6. Representing Rotation

Transcription

1 6. Representing Rotation Mechanics of Manipulation Matt Mason Carnegie Mellon Lecture 6. Mechanics of Manipulation p.1

2 Lecture 6. Representing Rotation. Chapter 1 Manipulation Case 1: Manipulation by a human Case 2: An automated assembly system Issues in manipulation A taxonomy of manipulation techniques Bibliographic notes 8 Exercises 8 Chapter 2 Kinematics Preliminaries Planar kinematics Spherical kinematics Spatial kinematics Kinematic constraint Kinematic mechanisms Bibliographic notes 36 Exercises 37 Chapter 3 Kinematic Representation Representation of spatial rotations Representation of spatial displacements Kinematic constraints Bibliographic notes 72 Exercises 72 Chapter 4 Kinematic Manipulation Path planning Path planning for nonholonomic systems Kinematic models of contact Bibliographic notes 88 Exercises 88 Chapter 5 Rigid Body Statics Forces acting on rigid bodies Polyhedral convex cones Contact wrenches and wrench cones Cones in velocity twist space The oriented plane Instantaneous centers and Reuleaux s method Line of force; moment labeling Force dual Summary Bibliographic notes 117 Exercises 118 Chapter 6 Friction Coulomb s Law Single degree-of-freedom problems Planar single contact problems Graphical representation of friction cones Static equilibrium problems Planar sliding Bibliographic notes 139 Exercises 139 Chapter 7 Quasistatic Manipulation Grasping and fixturing Pushing Stable pushing Parts orienting Assembly Bibliographic notes 173 Exercises 175 Chapter 8 Dynamics Newton s laws A particle in three dimensions Moment of force; moment of momentum Dynamics of a system of particles Rigid body dynamics The angular inertia matrix Motion of a freely rotating body Planar single contact problems Graphical methods for the plane Planar multiple-contact problems Bibliographic notes 207 Exercises 208 Chapter 9 Impact A particle Rigid body impact Bibliographic notes 223 Exercises 223 Chapter 10 Dynamic Manipulation Quasidynamic manipulation Brie y dynamic manipulation Continuously dynamic manipulation Bibliographic notes 232 Exercises 235 Appendix A Infinity 237 Lecture 6. Mechanics of Manipulation p.2

3 Outline. Generalities Axis-angle Rodrigues s formula Rotation matrices Euler angles Lecture 6. Mechanics of Manipulation p.3

4 Why representing rotations is hard. Rotations do not commute. The topology of spatial rotations does not permit a smooth embedding in Euclidean three space. Lecture 6. Mechanics of Manipulation p.4

5 Choices More than three numbers Rotation matrices Unit quaternions. (aka Euler parameters) Many-to-one Axis times angle (matrix exponential) Unsmooth and many-to-one Euler angles Unsmooth and many-to-one and more than three numbers Axis-angle Lecture 6. Mechanics of Manipulation p.5

6 Axis-angle Recall Euler s theorem: every spatial rotation leaves a line of fixed points: the rotation axis. Let O, ˆn, θ, be... Let rot(ˆn, θ) be the corresponding rotation. Many to one: rot( ˆn, θ) = rot(ˆn, θ) rot(ˆn, θ + 2kπ) = rot(ˆn, θ), for any integer k. When θ = 0, the rotation axis is indeterminate, giving an infinityto-one mapping. n Lecture 6. Mechanics of Manipulation p.6

7 Representation What do we want from a representation? For a start: Rotate points; Rodrigues s formula Compose rotations; Using axis-angle? Ugh. (Convert to other representations.) Lecture 6. Mechanics of Manipulation p.7

8 Rodrigues s formula Others derive Rodrigues s formula using rotation matrices, missing the geometrical aspects. n n x n Given point x, decompose into components parallel and perpendicular to the rotation axis x = ˆn(ˆn x) ˆn (ˆn x) n n x n x x x Only x is affected by the rotation, yielding Rodrigues s formula: O x = ˆn(ˆn x)+sin θ (ˆn x) cos θ ˆn (ˆn x) A common variation: x = x+(sin θ) ˆn x+(1 cos θ) ˆn (ˆn x) Lecture 6. Mechanics of Manipulation p.8

9 Rotation matrices Choose O on rotation axis. Choose frame (û 1, û 2, û 3 ). Let (û 1, û 2, û 3) be the image of that frame. Write the û i vectors in û i coordinates, and collect them in a matrix: û 1 = a 11 a 21 = û 1 û 1 û 2 û 1 û 2 = a 31 a 12 a 22 = û 3 û 1 û 1 û 2 û 2 û 2 û 3 = a 32 a 13 a 23 = û 3 û 2 û 1 û 3 û 2 û 3 a 33 û 3 û 3 Lecture 6. Mechanics of Manipulation p.9

10 So many numbers A rotation matrix has nine numbers, but spatial rotations have only three degrees of freedom, leaving six excess numbers... There are six constraints that hold among the nine numbers. û 1 = û 2 = û 3 = 1 û 3 = û 1 û 2 i.e. the û i are unit vectors forming a right-handed coordinate system. Such matrices are called orthonormal or rotation matrices. Lecture 6. Mechanics of Manipulation p.10

11 Rotating a point Let (x 1, x 2, x 3 ) be coordinates of x in frame (û 1, û 2, û 3 ). Then x is given by the same coordinates taken in the (û 1, û 2, û 3) frame: x =x 1 û 1 + x 2 û 2 + x 3 û 3 =x 1 Aû 1 + x 2 Aû 2 + x 3 Aû 3 =A(x 1 û 1 + x 2 û 2 + x 3 û 3 ) =Ax So rotating a point is implemented by ordinary matrix multiplication. Lecture 6. Mechanics of Manipulation p.11

12 Rotating a point Let A and B be coordinate frames. Notation: x a point x A x a geometrical vector, directed from an origin O to the point x or, a vector of three numbers, representing x in an unspecified frame a vector of three numbers, representing x in the A frame Let B AR be the rotation matrix that rotates frame B to frame A. Then (see previous slide) B AR represents the rotation of the point x: B x = B AR B x Note presuperscripts all match. Both points, and xform, must be written in same coordinate frame. Lecture 6. Mechanics of Manipulation p.12

13 Coordinate transform There is another use for B A R: A x and B x represent the same point, in frames A and B resp. To transform from A to B: B x = B AR A x For coord xform, matrix subscript and vector superscript cancel. Rotation from B to A is the same as coordinate transform from A to B. Lecture 6. Mechanics of Manipulation p.13

14 Example rotation matrix B AR = = ( B x A B y A B z A ) How to remember what B A R does? Pick a coordinate axis and see. The x axis isn t very interesting, so try y: = Lecture 6. Mechanics of Manipulation p.14

15 Nice things about rotation matrices Composition of rotations: {R 1 ; R 2 } = R 2 R 1. ({x; y} means do x then do y.) Inverse of rotation matrix is its transpose B A R 1 = A B R = B A RT. Coordinate xform of a rotation matrix: B R = B AR A R A BR Lecture 6. Mechanics of Manipulation p.15

16 Converting rot(ˆn, θ) to R Ugly way: define frame with ẑ aligned with ˆn, use coordinate xform of previous slide. Keen way: Rodrigues s formula! x = x + (sin θ) ˆn x + (1 cos θ) ˆn (ˆn x) Define cross product matrix N: N = 0 n 3 n 2 n 3 0 n 1 n 2 n 1 0 so that Nx = ˆn x Lecture 6. Mechanics of Manipulation p.16

17 ... using Rodrigues s formula... Substituting the cross product matrix N into Rodrigues s formula: x = x + (sin θ)nx + (1 cos θ)n 2 x Factoring out x R = I + (sin θ)n + (1 cos θ)n 2 That s it! Rodrigues s formula in matrix form. If you want to you could expand it: n (1 n 2 1)cθ n 1 n 2 (1 cθ) n 3 sθ n 1 n 3 (1 cθ) + n 2 sθ n 1 n 2 (1 cθ) + n 3 sθ n (1 n 2 2)cθ n 2 n 3 (1 cθ) n 1 sθ n 1 n 3 (1 cθ) n 2 sθ n 2 n 3 (1 cθ) + n 1 sθ n (1 n 2 3)cθ where cθ = cos θ and sθ = sin θ. Ugly. Lecture 6. Mechanics of Manipulation p.17

18 Rodrigues s formula for differential rotations Consider Rodrigues s formula for a differential rotation rot(ˆn, dθ). so x =(I + sin dθn + (1 cos dθ)n 2 )x =(I + dθn)x dx =Nx dθ =ˆn x dθ It follows easily that differential rotations are vectors: you can scale them and add them up. We adopt the convention of representing angular velocity by the unit vector ˆn times the angular velocity. Lecture 6. Mechanics of Manipulation p.18

19 Converting from R to rot(ˆn, θ)... Problem: ˆn isn t defined for θ = 0. We will do it indirectly. Convert R to a unit quaternion (next lecture), then to axis-angle. Lecture 6. Mechanics of Manipulation p.19

20 Euler angles Three numbers to describe spatial rotations. ZY Z convention: (α, β, γ) rot(γ, ẑ ) rot(β, ŷ ) rot(α, ẑ) z z y Can we represent an arbitrary rotation? Rotate α about ẑ until ŷ ẑ ; Rotate β about ŷ until ẑ ẑ ; Rotate γ about ẑ until ŷ = ŷ. Note two choices for ŷ except sometimes infinite choices. x Lecture 6. Mechanics of Manipulation p.20 x y

21 From (α, β, γ) to R Expand rot(α, ẑ) rot(β, ŷ) rot(γ, ẑ) (Why is that the right order?) cα sα 0 sα cα 0 cβ 0 sβ cγ sγ 0 sγ cγ sβ 0 cβ cα cβ cγ sα sγ cα cβ sγ sα cγ cα sβ = sα cβ cγ + cα sγ sα cβ sγ + cα cγ sα sβ sβ cγ sβ sγ cβ (1) Lecture 6. Mechanics of Manipulation p.21

22 From R to (α, β, γ) the ugly way Case 1: r 33 = 1, β = π. α γ is indeterminate. R = cos(α + γ) sin(α + γ) 0 sin(α + γ) cos(α + γ) Case 2: r 33 = 1, β = π. α + γ is indeterminate. R = cos(α γ) sin(α γ) 0 sin(α γ) cos(α γ) For generic case: solve 3rd column for β. (Sign is free choice.) Solve third column for α and third row for γ.... but there are numerical issues... Lecture 6. Mechanics of Manipulation p.22

23 From R to (α, β, γ) the clean way Let Then σ = α + γ δ = α γ r 22 + r 11 = cos σ(1 + cos β) r 22 r 11 = cos δ(1 cos β) r 21 + r 12 = sin δ(1 cos β) r 21 r 12 = sin σ(1 + cos β) (No special cases for cos β = ±1?) Solve for σ and δ, then for α and γ, then finally β = tan 1 Lecture 6. (r 13 cos α + r 23 sin α, r 33 ) Mechanics of Manipulation p.23

24 Next: Quaternions. Chapter 1 Manipulation Case 1: Manipulation by a human Case 2: An automated assembly system Issues in manipulation A taxonomy of manipulation techniques Bibliographic notes 8 Exercises 8 Chapter 2 Kinematics Preliminaries Planar kinematics Spherical kinematics Spatial kinematics Kinematic constraint Kinematic mechanisms Bibliographic notes 36 Exercises 37 Chapter 3 Kinematic Representation Representation of spatial rotations Representation of spatial displacements Kinematic constraints Bibliographic notes 72 Exercises 72 Chapter 4 Kinematic Manipulation Path planning Path planning for nonholonomic systems Kinematic models of contact Bibliographic notes 88 Exercises 88 Chapter 5 Rigid Body Statics Forces acting on rigid bodies Polyhedral convex cones Contact wrenches and wrench cones Cones in velocity twist space The oriented plane Instantaneous centers and Reuleaux s method Line of force; moment labeling Force dual Summary Bibliographic notes 117 Exercises 118 Chapter 6 Friction Coulomb s Law Single degree-of-freedom problems Planar single contact problems Graphical representation of friction cones Static equilibrium problems Planar sliding Bibliographic notes 139 Exercises 139 Chapter 7 Quasistatic Manipulation Grasping and fixturing Pushing Stable pushing Parts orienting Assembly Bibliographic notes 173 Exercises 175 Chapter 8 Dynamics Newton s laws A particle in three dimensions Moment of force; moment of momentum Dynamics of a system of particles Rigid body dynamics The angular inertia matrix Motion of a freely rotating body Planar single contact problems Graphical methods for the plane Planar multiple-contact problems Bibliographic notes 207 Exercises 208 Chapter 9 Impact A particle Rigid body impact Bibliographic notes 223 Exercises 223 Chapter 10 Dynamic Manipulation Quasidynamic manipulation Brie y dynamic manipulation Continuously dynamic manipulation Bibliographic notes 232 Exercises 235 Appendix A Infinity 237 Lecture 6. Mechanics of Manipulation p.24