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zg`pUR{V*{C2jLJLhQo0rMx*{4ME~zG17~3p&cRfihv}G!`j3U>xDs=5wKSI6N8kSg XMxz2Qms76GAqp@uISM5uMNdWwFxpwQ diff --git a/doc/rcfs.tex b/doc/rcfs.tex index d40dc55..e1f0625 100644 --- a/doc/rcfs.tex +++ b/doc/rcfs.tex @@ -150,20 +150,32 @@ Similarly, the solution of a cost function composed of several quadratic terms \bm{\hat{\mu}} = \arg\min_{\bm{x}} \sum_{k=1}^K {(\bm{x} - \bm{\mu}_k)}^\trsp \bm{W}_k (\bm{x} - \bm{\mu}_k) \label{eq:quad_costs_multiple} \end{equation} -can be seen as a product of Gaussians $\prod_{k=1}^K\mathcal{N}(\bm{\mu}_k,\bm{W}_k^{-1})$, with centers $\bm{\mu}_k$ and covariance matrices $\bm{W}_k^{-1}$. The Gaussian $\mathcal{N}(\bm{\hat{\mu}},\bm{\hat{W}}^{-1})$ resulting from this product has parameters +can be seen as a product of Gaussians $\prod_{k=1}^K\mathcal{N}(\bm{\mu}_k,\bm{W}_k^{-1})$, with centers $\bm{\mu}_k$ and covariance matrices $\bm{\Sigma}_k=\bm{W}_k^{-1}$. The Gaussian $\mathcal{N}(\bm{\hat{\mu}},\bm{\hat{W}}^{-1})$ resulting from this product has parameters \begin{equation*} \bm{\hat{\mu}} = {\left(\sum_{k=1}^K \bm{W}_k\right)}^{\!-1} \left(\sum_{k=1}^K\bm{W}_k \bm{\mu}_k \right), \quad \bm{\hat{W}} = \sum_{k=1}^K \bm{W}_k. \end{equation*} -$\bm{\hat{\mu}}$ and $\bm{\hat{W}}$ are the same as the solution of~\eqref{eq:quad_costs_multiple} and its Hessian, respectively. Viewing the quadratic cost probabilistically can capture more information about the cost function in the form of the covariance matrix $\bm{\hat{W}}^{-1}$. +$\bm{\hat{\mu}}$ and $\bm{\hat{W}}$ are the same as the solution of~\eqref{eq:quad_costs_multiple} and its Hessian, respectively. Viewing the quadratic cost probabilistically can capture more information about the cost function in the form of the covariance matrix $\bm{\hat{\Sigma}}= \bm{\hat{W}}^{-1}$. -In case of Gaussians close to singularity, to be numerically more robust when computing the product of two Gaussians $\mathcal{N}(\bm{\mu}_1, \bm{\Sigma}_1)$ and $\mathcal{N}(\bm{\mu}_2, \bm{\Sigma}_2)$, the Gaussian $\mathcal{N}(\bm{\hat{\mu}}, \bm{\hat{\Sigma}})$ resulting from the product can be alternatively computed with +There are also computation alternatives in case of Gaussians close to singularity. To be numerically more robust when computing the product of two Gaussians $\mathcal{N}(\bm{\mu}_1, \bm{\Sigma}_1)$ and $\mathcal{N}(\bm{\mu}_2, \bm{\Sigma}_2)$, the Gaussian $\mathcal{N}(\bm{\hat{\mu}}, \bm{\hat{\Sigma}})$ resulting from the product can indeed be computed with \begin{equation*} \bm{\hat{\mu}} = \bm{\Sigma}_2 {\left(\bm{\Sigma}_1+\bm{\Sigma}_2\right)}^{-1} \bm{\mu}_1 + \bm{\Sigma}_1 {\left(\bm{\Sigma}_1+\bm{\Sigma}_2\right)}^{-1} \bm{\mu}_2 , \quad - \bm{\hat{\Sigma}} = \bm{\Sigma}_1 {\left(\bm{\Sigma}_1+\bm{\Sigma}_2\right)}^{-1} \bm{\Sigma}_2. + \bm{\hat{\Sigma}} = \bm{\Sigma}_1 {\left(\bm{\Sigma}_1+\bm{\Sigma}_2\right)}^{-1} \bm{\Sigma}_2, \end{equation*} +by exploiting the fact that $\bm{\Sigma}_1\bm{\Sigma}_1^{-1}\!=\!\bm{I}$ and $\bm{\Sigma}_2^{-1}\bm{\Sigma}_2\!=\!\bm{I}$, we can observe that +\begin{align*} + \bm{W}_1+\bm{W}_2 &= \bm{\Sigma}_2^{-1}+\bm{\Sigma}_1^{-1}\\ + &= \bm{\Sigma}_2^{-1}\bm{\Sigma}_1\bm{\Sigma}_1^{-1}+\bm{\Sigma}_2^{-1}\bm{\Sigma}_2\bm{\Sigma}_1^{-1}\\ + &=\bm{\Sigma}_2^{-1} (\bm{\Sigma}_1+\bm{\Sigma}_2) \bm{\Sigma}_1^{-1}, +\end{align*} +so that +\begin{align*} +\bm{\hat{\Sigma}} &= {(\bm{W}_1+\bm{W}_2)}^{-1} = \bm{\Sigma}_1 {(\bm{\Sigma}_1+\bm{\Sigma}_2)}^{-1} \bm{\Sigma}_2,\\ +\bm{\hat{\mu}} &= {(\bm{W}_1+\bm{W}_2)}^{-1} (\bm{\Sigma}_1^{-1}\bm{\mu}_1+\bm{\Sigma}_2^{-1}\bm{\mu}_2) += \bm{\Sigma}_2 {\left(\bm{\Sigma}_1+\bm{\Sigma}_2\right)}^{-1} \bm{\mu}_1 + \bm{\Sigma}_1 {\left(\bm{\Sigma}_1+\bm{\Sigma}_2\right)}^{-1} \bm{\mu}_2. +\end{align*} \begin{figure} \centering diff --git a/python/IK_manipulator.py b/python/IK_manipulator.py index 7e955be..46710ba 100644 --- a/python/IK_manipulator.py +++ b/python/IK_manipulator.py @@ -71,9 +71,9 @@ plt.scatter(fh[0], fh[1], color='r', marker='.', s=10**2) #Plot target for t in range(param.nbData): f = fkin(x, param) # Forward kinematics (for end-effector) J = Jkin(x, param) # Jacobian (for end-effector) -# x += np.linalg.pinv(J) @ (fh - f) * 10 * param.dt # Update state - x += np.linalg.pinv(J) @ logmap(fh, f) * 10 * param.dt # Update state - + u = np.linalg.pinv(J) @ logmap(fh, f) * 0.1 / param.dt # Velocity command (u=delta_x/dt) + x += u * param.dt # Update state + f_rob = fkin0(x, param) # Forward kinematics (for all articulations, including end-effector) plt.plot(f_rob[0,:], f_rob[1,:], color=str(1-t/param.nbData), linewidth=2) # Plot robot -- GitLab