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Update test examples and Readme.md
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‎Readme.md

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# Class for calculating Instantaneous Axis of Rotation
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The instantaneous axis of rotation between two bodies is a useful concept in
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biomechanics. In this post, we will dig into how to calculate the instantaneous
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axis of rotation and show off an AnyScript `class_template` to do this
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calculation and plot the axis between any two reference frames in the AnyBody
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Modeling System.
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The are several papers that discusses the
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importance of the instantaneous axis of rotation in biomechanics. See for
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example XX and XX. But first, let us look briefly at the math behind.
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## 3D Rotations
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We can view any displacement of a body in a three-dimensional space as rotation
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around a single axis and a translation along that axis. This gives rise to the
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idea of a screw motion in 3D along what is also called the helical axis. If we
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spit the movement up into infinitesimally small movements each of these will
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have an rotation axis and a linear velocity along that axis. This is the
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instantaneous axis of rotation, and as the name indicate will change direction
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and location as the object moves.
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If we know the angular and linear velocity of any point on a rigid body, we can
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calculate the properties of the screw motion and the instantaneous axis of
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rotation.
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From a mathematical point of view the rate of change of a 3x3 rotation matrix
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$R$ is:
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$$ \frac{{\mathrm d}}{{\mathrm d}t}R = \vec{\omega} \times R$$
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where $\vec{\omega}$ is called the angular velocity vector. This angular
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velocity is quantity shared by all points on the rigid body (including those on
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the rotation axis). Using $\vec{\omega}$ we can write the linear velocity
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$\vec{v}$ of any point at some distance $\vec{r} from the instantaneous axis of
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rotation as:
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$$\vec{v} = \vec{\omega} \times \vec{r}$$ (1)
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Similarly we can also calculate backwards and find the intantanous axis of
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rotation at some distance $-\vec{r}$ from any point if we know the linear
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velocity of the point and the angular velocity:
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This is given by:
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$$
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-\vec{r}= \frac{\vec{\omega} \times \vec{v} }{ \vec{\omega}\cdot\vec{\omega} }
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$$
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Here is short proof. Skip it if you like:
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>Start with $\vec{\omega} \times \vec{v}$, insert (1) and use the tripple product to expand it:
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>$$
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>\vec{\omega} \times \vec{v} = \vec{\omega} \times (\vec{\omega} \times \vec{r})=\vec{\omega}(\vec{\omega}\cdot\vec{r})-\vec{r}(\vec{\omega}\cdot\vec>{\omega})
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>$$
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>
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>The shortest vector $\vec{r}$ from the rotation axis to any point is by defintion perpendicular to angular velocity $\vec{\omega}$, hence:
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>
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>$$
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>\require{cancel}\vec{\omega} \times \vec{v} =\vec{\omega}(\cancel{\vec{\omega}\cdot\vec{r}})-\vec{r}(\vec{\omega}\cdot\vec{\omega})=-r(\vec{\omega}>\cdot\vec{\omega})
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>$$
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Thus if know the position of a point $\vec{r}_P$ we can find closest point
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$\vec{r}_C$ on the rotation axis as:
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$$\vec{r}_C = \vec{r}_P - \vec{r} = \vec{r}_P + \frac{\vec{\omega}\times\vec{v}_P}{\vec{\omega}\cdot\vec{\omega}}$$
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## Properties
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If for example we have a rigid body with angular velocity $\vec{\omega}$ and
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some point $P$ with position $\vec{r}_P$ and velocity $\vec{v}_p$ then we can
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define the different properties:
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1. The magintude of agular rotation:
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$$ \omega = \|\vec{\omega}\|$$
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2. The direction of the intantanous axis of rotation:
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$$ \vec{v}_{IOAR} = \frac{\vec{\omega}}{\omega}$$
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3. The point C on the intantanous axis of rotation:
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$$\vec{r}_C = \vec{r}_P + \frac{\vec{\omega}\times\vec{v}_P}{\vec{\omega}\cdot\vec{\omega}}$$
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4. Ratio of angular to linear angular velocity:
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$$ h = \frac{\vec{\omega} \cdot \vec{v}_P}{\vec{\omega}\cdot\vec{\omega}}$$
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5. The linear velocity at point $C$ along the screw axis is then given by
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$$\vec{v}_C = h\,\vec{\omega}$$
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All we need to know is the rotation velocity of a body and the velocity of any point to find the instantaneous axis of rotation.
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## AnyScript implementation
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In AnyScript we can easily find the angular velocity between two reference frames using the class `AnyKinRotational` and the linear velocity using the class `AnyKinLinear`.
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We have created a custom class_template that makes it easy to add all this code to a model and display the instantaneous axis of rotation. Here is a short example on how to use the class.
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First add the following to the top of your main file.
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The class template is called `InstantaneousAxisOfRotation`. does exactly this, and then calculate the properties listed above.
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------------------
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```c++
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#include "<ANYBODY_PATH_TOOLBOX>/Rotations/InstantaneousAxisOfRotation.any"
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```
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Then add the `InstantaneousAxisOfRotation` class in one of your studies.
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EXAMPLE:
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```c++
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#include "<ANYBODY_PATH_TOOLBOX>/Rotations/InstantaneousAxisOfRotation.any"
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// This is a simple example of using the InstantaneousAxisOfRotation class
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// to the calculate and vizualize the instantaneous axis of rotation
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// between Body1 and Body2
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InstantaneousAxisOfRotation IAOR (AnyRefFrame& Body1, AnyRefFrame& Body2 ) =
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InstantaneousAxisOfRotation IAOR (AnyRefFrame& Body1, AnyRefFrame& Body2) =
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{
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// The sphere drawn at the mid point of the vizualized axis.
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#var AnyVar SphereSize=0.02;
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};
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```
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## Output
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The object will IAOR object will plot the axis, and provide the following outputs:
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|---------------------|--------------|------------------------------------------------|
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| Output variable | Type | Desciption |
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|---------------------|--------------|------------------------------------------------|
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| MidPoint | AnyVec3 | The mid point on axis in global coordinates |
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| Direction | AnyVec3 | The direction of the axis in global coordinates|
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|---------------------|--------------|------------------------------------------------|
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|-----------------------|--------------|------------------------------------------------|
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| Output variable | Type | Desciption |
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|:----------------------|--------------|:-----------------------------------------------|
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| RotationAxisDirections| AnyVec3 | The mid point on axis in Body1 coordinates |
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| RotationMagnitude | AnyVec3 | The direction of the axis in Body1 coordinates |
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| rc_2 | AnyVec3 | Point on axis closest to Body2 |
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| rc_1 | AnyVec3 | Point on axis closest to Body1 |
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| MotionPitch | AnyVar | The ratio of linear motion to angular motion |
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| rc_dot | AnyVec3 | The linear velocity along the axis of rotation |
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| omega | AnyVec3 | Anuglar velocity vector of Body2 wrt. Body1 |
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| r | AnyVec3 | Position of Body2 wrt. Body1 |
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| r_dot | AnyVec3 | Linear velocity of Body2 wrt. Body1 |
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|-----------------------|--------------|------------------------------------------------|

‎Test_rolling_without_slipping.any

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// Todo: Write a small description of your model here
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#include "InstantaneousAxisOfRotation.any"
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Main = {
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// The actual body model goes in this folder
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AnyFolder MyModel = {
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// Global Reference Frame
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AnyFixedRefFrame GlobalRef = {
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AnyDrawRefFrame drw={};
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}; // Global reference frame
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AnyFolder Segments = {
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AnySeg Ground = {
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Mass = 0;
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Jii = {0, 0, 0};
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AnyRefNode jnt ={ARel = RotMat(0.3*pi,y)*RotMat(0.3*pi,x);};
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AnyDrawRefFrame drw1 ={ RGB={0,0,1};};
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};
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AnySeg Ball = {
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Mass = 0;
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Jii = {0, 0, 0};
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AnyDrawRefFrame drw1 ={ RGB={0,0,1};};
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AnyDrawSphere drw2={ScaleXYZ={1,1,1}*0.5;};
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};
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};
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AnyFolder Joints = {
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AnyKinLinear Lin1 ={
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AnyRefFrame &ref1=..GlobalRef;
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AnyRefFrame &ref2=..Segments.Ground;
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};
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AnyKinRotational Rot1 ={
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AnyRefFrame &ref1=..GlobalRef;
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AnyRefFrame &ref2=..Segments.Ground;
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Type=RotAxesAngles;
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};
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AnyKinEqSimpleDriver drv1={
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AnyKinMeasure &ref1=.Lin1;
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AnyKinMeasure &ref2=.Rot1;
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DriverPos ={0,0,0,0,0.25*pi,0};
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DriverVel ={0,0,0,0,0,0};
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};
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AnyKinLinear Lin2 ={
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Ref=0;
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AnyRefFrame &ref1=..Segments.Ground.jnt;
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AnyRefFrame &ref2=..Segments.Ball;
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};
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AnyKinRotational Rot2 ={
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AnyRefFrame &ref1=..Segments.Ground.jnt;
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AnyRefFrame &ref2=..Segments.Ball;
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Type=RotAxesAngles;
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};
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AnyKinEqSimpleDriver drv2={
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AnyKinMeasure &ref1=.Lin2;
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AnyKinMeasure &ref2=.Rot2;
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AnyVar Radius =0.5;
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DriverPos ={0,0.5,0,0,0,0};
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DriverVel ={2*pi*Radius,0,0,-2*pi,0,0};
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};
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};
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// AnyFolder InstantenousAxisCal ={
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InstantaneousAxisOfRotation IAOR (Body1=.Segments.Ground, Body2=.Segments.Ball) =
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{
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AxisThickness = 0.05;
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};
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}; // MyModel
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// The study: Operations to be performed on the model
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AnyBodyStudy MyStudy = {
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AnyFolder &Model = .MyModel;
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Gravity = {0.0, -9.81, 0.0};
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nStep = 200;
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tStart = 0;
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tEnd = 2;
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};
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};
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‎Test_two_reference_frames.any

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#include "InstantaneousAxisOfRotation.any"
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Main = {
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AnyFolder Model ={
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AnyFixedRefFrame Frame ={ AnyDrawRefFrame drw={}; };
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AnySeg Object1 ={ Mass=1; Jii={0,0,0}; };
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AnyStdJoint StdJoint = {
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AnyFixedRefFrame &ref1=.Frame;
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AnySeg &ref2=.Object1;
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};
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AnySeg Object2 ={ Mass=1; Jii={0,0,0}; AnyDrawRefFrame drw={}; };
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AnyKinLinear lin ={
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AnyFixedRefFrame &ref1=.Frame;
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AnySeg &ref2=.Object2;
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};
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AnyKinRotational rot ={
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AnyFixedRefFrame &ref1=.Frame;
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AnySeg &ref2=.Object2;
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Type =RotAxesAngles;
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};
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AnyKinEqInterPolDriver Lindrv ={
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Type =Bspline;
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AnyKinLinear &ref=.lin;
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T={0,0.25,0.5,1};
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Data={1*{0,0.5,0.75,1},1*{0,0.5,0.75,1},1*{0,0.5,0.75,1}};
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};
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AnyKinEqInterPolDriver rotdrv ={
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AnyKinRotational &ref=.rot;
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Type =Bspline;
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T={0,0.25,0.5,1};
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Data=1*{0*{0,0.5,0.75,1},2*{0,0.5,0.75,1},10*{0,0.5,0.75,1}};
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//Data=1*{0*{0,0.5,0.75,1},0*{0,0.5,0.75,1},10*{0,0.5,0.75,1}};
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};
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InstantaneousAxisOfRotation IAOR (Body1=.Object1, Body2=.Object2) ={};
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};
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AnyBodyStudy study ={
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AnyFolder &ref=.Model;
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Gravity ={0,0,-9.81};
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tEnd=0.99;
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};
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};

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