200

Journal of Applied Biomechanics, 2012, 28, 200-209 © 2012 Human Kinetics, Inc.

Gianluca Gatti is with the Department of Mechanical Engineer-ing, University of Calabria, Cosenza, Italy.

On the Estimate of the Two Dominant Axes of the Knee Using an Instrumented Spatial Linkage

Gianluca GattiUniversity of Calabria

This article presents the validation of a technique to assess the appropriateness of a 2 degree-of-freedom model for the human knee, and, in which case, the dominant axes of flexion/extension and internal/external longitudinal rotation are estimated. The technique relies on the use of an instrumented spatial linkage for the accurate detection of passive knee kinematics, and it is based on the assumption that points on the longitudinal rotation axis describe nearly circular and planar trajectories, whereas the flexion/extension axis is perpendicular to those trajectories through their centers of rotation. By manually enforcing a tibia rotation while bending the knee in flexion, a standard optimization algorithm is used to estimate the approximate axis of longitudinal rotation, and the axis of flexion is estimated consequently. The proposed technique is validated through simu-lated data and experimentally applied on a 2 degree-of-freedom mechanical joint. A procedure is proposed to verify the fixed axes assumption for the knee model. The suggested methodology could be possibly valuable in understanding knee kinematics, and in particular for the design and implant of customized hinged external fixators, which have shown to be effective in knee dislocation treatment and rehabilitation.

Keywords: knee kinematics, flexion axis, tibia rotation, passive knee, hinged external fixator

Kinematic skeletal models of the knee are often needed for a variety of applications, such as human movement analysis, rehabilitation devices design, and prostheses implantation. Such models may be adapted to specific subjects by adjusting the parameters defin-ing joint location. Although knee joint kinematics is three-dimensional and requires the description and measurement of six motion components (Chao, 1980; Grood & Suntay, 1983), simplified models have been proposed in the literature, which only retain the main functional joint axes. For instance, simplified models have been used for biomedical assessment (Churchill et al., 1998; Most et al., 2004; Piazza & Cavanagh, 2000; Asano et al., 2005; Schache et al., 2006), where the knee was modeled as a single degree-of-freedom (df ) joint, having one axis of rotation. To estimate this axis, several procedures were also proposed, the majority of which are based on measurements of surface markers (Halvorsen et al., 1999; Halvorsen, 2003; Schwartz & Rozumalski, 2005). Although knee motion is dominated by the rota-tion around the flexion/extension axis, movements about secondary axes occur, and in these cases, functional meth-ods to robustly estimate a fixed average axis of rotation have been proposed, such as by Gamage and Lasenby (2002), Cerveri et al. (2005), and Chang and Pollard (2007). A comparison among some of previously reported

techniques for determining the axis of rotation of human joints from marker measurements was presented by Ehrig et al. (2007), and more recently by MacWilliams (2008).

Despite the well-established approach, the main disadvantage of using noninvasive determination of joint motion from reflective markers is due to the relative movements between the markers and the bone, which are the main causes of artifacts (Leardini et al. 2005; Taylor et al., 2005; Stagni et al., 2005).

More reliable results, despite the related ethical aspects, may be achieved by using direct fixation to bony segments. In these cases, instrumented spatial linkages (ISLs) have demonstrated their ability to accurately detect full three-dimensional joint kinematics. They were used for either in vitro (Sorger et al., 1997; Lewandowski et al., 1997; Jan et al., 2002) or in vivo (Townsend et al., 1977; Shiavi et al., 1987; Salvia et al., 2000) measurement, although these latter are complicated by the requirements for adequate fixation. Invasive techniques, through direct fixation to bones, have been used by Gardner et al. (1996) and Ishii et al. (1997).

In this work, a technique that employs an ISL for simultaneously estimating both the functional flexion/extension axis and the longitudinal internal/external rotation axis of the human knee is presented and vali-dated. The proposed procedure retains the main concept of the mechanical axis finder developed by Hollister et al. (1993), and recently improved by Roland et al. (2010), which is based on the assumption that points on the longitudinal axis describe nearly circular and planar

Two Dominant Knee Axes and Instrumented Spatial Linkage 201

trajectories, whereas the flexion axis is approximately perpendicular to those trajectories through their centers of rotation. One main advantage of the proposed technique is that both axes are identified simultaneously, from a one-step process. Moreover, a general procedure is introduced to verify, in a quantitative manner, if a simplified 2 df model well suits a three-dimensional knee kinematics. A verification of the assumption of fixed axes of flexion and longitudinal rotation on cadaveric specimens would then support the understanding of knee kinematics and further investigate the hypothesis of fixed axis of rotation vs. instant center of rotation, which was formulated by some researchers, such as Smith et al. (2003) and Wil-liams and Logan (2004).

MethodsInstrumented Spatial LinkageThe mechanical linkage adopted for measuring knee kinematics is illustrated in Figure 1 and described in detail in Gatti et al. (2010). It consists of a serial chain of seven aluminum links directly connected to each other by revolute optical encoders, with a resolution of about 0.044 deg. It is used to estimate the pose of its distal link (assumed to be rigidly connected to the tibia) relative to the proximal (assumed to be rigidly connected to the femur), so as to detect the envelope of knee motion. Its accuracy is estimated to be 0.183 ± 0.081 mm and 0.085 ± 0.036 deg after kinematic calibration and geometrical

error compensation, as reported in Gatti and Danieli (2007). To mathematically describe the acquired knee motion, two orthogonal frames, {B} and {E}, are attached to the femur and tibia, respectively, as depicted in Figure 2a, and the transformation matrix TE

B —describing the pose of frame {E} on the tibia relative to frame {B} on the femur—is estimated from the transducer readings and the linkage geometry through the direct kinematic model.

Data Collection and Acquisition Protocol

To estimate the dominant axes of knee motion, a data collection procedure is performed, where a reciprocating tibia rotation is enforced during flexion and extension. The identification technique is based on the assumption that knee kinematics is approximated by a functional flex-ion/extension (FE) and an auxiliary longitudinal rotation (LR) of the tibia respect to the femur (Blankevoort et al., 1988; Wilson et al., 2000). The LR axis is identified in frame {E} by two points (referred to as A and B in the text) fixed to the tibia, which perform the most-likely planar trajectories during motion. The FE axis is then identified in frame {B} on the base of the trajectories of points A and B. Figure 2a and the animation in the movie attached to the electronic version show the trajectories of four points A, B, C, and D attached to the tibia, as the latter performs a combined movement of flexion and reciprocating longitudinal rotation around fixed axes. It is noted that only points A and B, aligned to the longitudinal axis, describe coplanar and circular trajectories.

Figure 1 — Picture of the actual ISL assembled on an artificial leg system (MITA Endo Leg, Medical Models Ltd. UK).

202 Gatti

To practically apply the identification procedure, the femur is grounded and the tibia is manually bent and extended, while a reciprocating longitudinal internal/external rotation is imposed. The envelope of motion is measured by the ISL assembled as shown in Figure 1.

Estimate of the Dominant Axes

With reference to Figures 2a and b, the two unknown points—A and B, identifying the LR axis—are defined

Figure 2 — (a) Virtual model of the system with embedded frames {B} and {E} fixed to the femur and tibia, respectively. Points A, B, C, and D are also indicated, along with their trajec-tories when a combined movement of flexion and reciprocating longitudinal rotation (around fixed axes) is enforced. (b) Detail of the location of points A, B, C, and D in frame {E}, where their coordinates are given according to Eqs. (1a, b) and (2a, b).

relative to points C and D, whose location is conveniently identified by the following homogeneous coordinates (in millimeters)

EC = 0 0 0 1⎡⎣ ⎤⎦T

E D = 0 100 0 1⎡⎣ ⎤⎦T (1a, b)

where the superscripts on the left-hand side indicate the frame relative to which coordinates are given. Points A and B are thus defined as

E A = EC + −RA sin θ A( ) 0 − RA cos θ A( ) 1⎡⎣ ⎤⎦TT

E B = E D + −RB sin θB( ) 0 − RB cos θB( ) 1⎡⎣ ⎤⎦T (2a, b)

The trajectories described by A and B can be then computed in the absolute frame {B} as

BA = TE

B ⋅ E ABB = TE

B ⋅ EB (3a, b)

where TEB is the transformation matrix describing knee

motion and it is measured by the ISL, as discussed above.To estimate the location of points A and B in frame

{E}, a plane equation is fitted to each of their absolute trajectories using a standard least-squares algorithm, and a nonlinear optimization technique (“fminsearch” from MATLAB) is used to identify the best set of parameters RA, θA, RB, θB in Eq. (2), which minimizes the residual errors. The two planes obtained in this way are referred to as the optimum planes for points A and B.

The LR axis may be then defined in frame {E} as the axis through point A with a direction identified by the unit vector nLR = B – A( ) B – A .

The FE axis is defined in frame {B} as follows. If nA and nB are the unit normal vectors to the optimum planes for points A and B, respectively, then the direction of the FE axis may be estimated as the “average” unit vector between those two, as

nFE = nA + nB( ) nA + nBTo identify the location of the FE axis, the trajectories

of points A and B are projected onto their correspond-ing optimum planes and standard circle fitting is applied to determine their approximate centers of rotation. The midpoint between these two is assumed to belong to the FE axis.

Identification of the Simplified 2 df Knee ModelOnce the FE axis and the LR axis are estimated, a 2 df kinematic model of the knee joint can be defined, as sketched in Figure 3a, by adopting the Denavit–Harten-berg (DH) convention, which describes the relative loca-tion between two consecutive axes in space (Craig, 1989). The kinematic model of the approximate 2 df knee joint defines the relative pose TE

B between frame {B} on the

(2a, b)

Two Dominant Knee Axes and Instrumented Spatial Linkage 203

femur and frame {E} on the tibia, and it is given by the following consecutive transformation matrices

TEB = T1

B ⋅T21 ⋅T3

2 ⋅TE3 (4)

where frames {1}, {2}, and {3} are indicated in Figure 3a, and each transformation matrix in Eq. (4) is given as

T1B = Rot z,θ1( ) ⋅Trans z,d1( ) ⋅

Trans x,a1( ) ⋅Rot x,α1( ) (5a)

T21 = Rot z,θ2 +ϕFE( ) ⋅Trans z,d2( ) ⋅

Trans x,a2( ) ⋅Rot x,α 2( ) (5b)

T32 = Rot z,θ3 +ϕLR( ) ⋅Trans z,d3( ) ⋅

Trans x,a3( ) ⋅Rot x,α3( ) (5c)

TE3 = Rot z,θ4( ) ⋅Trans z,d4( ) (5d)

In Eqs. (5a–d) Trans i,•( ) / Rot i,•( ) is the elemen-tary translational/rotational transformation matrix along/

around the axis i = x, y,z of the displacement/angle “• ”, respectively (Craig, 1989); θ i ,di ,ai ,α i are the DH param-eters of the 2 df knee model in Figure 3a; ϕFE ,ϕLR are the angles of flexion and longitudinal rotation, respectively.

It is worth noting that all parameters, except for d2 ,a2 ,α 2 ,d3 are constant by definition, since they define the location of either the FE axis relative to frame {B} or the LR axis relative to frame {E}. In either case, as discussed above, such location is assumed to be fixed, once those two axes have been estimated. Parameters d2 ,a2 ,α 2 ,d3 , instead, define the relative location between the FE axis and the LR axis. Their values may be vari-able, because the actual knee motion and, hence, the instantaneous location of points A and B (as measured in the absolute reference frame) is affected by measurement errors and by the fact that the FE axis and the LR axis may not be fixed, as assumed. The identification of these parameters is thus not unique, but a single approximate estimate may be determined by the corresponding average value throughout the acquired knee motion. This is the reason why the symbol “~” has been adopted in the transforma-tion matrix of Eq. (4), which is, in fact, an approximation of the measured transformation matrix in Eq. (3).

Figure 3 — (a) Kinematic model of the 2 df knee joint as identified by the proposed procedure. Frames of reference are assigned according to the Denavit–Hartenberg convention. (b) Kinematic model of a 6 df linkage consisting of the 2 df model shown in (a) attached to a four joint additional chain (thinner lines). This is used to assess the validity of the 2 df knee model assumption.

204 Gatti

Validation of the Identification Procedure

To validate the procedure described above, computer simulated data are used at first. They are generated by means of a 2 df kinematic model of the knee joint, consist-ing of two nonintersecting and fixed axes of rotation, as that represented in Figure 3a. A 2 df mechanical hinged joint is finally used for an experimental test.

The simulated range of motion was based on the literature with reference to the passive knee joint motion (Blankevoort et al., 1988; Wilson et al., 2000), and con-sisted of a maximum of 90 deg of flexion (followed by extension), and a maximum of 30 deg of reciprocating internal/external rotation of the tibia. The number of reciprocating cycles in the tibia longitudinal rotation was fixed to four per flexion/extension. The ideal relative pose between frame {B} and {E} is then computed in terms of the TE

B transformation matrix.To simulate the application of the ISL and the

unavoidable measurement errors, the inverse kinematics of the ISL is used to determine the transducer angles from simulated data, which are then corrupted by random errors within the transducers resolution. The “measured” trans-formation matrix TE

B is then estimated by feeding these corrupted angles into the ISL direct kinematic model.

The performance of the identification procedure is given by the error between the identified axes of motion and the corresponding ones used to generate data. In particular, for each axis, the error is specified in terms of a linear and an angular component, respectively computed as the distance between two corresponding points on the axes and as the angle between these latter.

Verification of the AssumptionThe approach described so far assumes that the axes of rotation of the knee are fixed (with respect to bones), so that the above-mentioned errors may be effectively calculated and adopted to quantify the performance of the identification procedure.

In the more general case, where those axes are not fixed, the proposed procedure still estimates two fixed average axes, the motion around which is thus an approxi-mation of the actual knee motion.

To verify whether a 2 df model is a good approxima-tion for the measured knee motion, a novel procedure is adopted. To this end, the 2 df knee model is attached to a 4 df additional kinematic chain consisting of three translational and one rotational joints, as indicated in Figure 3b. The inverse kinematics of the resulting 6 df auxiliary linkage is solved on the base of the measured knee motion (i.e., from theTE

B matrix), and the root mean squared error (RMSE) of the values of the four additional joint variables is computed and given as indication of the deviation of the 2 df model from the measured motion. It is evident that in the case of a knee joint behaving as a pure 2 df system and in the ideal error-free case, the values of those four additional joint variables are zeros. On the contrary, in a more general case, the more distant the measured knee motion is from a 2 df kinematics, the

further from zero the RMSE values of those four joint variables will be.

As a remark, it is worth noting that the linkage shown in Figure 3b is not meant to be a 6 df model of the human knee joint, but it is only used to assess the validity of the 2 df knee model assumption.

ResultsThe procedure for the identification of the two fixed axes of rotation was validated by running a number of 100 simulations. Each time, the parameters defining knee dimensions—as given by Eq. (5)—were varied to simulate different knee configurations and a number of 50 data samples were generated for each trial. The ranges of parameter used in simulation were defined according to the values listed in Table 1. The error in the estimate of each axis is given by the root mean square of the corresponding error distribution. The LR axis was thus identified with a RMSE of about 0.67 mm and 0.24 deg, and the FE axis was identified with a RMSE of about 0.12 mm and 0.04 deg.

Two further simulations were run, as indications, to verify the assumption of the fixed axes of rotation. In these cases, the knee dimensions were set to the baseline values indicated in Table 1. In a first simulation, only mea-surement errors were supplied to the model generating data, and in this particular case, the LR axis was identified with an error of 0.20 mm and 0.15 deg, while the FE axis was identified with an error of 0.10 mm and 0.03 deg. By then applying the inverse kinematics to the 6 df linkage in Figure 3b, an indication of the fixed axes assumption was inferred by the values assumed by the corresponding first four additional joint variables, as illustrated in Figure 4.

Table 1 Knee model parameters used for simulations

ParameterBaseline

ValueRange of Variation

d1 = a3 [mm] 0 –20 to 20

θ1 = θ4 [deg] 0 –5 to 5

a1 = d3 [mm] 160 –20 to 20

α1 = θ2 =α 2 [deg] –90 –5 to 5

d2 [mm] –60 –20 to 20

a2 [mm] –10 –20 to 20

θ3 [deg] 120 –5 to 5

α3 [deg] 90 –5 to 5

d4 [mm] 60 –20 to 20

Two Dominant Knee Axes and Instrumented Spatial Linkage 205

A second simulation was performed, where the axis of FE was not fixed, but simultaneously translating of 10 mm and rotating of 5 deg in a linear fashion during motion. In this particular case, the errors in the estimate of the FE and LR axes were meaningless (since they were not con-stant), and the values assumed by the four additional joint variables in the 6 df linkage are illustrated in Figure 5.

Finally, the whole procedure described above was experimentally applied to the simple 2 df mechanical joint illustrated in Figure 6, which was designed to have nominally fixed axes of FE and LR. Mechanical com-ponents were aluminum made and no bearing was used in assembling moving components: simple screw-bolt connections were adopted, so that small clearances in the joints were expected. The ISL was attached to the mechanical joint through the pairs of pins illustrated in the figure, and for this reason, the exact location of the FE and LR axes could not be known a priori (as in practical situations). Nevertheless, the identification procedure estimated the position of points A and B in frame {6} as

E A = 23.11,0,134.30⎡⎣ ⎤⎦ ,EB = 23.58,100,135.02⎡⎣ ⎤⎦

and the orientation of the FE axis with an error of about 1.58 deg respect to its nominal direction, aligned to the reference axis yB. The values assumed by the four addi-tional joint variables in the 6 df linkage are illustrated in Figure 7, as for verification of the 2 df model assumption.

Discussion

The validation of a methodology to estimate the dominant axes of FE and LR of the human knee has been presented in this paper.

The procedure is based on previous observations and assumptions already adopted in the works by Hollister et al. (1993) for the development of their mechanical axis finder, and by Roland et al. (2010) who proposed their virtual axis finder. Both these systems allowed for the identification of one single axis at a time, while the procedure described in this paper does allow for the identification of both the FE axis and the LR axis simul-taneously. A simplified 2 df model of the knee is then obtained from the estimated location and orientation of these two axes of rotation.

The results from validation show that the proposed procedure is able to estimate the two rotational axes in a 2 df model with an accuracy comparable to the methodol-ogy proposed by Roland et al. (2010), without the need to perform a two-step process. In particular, the errors in identifying the position and orientation of the FE and LR axes were evaluated, respectively, as 0.67 mm and 0.24 deg, and 0.12 mm and 0.04 deg. However, it is worth noting that, here, the random errors introduced into the model to simulate real measurements are added at ISL joints level, while in the procedure described by Roland et al. (2010) they were added in the Cartesian position of the optical markers used for measurement.

Compared also to some other earlier works by Halvorsen et al. (1999) and Gamage and Lasenby (2002) or more recently by Schache et al. (2006), Chang and Pol-lard (2007) and Ehrig et al. (2007), which were mainly focused on the estimate of a single axis of rotation from skin markers data, the proposed procedure is based on the use of an ISL for data collection, which may be directly fixed to the bony segments, reducing the effect of artifacts due to skin movements, and thus improving the measurement accuracy at source.

Since the proposed identification procedure assumes that both axes of FE and LR of the human knee remain fixed to bones or vary slightly during motion, a novel approach is also formulated to verify such an assumption, which may not hold completely in clinical situations. To this end, the 2 df identified knee model is attached to a 4 df kinematic chain, to come up with a 6 df linkage, as illustrated in Figure 3b. By solving the inverse kinematics of such linkage from measured data, an indication of the deviation of the actual motion from that achievable by the simplified 2 df model is obtained. This is performed both in simulations and in an experimental test where a 2 df mechanical joint is adopted.

The results of this procedure are illustrated in Figures 4 and 5 for the simulated motion, and in Figure 7 for the experimental system. In particular, it is observed that, in the case of Figure 4, which refers to the simulation with fixed axes of rotation, the RMSEs of the first four addi-tional joint variables of the 6 df linkage, during motion, are quite low, and consistent to the simulated measure-ment errors. In the case of Figure 5, instead, which refers to the simulation where the FE axis is moved both in posi-tion and orientation during motion, the RMSEs of those four joint variables increase greatly respect to the case of Figure 4, and they all show a slight linear trend, which is consistent to the linear changes in the axis location. This indicates that a simplified 2 df model does not properly match the actual motion data. In the case of Figure 7, which refers to the experimental system with nominally fixed axes of motion, the RMSEs of the four additional joint variables in the 6 df linkage seem to be consistent with the accuracy of the actual ISL used for measurement and with the accuracy of the assembled mechanical joint. In this case, however, the exact location of the axes of motion was not known a priori (due to the mechanical coupling of the ISL), so that an error in identifying those axes was not possible to be evaluated with confidence. Despite this, the LR axis is estimated by the position of points A and B in the reference frame attached to the tibia, and this axis is found to be approximately parallel to the reference axis yE, as from nominal design specifications. Furthermore, the orientation of the FE axis is estimated to be approximately parallel to the yB axis, again as from nominal design specifications. Within the accuracy of the experimental rig, a 2 df model is thus effectively consistent with the mechanical joint.

In conclusion, the proposed identification proce-dure is able to estimate an approximate location for the two quasi-fixed axes of FE and LR, while a verification

206

Figure 4 — Assessment of the 2 df knee model assumption for a simulated knee with fixed axes of FE and LR, at the baseline con-ditions and with data affected by ISL transducer resolutions. Trends of the four additional joints in the six DOF linkage of Fig. 3b: (a) q1 (solid line), q2 (dash line) and q3 (dot line); (b) q4. RMSE (q1) = 0.067 mm, RMSE (q2) = 0.049 mm, RMSE (q3) = 0.053 mm, RMSE (q4) = 0.049 deg.

Figure 5 — Assessment of the 2 df knee model assumption for a simulated knee with moving FE axis, which is linearly translated and rotated of 10 mm and 5 deg, respectively, throughout motion. Simulated data are also affected by ISL transducer resolutions. Trends of the four additional joints in the 6 df linkage of Fig. 3b: (a) q1 (solid line), q2 (dash line) and q3 (dot line); (b) q4. RMSE (q1) = 13.89 mm, RMSE (q2) = 1.46 mm, RMSE (q3) = 10.34 mm, RMSE (q4) = 1.03 deg.

207

Figure 7 — Assessment of the 2 df knee model assumption for the mechanical joint in Fig. 6. Trends of the four additional joints in the6 df linkage of Fig. 3b: (a) q1 (solid line), q2 (dash line) and q3 (dot line); (b) q4. RMSE (q1) = 0.75 mm, RMSE (q2) = 0.57 mm, RMSE (q3) = 1.04 mm, RMSE (q4) = 0.19 deg.

Figure 6 — Mechanical joint used for the experimental test. This is manufactured to nominally behave as a pure 2 df system with fixed axes of flexion and longitudinal rotation.

208 Gatti

procedure is used to give a quantitative indication of the compatibility of the actual knee motion with the identi-fied 2 df kinematics: this allows assessing whether the two estimated axes of rotation are meaningful or not. The indication of how well a 2 df model would match real knee kinematic measurements could be then particu-larly valuable in clinical situations, where the proposed technique could be also potentially adopted for the cor-rect design and implantation of hinged external fixators (Sommers et al., 2004; Fragomeni et al., 2006), which are by themselves directly fixed to bones, and demonstrated their effectiveness in knee dislocation treatment and rehabilitation (Richter & Lobenhoffer, 1998; Simonian et al., 1998; Stannard et al., 2003; Fitzpatrick et al., 2005).

ReferencesAsano, T., Akagi, M., & Nakamura, T. (2005). The Functional

Flexion-Extension Axis of the Knee Corresponds to the Surgical Epicondylar Axis. The Journal of Arthroplasty, 20, 1060–1067. PubMed doi:10.1016/j.arth.2004.08.005

Blankevoort, L., Huiskes, A., & de Lange, A. (1988). The enve-lope of passive knee joint motion. Journal of Biomechan-ics, 21, 705–709, 711–720. PubMed doi:10.1016/0021-9290(88)90280-1

Cerveri, P., Lopomo, N., Pedotti, A., & Ferrigno, G. (2005). Derivation of centers and axes of rotation for wrist and fingers in a hand kinematic model: methods and reliability results. Annals of Biomedical Engineering, 33, 402–412. PubMed doi:10.1007/s10439-005-1743-9

Chang, L.Y., & Pollard, N.S. (2007). Robust estimation of dominant axis of rotation. Journal of Biomechanics, 40, 2707–2715. PubMed doi:10.1016/j.jbiomech.2007.01.010

Chao, E.Y. (1980). Justification of triaxial goniometer for the measurement of joint rotation. Journal of Biome-chanics, 13, 989–1006. PubMed doi:10.1016/0021-9290(80)90044-5

Churchill, D.L., Incavo, S.J., Johnson, C.C., & Beynnon, B.D. (1998). The transepicondylar axis approximates the optimal flexion axis of the knee. Clinical Ortho-paedics and Related Research, 356, 111–118. PubMed doi:10.1097/00003086-199811000-00016

Craig, J.J. (1989). Introduction to Robotics Mechanics and Control. New York: Addison-Wesley.

Ehrig, R.M., Taylor, W.R., Duda, G.N., & Hellera, M.O. (2007). A survey of formal methods for determining functional joint axes. Journal of Biomechanics, 40, 2150–2157. PubMed doi:10.1016/j.jbiomech.2006.10.026

Fitzpatrick, D.C., Sommers, M.B., Kam, B.C.C., Marsh, J.L., & Bottlang, M. (2005). Knee stability after articulated external fixation. American Journal of Sports Medicine, 33, 1735–1741. PubMed doi:10.1177/0363546505275132

Fragomeni, G., Mundo, D., Gatti, G., Moschella, D., & Dan-ieli, G.A. (2006). Preliminary design of a knee external fixator based on planar geometric synthesis. Transactions of the Canadian Society for Mechanical Engineering, 30, 567–579.

Gamage, S.S.H.U., & Lasenby, J. (2002). New least squares solutions for estimating the average centre of rotation and the axis of rotation. Journal of Biomechanics, 35, 87–93. PubMed doi:10.1016/S0021-9290(01)00160-9

Gardner, T.N., Evans, M., & Kyberd, P.J. (1996). An instru-mented spatial linkage for monitoring relative three-

dimensional motion between fracture fragments. Journal of Biomechanical Engineering, 118, 586–594. PubMed doi:10.1115/1.2796048

Gatti, G., & Danieli, G. (2007). Validation of a calibration tech-nique for 6-DOF instrumented spatial linkages. Journal of Biomechanics, 40, 1455–1466. PubMed doi:10.1016/j.jbiomech.2006.06.021

Gatti, G., Mundo, D., & Danieli, G. (2010). Kinematic analysis and performance evaluation of 6R instrumented spatial linkages. Transactions of the Canadian Society for Mechanical Engineering, 34, 57–73.

Grood, E.S., & Suntay, W.J. (1983). A joint coordinate system for the clinical description of three-dimen-sional motions: application to the knee. Journal of Biomechanical Engineering, 105, 136–144. PubMed doi:10.1115/1.3138397

Halvorsen, K., Lesser, M., & Lundberg, A. (1999). A new method for estimating the axis of rotation and the center of rotation. Journal of Biomechanics, 32, 1221–1227. PubMed doi:10.1016/S0021-9290(99)00120-7

Halvorsen, K. (2003). Bias compensated least squares estimate of the center of rotation. Journal of Biomechanics, 36, 999–1008. PubMed doi:10.1016/S0021-9290(03)00070-8

Hollister, A.M., Jatana, S., Singh, A.K., Sullivan, W.W., & Lupichuk, A.G. (1993). The axes of rotation of the knee. Clinical Orthopaedics and Related Research, 290, 259–268. PubMed

Ishii, Y., Terajima, K., Terashima, S., & Koga, Y. (1997). Three-dimensional kinematics of the human knee with intra-cortical pin fixation. Clinical Orthopaedics and Related Research, 343, 144–150. PubMed doi:10.1097/00003086-199710000-00024

Leardini, A., Chiari, L., Della Croce, U., & Cappozzo, A. (2005). Human movement analysis using stereophoto-grammetry. Part 3. Soft tissue artefact assessment and compensation. Gait & Posture, 21, 212–225. PubMed doi:10.1016/j.gaitpost.2004.05.002

Lewandowski, P.J., Askew, M.J., Lin, D.F., Hurst, F.W., & Melby, A. (1997). Kinematics of posterior cruciate ligament-retaining and -sacrificing mobile bearing total knee arthroplasties: An in vitro comparison of the New Jersey LCS meniscal bearing and rotating platform pros-theses. The Journal of Arthroplasty, 12, 777–784. PubMed doi:10.1016/S0883-5403(97)90008-5

MacWilliams, B.A. (2008). A comparison of four functional methods to determine centers and axes of rotations. Gait & Posture, 28, 673–679. PubMed doi:10.1016/j.gait-post.2008.05.010

Most, E., Axe, J., Rubash, H., & Li, G. (2004). Sensitivity of the knee joint kinematics calculation to selection of flexion axes. Journal of Biomechanics, 37, 1743–1748. PubMed doi:10.1016/j.jbiomech.2004.01.025

Piazza, S.J., & Cavanagh, P.R. (2000). Measurement of the screw-home motion of the knee is sensitive to errors in axis alignment. Journal of Biomechanics, 33, 1029–1034. PubMed doi:10.1016/S0021-9290(00)00056-7

Richter, M., & Lobenhoffer, P. (1998). Chronic posterior knee dislocation: treatment with arthrolysis, posterior cruci-ate ligament reconstruction and hinged external fixation device. Injury, 29, 546–549. PubMed doi:10.1016/S0020-1383(98)00095-3

Roland, M., Hull, M.L., & Howell, S.M. (2010). Virtual axis finder: a new method to determine the two kinematic axes of rotation for the tibio-femoral joint. Journal of Biome-chanical Engineering, 132(011009). PubMed

Two Dominant Knee Axes and Instrumented Spatial Linkage 209

Salvia, P., Woestyn, L., David, J.H., Feipel, V., Van Sint Jan, S., Klein, P., et al. (2000). Analysis of helical axes, pivot and envelope in active wrist circumduction. Clinical Biomechanics (Bristol, Avon), 15, 103–111. PubMed doi:10.1016/S0268-0033(99)00055-8

Schache, A.G., Baker, R., & Lamoreux, L.W. (2006). Defin-ing the knee joint flexion–extension axis for purposes of quantitative gait analysis: An evaluation of methods. Gait & Posture, 24, 100–109. PubMed doi:10.1016/j.gaitpost.2005.08.002

Shiavi, R., Limbird, T., Frazer, M., Stivers, K., Strauss, A., & Abramovitz, J. (1987). Helical motion analysis of the knee. I. Methodology for studying kinematics during locomo-tion. Journal of Biomechanics, 20, 459–469. PubMed doi:10.1016/0021-9290(87)90247-8

Simonian, P.T., Wickiewicz, T.L., Hotchkiss, R.N., & Warren, R.F. (1998). Chronic knee dislocation: Reduction, reconstruction, and application of a skeletally fixed knee hinge - A report of two cases. American Journal of Sports Medicine, 26, 591–596. PubMed

Smith, P.N., Refshauge, K.M., & Scarvell, J.M. (2003). Devel-opment of the concepts of knee kinematics. Archives of Physical Medicine and Rehabilitation, 84, 1895–1902. PubMed doi:10.1016/S0003-9993(03)00281-8

Sommers, M.B., Fitzpatrick, D.C., Kahn, K.M., Marsh, J.L., & Bottlang, M. (2004). Hinged external fixation of the knee - Intrinsic factors influencing passive joint motion. Journal of Orthopaedic Trauma, 18, 163–169. PubMed doi:10.1097/00005131-200403000-00007

Sorger, J.I., Federle, D., Kirk, P.G., Grood, E., Cochran, J., & Levy, M. (1997). The posterior cruciate ligament in total knee arthroplasty. The Journal of Arthroplasty, 12, 869–879. PubMed doi:10.1016/S0883-5403(97)90156-X

Stannard, J.P., Sheils, T.M., McGwin, G., Volgas, D.A., & Alonso, J.E. (2003). Use of a hinged external knee fix-

ator after surgery for knee dislocation. Arthroscopy—The Journal of Arthroscopy and related. Surgery, 19, 626–631.

Schwartz, M.H., & Rozumalski, A. (2005). A new method for estimating joint parameters from motion data. Journal of Biomechanics, 38, 107–116. PubMed doi:10.1016/j.jbiomech.2004.03.009

Stagni, R., Fantozzi, S., Cappello, A., & Leardini, A. (2005). Quantification of soft tissue artefact in motion analysis by combining 3D fluoroscopy and stereophotogram-metry: a study on two subjects. Clinical Biomechanics (Bristol, Avon), 20, 320–329. PubMed doi:10.1016/j.clinbiomech.2004.11.012

Taylor, W.R., Ehrig, R.M., Duda, G.N., Schell, H., Seebeck, P., & Heller, M.O. (2005). On the influence of soft tissue cov-erage in the determination of bone kinematics using skin markers. Journal of Orthopaedic Research, 23, 726–734. PubMed doi:10.1016/j.orthres.2005.02.006

Townsend, M.A., Izak, M., & Jackson, R.W. (1977). Total motion knee goniometry. Journal of Biomechanics, 10, 183–193. PubMed doi:10.1016/0021-9290(77)90057-4

Van Sint Jan, S., Salvia, P., Hilala, I., Sholukhaa, V., Roozea, M., & Clapworthy, G. (2002). Registration of 6-DOFs electrogoniometry and CT medical imaging for 3D joint modelling. Journal of Biomechanics, 35, 1475–1484. PubMed doi:10.1016/S0021-9290(02)00074-X

Williams, A., & Logan, M. (2004). Understanding tibio-femoral motion. The Knee, 11, 81–88. PubMed doi:10.1016/j.knee.2003.12.004

Wilson, D.R., Feikes, J.D., Zavatsky, A.B., & O’Connor, J.J. (2000). The components of passive knee movement are coupled to flexion angle. Journal of Biomechanics, 33, 465–473. PubMed doi:10.1016/S0021-9290(99)00206-7