「研究ノート/最適化」の編集履歴（バックアップ）一覧はこちら

「研究ノート/最適化」(2008/07/02 (水) 01:42:19) の最新版変更点

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***Objective function
$$ \sum_{i = 1}^{n - 2} \sum_{j = 0}^{m - 1} \omega^j \left[ \alpha \left( \theta_i^j - \theta_{i - 1}^j \right)^2 + \left( 1 - \alpha \right) \left( \theta_i^j - \Theta_i^j \right)^2 \right] $$
$$ \omega $$ : the weight of some DOF, $$ \textstyle \sum_{j = 0}^{m - 1} \omega^j = 1 $$
$$ \alpha $$ : the weight of continuity, $$ 0 \le \alpha \le 1 $$
$$ \theta $$ : some DOF of the motion
$$ \Theta $$ : some DOF of the original motion

***Gradient
$$ \frac {\partial F} {\partial \theta_i^j} = \frac {\partial \left[ \left( \theta_i^j - \theta_{i-1}^j \right)^2 + \left( \theta_i^j - \Theta_i^j \right)^2 + \left( \theta_{i+1}^j - \theta_i^j \right)^2 + \left( \theta_{i+1}^j - \Theta_{i+1}^j \right)^2 \right]} {\partial \theta_i^j} $$

$$ = 2 \left( \theta_i^j - \theta_{i-1}^j \right) + 2 \left( \theta_i^j - \Theta_i^j \right) - 2 \left( \theta_{i+1}^j - \theta_i^j \right) $$

$$ = 2 \left( 3\theta_i^j - \theta_{i-1}^j - \theta_{i+1}^j - \Theta_i^j \right) $$

***Objective function
$$ \sum_{i = 1}^{n - 2} \sum_{j = 0}^{m - 1} \omega^j \left[ \alpha \left( \theta_i^j - \theta_{i - 1}^j \right)^2 + \left( 1 - \alpha \right) \left( \theta_i^j - \Theta_i^j \right)^2 \right] $$
$$ \omega $$ : the weight of some DOF, $$ \textstyle \sum_{j = 0}^{m - 1} \omega^j = 1 $$
$$ \alpha $$ : the weight of continuity, $$ 0 \le \alpha \le 1 $$
$$ \theta $$ : some DOF of the motion
$$ \Theta $$ : some DOF of the original motion

***Gradient
$$ \frac {\partial F} {\partial \theta_i^j} = \frac {\partial \omega^j \left[ \left( \theta_i^j - \theta_{i-1}^j \right)^2 + \left( \theta_i^j - \Theta_i^j \right)^2 + \left( \theta_{i+1}^j - \theta_i^j \right)^2 + \left( \theta_{i+1}^j - \Theta_{i+1}^j \right)^2 \right]} {\partial \theta_i^j} $$

$$ = \omega^j \left[ 2 \left( \theta_i^j - \theta_{i-1}^j \right) + 2 \left( \theta_i^j - \Theta_i^j \right) - 2 \left( \theta_{i+1}^j - \theta_i^j \right) \right] $$

$$ = 2\omega^j \left( 3\theta_i^j - \theta_{i-1}^j - \theta_{i+1}^j - \Theta_i^j \right) $$