~~NOTOC~~

===== Homework 07 - Inverse Kinematics by Gröbner Basis Computation =====

The goal of this task is to solve the inverse kinematic task for a general 6R mechanism using a general Gröbner basis computation. This consists of the six main elements

  - Provide mechanism parameters ($\sin \alpha_i$, $\cos \alpha_i$, $a_i$, $d_i$) as  rational numbers satisfying identities on cosines and sines.
  - Provide the end effector pose as a rational matrix with rotation matrix satisfying the rotation identities (exactly).
  - Formulate the algebraic equations describing the inverse kinematics task.
  - Find a Gröbner basis of the formulated IKT equations w.r.t. lexicographic monomial ordering (via Maple).
  - Recover real solutions for sines and cosines from a Gröbner basis using back-substitution.
  - Recover the joint angles from the computed sines and cosines.

Gröbner basis computation is done in exact arithmetics over rational numbers and therefore input must be provided in rational numbers. At the same time, the input must satisfy all identities on rotations, otherwise the systems would have no solution.


=== Task ===

Steps to setup:
  - Install [[https://download.cvut.cz/maple-2022-for-students/ | Maple 2022 for students]] and add the path to the binary **maple** (for Linux, Mac OS) or **cmaple** (for Windows) into the PATH variable (so that Maple can be run from terminal by the command ''maple'' (Linux, Mac OS) or ''cmaple'' (Windows)).
  - Study the [[https://cw.fel.cvut.cz/b221/_media/courses/pkr/labs/hw07.pdf|solution to IKT by GB computation]].
  - Download the project [[https://cw.fel.cvut.cz/b221/_media/courses/pkr/labs/hw07.zip | hw07.zip]] and implement the functions described below.

<note warning>
**Allowed libraries**: numpy, sympy, fractions, os

**Forbidden modules/methods**: sympy.solvers, fractions.Fraction.limit_denominator
</note>

**a.** Implement function ''rat_approx(n, tol)'' that returns a rational approximation of a given number ''n''.

I/O specifications for ''rat_approx'':
  - ''n'': float number
  - ''tol'' : positive float number
  - **Return value**: ''Fraction'' number ''f'' such that
$$|f - n| < tol$$

**b.** Implement function ''rational_cs(angle, tol)'' that returns a rational point on the unit circle, which is close to a given point defined by ''angle''.

I/O specifications for ''rational_cs'':
  - ''angle'': float number from the interval $(-\pi; \pi]$
  - ''tol'' : positive float number
  - **Return value**: list ''[c, s]'' of ''Fraction'' approximations of $\cos\theta$ and $\sin\theta$, such that 
$$c^2 + s^2 = 1 \textrm{ (exactly) }$$

**c.** Implement function ''rational_rot(q, tol)'' that returns a rational rotation close to the one given by quaternion ''q''.

I/O specifications for ''rational_rot'':
  - ''q'' : 1D ''np.ndarray'' of 4 float numbers of (approximately) unit norm
  - ''tol'' : positive float number
  - **Return value**: 3x3 matrix ''r'' of type ''np.ndarray'' with ''Fraction'' numbers inside, such that 
$$ r^\top r = \mathrm{I}, \;\; \det{r} = 1 \textrm{ (exactly) } \quad \& \quad ||r - R(q)||_{\mathrm{F}} < tol $$
where $||\cdot||_{\mathrm{F}}$ denotes the Frobenius norm.

**d.** Implement function ''rational_mechanism(mechanism, tol)'' which converts the input mechanism to the rational one.

I/O specifications for ''rational_mechanism'':
  - ''mechanism'': dictionary with 4 keys ''"theta offset"'', ''"d"'', ''"a"'', ''"alpha"''.
  - ''tol'': positive float number
  - **Return value**: dictionary with 5 keys ''"theta offset"'', ''"d"'', ''"a"'', ''"sin alpha"'', ''"cos alpha"''. The value for the key ''"theta offset"'' remains unchanged. The values for all the other keys are lists of ''Fraction'' numbers.

**e.** Implement function ''rational_pose(pose, tol)'' which converts the input pose to the rational one.

I/O specifications for ''rational_pose'':
  - ''pose'': dictionary with 2 keys ''"q"'' and ''"t"'', whose values are the (approximately) unit quaternion (1D ''np.ndarray'' of 4 float numbers) and the translation (1D ''np.ndarray'' of 3 float numbers) of the end effector, respectively.
  - ''tol'': positive float number
  - **Return value**: 4х4 homogeneous transformation matrix of type ''np.ndarray'' with ''Fraction'' numbers inside that is the approximation of the input pose.

**f.** Implement function ''ikt_eqs(mechanism, pose, tol)'' which returns the polynomial equations for the rationalized inverse kinematic task.

I/O specifications for ''ikt_eqs'':
  - ''mechanism'': dictionary with 4 keys ''"theta offset"'', ''"d"'', ''"a"'', ''"alpha"''.
  - ''pose'': dictionary with 2 keys ''"q"'' and ''"t"'', whose values are the (approximately) unit quaternion (1D ''np.ndarray'' of 4 float numbers) and the translation (1D ''np.ndarray'' of 3 float numbers) of the end effector, respectively.
  - ''tol'': positive float number
  - **Return value**: list of ''Poly'' objects with rational coefficients in variables ''c1'', ''s1'', $\dots$, ''c6'', ''s6'' describing the polynomial equations of IKT for the rationalized mechanism and pose.

**g.** Implement function ''solve_ikt_gb_lex(gb)'' which returns the list of real solutions to a lexicographic Gröbner basis of the instance of IKT.

I/O specifications for ''solve_ikt_gb_lex'':
  - ''gb'': list of ''Poly'' objects with float coefficients describing the reduced lexicographic (the order of variables is ''c1'' > ''s1'' > $\dots$ > ''c6'' > ''s6'') Gröbner basis of some instance of IKT. The order of polynomials in ''gb'' is the following: ''gb[0]'' is a polynomial in ''s6'' only, ''gb[1]'' is a polynomial in ''c6'' and ''s6'', $\dots$, ''gb[11]'' is a polynomial in ''c1'' and ''s6''.
  - **Return value**: list of real solutions to the given polynomial system ''gb''. Every solution is a list of float numbers corresponding to ''c1'', ''s1'', $\dots$, ''c6'', ''s6''.

<note>
Use ''numpy.roots'' method to compute the roots of the univariate polynomial, since implementing it via companion matrix may not give suitable results for some polynomials with huge coefficients.
</note>
   
=== Upload ===

Upload a zip archive ''hw07.zip'' containing
  - ''hw07.json'' - json file, containing the real solutions for the mechanism and the pose specified for you in BRUTE. Consider the tolerance $tol = 10^{-5}$. (If the automatic evaluation shows $-1$ as the error, this means that you have the wrong number of solutions.)
  - ''hw07.py'' - python script containing the implemented functions ''rat_approx'', ''rational_cs'', ''rational_rot'', ''rational_mechanism'', ''rational_pose'', ''ikt_eqs'', ''solve_ikt_gb_lex''

**Creating** ''hw07.json'':

The value stored in ''hw07.json'' will be a list of all real joint solutions for the mechanism and the pose specified for you in BRUTE. Every solution must be represented as a list. **Every angle must belong to the interval $(-\pi, \pi]$**. An example looks as follows:

<code>
real_sols = [[1.1, 1.2, 1.3, 1.4, 1.5, 1.6], 
             [2.1, 2.2, 2.3, 2.4, 2.5, 2.6]]
import json
with open("hw07.json", "w") as outfile:
    json.dump(real_sols, outfile)
</code>

<note tip>

**Example of how to use ''eqs2gb''**:
<code>
from sympy import symbols
x, y = symbols('x, y')
eqs = [poly(x*y - 1), poly(x**2 - y)]
gb = eqs2gb(eqs, [x, y], "lex")
print(gb)
</code>

The output of the ''print(gb)'' command is
<code>
[Poly(1.0*y**3 - 1.0, y, domain='RR'), Poly(1.0*x - 1.0*y**2, x, y, domain='RR')]
</code>

</note>

<note>
The function ''eqs2gb'' works only locally on your computer, don't try to upload the code with it to the upload system, it will fail.
</note>