Transformation. Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering
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1 RBE 550 MOTION PLANNING BASED ON DR. DMITRY BERENSON S RBE 550 Transformation Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering
2 Announcement Project Presentation # individual project VS # team project All or some?
3 Representation of Rigid-body Configurations Parameterization matter!
4 Representation of Rigid-body Configurations Many representations Euler angle Homogeneous transformation Quaternions Different representations have pros and cons
5 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
6 Right-handed Coordinate System BASED ON PROF. JACOB ROSEN MAE 263
7 Representation for a Position 3X1 position vector in a reference coordinate system
8 Representation for a Rotation Attach a frame to the Body B Rotation of Body B with respect to Frame A
9 Representation of a Frame Complete specification of a 3D object Position + Orientation
10 Translation of Frame Translation of Frame B with respect to Frame A
11 Rotation of a Frame Rotation of Frame B with respect to Frame A
12 Inverse a Rotation Frame Given A Rotation matrix from Frame B with respect to Frame A B R Calculate B Rotation matrix from Frame A with respect to Frame B A R
13 Inverse a Rotation Frame BASED ON PROF. JACOB ROSEN MAE 263
14 Rotation Frames - Example Given Calculate Solution
15 Rotation Frames - Example BASED ON PROF. JACOB ROSEN MAE 263
16 Rotation Frames - Example BASED ON PROF. JACOB ROSEN MAE 263
17 Rotation about X, Y, and Z-axis Rotation matrices Rotation about A fixed reference frame A moving reference frame Euler Angle
18 Rotation about a Fixed Frame Each rotation of Frame B takes place about an axis in Frame A
19 Rotation about a Fixed Frame Compute rotation with respect to (World) Frame A
20 Rotation about a Fixed Frame Special case
21 Rotation about a Moving Frame Euler Angle Each rotation of Frame B takes place about an axis in Frame B
22 Rotation about a Moving Frame Euler Angle Compute rotation with respect to (local) Frame B
23 Fixed Angle VS Euler Angle Same result, but different operation order
24 Gimbal Lock Because rotations are performed in orders about moving axis Previous operation affects the next operation
25 Gimbal Lock BASED ON PROF. JACOB ROSEN MAE 263
26 Gimbal Lock Singularity Example RBE 550 MOTION PLANNING BASED ON DR. DMITRY BERENSON S RBE 550 Singularities Multiple Euler Angles 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 Homogeneous Transformation
28 General Frame = Translation + Rotation Frame B with respect to Frame A
29 Homogeneous Transformation Translation and Rotation in a single matrix Which one is performed first?
30 First Rotation, Then Translation BASED ON PROF. JACOB ROSEN MAE 263
31 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 Frame A
32 Example BASED ON PROF. JACOB ROSEN MAE 263
33 Compound Transformation Given Calculate
34 Inverted Transformation Given Calculate
35 Example Given Manipulation Rotate Frame B about Z-axis of Frame A by 30 deg, then Translate 4 units along X-axis and 3 units along Y-axis, in Frame A Find
36 Example BASED ON PROF. JACOB ROSEN MAE 263
37 Transformation Equations Given Calculate
38 Quaternion BASED ON PROF. JACOB ROSEN MAE 263
39 Quaternions Generalizations of complex numbers Identities
40 Intuition from Complex Numbers Use a second imaginary dimension Permits manipulation of rotations like a vector i Rotation about 3 axes θ
41 Notation 4-tuples Hyper-complex number Real + Imaginary Ordered doublet Exponential
42 Operation Addition Multiplication Non-commutative
43 Operation Conjugate just like a complex number Dot product Norm Product with conjugate = dot product Another way to compute norm
44 Operation Inverse Therefore Note that for unit quaternion,
45 Rotation of a Vector Convert a vector to quaternion given Rotation axis + Rotation angle polar decomposition Rotating a vector sandwich Composite rotation
46 Quaternion to Rotation Matrix For a quaternion Rotation matrix
47 Comparison Euler angle Intuitive Rotation and translation are separate Gimbal lock - Transformation is not unique at singularity Homogenous Transform 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
48 Readings BASED ON PROF. JACOB ROSEN MAE 263
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