Notes in Week 4- Jacobian and Dynamics

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Status	Last Update	Fields
Published	11/26/2024	{{c1::Velocity kinematics}} establishes the relationship between the linear and angular velocities of the end-effector and the joint velocities.
Published	11/26/2024	The Jacobian matrix generalizes the notion of the {{c1::derivative}} to relate joint rates to end-effector velocities.
Published	11/26/2024	In velocity kinematics, the Jacobian matrix is derived from the {{c1::forward kinematic}} equations.
Published	11/26/2024	The {{c1::linear velocity}} of any point on a rigid body in rotation is given by \( \mathbf{v} = \mathbf{\omega} \times \mathbf{r} \).
Published	11/26/2024	The angular velocity of a rigid body rotating about a fixed axis is given by {{c1::\( \mathbf{\omega} = \dot{\theta} \mathbf{k} \),}} where \( \mathbf…
Published	11/26/2024	The objective in velocity kinematics is to relate the {{c1::linear}} and {{c2::angular}} velocities of the end-effector to joint velocities.
Published	11/26/2024	For an \( n \)-link manipulator, the {{c1::Jacobian}} \( \mathbf{J} \) is a matrix mapping joint velocities \( \dot{\mathbf{q}} \) to end-ef…
Published	11/26/2024	The angular velocity of the end-effector can be written as {{c2::\( \mathbf{\omega} = \mathbf{J}_\omega \dot{\mathbf{q} } \),}} with \( \mathbf{J…
Published	11/26/2024	The combined expression {{c1::\( \mathbf{\nu} = \mathbf{J} \dot{\mathbf{q} } \)}} involves both linear and angular velocities through the composite Ja…
Published	11/26/2024	For a rigid body in rotation, every point moves in a circle centered on the {{c1::axis of rotation}}.
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