Rotating Kinematics

Motion Analysis of the Robotic Arm

Assumptions:

• The arm is considered a point mass.
• The arm is initially aligned with the reference line, which is the zero angular position.

Kinematic Equations:

The following equations will be used in all steps to determine the required angular acceleration.

Angular Displacement:

\[\Delta \theta = \theta_{\text{target}} - \theta_{\text{initial}}\]

Angular Velocity:

The time taken to complete the angular displacement ( \Delta \theta ) is ( t ), and the angular velocity at any instant is:

\[\omega_{\text{final}} = \frac{d \theta}{dt}\]

Angular Acceleration:

The rate of change of angular velocity:

\[\alpha = \frac{d \omega}{dt}\]

Kinematic Equation for Angular Displacement:

\[\theta_{\text{final}} = \theta_{\text{initial}} + \omega_{\text{initial}} \cdot t + \frac{1}{2} \alpha \cdot t^2\]

Solving for the required angular acceleration:

\[\alpha_{\text{required}} = \frac{2}{t^2} \cdot (\theta_{\text{final}} - \theta_{\text{initial}} - \omega_{\text{initial}} \cdot t)\] \[\alpha_{\text{required}} = \frac{2}{t^2} \cdot \Delta \theta\]

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Step 1: Rotate to the Right Side (No Payload)

Given:

• The object is at rest initially \(\omega\_{\text{initial}} = 0\)
• Initial angular position \(\theta\_{\text{initial}} = 0\)
• The arm must rotate clockwise to reach the object at \(\theta\_{\text{target}}\)

The above kinematic equations are used to compute the required angular acceleration.

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Step 2: Lower the Arm

To visualize on a 2D plane, we can consider the Y-Z plane and adjust the view to clearly represent the circular rotation.

This involves rotation at the joint. The arm segment must rotate clockwise to lower the clamp.

Given:

• The arm is initially at \(\theta*{\text{initial}}\) and The arm must rotate to reach the object at: \(\theta*{\text{target}}\)

The same kinematic equations are used to calculate the required angular acceleration.

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Step 3: Rotate to the Left Side (With Payload) to the Container Location

Once the object is picked up, the arm must rotate counterclockwise to place it in the container.

Key Change:

• The arm now carries the payload, so the moment of inertia of the arm changes.

The same kinematic equations are used to calculate the required angular acceleration, but now considering the increased moment of inertia.

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