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I'm currently solving the exercises of Modern Robotics book. I came across this problem but it is not clear to me how to approach it. I computed the analytic Jacobian which is

$$ \begin{bmatrix} \dot{x}\\\dot{y} \\ \dot{z}\end{bmatrix} = \left(\begin{array}{ccc} L\,s_{\theta_1}\,s_{\theta_2}\,s_{\theta_3}-L\,c_{\theta_2}\,s_{\theta_1}-L\,c_{\theta_2}\,c_{\theta_3}\,s_{\theta_1}-L\,c_{\theta_1} & -L\,c_{\theta_1}\,\left(s_{\theta_2}+\sin\left(\theta_2+\theta_3\right)\right) & -L\,c_{\theta_1}\,\sin\left(\theta_2+\theta_3\right)\\ L\,c_{\theta_1}\,c_{\theta_2}-L\,s_{\theta_1}+L\,c_{\theta_1}\,c_{\theta_2}\,c_{\theta_3}-L\,c_{\theta_1}\,s_{\theta_2}\,s_{\theta_3} & -L\,s_{\theta_1}\,\left(s_{\theta_2}+\sin\left(\theta_2+\theta_3\right)\right) & -L\,s_{\theta_1}\,\sin\left(\theta_2+\theta_3\right)\\ 0 & L\,\left(c_{\theta_2}+\cos\left(\theta_2+\theta_3\right)\right) & L\,\cos\left(\theta_2+\theta_3\right) \end{array}\right) \begin{bmatrix} \dot{\theta}_1\\\dot{\theta}_2 \\ \dot{\theta}_3\end{bmatrix} $$ At zero configuration $\theta_1=\theta_2=\theta_3=0.$, we get $$ \begin{bmatrix} \dot{x}\\\dot{y} \\ \dot{z}\end{bmatrix} = \begin{bmatrix} -L&0&0\\2L&0&0\\0&2L&L\end{bmatrix} \begin{bmatrix} \dot{\theta}_1\\\dot{\theta}_2 \\ \dot{\theta}_3\end{bmatrix} $$ The analytic Jacobian is not invertible at this configuration. My conclusion is the motion is not possible but it seems to me according to the question, there is a possibility. Any suggestions?

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Here is a top view of the robot, and the end effector velocity if $\dot{\theta}_1 > 0$:

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  • $\begingroup$ It is difficult to tell where the end-effector is heading before reaching to the zero-configuration. $\endgroup$
    – CroCo
    Jan 13 at 18:18

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