture 112 Transportation Theorem 11 11 33 22 AU 22 11 {A} {U} P r r 1 1 +r 2 2 + r 3 3 r = = r = r 1 1 + r 2 2 + r 3 3 + r 1 1 +r 2 2 + r 3 3 = + r 1 x 1 + r 2 x 2 + r 3 x 3 = + AU x r A V.I. dr dt A dr dt U dr dt A MMS I, Lecture 113 Transportation Theorem dr dt A = + AU x r A dr dt U d dt d dt = + AU x + AU x r A + AU x r A + AU x ( AU x r A ) = r A + 2 AU x r A + AU x r A + AU x ( AU x r A ) dr dt A d dt dr dt A