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Rigid-Scales

Quick notes on Rigid-Scale kinematics.

  1. Affine Transformation
    1. Mapping
    2. Jacobian
    3. Hessian
    4. Geometric Stiffness
  2. Deformation Field
    1. Mapping
    2. Jacobian
    3. Hessian
    4. Geometric Stiffness

Affine Transformation

\[f: \RR^3 \times \RR^3 \times \RR^3 \to \mathrm{GA}(3)\]

Mapping

\[f(\omega, s, t) = \block{\exp(\omega)RS, t}\]

where \(S = \diag(s)\).

Jacobian

\[\dd f = \block{\dd \exp(\omega).\dd \omega\ RS + \exp(\omega)R\dd S, \dd t}\] \[\omega = 0: \quad \dd f = \block{\dd \omega\ RS + R\dd S, \dd t}\]

Hessian

\[\dd^2 f = \block{\dd^2 \exp(\omega).\dd \omega_2.\dd \omega_1\ RS + \dd \exp(\omega).\dd \omega_1 R \dd S_2 + \dd \exp(\omega).\dd \omega_2 R\dd S_1, 0}\] \[\newcommand{\ddexp}[2]{\frac{\hat{#1} \hat{#2} + \hat{#2} \hat{#1}}2}\] \[\omega = 0: \quad \dd^2 f = \block{\ddexp{\dd \omega_1}{\dd \omega_2}RS + \dd \omega_1 R \dd S_2 + \dd \omega_2 R \dd S_1, 0}\]

Geometric Stiffness

\[\trace{\block{\lambda, \lambda_t}^T \dd^2 f}_{\omega=0} = \trace{\lambda^T \ddexp{\dd \omega_1}{\dd \omega_2}RS} + \trace{\lambda^T\dd \omega_1 R \dd S_2} + \trace{\lambda^T\dd \omega_2 R \dd S_1}\]

Since:

\[\ddexp{\dd \omega_1}{\dd \omega_2} = \frac{\dd \omega_1 \dd \omega_2^T + \dd \omega_2 \dd \omega_1^T}{2} - \dd \omega_1^T \dd \omega_2I\]

We get:

\[\begin{align} \trace{\lambda^T \ddexp{\dd \omega_1}{\dd \omega_2}RS} &= \trace{\underbrace{RS\lambda^T}_{K^T} \ddexp{\dd \omega_1}{\dd \omega_2}} \\ &= \trace{K^T \frac{\dd \omega_1 \dd \omega_2^T + \dd \omega_2 \dd \omega_1^T}{2}} - \trace{K^T\dd \omega_1^T \dd \omega_2I} \\ &= \dd \omega_2^T \frac{K + K^T}{2} \dd \omega_1 - \dd \omega_2^T \trace{K} \dd \omega_1 \\ \end{align}\]

Finally:

\[\begin{align} \trace{\lambda^T\dd \omega_1 R \dd S_2} &= \trace{\lambda^TRR^T\dd \hat{\omega}_1 R \dd S_2} \\ &=\trace{\lambda^T R \hat{R^T \dd \omega_1}\dd S_2} \\ &= \sum_i e_i^T\block{\lambda^T R \hat{R^T \dd \omega_1}\dd S_2}e_i\\ &= \sum_i e_i^T\block{\lambda^T R \hat{R^T \dd \omega_1}e_i \dd S_{2_i}}\\ &= -\sum_i e_i^T\block{\lambda^T R \hat{e_i}R^T \dd \omega_1 \dd S_{2_i}}\\ &= -\sum_i e_i^T\block{\lambda^T \hat{Re_i} \dd \omega_1 \dd S_{2_i}}\\ &= -\block{\sum_i \dd S_{2_i}e_i^T\block{\lambda^T \hat{Re_i}}} \dd \omega_1\\ &= \block{\sum_i \dd S_{2_i}\block{\lambda_i^T \times R_i}^T} \dd \omega_1\\ &= \dd s_2^T \mat{\block{\lambda_i^T \times R_i}^T\\ \vdots} \dd \omega_1\\ \end{align}\]

(phhew!)

Deformation Field

\[g: \RR^3 \times \RR^3 \times \RR^3 \times \RR^3 \to \RR^3\]

Mapping

\[g(\omega, s, t, x) = \exp(\omega)RSx + t = f(\omega, s, t)\mat{x\\ 1}\]

Jacobian

\[\begin{align} \dd g &= \dd f \mat{x\\ 1} + f(\omega, s, t) \mat{\dd x\\ 0} \\ &=\block{\dd \omega\ RS + R\dd S}x + \dd t + RS \dd x \\ &= -\hat{RSx}\ \dd \omega + R\diag(x)\ \dd s + \dd t + RS\ \dd x \\ \end{align}\]

Hessian

\[\dd^2 g = \dd^2 f \mat{x\\ 1} + \dd f_1 \mat{\dd x_2\\ 0} + \dd f_2 \mat{\dd x_1\\ 0}\]

Geometric Stiffness

\[\begin{align} \trace{\lambda^T \dd^2 g} &= \trace{\lambda^T \dd^2 f \mat{x\\ 1}} + \trace{\lambda^T \dd f_1 \mat{\dd x_2\\ 0}} + \trace{\lambda^T \dd f_2 \mat{\dd x_1\\ 0}} \\ &= \trace{x\lambda^T \dd^2 f_{\omega, s}} + \trace{\lambda^T \dd f_1 \mat{\dd x_2\\ 0}} + \trace{\lambda^T \dd f_2 \mat{\dd x_1\\ 0}} \end{align}\] \[\begin{align} \trace{\lambda^T \dd f_2 \mat{\dd x_1\\ 0}} &= \trace{\lambda^T\block{\dd \hat{\omega_2} RS + R\dd S_2}.\dd x_1} \\ &= -\dd \omega_2^T \hat{\lambda}RS\dd x_1 + \dd s_2^T \diag\block{R^T \lambda} \dd x_1 \\ \end{align}\]