Update Content - 2021-05-30

content/zettels/jacobian.md

@@ -10,18 +10,18 @@ Tags
 
 ## Jacobian Matrices of a Parallel Manipulator {#jacobian-matrices-of-a-parallel-manipulator}
 
-From ([Taghirad 2013](#org1eb570f)):
+From ([Taghirad 2013](#orgdcd348b)):
 
 > The Jacobian matrix not only reveals the **relation between the joint variable velocities of a parallel manipulator to the moving platform linear and angular velocities**, it also constructs the transformation needed to find the **actuator forces from the forces and moments acting on the moving platform**.
 
-([Merlet 2006](#org77b3718))
+([Merlet 2006](#org3c93a40))
 
 
 ## Computing the Jacobian Matrix {#computing-the-jacobian-matrix}
 
-How to derive the Jacobian matrix is well explained in chapter 4 of ([Taghirad 2013](#org1eb570f)) ([notes]({{< relref "taghirad13_paral" >}})).
+How to derive the Jacobian matrix is well explained in chapter 4 of ([Taghirad 2013](#orgdcd348b)) ([notes]({{< relref "taghirad13_paral" >}})).
 
-Consider parallel manipulator shown in Figure [1](#orgf2877e3) (it represents a Stewart platform).
+Consider parallel manipulator shown in Figure [1](#orgea8dcb1) (it represents a Stewart platform).
 
 Kinematic loop closures are:
 
@@ -50,7 +50,7 @@ By taking the time derivative, we obtain the following **Velocity Loop Closures*
 {}^A\hat{\bm{s}}\_i {}^A\bm{v}\_p + ({}^A\bm{b}\_i \times \hat{\bm{s}}\_i) {}^A\bm{\omega} = \dot{l}\_i \label{eq:velocity\_loop\_closure}
 \end{equation}
 
-<a id="orgf2877e3"></a>
+<a id="orgea8dcb1"></a>
 
 {{< figure src="/ox-hugo/jacobian_geometry.png" caption="Figure 1: Example of parallel manipulator with defined frames and vectors" >}}
 
@@ -83,7 +83,7 @@ And therefore \\(\bm{J}\\) then **depends only** on:
 -   \\({}^A\hat{\bm{s}}\_i\\) the orientation of the limbs
 -   \\({}^A\bm{b}\_i\\) the position of the joints with respect to \\(O\_B\\) and express in \\(\\{\bm{A}\\}\\).
 
-For the platform in Figure [1](#orgf2877e3), we have:
+For the platform in Figure [1](#orgea8dcb1), we have:
 
 \begin{equation}
 \begin{bmatrix} \dot{l}\_1 \\ \dot{l}\_2 \\ \dot{l}\_3 \\ \dot{l}\_4 \\ \dot{l}\_5 \\ \dot{l}\_6 \end{bmatrix} =
@@ -112,8 +112,9 @@ in which \\(\bm{\tau} = [f\_1, f\_2, \cdots, f\_6]^T\\) is the vector of actuato
 Note that it is here assumed that the forces are static and **along the limb axis** \\(\hat{\bm{s}}\_i\\).
 
 
+
 ## Bibliography {#bibliography}
 
-<a id="org77b3718"></a>Merlet, Jean-Pierre. 2006. “Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots.”
+<a id="org3c93a40"></a>Merlet, Jean-Pierre. 2006. “Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots.”
 
-<a id="org1eb570f"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.
+<a id="orgdcd348b"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.