Required Columns for Mass Properties

The Mass Property Table must contain the following columns. Other columns may exist and will remain unmodified.

• id unique identifier for each item (row)

• mass mass of the item (numeric)

• Cx \(x\)-component of center of mass (numeric)

• Cy \(y\)-component of center of mass (numeric)

• Cx \(z\)-component of center of mass (numeric)

• Ixx moment of inertia about the \(x\) axis (numeric)

• Iyy moment of inertia about the \(y\) axis (numeric)

• Izz moment of inertia about the \(z\) axis (numeric)

• Ixy product of inertia relative to the \(x\) and \(y\) axes (numeric)

• Ixz product of inertia relative to the \(x\) and \(z\) axes (numeric)

• Iyz product of inertia relative to the \(y\) and \(z\) axes (numeric)

• POIconv either ‘+’ or ‘-’, indicating the sign convention for products of inertia¹

• Ipoint logical indicator that this item is considered a point mass (i.e., its inertia contribution is negligible)²

Required Columns for Mass Properties Uncertainties

The following columns are required for uncertainty calculations:

• sigma_mass mass uncertainty (numeric)

• sigma_Cx \(x\)-component of center of mass uncertainty (numeric)

• sigma_Cy \(y\)-component of center of mass uncertainty (numeric)

• sigma_Cx \(z\)-component of center of mass uncertainty (numeric)

• sigma_Ixx moment of inertia about the \(x\) axis uncertainty (numeric)

• sigma_Iyy moment of inertia about the \(y\) axis uncertainty (numeric)

• sigma_Izz moment of inertia about the \(z\) axis uncertainty (numeric)

• sigma_Ixy product of inertia relative to the \(x\) and \(y\) axes uncertainty (numeric)

• sigma_Ixz product of inertia relative to the \(x\) and \(z\) axes uncertainty (numeric)

• sigma_Iyz product of inertia relative to the \(y\) and \(z\) axes uncertainty (numeric)

It is the caller’s responsibility to ensure that all values are expressed in appropriate and compatible units.

Tree

The tree is an igraph::graph with vertices named by identifiers in the mass properties table. It can be of arbitrary depth and shape as long as it satisfies certain well-formedness properties:

• it is connected and acyclic (as an undirected graph), i.e., it is a tree

• it is directed, with edge direction going from child to parent

• it contains neither loops (self-edges) nor multiple edges

• it contains a single root vertex (i.e., one whose out degree is zero)

Moments of Inertia

\[ \begin{align} I_{XX} & = \sum_{i=1}^{n} \left[ {I_{XX}}_i + w_i \left( {y}_i^2 + {z}_i^2 \right) - w_i \left( \bar{y}^2 + \bar{z}^2 \right) \right] & = \sum_{i=1}^{n} \left\{ {I_{XX}}_i + w_i \left[ \left( {y}_i - \bar{y} \right)^2 + \left( {z}_i - \bar{z} \right)^2 \right] \right\} \\ I_{YY} & = \sum_{i=1}^{n} \left[ {I_{YY}}_i + w_i \left( {x}_i^2 + {z}_i^2 \right) - w_i \left( \bar{x}^2 + \bar{z}^2 \right) \right] & = \sum_{i=1}^{n} \left\{ {I_{YY}}_i + w_i \left[ \left( {x}_i - \bar{x} \right)^2 + \left( {z}_i - \bar{z} \right)^2 \right] \right\} \\ I_{ZZ} & = \sum_{i=1}^{n} \left[ {I_{ZZ}}_i + w_i \left( {x}_i^2 + {y}_i^2 \right) - w_i \left( \bar{x}^2 + \bar{y}^2 \right) \right] & = \sum_{i=1}^{n} \left\{ {I_{ZZ}}_i + w_i \left[ \left( {x}_i - \bar{x} \right)^2 + \left( {y}_i - \bar{y} \right)^2 \right] \right\} \\ \end{align} \]

Products of Inertia

\[ \begin{align} I_{XY} & = \sum_{i=1}^{n} \left[ {I_{XY}}_i + w_i {{x}_i}{{y}_i} -w_i (\bar{x}\bar{y})\right] & = \sum_{i=1}^{n} \left[ {I_{XY}}_i + w_i ({x}_i - \bar{x})({y}_i - \bar{y})\right] \\ I_{XZ} & = \sum_{i=1}^{n} \left[ {I_{XZ}}_i + w_i {{x}_i}{{z}_i} -w_i (\bar{x}\bar{z})\right] & = \sum_{i=1}^{n} \left[ {I_{XZ}}_i + w_i ({x}_i - \bar{x})({z}_i - \bar{z})\right] \\ I_{YZ} & = \sum_{i=1}^{n} \left[ {I_{YZ}}_i + w_i {{y}_i}{{z}_i} -w_i (\bar{y}\bar{z})\right] & = \sum_{i=1}^{n} \left[ {I_{YZ}}_i + w_i ({y}_i - \bar{y})({z}_i - \bar{z})\right] \\ \end{align} \]

Matrix Formulation

Let \(I\) be the inertia tensor of the aggregate and \(I_i\) be that of part \(i\). The equations for products of inertia above clearly follow the positive integral convention, so

\[ I = \left[ \begin{matrix} I_{xx} & -I_{xy} & -I_{xz} \\ -I_{xy} & I_{yy} & -I_{yz} \\ -I_{xz} & -I_{yz} & I_{zz} \end{matrix} \right] \]

and similarly for \(I_i\).

Noting the repeated appearance of terms of the form \(({x}_i - \bar{x})({y}_i - \bar{y})\), we form the outer product

\[ \boxed{ \begin{align} \boldsymbol{d}_i & = (({x}_i - \bar{x}) \quad ({y}_i - \bar{y}) \quad ({z}_i - \bar{z}))^T \\ \boldsymbol{Q}_i & = \boldsymbol{d}_i {\boldsymbol{d}_i}^T \end{align} } \] Then

\[ \begin{align} \boldsymbol{Q}_i & = \begin{bmatrix} ({x}_i - \bar{x})^2 & ({x}_i - \bar{x})({y}_i - \bar{y}) & ({x}_i - \bar{x})({z}_i - \bar{z}) \\ ({y}_i - \bar{y})({x}_i - \bar{x}) & ({y}_i - \bar{y})^2 & ({y}_i - \bar{y})({z}_i - \bar{z}) \\ ({z}_i - \bar{z})({x}_i - \bar{x}) & ({z}_i - \bar{z})({y}_i - \bar{y}) & ({z}_i - \bar{z})^2 \\ \end{bmatrix} \end{align} \]

Let \(\boldsymbol{s}_i\) be the matrix of inertia tensor summands from the reference equations. That is,

\[ \boldsymbol{I} = \sum_{i=1}^{n} \boldsymbol{s}_i \]

where \[ \begin{align} \boldsymbol{s}_i & = \boldsymbol{I}_i \\ & - w_i \begin{bmatrix} -({y}_i - \bar{y})^2 - ({z}_i - \bar{z})^2 & ({x}_i - \bar{x})({y}_i - \bar{y}) & ({x}_i - \bar{x})({z}_i - \bar{z}) \\ ({x}_i - \bar{x})({y}_i - \bar{y}) & -({x}_i - \bar{x})^2 - ({z}_i - \bar{z})^2 & ({y}_i - \bar{y})({z}_i - \bar{z}) \\ ({x}_i - \bar{x})({z}_i - \bar{z}) & ({y}_i - \bar{y})({z}_i - \bar{z}) & -({x}_i - \bar{x})^2 - ({y}_i - \bar{y})^2 \\ \end{bmatrix} \\ & = \boldsymbol{I}_i \\ & - w_i \begin{bmatrix} ({x}_i - \bar{x})^2 & ({x}_i - \bar{x})({y}_i - \bar{y}) & ({x}_i - \bar{x})({z}_i - \bar{z}) \\ ({x}_i - \bar{x})({y}_i - \bar{y}) & ({y}_i - \bar{y})^2 & ({y}_i - \bar{y})({z}_i - \bar{z}) \\ ({x}_i - \bar{x})({z}_i - \bar{z}) & ({y}_i - \bar{y})({z}_i - \bar{z}) & ({z}_i - \bar{z})^2 \\ \end{bmatrix} \\ & - w_i \begin{bmatrix} -({x}_i - \bar{x})^2 - ({y}_i - \bar{y})^2 - ({z}_i - \bar{z})^2 & 0 & 0 \\ 0 & -({x}_i - \bar{x})^2 - ({y}_i - \bar{y})^2 - ({z}_i - \bar{z})^2 & 0 \\ 0 & 0 &-({x}_i - \bar{x})^2 - ({y}_i - \bar{y})^2 - ({z}_i - \bar{z})^2 \\ \end{bmatrix} \\ & = \boldsymbol{I}_i - w_i \left( \boldsymbol{Q}_i - \mathrm{tr}(\boldsymbol{Q}_i) \boldsymbol{1}_3 \right) \end{align} \]

where \(\boldsymbol{1}_3\) is the 3 ⨉ 3 identity matrix. Therefore

\[ \boxed{ \boldsymbol{I} = \sum_{i=1}^{n} \left( \boldsymbol{I}_i - w_i \boldsymbol{M}_i \right) } \] where

\[ \boxed{ \boldsymbol{M}_i = \boldsymbol{Q}_i - \mathrm{tr}(\boldsymbol{Q}_i) \boldsymbol{1}_3 } \]

The corresponding R code is

  r$inertia <- Reduce(`+`, Map(
    f  = function(v) {
      d <- r$center_mass - v$center_mass
      M <- outer(d, d) - sum(d^2) * diag(3)
      if (v$point) -v$mass * M else v$inertia - v$mass * M
    },
    vl
  ))

Moments of Inertia Uncertainties

\[ \begin{align} \sigma_{I_{XX}} & = \sqrt{ \sum_{i=1}^n \big\{ \sigma_{{I_{XX}}_i}^2 + \big[ 2 w_i ({y}_i - \bar{y}) \sigma_{{y}_i} \big]^2 + \big[ 2 w_i ({z}_i - \bar{z}) \sigma_{{z}_i} \big]^2 + \big[ \big(({y}_i - \bar{y})^2 + ({z}_i - \bar{z})^2 \big)\sigma_{w_i}\big]^2 \big\} } \\ \sigma_{I_{YY}} & = \sqrt{ \sum_{i=1}^n \big\{ \sigma_{{I_{YY}}_i}^2 + \big[ 2 w_i ({x}_i - \bar{x}) \sigma_{{x}_i} \big]^2 + \big[ 2 w_i ({z}_i - \bar{z}) \sigma_{{z}_i} \big]^2 + \big[ \big(({x}_i - \bar{x})^2 + ({z}_i - \bar{z})^2 \big)\sigma_{w_i}\big]^2 \big\} } \\ \sigma_{I_{ZZ}} & = \sqrt{ \sum_{i=1}^n \big\{ \sigma_{{I_{ZZ}}_i}^2 + \big[ 2 w_i ({x}_i - \bar{x}) \sigma_{{x}_i} \big]^2 + \big[ 2 w_i ({y}_i - \bar{y}) \sigma_{{y}_i} \big]^2 + \big[ \big(({x}_i - \bar{x})^2 + ({y}_i - \bar{y})^2 \big)\sigma_{w_i}\big]^2 \big\} } \\ \end{align} \]

Products of Inertia Uncertainties

\[ \begin{align} \sigma_{I_{XY}} & = \sqrt{ \sum_{i=1}^n \big\{ \sigma_{{I_{XY}}_i}^2 + \big[ ({x}_i - \bar{x}) w_i \sigma_{{y}_i}\big]^2 + \big[ ({x}_i - \bar{x})({y}_i - \bar{y})\sigma_{w_i} \big]^2 + \big[ ({y}_i - \bar{y}) w_i \sigma_{{x}_i} \big]^2 \big\} } \\ \sigma_{I_{XZ}} & = \sqrt{ \sum_{i=1}^n \big\{ \sigma_{{I_{XZ}}_i}^2 + \big[ ({x}_i - \bar{x}) w_i \sigma_{{z}_i}\big]^2 + \big[ ({x}_i - \bar{x})({z}_i - \bar{z})\sigma_{w_i} \big]^2 + \big[ ({z}_i - \bar{z}) w_i \sigma_{{x}_i} \big]^2 \big\} } \\ \sigma_{I_{YZ}} & = \sqrt{ \sum_{i=1}^n \big\{ \sigma_{{I_{YZ}}_i}^2 + \big[ ({y}_i - \bar{y}) w_i \sigma_{{z}_i}\big]^2 + \big[ ({y}_i - \bar{y})({z}_i - \bar{z})\sigma_{w_i} \big]^2 + \big[ ({z}_i - \bar{z}) w_i \sigma_{{y}_i} \big]^2 \big\} } \\ \end{align} \]

Matrix Formulation

Let

\[ \boxed{ \begin{align} \boldsymbol{d}_i & = (({x}_i - \bar{x}) \quad ({y}_i - \bar{y}) \quad ({z}_i - \bar{z}))^T \\ {\boldsymbol{\sigma_c}}_i & = ({\sigma_x}_i \quad {\sigma_y}_i \quad {\sigma_z}_i)^T \\ \boldsymbol{P}_i & = \boldsymbol{d}_i {\boldsymbol{\sigma_c}}_i^T \\ \boldsymbol{Q}_i & = \boldsymbol{d}_i {\boldsymbol{d}_i}^T \end{align} } \]

Then

\[ \begin{align} \boldsymbol{P}_i & = \begin{bmatrix} (x_i - \bar{x})\sigma_{x_i} &({x}_i - \bar{x})\sigma_{y_i} &({x}_i - \bar{x})\sigma_{z_i} \\ ({y}_i - \bar{y})\sigma_{x_i} & ({y}_i - \bar{y})\sigma_{y_i} & ({y}_i - \bar{y})\sigma_{z_i} \\ ({z}_i - \bar{z})\sigma_{x_i} & ({z}_i - \bar{z})\sigma_{y_i} & ({z}_i - \bar{z})\sigma_{z_i} \\ \end{bmatrix} \\ \\ \boldsymbol{Q}_i & = \begin{bmatrix} (x_i - \bar{x})^2 &({x}_i - \bar{x})({y}_i - \bar{y}) &({x}_i - \bar{x})({z}_i - \bar{z}) \\ ({y}_i - \bar{y})(x_i - \bar{x}) & ({y}_i - \bar{y})^2 & ({y}_i - \bar{y})({z}_i - \bar{z}) \\ ({z}_i - \bar{z})(x_i - \bar{x}) & ({z}_i - \bar{z})({y}_i - \bar{y}) & ({z}_i - \bar{z})^2 \\ \end{bmatrix} \end{align} \]

Let \(\boldsymbol{s}_i^2\) be the matrix of inertia tensor uncertainty summands in the standard formulas for a given subcomponent \(i\) above. That is,

\[ \boldsymbol{\sigma_I}^2 = \sum_{i=1}^{} \boldsymbol{s}_i^2 \]

Let \({p_X}_i\), \({p_Y}_i\), and \({p_Z}_i\) be the respective diagonal elements of \(P_i\). Let \(\boldsymbol{1}_3\) be the 3 ⨉ 3 identity matrix. If we interpret squaring a matrix as the Hadamard (element-wise) product with itself, then \[ \begin{align} \boldsymbol{s}_i^2 & = {\boldsymbol{\sigma_I}}_i^2 \\ & + \begin{bmatrix} 2 w_i ({y}_i - \bar{y}) \sigma_{y_i} & w_i({x}_i - \bar{x}) \sigma_{y_i} & w_i({x}_i - \bar{x}) \sigma_{z_i} \\ w_i({x}_i - \bar{x}) \sigma_{y_i} & 2 w_i({x}_i - \bar{x}) \sigma_{x_i} & w_i ({y}_i - \bar{y}) \sigma_{z_i} \\ w_i({x}_i - \bar{x}) \sigma_{z_i} & w_i ({y}_i - \bar{y}) \sigma_{z_i} & 2 w_i({x}_i - \bar{x}) \sigma_{x_i} \end{bmatrix}^2 \\ & + \begin{bmatrix} 2 w_i ({z}_i - \bar{z}) \sigma_{z_i} & w_i ({y}_i - \bar{y}) \sigma_{x_i} & w_i ({z}_i - \bar{z}) \sigma_{x_i} \\ w_i ({y}_i - \bar{y}) \sigma_{x_i} & 2 w_i ({z}_i - \bar{z}) \sigma_{z_i} & w_i ({z}_i - \bar{z}) \sigma_{y_i} \\ w_i ({z}_i - \bar{z}) \sigma_{x_i} & w_i ({z}_i - \bar{z}) \sigma_{y_i} & 2 w_i ({y}_i - \bar{y}) \sigma_{y_i} \end{bmatrix}^2 \\ & + \begin{bmatrix} (({y}_i - \bar{y})^2 + ({z}_i - \bar{z})^2)\sigma_{w_i} &({x}_i - \bar{x})({y}_i - \bar{y})\sigma_{w_i} &({x}_i - \bar{x})({z}_i - \bar{z})\sigma_{w_i} \\ ({y}_i - \bar{y})(x_i - \bar{x})\sigma_{w_i} & ((x_i - \bar{x})^2 + ({z}_i - \bar{z})^2)\sigma_{w_i} & ({y}_i - \bar{y})({z}_i - \bar{z})\sigma_{w_i} \\ ({z}_i - \bar{z})(x_i - \bar{x})\sigma_{w_i} & ({z}_i - \bar{z})({y}_i - \bar{y})\sigma_{w_i} & ((x_i - \bar{x})^2 + ({y}_i - \bar{y})^2)\sigma_{w_i} \\ \end{bmatrix}^2 \\ \\ & = {\boldsymbol{\sigma_I}}_i^2 \\ & + w_i^2 \left( \boldsymbol{P}_i - \begin{bmatrix} (x_i - \bar{x})\sigma_{x_i} - 2 ({y}_i - \bar{y})\sigma_{y_i} & 0 & 0 \\ 0 & ({y}_i - \bar{y})\sigma_{y_i} - 2({x}_i - \bar{x})\sigma_{x_i} & 0 \\ 0 & 0 & ({z}_i - \bar{z})\sigma_{y_i} - 2({x}_i - \bar{x})\sigma_{x_i} \\ \end{bmatrix} \right) ^2 \\ & + w_i^2 \left( \boldsymbol{P}_i^T - \begin{bmatrix} (x_i - \bar{x})\sigma_{x_i} - 2 ({z}_i - \bar{z})\sigma_{y_i} & 0 & 0 \\ 0 & ({y}_i - \bar{y})\sigma_{y_i} - 2 ({z}_i - \bar{z})\sigma_{z_i} & 0 \\ 0 & 0 & ({z}_i - \bar{z})\sigma_{y_i} - 2 ({y}_i - \bar{y})\sigma_{y_i} \\ \end{bmatrix} \right) ^2 \\ & + \sigma_{w_i}^2 \left( \boldsymbol{Q}_i - \mathrm{tr}(\boldsymbol{Q}_i)\boldsymbol{1}_3 \right)^2 \\ \\ & = {\boldsymbol{\sigma_I}}_i^2 \\ & + w_i^2 \left( \boldsymbol{P}_i - \begin{bmatrix} {p_X}_i - 2 {p_Y}_i & 0 & 0 \\ 0 & {p_Y}_i - 2 {p_X}_i & 0 \\ 0 & 0 & {p_Z}_i - 2 {p_X}_i \\ \end{bmatrix} \right) ^2 \\ & + w_i^2 \left( \boldsymbol{P}_i^T - \begin{bmatrix} {p_X}_i - 2{p_Z}_i & 0 & 0 \\ 0 & {p_Y}_i - 2 {p_Z}_i & 0 \\ 0 & 0 & {p_Z}_i - 2 {p_Y}_i \\ \end{bmatrix} \right) ^2 \\ & + \sigma_{w_i}^2 \left( \boldsymbol{Q}_i - \mathrm{tr}(\boldsymbol{Q}_i)\boldsymbol{1}_3 \right)^2 \end{align} \]

Finally,

\[ \boldsymbol{\sigma_I} = \sqrt{ \sum_{i=1}^{n} {\left\{ {\boldsymbol{\sigma_I}}_i^2 + w_i^2 \left( {{\boldsymbol{M}_1}_i}^2 + {{\boldsymbol{M}_2}_i}^2 \right) + \left( \sigma_{w_i} {\boldsymbol{M}_3}_i \right)^2 \right\}} } \]

where

\[ \boxed{ \begin{align} {\boldsymbol{M}_1}_i & = \boldsymbol{P}_i - \begin{bmatrix} {p_X}_i - 2 {p_Y}_i & 0 & 0 \\ 0 & {p_Y}_i - 2 {p_X}_i & 0 \\ 0 & 0 & {p_Z}_i - 2 {p_X}_i \\ \end{bmatrix} \\ {\boldsymbol{M}_2}_i & = \boldsymbol{P}_i^T - \begin{bmatrix} {p_X}_i - 2{p_Z}_i & 0 & 0 \\ 0 & {p_Y}_i - 2 {p_Z}_i & 0 \\ 0 & 0 & {p_Z}_i - 2 {p_Y}_i \\ \end{bmatrix} \\ {\boldsymbol{M}_3}_i & = \boldsymbol{Q}_i - \mathrm{tr}(\boldsymbol{Q}_i)\boldsymbol{1}_3 \end{align} } \]

The corresponding R code is

  r$sigma_inertia = sqrt(Reduce(`+`, Map(
    f = function(v) {

      d <- v$center_mass - r$center_mass

      P <- outer(d, v$sigma_center_mass)
      p <- as.list(diag(P))

      M1 <-   P  - diag(c(p$x - 2 * p$y, p$y - 2 * p$x, p$z - 2 * p$x))
      M2 <- t(P) - diag(c(p$x - 2 * p$z, p$y - 2 * p$z, p$z - 2 * p$y))
      M3 <- outer(d, d) - sum(diag(d^2)) * diag(3)

      M4 <- v$mass^2 * (M1^2 + M2^2) + (v$sigma_mass * M3)^2
      if (v$point) M4 else v$sigma_inertia^2 + M4
    },
    vl
  )))