19.1 A Simple Example

Inverse dynamics is very much an engineering approach to biomechanical simulation. Let us consider the simplified example of a foot on an ankle joint shown in the free body diagram of Fig. 19.1. Despite the anatomical reality, we shall consider the foot as a single rigid segment and the ankle as a hinge joint. In this two-dimensional case, two reaction force components, R_x and R_y , are working in the ankle joint. The ball of the foot is loaded by a vertical ground reaction force, F_g . We consider initially only the muscle force, F_m , from the soleus muscle working through the Achilles tendon. The mass of the foot is disregarded.

We start by the moment equilibrium about the ankle:

Horizontal equilibrium quickly reveals that

All that remains now is to determine R_y from vertical equilibrium:

Let us insert some plausible numbers:

s ₁ = 0.1 m, s ₂ = 0.04 m, F_g = 1,600 N

This leads to the following forces in the system:

F_m = 4,000 N

R_y = 5,600 N

Please notice that the modest external force gives rise to large internal forces. This is typical for musculoskeletal systems. We are generally unaware of the magnitude of the forces working inside our bodies and the strength of our tissues. When a basketball player ruptures a knee ligament or a sprinter ruptures an Achilles tendon, we can rest assured that really substantial forces are at play. It is not difficult to get an impression of the forces by palpation of the tendons, for instance inside the elbow during elbow flexion.

Despite this example being static, it is in reality inverse dynamics in its simplest form: We know the posture and the velocity (in this case the velocity is zero) of the elements in the system, and we also know the external forces acting on the system. With this input, we can compute the internal forces (i.e., the muscle force and the joint reaction force).

In practical use of inverse dynamics, the systems tend to be much more complicated than the simple foot example, so we prefer to have computers solve the equations for us. Computers are generally excellent equation solvers, but they are not very intelligent and not good at making smart decisions about how to form the equations to simplify the solution procedure by decoupling the equations, as we did in the example. Therefore, they generally have to solve many simultaneous equations, and their formation requires a stringent approach. Let us review the structure of the problem in the case where we formulate it as three equations with three unknowns:

We can cast these equations into a matrix form:

This type of formulation is particularly well suited for computer processing, and there are very efficient numerical methods available for handling of matrices and linear equations. Despite the fact that anatomically realistic models are usually much more complicated than this simple example, it turns out that it is always possible to cast the equilibrium equations into the form of equation (19.6), even in the case of the full human body with 200 bones and three-dimensional conditions. Because of this general feature, it is convenient to introduce a formalism that can be used later:

where

In this system, matrix C contains only constants that can be found prior to the solution of the problem, f is a vector representing the unknown internal forces, and r contains the external forces. The system of equations therefore fundamentally expresses that the external forces on the system must be balanced by internal forces. This is the fundamental mathematical structure of an inverse dynamics problem.

19.2 Redundancy and Muscle Recruitment

Although the basic structure of the equilibrium equations is always as displayed previously, some important complications arise for realistic musculoskeletal systems. One of the problems is redundancy in the system, which arises because the body has many more muscles than degrees of freedom. To illustrate this point, we shall extend the previous example a little, as shown in Fig. 19.2.

The only difference from the previous example is that we now have two muscle forces, F _m1 and F _m2, performing plantarflexion of the ankle.

With the additional muscle force, the equilibrium equations are:

which in matrix form becomes:

The coefficient matrix C is now rectangular, indicating that there are more unknowns than equations in the problem. The mathematical consequence of this is that the system of equations does not have a unique solution, but rather infinitely many different solutions. Physically, we can understand this property by noticing that the ground reaction force, F_g , can be balanced by F _m1 alone, by F _m2 alone, or by many different combinations of the two muscle forces, as long as the two muscle forces in concert produce a moment about the ankle joint of the same size and opposite the moment produced by F_g .

For instance, when we walk, we are simultaneously activating several muscles about the ankle joint in a carefully tuned pattern to produce gait of the desired speed and direction, and we have no perception of the fact that we are instantly choosing between an infinite number of possible activations of the muscles. It even turns out that, if we measure the activity in the muscles in cyclic movements by electromyography, the same muscle activation patterns seem to be repeated over and over again, although a perfectly repetitive movement could theoretically be produced by different muscle activation patterns in each cycle. This indicates that the body does not choose muscle activations randomly but employs a rational criterion.

From experimental investigations, some criteria can be identified:

• 
  

  Muscles spanning the same joint tend to help each other. This phenomenon is called synergism, and muscles helping each other are called synergistic muscles. Referring back to the example, we can therefore rule out the option that only one of the two muscles may be doing all of the work.

  
  
• 
  

  When the externally applied moment over a joint is increased, then so is the activity in all the synergistic muscles. 1

  
  
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  Certain muscles can be observed to work “against the movement” (that last phrase is in quotes because it turns out that it is not completely trivial to define precisely what it means). Such muscles are called antagonistic muscles or simply antagonists, and their presence has been recognized for many years and is known as Lombard′s paradox. 2

The observation that muscles tend to work in systematic patterns seems logical in an evolutionary thinking, where optimal muscle coordination may have resulted from natural selection (i.e., species and individuals with good muscle coordination have an advantage and may propagate their genes). Mathematically, such an idea can be formulated as an optimization problem, as follows.

subject to

This is a so-called mathematical program, and its solution is a set of muscle forces, f ^(M), that minimize the objective function G, while honoring the constraints. Notice that one of the constraints is the set of equilibrium equations (i.e., feasible solutions must be in equilibrium with the external forces).

Notice that the vector of internal forces, f, has been divided into two parts:

The first part, f ^(M), contains muscle forces, while the latter, f ^(R), contains the joint reactions. This division reflects that muscle forces require metabolism and therefore energetic resources, whereas the joint reactions come for free. It is therefore plausible that the cost function, G, depends on the muscle forces but not necessarily on the joint reactions. Finally, please notice the additional nonnegativity condition for each element of f ^(M). This is the mathematical way of requiring that muscles cannot push.

These points all had physiological motivations, but what about the cost function G? Which function is the right one, and what might nature in fact be trying to minimize in its selection of muscle recruitment patterns? The set of possible cost functions is of course endless, so it is useful to limit the scope a little, and this may be done with mathematical arguments. One of the obvious systematizations is to limit the possible objective functions to polynomial sums of muscle forces:

If we believe that the right objective function can be found among this class of functions, then the problem is reduced to identifying the normalizers, N_i , and the degree of the polynomial, p. We are fairly certain that large muscles should pull more load than small muscles, and this indicates that N_i should somehow express the strength of the i′th muscle (i.e., the cross-sectional area of the muscle). There is good experimental evidence for the notion that we will not get realistic predictions of muscle forces if we do not consider the mutual strengths of the muscles in the formulation of the problem. 3

The degree, p, of the polynomial is more disputable except for the general agreement that p = 1 can be ruled out because it does not produce the synergism between the muscles that can be observed experimentally. Any value of p > 1 will result in synergy between the muscles and also some extent of antagonistic muscle forces in complex systems. It is possible to show that this antagonism arises as a result of either biarticular muscles or of three-dimensional joints. It is also possible to show mathematically that the amount of synergism between the muscles increases with p. A special case occurs when p goes to infinity (p → ∝). In this case, the muscle recruitment problem becomes equivalent to the following.

subject to

The physiological explanation for this formulation is minimization of fatigue. The objective function focuses on the single muscle in the system that has the largest relative load; all muscles are recruited to minimize this load. This means that all of the muscles end up helping each other as much as possible. Numerical experiments with the different options indicate that smaller values of p, for instance p = 2 or p = 3 yield good results for submaximal loads, whereas the minimum fatigue criterion is useful for larger loads. However, compared with the biological variation between individuals, all values of p > 1 show similar trends in the muscle recruitment, and it is impossible to determine conclusively whether one or the other is the correct criterion, 4 although recent investigations favor the minimum fatigue criterion. 5