CSCE689 | Minseo Park

asdsad

1-linked system

as $f (θ)$ .

\begin{array}{r} f (θ) = \frac{1}{2} w_{t a r g e t} ‖ Δ \vec{p} ‖_{2}^{2} + \frac{1}{2} w_{r e g u l a r} θ^{2} \\ Δ \vec{p} = \vec{p} (θ) - {\vec{p}}_{t a r g e t} \\ g (θ) = \frac{d f}{d θ} = \frac{d}{d θ} {\frac{1}{2} w_{t a r g e t} Δ {\vec{p}}^{⊺} \cdot Δ \vec{p}} + \frac{d}{d θ} {\frac{1}{2} w_{r e g u l a r} θ^{2}} \\ = w_{t a r g e t} Δ {\vec{p}}^{⊺} \cdot Δ {\vec{p}}^{″} + w_{r e g u l a r} θ \\ Δ {\vec{p}}^{'} = \frac{d \vec{p}}{d θ} - \frac{d {\vec{p}}_{t a r g e t}}{d θ} \\ = {\vec{p}}^{'} = w_{t a r g e t} Δ {\vec{p}}^{⊺} \cdot {\vec{p}}^{'} + w_{r e g u l a r} θ \\ H (θ) = w_{t a r g e t} ({\vec{p}}^{' ⊺} \cdot {\vec{p}}^{'} + Δ {\vec{p}}^{⊺} \cdot {\vec{p}}^{″}) + w_{r e g u l a r} \\ R (θ) = [\begin{array}{c} \cos θ & - \sin θ & 0 \\ \sin θ & \cos θ & 0 \\ 0 & 0 & 0 \end{array}] \\ R (θ)^{'} = [\begin{array}{c} - \sin θ & - \cos θ & 0 \\ \cos θ & - \sin θ & 0 \\ 0 & 0 & 0 \end{array}] \\ R (θ)^{″} = [\begin{array}{c} - \cos θ & \sin θ & 0 \\ - \sin θ & - \cos θ & 0 \\ 0 & 0 & 0 \end{array}] \end{array}

$\vec{θ} = (θ_{1}, θ_{2}, θ_{3}, θ_{4})^{⊺}$

min_{\vec{θ}} f (\vec{θ}), f (\vec{θ}) = \frac{1}{2} w_{t a r g e t} ‖ \vec{p} (θ) - {\vec{p}}_{t a r g e t} ‖ + \frac{1}{2} {\vec{θ}}^{⊺} w_{r e g u l a r} \vec{θ} \vec{g} (\vec{θ}) = R^{4}, H (\vec{θ}) = R^{4 \times 4}