J_b is partitioned as:
\[ \mathbf{J}_{b} = \begin{bmatrix} \mathbf{J}_{bw} \\ \mathbf{J}_{bv} \end{bmatrix} \]
What is the dimension of \( \mathbf{J}_{bv} \mathbf{J}_{bv}^T \), which is used to generate the linear component of the manipulability ellipsoid?
Options:
- 3 x 3
- 6 x 6
- n x n
Answer (1)
J b is a matrix that is partitioned into two submatrices: J b w and J b v . Given the expression J b v J b v T , we want to determine its dimension.
In a typical robotics context, J b v represents the Jacobian matrix associated with the velocity components of a robotic manipulator, where columns correspond to different joints and rows represent different velocity components in space.
Dimensions of J b v
The dimension of J b v is generally expressed as m × n , where:
m is the number of velocity components (often 3 for translational velocity and 3 for angular velocity in 3D space).
n is the number of joints or degrees of freedom of the manipulator.
Calculating the Dimension of J b v J b v T
The matrix J b v J b v T is the result of multiplying J b v by its transpose.
If J b v is of dimension m × n , then J b v T will be of dimension n × m .
Thus, J b v J b v T results in a matrix of dimension m × m .
Conclusion
Considering that J b v usually has 3 rows corresponding to velocity components in a typical 3D application, J b v J b v T would therefore be a 3 × 3 matrix. This reflects the linear component of the manipulability ellipsoid in the space of velocities.
Chosen Option: 3 x 3
