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Autonomous Mobile Manipulation

Transformations

C. Papachristos

Robotic Workers (RoboWork) Lab

University of Nevada, Reno

CS-791

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Coordinate Systems

  • In Manipulation, Guidance, Navigation and Control, reference frames are fundamental

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  • Describe the relative position and orientation of:
    • Robot Base frame relative to the Inertial frame
    • Arm Base frame relative to the Robot Base frame
    • End-effector relative to Arm Base frame
    • Camera relative to the End-effector
    • Robot relative to a manipulation Target

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  • Some expressions are easier to formulate in specific frames:
    • Newton’s law
    • End-effector pose
    • Externally applied forces/moments
    • Camera frames
    • Inertial sensor data
    • GPS coordinates

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ENU & NED Conventions

  • Ground Robots and ROS use East-North-Up (ENU):
    • Z is up.
    • X is aligned with the East.
    • Right-hand coordinate system.
  • UAVs and IMUs typically use aircraft conventions

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  • North-East-Down (NED):
    • Z is the direction of gravity.
    • Vehicle and Inertial frame have the same orientation.
    • Vehicle frame is fixed at the Center of Mass (CoM).
  • Computer Vision uses�Camera Coordinates:
    • Z away from camera.
    • XY aligned with image projection plane.

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Vectors

 

 

 

 

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Vectors

 

 

 

 

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Rotations

 

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Rotations

 

 

Specifies relative orientation of {B} as seen from frame{S}

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Rotations

 

 

 

 

But from previous:

 

 

 

and

 

, so:

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Rotations

 

 

 

 

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Rotations

 

 

 

 

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ENU Convention

NED Convention

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Special Orthogonal Group SO(3)

 

More on these later on…

 

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Axis-Angle

 

 

 

 

 

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Euler Angles

  • Sequence of single-axis rotations

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  • ψ is the “Yaw” angle about the k (or z) axis:

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​

 

 

 

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Euler Angles

  • Sequence of single-axis rotations

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  • θ is the “Pitch” angle about the j (or y) axis:

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​

 

 

 

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Euler Angles

  • Sequence of single-axis rotations

  • φ is the “Roll” angle about the i (or x) axis:

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Euler Angles

  • Rotation Hierarchy:
    • Yaw, Pitch, Roll (Others also exist)

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  • Yaw/Pitch/Roll Rotation Composition:
    • Post-multiply (sequence of rotations w.r.t. Body-frame) one after the other

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​

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  • So to transform a vector:

 

 

 

 

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Orientation Representations

 

 

 

Axis-Angle

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Advantages:

  • Infinitesimal rotation intuition

Disadvantages:

  • Cannot be composed directly
  • Cannot directly transform points

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Quaternions

 

 

Scalar part

Vector part

 

 

Note: Required to make division� algebra associative.� (but commutativity is still lost…)

(with respect to Hamilton Product)

Note: The Hamilton Product� of two quaternions is also a quaternion

 

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Rotation Quaternions

 

 

therefore:

 

 

 

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Quaternions

  • A conversion also exists between this representation and the Rotation matrix one:

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  • Conversions also exist between this representation and the Euler angles one:

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Rotation Quaternions

 

 

 

 

 

 

 

 

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Quaternions

 

 

 

  • Disadvantages

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Configuration / Pose

 

 

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Homogeneous Representation

 

 

 

 

 

 

 

 

 

and:

 

Note(!):

 

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Rotation – Translation

 

1)

2)

 

 

composed as:

 

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Rotation – Translation

 

 

composed as:

 

1)

2)

 

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Special Euclidian Group SE(3)

 

More on these later on…

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Resources

  • Equivalent Representations and Transformations – a CheatSheet

http://people.csail.mit.edu/jstraub/download/straubTransformationCookbook.pdf

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Resources

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Resources

  • Thorough academic resources:

Modern Robotics: Mechanics, Planning, and Control�Kevin M. Lynch and Frank C. Park, Cambridge University Press, 2017

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Resources

  • Python toolbox library for homogeneous matrix transformations & quaternions

https://pypi.org/project/transforms3d/

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Resources

  • ROS’ Transformation(s) library: http://wiki.ros.org/tf2

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Resources

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Time for Questions !

CPE-791

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