The document discusses motion planning in robotics, focusing on rigid-body transformations and their various representations, including Euler angles, homogeneous transformations, and quaternions. It emphasizes the importance of understanding rotations and translations in both fixed and moving frames, and highlights the limitations such as gimbal lock. Different transformation techniques are compared for their efficiency and applicability in motion planning contexts.

1. RBE 550 MOTION PLANNING
BASED ON DR. DMITRY BERENSON’S RBE
550
Jane Li
Assistant Professor
Mechanical Engineering & Robotics Engineering
http://users.wpi.edu/~zli11
Transformation

2. Announcement
 Project Presentation
 # individual projectVS # team project
 All or some?

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Representation of Rigid-body Configurations
 Parameterization matter!

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Representation of Rigid-body Configurations
 Many representations
 Euler angle
 Homogeneous transformation
 Quaternions
 Different representations have pros and cons

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Transformations
 An understanding of 2D and 3D rigid-body transformations is key
to motion planning
 There is no “best” representation
 Each representation is useful in a different way

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Right-handed Coordinate System

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Representation for a Position
 3X1 position vector in a reference coordinate system

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Representation for a Rotation
 Attach a frame to the Body B
 Rotation of Body B with respect to Frame A

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Representation of a Frame
 Complete specification of a 3D object
 Position + Orientation

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Translation of Frame
 Translation of Frame B with respect to Frame A

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Rotation of a Frame
 Rotation of Frame B with respect to Frame A

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Inverse a Rotation Frame
 Given
 Rotation matrix from Frame B with respect to Frame A 𝑅
𝐵
𝐴
 Calculate
 Rotation matrix from Frame A with respect to Frame B 𝑅
𝐴
𝐵

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Inverse a Rotation Frame

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Rotation Frames - Example
 Given
 Calculate
 Solution

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Rotation Frames - Example

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Rotation Frames - Example

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Rotation about X,Y, and Z-axis
 Rotation matrices
 Rotation about
 A fixed reference frame
 A moving reference frame – Euler Angle

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Rotation about a Fixed Frame
 Each rotation of Frame B takes place about an axis in Frame A

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Rotation about a Fixed Frame
 Compute rotation with respect to (World) Frame A

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Rotation about a Fixed Frame
 Special case

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Rotation about a Moving Frame – Euler Angle
 Each rotation of Frame B takes place about an axis in Frame B

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Rotation about a Moving Frame – Euler Angle
 Compute rotation with respect to (local) Frame B

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Fixed AngleVS Euler Angle
 Same result, but different operation order

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Gimbal Lock
 Because rotations are performed in orders about moving axis
 Previous operation affects the next operation

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Gimbal Lock

26. RBE 550 MOTION PLANNING
BASED ON DR. DMITRY BERENSON’S RBE
550
Gimbal Lock – Singularity Example
 Singularities
 Multiple EulerAngles map to one rotation (Gimbal Lock)
 Let’s say this is our convention:
 Lets set β = 0
 Multiplying through, we get:
 Simplify: α and γ do the same thing!
We have lost a degree
of freedom!

27. HomogeneousTransformation

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General Frame =Translation + Rotation
 Frame B with respect to Frame A

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HomogeneousTransformation
 Translation and Rotation in a single matrix
 Which one is performed first?

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First Rotation,ThenTranslation

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Example
 Given
 Manipulation
 Rotate Frame B about Z-axis of Frame A by 30 deg, then
 Translate 10 units along X-axis and 5 units along Y-axis, in Frame A
 Find
 in FrameA

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Example

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CompoundTransformation
 Given
 Calculate

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InvertedTransformation
 Given
 Calculate

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Example
 Given
 Manipulation
 Rotate Frame B about Z-axis of FrameA by 30 deg, then
 Translate 4 units along X-axis and 3 units alongY-axis, in FrameA
 Find

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Example

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Transformation Equations
 Given
 Calculate

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Quaternion

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Quaternions
 Generalizations of complex numbers
 Identities

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Intuition from Complex Numbers
 Use a second “imaginary” dimension
 Permits manipulation of rotations like a vector
i
θ
Rotation about 3 axes

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Notation
 4-tuples
 Hyper-complex number
 Real + Imaginary
 Ordered doublet
 Exponential

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Operation
 Addition
 Multiplication
Non-commutative

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Operation
 Conjugate – just like a complex number
 Dot product
 Norm
 Product with conjugate = dot product
Another way to compute norm

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Operation
 Inverse
 Therefore
 Note that for unit quaternion,

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Rotation of aVector
 Convert a vector to quaternion
given
 Rotation axis + Rotation angle
 Rotating a vector – sandwich
 Composite rotation
polar decomposition

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Quaternion to Rotation Matrix
 For a quaternion
 Rotation matrix

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Comparison
 Euler angle
 Intuitive
 Rotation and translation are separate
 Gimbal lock -Transformation is not unique at singularity
 HomogenousTransform
 Intuitive
 Rotation +Translation in one shot
 Not compact
 Quaternion
 Rotation and translation are separate, but in the same format
 Fancy math, less intuitive
 Compact format – efficient for the computation of some problems

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Readings