Typical Moments of Inertia

This section explains how to calculate the moment of inertia about the center of mass for a uniform-density object with mass \(m\). Use this when creating a URDF.

Rectangular prism with side lengths \(x\), \(y\), and \(z\)

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\[ \begin{aligned} I_{xx} &= \frac{m (y^2 + z^2)}{12} \\ I_{yy} &= \frac{m (z^2 + x^2)}{12} \\ I_{zz} &= \frac{m (x^2 + y^2)}{12} \\ I_{xy} &= I_{yz} = I_{zx} = 0 \\ \end{aligned} \]

<xacro:macro name="box_inertia" params="x y z m">
  <inertia ixx="${m * (y*y + z*z) / 12}" ixy="0.0" ixz="0.0" iyy="${m * (z*z + x*x) / 12}" iyz="0.0" izz="${m * (x*x + y*y) / 12}"/>
</xacro:macro>

Cylinder with radius \(r\) and height \(h\)

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\[ \begin{aligned} I_{xx} &= I_{yy} = m (\frac{r^2}{4} + \frac{h^2}{12}) \\ I_{zz} &= \frac{m r^2}{2} \\ I_{xy} &= I_{yz} = I_{zx} = 0 \\ \end{aligned} \]

<xacro:macro name="cylinder_inertia" params="r h m">
  <inertia ixx="${m*r*r/4 + m*h*h/12}" ixy="0.0" ixz="0.0" iyy="${m*r*r/4 + m*h*h/12}" iyz="0.0" izz="${m*r*r/2}"/>
</xacro:macro>

Sphere with radius \(r\)

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\[ \begin{aligned} I_{xx} &= I_{yy} = I_{zz} = \frac{2}{5} m r^2 \\ I_{xy} &= I_{yz} = I_{zx} = 0 \\ \end{aligned} \]

<xacro:macro name="sphere_inertia" params="r m">
  <inertia ixx="${0.4 * m * r*r}" ixy="0.0" ixz="0.0" iyy="${0.4 * m * r*r}" iyz="0.0" izz="${0.4 * m * r*r}"/>
</xacro:macro>