Leg IK in UE4

It seems that Leg IK has been existed in UE4 for quite a long time, and it seems necessary to dig into the code and try to understand what is going on underneath.

And btw, it seems to be very convenient to use geogebra to draw 3D diagrams. This article would use diagrams created by this website.

Leg bones

Since this node is designed for multi-bone legs IK calculation, but how does it handle multi-bone legs? CCD or FABRIK? Or some other black magic?

void FIKChain::ReachTarget(const FVector& InTargetLocation, float InReachPrecision, int32 InMaxIterations)
{
	if (!bInitialized)
	{
		return;
	}

	const FVector RootLocation = Links.Last().Location;

	// If we can't reach, we just go in a straight line towards the target,
	if ((NumLinks <= 2) || (FVector::DistSquared(RootLocation, InTargetLocation) >= FMath::Square(GetMaximumReach())))
	{
		const FVector Direction = (InTargetLocation - RootLocation).GetSafeNormal();
		OrientAllLinksToDirection(Direction);
	}
	// Two Bones, we can figure out solution instantly
	else if (NumLinks == 3 && (CVarAnimLegIKTwoBone.GetValueOnAnyThread() == 1))
	{
		SolveTwoBoneIK(InTargetLocation);
	}
	// Do iterative approach based on FABRIK
	else
	{
		SolveFABRIK(InTargetLocation, InReachPrecision, InMaxIterations);
	}
}

It is easy to tell that for a three-bone(thigh->leg->foot) leg, this solver would eventually fallback to a simple two-bone IK.

But as for a leg that has more than 3 bones(spider leg or octopus leg), it would perform a FABRIK. Since I’ve already wrote a blog about FABRIK on csdn and I am not going to rewrite it again.

How does Leg IK work?

You can define your legs in this node. Each leg can be defined using FKFoot Bone and Num Bones in Limb, which defines a leg from toe to hip.

And you need an IKFoot Bone in your skeleton if you want to use this node. And during evaluation, this node would try to put your toe(in another word, FKFoot Bone) to the IKFoot Bone.

!

let’s check out the whole algorithm:

1. Leg transforms would firstly be aligned with the ik target. Delta normal is calculated between InitialDir(From Hip to FootFKLocation) and TargetDir(From Hip to FootIKLocation).

1. After being aligned with IKFoot Bone, try to reach for ik effector. Either Two bone IK or FABRIK is used for this purpose. For a Two bone IK, Hinge Rotation Axis is used for bending direction, and for a FABRIK, Min Rotation Angle is used for rotation limit, if enabled.
2. Knee twist would be adjusted if Enable Knee Twist Correction is checked. You need to define the Foot Bone Forward Axis for this feature. This axis is used for determining how much angle the foot has been twisted. By Bone Forward Axis it means the axis pointing to the ground.

Rotation limit for FABRIK

Something still needs to be done for constraints in FABRIK, when used for leg ik calculation. And hence we would go through those extra work here.

Plane normal&LinkAxisZ

Since we have just mentioned that legs would be aligned to the target effector, thus bone rotation limits are calculated in a 2-D space.

First thing that needs to be done is to get a valid plane normal for the whole leg.

By traverse vectorA->B', A->C' and A->FKFootBone' until we get a valid cross product result with vector A->IKFootBone.

After getting a valid plane normal, we still need a valid LinkAxisZ vector for each leg bone. The reason is that sometimes we may have a leg inner angle larger than pi.

In this case, we need to flip plane normal, and this is what LinkAxisZ do during limiting rotation, See LinkAxisZVector in the following image:

Apply rotation limitation

For a simple FABRIK without rotation limitation, the whole leg would loot like this after a forwardpass:

As for a leg, it looks terrible. Thus we need to apply rotation limit to it. Rotation limit is applied after each bone’s reaching stage, both forward and backward.

Let’s take a look at bone C'' rotation limit as an example:

1. First childAxisYC'' is calculated using a cross product between LinkAxisZ and childAxisXC'', which point from C'' to its child bone.

1. Then dot product between parentAxisC'', which points from C'' to its parent bone, and childAxisXC'' and childAxisYC'' is calculated. By this we could get the cos and sin value of current angle. We use this to determine whether current angle has already reached the limit.

1. If the sin value if less than 0(which means the the leg inner angle is larger than pi and the leg needs to be flipped), or the cos value is larger than the cos(MinRotationAngleRadians)(which means that the angle has already reached the angle limit), a rotation limitation should be performed. The current angle in the image above is 56.15 degrees, and our MinRotationAngleRadians is 60 degrees, which is larger than current degree.

2. As a result, we need to apply rotation limitation now. Since this is a forward reach pass, we need to adjust the parent bone location by code like this:

   ParentLink.Location = CurrentLink.Location + CurrentLink.Length * (FMath::Cos(IKChain.MinRotationAngleRadians) * ChildAxisX + FMath::Sin(IKChain.MinRotationAngleRadians) * ChildAxisY);

   And here is the result.

   

Some extra work

There is still some extra work that this anim node does for a better effect.

Distribute pull to re-position limb

The relevant code is:

// Re-position limb to distribute pull
const FVector PullDistributionOffset = PullDistributionAlpha * (InTargetLocation - Links[0].Location) + (1.f - PullDistributionAlpha) * (RootTargetLocation - Links.Last().Location);
for (int32 LinkIndex = 0; LinkIndex < NumLinks; LinkIndex++)
{
	Links[LinkIndex].Location += PullDistributionOffset;
}

There is a CVarAnimLegIKPullDistribution(Console variable a.AnimNode.LegIK.PullDistribution), whose default value is 0.5f.

And this value means whether we care more about the foot, or the hip.

As a result, the bone in the whole leg seems to move a delta offset.

But RootTargetLocation is totally the same as Links.Last().Location:

const FVector RootTargetLocation = Links.Last().Location;

So what are you exactly trying to do?

Average pull

The relevant code is:

			// Pull averaging only has a visual impact when we have more than 2 bones (3 links).
			if ((NumLinks > 3) && (CVarAnimLegIKAveragePull.GetValueOnAnyThread() == 1) && (Slop > 1.f))
			{
				FIKChain ForwardPull = *this;
				FABRIK_ForwardReach(InTargetLocation, ForwardPull);

				FIKChain BackwardPull = *this;
				FABRIK_BackwardReach(RootTargetLocation, BackwardPull);

				// Average pulls
				for (int32 LinkIndex = 0; LinkIndex < NumLinks; LinkIndex++)
				{
					Links[LinkIndex].Location = 0.5f * (ForwardPull.Links[LinkIndex].Location + BackwardPull.Links[LinkIndex].Location);
				}

#if ENABLE_ANIM_DEBUG
				if (bDrawDebug)
				{
					DrawDebugIKChain(ForwardPull, FColor::Green);
					DrawDebugIKChain(BackwardPull, FColor::Blue);
				}
#endif
			}

To be honest, I truly have no idea how pull averaging eventually work somehow(cannot find any paper about it). Nor do I know why does pull averaging only has a visual impact when we have more than 2 bones.

If average pull is enabled, instead of modify the original location directly, forward and backward pass would be stored separately in two FIKChain. The final location would be the average value of the forward and the backward value.

Can it be better?

• Currently there is only a MinRotationAngleRadians for us to control the leg constraint. But a MaxRotationAngleRadians should always be concerned.
• It would be much better if we can set the MinRotationAngleRadians and MaxRotationAngleRadians for each joint.