Inertia Calculation

name: inertia-calculation description: 'Calculate inertial properties for URDF. Use when computing mass, center of mass, and inertia tensors for box, cylinder, or sphere links.'

Inertia Calculation

Why Correct Inertias Matter

Gazebo's physics engine requires valid inertial parameters for every link in the simulation. Incorrect inertias cause:

• Simulation instability (robot flies away, oscillates, or explodes)
• Unrealistic dynamics (robot slides, tips, or doesn't respond to forces)
• Solver divergence (Gazebo crashes or produces NaN values)

The <inertial> Element

<inertial>
  <mass value="2.5"/>
  <origin xyz="0 0 0.03" rpy="0 0 0"/>  <!-- center of mass -->
  <inertia ixx="0.0108" ixy="0.0" ixz="0.0"
           iyy="0.0417" iyz="0.0"
           izz="0.0483"/>
</inertial>

• mass: in kilograms
• origin: center of mass relative to the link's origin frame
• inertia: 3×3 symmetric inertia tensor (6 unique values), in kg·m², about the center of mass

The inertia tensor matrix:

$$ I = \begin{bmatrix} i_{xx} & i_{xy} & i_{xz} \ i_{xy} & i_{yy} & i_{yz} \ i_{xz} & i_{yz} & i_{zz} \end{bmatrix} $$

Standard Shape Formulas

Box (dimensions $l_x$, $l_y$, $l_z$, mass $m$)

$$ i_{xx} = \frac{m}{12}(l_y^2 + l_z^2), \quad i_{yy} = \frac{m}{12}(l_x^2 + l_z^2), \quad i_{zz} = \frac{m}{12}(l_x^2 + l_y^2) $$

All off-diagonal terms = 0 (when axes align with principal axes).

Cylinder (radius $r$, height $h$ along z-axis, mass $m$)

$$ i_{xx} = i_{yy} = \frac{m}{12}(3r^2 + h^2), \quad i_{zz} = \frac{mr^2}{2} $$

Sphere (radius $r$, mass $m$)

$$ i_{xx} = i_{yy} = i_{zz} = \frac{2mr^2}{5} $$

Xacro Inertia Macros

Box Inertia Macro

<xacro:macro name="box_inertia" params="m x y z">
  <inertial>
    <mass value="${m}"/>
    <inertia ixx="${m/12*(y*y + z*z)}" ixy="0" ixz="0"
             iyy="${m/12*(x*x + z*z)}" iyz="0"
             izz="${m/12*(x*x + y*y)}"/>
  </inertial>
</xacro:macro>

<!-- Usage: chassis 0.4m x 0.3m x 0.1m, 5kg -->
<link name="base_link">
  <xacro:box_inertia m="5.0" x="0.4" y="0.3" z="0.1"/>
  <visual>
    <geometry><box size="0.4 0.3 0.1"/></geometry>
  </visual>
</link>

Cylinder Inertia Macro

<xacro:macro name="cylinder_inertia" params="m r h">
  <inertial>
    <mass value="${m}"/>
    <inertia ixx="${m/12*(3*r*r + h*h)}" ixy="0" ixz="0"
             iyy="${m/12*(3*r*r + h*h)}" iyz="0"
             izz="${m*r*r/2}"/>
  </inertial>
</xacro:macro>

Sphere Inertia Macro

<xacro:macro name="sphere_inertia" params="m r">
  <inertial>
    <mass value="${m}"/>
    <inertia ixx="${2*m*r*r/5}" ixy="0" ixz="0"
             iyy="${2*m*r*r/5}" iyz="0"
             izz="${2*m*r*r/5}"/>
  </inertial>
</xacro:macro>

Parallel Axis Theorem

When combining shapes or when the center of mass is offset from the link origin, use the parallel axis theorem:

$$ I_{new} = I_{cm} + m \cdot d^2 $$

For a full 3D shift by $(d_x, d_y, d_z)$:

$$ i_{xx}' = i_{xx} + m(d_y^2 + d_z^2) $$ $$ i_{yy}' = i_{yy} + m(d_x^2 + d_z^2) $$ $$ i_{zz}' = i_{zz} + m(d_x^2 + d_y^2) $$ $$ i_{xy}' = i_{xy} - m \cdot d_x \cdot d_y $$

This is needed when the <origin> in <inertial> is non-zero, or when approximating a complex body as a combination of primitives.

CAD Export

Fusion 360 and SolidWorks can compute inertial properties directly:

• Fusion 360: Inspect → Physical Properties → shows mass, center of mass, and inertia tensor
• SolidWorks: Evaluate → Mass Properties

Ensure the coordinate frame in the CAD tool matches the URDF link frame. Export values and paste into the URDF.

Sanity Checks

Before running simulation, verify:

1. All diagonal components positive: $i_{xx} > 0$, $i_{yy} > 0$, $i_{zz} > 0$
2. Triangle inequality: $i_{xx} + i_{yy} \geq i_{zz}$ (and cyclic permutations)
3. Off-diagonals bounded: $|i_{xy}| \leq \sqrt{i_{xx} \cdot i_{yy}}$
4. Reasonable magnitudes: for a 1 kg, 0.1 m object, inertias ~0.001 kg·m²
5. No huge ratios: adjacent links shouldn't have inertia ratios > 100:1

Common Mistakes

Quick Estimation Script

def box_inertia(m, x, y, z):
    return {
        'ixx': m/12*(y**2 + z**2),
        'iyy': m/12*(x**2 + z**2),
        'izz': m/12*(x**2 + y**2),
    }

def cylinder_inertia(m, r, h):
    return {
        'ixx': m/12*(3*r**2 + h**2),
        'iyy': m/12*(3*r**2 + h**2),
        'izz': m*r**2/2,
    }

# Example: the robot chassis ~5kg, 0.4m x 0.3m x 0.1m
print(box_inertia(5.0, 0.4, 0.3, 0.1))
# {'ixx': 0.0417, 'iyy': 0.0708, 'izz': 0.1042}