Case 1: Rotating Inertia w.r.t Base Motor

Vertical Rotation

Segment 1

  • The rotation axis passes through the natural axis of the segment.
\[I = \frac{1}{2} M R^2\]

Segment 2

  • The rotation axis is perpendicular to the surface.
\[I = \frac{1}{4} M R^2 + \frac{1}{12} M L^2\]

Segment 3

  • The rotation axis passes through the natural axis of the segment.
\[I = \frac{1}{2} M R^2\]

Hand Assembly

Base Slab

  • The rotation axis is perpendicular to the surface.
\[I_{\text{base}} = \frac{1}{12} M (a^2 + b^2)\]

Holder Slabs

  • The rotation axis is parallel to the surface and along the length of the slabs but does not pass through the center of mass.
\[I_{\text{holder, COM.parallel}} = \frac{1}{12} M a^2\]
  • Using the Parallel Axis Theorem:
\[I_{\text{holder}} = I_{\text{holder, COM.parallel}} + M l^2\]
  • For both holders:
\[I_{\text{holders}} = 2 \times \left( \frac{1}{12} M a^2 + M l^2 \right)\]

Total Moment of Inertia for the Hand Assembly

\[I_{\text{total}} = I_{\text{base}} + I_{\text{holders}}\]

Case 2: Rotating Inertia w.r.t Joint Motor

Horizontal Rotation

Segment 3

  • The rotation axis is perpendicular to the surface and does not pass through the center of mass.
\[I = \frac{1}{4} M R^2 + \frac{1}{12} M L^2\]

Hand Assembly

Base Slab

  • The rotation axis is parallel to the surface and along the length of the slab but does not pass through the center of mass.
\[I_{\text{base, COM.parallel}} = \frac{1}{12} M a^2\]
  • Using the Parallel Axis Theorem:
\[I_{\text{base}} = I_{\text{base, COM.parallel}} + M l^2\]

Holder Slabs

  • The rotation axis is perpendicular to the surface but does not pass through the center of mass.
\[I_{\text{holder, COM.perpendicular}} = \frac{1}{12} M (a^2 + b^2)\]
  • Using the Parallel Axis Theorem:
\[I_{\text{holder}} = I_{\text{holder, COM.perpendicular}} + M l^2\]
  • For both holders:
\[I_{\text{holders}} = 2 \times \left( \frac{1}{12} M (a^2 + b^2) + M l^2 \right)\]

Total Moment of Inertia for the Hand Assembly

\[I_{\text{total}} = I_{\text{base}} + I_{\text{holders}}\]

Payload Contribution: The object exhibits a spherical geometry

When the the clamp secures the payload, the moment of inertia of the payload is added to the total moment of inertia.

\[I = \frac{2}{5} M R^2\]