Fuzzy clustering of gait patterns of patients after ankle arthrodesis based on kinematic parameters

Medical Engineering & Physics 23 (2001) 83–90 www.elsevier.com/locate/medengphy

Fuzzy clustering of gait patterns of patients after ankle arthrodesis based on kinematic parameters Fong-Chin Su a

a,*

, Wen-Lan Wu

a, b

, Yuh-Min Cheng c, You-Li Chou

a

Institute of Biomedical Engineering, National Cheng Kung University, Tainan 701, Taiwan b Department of Physical Therapy, Foo-Yin Institute of Technology, Kaohsiung, Taiwan c Department of Orthopaedic Surgery, Kaohsiung Medical University, Kaohsiung, Taiwan

Received 23 February 2000; received in revised form 5 February 2001; accepted 15 February 2001

Abstract Kinematic parameters for 10 normal subjects and 10 patients with ankle arthrodesis are grouped using the fuzzy cluster paradigm. The features chosen for clustering are Euler angles of the sagittal plane in the hindfoot, the forefoot and combined hindfoot and forefoot joints. Gait patterns are identified using information provided by cluster validity techniques, giving three, three and two clusters for the hindfoot, forefoot and combined hindfoot and forefoot joints, respectively. The cluster centers represent distinct walking strategies adopted by normal subjects and patients after ankle arthrodesis. Utilizing angle values normalized by gait cycle, it is possible to classify any subject and to generate an individual’s membership value for each of the clusters. The clinical utility of the fuzzy clustering approach is demonstrated with data for subjects with ankle arthrodesis, where changes in membership of the clusters provide an objective technique for measuring changes of gait pattern after ankle arthrodesis. This approach can be adopted to study other clinical entities where different cluster centers would be established using the algorithm provided in this study.  2001 IPEM. Published by Elsevier Science Ltd. All rights reserved. Keywords: Ankle arthrodesis; Gait; Kinematic; Fuzzy c-means; Cluster

1. Introduction Human gait is a complex phenomenon. A detailed specification of gait would have several applications: among them are monitoring the effects of orthopedic surgery, improving athletic performance, etc. However, in many studies the methods used have depended on qualitative descriptions. It is important to establish a quantitative method of analysis [1–12]. Many descriptors are needed to completely describe gait in terms of the biomechanics involved. The descriptors, when expressed as a function of the gait cycle, are complex waveforms [13–16]. For each of these variables, a single “normal” pattern with bands of deviation has generally been accepted as a reference in clinical/research use to explain the abnormalities in a patient’s walking pattern. In fact, one observes many “normal” patterns, and a body of

* Corresponding author. Tel.: +886-6-2760665; fax: +886-62343270. E-mail address: [email protected] (F.-C. Su).

research has been devoted to explaining the differences between these patterns in terms of walking speed, age, cadence, sex, etc. It would be simpler in one sense to start with the fact that different people walk with different patterns, not one pattern with bands of deviation. Numerical representation of the waveforms simplifies the analysis and interpretation of waveform data and facilitates comparison between subjects or groups of subjects. When combined with pattern recognition techniques, it is also useful for identifying subpatterns within a group. Classical (crisp) clustering algorithms generate partitions such that each object is assigned to exactly one cluster. Often, however, objects cannot adequately be assigned to strictly one cluster (because they are located “between” clusters). In these cases fuzzy clustering methods provide a much more adequate tool for representing real-data structures. Pattern recognition is one of the oldest and most obvious application areas of fuzzy set theory. The term pattern recognition embraces a very large and diversified literature. It includes research in the areas of artificial

1350-4533/01/$20.00  2001 IPEM. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 0 - 4 5 3 3 ( 0 1 ) 0 0 0 2 0 - 0

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intelligence, computers with interactive graphics, computer-aided design, psychological and biological pattern recognition, linguistic and structural pattern recognition, and a variety of other research topics. The following are references for the application of classical clustering algorithms or fuzzy pattern recognition to medical diagnosis, in particular for pattern searching in multidimensional gait problems. In 1997, Dickerson et al. [1] showed how multiple causes interact to cause patient diagnosis by clusters in the diagnostic categories. Many studies showed how multidimensional statistical methods (e.g., principal-components analysis, k-means cluster) may provide insight into gait data [2–6]. Temporal–distance parameters of cerebral palsy children were grouped using the fuzzy clustering paradigm [7]. Studies by Kosanovic et al. showed how well the dynamic profiles of physical processes can be estimated using a fuzzy c-means (FCM) algorithm [8–12]. In 1979, the long-term problem of post-ankle arthrodesis was investigated with gait analysis by Mazur et al. [13]. In their model, the ankle–foot complex was constructed with three markers around the foot and the method of two-dimensional projected coordinates of each point was used to establish the local coordinate system. The results of gait analysis were then reviewed, combined with the roentgenograms of a fused ankle with dorsiflexion and plantar-flexion moment applied. The analysis showed that the lost ankle motion of patients who had undergone ankle fusion was compensated by motion of the small joints of the ipsilateral foot, and resulted in altered motion of the ankle in the contralateral limb. In 1987, the problem of optimum positioning for arthrodesis of the ankle was investigated again by Buck et al. [14]. Three-dimensional electrogoniometers were used to measure the motion of the ankle–foot complex and knee joint. It had been found that a valgus position of the arthrodesis is more advantageous than a varus position and provides a more normal gait. However, it is well known that the foot is a complex, dynamic structure consisting of 26 bones, 30 muscles, 21 functional joints and over 100 ligaments. It is apparent that it is inappropriate to regard the foot as a rigid lever in analysis. To provide a more detailed analysis of normal and pathological foot function, a three-rigid-bodies model was used in our study. The ankle joint complex provides stability and energy transfer in human walking. However, because of the complex geometry of the articulating surfaces and the numerous ligaments surrounding the joints, the kinematics and kinetics of all ankle and foot joints must be highly linked. This was confirmed in the study of Astion et al. [15]. They used a three-dimensional (3D) magnetic space-tracking system to evaluate the motion of the hindfoot after simulated arthrodesis. The results revealed that any combination of simulated arthrodeses that included

the talonavicular joint severely limited the motion of the remaining joints to about 2°. Our recent gait study [16] on one normal subject who underwent a pin insertion over tibia, talus and calcaneus verified that the motion in the sagittal and coronal planes showed little difference between the tibiohindfoot joint and the sum of talocrural and subtalar joints during the stance phase. However, at the end of the stance phase, the discrepancy increased gradually in motion of all three planes, in particular for motion in the coronal and transverse planes. Additionally, it is interesting to discover how the hindfoot and forefoot joints can reflect the combined motion of the ankle joint complex, both in normal subjects and in patients after ankle arthrodesis. Therefore, the purpose of this study was to objectively group the ambulation of normal subjects and patients after ankle arthrodesis using the fuzzy clustering technique. In addition, the coupled motion relationship between the rear and fore foot segments was determined and analyzed.

2. Fuzzy logic system Fuzzy logic systems are numerical model-free estimators and dynamical systems. They offer the ability to improve the intelligence of systems working in an uncertain, imprecise and noisy environment. Since fuzzy expressions contain human knowledge and experience, fuzzy logic systems can help us deal with those that are difficult to solve by mathematical or conventional pattern recognition methods. Fuzzy logic pattern recognition techniques are superior to conventional pattern recognition ones by their human-like decision and adaptive pattern recognition abilities.

3. Mathematical algorithms The fuzzy c-means algorithm is measured for each c, the number of clusters, by an objective function [17]. Typically, local extrema of the objective function are defined as optimal clusterings. Many different objective functions have been suggested for clustering (crisp clustering as well as fuzzy clustering). We used one of the frequently used criteria, the so-called variance criterion, in this study to improve an initial partition. This criterion measures the dissimilarity between the points in a cluster and its cluster center by the Euclidean distance. The Euclidean distance (储xk⫺ni储2G) was used for the inner product metric 储xk⫺ni储2G⫽(xk⫺ni)TG(xk⫺ni),

(1)

where G equals the identity matrix, xk is the kth p-dimensional feature vector, ni is the centroid of the ith cluster, and the fuzzy c-means algorithm is a fuzzy partitioning

F.-C. Su et al. / Medical Engineering & Physics 23 (2001) 83–90

of the sample data into c classes. It is based on minimization of the following objective function with respect to ˜ , a fuzzy c-partition of the data set, and to n, a set of U c centroids,

冘冘 n

˜ ; n)⫽ min zm(U

c

(mik)m储xk⫺ni储2G,

c⬍n

3.1. Iterative steps

1. Choose c (2ⱕcⱕn), m (1⬍m⬍⬁) and the symmetric ˜ (l), set l=0. and positive-definite matrix G. Initialize U (l) 2. Calculate the c fuzzy cluster centers {ni } by using ˜ (l) from the equation shown below: U

冘

Ij

冉冊

(⫺1)k+1

k⫽1

c

k

册冎

(1⫺kmj )c−1 .

In this algorithm, the only values from the matrix U that are used are the n maximum entries from each of its columns. The proportion of possible c-dimensional membership functions is given by

冘

冉冊

(⫺1)k+1

k⫽1

c

k

(1⫺km)c−1,

where I is the maximal integer in 1/m. In other words, the interpretation of these results is dependent on c. If the sample size is very large, the proportion for the matrix is likely to be very small indeed. It is for this reason that the final computation is performed, namely, taking the negative logarithm (base 2).

n

1

n

(mik)

j⫽1

I

Fuzzy clustering was carried out through the following iterative steps.

冘

再 冋冘 n

˜ )⫽⫺log2 ⌸ P(U

where m is any number greater than one, mik is the degree of membership of xk in the ith cluster, n is the number of data points and c is the number of clusters. The parameter m is the weighting exponent for mik and controls the “fuzziness” of the resulting clusters.

ni⫽

clusters most likely to exist within a given data set. The proportion exponent is introduced as a measure of the validity of the clustering obtained for a data set using a fuzzy clustering algorithm. The proportion exponent ˜ ) of U ˜ is defined by P(U

(2)

k⫽1i⫽1

85

(mik)mxk,

i⫽1, …, c

(3) 4. Motion analysis

mk⫽1

k⫽1

4.1. Kinematic model ˜ (l+1) by using 3. Calculate the new membership matrix U } from the equation shown below: {n(l) i mik⫽

(1/储xk−ni储2G)1/(m−1)

冘 c

,

i⫽1, …, c; k

(4)

(1/储xk−n 储 )

2 1/(m−1) j G

j⫽1

⫽1, …, n 4. Choose a suitable matrix norm and calculate ˜ (l)储G. If ⌬⬎e set l=l+1 and go to step (2). ˜ (l+1)⫺U ⌬=储U Otherwise, stop. We chose the exponent m=2 for the partition matrix ˜ , and a minimal amount of improvement, ⌬=10⫺5, in U this study. The clustering process stops when the objective function improvement between two consecutive iterations is less than the minimal amount of improvement specified. The summation of each column of the gener˜ is equal to unity, as required by fuzzy c-means ated U clustering. The ideal validity function should be independent of as many parameters as possible; it is the partial achievement of this goal that motivates the introduction of the proportion exponent. The cluster validity technique [18] has therefore been used here to estimate the number of

This paper used a three-segment rigid-body model to describe the motion of the foot and ankle [19,20]. The segments consisted of the tibial/fibular, hindfoot/midfoot and forefoot. Due to the fixed ankle joint, the hindfoot/midfoot segment included the calcaneus and navicular in our study, while the forefoot included the cuneiforms, cuboid and metatarsals. Motion of each segment was expressed in relation to the next proximal segment in terms of Euler angles. We assumed that the ankle joint consisted of the tibial/fibular and hindfoot/midfoot, and the midtarsal joint consisted of the hindfoot/midfoot and forefoot. In addition, we assume all surface markers are rigid bodies and have no movement on the skin surface. 4.2. Gait measurements Ten patients were recruited for this study (mean±one standard deviation: age 28.8±3.8 years, body mass 61.2±11.2 kg, height 156.9±14.6 cm). They included seven males and three females, with single-side solid arthrodesis of the ankle due to trauma, degenerative osteoarthritis or rheumatoid arthritis. The mean duration of follow-up after arthrodesis was 1.7 years (range, 0.5 to 4 years). Ten normal subjects (mean age 39.6±15.3 years, body mass 61.5±9.9 kg, height 164.3±7.3 cm)

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served as controls for comparison. Informed consent was written before each test. Subjects were asked to walk barefoot at a self-selected velocity along the walkway. The ExpertVision Hi-Res motion analysis system (Motion Analysis Corp., CA, USA) was used to collect the position of all reflective markers at a sampling frequency of 60 Hz. To characterize the motion of hindfoot and forefoot, reference frames were defined. Eight markers on four iron sticks fixed on the force plate were used to calibrate the space [19,20]. The capture volume dimensions were 0.34 m wide by 0.5 m high by 1.08 m long. The forward direction of the walkway is the positive x-axis, positive yaxis to the left, and the positive z-axis is vertical taking into consideration any upward movements. Eleven spherical retroreflective skin markers (diameter 1.2 cm) were placed on the subject’s foot and ankle to determine the kinematic behavior of the foot and ankle. They were attached to the skin over the medial, lateral and rear calcaneus, head and base of the first and fifth metatarsal, lateral and medial tibial condyles and malleolus. In addition, local coordinate systems were defined for the tibial/fibular, hindfoot/midfoot and forefoot segments [19]. After tracking 3D video data, this study used the GCVSPL method [21] to smooth these markers’ trajectories, then to calculate kinematic parameters using three sequential Euler rotations. Euler angles are employed to describe the relative angular motion of foot and ankle in this study.

5. Results Motions of the hindfoot, forefoot and the forefoot to tibia for level walking are shown in Fig. 1 and Table 1. For the arthrodesis group, the hindfoot joint had a mean range of 10.8° in dorsiflexion/plantar flexion. The forefoot motion had a mean range of 18.8° in dorsiflexion/plantar flexion. For normal subjects, the mean ranges were 16.3° in the hindfoot joint and 13.4° in the forefoot joint. Sagittal plane motion in the hindfoot decreased significantly in the patient group compared with normal subjects (Mann–Whitney test, P⬍0.01). The kinematic data explicitly indicated generalized stiffness of the hindfoot on the involved foot in the sagittal plane. In contrast, sagittal plane motion in the forefoot demonstrated relatively larger values in patients than controls (Mann–Whitney test, P⬍0.01). This indicated that the forefoot joint compensates for the angle limitation of the hindfoot joint in the sagittal plane. We found that the mean relative movement of the forefoot to the tibia (ankle complex) was 28.8° in the sagittal plane, for normal subjects walking on level ground. For patients with ankle arthrodesis, the value was 19.1° in the sagittal plane. The range of motion of normal subjects in the sagittal plane (28.8°) was larger

than that for the patients (19.1°), which combines the motion in the hindfoot and forefoot joints in this study. The fuzzy c-means algorithm requires the choice of a “fuzziness” factor (m⬎1). For m=1, the algorithm produces a hard partition and for large m, every subject becomes an equal member of all clusters. The value of m=2 is used here because it is a compromise and in common use in other studies. Fig. 2 shows cluster validity using the proportion exponent, which is a maximum for valid clusters and has a local maximum at three clusters for the hindfoot joint, three clusters for the forefoot joint and two clusters for the combined hind and forefoot joint. Therefore, this analysis suggests that three, three and two are the most valid numbers of clusters for hindfoot, forefoot and combined hind and forefoot, respectively. All of the data pooled from all patients and normal subjects were used to determine the numbers of clusters, c, and then the center of each cluster was generated by through minimization of the objective function. Fig. 3(a) and Table 2 show the resulting three cluster centers A, B1, B2 for the hindfoot joint. The angle contour ‘A’ is 95% similar to the control group’s angle. The other two angle contours ‘B1’ and ‘B2’ are 67% and 86% similar, respectively, to the patient group’s angle (using the correlation method). Therefore, we can characterize Type ‘A’ as a “nearly normal” pattern. Type ‘B1’ is defined as a pathological pattern with mild stiffness of the hindfoot, and Type ‘B2’ is a pathological pattern with severe stiffness of the hindfoot. Fig. 3(b) and Table 2 show the resulting three cluster centers A, B1 and B2 for the forefoot joint. The angle contour ‘A’ is 93% similar to the control group’s angle. The other two angle contours ‘B1’ and ‘B2’ are 89% and 91% similar, respectively, to the patient group’s angle. Therefore, we can characterize Type ‘A’ as a “nearly normal” pattern. Type ‘B1’ is defined as a pathological pattern with large compensation of the forefoot, and Type ‘B2’ is a pathological pattern with mild compensation of the forefoot. Fig. 3(c) and Table 2 show the two resulting cluster centers, A and B, for the combined fore and hindfoot joint. The angle contour ‘A’ is 99% similar to the control group’s angle. The angle contour ‘B’ is 98% similar to the patient group’s angle. Therefore, we can characterize Type ‘A’ as a nearly normal pattern, and Type ‘B’ fully represents the pathological gait pattern. Based on the largest membership values, the patients were clustered into four groups (Fig. 4). The compensation mechanisms in Cases 4 and 6 are mild stiffness hindfoot vs. mild compensated forefoot (Group I) and in Cases 1, 2, 3, 5 and 7–10 are severe stiffness hindfoot vs. large compensated forefoot (Group III). In the assessment of patient gait the possibility exists that the patients’ data may not cluster together. They may differ from the normal model in different “direc-

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Fig. 1. Joint angular motion (in °) for patients with ankle arthrodesis. Mean±one standard deviation forms the normal bandwidth (shadow) for normal subjects. Saggital plane motion in the hindfoot joint (a), the forefoot joint (b) and combined fore and hindfoot joints (c). The changes within the patients and controls are detected with the fuzzy c-means algorithm. Table 1 Summary of angular ranges of motion (mean±one standard deviation) during level walking in ankle arthrodesis studies Joint movement (°) in sagittal plane

Hindfoot to tibia Forefoot to hindfoot Forefoot to tibia

Normal subjects

Ankle arthrodesis patients

16.3±3.7 13.4±3.8 28.8±4.4

10.8±4.8 18.8±3.9 19.1±4.9

tions”. Fig. 5 shows the memberships (mean±standard deviation) for controls and patients in the hindfoot, forefoot, and combined hind and forefoot. Relatively larger standard deviation is found in the hindfoot and forefoot joints than in the combined fore and hindfoot joint. This means that more than one pattern exists in the pathological forefoot and hindfoot movements. This may represent a non-homogeneity in ankle arthrodesis gait patterns. However, there is only one pattern found in the combined fore and hindfoot joint. This indicates the highly adaptive mechanism which occurs with the ankle–foot complex. Fig. 6 shows the membership values for each subject. Patient 1 is given as an example. For Patient 1 and the

˜ , P(U ˜ ), for the fuzzy clustering Fig. 2. Proportion exponent of U algorithm.

three memberships A, B1 and B2, values in the hindfoot are 0.09, 0.20 and 0.71, respectively. The corresponding values in the forefoot are 0.01, 0.96 and 0.03, respectively. The two membership values A and B in the combined fore and hindfoot are 0.05 and 0.96, respectively. We can see that the summation of membership values in any one joint of the foot is equal to 1. Therefore, the growth and decline of membership values in all joints could give clinicians a quantitative message regarding the changes in gait pattern.

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Table 2 The correlation coefficient between joint angle and resulting cluster centers Controls Hindfoot angle A 0.9455 B1 0.4028 B2 0.1103 Forefoot angle A 0.9334 B1 0.4485 B2 0.7646 Combined fore and hindfoot angle A 0.9870 B 0.9268

Patients

0.1474 0.6681 0.8609 0.7367 0.8928 0.9123 0.8924 0.9773

Fig. 4. Clustering the patients into four groups. A compensation mechanism was shown in all cases.

Fig. 3. The cluster center profiles and mean foot angles in the control group and patient group for the hindfoot (a), the forefoot (b) and combined fore and hindfoot (c).

6. Discussion Based on Fig. 4, we can conclude that almost 100% of the gait pattern follows the adaptation mechanism

Fig. 5. Memberships for controls and patients in the hindfoot (a), forefoot (b) and combined hind and forefoot (c).

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Fig. 6. Membership changes for all subjects [10 normal subjects (abbreviated as c) and 10 patients after ankle arthrodesis (abbreviated as p)].

because none falls in Group II or Group IV. Each group was then studied and the most consistent identifying or discriminating clinical features were identified. Group I includes two cases. These patients walk with a more natural pattern than Group III (eight cases). For the patients with severe movement limitation after ankle arthrodesis (Group III), the motions in the hindfoot are significantly limited. They are represented by a flattened ankle rocker (defined as dorsiflexion at the ankle that contributes to limb progression) compared with Group I cases. In contrast, the ranges of motion in the forefoot in this group are significantly larger than in the Group I cases, both in heel rocker stage (defined as progression of the limb while the heel is the pivotal area of support)

89

and ankle rocker stage (Fig. 1). Furthermore, there is delayed action of the forefoot rocker (defined as progression of the limb while the forefoot is in the pivotal area of support) in the forefoot joint. Through stance, momentum is preserved by the pivotal system created by the foot and ankle to accomplish the forward movement. In serial fashion the heel, ankle and forefoot serve as rockers that allow the body to advance. For ankle arthrodesis patients, progression contributed by different degrees of combination of forefoot and hindfoot joints occurs, because the lever arms of ankle muscles have been changed. Generally, the movement of the heel rocker in the hindfoot for patients with ankle arthrodesis is significantly larger than the value in controls (Fig. 1). For patients, there is a flattened and early occurring ankle rocker in the hindfoot joint. Also, the amplitudes of heel and ankle rockers measured in the forefoot joint increase significantly compared with the patient group. The patterns of the forefoot rocker observed in the hindfoot and forefoot joints are similar to those seen with the controls. In this paper, we introduced the concept of the “combined fore and hindfoot” to investigate the coupling relationship of the foot joints. Understanding the synergy of forefoot and hindfoot patterns is useful in determining the cause and effect in the resulting kinematics. The fuzzy c-means method is exploited to cluster the training data. Since the results inferred from the fuzzy model may not coincide with the desired outputs, genetic algorithms could be used to optimize the membership functions [22], cluster centers [23] or search for an appropriate definition of distance [24]. In the future, we hope to integrate genetic algorithms in our model and estimate the mathematical relationship between the hindfoot and forefoot. We also wish to model these timevarying systems as dynamic fuzzy sets. In 3D human movement analysis using close-range photogrammetry, surface markers deform and move rigidly relative to the underlying bone. This introduces an important artefact (skin movement artefact) which propagates into joint kinematics estimate. In this study, skin movement artefacts have been ignored in analyzing the data. However, they may be significant in membership values for each subject. Further consideration of the artefacts contributed in these membership values may provide additional and more precise information.

7. Conclusion Gait analysis yields redundant information that is often difficult to interpret. The fuzzy c-means method simplifies the analysis and interpretation of waveform data and facilitates comparison between subjects. The changes in membership of the clusters provide an objective technique for measuring changes of gait pattern after

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ankle arthrodesis. This new method will characterize the patients’ pathology and the changes of gait pattern after ankle arthrodesis in greater and more specific detail. Acknowledgements This work was supported by the National Science Council grant NSC 87-2314-B-006-110-M08, Taiwan. References [1] Dickerson JA, Daaboul Y, Jobe TH, Helgason CM, Isik C, Cross V. Analysis of concomitant mechanisms in stroke pathogenesis using fuzzy clustering techniques. In: Proceedings of Annual Meeting of the North American Fuzzy Information Processing Society, 1997:211–6. [2] Olney SJ, Griffin MP, McBride ID. Multivariate examination of data from gait analysis of persons with stroke. Phys Ther 1998;78(8):814–28. [3] Nieuwboer A, Weerdt WD, Dom R, Lesaffre E. A frequency and correlation analysis of motor deficits in Parkinson patients. Disabil Rehabil 1998;20(4):142–50. [4] Yamamoto S, Suto Y, Kawamura H, Hashizume T, Kakurai S. Quantitative gait evaluation of hip diseases using principal component analysis. J Biomech 1983;16(9):717–26. [5] Loslever P, Laassel EM, Angue JC. Combined statistical study of joint angles and ground reaction forces using component and multiple correspondence analysis. IEEE Trans Biomed Eng 1994;41(12):1160–7. [6] O’Byrne JM, Jenkinson A, O’Brien TM. Quantitative analysis and classification of gait patterns in cerebral palsy using a threedimensional motion analyzer. J Child Neurol 1998;13(3):101–8. [7] O’Malley MJ, Abel MF, Damiano DL, Vaughan CL. Fuzzy clustering of children with cerebral palsy based on temporal–distance gait parameters. IEEE Trans Rehabil Eng 1997;5(4):300–9. [8] Kosanovic BR, Chaparro LF, Sclabassi RJ. Signal modeling with dynamic fuzzy sets. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 1996;5:2829-2832 [9] Kosanovic BR, Chaparro LF, Sclabassi RJ. On estimation of temporal fuzzy sets for signal analysis: FCM vs. FMLE approaches. In: NAFIPS, Uncertainty Modeling and Analysis, Annual Conference of the North American Fuzzy Information Processing Society. Proceedings of ISUMA on Third International Symposium, 1995:583–8.

[10] Kosanovic BR, Chaparro LF, Sclabassi RJ. Hidden process modeling. In: Proceedings of International Conference on Acoustics, Speech, and Signal Processing, 1995;5:2935-2938 [11] Kosanovic BR, Chaparro LF, Sclabassi RJ. Modeling of quasistationary signals using temporal fuzzy sets and time–frequency distributions. In: Time–Frequency and Time-Scale Analysis, Proceedings of the IEEE-SP International Symposium, 1994:425–8. [12] Kosanovic BR, Chaparro LF, Mingui S, Sclabassi RJ. Physical system modeling using temporal fuzzy sets. In: Proceedings of the First International Joint Conference of Fuzzy Information Processing Society Biannual Conference, Industrial Fuzzy Control and Intelligent Systems Conference, and the NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, 1994:429–33. [13] Mazur JM, Tschwartz E, Simon SR. Ankle arthrodesis — long term follow-up with gait analysis. J Bone Joint Surg 1979;61A(7):964–75. [14] Buck P, Morrey BF, Chao EYS. The optimum position of arthrodesis of the ankle. J Bone Joint Surg 1987;69A(7):1052–62. [15] Astion DJ, Deland JT, Otis JC, Kenneally S. Motion of the hindfoot after simulated arthrodesis. J Bone Joint Surg 1987;79A(2):241–6. [16] Chou CK. Kinematic analysis in ankle joint. Master thesis. Taiwan: Institute of Biomedical Engineering, National Cheng Kung University, 1998. [17] Bezdek JC. Pattern recognition with fuzzy objective function algorithms. New York/London: Plenum Pub Corp., 1981. [18] Windham MP. Cluster validity for the fuzzy clustering algorithms. Fuzzy Systems 1981;5:177–85. [19] Su FC, Wu WL, Wang TL. Repeatability of kinematic data of the foot and ankle in gait. Biomed Eng Appl Basis Comm 2000;12(1):1–8. [20] Wu WL, Su FC, Cheng YM, Huang PJ, Chou YL, Chou CK. Gait analysis after ankle arthrodesis. Gait & Posture 2000;11:54–61. [21] Woltring HJ. A FORTRAN package for generalized, cross-validatory spline smoothing and differentiation. Adv Eng Software 1986;8:104–13. [22] Huang YP, Chen YR. Modeling a fuzzy system by the integrated virtual and genetic algorithms. In: Fuzzy Systems, Proceedings of the Sixth IEEE International Conference, 1997;1:521-526 [23] Hall LO, Bezdek JC, Boggavarpu S, Bensaid A. Genetic fuzzy clustering. In: Proceedings of the First International Joint Conference of Fuzzy Information Processing Society Biannual Conference, Industrial Fuzzy Control and Intelligent Systems Conference, and the NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, 1994:411–5. [24] Yuan B, Klir GJ, Swan-Stone JF. Evolutionary fuzzy c-means clustering algorithm. In: Proceedings of 1995 IEEE International Conference on Fuzzy System, 1995;4:2221-2226