#### Describing rotation in 2D

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We learn how to describe the orientation of an object by a 2×2 rotation matrix which has some special properties.

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We learn how to describe the orientation of an object by a 2×2 rotation matrix which has some special properties.

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The orientation of a body in 3D can be described by three angles, examples of which are Euler angles and roll-pitch-yaw angles. Note that in the MATLAB example at 8:24 note that recent versions of the Robotics Toolbox (9.11, 10.x) give a different result: >> rpy2r(0.1,0.2,0.3)ans = 0.9363 -0.2751 0.2184 0.2896 0.9564 -0.0370 -0.1987 0.0978 […]

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We resume our analysis of the 6-link robot Jacobian and focus on the rotational velocity part.

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We learn the mathematical relationship between angular velocity of a body and the time derivative of the rotation matrix describing the orientation of that body.

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We learn how to create smoothly varying orientation in 3D by interpolating Euler angles and Quaternions. In the MATLAB example starting at 5:44 I use the Quaternion class. For Toolbox version 10 (2017) please use UnitQuaternion instead.

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The orientation of a body in 3D can also be described by a single rotation about a particular axis in space.

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If we apply a sequence of 3D rotations to an objects we see that the order in which they are applied affects the final result.

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We learn how to describe the orientation of an object by a 3×3 rotation matrix which has some special properties.

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The orientation of a body in 3D can also be described by a unit-Quaternion, an unusual but very useful mathematical object. In the MATLAB example starting at 3:48 I use the Quaternion class. For Toolbox version 10 (2017) please use UnitQuaternion instead.

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The orientation of a body in 3D can also be described by two vectors, often called the approach and orientation vectors.