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A stiffness-varying model of human gait
PII: S1350-4533(97)00022-2
Med. Eng. Phys. Vol. 19, No. 6, pp. 518–524, 1997 1997 IPEM. Published by Elsevier Science All rights reserved. Printed in Great Britain 1350–4533/97 $17.00 + 0.00
A stiffness-varying model of human gait X. H. Duan*, R. H. Allen† and J. Q. Sun‡ *Institute of Gerontology, University of Michigan, 300 North Ingalis, Ann Arbor, MI 48109, USA; †Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA; ‡Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA Received 3 September 1996, accepted 1 April 1997
ABSTRACT We report on a conceptual two degrees of freedom (2 DOF) human gait model, which incorporates nonlinear joint stiffness as a stabilizing agent. Specifically, muscle spring-like property provides inherent stability during gait movement using a nonlinear angular spring and dash pot at each joint. The instability problem of the gait model in direct dynamic analysis is overcome by simulating the human co-contraction muscle function. By developing dynamic system stability requirements and hypothesizing a minimum joint stiffness criterion, we determine time-varying joint stiffness. Optimum joint stiffnesses are present for varying gait pattern, stride lengths and cadences. We conclude that nonlinear joint stiffness can be incorporated into gait models to overcome stability problems inherent in such linkage models. 1997 IPEM. Published by Elsevier Science Ltd Keywords: Model, gait, joint stifness Med. Eng. Phys., 1997, Vol. 19, 518–524, September
1. INTRODUCTION One of the problems in using the Direct Dynamics method to analyze human gait is the instability of current widely used linkage models. Small errors, even those that originate from the model, can cause the simulated results to diverge from the expected trajectories1–3. To solve this problem, researchers have attempted to simulate the internal stability function of the human body. Bizzi et al., Hogan and Iqbal et al. have shown the phenomena that human joint movement in the absence of proprioceptive feedback can exhibit stable equilibrium behavior4–6. Iqbal et al.6 hypothesized the viscoelastic function of the musculoskeletal system provides a linear controller for the system, and such an intrinsic controller can effectively duplicate simple well-learned tasks in the absence of higher level CNS feedback. Guo et al.7 show that muscle co-contraction provides sufficient position and velocity feedback for stability, and no extrinsic feedback is needed in the stability analysis of the human torso. In neurophysiology, much research has focused on muscle spring-like property for movement execution. The dependence of muscle force on muscle length is shown to play an important role in the execution of single joint posture and movement8–10. Hogan11–13 extends this concept into the Correspondence to: Dr X. H. Duan, Institute of Gerontology, University of Michigan, 300 North Ingalis, Ann Arbor, MI 48109-2007, USA.
multi-joint movement analysis and has shown that the spring-like behavior of single muscle can be extended to the multiple variable case. The Equilibrium Point (EP) hypothesis suggests that the CNS may execute a movement by smoothly shifting a joint’s equilibrium point while maintaining proper joint stiffnesses14–16. For slow movement, the joint stiffness remains nearly constant throughout the movement17. For fast movement, the joint stiffness is dynamically controlled by the CNS18 and is time-varying19 and is also constrained by skeleton system dynamics. The analysis of joint stiffness may be achieved using two approaches: one is a top-down procedure, that is, from CNS planning and control to deduce the joint stiffness. Another is an inverse procedure, that is, from the skeleton system performance to deduce the required joint stiffness. Most work follows the top-down procedure and has focused on the analysis of CNS planning and control20,21. The second method has been used by Scheiner et al.3 In their model, active joint stiffness and dampers are considered at each joint. Those active parameters are attributed to co-contraction and the inherent length-tension and force–velocity characteristics of muscle. The effects of joint stiffness are emulated as proportional-derivative (PD) controllers22,23 and their effect on android stability is examined. Because of the complexity of the system, the joint stiffness and damping are kept constant within each cycle and are varied from cycle to cycle in the examining process.
A stiffness-varying model of human gait: X. H. Duan et al.
Prior research shows that muscle joint stiffness plays an important function in the process of stabilizing and controlling joint movement. Some quantitative analyses of its time-varying process have been made following a top-down procedure. The muscle joint stiffness deduced from this method depends on having a CNS planning and controlling procedure. In the analysis of the skeleton system mechanics’ requirement to determine the joint stiffness, the function of muscle stiffness is simulated as a feedback controller. The complexity of the controller in the gait model limits the analysis of its time-varying property. So far, no attempt has been made to include the muscle joint stiffness as a basic component of the skeleton system gait model, and to analyze stiffness timevarying properties. That is the purpose of this study. 2. MATHEMATICAL MODEL Our idealized model, shown in Figure 1, consists of two rigid bars, which represent the stand leg and upper bodym respectively, and two masses associated with each bar. The mass of the body segment is concentrated at one point, the center of gravity of the body segment. The dimensions and mass of the foot are much smaller than that of the leg and upper body, so they are neglected. We assume that the ankle is hinged directly on the ground through a revolute. Although this is not an accurate model of the stand leg, our purpose here is to demonstrate the method with a simple model; a movable ankle is coincided in a separated 8DOF Model24. Numerical analysis reveals that the fix ankle does not affect the final results significantly. In our model, on both the hip and ankle joints, there is an actuator, which provides the driving moments necessary to achieve Mb
θ2
motion. Across each joint, there are an angular spring and a dash pot, which represent the joint passive stiffness and viscosity, as well as active stiffness and viscosity produced by co-contraction muscles. We chose the vertical direction as the neutral position and let 0 = 0. In this model, we assume that: • Co-contraction muscles that stabilize the joint have spring-like characteristics and can be simulated by the angular spring mounted on the joint. • Human walking is driven by varying pseudo moments and forces acting on the joints in a feed-forward manner. • Motion consists of one swing leg and one support leg. The double support phase is not considered. • The swing leg does not have significant effects on the stand leg joint stiffness and motion, and it is neglected in the model. • The stand foot remains in contact with the ground throughout the swing phase, and there is no horizontal slippage between the stand foot and the ground. After normalizing the segment mass with the total body mass and segment length with leg length, the equations of motion are derived with the Langrange mechanics analysis method. In matrix form, they are: · · [A()]{¨ } + [C()]{ } + [B()]{ 2} + [K()]{} + [G]{sin} = [T(t)]
(1)
where [A()] is a symmetrical inertial matrix. [B()] is an asymmetrical matrix, [C()] is a symmetrical viscosity matrix, [K()] is a symmetrical stiffness matrix and [G] is a diagonal gravity force matrix. All the matrices are two by two. {} is a second-order vector that represents the absolute angles of links and {T(t)} is a second-order pseudo joint moment vector. Individual terms for each of the matrices are provided in appendix A. 3. SYSTEM STABILITY ANALYSIS The equations of motion are first linearized about an arbitrary configuration {0}. Let ˜ = ˜ 0 + ⌬ ˜ . Where {⌬} is a small disturbance, the linearized equation is given by: ˆ (0)]{⌬ · } + [K ˆ (0)]{⌬ } = 0 [A(0)]{⌬ ¨ } + [C (2)
Hip K2 C2
M1
θ1
C1 Ankle K1
Figure 1 Two-link conceptual body model.
Equation (2) represents a second-order dynamic system. An important assumption made in stability ˆ (0)] is positive or analysis is that the matrix [C semi-positive definite. Under this assumption, the condition of system being stable is that the eigen ˆ (0)] have values of the matrix ⫺ [A(0)]⫺1[K non-positive real parts. Hence, the solutions of the characteristic equation ˆ (0)] ⫺ [I]) = 0 det( ⫺ [A(0)]⫺1[K (3) lie on the left side of the phase plane or on the imaginary axis. To simplify the expression, we re-
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A stiffness-varying model of human gait: X. H. Duan et al.
express 0 as , then the characteristic equation can be written as: · · g1(K1,K2,,,¨ ) g2(K1,K2,,,¨ ) 2 + + =0 (4) g3() g3() where g1, g2 and g3 are functions originating from Equation (3). The list is given in appendix A. In the system parameters range we used in this study g3() ⬎ 0 (5) ensures the following Routh’s stability conditions 25: · g1(K1,K2,1,1,¨ 1) ⱖ 0 (6) · ¨ g2(K1,K2,1,1,1) ⱖ 0 (7) From these conditions, we obtain · K1 ⱖ f1(K2,,,¨ ) (8) and · K1 ⱖ f2(K2,1,1,¨ ) · for K2 ⬎ Ms2g ⫺ Ms2Hl(cos(1) 21 + sin(1)¨ 1) (9a) or · K1 ⱕ f2(K2,1,1,¨ ) · for K2 ⬍ Ms2g ⫺ Ms2Hl(cos(1) 21 + sin(1)¨ 1) (9b) These bounds on K1 and K2 ensure stability and they are determined by body segment positions, velocity and accelerations. Because the upper body remains in the nearly vertical position during gait, the effect of upper body’s small rotation on stability is neglected, and the special case of 2 = 0 is considered. The effects of other kinematic data are analyzed separately, and the results are plotted in Figure 2. From it, we can see that there is a jump phenomenon in the system’s stable regions when · K2 = Ms2g ⫺ Ms2Hl(cos(1) 21 + sin(1)¨ 1) ⬅ K2c where K2c is the critical point of K2. On or near the jump point, the value of K1 must keep increasing to keep the system stable. These values are beyond the limitation of the human joint. The value of K2 can not cross the jump point during
f2=0 K2 N/ (Kg m) 100 50 0
–10
–50
–5 f1=0
–100 20
0 θ1 (1/ s)
15 5
10 K2 N/ (Kg m)
5
Figure 2 The system stability regions.
520
0 10
· its varying process because the values of 1,1,¨ 2 are varying in a continuous fashion in the walking process. The value of K should lie either in one region of K2 ⬍ K2c, or in the other region of K2 ⬎ K2c, not in both, during the whole walking cycles. In the region of K2 ⬍ K2c, there is no region which can satisfy both conditions (8) and (9b) throughout the continuously varying range of · 1,1,¨ 2, it is an unstable region. In the region of K2 ⬎ K2c, the conditions of (8) and (9a) can be both satisfied in some sub-region, so the system stable region is found in here. In the sub-region of f1 ⬎ f2, condition (8) dominates system stability. In the sub-region of f1 ⬍ f2, condition (9a) dominates system stability. Based on the above analysis, the system’s stable region is determined by conditions (11), (12a) and: · K2 ⬎ Ms2g ⫺ Ms2Hl(cos(1) 21 + sin(1)¨ 1). (10) 4. THE OPTIMUM STIFFNESS Previous studies have shown that the movement planing and control can be best characterized with the minimal effort criterion20,21. In this study, we propose the minimal muscle stiffness criterion, because the moments caused by voluntarily contracting muscles have a proportional relationship with muscle stiffness. The penalty function of this performance measurement for the joint stiffness can be represented as:
冕 T
· J = {K 21 + K 22 + 1(t)g1(K1,K2,1,1,¨ 1) 0
· +2(t)g2(K1,K2,1,1,¨ 1)}dt
冕 T
= F(K1,K2)dt
(11)
0
where 1 and 2 are Langrange multipliers. This criterion penalizes unnecessarily high joint stiffness during movement, and thus limits excessive co-contraction. Within the boundary of K2 ⬎ K2c, the necessary conditions of optimum K1 and K2 are · ∂F ∂g1(K1,K2,1,1,¨ 1) = 2K1 + 1(t) ∂K1 ∂K1 · ¨ ∂g2(K1,K2,1,1,1) + 2(t) =0 (12) ∂K1 · ∂F ∂g2(K1,K2,1,1,¨ 1) = 2K2 + 1(t) ∂K2 ∂K2 · ¨ ∂g2(K1,K2,1,1,1) + 2(t) =0 (13) ∂K2 Let us · define K2* as · the point at which f 1(K2*,1,1,¨ 1) = f2(K2*,1,1,¨ 1). Then the constraint equations are: 2(t) = 0 (14) for K2 ⬍ K2* · g1(K1,K2,1,1,¨ 1) = 0
再
再
A stiffness-varying model of human gait: X. H. Duan et al.
1(t) = 0
· g2(K1,K2,1,1,¨ 1) = 0
for K2 ⬎ K2*
(15)
and
再
· g1(K1,K2,1,1,¨ 1) = 0 for K2 = K2* · g2(K1,K2,1,1,¨ 1) = 0
(16)
5. RESULTS The dynamic regulating procedure of joint stiffness in the human gait process is dependent on body segment position, velocity and acceleration. Here we analyze those relations. Because no knee is included in our 2DOF model, actual walking trajectories cannot be used, and some artificial paths are used in our analysis. Along those paths, the simulated stand leg moves from time t = 0, which is the moment of swing leg toe off, to the time of t = T, which is the moment of heel contact. The period, T, defines the time from swing leg toe off to heel contact. The paths that can help us isolate and recognize the effects of different parameters are chosen seperately. One type of idealized path is that the stand leg moves with constant velocity. When we use this kind of path, the effects of stride length and cadence on the stiffness can be isolated and analyzed separately without affecting other parameters. Another type of path has different position trajectory shapes. In this type of path, the velocity and acceleration keep changing with time according to their position trajectory profile. By using these paths, the dynamic regulation of stiffness relative to the trajectory pattern can be analyzed. 5.1. Effect of joint stiffness on system stability A path of 1 = cos(/2t + /4) is specified here, and along it, 1 changes from 0.7 rad to ⫺ 0.7 rad while 2 keeps to zero. To demonstrate the system’s inherent instability, we assume K1 = K2 = 0. The net moments needed to be provided by muscles are derived by using an inverse dynamics method that uses the assumed trajectories as the input of Equation (1) and pseudo moments as the output of the model. Then Equation (1) is solved again via the direct dynamics method by using those pseudo moments as an input, and the system motion kinematics data are obtained. As expected, the results show that the 1 and 2 diverge from the expected paths (see Figure 3). The reason for the direct dynamic solution divergence is that the model without considering the joint stiffness is an unstable model. This model does not have the ability to withstand disturbance, even the numeral error. For the above specified path, the extremum joint stiffnesses are obtained by solving Equations (12) and (13). The results are plotted in Figure 4, which shows the variations in K1 and K2 as a function of positions. For a specified trajectory, the values of K1 are nearly a constant and K2 has about 20% variation, the ankle spring is nearly twice as
Figure 3 Optimum joint stiffness for trajectories: 1 = cos(/2t + /4) and 2 = 0.
Figure 4 The values of performance measurement F with varying stiffness for trajectories 1 = cos(/2t + /4) and 2 = 0.
stiff as the hip joint spring. Figure 5 shows that at the extremum point of joint stiffness, the performance measurement function F has the minimum values, which shows that the results we obtained are the points of minimum joint stiffness. Because less joint stiffness implies less muscle effort, these are the optimum joint stiffnesses. To analyze human gait by the direct dynamic method based on a stable (non-zero Ki) model, for the same motion trajectory described above,
Figure 5 ness.
Direct dynamic solutions for the model without joint stiff-
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A stiffness-varying model of human gait: X. H. Duan et al.
we considered the optimum joint stiffness derived earlier and a linear relation Ci = 0.03 Ki21. These ˆ ] is positive definite. values ensure that matrix [C The same analyzing procedure used for the zero stiffness model is used, and the results are shown in Figure 6. From it, we can see that direct dynamic analysis results follow the expected path within 1%. From the above results, we can conclude that the model considering optimal joint stiffnesses is a stable one. This kind of model reflects the human’s ability to withstand disturbance and keep stable during the normal walking process. 5.2. Effect of walking speed on optimum joint stiffness Human walking velocity is dependent on cadence and stride length. Their relations can be expressed as Walking Velocity ⬀ Cadence × Stride Length The cadence can be measured by the period T. The stride length is determined by the distance between hip initial and final position in one stride. In our model, this can be calculated from the initial and final angle of the stand leg in the swing phase. To show how the joint stiffness is adjusted according to the stride length, we choose the paths 1 = S/2Hl ⫺ t. When moving along these paths, the stand leg’s angular velocity and acceleration are not altered ( ⫺ 1 rad/s and 0 rad/s2). The only changeable parameter is the step length S. The joint stiffnesses for those three different step lengths are shown in Figure 7, where s = 1.38 m, s = 1.51 m and s = 1.64 m are slow, natural and fast walking stride lengths, respectively. From it we can see that the stiffness increase is about 1–2% with the step length. To see the effect of the cadence on the joint stiffness, here we choose the path 1 = S/2Hl(1 ⫺ 2t/T) and fix the step length S. Obtaining three different cadences by using discrete values of the period, T, the results are shown in Figure 8. From it, we can see that the cadence has a much larger effect on the stiffness than step length. When the
Figure 6 Direct dynamic solutions for the model with optimum joint stiffness.
522
Figure 7
Optimum joint stiffness for different step lengths.
Figure 8
Optimum joint stiffness for different cadences.
cadence increases, there is a 10% increase in the joint stiffness. Based on the relationship of walking speed with the step length and cadence, we can conclude that for faster walking speed, higher joint stiffnesses are needed to keep the body stable. 5.3. Effect of walking pattern on desirable joint stiffness We now choose the paths where velocity and acceleration change with time. For the given step length S and period T, there are different walking patterns according to their moving trajectory. These paths’ trajectories can be described as 1 = Acos( ⫺ 2/T t + ), where A = sin⫺1(S/2Hl)/cos. We constrain step length and period as constant by assigning S = 1.51 and T = 0.84. The different trajectories, which represent different walking patterns, can be obtained by changing phase angle, . The joint stiffnesses corresponding to four different walking patterns are shown in Figure 9. From it we can see that the optimum joint stiffness is affected by the gait pattern. The joint stiffness in the gait process is dynamically controlled by the walking kinematic
A stiffness-varying model of human gait: X. H. Duan et al.
4. 5. 6. 7.
8. Figure 9 Optimum joint stiffness for different gait pattern.
parameters, which compare with the conclusions of Latash and Gottlieb18. 6. DISCUSSION AND CONCLUSIONS We describe an idealized 2DOF human gait model that incorporates nonlinear joint stiffness. Using stability requirement and minimum joint stiffness criteria, we present an optimum joint stiffness for gait patterns, stride lengths and cadence under three walking pattern: slow, natural and fast. We observe little effect of stride length on stiffness, while cadence affects joint stiffness up to 20%. Similarly, gait pattern also affects joint stiffness considerably. Our two principal conclusions are these: • Inherent instability in this gait model can be overcome by using nonlinear stiffness without necessarily resorting to feedback models. • Stiffness variation is affected by gait pattern and cadence, but is little affected by stride length. In this simple concept model, the ankle is fixed, no knee is included and the real walking trajectories cannot be used in this analysis. More complex models are needed to analyze the joint stiffness response on practical gait excitations; that is the focus of current work.
9. 10.
11. 12. 13. 14. 15. 16.
17.
18. 19. 20.
ACKNOWLEDGEMENTS The support of The Department of Mechanical Engineering at The University of Delaware is acknowledged.
21. 22.
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gait. Ph.D. Thesis, University of Delaware, Newark, Delaware, 1997. 25. Franklin, G. F., Powell, J. D. and Emami-Naeini, A. Feedback Control of Dynamic Systems. Addison-Wesley, New York, 1994.
APPENDIX
冋
[A()] =
Ms2Hl cos(2 ⫺ 1)
Ms2Hlcos(2 ⫺ 1)
Is2
[B()] =
冋
册
Is1
册
0
⫺ Ms2Hl sin(2 ⫺ 1)
Ms2H1sin(2 ⫺ 1)
0
冋
[K()] =
K1 + K2 ⫺ K2 ⫺ K2
K2
册
[B⬘(0)] =
冋
· ⫺ 2Ms2Hl sin(20 ⫺ 10)20
0 · 2Ms2Hl sin(20 ⫺ 10) 10
[K⬘(0)] =
冋
K⬘11 K⬘12
K⬘21 K⬘22
0
册
册
K⬘11 = K1 + K2 ⫺ Ms1g cos10 + Ms2Hl(sin(20 · ⫺ 10)¨ 20 + cos(20 ⫺ 10)220) K⬘12 = ⫺ K2 ⫺ Ms2Hl (sin(20 ⫺ 10)¨ 20 · + cos(20 ⫺ 10) 220) K⬘21 = ⫺ K2 ⫺ Ms2Hl(sin(20 ⫺ 10)¨ 10 ⫺ cos(20 · ⫺ 10) 210) K⬘22 = K2⫺Ms2g cos20⫺Ms2Hl (sin(20 ⫺ 10)¨ 10 · ⫺ cos(20 ⫺ 10) 210) [G] = [T] =
冋 冋
0
0
⫺ Ms2g
册
Tp ⫺ Ta ⫺ Tp
Is1 =
Il + mlH 212 + mb H2l mH2l
Is2 =
Ib + mbH 2bc Mh2l
524
册
⫺ Ms1g
mlH12 + mbHl Mhl mbHbc Ms2 = Mhl m: The mass of total body, 64.9 kg; mb: The mass of trunk, 43.08 kg; mll: The mass of whole leg; 10.14 kg; Ib: The inertia moment of trunk, 3.2 kgm2; Il: The inertia moment of whole leg, 0.3 kgm2; Hbc: The distance between mass center of trunk and hip joint, 0.3 m; Hl: The distance between ankle joint and hip joint, 0.895 m; Hl1: The distance between hip joint and mass center of leg, 0.399 m; Hl2: The distance between ankle joint and mass center of leg, 0.496 m; K1: Normalized ankle joint sprint stiffness, N/(kg m); K2: Normalized hip joint sprint stiffness, N/(kg m); Ta: Normalized driving moment acted at ankle joint, N/(kg m); Tp: Normalized driving moment acted at hip joint, N/(kg m); ˜· g1(K1,K2,˜ ,,˜¨ ) = Is2K1 + (Is1 + Is2 + 2Ms2Hl cos(2 ⫺ 1))K2 Ms1 =
⫺ Is1Ms2g cos2 ⫺ Is2Ms1g cos1 · + Is2Ms2Hl(cos(2 ⫺ 1) 22 + sin(2 ⫺ 1)¨ 2) · + Is1Ms2Hl(cos(2 ⫺ 1) 21 ⫺ sin(2 ⫺ 1)¨ 1) · · + M 2s2H2l cos(2 ⫺ 1)(cos(2 ⫺ 1)( 22 ⫺ 21) + sin(2 ⫺ 1)(¨ 2 + ¨ 1)) · g2(K1,K2,˜ ,˜·,˜¨ ) = K1K2 + Ms2[Hl(cos(2 ⫺ 1)21 ⫺ sin(2 ⫺ 1)¨ 1 · ⫺ g cos2]K1 + Ms2[2Hl(cos(2 ⫺ 1) 21 ⫺ sin(2 ⫺ 1)¨ 1) ⫺ g(cos1 · + cos2)]K2 + 2M2s2H2l [(cos(2 ⫺ 1)22 · + sin(2 ⫺ 1)¨ 2)(cos(2 ⫺ 1)21 ⫺ sin(2 ⫺ 1)¨ 1)] ⫺ Ms1Ms2Hlg cos1((cos(2 · ⫺ 1)21 + sin(2 ⫺ 1)¨ 1) · ⫺ M2s2Hlg cos2((cos(2 ⫺ 1) 22 + sin(2 ⫺ 1)¨ 2) + Ms1Ms2g2cos1cos2 g3(˜ ) = Is1Is2 ⫺ M 2s2H2l cos2(2 ⫺ 1)
Med. Eng. Phys. Vol. 19, No. 6, pp. 518–524, 1997 1997 IPEM. Published by Elsevier Science All rights reserved. Printed in Great Britain 1350–4533/97 $17.00 + 0.00
A stiffness-varying model of human gait X. H. Duan*, R. H. Allen† and J. Q. Sun‡ *Institute of Gerontology, University of Michigan, 300 North Ingalis, Ann Arbor, MI 48109, USA; †Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA; ‡Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA Received 3 September 1996, accepted 1 April 1997
ABSTRACT We report on a conceptual two degrees of freedom (2 DOF) human gait model, which incorporates nonlinear joint stiffness as a stabilizing agent. Specifically, muscle spring-like property provides inherent stability during gait movement using a nonlinear angular spring and dash pot at each joint. The instability problem of the gait model in direct dynamic analysis is overcome by simulating the human co-contraction muscle function. By developing dynamic system stability requirements and hypothesizing a minimum joint stiffness criterion, we determine time-varying joint stiffness. Optimum joint stiffnesses are present for varying gait pattern, stride lengths and cadences. We conclude that nonlinear joint stiffness can be incorporated into gait models to overcome stability problems inherent in such linkage models. 1997 IPEM. Published by Elsevier Science Ltd Keywords: Model, gait, joint stifness Med. Eng. Phys., 1997, Vol. 19, 518–524, September
1. INTRODUCTION One of the problems in using the Direct Dynamics method to analyze human gait is the instability of current widely used linkage models. Small errors, even those that originate from the model, can cause the simulated results to diverge from the expected trajectories1–3. To solve this problem, researchers have attempted to simulate the internal stability function of the human body. Bizzi et al., Hogan and Iqbal et al. have shown the phenomena that human joint movement in the absence of proprioceptive feedback can exhibit stable equilibrium behavior4–6. Iqbal et al.6 hypothesized the viscoelastic function of the musculoskeletal system provides a linear controller for the system, and such an intrinsic controller can effectively duplicate simple well-learned tasks in the absence of higher level CNS feedback. Guo et al.7 show that muscle co-contraction provides sufficient position and velocity feedback for stability, and no extrinsic feedback is needed in the stability analysis of the human torso. In neurophysiology, much research has focused on muscle spring-like property for movement execution. The dependence of muscle force on muscle length is shown to play an important role in the execution of single joint posture and movement8–10. Hogan11–13 extends this concept into the Correspondence to: Dr X. H. Duan, Institute of Gerontology, University of Michigan, 300 North Ingalis, Ann Arbor, MI 48109-2007, USA.
multi-joint movement analysis and has shown that the spring-like behavior of single muscle can be extended to the multiple variable case. The Equilibrium Point (EP) hypothesis suggests that the CNS may execute a movement by smoothly shifting a joint’s equilibrium point while maintaining proper joint stiffnesses14–16. For slow movement, the joint stiffness remains nearly constant throughout the movement17. For fast movement, the joint stiffness is dynamically controlled by the CNS18 and is time-varying19 and is also constrained by skeleton system dynamics. The analysis of joint stiffness may be achieved using two approaches: one is a top-down procedure, that is, from CNS planning and control to deduce the joint stiffness. Another is an inverse procedure, that is, from the skeleton system performance to deduce the required joint stiffness. Most work follows the top-down procedure and has focused on the analysis of CNS planning and control20,21. The second method has been used by Scheiner et al.3 In their model, active joint stiffness and dampers are considered at each joint. Those active parameters are attributed to co-contraction and the inherent length-tension and force–velocity characteristics of muscle. The effects of joint stiffness are emulated as proportional-derivative (PD) controllers22,23 and their effect on android stability is examined. Because of the complexity of the system, the joint stiffness and damping are kept constant within each cycle and are varied from cycle to cycle in the examining process.
A stiffness-varying model of human gait: X. H. Duan et al.
Prior research shows that muscle joint stiffness plays an important function in the process of stabilizing and controlling joint movement. Some quantitative analyses of its time-varying process have been made following a top-down procedure. The muscle joint stiffness deduced from this method depends on having a CNS planning and controlling procedure. In the analysis of the skeleton system mechanics’ requirement to determine the joint stiffness, the function of muscle stiffness is simulated as a feedback controller. The complexity of the controller in the gait model limits the analysis of its time-varying property. So far, no attempt has been made to include the muscle joint stiffness as a basic component of the skeleton system gait model, and to analyze stiffness timevarying properties. That is the purpose of this study. 2. MATHEMATICAL MODEL Our idealized model, shown in Figure 1, consists of two rigid bars, which represent the stand leg and upper bodym respectively, and two masses associated with each bar. The mass of the body segment is concentrated at one point, the center of gravity of the body segment. The dimensions and mass of the foot are much smaller than that of the leg and upper body, so they are neglected. We assume that the ankle is hinged directly on the ground through a revolute. Although this is not an accurate model of the stand leg, our purpose here is to demonstrate the method with a simple model; a movable ankle is coincided in a separated 8DOF Model24. Numerical analysis reveals that the fix ankle does not affect the final results significantly. In our model, on both the hip and ankle joints, there is an actuator, which provides the driving moments necessary to achieve Mb
θ2
motion. Across each joint, there are an angular spring and a dash pot, which represent the joint passive stiffness and viscosity, as well as active stiffness and viscosity produced by co-contraction muscles. We chose the vertical direction as the neutral position and let 0 = 0. In this model, we assume that: • Co-contraction muscles that stabilize the joint have spring-like characteristics and can be simulated by the angular spring mounted on the joint. • Human walking is driven by varying pseudo moments and forces acting on the joints in a feed-forward manner. • Motion consists of one swing leg and one support leg. The double support phase is not considered. • The swing leg does not have significant effects on the stand leg joint stiffness and motion, and it is neglected in the model. • The stand foot remains in contact with the ground throughout the swing phase, and there is no horizontal slippage between the stand foot and the ground. After normalizing the segment mass with the total body mass and segment length with leg length, the equations of motion are derived with the Langrange mechanics analysis method. In matrix form, they are: · · [A()]{¨ } + [C()]{ } + [B()]{ 2} + [K()]{} + [G]{sin} = [T(t)]
(1)
where [A()] is a symmetrical inertial matrix. [B()] is an asymmetrical matrix, [C()] is a symmetrical viscosity matrix, [K()] is a symmetrical stiffness matrix and [G] is a diagonal gravity force matrix. All the matrices are two by two. {} is a second-order vector that represents the absolute angles of links and {T(t)} is a second-order pseudo joint moment vector. Individual terms for each of the matrices are provided in appendix A. 3. SYSTEM STABILITY ANALYSIS The equations of motion are first linearized about an arbitrary configuration {0}. Let ˜ = ˜ 0 + ⌬ ˜ . Where {⌬} is a small disturbance, the linearized equation is given by: ˆ (0)]{⌬ · } + [K ˆ (0)]{⌬ } = 0 [A(0)]{⌬ ¨ } + [C (2)
Hip K2 C2
M1
θ1
C1 Ankle K1
Figure 1 Two-link conceptual body model.
Equation (2) represents a second-order dynamic system. An important assumption made in stability ˆ (0)] is positive or analysis is that the matrix [C semi-positive definite. Under this assumption, the condition of system being stable is that the eigen ˆ (0)] have values of the matrix ⫺ [A(0)]⫺1[K non-positive real parts. Hence, the solutions of the characteristic equation ˆ (0)] ⫺ [I]) = 0 det( ⫺ [A(0)]⫺1[K (3) lie on the left side of the phase plane or on the imaginary axis. To simplify the expression, we re-
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A stiffness-varying model of human gait: X. H. Duan et al.
express 0 as , then the characteristic equation can be written as: · · g1(K1,K2,,,¨ ) g2(K1,K2,,,¨ ) 2 + + =0 (4) g3() g3() where g1, g2 and g3 are functions originating from Equation (3). The list is given in appendix A. In the system parameters range we used in this study g3() ⬎ 0 (5) ensures the following Routh’s stability conditions 25: · g1(K1,K2,1,1,¨ 1) ⱖ 0 (6) · ¨ g2(K1,K2,1,1,1) ⱖ 0 (7) From these conditions, we obtain · K1 ⱖ f1(K2,,,¨ ) (8) and · K1 ⱖ f2(K2,1,1,¨ ) · for K2 ⬎ Ms2g ⫺ Ms2Hl(cos(1) 21 + sin(1)¨ 1) (9a) or · K1 ⱕ f2(K2,1,1,¨ ) · for K2 ⬍ Ms2g ⫺ Ms2Hl(cos(1) 21 + sin(1)¨ 1) (9b) These bounds on K1 and K2 ensure stability and they are determined by body segment positions, velocity and accelerations. Because the upper body remains in the nearly vertical position during gait, the effect of upper body’s small rotation on stability is neglected, and the special case of 2 = 0 is considered. The effects of other kinematic data are analyzed separately, and the results are plotted in Figure 2. From it, we can see that there is a jump phenomenon in the system’s stable regions when · K2 = Ms2g ⫺ Ms2Hl(cos(1) 21 + sin(1)¨ 1) ⬅ K2c where K2c is the critical point of K2. On or near the jump point, the value of K1 must keep increasing to keep the system stable. These values are beyond the limitation of the human joint. The value of K2 can not cross the jump point during
f2=0 K2 N/ (Kg m) 100 50 0
–10
–50
–5 f1=0
–100 20
0 θ1 (1/ s)
15 5
10 K2 N/ (Kg m)
5
Figure 2 The system stability regions.
520
0 10
· its varying process because the values of 1,1,¨ 2 are varying in a continuous fashion in the walking process. The value of K should lie either in one region of K2 ⬍ K2c, or in the other region of K2 ⬎ K2c, not in both, during the whole walking cycles. In the region of K2 ⬍ K2c, there is no region which can satisfy both conditions (8) and (9b) throughout the continuously varying range of · 1,1,¨ 2, it is an unstable region. In the region of K2 ⬎ K2c, the conditions of (8) and (9a) can be both satisfied in some sub-region, so the system stable region is found in here. In the sub-region of f1 ⬎ f2, condition (8) dominates system stability. In the sub-region of f1 ⬍ f2, condition (9a) dominates system stability. Based on the above analysis, the system’s stable region is determined by conditions (11), (12a) and: · K2 ⬎ Ms2g ⫺ Ms2Hl(cos(1) 21 + sin(1)¨ 1). (10) 4. THE OPTIMUM STIFFNESS Previous studies have shown that the movement planing and control can be best characterized with the minimal effort criterion20,21. In this study, we propose the minimal muscle stiffness criterion, because the moments caused by voluntarily contracting muscles have a proportional relationship with muscle stiffness. The penalty function of this performance measurement for the joint stiffness can be represented as:
冕 T
· J = {K 21 + K 22 + 1(t)g1(K1,K2,1,1,¨ 1) 0
· +2(t)g2(K1,K2,1,1,¨ 1)}dt
冕 T
= F(K1,K2)dt
(11)
0
where 1 and 2 are Langrange multipliers. This criterion penalizes unnecessarily high joint stiffness during movement, and thus limits excessive co-contraction. Within the boundary of K2 ⬎ K2c, the necessary conditions of optimum K1 and K2 are · ∂F ∂g1(K1,K2,1,1,¨ 1) = 2K1 + 1(t) ∂K1 ∂K1 · ¨ ∂g2(K1,K2,1,1,1) + 2(t) =0 (12) ∂K1 · ∂F ∂g2(K1,K2,1,1,¨ 1) = 2K2 + 1(t) ∂K2 ∂K2 · ¨ ∂g2(K1,K2,1,1,1) + 2(t) =0 (13) ∂K2 Let us · define K2* as · the point at which f 1(K2*,1,1,¨ 1) = f2(K2*,1,1,¨ 1). Then the constraint equations are: 2(t) = 0 (14) for K2 ⬍ K2* · g1(K1,K2,1,1,¨ 1) = 0
再
再
A stiffness-varying model of human gait: X. H. Duan et al.
1(t) = 0
· g2(K1,K2,1,1,¨ 1) = 0
for K2 ⬎ K2*
(15)
and
再
· g1(K1,K2,1,1,¨ 1) = 0 for K2 = K2* · g2(K1,K2,1,1,¨ 1) = 0
(16)
5. RESULTS The dynamic regulating procedure of joint stiffness in the human gait process is dependent on body segment position, velocity and acceleration. Here we analyze those relations. Because no knee is included in our 2DOF model, actual walking trajectories cannot be used, and some artificial paths are used in our analysis. Along those paths, the simulated stand leg moves from time t = 0, which is the moment of swing leg toe off, to the time of t = T, which is the moment of heel contact. The period, T, defines the time from swing leg toe off to heel contact. The paths that can help us isolate and recognize the effects of different parameters are chosen seperately. One type of idealized path is that the stand leg moves with constant velocity. When we use this kind of path, the effects of stride length and cadence on the stiffness can be isolated and analyzed separately without affecting other parameters. Another type of path has different position trajectory shapes. In this type of path, the velocity and acceleration keep changing with time according to their position trajectory profile. By using these paths, the dynamic regulation of stiffness relative to the trajectory pattern can be analyzed. 5.1. Effect of joint stiffness on system stability A path of 1 = cos(/2t + /4) is specified here, and along it, 1 changes from 0.7 rad to ⫺ 0.7 rad while 2 keeps to zero. To demonstrate the system’s inherent instability, we assume K1 = K2 = 0. The net moments needed to be provided by muscles are derived by using an inverse dynamics method that uses the assumed trajectories as the input of Equation (1) and pseudo moments as the output of the model. Then Equation (1) is solved again via the direct dynamics method by using those pseudo moments as an input, and the system motion kinematics data are obtained. As expected, the results show that the 1 and 2 diverge from the expected paths (see Figure 3). The reason for the direct dynamic solution divergence is that the model without considering the joint stiffness is an unstable model. This model does not have the ability to withstand disturbance, even the numeral error. For the above specified path, the extremum joint stiffnesses are obtained by solving Equations (12) and (13). The results are plotted in Figure 4, which shows the variations in K1 and K2 as a function of positions. For a specified trajectory, the values of K1 are nearly a constant and K2 has about 20% variation, the ankle spring is nearly twice as
Figure 3 Optimum joint stiffness for trajectories: 1 = cos(/2t + /4) and 2 = 0.
Figure 4 The values of performance measurement F with varying stiffness for trajectories 1 = cos(/2t + /4) and 2 = 0.
stiff as the hip joint spring. Figure 5 shows that at the extremum point of joint stiffness, the performance measurement function F has the minimum values, which shows that the results we obtained are the points of minimum joint stiffness. Because less joint stiffness implies less muscle effort, these are the optimum joint stiffnesses. To analyze human gait by the direct dynamic method based on a stable (non-zero Ki) model, for the same motion trajectory described above,
Figure 5 ness.
Direct dynamic solutions for the model without joint stiff-
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A stiffness-varying model of human gait: X. H. Duan et al.
we considered the optimum joint stiffness derived earlier and a linear relation Ci = 0.03 Ki21. These ˆ ] is positive definite. values ensure that matrix [C The same analyzing procedure used for the zero stiffness model is used, and the results are shown in Figure 6. From it, we can see that direct dynamic analysis results follow the expected path within 1%. From the above results, we can conclude that the model considering optimal joint stiffnesses is a stable one. This kind of model reflects the human’s ability to withstand disturbance and keep stable during the normal walking process. 5.2. Effect of walking speed on optimum joint stiffness Human walking velocity is dependent on cadence and stride length. Their relations can be expressed as Walking Velocity ⬀ Cadence × Stride Length The cadence can be measured by the period T. The stride length is determined by the distance between hip initial and final position in one stride. In our model, this can be calculated from the initial and final angle of the stand leg in the swing phase. To show how the joint stiffness is adjusted according to the stride length, we choose the paths 1 = S/2Hl ⫺ t. When moving along these paths, the stand leg’s angular velocity and acceleration are not altered ( ⫺ 1 rad/s and 0 rad/s2). The only changeable parameter is the step length S. The joint stiffnesses for those three different step lengths are shown in Figure 7, where s = 1.38 m, s = 1.51 m and s = 1.64 m are slow, natural and fast walking stride lengths, respectively. From it we can see that the stiffness increase is about 1–2% with the step length. To see the effect of the cadence on the joint stiffness, here we choose the path 1 = S/2Hl(1 ⫺ 2t/T) and fix the step length S. Obtaining three different cadences by using discrete values of the period, T, the results are shown in Figure 8. From it, we can see that the cadence has a much larger effect on the stiffness than step length. When the
Figure 6 Direct dynamic solutions for the model with optimum joint stiffness.
522
Figure 7
Optimum joint stiffness for different step lengths.
Figure 8
Optimum joint stiffness for different cadences.
cadence increases, there is a 10% increase in the joint stiffness. Based on the relationship of walking speed with the step length and cadence, we can conclude that for faster walking speed, higher joint stiffnesses are needed to keep the body stable. 5.3. Effect of walking pattern on desirable joint stiffness We now choose the paths where velocity and acceleration change with time. For the given step length S and period T, there are different walking patterns according to their moving trajectory. These paths’ trajectories can be described as 1 = Acos( ⫺ 2/T t + ), where A = sin⫺1(S/2Hl)/cos. We constrain step length and period as constant by assigning S = 1.51 and T = 0.84. The different trajectories, which represent different walking patterns, can be obtained by changing phase angle, . The joint stiffnesses corresponding to four different walking patterns are shown in Figure 9. From it we can see that the optimum joint stiffness is affected by the gait pattern. The joint stiffness in the gait process is dynamically controlled by the walking kinematic
A stiffness-varying model of human gait: X. H. Duan et al.
4. 5. 6. 7.
8. Figure 9 Optimum joint stiffness for different gait pattern.
parameters, which compare with the conclusions of Latash and Gottlieb18. 6. DISCUSSION AND CONCLUSIONS We describe an idealized 2DOF human gait model that incorporates nonlinear joint stiffness. Using stability requirement and minimum joint stiffness criteria, we present an optimum joint stiffness for gait patterns, stride lengths and cadence under three walking pattern: slow, natural and fast. We observe little effect of stride length on stiffness, while cadence affects joint stiffness up to 20%. Similarly, gait pattern also affects joint stiffness considerably. Our two principal conclusions are these: • Inherent instability in this gait model can be overcome by using nonlinear stiffness without necessarily resorting to feedback models. • Stiffness variation is affected by gait pattern and cadence, but is little affected by stride length. In this simple concept model, the ankle is fixed, no knee is included and the real walking trajectories cannot be used in this analysis. More complex models are needed to analyze the joint stiffness response on practical gait excitations; that is the focus of current work.
9. 10.
11. 12. 13. 14. 15. 16.
17.
18. 19. 20.
ACKNOWLEDGEMENTS The support of The Department of Mechanical Engineering at The University of Delaware is acknowledged.
21. 22.
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APPENDIX
冋
[A()] =
Ms2Hl cos(2 ⫺ 1)
Ms2Hlcos(2 ⫺ 1)
Is2
[B()] =
冋
册
Is1
册
0
⫺ Ms2Hl sin(2 ⫺ 1)
Ms2H1sin(2 ⫺ 1)
0
冋
[K()] =
K1 + K2 ⫺ K2 ⫺ K2
K2
册
[B⬘(0)] =
冋
· ⫺ 2Ms2Hl sin(20 ⫺ 10)20
0 · 2Ms2Hl sin(20 ⫺ 10) 10
[K⬘(0)] =
冋
K⬘11 K⬘12
K⬘21 K⬘22
0
册
册
K⬘11 = K1 + K2 ⫺ Ms1g cos10 + Ms2Hl(sin(20 · ⫺ 10)¨ 20 + cos(20 ⫺ 10)220) K⬘12 = ⫺ K2 ⫺ Ms2Hl (sin(20 ⫺ 10)¨ 20 · + cos(20 ⫺ 10) 220) K⬘21 = ⫺ K2 ⫺ Ms2Hl(sin(20 ⫺ 10)¨ 10 ⫺ cos(20 · ⫺ 10) 210) K⬘22 = K2⫺Ms2g cos20⫺Ms2Hl (sin(20 ⫺ 10)¨ 10 · ⫺ cos(20 ⫺ 10) 210) [G] = [T] =
冋 冋
0
0
⫺ Ms2g
册
Tp ⫺ Ta ⫺ Tp
Is1 =
Il + mlH 212 + mb H2l mH2l
Is2 =
Ib + mbH 2bc Mh2l
524
册
⫺ Ms1g
mlH12 + mbHl Mhl mbHbc Ms2 = Mhl m: The mass of total body, 64.9 kg; mb: The mass of trunk, 43.08 kg; mll: The mass of whole leg; 10.14 kg; Ib: The inertia moment of trunk, 3.2 kgm2; Il: The inertia moment of whole leg, 0.3 kgm2; Hbc: The distance between mass center of trunk and hip joint, 0.3 m; Hl: The distance between ankle joint and hip joint, 0.895 m; Hl1: The distance between hip joint and mass center of leg, 0.399 m; Hl2: The distance between ankle joint and mass center of leg, 0.496 m; K1: Normalized ankle joint sprint stiffness, N/(kg m); K2: Normalized hip joint sprint stiffness, N/(kg m); Ta: Normalized driving moment acted at ankle joint, N/(kg m); Tp: Normalized driving moment acted at hip joint, N/(kg m); ˜· g1(K1,K2,˜ ,,˜¨ ) = Is2K1 + (Is1 + Is2 + 2Ms2Hl cos(2 ⫺ 1))K2 Ms1 =
⫺ Is1Ms2g cos2 ⫺ Is2Ms1g cos1 · + Is2Ms2Hl(cos(2 ⫺ 1) 22 + sin(2 ⫺ 1)¨ 2) · + Is1Ms2Hl(cos(2 ⫺ 1) 21 ⫺ sin(2 ⫺ 1)¨ 1) · · + M 2s2H2l cos(2 ⫺ 1)(cos(2 ⫺ 1)( 22 ⫺ 21) + sin(2 ⫺ 1)(¨ 2 + ¨ 1)) · g2(K1,K2,˜ ,˜·,˜¨ ) = K1K2 + Ms2[Hl(cos(2 ⫺ 1)21 ⫺ sin(2 ⫺ 1)¨ 1 · ⫺ g cos2]K1 + Ms2[2Hl(cos(2 ⫺ 1) 21 ⫺ sin(2 ⫺ 1)¨ 1) ⫺ g(cos1 · + cos2)]K2 + 2M2s2H2l [(cos(2 ⫺ 1)22 · + sin(2 ⫺ 1)¨ 2)(cos(2 ⫺ 1)21 ⫺ sin(2 ⫺ 1)¨ 1)] ⫺ Ms1Ms2Hlg cos1((cos(2 · ⫺ 1)21 + sin(2 ⫺ 1)¨ 1) · ⫺ M2s2Hlg cos2((cos(2 ⫺ 1) 22 + sin(2 ⫺ 1)¨ 2) + Ms1Ms2g2cos1cos2 g3(˜ ) = Is1Is2 ⫺ M 2s2H2l cos2(2 ⫺ 1)