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CS I400/B659: Intelligent Robotics

CS I400/B659: Intelligent Robotics. Rigid Transformations. 3D Rigid Objects. q = ( t x ,t y , q ) with q  [0,2 p ) Robot R 0  R 2 given in reference frame T 0 What’s the new robot R q ? { T q ( x,y ) | ( x,y )  R 0 } Define rigid transformation T q ( x,y ) : R 2  R 2.

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CS I400/B659: Intelligent Robotics

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  1. CS I400/B659: Intelligent Robotics Rigid Transformations

  2. 3D Rigid Objects

  3. q = (tx,ty,q) with q [0,2p) Robot R0R2 given in reference frame T0 What’s the new robot Rq?{Tq(x,y) | (x,y)  R0} Define rigid transformation Tq(x,y) : R2 R2 Rigid Transformation in 2D cos θ -sin θ sin θ cos θ x y tx ty Tq(x,y) = + 2D rotation matrix Affine translation

  4. q = (tx,ty,tz,R) with R a 3x3 rotation matrix Robot R0R3given in reference frame T0 What’s the new robot Rq?{Tq(x,y,z) | (x,y,z)  R0} Define rigid transformation Tq(x,y,z) : R3 R3 Rigid Transformation in 3D R x y z tx ty tz Tq(x,y,z) = + 3D rotation matrix Affine translation

  5. 3-D Rigid Rotations • det(A) = +1 • Orthogonal rows and columns: ATA=I, AAT=I • ||Ax|| = ||x|| for any x, L2 norm ||.|| • Product of two rotations is a rotation r11 r12 r13 r21 r22 r23 r31 r32 r33 (0,1,0) (r11,r21,r31) (Right-handed coordinate system) (r13,r23,r33) (1,0,0) (1,0,0) (r12,r22,r32) (0,0,1) (0,0,1)

  6. Coordinate Frames World frame Local frame (0,1,0) R (1,0,0) R (0,0,1) = RT R-1

  7. 3 representations • Euler angles REuler(f,q,y) • Axis angle RAA(v,q) • Quaternion RQuat(q) • All representations are “equivalent” but may have certain mathematical or computational advantages

  8. Axis-aligned rotations Rotate about: Z axis Y axis X axis cos θ -sin θ 0 sin θ cos θ 0 0 0 1 cos θ 0 sin θ 0 1 0 -sin θ 0 cos θ 1 0 0 0 cos θ -sin θ 0 sin θ cos θ

  9. Euler angles: (f,q,y) z z z z y f y q y y y x x x x Parameterization of SO(3) 1 2  3 4

  10. Euler Angles • Which axes to pick, and what order? • Convention A,B,C • REuler(f,q,y) = RA(f)RB(q)RC(y) • E.g., ZYZ, ZYX (roll-pitch-yaw), etc

  11. Euler Angles • Which axes to pick, and what order? • Convention A,B,C • REuler(f,q,y) = RA(f)RB(q)RC(y) • E.g., ZYZ, ZYX (roll-pitch-yaw), etc • Disadvantages • Must constrain to range of values • Singularities, e.g. ZYZ when q=0 or p (Gimbal lock) • Interpolation?

  12. Axis-Angle Representation • Every rotation matrix R can be obtained by rotating the identity matrix by some angle θ about some axis v! • R v = v

  13. Axis-Angle Representation • Axis v (||v||=1), angle θ • Rodrigues’ formula: rotate x about v -> x’ x’ = x cos θ + (v x x) sin θ + v (vT x) (1 - cos θ) Or in matrix form… RAA(θ,v) = cosθ I+ sin θ [v] + (1 - cosθ) vvT Cross product matrix 0 -vz vy vz 0 -vx -vy vx 0

  14. Recovering Axis and Angle from the Rotation Matrix • θ = Angle(R) = cos-1((r11+r22+r33-1)/2) = cos-1((tr(R)-1)/2) • v = Axis(R) = 1/(2 sin θ) r32-r23r13-r31r21-r12

  15. Properties of Axis-Angle • Disadvantages: • Non unique: RAA(θ,v)=RAA(-θ,-v) (can constrain θ to range [0,p]) • 4 parameters + one unit length constraint ||v||=1; dealing with this constraint is sometimes annoying, for example, in interpolation, optimization, or sampling • Unique encoding: vector w = θv • θ = ||w||, v = w/||w|| • “Exponential map” REM(w) = RAA( ||w|| , w/||w|| )

  16. Quaternion representation • Generalization of complex numbers • Complex z=z0+i z1, with |z|=1 can represent a 2D rotation. What’s the 3D analogue? q = q0+q1i + q2j +q3k, where i2 = j2 = k2 = -1i = jk = -kjj = ki = -ikk = ij = -ji Quaternions were a forerunner of vectors and were once mandatory of all students of physics and math!

  17. Unit quaternion representation • q = (q0,q1,q2,q3), ||q||=1 • Related to axis angle: • q = (cosq/2,vx sin q/2, vysin q/2, vzsin q/2) 2(q02+q12)-1 2(q1q2-q0q3) 2(q1q3+q0q2) 2(q1q2+q0q3) 2(q02+q22)-1 2(q2q3-q0q1) 2(q1q3-q0q2) 2(q2q3+q0q1) 2(q02+q32)-1 RQ(q) =

  18. Properties of Unit Quaternions • 4-sphere: 3D manifold in 4D space • Non-unique: Double cover of SO(3) • Advantages (widely used in computer animation): • Quaternion multiplication = rotation composition (slightly faster than matrix *) • Curve interpolation formulas (Shoemake, ‘85)

  19. Summary • Rotation matrix • 9 parameters Rij in range [-1,1] • Constraint: determinant +1 • Advantages: composition, inversion are easy • Euler angles • 3 parameters (f,q,y) in range [0,2p) • Axes picked by convention (e.g., Roll-Pitch-Yaw) • Advantages: no constraints, simple interpolation “works” • Disadvantages: multiple representation, singularities, composition & inversion not easy • Axis-angle • 4 parameters (q,v), q in [0,p] • Constraint: |v|=1 • Advantages: inversion is easy • Disadvantages: constraint, composition & interpolation not easy , multiple representation at p • Moment representation (aka exponential map) • 3 parameters w (= q*v), ||w||<=p • Advantages: no constraints, inversion easy • Disadvantages: composition & interpolation not easy, multiple representation at p • Quaternion • 4 parameters q=(q0,q1,q2,q3) • Constraint: |q|=1 • Advantages: composition & interpolation formulas, inversion easy • Disadvantages: constraint, multiple representation R(q) = R(-q)

  20. Notion of Distance • θ(R1TR2) = cos-1 ((tr(R1TR2)-1)/2) measures the minimum angle needed to rotate from frame R1 to R2 • Makes a good distance metric

  21. Rotation in Motion • Interpolating between two rotation matrices A and B • We want a path X(s) : [0,1] -> SO(3) with R(0) = A and R(1) = B • Let v = Axis(ATB), θ = Angle(ATB) • Verify that X(s) = A RAA(sθ,v) satisfies the desired properties Actually a geodesic in SO(3)!

  22. Angular Velocities in 2D • For a rotation trajectory parameterized by time, we can show: • => is the speed of rotation • For a point in world coordinates , we have

  23. Angular Velocities in 3D • For parameterized rotation trajectory REM(wt), we can show: d/dt REM(wt) = [w] REM(wt) • => |w| is the speed of rotation • [w] is the cross product matrix • => w x x is the velocity of some point x, specified in world coordinates, attached to the frame as it rotates • => w is the angular velocity

  24. Recap • Multiple representations of SO(3) • All four implemented robustly in C++ in KrisLibrary • Rotation matrix, axis angle, moment representations are implemented in Python klampt.so3 module • Notion of distance, geodesic, speed in SO(3)

  25. Next Weel • Read Principles Ch. 3.8 • Optional: A Mathematical Introduction to Robotic Manipulation, Ch. 3.1-4 • http://www.cds.caltech.edu/~murray/mlswiki/?title=First_edition

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