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Rotation representation and usage

Rotation representation and usage. KH Wong. A study of Rcam. % Examine the characteristics of Rcam for Pc=Rcam -1 (Pw* X-c ) % a camera has rotated Rcam = inv ( Rc ) and translated c % Pw is a point in world coordinates % Pc is the same point (Pw) but in camera coordinates.

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Rotation representation and usage

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  1. Rotation representation and usage KH Wong Ch2. Cameras v.9e2

  2. A study of Rcam % Examine the characteristics of Rcamfor Pc=Rcam-1(Pw*X-c) % a camera has rotated Rcam= inv(Rc) and translated c % Pw is a point in world coordinates % Pc is the same point (Pw) but in camera coordinates Ch2. Cameras v.9e2

  3. Part1: Rotation of a vector Vz=[X,Y,Z]T(assume it is the Z-axis-component of the principal axis of the camera , so may take this rotation as Rcam_z) • Assume a vector Vz=[X,Y,Z]Tof length r in the world coordinate is being rotated (Rcam_z) around the Z-axis only by z • Ref: https://www.onlinemathlearning.com/sine-cosine-addition-formula.html • https://en.wikipedia.org/wiki/Rotation_matrix Camera principal axis after rotation(Vz’) Y-axis Camera principal axis before rotation (Vz) r z Y’ r Y  X-axis X’ X Ch2. Cameras v.9e2 Z-axis is pointing out of the paper

  4. Part2: Rotation of a vector Vy=[X,Y,Z]T(assume it is the y-axis-component of the principal axis of a camera, so may take this rotation as Rcam_y) • Assume a vector Vy=[X,Y,Z]Tof length r in the world coordinate is being rotated (Rcam_y) around the Z-axis only by y • Ref: https://www.onlinemathlearning.com/sine-cosine-addition-formula.html • https://en.wikipedia.org/wiki/Rotation_matrix Camera principal axis after rotation(Vy’) X-axis Camera principal axis before rotation (Vy) r y X’ r X  Z-axis Z’ Z Ch2. Cameras v.9e2 Y-axis is pointing out of the paper

  5. Part3: Rotation of a vector Vx=[X,Y,Z]T(assume it is the y-axis-component of the principal axis of a camera, so may take this rotation as Rcam_x) • Assume a vector Vx=[X,Y,Z]Tof length r in the world coordinate is being rotated (Rcam_x) around the X-axis only by x • Ref: https://www.onlinemathlearning.com/sine-cosine-addition-formula.html • https://en.wikipedia.org/wiki/Rotation_matrix Camera principal axis after rotation(Vx’) Z-axis Camera principal axis before rotation (Vx) r x Z’ r Z  Y-axis Y’ Y Ch2. Cameras v.9e2 X-axis is pointing out of the paper

  6. Combining Rcam_x,Rcam_y, Rcam_y http://scipp.ucsc.edu/~haber/ph216/rotation_12.pdf • Rcam_xyz, rotate around x-axis first, then y-axis , finally z-axis Ch2. Cameras v.9e2

  7. A study of Rc % Examine the characteristics of Rc for Pc=Rc(Pw*X-c) % a camera has rotated Rcam= inv(Rc) and translated c % Pw is a point in world coordinates % Pc is the same point (Pw) but in camera coordinates Ch2. Cameras v.9e2

  8. The camera moves to a new position :Rotate of the camera Rcam, and translate (TC) to a new position. V’c=Rcam*Vz+TC,(Note Vw and Vc are in the same world coordinates) Pw Yw Vz=Zw Camera coordinates Yc Vz’=Zc Rc Xc TCam= TC =camera translation in the world coordinates Rcam Rotation World center Xw Camera center World coordinates (reference) Ch2. Cameras v.9e2

  9. Recall: If you rotate the camera principal axis (Vz) of the camera from the world coordinate to a new position (Vz’) using Rcam (Vz and Vz’ are in the same world coordinates) . • Assume a point [X,Y,Z]w ora vector Vzof length r in the world coordinate is being rotated (Rcam) around the Z-axis only by z Camera principal axis after rotation(Vz’) Y-axis Camera principal axis before rotation (Vz) r z Y’ r Y  X-axis X’ X Ch2. Cameras v.9e2

  10. Here, rotate the coordinate system (around Z-axis) so a vector Pw (in world coordinates) is the same as a vector Pc (in camera coordinates) • Camera coordinates rotated z about Z-axis relative to the world coordinates • A vector Pw=[Xw,Yw,Zw]’ is in the world coordinates (blue/solid_axes) is the same vector [Xc,Yc,Zc]’ in the camera coordinates (red/dash_axes) • Z-axis is facing you Concentrate on Yc Yc= -Xwsin(z)+Ywcos(z) Yc Concentrate on Xc Yw z Yw z z z Xwcos(z) Xw Xw z Xwsin(z) Pw=Pc z Xc Pw=Pc Xc=Xwcos(z)+Ywsin(z) Ywcos(z) Ywsin(z) Ch2. Cameras v.9e2

  11. Here, rotate the coordinate system (around Y-axis) so a vector Pw (in world coordinates) is the same as a vector Pc (in camera coordinates) • Camera coordinates rotated -y about Y-axis relative to the world coordinates • A vector Pw=[Xw,Yw,Zw]’ is in the world coordinates (green/dotted_axes) is the same vector [Xc,Yc,Zc]’ in the camera coordinates (red/dash_axes) • Y-axis is facing away from you Concentrate on Zc Zc= -Xwsin(-z)+Zwcos(-z) Concentrate on Xc Zw -z Zw -y -z -y Xwcos(-y) Xw Xw -z Xwsin(-z) Pw -y Pw Xc Xc=Xwcos(y)+Zwsin(y) Zwcos(-z) Zwsin(-y) Ch2. Cameras v.9e2 cos(-θ) = cos θ, sin(-θ) = - sin θ.

  12. Here, rotate the coordinate system (around X-axis) so a vector Pw (in world coordinates) is the same as a vector Pc (in camera coordinates) • Camera coordinates rotated x about X-axis relative to the world coordinates • A vector Pw=[Xw,Yw,Zw]’ is in the world coordinates (purple/dotted_axes) is the same vector [Xc,Yc,Zc]’ in the camera coordinates (red/dash_axes) • X-axis is facing you Concentrate on Zc Zc= -Ywsin(x)+Zwcos(x) Concentrate on Yc Zw x Zw x x x Ywcos(x) Yw Yw x Ywsin(x) Pw=Pc x Xc Pw=Pc Yc=Ywcos(x)+Zwsin(x) Zwcos(x) Zwsin(x) Ch2. Cameras v.9e2

  13. Relating Rcam and Rc • Exercise: Think clearly what is the meaning of Rcam and Rc? • And how do they related? • Note: we only look at rotation , translation is considered separately Pw V’z Vz Camera center Rcam Rotation World center Ch2. Cameras v.9e2

  14. Precise definition of Rc, Tc and Rcam • Rc: A vector Pw will appear as Pc in camera coordinates using Rc • Pc=Rc*Pw, where a vector Pw in world coordinates is the same vector Pc in camera coordinates . Pw does not change, it is only the coordinate system changes • Rcam is the rotational matrix that rotates the principal axis (Vz) of the camera position to a new position in the world coordinates. Rcam will change Vz, but it is always in the world coord. Sys. before and after the change. • Pc=Rc*Pw,where a vector Pw in world coordinates is the same vector Pc in camera coordinates • Rcam = (Rc) -1(Note R-1=RT for rotation matrices) • Tcam=Tc is the translation of the camera center in the world coordinate system. • (-Tc) takes the camera center back to the world center in the world coordinate system. Ch2. Cameras v.9e2

  15. Study the rotation matrix RcRelate world 3D to camera 3D coordinates • We study pure camera rotation first • Assume Translation (Tc) =0 (from world center to camera center) • Pw=a stationary vector in World 3D coordinates Pw=[Xw,Yw,Zw]T • Pc=a point in camera 3D coordinates =[Xc,Yc,Zc] T • Rotation of camera is Rcam , • We introduce a transform Rc , such that Rc=(Rcam)-1 • Pc=Rc*Pw where a vector Pw in world coordinates is the same vector Pc in camera coordinates • That means a vector Pw in the world coordinate system will appear as Pc=Rc*Pw in the camera coordinate system • Rotation and Translation can be treated independently • We will study Rc in the following slides Yw Pw Zw Rcam =Rc-1 Xw World coordinates (reference) Camera rotated Rcam Ch2. Cameras v.9e2

  16. Explanation for the previous slide • Assume the camera coordinate system is rotated by z relative to the world coordinate system. A vector in the world coordinates (large blue) is (Xw,Yw) . We are not interested in Zw , because it will not be changed by rotation in the Z-axis. • Note: The large blue vector Pw is not changed, the only change here is the camera coordinate system (rotated z against the z-axis). And the same (large blue) Pw vector as observed in the camera coordinate system is (Xc,Yc) • The left-hand diagram shows that the relation between Xc and (Xw,Yw, and z), which is Xc=Xwcos(z)+Ywsin(z) • The right-hand diagram shows thatYc= -Xwsin(z)+Ywcos(z) • And Zw = Zc , because there is no change of value in the z-axis. • So it generates the matrix formula at the top • Copied here for your reference: • That means a vector Pw in the world coordinate system will appear as Pc=Rc*Pw in the camera coordinate system. • Or it means [Xc,Yc,Zc]’=Rz*[Xw,Yw,Zw]’ in the camera coordinate system refers to the same vector as [Xw,Yw,Zw]’ in the world coordinate system. • Remember that Pc=[Xc,Yc,Zc]’ is a vector in the camera coordinate system and Pw=[Xw,Yw,Zw]’ is a vector in the world coordinate system. Ch2. Cameras v.9e2

  17. Given rotation angles of a camera find Rc=RxRyRzA point in World Pw=[Xw,Yw,Zw]’ is the same as Pc=[Xc,Yc,Zc]’ in the camera coordinates by these transformations Proved in the previous slide You may prove the other 2 cases as an exercise http://ccar.colorado.edu/ASEN5070/handouts/coordsys.htm Ch2. Cameras v.9e2 http://www.google.com.hk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCwQFjAA&url=http%3A%2F%2Fwww.colorado.edu%2FASEN%2Fasen3200%2Fhandouts%2FCoordinateTransformations.doc&ei=nAfdUfTzBqWSiQeo-oCYDw&usg=AFQjCNGp5O37pYqGghl5Fft1skUmSXJ2-A&sig2=6q91UADC3ZZreVm_frd7rg&bvm=bv.48705608,d.aGc&cad=rja

  18. Combine rotations in 3 axes:The camera has rotated (x, y, z), Rc brings a vector in world coordinates to the camera coordinates Convention used: In this chapter we rotate the camera an_x(angle around x-axis) first, then an_y and finally an_x. So Rcam=Rcam_z*Rcam_y*Rcam_x Rc=transpose(Rcam)=(Rcam_z*Rcam_y*Rcam_x)T =(Rcam_x)T*(Rcam_x)T*(Rcam_x)T , since (AB)T=BT*AT =Rc_x*Rc_y*Rc_z, because Rc_x=(Rcam_x)T etc. Ch2. Cameras v.9e2

  19. Matlab program for RcUsed the Matlab/symbolic processor to create the transformation matrix • % ---- matlab demo: camera position change ---------------------- • %Init position of the camera • %A camera is normally at eye(3)(Identity matrix of 3x3), no rotate in world coord.sys, • %rotated vs 3 axes: an_x,an_y,an_z (an_z first, an_y second, an_x third here),to a new (camera) coord sys. • %Rc will bring a vector in world coord. to camera coord. Such that • %so Pc=Rc*Pw, Pw is a vector in world, Pc is the same vector in cam. sys. • symsan_xan_yan_z • Rz=[cos(an_z) sin(an_z) 0 • -sin(an_z) cos(an_z) 0 • 0 0 1] • Ry=[cos(an_y) 0 -sin(an_y) • 0 1 0 • sin(an_y) 0 cos(an_y)] • Rx=[1 0 0 • 0 cos(an_x) sin(an_x) • 0 -sin(an_x) cos(an_x)] • Rc = Rx*Ry*Rz % For Rcam, the camera moves around x-axis first, then y-axis, finally z-axis • % (assume extrinsic rotation), therefore Rcam= *Rcam_z*Rcam_y*Rcam_x, since Rc=inverse(Rcam) , • % so Rc= Rx*Ry*Rz • %Properties transpose(R)*R=I, inverse(R)*R=I, inverse(R)=transpose(R), det(R)=I . • Rc= • [ cos(an_y)*cos(an_z), cos(an_y)*sin(an_z), -sin(an_y)] • [ cos(an_z)*sin(an_x)*sin(an_y) - cos(an_x)*sin(an_z), cos(an_x)*cos(an_z) + sin(an_x)*sin(an_y)*sin(an_z), cos(an_y)*sin(an_x)] • [ sin(an_x)*sin(an_z) + cos(an_x)*cos(an_z)*sin(an_y), cos(an_x)*sin(an_y)*sin(an_z) - cos(an_z)*sin(an_x), cos(an_x)*cos(an_y)] Ch2. Cameras v.9e2

  20. Test code • %%%% khwong 2019.3.17 ver.9b • % Examine the characteristics of Rc for Pc=Rc(Pw*X-c) • % a camera has rotated Rcam= inv(Rc) and translated c • % Pw is a point in world coordinates • % Pc is the same point (Pw) but in camera coordinates • %%%%%%%%%%%%%%%%%%%%% • symsan_xan_yan_z real • %%%%%%%%%%%%%%%%%%% • Rc_z=[cos(an_z) sin(an_z) 0 • -sin(an_z) cos(an_z) 0 • 0 0 1]; • Rc_y=[cos(an_y) 0 -sin(an_y) • 0 1 0 • sin(an_y) 0 cos(an_y)]; • Rc_x=[1 0 0 • 0 cos(an_x) sin(an_x) • 0 -sin(an_x) cos(an_x)]; • %%%%%%%%%%%%%%%%%% • Rcam_z=[cos(an_z) -sin(an_z) 0 • sin(an_z) cos(an_z) 0 • 0 0 1]; • Rcam_y=[cos(an_y) 0 sin(an_y) • 0 1 0 • -sin(an_y) 0 cos(an_y)]; • Rcam_x=[1 0 0 • 0 cos(an_x) -sin(an_x) • 0 sin(an_x) cos(an_x)]; • %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% • 'transpose(Rc_z), and Rcam_z are shown below, they should be the same' • Rc_z' • Rcam_z • 'transpose(Rc_y), and Rcam_y are shown below, they should be the same' • Rc_y' • Rcam_y • 'transpose(Rc_x), and Rcam_x are shown below, they should be the same' • Rc_x' • Rcam_x • 'finally display Rcam_xyz' • %Rcam_xyz=Rcam_z*Rcam_y*Rcam_x • Rcam_zyx=(Rcam_x*Rcam_y*Rcam_z)' Ch2. Cameras v.9e2

  21. Summary % Pw is a point in world coordinates % Pc is the same point (Pw) but in camera coordinates % Pc=Rc(Pw*X-c) % A camera has rotated Rcam and translated c, and Rc=Inv(Rcam) % %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% Rcam_z=[cos(an_z) -sin(an_z) 0 sin(an_z) cos(an_z) 0 0 0 1]; Rcam_y=[cos(an_y) 0 sin(an_y) 0 1 0 -sin(an_y) 0 cos(an_y)]; Rcam_x=[1 0 0 0 cos(an_x) -sin(an_x) 0 sin(an_x) cos(an_x)]; %%%%%%%%%%%%%%%%%% Rc_z=[cos(an_z) sin(an_z) 0 -sin(an_z) cos(an_z) 0 0 0 1]; Rc_y=[cos(an_y) 0 -sin(an_y) 0 1 0 sin(an_y) 0 cos(an_y)]; Rc_x=[1 0 0 0 cos(an_x) sin(an_x) 0 -sin(an_x) cos(an_x)]; Ch2. Cameras v.9e2

  22. Introduction • meaning • Representation • Euler angles and Tait–Bryan angles • Axis angles • quaternions Ch2. Cameras v.9e2

  23. Fromhttps://en.wikipedia.org/wiki/Euler_angles • Any orientation can be achieved by composing three elemental rotations, i.e. rotations about the axes of a coordinate system. Euler angles can be defined by three of these rotations. They can also be defined by elemental geometry and the geometrical definition demonstrates that three rotations are always sufficient to reach any frame. • The three elemental rotations may be  • extrinsic (rotations about the axes xyz of the original coordinate system, which is assumed to remain motionless), or  • intrinsic(rotations about the axes of the rotating coordinate system XYZ, solidary with the moving body, which changes its orientation after each elemental rotation). • Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) • Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z). • Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called "Euler angles". In that case, the sequences of the first group are called proper or classic Euler angles. Ch2. Cameras v.9e2

  24. Tait–Bryan angles v.s. Proper Euler angleshttps://en.wikipedia.org/wiki/Euler_angles • The only difference is that Tait–Bryan angles represent rotations about three distinct axes (e.g. x-y-z, or x-y′-z″), while proper Euler angles use the same axis for both the first and third elemental rotations (e.g., z-x-z, or z-x′-z″). • Rotation matrix[edit] • Any orientation can be achieved by composing three elemental rotations, starting from a known standard orientation. Equivalently, any rotation matrix R can be decomposed as a product of three elemental rotation matrices. For instance: • is a rotation matrix that may be used to represent a composition of extrinsic rotations about axes z, y, x, (in that order), or a composition of intrinsic rotations about axes x-y′-z″ (in that order).  Rotation matrix order reversed for intrinsic rotations (mobile coordinate frame) Ch2. Cameras v.9e2

  25. See the demohttps://www.mecademic.com/resources/Euler-angles/Euler-angles • X,Y,Z for fixed coordinate frame (extrinsic rotation) • X’Y’Z’ for mobile coordinate frame (intrinsic rotation) • Rotation matrix order reversed for intrinsic rotations (mobile coordinate frame) see http://web.cs.iastate.edu/~cs577/handouts/rotation.pdf Ch2. Cameras v.9e2

  26. Systems of Euler angles • Proper (classic) Euler angles combining 3 rotations : (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) • Mobile coordinate frame (intrinsic rotation) • Fixed coordinate frame (extrinsic rotation) • Tait–Bryan angles (or yaw, pitch, and roll) combining 3 rotations : (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z). • Mobile coordinate frame (intrinsic rotation) • Fixed coordinate frame (extrinsic rotation) Ch2. Cameras v.9e2

  27. Elementary rotations around, z,y,x axis • They can be combined to build multiple rotations in Euler angles angles • Rz(an_z)=[cos(an_z) -sin(an_z) 0 • sin(an_z) cos(an_z) 0 • 0 0 1]; • Ry(an_y)=[cos(an_y) 0 sin(an_y) • 0 1 0 • -sin(an_y) 0 cos(an_y)]; • Rx(an_x)=[1 0 0 • 0 cos(an_x) -sin(an_x) • 0 sin(an_x) cos(an_x)]; Ch2. Cameras v.9e2

  28. Proper Euler angles(use rotation around 2 axes only) • Combing 3 elementary rotations , depending on the sequence, it can be (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) totally 6 methods. • E.g.z-x-z, a rotation angle a1 around z first, them a2 around x, a3 finally around z again. • For fixed coordinate frame (extrinsic rotation) • Rf=Rz(a3)*Ry(a2)*Rz(a1) • For mobile coordinate frame (intrinsic rotation) • Rm=R’z(a1)*R’y(a2)*R’z(a3) • The order is reversed, i.e. the last rotation a3 will operate first (on the right hand side). Why? Ch2. Cameras v.9e2

  29. Tait–Bryan angles (yaw, pitch, roll)(use rotation around 3 axes) • Combing 3 elementary rotations , depending on the sequence, it can be (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z) totally 6 methods. • E.g. z-x-y, a rotation angle a1 around z first, them a2 around x, a3 finally around y . • For fixed coordinate frame (extrinsic rotation) • Rf=R_x(a3)*R_y(a2)*R_z(a1) • For mobile coordinate frame (intrinsic rotation) • Rm=R’z(a1)*R’y(a2)*R’x(a3) • The order is reversed, i.e. the last rotation a3 will operate first (on the right hand side). Why? Ch2. Cameras v.9e2

  30. Important idea • fixed coordinate frame (extrinsic) • The coordinate system is fixed. No change after each induvial axis rotation • mobile coordinate frame (intrinsic) • The coordinate system is changed after each individual axis rotation. • Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice versa.  • Try it yourself https://www.mecademic.com/resources/Euler-angles/Euler-angles Ch2. Cameras v.9e2

  31. Proper Euler angles (rotates 2 axes) , fixed coordinate frame (extrinsic) • The coordinates during each individual rotation on an axis does not change. • V’=Rf*V, a combined rotation Rf brings V to V’. • So e.g. z-x-z, a rotation angle a1 around z first, them a2 around x, a3 finally around z again. • Rf=Rz(a3)*Ry(a2)*Rz(a1) : [extrinsic rotation] • Since rotate a1 around z axis operate first, so will R_z(a1) is closer to the vector V (in the eauation), and so on, hence • V’=Rf*V=Rz(a3)*Ry(a2)*Rz(a1)*V • V’ and V are in the same world coordinate system • It is the case of R_cam we studied earlier. • Same applies to all 6 (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) different rotations. Ch2. Cameras v.9e2

  32. Proper Euler angles (rotates 2 axes) , mobile coordinate frame (intrinsic) • The coordinates change after each individual rotation V’=Rm*V, a combined rotation Rm brings V’ to V in the original (world) coordinate system before any rotation occurs. • So e.g. z-x-z, a rotation angle a1 around z first, them a2 around x, a3 finally around z again. • Rm=R’z(a1)*R’y(a2)*R’z(a3): [Intrinsic rotation] • Why Rz, Ry are arranged in this order? • It is a little complicated than the fixed frame system, so we will explain the reason. Ch2. Cameras v.9e2

  33. Proper Euler angles (rotates 2 axes) , mobile coordinate frame (intrinsic) • We want to prove: Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice versa. • Example: Use z-y-z, a rotation angle a1 around z-axis first, them a2 around y, a3 finally around z again. Notation :R (extrinsic), R’(intrinsic) elemental rotations • Rm=R’z(a1)*R’y(a2)*R’z(a3) ---------------------------------(i) (intrinsic ) • But if we reverse the order rotation angles: a3 around z first, then a2 around y, a1 around z again • Relating R(extrinsic) and R’(intrinsic), • R’z(a3)=Rz(a3)----------------------------------------------------(ii) now R’z(a3) around z first • R’y(a2)=Rz(a3)*Ry(a2)*Rz(a3)-1-----------------------------(iii) second rotation • In (iii) since the coordinates sys. has changed, we need to rotate backward Rz(a3) first, then Ry(a2) is the same as R’y(a2), but need to move back to the second coordinates by Rz(a3)* at the end • R’z(a1)=Rz(a3)*Ry(a2)*Rz(a1)*Ry(a2)-1*Rz(a3)-1 ------(iv) , same logic • Recall Rm=R’z(a1)*R’y(a2)*R’z(a3) , put (ii) (iii), (iv) to here • ={Rz(a3)*Ry(a2)*Rz(a1)*Ry(a2)-1*Rz(a3)-1 }* {Rz(a3)*Ry(a2)*Rz(a3)-1} * Rz(a3) • =Rz(a3)*Ry(a2)*Rz(a1)-----------------------------------------(v) • For this reversed rotation order of Rm: a3 around z first, then a2 around y, a1 around z again • Is the same Rf for the order: angle a1 around x-axis first, them a2 around y, a3 finally around z • Rf is = Rz(a3)*Ry(a2)*Rz(a1), same as (v) • Proved! Ch2. Cameras v.9e2

  34. Add another proof for roll pitch yaw • We want to Proof: Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice versa. • Can add a proof using Rc and Rcam concept later. Ch2. Cameras v.9e2

  35. Tait–Bryan angles (rotates 3 axes) , mobile coordinate frame (intrinsic) • We want to prove: Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice versa. • Example: Use x-y-z, a rotation angle a1 around x-axis first, them a2 around y, a3 finally around z again. Notation :R (extrinsic), R’(intrinsic) elemental rotations • Rm=R’x(a1)*R’y(a2)*R’z(a3) ---------------------------------(i) (intrinsic ) • But if we reverse the order rotation angles: a3 around z first, then a2 around y, a1 around x again • Relating R(extrinsic) and R’(intrinsic), • R’z(a3)=Rz(a3)----------------------------------------------------(ii) now a3 around z first: • R’y(a2)=Rz(a3)*Ry(a2)*Rz(a3)-1-----------------------------(iii) second rotation • In (iii) since the coordinates sys. has changed, we need to rotate backward Rz(a3) first, then Ry(a2) is the same as R’y(a2), but need to move back to the second coordinates by Rz(a3)* at the end • R’x(a1)=Rz(a3)*Ry(a2)*Rz(a1)*Ry(a2)-1*Rz(a3)-1 ------(iv) , same logic • Recall Rm=R’x(a1)*R’y(a2)*R’z(a3) , put (ii) (iii), (iv) to here • ={Rz(a3)*Ry(a2)*Rz(a1)*Ry(a2)-1*Rz(a3)-1 }* {Rz(a3)*Ry(a2)*Rz(a3)-1} * Rz(a3) • =Rz(a3)*Ry(a2)*Rz(a1)----------------------------------------(v) • For this reversed rotation order: a3 around z first, then a2 around y, a1 around z again • It is the same as the Rf: rotate a1 around x-axis first, then a2 around y, finally a3 around z • Rf is = Rz(a3)*Ry(a2)*Rz(a1), same as (v) • Proved! Ch2. Cameras v.9e2

  36. Proper Euler angles , mobile coordinate frame (intrinsic) • Example: Use z-x-z, a rotation angle a1 around z first, them a2 around x, a3 finally around z again. • The first rotation a1 around Z-axis brings coordinates from s0 to a new system s1, V now becomes V’ but it is actually the same point space. • V= Rz(a1)-1 * V’ • The second rotation a2 around x-axis brings the vector v’ to v’’ in the coordinate system s2. • V’= Rx(a2) -1 *V’’ • The third rotation a3 around z-axis brings the vector v’’ to v’’’ in the coordinate system s3. • V’’= Rz(a3) -1 * V’’’ • So V= Rc_z(a1)-1* Rc_x(a2) -1* Rc_z(a3)-1*V’’’ • =V’’’=(Rc_z(a1) -1 * Rc_x(a2) -1 * Rc_z(a3) -1 )-1*V • hence V’’’= Rz(a3)-1 * Rc_x(a2) -1* Rc_z(a1) -1*V Ch2. Cameras v.9e2

  37. Quaternions • Quaternions Ch2. Cameras v.9e2

  38. Axis-angle • Axis-angle Ch2. Cameras v.9e2

  39. Gimbal lock • A gimbal is a ring that is suspended so it can rotate about an axis. • When angles of 2 axes are the same it is a problem and is called the Gimbal lock problem • See • https://www.youtube.com/watch?v=oj7v3MXJL3M • https://www.youtube.com/watch?v=zc8b2Jo7mno Ch2. Cameras v.9e2

  40. References • https://en.wikipedia.org/wiki/Euler_angles • How is orientation in space represented with Euler angles? https://www.mecademic.com/resources/Euler-angles/Euler-angles • A Tutorial on Euler Angles and Quaternions http://www.weizmann.ac.il/sci-tea/benari/sites/sci-tea.benari/files/uploads/softwareAndLearningMaterials/quaternion-tutorial-2-0-1.pdf • https://en.wikipedia.org/wiki/Davenport_chained_rotations Ch2. Cameras v.9e2

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