Minimal representations of orientation

Transcription

1 Robotics 1 Minimal representations of orientation (Euler and roll-pitch-yaw angles) Homogeneous transformations Prof. lessandro De Luca Robotics 1 1

2 Minimal representations rotation matrices: 9 elements - 3 orthogonality relationships - 3 unitary relationships = 3 independent variables direct problem inverse problem sequence of 3 rotations around independent axes fixed (a i ) or moving/current (a i ) axes generically called Roll-Pitch-Yaw (fixed axes) or Euler (moving axes) angles possible different sequences (e.g., XYX) actually, only 1 since {(a 1 α 1 ), (a α ), (a 3 α 3 )} { (a 3 α 3 ), (a α ), (a 1 α 1 )} Robotics 1

3 x z z R z (ψ) = ZX Z Euler angles 1 R z (φ) = RF φ φ x RF y y cos φ sin φ 0 sin φ cos φ θ z x x z RF cos ψ sin ψ 0 sin ψ cos ψ z z ψ y θ y x ψ R x (θ) = cos θ sin θ 0 sin θ cos θ x y y RF Robotics 1 3

4 ZX Z Euler angles direct problem: given φ, θ, ψ ; find R R ZX Z (φ, θ, ψ) = R Z (φ) R X (θ) R Z (ψ) order of definition in concatenation = cφ cψ - sφ cθ sψ - cφ sψ - sφ cθ cψ sφ sθ sφ cψ + cφ cθ sψ - sφ sψ + cφ cθ cψ - cφ sθ sθ sψ sθ cψ cθ given a vector v = (x,y,z ) expressed in RF, its expression in the coordinates of RF is v = R ZX Z (φ, θ, ψ) v the orientation of RF is the same that would be obtained with the sequence of rotations: ψ around z, θ around x (fixed), φ around z (fixed) Robotics 1 4

5 ZX Z Euler angles inverse problem: given R = {r ij }; find φ, θ, ψ r 11 r 1 r 13 r 1 r r 3 r 31 r 3 r 33 = cφ cψ - sφ cθ sψ - cφ sψ - sφ cθ cψ sφ sθ sφ cψ + cφ cθ sψ - sφ sψ + cφ cθ cψ - cφ sθ sθ sψ sθ cψ cθ r 13 + r 3 = s θ, r 33 = cθ θ = TN{ ± r 13 + r 3, r 33 } if r 13 + r 3 0 (i.e., sθ 0) r 31 /sθ = sψ, r 3 /sθ = cψ ψ = TN{ r 31 /sθ, r 3 /sθ } similarly... φ = TN{ r 13 /sθ, -r 3 /sθ } there is always a pair of solutions there are always singularities (here θ = 0, ± π) two values differing just for the sign Robotics 1 5

6 x x z Roll-Pitch-Yaw angles 1 ROLL z z R X (ψ) = ψ ψ y y cos ψ sin ψ 0 sin ψ cos ψ C R Z (φ)c T with R Z (φ) = z 3 θ x x YW cos φ sin φ 0 sin φ cos φ θ z y y z y x PITCH φ C 1 R Y (θ)c 1 T φ with R Y (θ) = cos θ 0 sin θ sin θ 0 cos θ Robotics 1 6 z x y y

7 Roll-Pitch-Yaw angles (fixed XYZ) direct problem: given ψ, θ, φ ; find R R RPY (ψ, θ, φ) = R Z (φ) R Y (θ) R X (ψ) order of definition cφ cθ cφ sθ sψ - sφ cψ cφ sθ cψ + sφ sψ = sφ cθ sφ sθ sψ + cφ cψ sφ sθ cψ - cφ sψ - sθ cθ sψ cθ cψ inverse problem: given R = {r ij }; find ψ, θ, φ note the order of products! r 3 + r 33 = c θ, r 31 = -sθ θ = TN{-r 31, ± r 3 + r 33 } if r 3 + r 33 0 (i.e., cθ 0) for r 31 <0, two symmetric values w.r.t. π/ r 3 /cθ = sψ, r 33 /cθ = cψ ψ = TN{r 3 /cθ, r 33 /cθ} similarly... φ = TN{ r 1 /cθ, r 11 /cθ } singularities for θ = ± π/ Robotics 1 7

8 why this order in the product? R RPY (ψ, θ, φ) = R Z (φ) R Y (θ) R X (ψ) order of definition reverse order in the product (pre-multiplication ) need to refer each rotation in the sequence to one of the original fixed axes use of the angle/axis technique for each rotation in the sequence: C R(α) C T, with C being the rotation matrix reverting the previously made rotations (= go back to the original axes) concatenating three rotations: [ ] [ ] [ ] (post-multiplication ) R RPY (ψ, θ, φ) = [R X (ψ)] [R XT (ψ) R Y (θ) R X (ψ)] [R XT (ψ) R YT (θ) R Z (φ) R Y (θ) R X (ψ)] = R Z (φ) R Y (θ) R X (ψ) Robotics 1 8

9 Homogeneous transformations P B p p RF B p B affine relationship O B RF O p = p B + R B B p p hom = p R B p B B p = = T B B p hom linear relationship vector in homogeneous coordinates 4x4 matrix of homogeneous transformation Robotics 1 9

10 Properties of T matrix describes the relation between reference frames (relative pose = position & orientation) transforms the representation of a position vector (applied vector starting from the origin of the frame) from a given frame to another frame it is a roto-translation operator on vectors in the three-dimensional space it is always invertible ( T B ) -1 = B T can be composed, i.e., T C = T B B T C note: it does not commute! Robotics 1 10

11 Inverse of a homogeneous transformation p = p B + R B B p B p = B p B + B R p = - R B T p B + R B T p R B p B B R B p B = R B T - R B T p B T B B T ( T B ) -1 Robotics 1 11

12 Defining a robot task y E RF E absolute definition of task task definition relative to the robot end-effector z E RF B RF T 1 3 W T T = W T B B T E E T T known, once the robot direct kinematics of the is placed robot arm (function of q) RF W B T E (q) = W T B -1 W T T E T T -1 = constant Robotics 1 1

13 Final comments on T matrices they are the main tool for computing the direct kinematics of robot manipulators they are used in many application areas (in robotics and beyond) in positioning/orienting a vision camera (matrix b T c with extrinsic parameters of the camera pose) in computer graphics, for the real-time visualization of 3D solid objects when changing the observation point T B = R B α x α y α z p B σ all zero in robotics coefficients of perspective deformation scaling coefficient always unitary in robotics Robotics 1 13