1. 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

2. MMS I, Lecture 12 Area of use Roll Pitch Yaw

3. 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

4. MMS I, Lecture 14 Basic Geometry P ê3ê3 ê2ê2 ê1ê1 O x3x3 x2x2 x1x1 Vectors R 3 OP = ( p 1, p 2, p 3 ) T = (x 1,x 2,x 3 ) T x1x2x3x1x2x3 x = [ê 1 ê 2 ê 3 ] For ortogonal coordinat cystems: ê i · ê i = 1 ; ê 1 xê 2 = ê 3 ê i xê i = 0 ê 1 xê 3 = - ê 2 ê 2 xê 3 = ê 1 {A} x x = x 1 ê 1 + x 2 ê 2 + x 3 ê 3 ≡ ∑ x i ê i x i = x · ê i i=1 3

5. 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??

6. 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 ω(t) · Dynamic F N v(t) ω(t) Kinematics s(t) θ(t) θ(t) θ(t) a(t) ω(t) · ···

7. MMS I, Lecture 17 Rotation matrix Direct cosine â3â3 â2â2 â1â1 {A} {U} û3û3 û2û2 û1û1 â 1 = C 11 û 1 + C 12 û 2 + C 13 û 3 â 2 = C 21 û 1 + C 22 û 2 + C 23 û 3 â 3 = C 31 û 1 + C 32 û 2 + C 33 û 3 â 1 C 11 C 12 C 13 â 2 = C 21 C 22 C 23 â 3 C 31 C 32 C 33 û1û2û3û1û2û3 = C AU û1û2û3û1û2û3 â 1 · û 1 â 1 · û 2 â 1 · û 3 â 2 · û 1 â 2 · û 2 â 2 · û 3 â 3 · û 1 â 3 · û 2 â 3 · û 3 C AU = r r= r 1 û 1 + r 2 û 2 + r 3 û 3 = r’ 1 â 1 + r’ 2 â 2 + r’ 3 â 3 C AU is the rotationsmatrix fra A U

8. 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 ) = + - 1 4. (â i · û 1 ) 2 + (â i · û 2 ) 2 + (â i · û 3 ) 2 = 1 i = (1,2,3,) T â 1 · û 1 â 1 · û 2 â 1 · û 3 â 2 · û 1 â 2 · û 2 â 2 · û 3 â 3 · û 1 â 3 · û 2 â 3 · û 3 T T = C AU C UA = û 1 · â 1 û 1 · â 2 û 1 · â 3 û 2 · â 1 û 2 · â 2 û 2 · â 3 û 3 · â 1 û 3 · â 2 û 3 · â 3 ↨ C AU = C UA T

9. MMS I, Lecture 19 Euler angels (3-2-1) 1 2 3 θ3θ3 1 2 θ2θ2 1 θ1θ1 con θ 3 sin θ 3 0 – sin θ 3 con θ 3 0 0 0 1 C 3 (θ 3 ) = con θ 2 0 – sin θ 2 0 1 0 sin θ 2 0 con θ 2 C 2 (θ 2 ) = 1 0 0 0 con θ 1 sin θ 1 0 – sin θ 1 con θ 1 C 1 (θ 1 ) = C UA = C UV C VW C WA = C 1 (θ 1 )·C 2 (θ 2 )·C 3 (θ 3 ) {A} {W} {V} {U}

10. 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 ψ c φ - s ψ s φ c θ s φ c ψ +c φ c θ s ψ s θ s ψ -c φ s ψ -s φ c θ c ψ -s φ s ψ +c φ c θ c ψ s θ c ψ s φ s θ -c φ s θ c θ 1 1 3 3 ψφ θ C ψ C θ C φ = 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 θ 1 C θ 3 C θ 2 = C θ 1 C θ 2 C θ 3 =

11. MMS I, Lecture 111 Vector differentiation Angular velocity: ê1ê1 ê2ê2 θ ω x P O ω = dθ dt = ω x â i = â i i = 1,2,3 dâiâi dt â1â1 â1â1 û3û3 â2â2 ω û2û2 û1û1 {A} {U} ω = ω 1 â 1 +ω 2 â 2 + ω 3 â 3 U · Something rotten!

12. MMS I, Lecture 112 Transportation Theorem â1â1 â1â1 û3û3 â2â2 ω AU û2û2 û1û1 {A} {U} P r r 1 â 1 +r 2 â 2 + r 3 â 3 r = = r = r 1 â 1 + r 2 â 2 + r 3 â 3 + r 1 â 1 +r 2 â 2 + r 3 â 3 = + r 1 ω x â 1 + r 2 ω x â 2 + r 3 ω x â 3 = + ω AU x r A V.I. dr dt A dr dt U · · · ·· · · dr dt A

13. MMS I, Lecture 113 Transportation Theorem dr dt A = + ω AU x r A dr dt U d dt d dt = + ω AU x + ω AU x r A + ω AU x r A + ω AU x (ω AU x r A ) = r A + 2 ω AU x r A + ω AU x r A + ω AU x (ω AU x r A ) dr dt A d dt dr dt A ·· · · · ·