# matrix

## Maya matrix nodes – Blending matrices

During the week, I got a comment on the first post in my Maya matrix nodes series – the matrix constraint one – about using the `wtAddMatrix` node to achieve the multiple targets with blending weights functionality similar to constraints. I have stumbled upon the `wtAddMatrix` node, but I think it is the fact that Autodesk have made it very fiddly to work with it – we need to show all attributes in the node editor and we have 0 access to setting the weight plug – that put me off ever using it. That being said, when RigVader commented on that post I decided I will give it a go. Since it actually works quite nicely, today I am looking at blending matrices in Maya.

Disclaimer: I will be using the matrix constraint setup outlined in the post I mentioned, so it might be worth having a look at that one if you have missed it.

Another disclaimer: In the comments, Harry Houghton flagged up the fact that the resulting orientation of the setup, when using weights other than half and half, is not comparable to the `parentConstraint` result, as the `wtAddMatrix` node does not have a way of controlling the interpolation type.

That being said, you might find that difference beneficial in certain cases, as the resulting rotation doesn’t flip when going beyond 180.

tl;dr: Using the `wtAddMatrix` we can blend between matrices before we plug the output into a matrix constraint setup to achieve having multiple targets with different weights.

Turns out, the `wtAddMatrix` is a really handy node. It gives us the chance to plug a number of matrices in the `.matrixIn` plugs of the `.wtMatrix` array attribute, and give them weight in the `.weightIn` plug. That, effectively lets us blend between them.

### Blending matrices for a matrix constraint setup

So, now that we know we can blend matrices, we just need to figure out exactly what do we need to blend.

Let us first have a look at the simpler case – not maintaining the offset.

The `group1` on the graph is the parent of `pCube1` and is used just so we convert the world matrix into a relative to the parent matrix, without using the `parentInverseMatrix`. The reason for that is we do not want to create benign cycles, which Raff sometimes talks about on the Cult of Rig streams. Other than that, everything seems to be pretty straightforward.

Bear in mind, the `wtAddMatrix` node does not normalize the weights, which means that we could have all of the targets fully influence our object. What is more, you could also push them beyond 1 or negate them, which would result in seemingly odd results, but that might just be what you need in some cases.

#### Maintaining the offset

Often we need to maintain the offset in order to achieve the desired behaviour, so the way we do that is we resort to the `multMatrix` node once more. I am not going in detail, as there are already a couple of ways you can do that outlined in the previous post, but let us see how it fits in our graph.

The two additional `multMatrix` nodes let us multiply the local offset for the current target by the world matrix of the current target, effectively constraining the object but also maintaining the initial offset.

Now, however clean and simple it may be, the graph gets to be a bit long. What this means is, it is probably getting a bit slower to evaluate as well. That is why, I thought I would do a bit of a performance test to see if there still is any benefit to using this setup over a `parentConstraint`.

### Performance

The way I usually do my tests is either loop a few hundred times in the scene and build the setup or build it once, save it in a file and then import the file a few hundred times and let it run with some dummy animation. Then I use Maya 2017’s Evaluation toolkit to Run a performance test, which gives us info about the performance in the different evaluation methods – DG, Serial and Parallel. Since, the results vary quite a bit, what I usually do is, run it three times and take the best ones.

In this case, I built the two setups in separate files, both with 2 target objects and maintain offset. Then I ran the tests on 200 hundred imported setups.

So here are the results.

##### Parent constraint
``````Playback Speeds
===============
DG = 89.8204 fps
EMS = 20.1613 fps
EMP = 59.2885 fps
``````
##### Matrix constraint
``````Playback Speeds
===============
DG = 91.4634 fps
EMS = 24.6305 fps
EMP = 67.2646 fps
``````

Bear in mind these tests are done on my 5 years old laptop, so the results you are going to get if you are to repeat this test are going to be significantly better.

As you can see even with the extended graph we are still getting about 7.5 fps increase by using the matrix constraint setup with blending matrices. Considering, we have 200 hundred instances in the scene (which is by no means a large number), that means we have about .0375 fps increase per setup, which in turn means that on every 26 setups we win a frame.

### Conclusion

So, there we have it, an even larger part of the `parentConstraint` functionality, can be implemented by just using matrix nodes. What this means is we can keep our outliner cleaner and get a better performance out of our rigs at the same time, which is a total win win.

Thanks to RigVader for pointing the `wtAddMatrix` node as a potential solution, it really works quite nicely!

## Maya matrix nodes – Part 3: Matrix rivet

This post is a part of a three post series, where I implement popular rigging functionalities with just using maya’s native matrix nodes.

Rivets are one of those things that blew my mind the first time I learned of them. Honestly, at the time, the ability to stick an object to the deforming components of a geometry seemed almost magical. Although, the more you learn about how geometries work in Maya, the more sense rivets start to make. The stigma around them, though, has always been that they are a bit slow, since they have to wait for the underlying geometry to evaluate and only then can they evaluate as well. And even though that is still the case, it seems that since parallel was introduced the performance has increased significantly.

It is worth trying to simplify and clean rivets up, considering how handy they are for rigging setups like:
– twist distributing ribbons
– bendy/curvy limbs
– sticking objects to geometries after squash and stretch
– sticking controls to geometries
– driving joints sliding on surfaces

and others.

When I refer to the classic rivet or the `aimConstraint` rivet, it is this one that I am talking about. I have seen it used by many riggers and also lots of lighters as well.

The purpose of this approach is to get rid of the `aimConstraint` that is driving the rotation of the rivet. Additionally, I have seen a `pointConstraint` used as well, in order to account for the parent inverse matrix, which would also be replaced by this setup. Even though we are stripping constraints, the performance increase is not very large, so the major benefit of the matrix rivet is a cleaner graph.

TL;DR: We are going to plug the information from a `pointOnSurfaceInfo` node directly into a `fourByFourMatrix` node, in attempt to remove constraints from our rigs.

Disclaimer: Bear in mind, I will be only looking at riveting an object to a NURBS surface. Riveting to poly geo would need to be done through the same old loft setup.

Limitations: Since we are extracting our final transform values using a `decomposeMatrix` node, we do not have the option to use any rotation order other than XYZ, as at the moment the `decomposeMatrix` node does not support other orders. A way around it, though, is taking the `outputQuat` attribute and pluging it into an `quatToEuler` node which actually supports different rotate orders.

### Difference between follicle and aimConstraint rivet

The locator is riveted using an `aimConstraint`. You can see there is a small difference in the rotations of the follicle and the locator. Why is that?

The classical rivet setup connects the `tangentV` and `normal` attributes of a `pointOnSurface` to the `aimConstraint`. The third axis is then the cross product of these two. But it seems like the follicle is actually using the `tangentU` vector for it’s calculations, since we get this difference between the two setups.

Choosing to plug the `tangentU` into the `aimConstraint`, instead of `tangentV`, results in the same `behaviour` as a follicle. To be honest, I am not sure which one would be preferable. In the construction of our matrix rivet, though, we have full control over that.

### Why not follicles?

As I already said, in parallel, follicles are fast! Honestly, for most of my riveting needs I wouldn’t mind using a follicle. The one aspect of follicles I really dislike though, is the fact that it operates through a shape node. I understand it was not meant for rigging, and having the objects clearly recognizable both in the outliner and the viewport is important, but in my case it is just adding up to clutter. Ideally, I like avoiding unnecessary DAG nodes, since they only get in the way.

Additionally, have you had a look at the follicle shape node? I mean, there are so many hair related attributes, it is a shame to use it just for `parameterU` and `parameterV`.

Therefore, if we could use a non-DAG network of simple nodes to do the same job without any added overhead, why should we clutter our rigs?

### Constructing the matrix rivet

So, the way matrices work in Maya is that the first three rows of the matrix describe the X, Y and Z axis and the fourth row is the position. Since, this is an oversimplification I would strongly suggest having a look at some matrix math resources and definitely watching the Cult of Rig streams, if you would like to learn more about matrices.

What this means to us, though, is that if we have two vectors and a position we can always construct a matrix out of them, since the cross product of the two vectors will give us a third one. So here is how our matrix construction looks like in the graph.

So, as you can see, we are utilizing the `fourByFourMatrix` node to construct a matrix. Additionally, we use the `vectorProduct` node set to Cross Product to construct our third axis out of the `normal` and the chosen tangent, in this case `tangentV` which gives us the same result as using the classic `aimConstraint` rivet. If we choose to use the `tangentU` instead, we would get the `follicle`‘s behaviour. Then, obviously we decompose the matrix and plug it into our riveted transform.

Optionally, similar to the first post in this series, we can use the `multMatrix` node to inverse the parent’s transform, if we so need to. What I usually do, though, is parent them underneath a transform that has it’s `inheritTransform` attribute turned off, so we can plug the world transforms directly.

It is important to note that in this case we are absolutely sure that the output matrix is orthogonal, since we know that the `normal` is perpendicular to both tangents. Thus, crossing it with any of the tangents, will result in a third perpendicular vector.

### Skipping the vector product

Initially, when I thought of building rivets like this, I plugged the `normal`, `tangentU` and `tangentV` directly from the `pointOnSurfaceInfo` to the `fourByFourMatrix`. What this means, is that we have a matrix that is not necessarily orthogonal, since the tangents might very well not be perpendicular. This results in a shearing matrix. That being said though, it was still giving me proper results.

Then, I added it to my modular system to test it on a couple of characters and it kept giving me steadily good results – 1 to 1 with the behaviour of a `follicle` or `aimConstraint` rivet, depending on the order I plug the tangents in.

What this means, then, is that the `decomposeMatrix` node separates all the shearing from the matrix and thus returns the proper rotation as if the matrix is actually orthogonal.

If that is the case, then we can safely skip the `vectorProduct` and still have a working rivet, considering we completely disregard the `outputShear` attribute of the `decomposeMatrix`.

Since, I do not understand how that shearing is being extracted, though, I will be keeping an eye on the behaviour of the rivets in my rigs, to see if there is anything dodgy about it. So far, it has proved to be as stable as anything else.

### Conclusion

If you are anything like me, you would really like the simplicity of the graph, as we literally are taking care of the full matrix construction ourselves. What is more, there are no constraints, nor follicle shapes in the outliner, which again, I find much nicer to look at.

This matrix series has been loads of fun for me to write, so I will definitely be trying to come up with other interesting functions we could use matrices for.

## Maya matrix nodes – Part 2: Node based matrix twist calculator

This post is a part of a three post series, where I implement popular rigging functionalities with just using maya’s native matrix nodes.

Calculating twist is a popular rigging necessity, as often we would rather smoothly interpolate it along a joint chain, instead of just applying it at the end of it. The classical example is limbs, where we need some twist in the forearm/shin area to support the rotation of the wrist or foot. Some popular implementations utilize ik handles or aim constraints, but I find them as a bit of an overkill for the task. Therefore, today we will have a look at creating a matrix twist calculator, that is both clean and quick to evaluate.

Other than matrix nodes I will be using a couple of quaternion ones, but I promise it will be quite simple, as even I myself am not really used to working with them.

tl;dr: We will get the matrix offset between two objects – relative matrix, then extract the quaternion of that matrix and get only the X and W components, which when converted to an euler angle, will result in the twist between the two matrices along the desired axis.

### Desired behaviour

Please excuse the skinning, I have just done a geodesic voxel bind

As you can see what we are doing is calculating the twist amount (often called roll as well from the yaw, pitch and roll notation) between two objects. That is, the rotation difference on the axis aiming down the joint chain.

### Limitations

An undesirable effect you can notice is the flip when the angle reaches 180 degrees. Now, as far as I am aware, there is no reasonable solution to this problem, that does not involve some sort of caching of the previous rotation. I believe, that is what the No flip `interpType` on constraints does. There was one, using an orient constraint between a no roll joint and the rolling joint and then multiplying the resulting angle by 2, which worked in simple cases, but I found it a bit unintuitive and not always predictable. Additionally, most animators are familiar with the issue, and are reasonable about it. In the rare cases, where this issue will be a pain in your production you can always add control over the twisting matrices, so the animators can tweak them.

Something else to keep in mind is to always use the first axis of the rotate order to calculate the twist in, since the other ones might flip at 90 degrees instead of 180. That is why, I will be looking at calculating the X twist, as the default rotate order is XYZ.

With that out of the way, let us have a look at the setup.

### Matrix twist calculator

I will be looking at the simple case of extracting the twist between two cubes oriented in the same way. Now, you might think that is too simple of an example, but in fact this is exactly what I do in my rigs. I create two locators, which are oriented with the X axis being aligned with the axis I am interested in. Then I parent them to the two objects I want to find the twist between, respectively. This, means that finding the twist on that axis of the locators, will give me the twist between the two objects.

Granted, I do not use actual locators or cubes, but just create matrices to represent them, so I keep my outliner cleaner. But, that is not important at the moment.

#### The relative matrix

Now, since we are going to be comparing two matrices to get the twist angle between them, we need to start by getting one of them in the relative space of the other one. If you have had a look at my Node based matrix constraint post or you were already familiar with matrices, you would know that we can do that with a simple multiplication of the child matrix by the inverse of the parent matrix. That will give us the matrix of the child object relative to that of the parent one.

The reason, we need that is because that relative matrix is now holding all the differences in the transformations between the two objects, and we are interested in exactly that, the difference on the aim axis.

Here is how that would look in the graph.

#### The quaternion

So, if we have the relative matrix, we can proceed to extracting the rotation out of it. The thing with rotations in 3D space is that they seem a bit messy, mainly because we usually think of them in terms of Euler angles, as that is what maya gives us in the `.rotation` attributes of transforms. There is a thing called a quaternion, though, which also represents a rotation in 3D space, and dare I say it, is much nicer to work with. Nicer, mainly because we do not care about rotate order, when working with quaternions, since they represent just a single rotation. What this gives us is a reliable representation of an angle along just one axis.

In practical terms, this means, that taking the X and W components of the quaternion, and zeroing out the Y and Z ones, will give us the desired rotation only in the X axis.

In maya terms, we will make use of the `decomposeMatrix` to get the quaternion out of a matrix and then use the `quatToEuler` node to convert that quaternion to an euler rotation, which will hold the twist between the matrices.

Here is the full graph, where the `.outputRotateX` of the `quatToEuler` node is the actual twist value.

### Conclusion

And that is it! As you can see, it is a stupidly simple procedure, but has proved to be giving stable results, which in fact are 100% the same as using an ik handle or an aim constraint, but with little to no overhead, since matrix and quaternion nodes are very computationally efficient.

Stay tuned for part 3 from this matrix series, where I will look at creating a rivet by using just matrix nodes.

## Maya matrix nodes – Part 1: Node based matrix constraint

This post is a part of a three post series, where I will try to implement popular rigging functionalities by only using maya’s native matrix nodes.

Following the Cult of rig lately, I realized I have been very wasteful in my rigs in terms of constraints. I have always known that they are slower than direct connections and parenting, but then I thought that is the only way to do broken hierarchy rigs. Even though I did matrix math at university, I never used it in maya as I weirdly thought the matrix nodes are broken or limited. There was always the option of writing my own nodes, but since I would like to make it as easy for people to use my rigs, I would rather keep everything in vanilla maya.

Therefore, when Raffaele used the `matrixMult` and `decomposeMatrix` nodes to reparent a transform, I was very pleasantly inspired. Since then, I have tried applying the concept to a couple of other rigging functionalities, such as the twist calculation and rivets and it has been giving me steadily good results. So, in this post we will have a look at how we can use the technique he showed in the stream, to simulate a parent + scale constraint, without the performance overhead of constraints, effectively creating a node based matrix constraint.

### Limitations

There are some limitations with using this approach, though. Some of them are not complex to go around, but the issue is that this adds extra nodes to the graph, which in turn leads to performance overhead and clutter. That being said, constraints add up to the outliner clutter, so I suppose it might be a matter of a preference.

#### Joints

Constraining a joint with `jointOrient` values, will not work, as the `jointOrient` matrix is applied before the rotation. There is a way to get around this, but it involves creating a number of other nodes, which add some overhead and for me are making it unreasonable to use the setup instead of an orient constraint.

If you want to see how we go around the `jointOrient` issue just out of curiosity, have a look at the joint orient section.

#### Weights and multiple targets

Weights and multiple targets are also not entirely suitable for this approach. Again, it is definitely not impossible, since we can always blend the output values of the matrix decomposition, but that will also involve an additional `blendColors` node for each of the transform attributes we need – `translate`, `rotate` and `scale`. And similarly to the previous one, that means extra overhead and more node graph clutter. If there was an easy way to blend matrices with maya’s native nodes, that would be great.

#### Rotate order

Weirdly, even though the decompose matrix has a `rotateOrder` attribute, it does not seem to do anything, so this method will work with only the `xyz` rotate order. Last week I received an email from the maya_he3d mailing list, about that issue and it seems like it has been flagged to Autodesk for fixing, which is great.

### Construction

The construction of such a node based matrix constraint is fairly simple both in terms of nodes and the math. We will be constructing the graph as shown in the Cult of Rig stream, so feel free to have a look at it for a more visual approach. The only addition I will make to it is supporting a maintainOffset functionality. Also, Raffaele talks a lot about math in his other videos as well, so have a look at them, too.

All the math is happening inside the `matrixMult` node. Essentially, we are taking the `worldMatrix` of a target object and we are converting it to relative space by multiplying by the `parentInverseMatrix` of the constrained object. The `decomposeMatrix` after that is there to break the matrix into attributes which we could actually connect to a transform – `translate`, `rotate`, `scale` and `shear`. It would be great if we could directly connect to an input matrix attribute, but that would probably create it’s own set of problems.

That’s the basic node based matrix constraint. How about maintaining the offset, though?

### Maintain offset

In order to be able to maintain the offset, we need to just calculate it first and then put it in the `multMatrix` node before the other two matrices.

#### Calculating offset

The way we calculate the local matrix offset is by multiplying the `worldMatrix` of the object by the `worldInverseMatrix` of the parent (object relative to). The result is the local matrix offset.

##### Using the multMatrix node

It is entirely possible to do this using another `matrixMult` node, and then doing a `getAttr` of the output and set it in the main `matrixMult` by doing a `setAttr` with the `type` flag set to `"matrix"`. The local `matrixMult` is then free to be deleted. The reason we get and set the attribute, instead of connecting it, is that otherwise we create a cycle.

##### Using the Maya API

What I prefer doing, though, is getting the local offset via the API, as it does not involve creating nodes and then deleting them, which is much nicer when you need to code it. Let’s have a look.

``````import maya.OpenMaya as om

def getDagPath(node=None):
sel = om.MSelectionList()
d = om.MDagPath()
sel.getDagPath(0, d)
return d

def getLocalOffset(parent, child):
parentWorldMatrix = getDagPath(parent).inclusiveMatrix()
childWorldMatrix = getDagPath(child).inclusiveMatrix()

return childWorldMatrix * parentWorldMatrix.inverse()
``````

The `getDagPath` function is just there to give us a reference to an `MDagPath` instance of the passed object. Then, inside the `getLocalOffset` we get the `inclusiveMatrix` of the object, which is the full world matrix equivalent to the `worldMatrix` attribute. And in the end we return the local offset as an `MMatrix` instance.

Then, all we need to do is to set the `multMatrix.matrixIn[0]` attribute to our local offset matrix. The way we do that is by using the `MMatrix`‘s `()` operator which returns the element of the matrix specified by the row and column index. So, we can write it like this.

``````localOffset = getLocalOffset(parent, child)
mc.setAttr("multMatrix1.matrixIn[0]", [localOffset(i, j) for i in range(4) for j in range(4)], type="matrix")
``````

Essentially, we are calculating the difference between the `parent` and `child` objects and we are applying it before the other two matrices in the `multMatrix` node in order to implement the `maintainOffset` functionality in our own node based matrix constraint.

### Joint orient

Lastly, let us have a look at how we can go around the joint orientation issue I mentioned in the Limitations section.

What we need to do is account for the `jointOrient` attribute on joints. The difficulty comes from the fact that the `jointOrient` is a separate matrix that is applied after the `rotation` matrix. That means, that all we need to do is, in the end of our matrix chain rotate by the inverse of the `jointOrient`. I tried doing it a couple of times via matrices, but I could not get it to work. Then I resolved to write a node and test how I would do it from within. It is really simple, to do it via the API as all we need to do is use the `rotateBy` function of the `MTransformationMatrix` class, with the inverse of the `jointOrient` attribute taken as a `MQuaternion`.

Then, I thought that this should not be too hard to implement in vanilla maya too, since there are the quaternion nodes as well. And yeah there is, but honestly, I do not think that graph looks nice at all. Have a look.

As you can see, what we do is, we create a quaternion from the joint orientation, then we invert it and apply it to the calculated output matrix of the `multMatrix`. The way we apply it is by doing a quaternion product. All we do after that is just convert it to euler and connect it to the rotation of the joint. Bear in mind, the `quatToEuler` node supports rotate orders, so it is quite useful.

Of course, you can still use the `maintainOffset` functionality with this method. As I said though, comparing this to just an orient constraint it seems like the orient constraint was performing faster every time, so I see no reason of doing this other than keeping the outliner cleaner.

Additionally, I am assuming that there is probably an easier way of doing this, but I could not find it. If you have something in mind, give me a shout.

### Conclusion

Using this node based constrain I was able to remove parent, point and orient constraints from my body rig, making it perform much faster than before, and also the outliner is much nicer to look at. Stay tuned for parts 2 and 3 from this matrix series, where I will look at creating a twist calculator and a rivet by using just matrix nodes.