
The functional method is useful in locating the center of rotation of a ballandsocket joint or the axis of rotation of a hinge joint. It computes the center/axis of rotation numerically rather than relying on a set of predetermined ratios or prediction equations. The hip joint is a ballandsocket joint and has 3 degrees of freedom. In an ideal situation, markers attached to the thigh will form spheres with the hip joint being the center if the pelvis is stationary. Thus, it is possible to numerically compute the center of the spheres based on the marker coordinates. Four methods (algorithms) are included in this page: 2 closedform least square algorithms by HU Gamage & Lasenby (2002), and Halvorsen et al. (1999), respectively, the NelderMead downhill simplex method, and the Newton's method. Acknowledgement: The author is grateful to Dr. Nels Madsen of Auburn University for his invaluable contribution to this page. He kindly provided a detailed rundown of the equations. Fig. 1 shows the geometric relationship between the hip joint and the markers attached to the thigh. Each marker on the thigh will form a sphere with the hip joint being its center. Vector in Fig. 1 is the position of the hip joint center while vector is the position of marker j at frame i.
The error function can be defined as
where R_{j} = radius of the sphere formed by marker j. Thus, the cost function for the least square approximation of the hip joint center becomes
The cost function U in [1] must be minimized to obtain the position of the center of the spheres (hip joint). Now, let's define the mean position of marker j shown in Fig. 1 as:
The relative position of the marker to the mean position at frame i can be expressed in terms of the mean position:
where
Vector shown in Fig. 1 is the relative position of the joint center to the mean position of marker j:
Therefore:
The cost function shown in [1] can be then rewritten as
Minimization of the cost function U yields
and
From [9]:
where . From [10]:
Note that [5] and [11] were used in deriving [12]. From [12]:
Rewrite [13] in the matrix form and divide both sides by n:
where { } = column matrix of the vector. [14] basically has the following form:
where
and
A_{j} = (3 x 3) while B_{j} = (3 x 1). From [15], [16], and [17]:
[18] is basically in the form of
where
Expand [19] and [20] to all markers:
where C = (3 x 3) while D = (3 x 1). [19] is a linear system of equations of the position of the hip joint. From [21]:
This method was introduced by H. U. Gamage and Lasenby (2002). The advantage of this method is that it can handle multiple markers at once. The basic assumption is that the spheres formed by the markers have a common center (hip joint). NelderMead Downhill Simplex Method Another algorithm that can be used in finding the hip joint center is the socalled downhill simplex method following Nelder and Mead. This method performs multidimensional minimization of the given function. In other words, it finds the minimum of the given cost function and the corresponding position of the hip joint center. Since this method does not involve derivatives, the cost function can be written as either
or
Although [24] looks more complex than [23], it requires less computations since the magnitude of a vector is typically computed from its square. that minimizes U is the position of the hip. The uniqueness of this method is in the method of systematic perturbation of through iterations in multi dimensions to obtain the minimum of function U. It uses the socalled simplex for this and tetrahedron defined by 4 vertices (points) is the simplex in three dimension. In other words, four position vectors instead of one will be used in obtaining the position of the hip joint center. If , the initial guess of , is available, the four vertices of the starting simplex can be given as
where = constant, and i, j, & k = unit vectors of the axes. For example, the overall mean position of the markers can be used as the initial guess. Each iteration consists of the following steps:
There are 6 different types of simplex manipulations possible:
Through the different simplex manipulations described above, the method approaches to the 'valley floor' and provides minimum of the cost function and location of the hip joint center. See Press et al. (1986) for details of the downhill simplex method and the sample function Amoeba. The iterations can be stopped if the change in the function value is fractionally smaller than a given tolerance:
where j = high point, k = low point, and U = cost function. It is customary to start the iteration once again w/ the low point obtained from the previous session being one of the starting vertices to make sure that the converged low point is the true solution. This method is based on a simple geometric relationship between the marker positions at two different instants. If the pelvis is stationary, the thigh markers always lie on the spheres they form with the hip joint being the center. In this case the line connecting two different positions of the same marker at two different instants is always perpendicular to the line drawn from the hip joint to their midpoint (Fig.3):
The error function is then
Let the cost function U be
which needs to be minimized. Minimization of the cost function U shown above has two meanings: (a) keep the angle between the two vector as close to 90 degrees as possible, and (b) keep the magnitude of the vector drawn from the joint center to the midpoint as small as possible. From [33]:
Covert [36] to the equivalent matrix form:
where {} = column matrix operator. Expand [37] to m markers:
where,
and
Form [38]:
Theoretically, [41] can be expanded to multiple twoframe combinations to increase the redundancy. One issue in this method is to determine the frames to be used in the computation. For a given marker, let the random error function for marker j at frame i be
For n frames and m markers, we will have a nonlinear system of m x n equations. The partial derivatives of function are
where
Therefore, the Jacobian matrix of the nonlinear system of equations of interest can be written as
The Newton's method involves iterations and each iteration consists of three steps. The first step is to solve the following linear system of equations:
where, k = the current iteration, = the solution from the previous iteration,
and
In order for [46] to work, the initial guess of r_{c} and R_{j}'s, , must be provided. The mean coordinates of the marker may be used as the initial value. The second step in each iteration is to update r_{c} and R_{j}'s using the solution obtained in the previous step:
The last step is to check if the solution is sufficiently converged:
where TOL = tolerance. The iteration will stop if the solution is sufficiently converged. Acknowledgement: This method was suggested by Dr. YongSan Yoon of Korea Advanced Institute of Science and Technology (KAIST). From [1]:
[51] can be rewritten as
where
[52] is basically a linear equation of r_{c} and . Expand [52] to n frames and m markers to obtain
Use the typical least square approach to solve the system of equations. R_{j}'s of the markers can be computed afterward using [53], if necessary. References and Related Literature Gander, W., Golub, G. H., & Strebel, R. (1994). Fitting of circles and ellipses: least squares solution. Tech Report, Departement Informatik, ETH Zurich. Halvorsen, K., Lesser, M., & Lundberg, A. (1999). A new method for estimating the axis of rotation and the center of rotation. J. Biomechanics 32, 12211227. Hiniduma Udugama Gamage, S.S., & Lasenby, J. (2002). New least squares solutions for estimating the average center of rotation and the axis of rotation. J. Biomechanics 35, 8793. Press, W.H., Flannery, B.P., Teukolsky, S.A., & Vetterling, W.T. (1986). Numerical recipes: the art of scientific computing. New York, NY: Cambridge University Press. Thompson, M.S., Dawson, T., Kuiper, J.H., NorthmoreBall, M.D., & Tanner, K.E. (2000). Acetabular morphology and resurfacing design. J. Biomechanics 33, 16451653. 
© YoungHoo Kwon, 1998 