Kjartan Halvorsen

07-17-2007, 03:19 PM

Dear colleagues,

Good remarks have been offered on this interesting topic! At the ISB meeting in Taipei I gave a talk on this exact topic, in which results were presented on a new approach to finding the directions of the talar and subtalar axes. Unlike the method described in Ton van den Bogert et als 1994 paper, the problem was formulated in order to avoid having to estimate the actual rotations occurring at the two joints. The formulation was based on the observations that the two axes - assuming hinge axes - would be fixed with respect to each other during the recorded movement. This lead to an optimization problem. The method was evaluated using data from bone-anchored markers (data collected by Karolinska Institute, Salford University and ETH Zürich).

Our approach turned out not to be fruitful, at least not with the problem formulation we explored. Simulation tests showed that the minimum of the criterion function shifted remarkably with only random and modest noise in the data (simulated noise was random rotations of the tibia and calcaneus), indicating that the errors combined in a destructive manner. Thus, Ton's comment on the power of combining a large number of noisy measurements is true as long as the noise combines in ways (linear, mostly) that tend to cancel the noise.

We concluded that with the methods currently available, the best way to obtain reasonable estimates of the talar and subtalar axes is to isolate as good as possible the movement to each of the two joints in turn, and compute the functional joint axis using the method recommended by Ehrig et al (2007). Apparently, Lewis, Kirby and Pizza (2007) reached the same conclusion.

But, going a step further, we asked the question whether (talar and subtalar) joint rotations could be reasonably estimated from the movement of tibia and calcaneus using a two-hinge model with large errors (20 degree error in direction of subtalar axis, 5 degree error in talar axis). A recursive filter (extended Kalman filter) was applied to estimate the joint rotations (Halvorsen et al 2007).The subjects (same data as above) performed movements with a loaded ankle joint, moving the talar and subtalar joints with a range of motion of about 40 and 35 degrees, respectively. It turned out that the RMS error in estimated rotations varied from 2.4 to 6.2 degrees across the four subjects.

There is both a positive and a negative message in the result that good estimates can be obtained with a bad model, i.e. that the estimates are insensitive (to some) model errors. The positive message is that a crude model can be good enough. The negative message is that if the goal is to estimate model parameters, then, since the results are insensitive to (some of the) model parameters, cost functions to be minimized are likely to be very flat around the optimal point.

Sincerely,

Kjartan Halvorsen

GIH

The Swedish School of Sport and Health Sciences,

Stockholm

Sweden

References:

Ehrig RM, Taylor WR, Duda GN, Heller MO. 2007 "A survey of formal methods for determining functional joint axes."

J Biomech. 40(10):2150-7

Lewis GS, Kirby KA, Piazza SJ 2007 "Determination of subtalar joint axis location by restriction of talocrural joint motion"

Gait & Posture 25: 63-69.

van den Bogert AJ, Smith GD, Nigg BM. 1994 "In-vivo determination of the anatomical axes of the ankle joint complex: an optimization approach."

J Biomech. 27(12):1477-88

Halvorsen K, Johnston C, Back W, Stokes V, Lanshammar H. 2007 "Tracking the motion of hidden segments using kinematic constraints and kalman filtering." J Biomech Engn. Accepted for publication.

Good remarks have been offered on this interesting topic! At the ISB meeting in Taipei I gave a talk on this exact topic, in which results were presented on a new approach to finding the directions of the talar and subtalar axes. Unlike the method described in Ton van den Bogert et als 1994 paper, the problem was formulated in order to avoid having to estimate the actual rotations occurring at the two joints. The formulation was based on the observations that the two axes - assuming hinge axes - would be fixed with respect to each other during the recorded movement. This lead to an optimization problem. The method was evaluated using data from bone-anchored markers (data collected by Karolinska Institute, Salford University and ETH Zürich).

Our approach turned out not to be fruitful, at least not with the problem formulation we explored. Simulation tests showed that the minimum of the criterion function shifted remarkably with only random and modest noise in the data (simulated noise was random rotations of the tibia and calcaneus), indicating that the errors combined in a destructive manner. Thus, Ton's comment on the power of combining a large number of noisy measurements is true as long as the noise combines in ways (linear, mostly) that tend to cancel the noise.

We concluded that with the methods currently available, the best way to obtain reasonable estimates of the talar and subtalar axes is to isolate as good as possible the movement to each of the two joints in turn, and compute the functional joint axis using the method recommended by Ehrig et al (2007). Apparently, Lewis, Kirby and Pizza (2007) reached the same conclusion.

But, going a step further, we asked the question whether (talar and subtalar) joint rotations could be reasonably estimated from the movement of tibia and calcaneus using a two-hinge model with large errors (20 degree error in direction of subtalar axis, 5 degree error in talar axis). A recursive filter (extended Kalman filter) was applied to estimate the joint rotations (Halvorsen et al 2007).The subjects (same data as above) performed movements with a loaded ankle joint, moving the talar and subtalar joints with a range of motion of about 40 and 35 degrees, respectively. It turned out that the RMS error in estimated rotations varied from 2.4 to 6.2 degrees across the four subjects.

There is both a positive and a negative message in the result that good estimates can be obtained with a bad model, i.e. that the estimates are insensitive (to some) model errors. The positive message is that a crude model can be good enough. The negative message is that if the goal is to estimate model parameters, then, since the results are insensitive to (some of the) model parameters, cost functions to be minimized are likely to be very flat around the optimal point.

Sincerely,

Kjartan Halvorsen

GIH

The Swedish School of Sport and Health Sciences,

Stockholm

Sweden

References:

Ehrig RM, Taylor WR, Duda GN, Heller MO. 2007 "A survey of formal methods for determining functional joint axes."

J Biomech. 40(10):2150-7

Lewis GS, Kirby KA, Piazza SJ 2007 "Determination of subtalar joint axis location by restriction of talocrural joint motion"

Gait & Posture 25: 63-69.

van den Bogert AJ, Smith GD, Nigg BM. 1994 "In-vivo determination of the anatomical axes of the ankle joint complex: an optimization approach."

J Biomech. 27(12):1477-88

Halvorsen K, Johnston C, Back W, Stokes V, Lanshammar H. 2007 "Tracking the motion of hidden segments using kinematic constraints and kalman filtering." J Biomech Engn. Accepted for publication.