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Fixes linkrobot example 11 24 #29

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Fixes linkrobot example 11 24 #29

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9cf1608
Cleanup of Codeexamples
MBN04 Nov 26, 2024
f1721c8
Fixed Typos
MBN04 Nov 29, 2024
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6 changes: 2 additions & 4 deletions content/examples/6Rlinkrobot.md
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Original file line number Diff line number Diff line change
Expand Up @@ -29,9 +29,7 @@ Let us denote by $z_1,\ldots,z_6$ the unit vectors that point in the direction o

for $\alpha=(\alpha_1\ldots, \alpha_5)$ and $a=(a_1,\ldots,a_9)$ and $p=(p_1,p_2,p_3)$.
The $\alpha_i$ are the "twist angle" between joints, the $a_i$ are the "link length" between joint axes
and the $p$ encodes the position of the hand. Here $\times$ is the cross product in $\mathbb{R}^3$.

see the above reference for a detailed explanation on how these numbers are to be interpreted). Here $\times$ is the cross product in $\mathbb{R}^3$.
and the $p$ encodes the position of the hand (see the above reference for a detailed explanation on how these numbers are to be interpreted). Here $\times$ is the cross product in $\mathbb{R}^3$.


In this notation the forward problem consists of computing $(\alpha,a)$ given the $z_i$ and $p$ and the backward problem consists of computing $z_2,\ldots,z_5$ that realize some fixed $(\alpha,a,p,z_1,z_6)$ (knowing $z_1,z_6$ means that the position where the robot is attached to the ground and the position where its hand should be are fixed).
Expand Down Expand Up @@ -70,7 +68,7 @@ Result with 16 solutions
```


We find 16 solutions, which is the correct number of solutions for these type of systems.
We find 16 solutions, which is the correct number of solutions for this type of systems.

But if we study the problem a little bit closer, we can see that the equations are bi-homogeneous with respect to the variable groups $\\{z_2, z_4\\}$ and $\\{z_3, z_5\\}$.
The multi-homogeneous Bezout number with respect ot this variable group is
Expand Down
12 changes: 6 additions & 6 deletions content/examples/bacillus-subtilis.md
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Expand Up @@ -110,7 +110,7 @@ F = System(SteadyStates,
Now, we solve `F=0` for the parameter values `p`.

```julia
julia> S = solve(F, target_parameters = p)
S = solve(F, target_parameters = p)
Result with 44 solutions
========================
• 76 paths tracked
Expand All @@ -122,7 +122,7 @@ Result with 44 solutions
Only real positive zeros are physically meaningful. Using our implementation we can certify that there are 12 real zeros:

```julia
julia> cert = certify(F, S, target_parameters = p)
cert = certify(F, S, target_parameters = p)
CertificationResult
===================
• 44 solution candidates given
Expand All @@ -133,16 +133,16 @@ CertificationResult
We can also certify that among them there is a unique positive one.

```julia
julia> c = certificates(cert)
julia> pos_real = c[findall(is_positive.(c))]
julia> length(pos_real)
c = certificates(cert)
pos_real = c[findall(is_positive.(c))]
length(pos_real)
1
```

The positive real solution has the following values:

```julia
juila> certified_solution_interval(pos_real[1])
certified_solution_interval(pos_real[1])
10×1 Arblib.AcbMatrix:
[0.00406661084 +/- 5.50e-12] + [+/- 2.45e-12]im
[0.0557971948 +/- 5.02e-11] + [+/- 2.14e-11]im
Expand Down