1 / 13

Course content

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

judd
Télécharger la présentation

Course content

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related