Inertial Measurement Units II!

Transcription

1 ! Inertial Measurement Units II! Gordon Wetzstein! Stanford University! EE 267 Virtual Reality! Lecture 10! stanford.edu/class/ee267/!!

2 wikipedia! Polynesian Migration!

3 Lecture Overview! short review of coordinate systems, tracking in flatland, and accelerometer-only tracking! rotations: Euler angles, axis & angle, gimbal lock! rotations with quaternions! 6-DOF IMU sensor fusion with quaternions!

4 primary goal: track orientation of head or device! inertial sensors required to determine pitch, yaw, and roll! oculus.com!

5 from lecture 2:! vertex in clip space! vertex! v clip = M proj M view M model v oculus.com!

6 from lecture 2:! vertex in clip space! vertex! v clip = M proj M view M model v projection matrix! view matrix! model matrix! rotation! translation! M view = R T ( eye) oculus.com!

7 Euler angles! θ y oculus.com! θ z θ x roll! rotation! translation! M view = R T ( eye) R = R z ( θ z ) R x ( θ x ) R y θ y pitch! yaw!

8 Euler angles! θ y oculus.com! θ z θ x 2 important coordinate systems:! body/sensor frame! world/inertial frame! M view = R T eye R = R z ( θ z ) R x ( θ x ) R y θ y roll! pitch! yaw!

9 ! Gyro Integration aka Dead Reckoning! from gyro measurements to orientation use Taylor expansion! have: angle at! last time step! have:! time step! θ ( t) + t θ t θ t + Δt want: angle at! current time step! = ω have: gyro measurement! (angular velocity)! Δt + ε, ε O Δt 2 approximation error!!

10 Orientation Tracking in Flatland! problem: track 1 angle in 2D space! sensors: 1 gyro, 2-axis accelerometer! sensor fusion with complementary filter, i.e. linear interpolation:! θ t + 1 α = α θ ( t 1) +!ω Δt atan2!a x,!a y IMU VRduino no drift, no noise!

11 Tilt from Accelerometer! assuming acceleration points up (i.e. no external forces), we can compute the tilt (i.e. pitch and roll) from a 3-axis accelerometer! â =!a 0!a = R 1 0 = R z ( θ z ) R x ( θ x ) R y θ y = cos( θ x )sin θ z cos( θ x )cos θ z sin θ x θ x = atan2 â z,sign( â y ) â 2 2 x + â y θ z = atan2( â x,â y ) both in rad

12 Euler Angles and Gimbal Lock! so far we have represented head rotations with Euler angles: 3 rotation angles around the axis applied in a specific sequence! problematic when interpolating between rotations in keyframes (in computer animation) or integration à singularities!

13 Gimbal Lock! The Guerrilla CG Project, The Euler (gimbal lock) Explained see youtube.com!

14 Rotations with Axis-Angle Representation and Quaternions!

15 Rotations with Axis and Angle Representation! solution to gimbal lock: use axis and angle representation for rotation!! simultaneous rotation around a normalized vector v by angle! θ no order of rotation, all at once around that vector!

16 ! Quaternions! think about quaternions as an extension of complex numbers to having 3 (different) imaginary numbers or fundamental quaternion units i,j,k! q = q w + iq x + jq y + kq z i j k i 2 = j 2 = k 2 = ijk = 1 ij = ji = k ki = ik = j jk = kj = i

17 Quaternions! think about quaternions as an extension of complex numbers to having 3 (different) imaginary numbers or fundamental quaternion units i,j,k! q = q w + iq x + jq y + kq z quaternion algebra is well-defined and will give us a powerful tool to work with rotations in axis-angle representation in practice!

18 Quaternions! axis-angle to quaternion (need normalized axis v)! = cos θ 2 q θ,v!" # $# + i v x sin θ 2!# " $# + j v y sin θ 2!# " $# + k v z sin θ 2!# " $# q w q x q y q z

19 Quaternions! axis-angle to quaternion (need normalized axis v)! = cos θ 2 q θ,v!" # $# + i v x sin θ 2!# " $# + j v y sin θ 2!# " $# + k v z sin θ 2!# " $# q w q x q y q z valid rotation quaternions have unit length! q = q w 2 + q x 2 + q y 2 + q z 2 = 1

20 ! Two Types of Quaternions! vector quaternions represent 3D points or vectors u=(u x,u y,u z ) can have arbitrary length! q u = 0 + iu x + j u y + k u z valid rotation quaternions have unit length! q = q w 2 + q x 2 + q y 2 + q z 2 = 1

21 !! quaternion addition:! q + p = q w + p w quaternion multiplication:! Quaternion Algebra! + i( q x + p x ) + j q y + p y qp = q w + iq x + jq y + kq z + k q z + p z ( p w + ip x + jp y + kp z ) + i( q w p x + q x p w + q y p z q z p y ) + j( q w p y q x p z + q y p w + q z p x ) + k( q w p z + q x p y q y p x + q z p w ) + = q w p w q x p x q y p y q z p z

22 Quaternion Algebra! quaternion conjugate:!! quaternion inverse:! q * = q w iq x jq y kq z q 1 = q* q 2 rotation of vector quaternion by q :! inverse rotation:! q u q' u = qq u q 1 q u = q 1 q' u q successive rotations by q 1 then q 2 :! q' u = q 2 q 1 q u q q 2

23 Quaternion Algebra! detailed derivations and reference of general quaternion algebra and rotations with quaternions in course notes! please read course notes for more details!

24 Quaternion-based! 6-DOF Orientation Tracking!

25 Quaternion-based Orientation Tracking! 1. 3-axis gyro integration! 2. computing the tilt correction quaternion! 3. applying a complementary filter!

26 !! Gyro Integration with Quaternions! start with initial quaternion:! q ( 0) = 1+ i0 + j0 + k0 convert 3-axis gyro measurements!ω =!ω x,!ω y,!ω z to instantaneous rotation quaternion as! avoid division by 0! integrate as! ( t+δt q ) ω = q ( t) q Δ q Δ = q Δt!ω, angle!!ω!ω rotation axis!

27 Gyro Integration with Quaternions! q ω t+δt integrated gyro rotation quaternion represents rotation from body to world frame, i.e.! = q ω t+δt q u world ( body q ) ( t+δt) u q 1 ω last estimate q ( t) is either from gyro-only (for dead reckoning) or from last complementary filter! integrate as! ( t+δt q ) ω = q ( t) q Δ

28 Tilt Correction with Quaternions! assume accelerometer measures gravity vector in body (sensor) coordinates!!a =!a x,!a y,!a z transform vector quaternion of into current estimation of world space as!!a ( world q ) ( t+δt a = q ) ( body ω q ) ( t+δt) a q 1 ω ( body q ) a = 0 + i!a x + j!a y + k!a z

29 Tilt Correction with Quaternions! assume accelerometer measures gravity vector in body (sensor) coordinates!!a =!a x,!a y,!a z transform vector quaternion of into current estimation of world space as!!a ( world q ) ( t+δt a = q ) ( body ω q ) ( t+δt) a q 1 ω if gyro quaternion is correct, then accelerometer world vector points up, i.e.! = 0 + i0 + j k0 q a world

30 Tilt Correction with Quaternions! gyro quaternion likely includes drift! accelerometer measurements are noisy and also include forces other than gravity, so it s unlikely that accelerometer world vector actually points up! if gyro quaternion is correct, then accelerometer world vector points up, i.e.! ( world q ) a = 0 + i0 + j k0

31 ! Tilt Correction with Quaternions! solution: compute tilt correction quaternion that would rotate into up direction! ( world) q a how? get normalized vector part of vector quaternion! ( world) q a v = ( world) q ax ( world) q a, q a y ( world) q a world, q a z ( world) q a world

32 ! Tilt Correction with Quaternions! solution: compute tilt correction quaternion that would rotate into up direction! v x v y v z i = cos φ n = v x v y v z q t = q φ, n n φ = cos 1 v y = v z 0 v x ( world) q a

33 Complementary Filter with Quaternions! complementary filter: rotate into gyro world space first, then rotate a bit into the direction of the tilt correction quaternion! = q ( 1 α )φ, n n q c t+δt q ω ( t+δt) 0 α 1 rotation of any vector quaternion is then! ( world q ) ( t+δt u = q ) ( body c q ) ( t+δt) u q 1 c

34 Integration into Graphics Pipeline! q c t+δt compute via quaternion complementary filter first! stream from microcontroller to PC! convert to 4x4 rotation matrix (see course notes)! ( t+δt q ) c R c! 1 set view matrix to M view = R c to rotate the world in front of the virtual camera

35 Head and Neck Model! y θ x l h θ z y l n IMU! z pitch around base of neck!! l n x roll around base of neck!!

36 Head and Neck Model! why? there is not always positional tracking! this gives some motion parallax! can extend to torso, and using other kinematic constraints! integrate into pipeline as! M view = T ( 0, l n, l h ) R T ( 0,l n,l h ) T ( eye)

37 Must read: course notes on IMUs!!