Quaternions and E*gen!

robotics focus group #3

Output a real-time map of surroundings

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Today:

• Discussion of hardware and software parts
• Quaternions
• Eigen

visual odometry: movement between camera frames

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3d rigid body motion: fundamentals of understanding how robot mathematically represents space

Why does all this rotation stuff matter?

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• We need to take our points from RPLidar to our software by transforming points from our laser coordinate system to our world coordinate system

rosrun tf static_transform_publisher 0 0 0 0 0 0 world laser 100

Robot Discussion

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Hardware

• Raspberry Pi Model 4B
• RPLidar by SlamTec (2D LIDAR points)
• Points are mapped to a 2D coordinate plane using r distance (time it takes for light beam to return back to LIDAR) and θ angle of rotation. Raw format is polar coordinate
• We need ROS (RViz) to access the laser mapping tools
• Chassis: Wheels, motor controllers

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Software

• RViz
• ROS1 (Noetic Nimjemys)

Euler Angles

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Problems:

• Gimbal Lock
• You lose one degree of freedom when two rotations are aligned, so you can’t represent some rotations

3x3 Rotation Matrix

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Each column represents result of rotation x/y/z axes, respectively

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0.32 = x-component of transformed x-axis in terms of original x-axis

0.46 = y-component of transformed y-axis in terms of original x-axis

-0.83 = z-component of transformed z-axis in terms of original x-axis

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Alternatives to Quaternions

3x3 Rotation Matrix (X/Y/Z)

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Euler Angles

Why Quaternions?

• Computationally more efficient to calculate when combining multiple orientations
• Avoids gimbal lock
• What are they?
• 3.23+9.46i+2.64j + 3.38k
• Let’s compare them with complex numbers!
• The product of 2 complex numbers is another complex number! That can be used to represent a rotation in a 2D plane

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z1=a+bi

z2=c+di

what is a quaternion?

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where

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it’s very similar to complex numbers that you are familiar with

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another representation

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“ordered pair”

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real quaternions

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pure quaternions

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binary form (uses unit quaternion)

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onto the fun stuff! rotations

rotating pure quaternions orthogonal to q is ok!

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but happens with quaternions not orthogonal to q?

we no longer get a pure quaternion :(

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genius method:

we post multiply the result by the inverse of q

this gives us a pure quaternion!

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but now we’ve rotated by double…

so we halve the angle, and use

interpolation

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quaternions overcome gimbal lock!

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we use something called SLERP, spherical linear interpolation

allows us to smoothly interpolate through space

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linear interpolation

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spherical interpolation

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we can use exactly this, just with quaternions

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we find the angle by using regular formula between vectors, just with two quaternions

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things to note:

• if dot-product is negative, we have the long way angle, so we just negate one of the directions
• if angular difference is small, sin becomes 0, then dividing gives undefined result
• so we use linear interpolation

conversion to and from Euler angles

Conversion_between_quaternions_and_Euler_angles

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cameras

in 3d engines, we can encode the direction (“look at”) as a quaternion

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we can transform world coordinates to camera coordinates and vice versa

by using quaternions!

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trivia

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Why Quaternions?

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• Let’s compare them with complex numbers!
• The product of 2 complex numbers is another complex number! That can be used to represent a rotation in a 2D plane
• Likewise, the product of 2 quaternion is a rotation in a 3D plane

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z1=a+bi

z2=c+di

Eigen!

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some helpful library links:

• geometry module docs
• 2 ways of representing rotations:
• not matrix compatible: angles/axis/quaternions

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AngleAxis<float> aa(angle_in_radian, Vector3f(ax,ay,az));

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• matrix compatible: projective/affine transformations

Quaternion<float> q; q = AngleAxis<float>(angle_in_radian, axis);

• euler angles? Not too convenient on Eigen

Most common Eigen structures for rotations

• Rotation matrix ( 3 × 3 ): Eigen::Matrix3d.� • Rotation vector ( 3 × 1 ): Eigen::AngleAxisd.� • Euler angle ( 3 × 1 ): Eigen::Vector3d.� • Quaternion ( 4 × 1 ): Eigen::Quaterniond.� • Euclidean transformation matrix ( 4 × 4 ): Eigen::Isometry3d. • Affine transform ( 4 × 4 ): Eigen::Affine3d.� • Perspective transformation ( 4 × 4 ): Eigen::Projective3d.

Let’s end off with an example

• Write a program to convert coordinates from robot 1’s coordinates to robot 2’s coordinate system
• Robot 1 and Robot 2 are located in the world coordinate system
• Pose of robot 1: (orientation as a quaternion & translation) q1 = [0.35, 0.2, 0.3, 0.1]T and t1 = [0.3, 0.1, 0.1]T
• Pose of robot 2: q2 = [−0.5, 0.4, −0.1, 0.2]T , t2 = [−0.1, 0.5, 0.3]T
• Robot 1 sees point: pR1 = [0.5, 0, 0.2]T . What are these coordinates in robot 2’s coordinate system?

transformation that preserves distances and angles

Iteratively applies the translations

T1W and T2W are instances of Isometry3d

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T1W is the transformation (rotation being q1 followed by translation by t1)

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P2 is the answer, resulting transformed point

Now, let’s take a look at some basic matrix things you can do with Eigen !

Robot Discussion

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Hardware

• Raspberry Pi Model 4B
• RPLidar by SlamTec (2D LIDAR points)
• Points are mapped to a 2D coordinate plane using r distance (time it takes for light beam to return back to LIDAR) and θ angle of rotation. Raw format is polar coordinate
• We need ROS (RViz) to access the laser mapping tools

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Software

• RViz
• ROS1 (Noetic Nimjemys)

Next few weeks.. the big picture

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• Install ROS1 Using Docker