http://dx.doi.org/10.5369/JSST.2017.26.6.371 pISSN 1225-5475/eISSN 2093-7563
Verification of Two Least-Squares Methods for Estimating Center of Rotation Using Optical Marker Trajectory
Jung Keun Lee1,2+
Abstract
An accurate and robust estimation of center of rotation (CoR) using optical marker trajectory is crucial in human biomechanics. In this regard, the performances of the two prevailing least-squares methods, the Gamage and Lasenby (GL) method, and the Chang and Pollard (CP) method, are verified in this paper. While both methods are sphere-fitting approaches in closed form and require no tuning parameters, they have not been thoroughly verified by comparison of their estimation accuracies. Furthermore, while for both methods, results for stationary CoR locations are presented, cases for perturbed CoR locations have not been investigated for any of them. In this paper, the estimation performances of the GL method and CP method are investigated by varying the range of motion (RoM) and noise amount, for both stationary and perturbed CoR locations. The difference in the estimation performance according to the variation in the amount of noise and RoM was clearly shown for both methods. However, the CP method outperformed the GL method, as seen in results from both the simulated and the experimental data. Particularly, when the RoM is small, the GL method failed to estimate the appropriate CoR while the CP method reasonably maintained the accuracy. In addition, the CP method showed a considerably better predictability in CoR estimation for the perturbed CoR location data than the GL method. Accordingly, it may be concluded that the CP method is more suitable than the GL method for CoR estimation when RoM is limited and CoR location is perturbed.
Keywords: Center of rotation, Least-squares method, Sphere-fitting, Optical maker trajectory.
1. INTRODUCTION
Modelling of human movement is often based on multi-link systems where links represent human limbs [1, 2]. As such links are connected by anatomical joints modeled as rotational joints, the study of human joint kinematics is of importance in biomechanics and human motion analysis [3]. In particular, the center of rotation (CoR) estimation of a joint is required for joint angle estimation and generation of skeletal kinematic chain.
Therefore, the CoR estimation from non-invasive surface measurements is a key component of biomechanical motion analysis [4-7]. In this paper, the CoR estimation using optical marker data from a motion capture system is discussed.
The CoR estimation methods can be largely categorized in two:
predictive methods and functional methods. The predictive methods calculate the CoR from empirical relations between specific anatomical landmarks [8]. However, these empirical relations typically depend on statistical approach, and not on the subject’s own data. Hence, the latest mainstream of CoR estimation is the functional method [7].
The underlying concept of the functional method is simple. For example, a hip joint is a spherical joint. 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. Specifically, functional methods use a cost function minimization to estimate the CoR based on the measured marker trajectory, i.e., often referred to as sphere-fitting.
The methods developed by Halvorsen [9] and Silaghi et al. [10]
are based on an iterative approach to estimate CoR. When we consider the CoR estimation as a step of calibration, calculation cost caused by applying an iterative approach may not be a critical issue in most cases. However, the main drawbacks of this iterative approach are the existence of local minimums of the cost function and the uncertainty of the solutions, i.e., the solution may vary depending on the values of optimization parameters [8]. Another
1Dept. of Mechanical Eng., Hankyong National Univ.
327 Jungang-ro, Anseong, Gyeonggi 456-749, Korea.
2Institute of Machine Convergence Tech., Hankyong Nat’l Univ.
327 Jungang-ro, Anseong, Gyeonggi 456-749, Korea.
+Corresponding author: [email protected]
(Received: Nov. 20, 2017, Revised: Nov. 24, 2017, Accepted: Nov. 25, 2017)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/
licenses/bync/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
approach to deal with the sphere-fitting problem is the least- squares method [8, 11, 12]. They are non-iterative in common.
The main differences between the existing least-squares methods are the cost functions and the weighting parameters.
This paper focuses on two prevailing least-squares methods proposed by Gamage and Lasenby [12] and Chang and Pollard [8]. The two main reasons why this paper focuses on the two methods are as follows: (i) both methods are in closed form and require no tuning parameters, which makes them useful in practice; and (ii) they are non-rigid sphere-fitting approaches such that the body does not have to be a rigid body, which makes them applicable to non-rigid human segments. In fact, Chang and Pollard made interesting comments in their article [8] on the effect of range of motion (RoM) on the estimation accuracy from the Gamage and Lasenby (GL) method. However, the Chang and Pollard (CP) method has not been thoroughly verified by other researchers in comparison to the GL method. Furthermore, while both methods presented results for stationary CoR locations, none have investigated cases for perturbed CoR locations. Note that, in real practice, the CoR location may be perturbed from the expected center. Accordingly, the contribution of this paper is twofold: the investigation of the estimation performances of the GL method and CP method (i) by varying the RoM and noise amount, and (ii) for both stationary and perturbed CoR locations.
In Sections 2 and 3, the GL method and the CP method are summarized, respectively. Section 4 presents the results of the estimation accuracy of the two methods using both simulated data and experimental motion capture data, for both stationary and perturbed CoR locations. Conclusions are drawn in Section 5.
2. GAMAGE AND LASENBY METHOD
The Gamage and Lasenby (GL) method for estimating CoR is a least-squares approach based on the assumption that the distance between markers attached to a body and the CoR is constant.
When is the vector from the ith marker in the jth time frame, is the CoR vector, and is the radius of the sphere formed by the ith marker, the least-squares cost function for the GL method is:
(1)
where m and n are the numbers of markers and frames, respectively. First, the differentiation of Eq. (1) with respect to yields:
. (2)
Next, the differentiation of Eq. (1) with respect to yields:
. (3)
Finally, the substitution of Eq. (2) into (3) yields:
(4) where
and (5)
. (6)
In Eqs. (5) and (6), , ,
where .
.
3. CHANG AND POLLARD METHOD
The Chang and Pollard (CP) method is also a non-rigid sphere- fitting approach like the GL method. Therefore, the spherical fit should have minimal radial variation. When a point
is near the surface of a sphere with center and radius , the radial variation can be modeled by the difference of the squared lengths:
. (7)
Eq. (7) expressed in terms of and can be rewritten to use the basis functions and their coefficients :
(8) where
and (9)
. (10)
In Eq. (9), w is the basis function . It should be noted that Eq. (8) is equivalent to Eq. (7) scaled by the coefficient a.
Expansion of Eq. (8) to n frames data of a single marker rij
rc ρi
fGL
2 2 2
1 1
( ) )
m n
GL ij c i
i j
f ρ
= =
⎡ ⎤
=∑∑⎣r r− − ⎦
ρi
2 1
1 n ( )
i ij c
n j
ρ
=
= ∑ r r−
rc
2 2 2
1 1
( ) ( ) )
m n
ij c ij c i
i j ρ
= =
⎡ ⎤
− ⎣ − − ⎦ =
∑∑ r r r r 0
c= K r γ
1 1 1 1
2m 1 n ij ijT i iT
i= n j=
⎛ ⎞
= ⎜ − ⎟
⎝ ⎠
∑ ∑
K r r a a
( 3 1 2)
1 m
i i i
i=
=∑ −
γ a a a
1 1
(n ) /
i ij
j
n
=
= ∑
a r 2 2
1
(n ) /
i ij
j
n
=
= ∑
a r
3 3 1
( n ) /
i ij
j
n
=
= ∑
a r rij3≡r rij ij2
[ , , ]x y zT
= r [ , , ]T
c= x y zc c c
r
ρ δ
2 2 2 2
( , ) (c x xc) (y yc) (z zc)
δ r ρ = − + − + − −ρ
rc ρ
b u
( ) T δ u =b u
[ , , , , 1]w x y z T
= b
[ , , , , ]a b c d eT
= c
2 2 2
x +y +z
trajectory yields:
(11) where
and (12)
. (13)
To find the spherical fit u, the cost function of the CP method to be minimized is:
. (14)
Noticeably, the CL method uses the following normalization constraint developed by Pratt [13] to avoid singularities in case of small RoM: , which can be rewritten in the following quadratic form:
(15) where the constraint matrix C is:
. (16)
Accordingly, the constrained minimization problem is
“Minimize subject to .”
The unconstrained minimization problem corresponding to the former constrained problem can be formulated by introducing the Lagrange multiplier λ as:
. (17)
Differentiation of Eq. (17) with respect to u yields:
(18) which can be solved with the QZ algorithm developed by Moler and Stewart [14]. In the Matlab implementation, the following code can be used for the QZ algorithm: [V,D]= eig(BTB, C, 'qz') where D is the diagonal matrix of eigenvalues and V is the matrix whose columns are the corresponding right eigenvectors. Once the coefficient vector u is chosen, the CoR and radius are
and , respectively.
Eq. (18) is based on the single marker trajectory and can be extended to cases of multiple markers by using the following augmented forms:
, (19)
, and (20)
. (21)
= δ Bu
1 2
[ , , , , ,δ δ δj δn]T
=
δ
[ , , , , ]1 j n T
=
B b b b
fCL
( ) ( )
T T T T
f =CL δ δ= Bu Bu =u B Bu
2 2 2 4 1
b + +c d − ae=
1
T =
u Cu
0 0 0 0 2
0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 2 0 0 0 0
⎡ − ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
=⎢ ⎥
⎢ ⎥
⎢− ⎥
⎣ ⎦
C
T T
u B Bu u CuT =1
_ T T ( T 1)
CL constrained
f =u B Bu−λu Cu−
T =λ
B Bu Cu
[ , , ] / 2T
c = −b c d
r ρ = r rc cT −( / )e a
1 2
( , , , , , ,..., )T
multi= a b c d e e em
u
1 multi
m
⎡ ⎤
⎢ ⎥
= ⎢ ⎥
⎢ ⎥
⎣ ⎦
B 1
B
B 1
0 2 2
2 0
2 0
multi
m m
m
− −
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
= ⎢ ⎥
⎢− ⎥
⎢ ⎥
⎢ ⎥
⎢− ⎥
⎣ ⎦
C
Fig. 1. Simulation data sets for ±30o RoM with 1 cm SD of random noise for M1 and M3 markers.
Fig. 2. Experimental setup: Flex 13 optical markers attached to a link constrained by a spherical joint at the bottom.
The CoR for the multiple markers is obtained by using the same equation with the single marker case and the individual radius is obtained by .
4. RESULTS
The estimation accuracy of the two methods was investigated using both simulated data and experimental motion capture data.
The CoR is estimated by each method for a given artificial noise and the RoM condition. The CoR error, the distance between the estimated CoR and the true CoR, was used for the comparison of estimation accuracy. Both methods were implemented with MATLAB (Mathworks Inc.).
4.1. Simulation Setup
There were two categories of simulations: (i) estimation of stationary CoR location, and (ii) estimation of perturbed CoR location.
First, the stationary CoR location was estimated by simulated marker trajectories for different RoMs and noise amounts. In terms of RoM, the simulated trajectories were generated starting from a nominal mean position and circling out to increasing angles relative to the mean position. Accordingly, the trajectories were located on the surface of cone formed by four different RoM angles, which were ±5o, ±15o, ±30o, and ±45o from the center. In terms of noise amplitudes, three different magnitudes of the random noises were normally distributed with standard deviations (SD) of 0.5 cm, 1 cm, and 2 cm and were imposed on each noise- free marker data. There were three markers: 30 cm (M1), 60 cm (M2), and 90 cm (M3) away from the true CoR. Each trial data set has 300 frames (see Fig. 1).
Next, the perturbed CoR location was estimated by simulated marker trajectories of different combinations of data set when each data set has a different CoR location. For example, input data sets of {(0,0,0) × 2 and (1,0,0) × 1} mean the combination of two data sets in which CoR of all markers are (0,0,0), and one data set in which CoR of all markers are (1,0,0), respectively.
The data sets were generated by a well-known sphere equation where the x-, y-, and z-
coordinates are , , and
. Each data set obtains 30 frames by varying θ (0o, 10o, 20o,····,90o) and (30o, 60o, 90o). Therefore, all markers are located in the x-y-z positive section. All markers have a radius of 30 cm, and no noise was imposed in the data.
[ , , ] / 2T
c= −b c d r
( / )
i c cT e ai
ρ = r r −
2 2 2 2
(x x− c) + −(y yc) + −(z zc) −ρ =0 cos sin c
x=ρ θ ϕ+x y=ρsin sinθ ϕ+yc
cos c
z=ρ ϕ+z ϕ
Table 1. Simulation results based on (a) M1 data, (b) M3 data, and (c) multiple data(=M1+M2+M3), for a stationary CoR location: (estimated xc, yc, zc) and distance from the true CoR (0,0,0).
(a) M1 marker (ρ = 30 cm)
RoM SD = 0.5 cm SD = 1.0 cm SD = 2.0 cm
±45o
GL (0.0,0.1,1.0)
1.0 (0.0,0.3,3.0), 3.0 (-0.1,-0.5,9.5), 9.5 CP (-0.1,0.1,0.3)
0.3 (0.0,0.4,-0.1), 0.4 (-0.1,-0.5,1.0), 1.2
±30o
GL (-0.1,-0.2,4.4) 4.4
(-0.1,-0.1,10.5), 10.5
(-0.8,0.1,19.9), 19.9 CP (-0.1,-0.2,1.3)
1.3 (-0.2,0.0,-1.5), 1.5 (-1.3,-0.2,-0.6), 1.4
±15o
GL (0.0,0.0,19.9) 19.9
(-0.1,-0.1,26.6), 26.6
(0.1,-0.2,28.1), 28.2 CP (-0.1,-0.1,2.5)
2.5
(-0.6,-0.4,-5.1), 5.1
(0.5,0.0,12.4), 12.4
±5o
GL (0.1,-0.1,29.7) 29.7
(0.0,-0.1,30.0), 30.0
(-0.4,-0.2,29.7), 29.7 CP (0.3,-0.1,9.9)
9.9
(-0.6,1.6,14.9), 15.0
(-0.6,-0.3,29.6), 29.6 (b) M3 marker (ρ = 90 cm)
RoM SD = 0.5 cm SD = 1.0 cm SD = 2.0 cm
±45o
GL (-0.2,-0.1,0.4) 0.4
(-0.1,-0.1,1.8) 1.8
(-0.2,0.2,5.0) 5.0 CP (-0.2,-0.1,0.1)
0.2
(-0.1,-0.1,0.6) 0.7
(-0.3,0.2,0.5) 0.6
±30o
GL (0.0,0.1,0.7) 0.7
(0.2,0.3,4.2) 4.2
(0.6,0.1,19.2) 19.2 CP (0.0,0.1,-0.7)
0.7
(0.2,0.3,-0.6) 0.7
(0.6,0.1,0.8) 1.1
±15o
GL (0.1,0.1,15.9) 15.9
(0.2,0.2,40.0) 40.0
(0.6,0.2,69.6) 69.6 CP (0.1,0.1,-0.3)
0.4
(0.3,0.2,2.2) 2.3
(0.7,0.5,4.4) 4.5
±5o
GL (0.0,0.1,84.8) 84.8
(0.1,-0.2,87.8) 87.8
(0.2,0.0,89.2) 89.2 CP (0.4,0.0,-4.3)
4.3
(0.8,-1.6,26.3) 26.3
(0.1,0.0,73.3) 73.3 (c) Multiple markers (=M1 + M2 + M3)
RoM SD = 0.5 cm SD = 1.0 cm SD = 2.0 cm
±45o
GL (-0.3,-0.1,0.6) 0.6
(-0.3,-0.3,2.4) 2.5
(-0.1,0.3,10.6) 10.6 CP (-0.2,0.0,-0.1)
0.2
(-0.2,-0.1,-0.2) 0.3
(-0.1,0.2,0.5) 0.5
±30o
GL (0.0,0.0,2.8) 2.8
(0.2,0.2,10.4) 10.4
(0.8,-0.1,35.2) 35.2 CP (0.0,0.0,0.3)
0.3
(0.1,0.1,-0.4) 0.5
(0.4,-0.1,0.8) 0.8
±15o
GL (0.2,0.2,30.8) 30.8
(0.2,0.4,60.3) 60.3
(0.8,0.3,78.9) 78.9 CP (0.1,0.1,0.1)
0.2
(0.0,0.3,-0.3) 0.4
(0.8,0.5,1.1) 1.4
±5o
GL (0.0,-0.1,86.4) 86.4
(0.8,-1.8,89.6) 89.7
(0.0,-2.6,95.0) 95.1 CP (0.1,-0.3,2.0)
2.0
(0.4,-0.7,47.0) 47.0
(-0.2,-0.4,33.5) 33.5
4.2. Experimental Setup
Similar to the simulated data, the stationary CoR location was estimated by empirical marker trajectories for different RoMs and noise amounts. An OptiTrack Flex13 camera motion capture system (from NaturalPoint, Inc. USA) was used to achieve the experimental marker data. For the stationary CoR location, an FX- 3460A monopod (from Horusbennu Inc., Korea) of which its end is a ball-and-socket joint was utilized. While the socket was fixed Fig. 3. CoR estimation errors for (a) M1 data (ρ = 30 cm) and (b) M3
data (ρ = 90 cm) in the simulation results.
Table 2. Simulation results for a perturbed CoR location. The expected CoRs are (0,0,0) for Cases 1 and 2 and (less than 2, 0,0) for Cases 3 and 4. The true radius is 30 cm for all cases.
Case GL CP
xc yc zc ρ xc yc zc ρ
Case 1 0.55 0.55 0.38 29.3 0.09 0.09 0.00 30.2 Case 2 0.68 0.68 0.47 29.1 0.09 0.09 0.00 30.2 Case 3 0.33 0.11 0.08 29.9 0.23 0.00 0.00 30.2 Case 4 0.49 0.25 0.19 29.7 0.37 0.00 0.00 30.2
* Data configuration
- Case 1: (0,0,0)×1 & (1,0,0)×1 & (0,1,0)×1 & (-1,0,0)×1
& (0,-1,0)×1
- Case 2: (1,0,0)×1 & (0,1,0)×1 & (-1,0,0)×1 & (0,-1,0)×1 - Case 3: (0,0,0)×8 & (1,0,0)×2
- Case 4: (0,0,0)×9 & (2,0,0)×1
Fig. 4. CoR estimation errors for (a) T1 marker data (ρ = 30 cm) and (b) T3 marker data (ρ = 90 cm) in the experimental results.
on the ground, three markers were attached to the monopod: 30 cm (T1), 60 cm (T2), and 90 cm (T3) away from the true CoR.
Three different RoM angles were applied by considering the limitations of the monopod diameter and allowable joint angle:
±10o, ±20o, and ±30o from the center. The RoM was controlled by a hole with a certain diameter installed at a certain height (see Fig.
2). The two different magnitudes of the random noises with standard deviations (SD) of 1 cm and 2 cm were imposed on each original marker data that had an SD of 10-4 cm noise. Each trial data set has 300 frames.
4.3. Simulation Results
Table 1 shows the simulation results for the stationary CoR location. In all trials, the CP method outperformed the GL method. The difference in estimation performance according to the change in noise amount and RoM was clearly shown in both methods. In particular, for the variations adopted in this paper, the performance difference according to RoM was more obvious than that according to noise amount (see Fig. 3). For cases where the radius estimation errors are large, it can be observed that the estimation errors are mostly due to the erroneous estimation of zc.
For the cone surface used in this paper, when the RoM is small, the change of the z-component in the marker data is very small compared to those of the x- or y-components. When a simulated random noise is applied to such a small change of the z- component, such a noise greatly degraded the estimation performance. In severe conditions, however, the CP method has much more robust performance than the GL method.
Table 2 shows the simulation results for the perturbed CoR locations. In Cases 1 and 2, several data sets around the expected CoR (0, 0, 0) were added into the input data. The radius estimation error from the GL method was 0.7 cm for Case 1 and 0.9 cm for Case 2, while that from the CP method was 0.2 cm for both cases. More importantly, in terms of CoR estimation, the results from the CP method are much closer to the expected CoR than those from the GL method. It is noticeable that the estimated zc of the CP results maintained at the zero position. This result exactly corresponds with our expectations because perturbations were made only to the x- and y-components, not to the z- components. Further, using this result, we can obtain a marker measurement strategy such that omni-directional movements will give us more reasonable estimations. This is important because, in real practice, it is impossible for CoR joints to keep stationary especially in the case of human joints. In Cases 3 and 4, in terms of radius estimation accuracy, both methods have similar accuracies. However, in terms of CoR estimation accuracy, the CP method outperformed the GL method. In particular, the estimated yc and zc of the CP results maintained at the zero position. This result exactly corresponds with our expectations because Table 3. Experimental results based on (a) T1 data, (b) T3 data, and
(c) multiple data (= T1 + T2 + T3), for a stationary CoR location: (estimated xc, yc, zc) and the distance from the true CoR (0,0,0).
(a) T1 marker (ρ = 30 cm)
RoM SD = 10-4 cm SD = 1.0 cm SD = 2.0 cm
±30o
GL (0.0,0.1,-0.1), 0.2
(-0.6,10.0,-0.5), 10.1
(-1.3,20.3,-1.4), 20.4 CP (0.0,0.1,-0.1),
0.1
(0.2,-1.9,0.2), 1.9
(0.2,0.2,-0.1), 0.3
±20o
GL (0.0,-0.3,-0.1), 0.3
(0.8,19.6,-0.5), 19.6
(1.1,24.8,-0.8), 24.8 CP (0.0,-0.3,-0.1),
0.4
(-0.1,-0.4,-0.2), 0.4
(0.1,0.4,-1.0), 1.0
±10o
GL (0.0,-1.0,-0.1), 1.0
(0.3,27.9,-0.6), 28.0
(0.4,28.5,-0.7), 28.5 CP (0.0,-1.1,-0.1),
1.1
(-0.2,-4.1,-1.2), 4.3
(0.4,24.2,-0.7), 24.2 (b) T3 marker (ρ = 90 cm)
RoM SD = 10-4 cm SD = 1.0 cm SD = 2.0 cm
±30o
GL (0.0,0.1,-0.1), 0.1
(-0.5,5.2,-0.2), 5.3
(-0.1,17.2,-0.1), 17.2 CP (0.0,0.1,0.1),
0.1
(-0.2,0.6,0.0), 0.7
(0.3,-0.6,0.7), 1.0
±20o
GL (0.0,-0.2,-0.1), 0.3
(0.3,17.9,-0.1), 17.9
(1.0,43.7,-0.3), 43.7 CP (0.0,-0.3,-0.1),
0.3
(-0.3,-2.2,0.1), 2.2
(0.1,7.2,0.1), 7.2
±10o
GL (0.0,0.0,-0.1), 0.1
(0.4,77.1,-0.6), 77.1
(0.6,83.1,-0.8), 83.1 CP (0.0,-0.1,-0.1),
0.2
(-0.2,-10.9,-0.1), 10.9
(0.3,30.6,-1.1), 30.6 (c) Multiple markers (=T1 + T2 + T3)
RoM SD = 10-4 cm SD = 1.0 cm SD = 2.0 cm
±30o
GL (0.0,0.0,-0.1), 0.1
(-0.5,6.3,-0.4), 6.3
(-1.0,18.3,-0.6), 18.4 CP (0.0,0.0,-0.1),
0.1
(-0.1,0.3,-0.1), 0.4
(0.4,-0.7,0.3), 0.9
±20o
GL (0.0,-0.3,-0.1), 0.3
(0.4,19.3,-0.2), 19.3
(1.1,38.6,-0.3), 38.6 CP (0.0,-0.3,-0.1),
0.3
(-0.2,-1.4,0.0), 1.5
(0.2,5.9,-0.1), 5.9
±10o
GL (0.0,0.6,-0.1), 0.6
(0.4,53.5,-0.5), 53.5
(0.5,57.3,-0.8), 57.3 CP (0.0,0.3,-0.1),
0.3
(0.3,3.2,-0.2), 3.2
(0.3,14.9,-0.9), 14.9
perturbations were made only to the x-components, not to the y- and z-components. Furthermore, let us compare Case 3 to Case 4.
Case 3 data have two erroneous data sets with a smaller amount of noise (i.e., 1) for each set, while Case 4 data have one erroneous data set with a larger amount of noise (i.e., 2) for each set. Estimation accuracy in Case 3 was higher than that in Case 4 in both methods. This implies that the purity of the input data is more important than the amount of data.
4.4. Experimental Results
Table 3 shows the experimental results for the stationary CoR location. The same tendency as in the case of the simulation was shown in the experiment. When the original marker data were used without any additional noise, the estimation accuracies of the two methods were almost the same, i.e., with high accuracy where the maximum CoR error was 1.1 cm. However, once the noises were imposed to the original data, a significant performance difference between the two methods was observed (see Fig. 4).
5. CONCLUSION AND DISCUSSION
An accurate and robust estimation of CoR using optical marker trajectory is crucial in human biomechanics. In this regard, this paper verified the performances of the GL and CP methods.
The difference in estimation performance according to the change in the amount of noise and RoM was clearly shown in both methods. However, the CP methods outperformed the GL method, in results from both the simulated and experimental data.
In particular, when the RoM is small, the GL method failed to estimate the appropriate CoR while the CP method reasonably maintained the accuracy. For example, the CoR errors, the distances between the estimated CoRs and the true CoR, were 0.2 cm from the CP method and 30.8 cm from the GL method for the RoM of ±15o; and 2.0 cm from the CP method and 86.4 cm from the GL method for the RoM of ±5o, when the amount of noise was 0.5 cm based on the multiple marker data. These results show that the application of normalization constraint made significant improvements in estimation robustness.
Furthermore, this paper investigated the effect of the perturbed CoR location data on the estimation accuracy. The predictability of the solution is important in practice. In the GL method, even if the marker data are perturbed in only one component (e.g., z- component), all components are affected by the perturbed data.
However, in the CP method, the effect of the perturbed input is
confined to the perturbed component. This solution is well- matched with users’ prediction. This predictability renders the CP method more suitable for the CoR estimation of human body segments.
Regarding the deterioration of the GL method for a small RoM, we need to think of the critical disadvantage of the GL method, i.e., a singularity problem. If a marker moves only on a plane (two-dimensional motion), the GL method cannot calculate the CoR because of this singularity issue. The following are the proofs: For Eq. (4) to have a solution, K should be invertible, i.e., non-zero determinant of K. The determinant of for the ith marker is obtained as
. (22)
For instance, if the z-component is not changed during measurement, is equal to . Therefore, the determinant becomes zero. Hence, markers associated with the GL method must be moved spatially during measurement in order to avoid the singularity. However, as the CP method has no plane singularity problem, it can be applied to find the CoR of a hinge joint. This is another advantage of the CP method over the GL method.
Finally, it may be concluded that the CP method is more suitable for CoR estimation than the GL method when RoM is limited.
ACKNOWLEDGMENT
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2015R1C1A1A02036373).
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Ki
2 2 2 2 2 2
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n = n = n =
= ∑ − ∑ − ∑ −
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2 1
(n j) /
j
z n
∑= z2
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