Analysis of human abnormal walking using a multi-body model: Joint models for abnormal walking and walking aids to reduce compensatory action

Journal of Biomechanics 33 (2000) 1405}1414

Analysis of human abnormal walking using a multi-body model: joint models for abnormal walking and walking aids to reduce compensatory action Yoshihiko Tagawa *, Naoto Shiba
, Shigeaki Matsuo, Tadashi Yamashita Kurume Institute of Technology, Kamitsu, Kurume 830-0052, Japan
Kurume University School of Medicine, Asashi, Kurume 830-0011, Japan Kurume Institute of Technology, Kamitsu, Kurume 830-0052, Japan Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan Accepted 5 May 2000

Abstract This paper proposes new models of diseased joints and evaluates the e!ectiveness of walking aids such as a cane and a brace for compensating for lost functions due to joint disorders. The ZMJ concept described in the previous work (Yamashita and Tagawa, 1988. In: Radharaman (Ed.), Robotics and Factories of the Future'87. Springer, New York, pp. 670}677) is modi"ed into three joint models as follows: a passive element joint (PEJ) which has a spring at the diseased joint; a constrained range joint (CRJ), the motion of which stays within some constrained relative angle; a partial moment joint (PMJ) which can produce a partial amount of the moment produced about the joint in normal walking. A cane can enlarge a supporting area and adjust the posture of the upper torso to be upright. An ischial weight-bearing brace is e!ective for conservative management of hip disorders by reducing a load to the joint (Shiba et al., 1998. Clinical Orthopaedics and Related Research 351, 149}157). Walking aids like a cane or brace have been conveniently used by the handicapped. Abnormal walking was simulated for each joint model. Dynamic e!ects of a cane and a brace on abnormal walking were examined by the multi-body walking model.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Simulation; Abnormal walking; Joint models; Compensatory action; Walking aids

1. Introduction Abnormal walking due to lost function at diseased joints has been mainly analyzed by experiments because of di$culties of modeling abnormalities. A basic concept for approaching abnormal walking by simulation has been proposed by Yamashita and Tagawa (1988), based on two facts. One fact is that many experimental results of normal walking have been obtained, but few results have been obtained involving abnormal walking on patients having similar walking conditions, de"ned clearly in physical terms. The second fact is that di!erences of characteristics between a normal model and its partly modi"ed (abnormal) model can be simulated under the same walking conditions. The di!erences which are quantitatively evaluated by simulation are considered to be compensatory actions caused by remaining unmodi-

* Corresponding author.

"ed sound joints. Model simulations are attractive because they are capable of giving predictive information about a system, including information that cannot be measured by experiments. In Yamashita and Tagawa(1988), the abnormal walking model has a zero moment joint (ZMJ) representing a loss of function at one joint of one leg: either the ankle, knee, or hip joint. The ZMJ is de"ned as an imaginary joint that can transmit a force to the adjacent links but cannot generate a moment during the stance phase. The ZMJ is an extremely idealized construct, compared with the real situation of lost function at a diseased joint. Three-joint models adjusting the characteristic of the ZMJ are proposed in this paper as follows: a passive element joint (PEJ) which has a spring at the diseased joint; a constrained range joint (CRJ) which can rotate within some constrained relative angle; a partial moment joint (PMJ) which can produce a limited amount of moment, as compared with a normal joint. The compensatory actions in the abnormal walking having one of

0021-9290/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 0 0 ) 0 0 1 1 1 - 1

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Y. Tagawa et al. / Journal of Biomechanics 33 (2000) 1405}1414

Nomenclature F FH G F) G F) H("F) !F) G G G\ "[FHR MHR]R) G G g J G l M G MH G p G S 6 ¹ ¹ " < e G n G Subscripts X, >, and Z r and l )

3D vector of linear force acting on the ith link at the ith joint 3D vector of net linear force acting on the ith link 6D spatial vector acting on the ith link at the ith joint, consists of linear force and moment"[FR MR ]R G G 6D spatial vector acting on the ith link, consists of force and moment at proximal and distal joints where the gravitational force is omitted by the assumption of the imaginary acceleration !g gravitational constant ith joint leg length ([M M M ], M "0 at the ZMP) 3D vector of moment acting on the ith link at the ith 6 7 8  joint 3D vector of net moment acting on the ith link 3D vector of position of the ith joint step length walking cycle time double support period which is a percentage of a walking cycle time walking speed 3D position vector from the (i!1)th joint to the center of gravity of the ith link 3D position vector from the (i!1)th joint to the ith joint

are components in inertial coordinate system O!X > Z    the right and left leg systems, respectively general symbol for the subscripts i and X,Y,Z. r is omitted for simplicity when an equation is obviously de"ned for the right leg.

these joint models are simulated. Also, dynamic e!ects of a cane and a brace on the abnormal walking are examined.

2. Abnormal walking model and functional roles of walking aids 2.1. ZMJ model Summing up our modeling method of walking including abnormal one in the previous paper (Yamashita and Tagawa, 1988), the following general results can be stated. (1) The proposed model can simulate both steady normal and abnormal walking on level ground. Characteristics of abnormal walking realized by a system having an ZMJ in the right leg have been obtained. The ZMJ is an expansion of the idea of the ZMP proposed by Vukobratovic et al. (1972, 1990). (2) Some variables in the walking system are predetermined to make the model walk essentially like a human, based on experimental results of other researchers. (3) To solve complicated nonlinear equations of human walking, linearization techniques are used.

In the abnormal walking having the ZMJ, the locus of the ZMP in the diseased leg side is decided in order to satisfy two zero-moment conditions, at the ZMJ and the ZMP, because the motions of lower extremities are assumed to be the same in both the normal and abnormal walking. The locus in the normal walking is predetermined based on experimental results by Yamashita and Katoh (1976). Assuming the ZMJ at the jth joint of the right leg of Fig. 1A, the force F is transmitted to H the adjacent links while the moment M is zero during H the stance phase. Thus the following three equations are derived. The position vector of the ZMJ H p "p #n # n , H   G G

(1)

where the unknown vectors are p and n .   The constrained equation of the motion of the ZMP and the dynamic equation of the upper torso are expressed, respectively, as H MH!(p !p );F "M "0, G H  H H G

(2)

Y. Tagawa et al. / Journal of Biomechanics 33 (2000) 1405}1414

Fig. 1. (A) Walking model. Our dynamic model consists of eight massive rigid links, which represent upper and lower torsos, thighs, shanks, and feet. Adjacent links are coupled by an ideal frictionless joint. The links are allowed to move in three-dimensional space. (B) Main symbols. Motions of each body are controlled by moments M , M , and 6G 7G M applied at the ith ideal joint, except the ZMJ. At the ZMJ, the 8G components of the moment about the joint are assumed to be zero during the stance phase. Link number 0 shows the ground being in contact with the foot; the number counts up to the upper links, i"1, 2, 3, 4, and 5; they correspond to the foot, the shank, the thigh, the lower torso, and the upper torso, respectively.





!F !F J HP F) H"!F) "   !p ;F !p ;F !M J J HP HP HP   ! F) H! F) H, (3) G GJ GH> G where the unknown quantities are F) and F) during the   stance phase, while F) is zero during the swing phase.  The unknown quantity F) H, which is the net force  acting on the upper torso and consists of inertia force and couple, is expressed as a function of the variable of the motion of the upper torso. Eliminating the unknown force F from Eq. (3), we have the equation of the angular H motion of the upper torso. The equation is solved to satisfy a periodic condition for one walking cycle starting from the heel contact of the right leg. 2.2. Modixed joint model The moment M at the jth joint can be modi"ed easily H to nonzero characteristics, which are closer to a real diseased situation than the ZMJ. As nonzero moment joints, three joint models corresponding to slight joint damage are proposed: the PEJ, the CRJ, and the PMJ. PEJ model: The PEJ, in which a spring is attached at the ZMJ, has coe$cients associated with the ankle, knee, and hip joints of 0.2, 0.1, and 0.1, respectively. These values are de"ned in a nondimensional form because in our simulation moments are calculated in nondimensional forms. The PEJ is based on the fact that the moment about the > (lateral)-axis during the singlesupport phase has a pattern similar to a relative angle at

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each leg joint in the sagittal plane. The X component of the moment at the PEJ is still zero. CRJ model: The CRJ can rotate within 403 of relative angle at the joint. The range was obtained by changing the relative angle between the foot and the ground at the toe o!. The CRJ idea tries to model the fact that rotational angles at diseased joints are usually limited within some range. The CRJ is combined with the other joint models during simulation. PMJ model: The PMJ model can generate a smaller moment about the joint than the moment magnitude in normal walking. The model may correspond to weakened muscles around the joint. The PMJ is assumed to accompany the CRJ in the joint, and the combined joint is represented as the (CR#PM)J. In our simulation, the joint is assumed to be able to produce a moment within 30 or 40% of the maximum value of moment of the normal walking. 2.3. Functional roles of walking aids Functional roles of a cane and a brace as walking aids are examined in the most severe abnormal walking with the (CR#PM)J at the right hip. Formulation of ewect of cane: For this purpose, the walking model with a cane is considered as shown in Fig. 2A. The e!ectiveness of a cane held in the hand opposite to the side possessing a diseased leg has been examined in stationary standing (Blount, 1956; Denham, 1959; Lehmann and de Latenr, 1990). This was con"rmed by the direct measurement of a joint force (Bergman et al., 1989) and of the #oor reaction force to the foot and/or the cane (Baxter et al., 1969, Shimada et al., 1988, Shimada and Arai, 1990) during walking. The force and moment components, including e!ects induced by the cane, are derived as follows:





!F !F !F J P ! F) H"!F) "   !p ;F !p ;F !p ;F !M J J P P J !   !F) H! F) H. (4)  GJ G The supporting period involving the cane starts at the instant of 20% of the walking cycle and ends at the 40% point in time. During such a short period, the displacement of the walking model is so small that the direction of the cane can be considered constant. Formulation of ewect of brace: For this purpose, an ischial partial weight-bearing brace is considered as shown in Fig. 2B. The load to the ischial seat in the simulation is predetermined based on experimental results (Shiba et al., 1998). The pre-load on the ischial seat is 10% of the weight of the walking model. The value of the load on the seat is de"ned as increasing proportionally to the extension of the thigh up to 17.5% of the weight until full extension.

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Y. Tagawa et al. / Journal of Biomechanics 33 (2000) 1405}1414

Fig. 2. (A) Cane walking. The walking model having the (CR#PM)J in the right leg is supported at its left shoulder joint J by a cane whose J end is gripped by the left hand. The following assumptions are considered in the simulation: the arm of the model and the cane are a massless rigid body; the cane makes a point contact with the ground; moments at the shoulder and the contact point of the cane are zero; a force F is transmitted to the shoulder. (B) Brace walking. Ischial ! partial weight-bearing brace is considered, consisting of three components: a proximal thigh part with an ischial seat; a condylar part that transfers the ischial load to the femoral condyle; a middle part that connects these two parts. The brace is used for patients with a diseased hip joint to reduce the load acting at the joint by distributing the load to the ischial seat. A force F that acts at the ischial seat is a distributed load from the hip joint. The brace is assumed to be massless in the simulation.

We de"ne that F is a reaction force vector from the ischium to the ischial seat and p is its position vector. The force and moment components, including e!ects induced by the brace, are obtained as follows:





!F !F !F J P F) H"!F) "   !p ;F !p ;F !p ;F !M J J P P   !F) H! F) H. (5)  GJ G 3. Simulation results 3.1. Compensatory actions for the ZMJ In our previous paper, Yamashita and Tagawa (1988), showed that the compensatory actions became greater as the ZMJ was located at a higher position of the leg joints. Fig. 3 shows the results of the most serious case, in which the ZMJ is located at the hip. The main results of simulation for one walking cycle, including both normal walking and abnormal walking with the ZMJ at the right hip, are compared. The normal walking conditions were de-

"ned as follows: a walking speed, <, of 1.5 m/s; a step length, S , of 0.74 m; a double support period, ¹ , of 6 " 10%; a parallel stance; an outward hip joint movement; and a distribution ratio of the #oor reaction forces to respective feet during the double stance phase, a common linear function of time in each component (see the appendix). The result shows that the upper torso inclines greatly over toward the right (diseased) leg and that great compensatory actions by sound joints in the right leg are required. The moment at the sound joints of the diseased leg is larger than that of the normal walking. The locus of the ZMP exceeds the area of the normal foot size, that is, a large supporting area like a ski is required. The most e!ective method for reducing the action was to slow down the speed and to shorten the step length (Tagawa and Yamashita, 1989). However, some compensatory actions of the abnormal walking having the ZMJ at the hip or the knee could not be reduced to acceptably small levels through modi"ed gait patterns alone. 3.2. Compensatory actions by new joint models The characteristics of the abnormal walking having one of our new joint models are compared with each other in Table 1. Improvements in compensatory actions produced by each new joint model can be summarized as follows. PEJ model: The PEJ improves posture of the upper torso in the sagittal plane (the > component) as compared with the case of the ZMJ, even though small moment values are generated by springs. CRJ model: The CRJ was combined with the ZMJ, abbreviated as (CR#ZM)J. The upper torso inclines forward more than in the ZMJ alone, due to the small angle of the foot at the toe o!. The CRJ is e!ective for reducing most of the moments as compared with the case of the ZMJ alone. The CRJ is superior to the PEJ in the reduction of the locus of the ZMP. PMJ model: The PMJ was combined with the CRJ, designated as (CR#PM)J, and moment within 30% of that of the normal walking was simulated. This combined model was most e!ective for reducing the compensatory movements of the torso. The motion of the upper torso in the frontal plane (the X component) remains more vertical and that in the sagittal plane (the > component) inclines forward more than in the other models. The X components of the moments at the joints located lower than the (CR#PM)J and at the waist joint become much smaller than in the other models. The locus of the ZMP stays almost within the normal foot size. (CR#PM ) J model with modixed walking conditions: An open stance in the aged or infants may be e!ective for reducing compensatory actions (Tagawa and Yamashita, 1989). Adding this walking condition to that of the

Y. Tagawa et al. / Journal of Biomechanics 33 (2000) 1405}1414

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Fig. 3. Compensatory actions of abnormal walking with ZMJ at hip. Normal and abnormal walking were simulated for the same normal walking conditions: a walking speed, <, of 1.5 m/s; a step length, S , of 0.74 m; a double support period, ¹ , of 10%; a parallel stance; an outward hip joint V " movement; and a distribution ratio of the #oor reaction forces to respective feet during the double stance phase, de"ned as a common linear function of time in each component (see the appendix). A stick diagram in top view is the projection of the model onto the horizontal plane at an interval of 5% cycle time. Components of the moment at each joint are given in time history curves, and divided by the product of the model weight and the leg length in order to be expressed in a normalized form. The locus of the ZMP in the right foot is normalized by the leg length.

(CR#PM)J walking, an excessive moment at the ankle joint having the (CR#PM)J at the hip is reduced from 4.57 to 1.69. The > component of the locus of the ZMP is also reduced from 1.15 to 0.47, and the locus stays within the normal foot size. So the abnormal walking having the (CR#PM)J and the open stance is an acceptable abnormal gait. Comparing this walking with the abnormal walking having the ZMJ at the hip under the normal walking conditions, the di!erences of the motions of the lower extremities and the upper torso are distinctive, as shown in two gait patterns in Fig. 4. In the (CR#PM)J walking with an open stance (Fig. 4A), the posture of the upper torso in the frontal plane inclines toward the supporting leg side and in the sagittal plane inclines forward during the entire walking cycle. The motion of the lower ex-

tremities stays within smaller angular displacements. On the other hand, under normal walking conditions, the posture of the upper torso for the ZMJ model in the frontal plane inclines toward the diseased leg side and in the sagittal plane oscillates back and forth during one cycle (Fig. 4B). 3.3. Functional roles of walking aids Dynamic e!ects of the cane and the brace on walking characteristics were examined in the most severe abnormal walking with the (CR#PM)J at the right hip, wherein the moment generated at the PMJ stays within 40% of normal walking. The simulation results for the cane and/or the brace are summarized in Table 2, in the same manner as in Table 1.

X Max. 12.8 Min. !10.1 Ave. 1.2

Max. Min. Ave.

Max. Min. AVe.

Component Walking with *** at ankle

Walking with *** at knee

Walking with *** at hip

> 0.00 0.66 0.38 0.38 0.72 0.00 0.44 0.71 1.27 1.10 0.00 0.66

X 0.00 0.56 0.65 0.81 3.99 0.00 0.43 1.13 12.25 1.13 0.00 1.87

PEJ > 0.23 0.70 0.40 0.46 0.69 0.26 0.43 0.54 1.31 1.12 0.06 0.68

X 0.00 0.48 0.55 0.78 3.27 0.00 0.43 1.13 9.21 0.85 0.00 1.64 > 0.00 0.52 0.21 0.40 0.57 0.00 0.43 0.54 0.66 0.72 0.00 0.48

X 0.30 0.47 0.54 0.72 1.25 0.30 0.32 1.13 4.57 0.39 0.30 1.05 > 0.30 0.57 0.27 0.47 0.68 0.27 0.33 0.54 0.67 0.27 0.19 0.54 6.00

1.68

X 0.46

(CR#ZM)J (CR#PM)J ZMJ

2.92

1.17

> 0.26

6.06

1.67

X 0.46

PEJ

Locus of ZMP (-)

2.91

1.18

> 0.26

1.85

0.38

X 0.13

2.65

1.05

> 0.16

0.97

0.98

X 0.98

1.15

0.33

> 0.03

(CR#ZM)J (CR#PM)J

Three characteristics are summarized for each joint model at three di!erent joints: the angular displacement of the upper torso is shown in the maximum, minimum, and averaged values. The moment values are the maximum magnitude at each joint which are divided by that of the moment at the joint in normal walking. As for the locus of the ZMP, the X (sagittal) component in the right foot is divided by the foot length and the > (lateral) component is divided by the foot width to compare the normal foot size. Characteristics for a speci"c joint model including its location can be read in the block which can be de"ned by crossing the column indicating the model and the row indicating its position. Symbol *** is substituted by a joint model to indicate its position; characters A, K, H, and W represent the ankle, knee, hip, and waist joints, respectively. Conditions of the abnormal walking are given as <"0.4 m/s, S "0.29 m, and the other conditions are the same as those in Fig. 3. 6

> X > X > X > 24.1 12.8 12.0 12.9 16.0 11.6 !3.9 A 4.2 !10.1 !2.8 !10.3 !3.2 !11.0 !12.2 K 13.6 1.2 5.1 1.2 3.8 0.2 !9.1 H W 23.5 !7.5 23.5 1.2 24.2 !14.2 15.5 !12.6 A !4.3 !18.3 !4.3 !9.9 !4.6 !24.0 !8.9 !16.1 K 9.4 !12.0 9.4 !3.9 9.6 !18.2 3.2 !14.3 H W 40.8 7.1 40.8 3.4 42.5 3.3 23.3 !12.6 A 4.6 0.8 4.6 !4.1 4.0 !8.4 !5.1 !16.2 K 22.2 2.8 22.2 !0.8 22.6 !2.2 8.9 !14.4 H W

X 0.00 0.56 0.65 0.82 3.99 0.00 0.43 1.12 12.24 1.13 0.00 1.87

ZMJ

(CR#ZM)J (CR#PM)J

ZMJ

Joint Model

PEJ

Moment (-)

Characteristics Angular displacement of upper torso (deg)

Table 1 Comparisons of compensatory actions in abnormal walking with each joint model

1410 Y. Tagawa et al. / Journal of Biomechanics 33 (2000) 1405}1414

Y. Tagawa et al. / Journal of Biomechanics 33 (2000) 1405}1414

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out cane. Table 2 also shows that the brace has almost the same function as the cane during (CR#PM)J walking. The load to hip joint is clearly reduced using the brace (Fig. 5B). However, simultaneous use of the brace and the cane does not give a good result.

4. Discussion

Fig. 4. (A) Walking with (CR#PM)J at right hip. Walking conditions of this abnormal walking are the same as in the case of Fig. 3 except <"0.4 m/s, S "0.29 m, and an open stance. The open stance was 6 de"ned as that each leg at the heel contact was located outside the vertical line from the hip joint by 53. (B) Walking with ZMJ at right hip. Walking conditions of this walking are the same as in the case of Fig. 3. Values in both the horizontal planes show a distance normalized by the leg length.

In Table 2, F shows each component of the cane force ! F acting at the left shoulder. At present, su$cient data ! for determining the values based on some rule are not available (Murray et al., 1969, Shimada et al., 1988). A constant value of F was selected by trials to make the ! torso upright during the cane's supporting period. The brace is used to reduce the force acting at the diseased hip joint. Each F in the table shows a value of

 the pre-load. The F during walking was determined based on our experimental results (Shiba et al., 1998) The NJ (normal joint) walking in Table 2 has distinct characteristics: a larger angular motion about the X-axis (in the frontal plane); a bigger X component of the moments at the ankle and waist joints. These values themselves change according to each gait pattern, from the aged a to the aged d. These simulation results suggest how each gait condition a!ects the walking characteristics. On the other hand, in the (CR#PM)J walking without a walking aid, the change of the walking condition from the aged a to the aged d was e!ective in reducing the moment at the ankle joint and to vertically adjust the upper body. Referring to these results, values of the cane force were determined. In the (CR#PM)J walking with a cane, the cane force provides a prop for vertical support and compensatory pushing forces in both directions of the progression and the right (diseased) leg. Comparing the characteristics of the (CR#PM)J walking between cases with and without a cane shows that the upper torso becomes more upright and the moment at the waist joint is reduced slightly in cane walking. The vertical component of the cane force lessens the load to the hip joint (Fig. 5A). However, in the (CR#PM)J walking with a cane, some moments at other joints located in positions lower than the damaged joint become greater. The locus of the ZMP in the > component becomes larger than in the walking with-

In the rehabilitation "eld, extensive measurement analyses of human motions have been carried out for various daily activities. However, some subjective and traditional treatments of patients should be replaced by modern methods if we can "nd some objective reasons for doing so. For example, computational approaches, aided by extensive case studies as proposed in the present paper, will o!er practical applications in such "elds by estimating inner forces and moments of human motions quantitatively. Since walking is a fundamental motion activity of human life, abnormality in walking must be thoroughly investigated. The ZMJ is certainly a model of a diseased joint, but zero moment at a joint would be an unrealistic model of lost functions at diseased joints. Therefore, three other joint models have been proposed: a passive element joint (PEJ) which has a spring at the ZMJ; a constrained range joint (CRJ), the motion of which remains within some constrained relative angle; a partial moment joint (PMJ) which can produce a partial amount of the moment, less than that produced during normal walking. These three joint models for diseased joints e!ectively reduce the excessive compensatory actions incurred by the lost function at the ZMJ model. The PEJ will be a useful method for reducing compensatory actions due to its simple mechanisms. In fact, springs are commonly used in braces for patients with insu$cient muscle powers. The PMJ that can generate a limited moment may correspond to weakened muscles around the joint; as remaining sound leg joints compensate for the loss of function incurred by the PMJ. Usually the region of motion of diseased joints becomes smaller than that of the normal joints. The CRJ model will correspond to such a case. A possible, more realistic, model for a diseased joint will be a combined model of the CRJ and the PMJ, the (CR#PM)J. This model corresponds to a limited region of motion and weakened muscles around the joint. Gait patterns seen in infants and the aged who have inferior muscles have been simulated to show their e!ectiveness in the (CR#PM)J walking, as seen in Fig. 4 and Table 2. Aids shown in Fig. 2 have been analyzed by simulation regarding their usefulness. A cane is used to enlarge the supporting area in a simultaneous support by the diseased leg and the cane (Fig. 2A). An ischial weightbearing brace is used to reduce the load acting at

F !6 F !7 F !8

!0.08 0.04 0.10

F 6 F 7 F 8

F 6 F 7 F 8 0.0 0.0 0.1

0.0 0.0 0.1 Max. Min. Ave.

Max. Min Ave.

Max. Min. Ave.

Max. Min. Ave.

Max. Min. Ave.

> X 2.1 8.5 !1.4 !8.5 0.7 0.0

The aged d

> X > 2.1 26.9 2.3 !1.4 !17.3 !2.2 0.7 5.0 0.7

The aged b

!3.6 2.4 !3.6 21.1 !3.2 !9.8 !11.6 !9.8 !24.2 !9.1 !6.6 !1.7 !6.6 !0.8 !6.4

9.4 !0.1 4.9

!3.1 5.0 !3.1 26.3 !2.6 !9.3 !10.5 !9.3 !23.6 !8.6 !6.0 !3.0 !6.0 1.4 !5.8

10.4 !7.7 6.6 !7.7 25.1 !8.3 0.4 !12.9 !10.5 !12.9 !24.2 !13.0 5.5 !10.5 !1.9 !10.5 0.6 !10.6

7.0 !0.7 3.2

14.3 !8.4 10.5 !8.4 28.9 !8.9 4.1 !13.3 !6.7 !13.3 !20.4 !13.7 9.4 !11.1 2.0 !11.1 4.4 !11.2

X 4.8 !4.8 0.0

The aged a

Angular displacement of upper torso (deg)

A K H W A K H W A K H W A K H W A K H W

X 0.82 0.89 0.93 0.79 5.16 0.48 0.40 1.15 6.80 0.37 0.40 1.12 4.71 0.59 0.40 1.03 4.89 0.61 0.40 1.04 > 0.81 0.75 0.80 0.37 0.79 0.36 0.40 0.86 0.85 0.32 0.40 0.82 0.80 0.37 0.40 0.86 0.86 0.34 0.40 0.82

The aged a

Moment (-)

X 1.30 0.78 0.76 1.25 1.48 0.42 0.40 1.27 3.15 0.46 0.40 1.29 1.78 0.52 0.40 1.27 4.16 0.76 0.40 1.30 > 0.81 0.75 0.80 0.73 0.79 0.36 0.40 0.86 0.85 0.32 0.40 0.82 0.80 0.37 0.40 0.86 0.86 0.34 0.40 0.82

The aged b X 1.24 0.37 0.37 1.79 1.02 0.36 0.39 1.56 0.99 0.28 0.39 1.53 1.66 0.46 0.39 1.46 2.58 0.48 0.39 1.51 > 0.81 0.75 0.70 0.75 0.79 0.36 0.40 0.82 0.85 0.33 0.40 0.77 0.79 0.37 0.40 0.81 0.86 0.34 0.40 0.77

The aged d

1.16

1.19

1.13

1.14

X 1.00

2.13

2.11

1.46

1.34

> 0.00

The aged a

1.16

1.19

1.13

1.14

X 1.00

1.42

1.39

0.95

0.51

> 0.00

The aged b

Locus of ZMP (-)

1.16

1.18

1.14

1.16

X 1.00

0.48

0.43

0.24

0.06

> 0.00

The aged d

The moment and the locus values are represented in the same manner as in Table 1. The (CR#PM)J is assumed at the hip, which is the most severe case. At this (CR#PM)J, the moment stays less than 40 % of the maximum moment of normal walking, and the relative angle stays within 403. The walking conditions of the aged a, b, d are as follows: for the aged a, the < and S are changed into 0.8 m/s and 0.39 m, respectively, in normal 6 walking conditions; for the aged b, a parallel stance of both feet in the conditions of the aged a is changed into an open stance. For the aged d, a lateral hip movement of the support leg in the aged b is changed into inward movement, and a ratio of the #oor reaction forces to respective feet during the double support phase in the aged b is changed in each force component. This ratio is de"ned so that the normal leg pushes o! the walking model strongly at toe o! and the diseased leg is put on the ground with some vertical force at heel-on (see the appendix). NJ means that every joint is normal.

(CR#PM)J walking with brace and cane

(CR#PM)J walking with brace

(CR#PM)J walking with cane

!0.08 !0.04 0.10

X, >, Z

Component NJ walking

F !6 F !7 F !8

The aged a,b,d

Walking condition

(CR#PM)J walking

Cane/brace force (-)

Characteristics

Table 2 Dynamic e!ects of walking aids in di!erent conditions on gait pattern

1412 Y. Tagawa et al. / Journal of Biomechanics 33 (2000) 1405}1414

Y. Tagawa et al. / Journal of Biomechanics 33 (2000) 1405}1414

1413

Fig. 5. E!ect of walking aid (A) cane or (B) brace on load at a diseased joint. A diseased joint is assumed at the right hip. The vertical load acting at the hip is compared by using the abnormal walking model with the (CR#PM)J. Walking conditions of <"0.8 m/s, S "0.39 m, and an open stance are 6 taken instead of the conditions in Fig. 3.

a damaged hip joint by partly distributing the load to the ischial seat of the brace (Fig. 2B). Dynamic e!ects of a cane and a brace on abnormal walking have previously been examined experimentally. Brand and Crowninshield (1980), Shimada et al. (1988), and Shimada and Arai (1990) estimate a force acting at a diseased hip joint in level walking with a cane. Davy et al. (1988) and Bergman et al. (1989) directly measure a joint force using a prosthesis "tted with sensors. Brand and Crowninshield (1980), Shimada et al. (1988), Shimada and Arai (1990), and Bergman et al. (1989) "nd that the cane contributes to the reduction of the force at the hip. Ducroquet (1973) observes that people with severely diseased joints sometimes use a walker or a cane which enlarges a supporting area, consequently adjusting the posture of the upper torso to be upright. As for a brace, Shiba et al. (1998) experimentally show an advantage of a new ischial weight-bearing type for conservative management of hip disorders. The new brace is designed in a manner that reduces the frontal component of moment about the hip joint. In walking tests, the maximum load taken up by the brace is 36.9% of the ground reaction force in the late stance phase. Also, the integrated electromyogram of abductor muscles while using the brace is reduced by 32.6% during the whole stance phase. Despite their long history of use, there are no standard approaches for making general quantitative evaluations of walking aids in advance of their actual employment. By using the abnormal walking model, e!ectiveness of the cane and the brace on walking characteristics was investigated by computer simulation in this paper. This kind of simulation can provide predictive information about the e!ectiveness of walking aids. The e!ectiveness of a cane or brace was simulated quantitatively by using the (CR#PM)J model with the open stance. There are no signi"cant di!erences between the e!ects of these two aids on walking characteristics.

However, the e!ect of the cane is more variable than that of the brace because e!ectiveness of the cane is more sensitive to the walking conditions. The walking aids used in our model decrease the Z (vertical) component of the load at the diseased hip joint directly. The relief of the load is favorable for patients having severe pain. The aids may enable patients and the aged to walk more smoothly, with greater stability and speed. It may also enable them to walk with less pain and fatigue.

5. Conclusion The compensatory actions due to lost function at a diseased joint were examined quantitatively by three-joint models for diseased joints. The e!ectiveness of walking aids, popular in the rehabilitation "eld, on abnormal walking was also examined. The force and moment induced by the walking aids could be formulated in the dynamic equation of the model of abnormal walking. Walking patterns similar to those of persons with inferior muscles were utilized in our attempt to reduce the compensations in abnormal walking. The walking aids in abnormal walking could not only reduce the force at the damaged joint, but also adjust the posture of the upper torso to be upright. The abnormal walking with the (CR#PM)J at the hip may be allowable under the walking pattern of the aged b or aged d with walking aids.

Appendix Ratios of distribution of #oor reaction forces to both feet during the double support phase, R (t) and P R (t) to right and left feet, respectively, are de"ned as J follows:

1414

Y. Tagawa et al. / Journal of Biomechanics 33 (2000) 1405}1414

distribution functions of time for the aged a, b; 0)t)¹

" R (t)"t/¹ , R (t)"1!t/¹ P " J " ¹ )t)¹/2 " R (t)"1, R (t)"0 P J ¹/2)t)¹/2#¹ " R (t)"1!(t!¹/2)/¹ , R (t)"(t!¹/2)/¹ P " J " ¹/2#¹ )t)¹ " R (t)"0, R (t)"1 P J distribution functions of time for the aged d; 0)t)¹ " R (t)"(t/¹ ), R (t)"1!(t/¹ ) 6P " 6J " R (t)"t/¹ , R (t)"1!t/¹ 7P " 7J " R (t)"0.2#0.8t/¹ , R (t)"0.8(1!t/¹ ) 8P " 8J " ¹ )t)¹/2 " R (t)"1, R (t)"0 P J ¹/2)t)¹/2#¹ " R (t)"1!(t!¹/2)/¹ , R (t)"(t!¹/2)/¹ 6P " 6J " R (t)"1!(t!¹/2)/¹ , R (t)"(t!¹/2)/¹ 7P " 7J " R (t)"1!0.8(t!¹/2)/¹ , R (t)"0.8(t!¹/2)/¹ 8P " 8J " ¹/2#¹ )t)¹ " R (t)"0, R (t)"1. P J For example, in normal walking the force vector of #oor reaction acting at the right foot R is expressed as R R by P P using the former case, where R"!F .  References

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