Basic Math Matrices & Transformations

Transcription

1 Basic Math Matrices & Transformations

2 Matrices An ordered table of numbers (or sub-tables) columns A{ 33 a a a a a a 0 2 a a a Rows Identity : I

3 Definition: a c b d Example: Matrix Addition (+) x u y v + a+x c+u b+y d+v Properties:. Closed result is always a matrix 2. Commutative A+B B+A 3. Associative (A+B)+C A+(B+C) 4. Identity A+0 A (null matrix) 5. Inverse A+A - 0 (negated matrix)

4 Matrix Multiplication (*) Definition: a c b d Example: x u y v * ax+cy au+cv bx+dy bu+dv Properties:. Closed result is always a matrix 2. NOT Commutative A*B! B*A 3. Associative (A*B)*C A*(B*C) 4. Identity A*I A (identity matrix) 5. Inverse A*A - I (difficult to obtain!) Distributive from either side (A+B)C AC + BC C(A+B) CA + CB

5 Examples: Vector is a special matrix v[x,y,z] row vector, x3 matrix v 2 column vector, 2x matrix

6 Vector addition: just like adding matrices Column vector a c b d x u y v + a+x c+u b+y d+v Row vector a c b d x u y v + a+x c+u b+y d+v

7 Matrix*Vector Multiplication: just like multiplying matrices v M x y a c b d Column vector on the right v 2 Mv a c bx d y ax cx by dy Result is a new column vector

8 Matrix*Vector Multiplication(example) v M v Mv 3 2 5**2 2* 3*2 7 8

9 Matrices as transformations Matrix-vector multiplication creates a new vector. New vector means new location in space. If all points of an object is multiplied by a matrix, then the whole object assumes a new position (and may be new shape and size, too). This is called transformation. v x y y M a c b d v v 2 v 2 Mv x

10 Notations in these course notes For space vectors and points we use column vectors. For transformation matrices use square matrices. To transform a point or vector, we left multiply the column vector with the matrix. The result is a column vector. v x y y M a c b d v v 2 v 2 Mv x

11 Matrices as transformations (examples) Transform 4 points [0,0], [,0],[,],[0,] using the matrix M. M v 0 2 Mv y v 0, 0 v v, v v y v, v v, v 2 v 2 x

12 v 2 Scaling S v S 5 0 S 2 v s x 0 0 s y *0*2 0* 3*2 S - /s x 0 0 /s y 5 6 S * S - I S s x s y s z 3D scaling matrix

13 Rotation about the Z axis y v 2 R a v v 2 v v 2 R -a a v x

14 Z-axis Rotation Matrix u and u _ u cos(a + b) sin(a + b) a b x cos(b) v v(x,y) and v ysin(b) cos(a) cos(b) sin(a) sin(b) sin(a) cos(b) + cos(a) sin(b)

15 Z-axis Rotation Matrix u and u _ u cos(a + b) sin(a + b) a b x cos(b) v v(x,y) and v ysin(b) x cos(a) cos(b) sin(a) sin(b) sin(a) cos(b) + cos(a) sin(b) y

16 Z-axis Rotation Matrix u and u _ u cos(a + b) sin(a + b) a b x cos(b) v v(x,y) and v ysin(b) x cos(a) cos(b) sin(a) sin(b) sin(a) cos(b) + cos(a) sin(b) y cos(a) x sin(a) y sin(a) x + cos(a) y cos(a) sin(a) sin(a) cos(a) x y _ v R a cos(a) sin(a) sin(a) cos(a) u R a v

17 Inverse of Z-axis Rotation Matrix R a cos(a) sin(a) sin(a) cos(a) R - a cos(- a) sin(- a) sin(- a) cos(- a) R - a cos(a) sin(a) -sin(a) cos(a) - R a R - a The proof: R a * R - a I Series of consecutive rotations R (abg) R a R b R c

18 Rotation Matrices around x, y and z Rotation by a around z axis in 3D notation, as 3x3 matrix R cos(a) sin(a) 0 a sin(a) cos(a) v 2 R a v When applied on vector v, it does not change z coordinate. Try it! Rotation matrices around x, y and z axes in 3D

19 The Generalized Rotation Matrix e e 2 e 3 are orthonormal base vectors v v 2 v 3 are orthonormal base vectors They are centered in a common point We look for the matrix that rotates one orthonormal vector base to the other about this common point e 3 e 2 e

20 Generalized rotation matrix Express v with the e e 2 e 3 base vectors e 3 e 2 (e 2 v ) e 2 (e 3 v ) e 3 e 2 v e 3 v e (e v ) e e v v (e v ) e + (e 2 v ) e 2 + (e 3 v ) e 3 (Eq. )

21 Generalized rotation matrix Similarly, express v v 2 v 3 vectors with e e 2 e 3 base vectors e 3 e 2 e v (e v ) e + (e 2 v ) e 2 + (e 3 v ) e 3 v 2 (e v 2 ) e + (e 2 v 2 ) e 2 + (e 3 v 2 ) e 3 v 3 (e v 3 ) e + (e 2 v 3 ) e 2 + (e 3 v 3 ) e 3 (Eq. 2)

22 Generalized rotation matrix P e 3 e 2 (Expressed v v 2 v 3 with (e e 2 e 3 ) orthonormal base vectors) e v (e v ) e + (e 2 v ) e 2 + (e 3 v ) e 3 v 2 (e v 2 ) e + (e 2 v 2 ) e 2 + (e 3 v 2 ) e 3 (Eq. 3) v 3 (e v 3 ) e + (e 2 v 3 ) e 2 + (e 3 v 3 ) e 3 Express P in both coordinate systems P r e + s e 2 + te 3 (r, s, t are coordinates in the e system) P x v + yv 2 + zv 3 (x, y, z are coordinates in the v system) (Eq. 4) (Eq. 5)

23 Generalized rotation matrix We replace v, v 2 and v 3 from Eq. 3 into Eq. 5: v (e v ) e + (e 2 v ) e 2 + (e 3 v ) e 3 v 2 (e v 2 ) e + (e 2 v 2 ) e 2 + (e 3 v 2 ) e 3 v 3 (e v 3 ) e + (e 2 v 3 ) e 2 + (e 3 v 3 ) e 3 P x v + yv 2 + zv 3 (x, y, z are coordinate values) Eq. 5 becomes the following P x(e v ) e + x(e 2 v ) e 2 + x(e 3 v ) e 3 + y(e v 2 ) e + y(e 2 v 2 ) e 2 + y(e 3 v 2 ) e 3 + z(e v 3 ) e + z(e 2 v 3 ) e 2 + z(e 3 v 3 ) e 3 Group the e, e 2 and e 3 members together P x(e v ) e + y(e v 2 ) e + z(e v 3 ) e + x(e 2 v ) e 2 + y(e 2 v 2 ) e 2 + z(e 2 v 3 ) e 2 + x(e 3 v ) e 3 + y(e 3 v 2 ) e 3 + z(e 3 v 3 ) e 3 (Eq. 6) (Eq. 7)

24 Generalized rotation matrix Group the scale factors for e e 2 and e 3 P (x(e v ) + y(e v 2 ) + z(e v 3 )) e + (x(e 2 v ) + y(e 2 v 2 ) + z(e 2 v 3 )) e 2 + (x(e 3 v ) + y(e 3 v 2 ) + z(e 3 v 3 )) e 3 (Eq. 8) Remember, how P was expressed in the (e e 2 e 3 ) system in Eq. 4 P r e + s e 2 + te 3 (r, s, t are coordinate values) In Eq.8 we recognize r, s, t as: r x(e v ) + y(e v 2 ) + z(e v 3 ) s x(e 2 v ) + y(e 2 v 2 ) + z(e 2 v 3 ) t x(e 3 v ) + y(e 3 v 2 ) + z(e 3 v 3 ) (Eq. 9)

25 Generalized rotation matrix Rearrange Eq 9. to column vector format r s t x(e v ) + y(e v 2 ) + z(e v 3 ) x(e 2 v ) + y(e 2 v 2 ) + z(e 2 v 3 ) x(e 3 v ) + y(e 3 v 2 ) + z(e 3 v 3 ) (Eq. 0) Recognize a matrix*vector product r s t (e v ) (e v 2 ) (e v 3 ) (e 2 v ) (e 2 v 2 ) (e 2 v 3 ) (e 3 v ) (e 3 v 2 ) (e 3 v 3 ) x y z (Eq. ) Recognize Eq. 4: P in the (e,e 2,e 3 ) system R ve Rotation matrix that takes P point from (v,v 2,v 3 ) system to (e,e 2,e 3 ) system Recognize Eq. 5: P in the (v,v 2,v 3 ) system

26 Generalized rotation matrix R ve what is it good for?? Rotates the (v,v 2,v 3 ) base vectors to (e,e 2,e 3 ) base vectors Transforms a point from (v,v 2,v 3 ) coordinate system to (e,e 2,e 3 ) coordinate system (assuming the v and e coordinate systems are centered in a common point. (If not, we will use translation to bring their centers together...)

27 Generalized rotation matrix R ve (e v ) (e v 2 ) (e v 3 ) (e 2 v ) (e 2 v 2 ) (e 2 v 3 ) (e 3 v ) (e 3 v 2 ) (e 3 v 3 ) (Eq. 2) For the reverse rotation, simply swap the corresponding v and e vectors R ev (v e ) (v e 2 ) (v e 3 ) (v 2 e ) (v 2 e 2 ) (v 2 e 3 ) (v 3 e ) (v 3 e 2 ) (v 3 e 3 ) (Eq. 3) Note that they are transposes of one another: R* ve R ev Note that they are inverses of one another: R - ve R ev

28 Direction cosigns in the generalized rotation matrix Recognize here the direction cosines R ve e 2 (e v ) (e v 2 ) (e v 3 ) (e 2 v ) (e 2 v 2 ) (e 2 v 3 ) (e 3 v ) (e 3 v 2 ) (e 3 v 3 ) where ( e v ) cos (e v )cos (a) ( e 2 v ) cos (e 2 v )cos (b) ( e i v j ) cos (e i v j ) cos (b ij ) e 2 v a b a e e v

29 Passage from coordinate system to another p e is a location expressed in the e system p v is the same location expressed in the v system e2 p e e3 p v _ t e 3. Translate 2. Rotate. Scale P v t ev + R ev S ev P e

30 Change of reference systems backwards (inverse transformation) p e RSp v + t p e t RSp v //left multiply by R - S - R - (p e t) p v //subtract t R - (p e t) Sp v //left multiply by S - Slight problem: mixes translation vector with matrices Solution: make t translation vector appear as a matrix T in multiplications. Then the equations will read as: p e (T(R(Sp v )))

31 Introduce 4x4 Translation Matrix T 0 0 d x 0 0 d y 0 0 d z v x y z Padding with (If I padded with 0 then inverse would not exist!) Tv 0 0 d x 0 0 d y 0 0 d z x y z x+d x y+d y z+d z Slight problem remains: rotation and scaling matrices must also be padded, so that we can multiply all 4x4 matrices.

32 R Homogeneous Transform Matrices Rotation matrix Scaling matrix S (e v ) (e v 2 ) (e v 3 ) 0 (e 2 v ) (e 2 v 2 ) (e 2 v 3 ) 0 (e 3 v ) (e 3 v 2 ) (e 3 v 3 ) s x s y s z v x y z p e (T(R(Sp v ))) Translation matrix T 0 0 d x 0 0 d y 0 0 d z Padding all vectors with

33 Passage from coordinate system to another With Homogeneous Coordinates e2 p e in F e e3 p v in F v _ t e 3. Translate (4x4) 2. Rotate (4x4). Scale (4x4) p v T ev R ev S ev p e P in v system (x4) P in e system (x4)

34 Coordinate system (aka frame) transformation compact form b p b p b T*R*S* p a p a p b F ab p a a F ab is often called frame transformation or coordinate system transformation.

35 Inverse of a frame transformation p b TRSp a Multiply from the left by T - T - p b RSp a Multiply from the left by R - R - T - p b Sp a Multiply from the left by S - S - R - T - p b p a

36 Series of frame transformations Most real systems involve complicated series of frames, where we traverse from frame to frame F ac p c F bc F ab p a We can undo them in reverse order: b F b c F a c p c p a F ba F cb p c p a F ab c Obviously F - abf ba F - cbf cb F - acf ca a

37 Series of frame transformations example Four-joint serial robot, with rotation in each joint and translation between each two joints. I am interested in the coordinates of a target point in end-effector frame if I know the target point in base frame. end-effector F 4 F 3 p 4 p 4 F 34 F 23 F 2 F 0 p 0 p 0 F F 2 robot base F 0

38 Rigid body transformation e3 A e e2 O e C e B e h 3 e h 2 O v A rigid body (such as the head) is moving during surgery ABC markers are affixed to the rigid head The markers are continuously localized (such as with optical stereo camera) in a home coordinate system We want to know the F ev transformation that takes the head from one pose (pose e) to another (pose v). This transformation will be expressed as a series of translations and rotations In each pose, we compute the frame defined by the ABC markers: e,e 2,e 3,O e v,v 2,v 3,O v O h Home coordinate system h

39 Rigid body transformation e3 A e e2 O e C e B e e O v Then we compute the following: T eh translation that takes the O e center of the e frame to the O h center of the home frame T hv translation that takes the O h center of the home frame to the O v center of the v frame. R ev rotation that rotates the e base vectors to v base vectors. h 2 To achieve the F ev transform of the rigid body from pose e to pose v, we concatenate the constituent transformations as: F ev T hv R ev T eh h 3 O h Home coordinate system h

40 Rotation about an arbitrary line v P h 3 Line h 2 The rigid body is defined by ABC markers The line is defined by P fixed point and v direction vector We want to rotate the body by some angle about the line. Method: Create a temporary frame, called line frame, that is centered in P on the line and in which one of the (v,v 2,v 3 ) base vectors is identical with the v direction vector of the line. (In the example we used v 2 ) Transform the body from home frame to line frame In lien frame, rotate the body about v 2 principal axis Transform the body from line frame back to home frame O h Home coordinate system h