# 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 `` Element ```xml ``` - **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 ```xml ``` ### Cylinder Inertia Macro ```xml ``` ### Sphere Inertia Macro ```xml ``` ## 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 `` in `` 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 | Mistake | Symptom | Fix | |---|---|---| | Zero inertia (all zeros) | Simulation crash or NaN | Use formulas above; even tiny links need nonzero inertia | | Inertia at link origin, not CoM | Asymmetric behavior | Set `` to actual center of mass | | Mass too small for link size | Link flies away under contact | Verify mass is physically reasonable | | Huge mass ratio between links | Solver instability | Keep mass ratios < 50:1 between connected links | | Forgetting `` entirely | Link treated as zero-mass (fixed to world in some solvers) | Always include for simulated links | ## Quick Estimation Script ```python 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} ```