3-D Kinematics

Position and Orientation of a Rigid BodyPosition and Orientation of a Rigid Body

Position and Orientation of a Rigid BodyPosition and Orientation of a Rigid Body

The component of each unit vector are the direction cosines of the axes of frame O’-x’y’z’

The position of origin O’ with respect to O-xyz is expressed by the relation

Rotation MatrixRotation Matrix

Orientation can be described by rotation matrix

R is orthogonal matrix

Elementary RotationsElementary Rotations

Rotation by an angle about axis z

Elementary RotationsElementary Rotations

Rotation by an angle about axis y

Rotation by an angle about axis x

Representation of a VectorRepresentation of a Vector

Representation of a VectorRepresentation of a Vector

Representation of p w.r.t O-xyz

Representation of p w.r.t O-x’y’z’

Rotation of a VectorRotation of a Vector

Equivalent Geometrical Meaningsof Rotation Matrix

Equivalent Geometrical Meaningsof Rotation Matrix

Composition of Rotation MatricesComposition of Rotation Matrices

Let Rij denote the rotation matrix of Frame i with respect to Frame j

Post-multiplication interpretation

Refer to current frame

Pre-multiplication interpretation

Refer to fixed frame

1112

01

0 nn

iin RRRRR

Euler AnglesEuler Angles

Minimal representation of orientation

Three parameters are sufficient

Euler Angles

Two successive rotations are not made about parallel axes

How many kinds of Euler angles are there?

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ZYZ AnglesZYZ Angles

The rotation described by ZYZ angles is

ZYZ AnglesZYZ Angles

ZYZ AnglesZYZ Angles

The rotation matrix is

ZYZ AnglesZYZ Angles

Inverse problem: determine the Euler angles corresponding to a given rotation matrix

Solution 1: theta is in the range (0, pi)

ZYZ AnglesZYZ Angles

y=1 x=1;

y=-1 x=1;

y=1 x=-1;

y=-1 x=-1;

ZYZ AnglesZYZ Angles

Solution 1: theta is in the range (0, pi)

ZYZ AnglesZYZ Angles

Solution 1: theta is in the range (0, pi)

ZYZ AnglesZYZ Angles

Solution 2: theta is in the range (-pi, 0)

ZYZ AnglesZYZ Angles

Solution 2: theta is in the range (-pi, 0)

ZYZ AnglesZYZ Angles

Solution 2: theta is in the range (-pi, 0)

ZYZ AnglesZYZ Angles

What will happen if sin(theta) = 0?

Matlab: eul2tr, tr2eul

Roll-Pitch-Yaw AnglesRoll-Pitch-Yaw Angles

Originate from (aero)nautical field

Roll-Pitch-Yaw AnglesRoll-Pitch-Yaw Angles

MATLAB: QUATDEMO

The rotation matrix is

Roll-Pitch-Yaw AnglesRoll-Pitch-Yaw Angles

Inverse problem: determine the Euler angles corresponding to a given rotation matrix

Solution 1: theta is in the range (-pi/2, pi/2)

Roll-Pitch-Yaw AnglesRoll-Pitch-Yaw Angles

Solution 2: theta is in the range (pi/2, 3pi/2)

Roll-Pitch-Yaw AnglesRoll-Pitch-Yaw Angles

What will happen if cos(theta) = 0?

Matlab: rpy2tr, tr2rpy

Roll-Pitch-Yaw AnglesRoll-Pitch-Yaw Angles

Non-minimal representation: four parameters

The unit vector of a rotation axis w.r.t O-xyz

The angle theta about the axis

Matlab: quatdemo

Angle and AxisAngle and Axis

Angle and AxisAngle and Axis

•Align r with z

•Rotate by theta about z

•Realign with the initial direction of r

Attention: always refer to the fixed frame

Angle and AxisAngle and Axis

The resulting rotation matrix is

Angle and AxisAngle and Axis

The inverse problem

Remember: the three component of r is not independent

Angle and AxisAngle and Axis

Problems:

solution is not unique

r is arbitrary when theta = 0

Unit QuaternionUnit Quaternion

Unit quaternion is defined as

Unit QuaternionUnit Quaternion

Inverse problem:

Matlab:quaternion, plot, quaternion.t