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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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You are correct in determining that the motion is not possible, but you don't need to check if the matrix is invertible or not, you just need to solve the equation at the zero configuration.

If you plug in $[\dot x, \dot y, \dot z]^T = [10, 0, 0]^T$ to your equation at the zero configuration, you will find that to achieve the desired linear velocity you need to satisfy both $\theta_1 = -10/L$ and $\theta_1 = 0$. Therefore, it is not possible.

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  • $\begingroup$ thanks for the answer. But why such a nice book says "if so, what are required input input velocities..." which suggests implicitly such a possibility. $\endgroup$
    – CroCo
    Commented Jan 24, 2022 at 9:44
  • $\begingroup$ This is pretty standard in engineering textbooks at the senior undergrad and graduate levels. Almost every textbook I've ever worked with has questions of the form: "Figure out whether X is true or not. If X is true, do something else." They sometimes also include "If X is not true, do some other thing." but not always. $\endgroup$
    – Brandon J. DeHart
    Commented Jan 24, 2022 at 15:43
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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
    Commented Jan 13, 2022 at 18:18

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