Int. J. Experimental and Computational Biomechanics, Vol. 3, No. 2, 2015
A subject-specific inverse-dynamics approach for estimating joint stiffness in sideways fall Yunhua Luo* and Masoud Nasiri-Sarvi Department of Mechanical Engineering, Faculty of Engineering, University of Manitoba, Winnipeg, Canada Email:
[email protected] Email:
[email protected] *Corresponding author Abstract: Sideways fall has been identified as the most critical situation leading to hip fracture in the elderly. The stiffness and damping property of the body joints are necessary for constructing effective biomechanical models to study fall dynamics. However, very little has been known about the joint behaviour when the body is in fall. We developed a subject specific inverse-dynamics approach to estimate the joint stiffness and damping property. The anthropometric parameters required for constructing the inverse-dynamics model was extracted from the subject’s whole body dual energy X-ray absorptiometry (DXA) image. The motion data of the body in sideways fall were obtained by protected fall tests using the same subject. The joints were represented by the Kelvin-Voigt model with undetermined stiffness and damping parameters, which were then determined by solving the inverse problems. For validation purpose, the obtained joint stiffness and damping parameters were substituted back into the dynamics equations and the forward problems were solved. The predicted fall kinematic variables were compared with those measured from the fall tests. Good agreements were observed, indicating that the proposed approach is reliable and reasonably accurate. Keywords: joint stiffness and damping; sideways fall; subject-specific; inverse dynamics model; protected fall test; anthropometric parameter. Reference to this paper should be made as follows: Luo, Y. and Nasiri-Sarvi, M. (2015) ‘A subject-specific inverse-dynamics approach for estimating joint stiffness in sideways fall’, Int. J. Experimental and Computational Biomechanics, Vol. 3, No. 2, pp.137–160. Biographical notes: Yunhua Luo is an Associate Professor at the Department of Mechanical Engineering and the Department of Biomedical Engineering, University of Manitoba, Canada. He holds a PhD in Computational Mechanics specialised in the finite element method. His current research interests include development of advanced finite element method, image-based computational biomechanics, injury analysis and prevention. He has produced about 100 peer-reviewed publications in international journals and conferences. Masoud Nasiri-Sarvi is currently a PhD student at the University of Manitoba, Canada. He obtained his Master’s degree from the Department of Mechanical Engineering, Sharif University of Technology, Iran, 2010. He holds one patent and has published several journal papers. His current research is focused on improvement of hip fracture risk assessment by developing subject-specific biomechanical models using clinical images.
Copyright © 2015 Inderscience Enterprises Ltd.
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Introduction
Computational simulation of fall dynamics has many important applications in injury analysis and prevention (Zhou et al., 2002; Hau et al., 2005). One such application is in the assessment of hip fracture risk among old people. Due to their decreased muscle strength and neural control ability, old people have a high incidence of accident fall (Rubenstein, 2006; Salvà et al., 2004). In addition, osteoporosis is a bone disease associated with aging and it greatly reduces bone strength (Bouxsein and Seeman, 2009). Therefore, old people with osteoporosis have a much higher probability of developing bone fracture once they fall. Sideways (or lateral) fall has been identified as the most critical situation for the elderly to develop hip fracture (Van den Kroonenberg et al., 1996; Kannus et al., 2006; Feldman and Robinovitch, 2007). The impact force applied to the great trochanter in sideways fall is a key parameter required for assessing hip fracture risk, and it is closely related to the fall kinematics of the body before hitting the ground (Bessho et al., 2007; Grassi et al., 2012). The body fall kinematics is in turn affected by the joint stiffness and damping of the subject. Although protected and controlled fall tests have been conducted to understand fall dynamics (Van den Kroonenberg et al., 1996; Van der Zijden et al., 2012), it is dangerous even for young volunteers and it is absolutely prohibited for the elderly. Therefore, computational simulation based on a valid dynamics model seems the only practical way to study fall dynamics of the elderly. Nevertheless, very limited research has been conducted on studying fall dynamics (Zhou et al., 2002; Van den Kroonenberg et al., 1995; Kim and Ashton Miller, 2009). The dynamics models developed by Van den Kroonenberg et al. (1995) represent the first effort to simulate sideways fall. Although the models are relatively simple, they are able to provide a certain insight into fall dynamics. Forward fall during gait was studied by Zhou et al. (2002) and Kim and Ashton Miller (2009) using their own dynamics models. Effectiveness of pre-impact segmental movements on the resulting impact injury risk was investigated by Lo and Ashton Miller (2008). However, the behaviour of the joints has been over-simplified in the existing models, as very little has been known about the stiffness and damping of the joints when the body is in fall. Experimental studies have shown that the joint stiffness and damping property have significant effect on the dynamics of the human body (Silder et al., 2007; Blackburn et al., 2004). Accurate identification of the joint property parameters related to fall is a challenging task, as the parameters are subject-dependent and they are also affected by a number of physiological and psychological factors. To the best knowledge of the authors, there is no reported research in the literature on how to determine the joint parameters when the body is in fall. Two over-simplified joint models have been considered in the literature, representing only two extreme situations. One is the so-called relaxed or free fall (Zhou et al., 2002), where the muscle forces and the joint moments are completely ignored. The other represents the active control of fall (Lo and Ashton Miller, 2008), where joints are modelled as actuators with full muscle strength. The behaviour of the joints in a real-life fall is something between the above two extremes. The objective of this study is to develop and validate an inverse-dynamics approach that can be applied to estimate the joint stiffness and damping parameters that are required in fall dynamics simulation; the impact force and hip fracture risk in sideways fall can thus be more accurately determined.
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Methods and materials
2.1 Inverse dynamics model The inverse-dynamics approach to be proposed in the following is in principle applicable to dynamics model of any complexity (Schiehlen, 2006; Liu et al., 2011). However, based on the consideration of balancing between model complexity and computational efficiency, a previously developed three-link dynamics model (Van den Kroonenberg et al., 1995) was improved and adopted in this study, as it is adequate for the study of sideways fall. First, two more degrees of freedom (DOFs), representing respectively the hip and knee flexion-extension, were added to the model to simulate more general body configurations in sideways fall. The two DOFs are constrained in the previous study (Van den Kroonenberg et al., 1995) by assuming that the body configuration does not have considerable changes in the DOFs, mainly to simplify the governing equations. However, considerable change of body configuration in the DOFs was observed in our experiment studies that will be described later in this paper. Second, the model was made subject-specific by extracting the required anthropometric information from the subject’s whole body dual energy X-ray absorptiometry (DXA) image. The dynamics model is shown in Figure 1. The three links labelled as 1, 2 and 3, respectively represent the shank, the thigh and the trunk. The rectangular coordinate system x-y-z is used in describing the motions of the links. The links are connected by two articulation joints (H and K) at respectively the hip and the knee(s). The model is connected to the ground by the ankle joint (F) with the assumption that there is no slippage between the feet and the ground during the fall. The knee joint (K) has a single degree of freedom representing the flexion extension rotation. Both the hip and the ankle joint have two DOFs, representing the rotations in the sagittal (x-z) and the coronal (y-z) plane. Therefore, five generalised coordinates, θ, φ, ψ, α and β as shown in Figures 1(a) and 1(b), were used in describing the body motion in fall. It should be noted that all the angles are projections of the link rotations on either the x-z or the y-z plane. The angular velocities of the links around the x- and y-axis can be expressed by the five generalised coordinates as, ω1x = −α , ω1 y = θ ω2 x = −α , ω2 y = − (φ − θ )
(1)
ω3 x = − ( β − α ) , ω3 y = ψ − (φ − θ )
where one-dot over a variable represents the first-order time derivative. The positive direction of the angular velocities is defined using the right-hand rule. The y-axis in Figure 1(a) and the x-axis in Figure 1(b) are pointing into the plane. The motion equations of sideways fall were established by the Lagrange dynamics (Kibble and Berkshire, 2004), ⎧ d ⎛ ∂T ⎞ ∂T ∂V − + = Fi , ⎪ dt ⎜⎝ ∂qi ⎟⎠ ∂qi ∂qi ⎨ ⎪q (0) = q , q (0) = q (i = 1, 2, " , 5) i0 i i0 ⎩ i
(2)
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In the above equations, q1 = θ, q2 = φ, q3 = ψ, q4 = α, q5 = β; T and V are respectively the kinetic and the potential energy. The energy expressions for the three-link model are provided in Appendix A. The expressions can be substituted into equation (2) to obtain the governing equations of the three-link model. Maple codes were developed to deal with the lengthy mathematical derivations. Figure 1
Three-link dynamics model of sideways fall of the human body, (a) sagittal view (b) coronal view (c) one view of the joint model
(a)
(b)
(c) Notes: Link 1 – the shank, Link 2 – the thigh, Link 3 – the trunk; A – the head; H – the hip; K – the knee; F – the ankle.
Fi (i = 1, 2, …, 5) in equation (2) are the joint moments. They are produced by the integral interaction of the muscles, tendons and ligaments across the joints, which is very complex and best described in the context of continuum solid mechanics (Doblare et al., 2007; Esat and Ozada, 2010). However, for the study of sideways fall, a simplified joint model is adequate and it is computationally more efficient. The Kelvin-Voigt model,
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which has been used to describe the mechanical behaviour of muscles (Anderson et al., 2002), has been adopted in this study to represent the joints. The joint model, as shown in Figure 1(c), consists of a torsional spring and a torsional damper working in parallel. The moment at a joint (i) can be expressed as Fi = ki qi + ci qi
(i = 1, 2, " , 5)
(3)
where ki and ci are the torsional stiffness and the torsional damping factor. They represent the resistance of the joint to the relative link rotations. These parameters were determined in this study by the proposed inverse dynamics approach. In calculating the kinetic and potential energy (see the expressions in Appendix A), anthropometric parameters such as the body segment length, mass, mass centre and mass moment of inertia are required. These parameters were obtained from the subject’s whole body DXA image using the recently developed method (Nasiri Sarvi and Luo, 2013; Burkhart et al., 2009). DXA is able to capture mass density information of different tissues, as the tissues have different attenuation coefficient of the X-ray. DXA has been accepted as a reliable method of estimating different tissues in living human body. The average between-measurer error is below 6% and the intraclass correlation coefficient is between 0.99 and 1.00 (Burkhart et al., 2009).
2.2 Acquisition of kinematics data via fall tests In an inverse dynamics model, the kinematic parameters (positions, velocities and accelerations) are known, and the kinetic variables (forces and moments) are to be determined. To construct subject-specific inverse-dynamics models in our study, three young volunteers were recruited under a human body research ethics approval. The subjects were first scanned by a clinic DXA scanner (Lunar Prodigy, GE Healthcare, USA). Their anthropometric and dynamics parameters were extracted from the DXA images. To obtain the body motion data in sideways fall, the volunteers then participated in protected and controlled fall tests. The configuration of the fall testing system is illustrated in Figure 2. The system consists of an electromagnetic release switch, nylon slings, a harness, a protection pad, a force plate (AMTI OR6-7MA, A-Tech Instruments Ltd., Canada) that is put under the pad, and a motion capture system with six infrared cameras (VICON, Vicon Motion Systems Ltd., UK). Reflective markers were put at the main joints of the subjects on both the left and right side. To avoid possible injury, the subjects were positioned by adjusting the slings so that the distance between the hip and the protection pad is about 50 cm. After the subject was released, the motion time histories of the reflective markers were automatically recorded by the motion capture system. Each subject had five trials. A data sampling rate of 200 frames per second was used in the tests. The motion data obtained from the experiments were then used in calculating the rotational angles (the generalised coordinates), angular velocities and angular accelerations of the links in a way that is consistent with (Van den Kroonenberg et al., 1995, 1996). First, the spatial coordinates of the left and the right markers were averaged to obtain the motions of the body central line. The spatial coordinates of the central line at the head, the hip, the knee and the ankle were used in calculating the rotational angles. The expressions for calculating the rotation angles from the spatial coordinates are given in Appendix B. Time derivatives of the angles were calculated used the high-order finite
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difference schemes (Chapra and Canale, 2006). To determine the joint moments, the above kinematic data, i.e., the angles, angular velocities and angular accelerations, were substituted into the left-hand side of equation (2). Figure 2
Protected and controlled fall testing system consisting of electromagnet release, nylon slings, harness, protective foam, force plate and motion capture system
2.3 Identification of joint stiffness and damping The joint stiffness and damping factor in equation (3) were then estimated as ⎡ ki −1 ⎤ T T ⎢ c = ( Q Q ) ( Q F )⎥ ⎣ i ⎦
(4)
with
F = ⎡⎣ Fi , Fi , " , Fi (1)
(2)
( n)
⎡ q (1) , qi(2) , " , qi( n ) ⎤ ⎤⎦ , Q = ⎢ i ⎥ (1) (2) (n) ⎣⎢ qi , qi , " , qi ⎦⎥
T
(5)
where n is the number of motion frames in the experimental data. To validate the inverse-dynamics approach, the joint stiffness and damping factor determined in equation (5) were substituted into equation (3), and then the forward dynamics problems defined by equations (2) and (3) were solved. The predicted motions were compared with those derived from the experimental data. To compare the joint moments derived from the experimental data and those predicted, the experimental data were first interpolated and then substituted into equation (2) to derive the experimental joint moments; the predicted body kinematic variable including link rotations and angular velocities were substituted into equation (3) to determine the predicted joint moments. To
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investigate differences between dynamics models with and without joint moments, forward problems with zero joint moments were also solved.
3
Results
The anthropometric and dynamics parameters of the three volunteers are provided in Table 1. The segment length, mass, mass, mass centre, and mass moment of inertia were determined from whole body DXA images of the subjects. Table 1
Anthropometric and dynamic parameters of the three volunteers determined from the subjects’ whole body DXA images Subject 1
Subject 2
Subject 3
Age
30
27
24
Total mass (kg)
77
72
64
1.73
1.72
1.74
Shank
0.44
0.42
0.42
Thigh
0.45
0.41
0.44
Trunk
0.84
0.89
0.88
Shank
7.3
6.9
6.8
Thigh
21.1
18.4
18.0
Trunk
47.1
45.1
39.8
Shank
0.25
0.23
0.22
Thigh
0.26
0.24
0.27
Trunk
0.37
0.38
0.38
Shank
0.14
0.14
0.15
Thigh
0.31
0.23
0.28
Trunk
2.63
2.33
1.78
Height (m) Segment length (m)
Segment mass (kg)
Segment mass centre* (m)
Segment mass moment of inertia
Note: *Measured from the distal end
The identified joint stiffness and damping parameters are listed in Tables 2 to 4. The initial kinematic conditions required in solving the forward dynamics problems were extracted from the experimental data and they are provided in Table 5. The predicted kinematic variables (the rotational angles, angular velocities and angular accelerations) of Subject 1 by the dynamics models with and without joint moments are plotted in Figures 3(a), 3(b), (3c) to 7(a), 7(b) and 7(c) together with the experimental data (Fall Test 5). The joint moments calculated from the experimental data and those derived from the predicted kinematic variables are plotted in Figures 3(d) to 7(d). In the figures, the dynamics model with joint moment is labelled as ‘DM (Kelvin)’ and the dynamics model without joint moment is denoted ‘DM (No joint force)’. The study results of Subjects 2 (with Fall Test 3) and Subject 3 (with Fall Test 1) are provided in Appendixes C and D.
144 Table 2
Fall 1 Fall 2 Fall 3 Fall 4 Fall 5
Table 3
Fall 1 Fall 2 Fall 3 Fall 4 Fall 5
Table 4
Fall 1 Fall 2 Fall 3 Fall 4 Fall 5
Y. Luo and M. Nasiri-Sarvi Joint stiffness and damping parameters corresponding to the five DOFs (θ, φ, ψ, α and β) of Subject 1 identified from his five fall trials
k (N/rad)
θ
φ
ψ
α
β
–3,491.9
5,881.3
–2,854.9
–953.4
3,673.5
c (N·s/rad)
183.7
–263.4
194.3
33.7
–173.8
k (N/rad)
2,169.2
–2,864.9
3,067.2
1,660.7
–118.1
c (N·s/rad)
–326.6
169.3
–142.6
–257.6
98.5
k (N/rad)
542.0
87.3
–284.5
328.5
–176.6
c (N·s/rad)
95.4
–188.4
73.6
–56.3
324.0
k (N/rad)
–176.4
–94.2
179.5
–37.5
183.8
c (N·s/rad)
–69.3
83.8
–73.2
–29.6
84.5
k (N/rad)
70.9
–25.8
–109.1
74.0
–97.9
c (N·s/rad)
–138.4
53.3
–131.4
63.1
–73.7
Joint stiffness and damping parameters corresponding to the five DOFs (θ, φ, ψ, α and β) of Subject 2 identified from his five fall trials θ
φ
ψ
α
β
2,956.2
81.3
–293.4
–1,734.6
782.4
c (N·s/rad)
–79.4
–569.3
183.3
683.4
183.4
k (N/rad)
1,298.3
838.3
–901.0
896.1
2,873.6
k (N/rad)
c (N·s/rad)
–763.5
–208.4
1,754.3
–653.5
–100.7
k (N/rad)
1,142.9
–102.3
–32.1
–112.6
–48.6
c (N·s/rad)
–393.3
139.7
–154.6
378.1
–28.2
k (N/rad)
83.5
–132.8
6.3
87.2
93.3
c (N·s/rad)
–10.4
93.8
–18.7
–35.6
–73.2
k (N/rad)
42.6
–75.2
–67.2
–92.6
–78.2
c (N·s/rad)
–65.7
129.9
–8.5
154.3
–27.5
Joint stiffness and damping parameters corresponding to the five DOFs (θ, φ, ψ, α and β) of Subject 3 identified from his five fall trials θ
φ
ψ
α
β
k (N/rad)
–2,851.8
139.4
–101.4
1,660.7
–118.1
c (N·s/rad)
875.1
–52.1
72.4
–527.6
98.5
k (N/rad)
1,481.6
5,881.3
–3,491.9
4,824.9
2,778.2
c (N·s/rad)
–583.7
–1,268.8
547.3
–1,329.1
–1,678.1
k (N/rad)
–241.3
21.3
–97.5
55.0
–104.7
c (N·s/rad)
–17.7
7.6
23.2
17.4
26.7
k (N/rad)
–31.7
163.1
–19.3
–92.9
11.5
c (N·s/rad)
–35.5
–92.3
–1.2
–110.7
15.4
k (N/rad)
–15.2
–119.0
6.0
–165.1
53.1
c (N·s/rad)
–53.4
–213.6
–16.0
–92.5
–65.1
A subject-specific inverse-dynamics approach for estimating joint stiffness Table 5
Initial kinematic conditions extracted from the beginning of the experimental motion data Subject 1
Subject 2
Subject 3
θ0 (rad)
0.83
0.11
0.57
θ0 (rad/s)
5.29
0.07
1.40
φ0 (rad)
2.81
1.07
1.47
φ0 (rad/s)
6.32
0.09
3.43
ψ0 (rad)
2.77
1.18
1.17
ψ 0 (rad/s)
2.66
0.04
1.26
α0 (rad)
1.22
0.63
0.53
α 0 (rad/s)
2.15
0.04
1.20
β0 (rad)
1.74
1.19
0.82
β0 (rad/s)
3.98
0.07
0.26
Figure 3
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Subject 1: ankle joint in the sagittal plane, (a) rotation θ(t) (b) angular velocity dθ/dt (c) angular acceleration dθ2/dt2 (d) joint moment Mθ(θ) (see online version for colours)
(a)
(b)
(c)
(d)
146 Figure 4
Figure 5
Y. Luo and M. Nasiri-Sarvi Subject 1: knee joint in the sagittal plane, (a) rotation φ(t) (b) angular velocity dφ/dt (c) angular acceleration dφ2/dt2 (d) joint moment Mφ(φ) (see online version for colours)
(a)
(b)
(c)
(d)
Subject 1: hip joint in the sagittal plane, (a) rotation ψ(t) (b) angular velocity dψ/dt (c) angular acceleration dψ2/dt2 (d) joint moment Mψ(ψ) (see online version for colours)
(a)
(b)
A subject-specific inverse-dynamics approach for estimating joint stiffness Figure 5
Subject 1: hip joint in the sagittal plane, (a) rotation ψ(t) (b) angular velocity dψ/dt (c) angular acceleration dψ2/dt2 (d) joint moment Mψ(ψ) (continued) (see online version for colours)
(c) Figure 6
147
(d)
Subject 1: ankle joint in the coronal plane, (a) rotation α(t) (b) angular velocity dα/dt (c) angular acceleration dα2/dt2 (d) joint moment Mα(α) (see online version for colours)
(a)
(b)
(c)
(d)
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Figure 7
4
Subject 1: hip joint in the coronal plane, (a) rotation β(t) (b) angular velocity dβ/dt (c) angular acceleration dβ2/dt2 (d) joint moment Mβ(β) (see online version for colours)
(a)
(b)
(c)
(d)
Discussions
The results reported in Figures 3 to 7, Figures C1 to C5, and Figures D1 to D5 show that the joint properties have important effect on the reliability and the accuracy of the dynamics models. If not considered, the dynamics models would have very poor prediction ability if examined against the experimental data. The dynamics models with the joint properties considered are able to more accurately predict the fall kinematics. The results also suggest that the proposed subject-specific inverse-dynamics approach is a viable and reliable method for estimating joint stiffness and damping properties related to fall. The use of subject-specific anthropometric parameters and experimental data is crucial in the approach, as all of them affect the accuracy of the identified joint parameters. There still exist considerable differences between the predicted and experimentally collected time histories of the kinematic/kinetic variables, as shown in the above mentioned figures. This may have been introduced by the assumed joint moment expression in equation (3). In this study, a linear relation was assumed between the joint moments and the stiffness/damping parameters, while the actual relation may be nonlinear and even time dependent.
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However, accurate determination of joint parameters related to fall is a challenging task, as the parameters are subject-dependent and they are also affected by a number of physiological and psychological factors. The reflex and instinctive response of the body is impossible to control or to repeat in fall tests. In Tables 2 to 4, there exist significant intra and inter-subject variations in the identified joint parameters, manifested in both their magnitudes and signs. The positive and negative signs in the parameters indicate that the joint muscles may act in either active or passive way during a fall. When the muscles are in an active status, the joint behaves like an actuator and it may have negative stiffness and damping effect; if the muscles are in a passive status, no active resisting moment is generated and the joint has positive stiffness and damping. The intra-subject differences in the repeated tests have been caused mainly by the reflexive and instinctive response of the human body. The generation of reflexive and instinctive response by the body is a very complicated neurophysical process that is neither controllable nor repeatable, although it can be re-shaped to some extent by training and experience (Angell, 1906; Franzisket, 1963). That explains why the magnitudes of the parameters decrease with the repeated tests, and the magnitudes from the first two tests are much larger than those from the last two, see Tables 2 to 4. The stiffness and damping at the ankle joint have not been considered in the existing fall dynamics models (Van den Kroonenberg et al., 1995). However, the results in Tables 2 to 4 indicate that they are significant and should not be ignored. With the stiffness and damping, two resistance moments can be generated at the ankle joint in sideways fall. The moment in the sagittal plane is produced by the uneven pressure acting on the feet; the moment in the coronal plane is generated due to the difference in the ground forces acting on the left and the right foot. The results suggest that in a sideways fall, the subject mainly uses the shank and the ankle to either recover from the fall or to find a more ‘comfortable’ configuration before hitting the ground. For the reasons discussed before, the joint parameters identified by the proposed approach from different subjects may be very diverse and should have a large scope. To determine the scope for different age and gender groups, a large number of volunteers should be recruited for conducting the fall tests. The main challenge is from old people, as the fall test used in this study is not viable for them. One possible solution is to use recorded videos from nursing homes of senior people. For example, a large-scale observation study of falls in elderly people has been recently conducted by Robinovitch et al.’s (2013) group, using digital videos captured in two long-term care facilities in British Columbia, Canada. These videos can be classified based on fall types. The videos of sideways fall can be used in replacement of the experimental data acquired by the fall tests. The whole body DXA images of the subjects can be either extracted from the clinic database if they are available or be scanned after the fall.
5
Conclusions
A subject-specific inverse-dynamics approach has been developed in this study to estimate joint stiffness and damping factor when the body is in sideways fall. Validation results with experimental data have shown that the approach is reliable and reasonably accurate. The study has also shown that the consideration of joint stiffness and damping factor in fall dynamics models is necessary for improving their prediction ability. The
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behaviour of joints in fall may have considerable intra- and inter-subject difference due to a number of physiological and psychological factors such as reflex and instinctive response.
6
Funding
The reported research has been supported by the Natural Sciences and Engineering Council (NSERC) and the Manitoba Health Research Council (MHRC) in Canada.
7
Ethical approval
The study was approved by the Health Research Ethics Board (HREB) at the University of Manitoba, Canada (Ethics Approval #: HS14084).
8
Conflict of interests
There is no conflict of interest involved in the reported study or in the published results.
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Appendix A Kinetic and potential energy expressions •
Kinetic energy 2 2 1 ⎡ 2 m1 ⎣( l1 cos α θ cos θ ) + ( l1 cos θα cos α ) + ( −l1 cos α θ sin θ − l1 cos θα sin α ) ⎤⎦ 2 1 2 2 + I1 ⎣⎡(α cos θ ) + ( θ cos α ) ⎦⎤ 2
T1 =
T2 =
{
2 1 m2 ⎡⎣ L1 cos α θ cos θ − l2 cos α (φ − θ ) cos(φ − θ ) ⎤⎦ 2
2 + [ L1 cos θα cos α + l2 cos(φ − θ )α cos α ] + [ − L1 cos α θ sin θ − L1 cos θα sin α
−l2 cos α (φ − θ ) sin(φ − θ ) − l2 cos(φ − θ )α sin α ⎤⎦
{
1 2 + I 2 [α cos(φ − θ ) ] + ⎡⎣(φ − θ ) cos α ⎤⎦ 2
T3 =
2
}
2
}
{
1 m3 ⎡⎣ L1 cos α θ cos θ − L2 cos α (φ − θ ) cos(φ − θ ) 2
+l3 cos ( ψ − (φ − θ ) ) ( β − α ) cos( β − α ) ⎤⎦ + [ L1 cos θα cos α + L2 cos(φ − θ )α cos α 2
−l3 cos( β − α ) ( ψ − (φ − θ ) ) cos ( ψ − (φ − θ ) ) ⎤⎦
2
+ ⎡⎣ − L1 cos α θ sin θ − L1 cos θα sin α − L2 cos α (φ − θ ) sin(φ − θ ) − L2 cos(φ − θ )α sin α − l3 cos ( ψ − (φ − θ ) ) ( β − α ) sin( β − α ) −l3 cos( β − α ) ( ψ − (φ − θ ) ) sin ( ψ − (φ − θ ) ) ⎤⎦
{
3
}
2 2 1 + I 3 ⎡⎣( β − α ) cos ( ψ − (φ − θ ) ) ⎤⎦ + ⎡⎣( ψ − (φ − θ ) ) cos( β − α ) ⎤⎦ 2
}
T = T1 + T2 + T1
•
Potential energy V = m1 gl1 cos θ cos α + m2 g [ L1 cos θ cos α + l2 cos(φ − θ ) cos α ] + m3 g ⎡⎣ L1 cos θ cos α + L2 cos(φ − θ ) cos α + l3 cos ( ψ − (φ − θ ) ) cos( β − α ) ⎤⎦
In the above expressions, the symbols are: mi mass of link Ii
mass moment of inertia
li
mass centre from the distal end
Li
Length of link i.
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Appendix B Given joint rectangular coordinates (x, y, and z), angular coordinates (θ, φ, ψ, α, and β) are calculated as zK − zF ⎛ θ = cos −1 ⎜ ⎜ ( xK − xF ) 2 + ( z K − z F ) 2 ⎝
⎞ ⎟ ⎟ ⎠
zH − zK
⎛
φ = θ + cos −1 ⎜
⎜ ( xH − xK ) + ( z H − z K ⎝ 2
)
2
⎞ ⎟ ⎟ ⎠
z A − zH ⎛ ψ = φ − θ + cos −1 ⎜ ⎜ ( x A − xH ) 2 + ( z A − z H ) 2 ⎝ zK − zF
⎛
α = cos −1 ⎜ ⎜ ⎝
⎞ ⎟ ⎟ ⎠
( yK − yF ) ⎜ ⎝
⎞ ⎟ + ( z K − z F ) ⎟⎠ 2
z A − zH
⎛
β = α + cos −1 ⎜
2
( y A − yH )
2
⎞ ⎟ + ( z A − z H ) ⎟⎠ 2
Appendix C Kinematic variables and joint moments of Subject 2 in sideways fall predicted by the subject-specific dynamics model and derived from the fall test (Fall Test 3). Figure C1
Subject 2: ankle joint in the sagittal plane, (a) rotation θ(t) (b) angular velocity dθ/dt (c) angular acceleration dθ2/dt2 (d) joint moment Mθ(θ) (see online version for colours)
(a)
(b)
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Figure C1
Subject 2: ankle joint in the sagittal plane, (a) rotation θ(t) (b) angular velocity dθ/dt (c) angular acceleration dθ2/dt2 (d) joint moment Mθ(θ) (continued) (see online version for colours)
(c) Figure C2
(d)
Subject 2: knee joint in the sagittal plane, (a) rotation φ(t) (b) angular velocity dφ/dt (c) angular acceleration dφ2/dt2; (d) joint moment Mφ(φ) (see online version for colours)
(a)
(b)
(c)
(d)
A subject-specific inverse-dynamics approach for estimating joint stiffness Figure C3
Figure C4
Subject 2: hip joint in the sagittal plane, (a) rotation ψ(t) (b) angular velocity dψ/dt (c) angular acceleration dψ2/dt2 (d) joint moment Mψ(ψ) (see online version for colours)
(a)
(b)
(c)
(d)
Subject 2: ankle joint in the coronal plane, (a) rotation α(t) (b) angular velocity dα/dt (c) angular acceleration dα2/dt2 (d) joint moment Mα(α) (see online version for colours)
(a)
(b)
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Figure C4
Subject 2: ankle joint in the coronal plane, (a) rotation α(t) (b) angular velocity dα/dt (c) angular acceleration dα2/dt2 (d) joint moment Mα(α) (continued) (see online version for colours)
(c) Figure C5
(d)
Subject 2: hip joint in the coronal plane, (a) rotation β(t) (b) angular velocity dβ/dt (c) angular acceleration dβ2/dt2 (d) joint moment Mβ(β) (see online version for colours)
(a)
(b)
(c)
(d)
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Appendix D Kinematic variables and joint moments of Subject 3 in sideways fall predicted by the subject-specific dynamics model and derived from the fall test (Fall Test 1). Figure D1
Subject 3: ankle joint in the sagittal plane, (a) rotation θ(t) (b) angular velocity dθ/dt (c) angular acceleration dθ2/dt2 (d) joint moment Mθ(θ) (see online version for colours)
(a)
(b)
(c)
(d)
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Figure D2
Figure D3
Subject 3: knee joint in the sagittal plane, (a) rotation φ(t) (b) angular velocity dφ/dt (c) angular acceleration dφ2/dt2 (d) joint moment Mφ(φ) (see online version for colours)
(a)
(b)
(c)
(d)
Subject 3: hip joint in the sagittal plane, (a) rotation ψ(t) (b) angular velocity dψ/dt (c) angular acceleration dψ2/dt2 (d) joint moment Mψ(ψ) (see online version for colours)
(a)
(b)
A subject-specific inverse-dynamics approach for estimating joint stiffness Figure D3
Subject 3: hip joint in the sagittal plane, (a) rotation ψ(t) (b) angular velocity dψ/dt (c) angular acceleration dψ2/dt2 (d) joint moment Mψ(ψ) (continued) (see online version for colours)
(c) Figure D4
(d)
Subject 3: ankle joint in the coronal plane, (a) rotation α(t) (b) angular velocity dα/dt (c) angular acceleration dα2/dt2 (d) joint moment Mα(α) (see online version for colours)
(a)
(b)
(c)
(d)
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Figure D5
Subject 3: hip joint in the coronal plane, (a) rotation β(t) (b) angular velocity dβ/dt (c) angular acceleration dβ2/dt2 (d) joint moment Mβ(β) (see online version for colours)
(a)
(b)
(c)
(d)
