Lesson 2: Adding FDK Features

Old tutorial:

This tutorial has not yet been updated to ver. 7 of the AnyBody Modeling System. Some concepts may have changed.

What we have now is a standard inverse dynamics AnyBody model capable of computing forces in a knee joint that is presumed to be a simple hinge. Real knees are unfortunately not as simple as that. Mechanically speaking, the difference between an idealized revolute knee and a real knee lies in the source of the forces which hold the joint together.

An idealized knee joint does not allow any deviations from its hinge motion, like if the tibia and femur were to start sliding on one another. This is accomplished by joint reaction forces which will enforce the zero sliding constraint, regardless of how large the required forces may be.

The real knee does not work like that. The cartilage cushioning the contact between the femoral condyles and the tibial plateau is elastic, and so are the ligaments and menisci stabilizing the knee against free sliding. Since the forces in these passive structures depend on their deformation, zero deformation implies the absence of any stabilizing force. In other words, the knee MUST deform a little to invoke these stabilizing forces and retain its integrity.

Let us begin the steps that will allow AnyBody to compute this deformation. We zoom in on the definition of the knee joint and change the definition of its reaction forces:

AnyRevoluteJoint KneeJoint = {
  AnyRefFrame &Shank = .Shank.KneeCenter;
  AnyRefFrame &Thigh = .Thigh.KneeCenter;
  
  // Prepare the joint for FDK: Define the reaction types in x and y
  // directions to be FDK-dependent. These reaction forces must then
  // be switched off and provided by some elastic element that we
  // define explicitly below.
  Constraints = {
    CType = {ForceDep, ForceDep, Hard, Hard, Hard};
    Reaction.Type={Off,Off,On,On,On};
  };
};

Here we redefine one of the default properties of the joint: the definition of constraints. As mentioned in “Getting Started: AnyScript Programming, Lesson 3: Connecting segments by joints”, connecting two completely independent rigid segments with a joint arrests some or all of the six degrees of relative motion freedom that existed between the two.

In this manner, a revolute joint imposes five constraints of which the first three are translational constraints (preventing relative sliding) whose violation is resisted by the joint reaction forces and the latter two are rotational constraints (preventing relative, out of plane rotation) enforced by reaction moments.

The shank’s KneeCenter node - which is the joint’s default coordinate system - has the y axis pointing proximally along the shank’s length axis and the x axis pointing forward. These are the two directions in which we’d like to introduce elastic stabilization of tibio-femoral translation, so the first two components of the CType vector are changed to the value ForceDep, which means that rather than being ‘Hard’ constraints, the forces are now defined by some elastic element, which we shall introduce later. We are thus switching off the usual reaction forces in those directions by setting the Reaction.Type vector.

Now let us add the necessary elasticity to the joint. This can be done anywhere in the model, but we might as well place it just below the joint:

// Knee joint. Notice that this is only going to be the nominal joint.
// The actual position of the knee joint center will depend on the forces
// acting upon it. Notice that we list the shank before the thigh. This
// defines the knee joint in the shank coordinate system and we can
// relate the reaction forces to the direction of the tibial plateau.
AnyRevoluteJoint KneeJoint = {
  AnyRefFrame &Shank = .Shank.KneeCenter;
  AnyRefFrame &Thigh = .Thigh.KneeCenter;
  // Prepare the joint for FDK: Define the reaction types in x and y
  // directions to be FDK-dependent. These reaction forces must then
  // be switched off and provided by some elastic element that we
  // define explicitly below.
  Constraints = {
    CType = {ForceDep, ForceDep, Hard, Hard, Hard};
    Reaction.Type={Off,Off,On,On,On};
  };
};
// Define springs in the knee, simulating the effect of cartilage
// and ligaments.
AnyForce KneeStiffness = {
  AnyKinLinear &lin = Main.MyModel.KneeJoint.Linear;
  F = {-1000*lin.Pos[0], -5000*lin.Pos[1], 0};
};

We are using the AnyForce class for this purpose. AnyForce in an abstract force that works on any kinematic measure we define inside it. In this case, we simply refer to the linear measure which tracks the distance between the two joint nodes on each segment. In an idealized joint, this measure will always be zero as long as AnyBody can successfully enforce all the translational constraints, however since the first two components of CType are set to ‘ForceDep’, they can now vary and become non-zero.

The x corresponds to sliding of the condyle along the tibial plateau. In this direction, we can perceive the elasticity as primarily being provided by the rim of the meniscus and the cruciate ligaments.

The y direction is along the shank’s long axis and in this direction, the elasticity is provided by the layer of cartilage between the tibial plateau and the femoral condyles. The z axis points laterally but since we are building a planar model of the knee, we leave it to be a conventional hard constraint.

It is therefore likely that the stiffness in the y direction is somewhat larger than in the x direction. We are going to define it that way and also choose values that are much smaller than in the real knee to get some nice, large deformations that are visually perceivable. So, the definition of the actual force inside the AnyForce object looks like this:

F = {-1000 * lin.Pos[0], -5000 * lin.Pos[1], 0};

As you can see, we simply specify the forces in the different directions as mathematical functions of the Pos property of the lin measure. Pos contains the actual linear displacements, and when we multiply those with -1000 and -5000 respectively, we are generating spring forces that are proportional and opposite to the translational deformation of the joint. As discussed earlier, we have made the y direction stiffness five times larger than the value for the x direction.

One of the beauties of the AnyScript language is that these expressions can be as complicated as you want. So if you happen to know more complex, realistic stiffness properties of the knee from a cadaver study or from a detailed finite element model, then you could just as well input those.

Let’s get the final part of the definition finalized. All that is remaining is to tell the solver in AnyBody that it should apply force-dependent kinematics to solve the problem. This is of course done in the study section:

AnyBodyStudy Study = {
  AnyFolder &Model = .MyModel;
  Gravity = {0.0, -9.81, 0.0};
  tStart = 1;
  tEnd = 10;
  nStep = 100;
  InverseDynamics.ForceDepKinOnOff=On;
};

That is all there is to it. The usual InverseDynamics operation will now compute elastic deformations in the knee joint resulting from the deformation of soft tissues in response to internal and external forces. Go ahead and try it out. If something does not work, you can download a functional model here.

TROUBLESHOOTING HELP: Inverse dynamics arrives at values of the force dependent degrees of freedom (corresponding to the flexible joint constraints) where the resulting passive stabilizing forces and computed muscle forces, place those degrees of freedom in static equilibrium. This is achieved by a gradient sensing optimizer which iteratively tries out different combinations of joint deformation and muscle force magnitudes which fulfil the equilibrium and optimization criteria.

It may therefore be necessary to adjust optimization settings of the AnyBodyStudy class such as “InverseDynamics.ForceDepKin.MaxNewtonStep” and “InverseDynamics.ForceDepKin.Perturbation”. For example, a large perturbation size implies a large finite-difference step for the knee translation values when the optimizer computes gradients numerically. If the knee stiffness was extremely non-linear, this gradient might not reflect the local behaviour of the functions which the optimizer is working with.

When using more anatomically realistic body models containing passive spring-like ligaments, it is good to ensure that the ligaments are calibrated to ensure that their resting length isn’t too short or long. You can read more on calibration in “Muscle Modeling, Lesson 7: Ligaments” and “Inverse Dynamics of Muscle Systems, Lesson 7: Calibration”.

In the next lesson we investigate the results in more detail.