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Course content. MM1 Basic geometry and rotations MM2 Rotation parameters and kinematics MM3 Rotational Dynamics MM4 Manipulator Kinematics MM5 Manipulator Dynamics. Area of use. Roll. Pitch. Yaw. Transformation T from one CS to another: T: R 3 R 3

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Course content

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  1. Course content • MM1 Basic geometry and rotations • MM2 Rotation parameters and kinematics • MM3Rotational Dynamics • MM4 Manipulator Kinematics • MM5 Manipulator Dynamics MMS I, Lecture 1

  2. Area of use Roll Pitch Yaw MMS I, Lecture 1

  3. Transformation T from one CS to another: T: R3 R3 Tv·Tw = v·w(preserve distance) Tv x Tw = v x w(preserve angle) T(v x w ) = v x w Content off to day • Vectors and coordinatsystems • Direct cosinus matrices (DCM) • Dirivitives in rotating coordinatsystems (Transport theorem) Ortogonal coordinat systems: MMS I, Lecture 1

  4. Vectors R3 OP = ( p1,p2, p3 )T = (x1,x2,x3)T 3 x = x1ê1 + x2ê2 + x3ê3 ≡ ∑ xiêi xi = x· êi i=1 P x1 x2 x3 x = [ê1ê2ê3 ] ê3 x3 ê2 ê1 O x1 x2 Basic Geometry {A} x For ortogonal coordinat cystems: êi· êi = 1 ; ê1xê2 = ê3 êixêi = 0 ê1xê3 = - ê2 ê2xê3 = ê1 MMS I, Lecture 1

  5. a(t) v(t) Finish f(s(t)) Kinematics Definition: ”Description of motion regardless of masses, forces and torques” ”Geometric description over time” Start no forces no torques Missing?? MMS I, Lecture 1

  6. a(t) Start v(t) Finish f(s(t)) v(t) F a(t) s(t) · Dynamic Kinematics · · ω(t) θ(t)θ(t) θ(t) N ω(t) ω(t) · · Dynamics Definition: ”Description of motion depending on masses M, inertia I, forces F and torques N ” F ω(t) I M N MMS I, Lecture 1

  7. r r {A} {U} â3 û1 û2 û3 û1 û2 û3 û1 û3 â2 â1 û2 â1 ·û1 â1 ·û2 â1·û3 â2 ·û1 â2 ·û2 â2 ·û3 â3 ·û1 â3 ·û2 â3 ·û3 = r1û1+ r2û2+r3û3 = r’1â1+ r’2â2+r’3â3 CAU = CAU is the rotationsmatrix fra A U Rotation matrix Direct cosine â1 = C11û1 + C12û2 + C13û3 â2 = C21û1 + C22û2 + C23û3 â3 = C31û1 + C32û2 + C33û3 â1C11 C12 C13 â2 = C21 C22 C23 â3C31 C32 C33 = CAU MMS I, Lecture 1

  8. â1 ·û1â1 ·û2â1·û3 â2 ·û1â2 ·û2â2 ·û3 â3 ·û1â3 ·û2â3 ·û3 û1 · â1û1 ·â2û1·â3 û2 ·â1û2 ·â2û2 ·â3 û3 ·â1û3 ·â2û3 ·â3 CUA= ↨ T CAU = CUA Direct cosine cont. Proporties ofCAU: T • CAU · CAU= I • CAU = CAU • det(CAUCAU ) = det I = det(CAU)2=1 ↔ det(CAU ) = + -1 • (âi ·û1)2 +(âi · û2)2 + (âi·û3)2 = 1 i = (1,2,3,) -1 T T = CAU MMS I, Lecture 1

  9. con θ3 sin θ3 0 – sin θ3 con θ3 0 0 0 1 C3 (θ3) = θ3 con θ2 0 – sin θ2 0 1 0 sin θ2 0 con θ2 2 C2 (θ2) = 1 2 1 0 0 0 con θ1 sin θ1 0 – sin θ1con θ1 C1 (θ1) = θ1 θ2 Euler angels (3-2-1) {A} 3 {V} {U} {W} 1 1 CUA = CUVCVWCWA = C1(θ1)·C2(θ2)·C3(θ3) MMS I, Lecture 1

  10. Euler angels (2-3-1) NASA c2c3 s3 - s2c3 -c1c2s3 + s1s2 c1c3 c1s2s3 + s1c2 s1c2s3 + c1s2 -s1c2 -s1s2s3 + c1c2 Cθ1Cθ3Cθ2 = Roll Pitch Yaw Euler angels (3-1-3) Orbit planes cψcφ - sψsφcθsφcψ+cφcθsψsθsψ -cφsψ-sφcθcψ-sφsψ+cφcθcψsθcψ sφsθ-cφsθcθ 3 CψCθCφ = 3 θ φ ψ 1 1 Euler angels (3-2-1) cont. c2c3 c2s3 -s2 s1s2c3 – c1c3 s1s2s3 – c1s3 s1c2 c1s2c3 + s1s3 c1s2s3 – s1c3 c1c2 Cθ1Cθ2Cθ3 = MMS I, Lecture 1

  11. ω {A} â2 {U} û3 â1 â1 û2 û1 ω = ω1â1+ω2â2 + ω3â3 · d d d d t θ t âi = ω x âi = âii = 1,2,3 U Vector differentiation Angular velocity: P x ê2 θ ω O ê1 ω = Something rotten! MMS I, Lecture 1

  12. ωAU A U A d d d d d d r t r r t t Transportation Theorem {A} â2 {U} û3 â1 P r â1 û2 û1 r = r1â1+r2â2 + r3â3 · · · · · · · = r = r1â1 + r2â2 + r3â3 + r1â1+r2â2 + r3â3 = + r1ω x â1 + r2ω x â2 + r3ω x â3 = + ωAU x r A V.I. MMS I, Lecture 1

  13. · · A A A U d d d d d d d d d d d d d d t r t t r t t r r t t Transportation Theorem = + ωAU x r A · · = + ωAU x + ωAU x r A + ωAU x r A + ωAU x (ωAU x r A) = rA +2 ωAU x r A + ωAU x r A + ωAU x (ωAU x r A) · · MMS I, Lecture 1

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