Center of Mass Jacobian

Previously we have covered the kinematic Jacobian which provided us with the following relationship

$\dot{x}=J\dot{\theta}$

In other words the relationship between the joint and tool tip velocities. The Center of Mass Jacobian(COM Jacobian) is a related concept which provides a relationship between the COM velocity and the joint velocities.

$\dot{x}_{com}=J^{COM}\dot{\theta}$

Now from a previous section, we know that the COM definition is.

$x_{com}=\frac{\sum_{i=0}^{N}x_{i}m_{i}}{\sum_{i=0}^{N}m_{i}}$

now if we define the partial COM. As the COM of the partial chain of links with respect to the base links coordinate frame.

$x_{com}^{*j}=R_{0}^{j}\frac{\sum_{i=j}^{N}x_{i}m_{i}}{\sum_{i=j}^{N}m_{i}}$

Note that we had to rotate the partial COM to get it into the base links coordinate frame.

Then the same method that was used to find the kinematic Jacobian can also be used to find the COM Jacobian. The joint axis doesn’t change but this time the distance from the current joint to the partial COM is used instead of the distance to the tool tip.

One final thing to note of is that the resultant linear velocity should be scaled by the mass of the partial COM because the COM is the average of the multi-mass system and high velocities on smaller masses play a lesser role on the total velocity of the COM.

$J_{n}^{com}=\dot{X}_{n}=\left[\begin{array}{c}\begin{array}{c}\frac{\sum_{i=n}^{N}m_{i}}{\sum_{i=0}^{N}m_{i}}(z_{n}\times x_{com}^{*n})\end{array}\\z_{n}\end{array}\right]\label{eq:com jacobian calc -1}$

So putting these individual joint Jacobian together, we get the total COM Jacobian.

Next: Linear Inverted Pendulum Model (LIPM)

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Center of Mass Jacobian

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