Consider a task to reach a constant position \(s_d\) for the end-effector.

Is the function \(r(q) = \sin(2\pi\|s_e(q)-s_d\|^2)\) a task function? Justify.

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Let \(r_1(q)\) and \(r_2(q)\) be task functions for the same task \(\mathcal{T}\), with the same number of components.

Can we say that \(r_3(q) = r_1(q)+r_2(q)\) is also a task function for \(\mathcal{T}\)? Justify.

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Let \(r_1(q)\) and \(r_2(q)\) be task functions for the same task \(\mathcal{T}\).

Can we say that \(r_3(q) = \|r_1(q)\|^2+\|r_2(q)\|^2\) is also a task function for \(\mathcal{T}\)? Justify.

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Let \(Q(q)\) and \(Q_d\) be rotation matrices representing the orientation of the end-effector and the desired orientation for the end-effector.

The function

$$r(q) = \det(Q(q)-Q_d)$$

is a task function? Justify.

Hint: note that (i) \(Q(q)-Q_d = (Q(q)Q_d^T - I_{3 \times 3})Q_d\), (ii) \(\det(UV) = \det(U)\det(V)\) (if \(U,V\) are square matrices) and (iii) every rotation matrix has eigenvalue 1.

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Let \(Q(q)\) and \(Q_d\) be rotation matrices representing the orientation of the end-effector and the desired orientation for the end-effector.

Let \(\|\cdot\|_F\) be the Frobenius norm of a matrix, that is, the square root of the sum of the squares of its elements.

The function

$$r(q) = \|I_{3 \times 3} - Q(q)Q_d^T\|_F^2$$

is a task function? Justify.

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Calculate the task Jacobian and the task feedforward for the task function of controlling the position of the end-effector:

$$r(q,t) = \|s_e(q)-s_d(t)\|^2.$$

Hint: use the fact that \(\|w\|^2 = w^Tw\) for a column vector \(w\).

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Consider the task function \(r(q)=s_e(q)-s_d\) to reach a constant position.

Suppose the specified dynamics \(\dot{r} = -Kr\) for a positive scalar \(K\) is perfectly satisfied.

Show that the path the robot will take in the workspace will be a straight line between the initial position and the desired position.

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Consider a non-negative one-dimensional task function, \(r(q) \in \mathbb{R}\), \(r(q) \geq 0\).

Consider that we impose the dynamics \(\dot{r} = -K\sqrt{r}\) with \(K > 0\) for the task function, and that it is perfectly satisfied.

Calculate how the function \(r(q(t))\) will be, and how long, in terms of \(K\) and \(r_0 = r(q(0))\), it will take to converge to \(r=0\).

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The kinematic controller fails in some situations. This happens when we specify the dynamics \(\dot{r} = F(r)\) and reach a configuration \(q\) where

$$J_r(q)^TF(r(q))=0_{n \times 1} \ \mbox{but} \ r(q) \not= 0_{g \times 1}.$$

In this case, the robot stops without having completed the task.

Consider the planar two-joint robot from Class 3 (see on the side), with \(L_1=L_2=1m\), with the position task function \(r(q) = s_e(q)-s_d\), with \(F(r)=-Kr\), \(K=1 s^{-1}\) and \(s_d = (0 \ 0 \ 1)^T m\).

Calculate a configuration \(q\) where, if we start from it, the controller fails.

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