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The goal of this task is to solve the inverse kinematic task for a general 6R mechanism using Newton's method.

**a.** Implement function `jac_r(mechanism, joints)`

that returns the rotational Jacobian for the given `mechanism`

and its configuration `joints`

.

I/O specifications for `jac_r`

:

`mechanism`

: dictionary with 4 keys`“theta offset”`

,`“d”`

,`“a”`

,`“alpha”`

.`joints`

: list of 6 float numbers that defines the configuration of the mechanism.**Return value**: 9×6 matrix of type`np.ndarray`

that defines the rotational Jacobian matrix evaluated at the given configuration, i.e.

$$\frac{\partial \mathrm{vec}(\mathbf{R})}{\partial \boldsymbol{\theta}}_{\big|\boldsymbol{\theta}_0}, \quad \text{where} \; \mathrm{vec}(\mathbf{R}) = \begin{bmatrix} r_{11} & r_{21} & r_{31} & r_{12} & r_{22} & r_{32} & r_{13} & r_{23} & r_{33} \end{bmatrix}^\top$$

**b.** Implement function `jac_t(mechanism, joints)`

that returns the positional Jacobian for the given `mechanism`

and its configuration `joints`

.

I/O specifications for `jac_t`

:

`mechanism`

: dictionary with 4 keys`“theta offset”`

,`“d”`

,`“a”`

,`“alpha”`

.`joints`

: list of 6 float numbers that defines the configuration of the mechanism.**Return value**: 3×6 matrix of type`np.ndarray`

that defines the positional Jacobian matrix evaluated at the given configuration, i.e.

$$\frac{\partial \mathbf{t}}{\partial \boldsymbol{\theta}}_{\big|\boldsymbol{\theta}_0}$$

**c.** Implement function `jac(mechanism, joints)`

that returns the full Jacobian for the given `mechanism`

and its configuration `joints`

.

I/O specifications for `jac`

:

`mechanism`

: dictionary with 4 keys`“theta offset”`

,`“d”`

,`“a”`

,`“alpha”`

.`joints`

: list of 6 float numbers that defines the configuration of the mechanism.**Return value**: 12×6 matrix of type`np.ndarray`

that defines the full Jacobian matrix evaluated at the given configuration, i.e.

$$ \begin{bmatrix} \frac{\partial \mathrm{vec}(\mathbf{R})}{\partial \boldsymbol{\theta}} \\ \frac{\partial \mathbf{t}}{\partial \boldsymbol{\theta}} \end{bmatrix}_{\big|\boldsymbol{\theta}_0} $$

**d.** Implement function `solve(mechanism, pose, joints, n_iter, tol)`

that applies Newton's method to solve IKT.

I/O specifications for `solve`

:

`mechanism`

: dictionary with 4 keys`“theta offset”`

,`“d”`

,`“a”`

,`“alpha”`

`pose`

: dictionary with 2 keys`“q”`

and`“t”`

that defines the desired pose of the mechanism's end effector`joints`

: list of 6 float numbers that defines the initial configuration of the mechanism`n_iter`

: number of iterations`tol`

: tolerance**Return value**: tuple consisting of the state of convergence and a list of 6 float numbers that defines the configuration of the mechanism that generates the given`pose`

with tolerance`tol`

or the algorithm failed to converge in`n_iter`

steps.

Upload a zip archive `hw08.zip`

containing

`hw08.py`

- python script containing the implemented functions`jac_r`

,`jac_t`

,`jac`

,`solve`

courses/pkr/labs/hw08.txt · Last modified: 2022/11/28 16:28 by korotvik