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Generating Pathological Gait Patterns via the Use of Robotic Locomotion Models∗ Anton Ephanov and Yildirim Hurmuzlu Mechanical Engineering Department Southern Methodist University Dallas, TX 75275 February 12, 2001

Abstract In this article we explore the feasibility of modeling normal and pathological human gait using a relatively simple five-element model. We use a robust, nonlinear control scheme to regulate the gait patterns of the model. Simulated gait patterns are generated through the use of five constraint relationships that depend on four gait parameters. Two pathological conditions due to muscle weaknesses were simulated by modifying the control torques at the joints. We demonstrate that the model successfully approximates the qualitative and quantitative dynamic trends that were observed in normal and pathological human locomotion.

1

Introduction

Developing human-like robotic bipedal locomotion systems and mathematical models of such systems has drawn the attention of many researches over the last few decades. A variety of robotic designs and control algorithms ∗

This article will appear in the journal of Technology and Health Care

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of the bipedal walking systems have been proposed and investigated (Yamaguchi 1990, Hurmuzlu 1993a, b, also see references within). Most of the previous work in the field focused on the design and development of models and prototypes of robotic biped walkers (Hirai et al., 1998, Minakata and Y. Hori, 1998). Fewer studies, however, have focused on developing locomotion models to simulate the dynamic characteristics of normal and pathological human gait (Koopman and Grootenboer, 1991). This is mainly due to the overwhelming complexity of the human body. Developing an advanced model of the musculoskeletal system involves the following six tasks (Zajac and Winters, 1990): 1. modeling body segments and joint kinematics 2. deriving dynamical equations of motion 3. modeling passive-tissue joint mechanics 4. specifying geometric joint transformations 5. modeling musculotendon force generation 6. modeling the neuromotor central nervous system (CNS) circuitry to control muscle excitation The musculoskeletal system control by CNS remains a subject of active research (Loeb and Levine, 1990). The control strategies that are used by humans in coordinating their moves remain to be mostly unknown (Winter et al., 1990a). Most approaches use dynamic optimization methods that mathematically optimize the solution according to some cost function (Hardt, 1978). One of the disadvantages of this method is that physiological conditions such as fatigue or metabolic costs are difficult to formulate. Another approach to modeling CNS commands is based on proportional and derivative (PD) type of control (Iqbal et al., 1993). From engineering point of view, the control system for the musculoskeletal model should be sufficiently robust to address nonlinearities, eliminate disturbances due to various external factors, and achieve acceptable control in the presence of modeling errors. Human’s ability to preserve or adjust their natural walking cycle to various surface and load conditions would be the ultimate goal for a perfect robust control system (Crago et al., 1990). It should be noted that control strategies 2

based on optimization methods are usually deficient in terms of robustness with respect to model uncertainty. In the present paper we seek to demonstrate that a relatively simple bipedal model may be able to capture some fundamental aspects of the dynamics of normal and pathological human gait. We use a five-element bipedal model that is driven by a robust, nonlinear control scheme. The controller is based on five dynamic constraint relationships that completely specify the gait pattern of the biped in terms of four gait parameters. These parameters are then used to create a “normal” gait pattern that has similar characteristics to that of the human gait. Subsequently, we modify the control scheme to simulate two well known “pathological” conditions of human gait. Humans are very inventive in their ability to adjust their gait in the presence of pathological conditions in order to preserve their locomotive functionality. They use, mostly on a subconscious level, various dynamical effects such as swinging a hip excessively to compensate for the effect of a weak knee joint. Yet, the overall locomotive functionality is maintained, but the gait pattern somehow deviates from normal (Winter et al., 1990b). The model presented here emulates the adjustments that individuals with pathologies use when they ambulate. The trends in the resulting gait patterns of the model is contrasted with the ones that arise in pathological human gait. The main objective of the present article is to assess the capability of a relatively simple model in capturing dynamics of pathological human gait by comparing the gait patterns of the model with the ones reported in the clinical literature.

2

The Five Element Model

In this section we define the main features of the five-link bipedal shown in Fig. (1). We consider gaits of the biped that include the single support phase only (i.e. only one of the lower limbs is on the ground surface at any given time). Although, we allow moments at the ankles, we neglect feet structures and assume point contacts between the lower limbs and the ground (see Table (1) for various dimensions, masses, and moments of inertia).

3

Mass mi (kg) 5 5 20 5 5

Member 1 2 3 4 5

Length li (m) 0.5 0.5 1.0 0.5 0.5

Mom. of Inertia Ji (kg.m2 ) 0.5 0.5 4.0 0.5 0.5

Table 1: Dimensions, masses and moments of inertia

Y

q3 q2

l3 = l / 2

l3 = l / 2

l4 = l

l2 = l l1 = l

− q1

q4

l5 = l

X

q5 Figure 1: Five Element Biped. The motion of a biped includes two stages. The first stage is the continuous forward motion during which the biped is pivoted on one limb (stance limb) and the other limb (swing limb) is moving in the forward direction. The second stage arises when the swing limb (leading limb at contact) comes 4

into a sudden contact with the walking surface. The pivot point transfers to the tip of the leading limb where contact occurs and the stance limb (trailing limb at contact) is lifted off. This event causes discontinuities in the generalized positions and velocities. The discontinuities in the generalized velocities is due to the impact phenomenon. Whereas, the transfer of pivot causes additional discontinuities in the mathematical model due to switching between the swing and stance sides from bipeds point of view. We assume that the impact is perfectly plastic (i.e. the post impact, normal velocity of the tip of the limb that contacts the walking surface is zero). We also assume that there is sufficient friction between the feet and the ground surface to prevent slippage. Equations of motion for this system my be found in Chang and Hurmuzlu (1993). During the continuous phase of the motion the equations of motion are given by the following general form: M(q)¨ q + C(q, q) ˙ q˙ + G(q) = T

(1)

where M(q) is the 5 x 5 symmetric, positive definite inertia matrix, C(q, q) ˙ q˙ is the 5 x 1 vector of centripetal and coriolis forces (C(q, q) ˙ is an 5 x 5 matrix), G(q) is the 5 x 1 vector representing gravitational forces, and T is the 5 x 1 vector of generalized forces applied at each joint. The 5-dimensional vectors q, q˙ and q ¨ represent the joint angles, angular velocities and accelerations respectively. Equations representing the impact and switching events can be found in Hurmuzlu (1993).

2.1

Motion Constraints

Now, we present five constraint equations that characterize the locomotion of the five-link biped in terms of four parameters, step length SL , overall progression speed Vp , maximum clearance of the tip of the swing limb Hm , and the stance knee bias σ. These relations will be used in conjunction with a variable structure control scheme (see section 3) to generate the desired gait patterns. The constraints can be enumerated as follows: 1. Erect body posture: One of the most important aspects of bipedal locomotion is that the biped should maintain an erect posture during locomotion. This requirement can be achieved for the present system when the net rotation of the upper body is kept to be zero at all times. 5

The condition that yields erect body posture can be written as q1 + q2 + q3 = 0

(2)

2. Overall progression speed: We define the “overall progression speed” as the linear velocity of the center of mass of the upper body in the positive x direction (see Fig. (1)). A steady progression speed can be maintained by letting (3) x˙ 3 = Vp where x˙ 3 is the velocity of the center of mass in the x direction and Vp is the desired progression speed. 3. Trajectory of the tip of the swing limb during the single support phase: The motion of the tip of the swing limb (in practice the swing foot) during the single support phase is the dominant factor in the trajectory planning process of a bipedal machine. One can generate various locomotion tasks such as stair climbing, walking on a flat or inclined surface by simply specifying the spatial trajectory of the tip of the swing limb. Accordingly, in the present study we let yT = c1 x2T + c2 xT + c3 .

(4)

where xT and yT are the x and y coordinates of the tip and the constants c1 , c2 and c3 are parameters that are yet to be determined in terms of step length and the maximum tip clearance. The quadratic form is chosen here because it is the simplest form that allows the selection of step length and step height independently. For walking on a level surface one can set 4Hm (5) c1 = − 2 SL c2 = 0 and

(6)

c3 = Hm .

(7)

4. Bias of the stance knee: We prevent the collapse of the stance knee during the single support phase by enforcing constant knee bias during this stage. The corresponding constraint is: q2 − σ = 0 6

(8)

5. Coordination of the motion of the limbs: A fifth constraint is necessary to prescribe the direction of the motion of the tip of the swing limb. Furthermore, the motion of the two lower limbs should be coordinated with each other. Therefore, we set xT = 2x3

(9)

where x3 is the x coordinate of the center of mass of the upper body. This relation also implies that the tip moves twice as fast as the center of mass of the upper body in the positive x direction. Thus, we also ensure that the swing limb arrives the contact point when the upper body is properly centered with respect to the two lower limbs. Having specified the form of the individual constraints, the vector of modified tracking errors (see Chang and Hurmuzlu, 1993) can be written as, 







˜ em em      −−  =  −−  − e˜l el            

(1 + α1 t)φ1 (q0 ) + tφ˙ 1 (q0 ) (1 + α2 t) [φ2 (q0 ) − Hm ] + tφ˙ 2 (q0 ) (1 + α3 t) [ φ3 (q0 ) − σ] + tφ˙ 3 (q0 ) (1 + α4 t)φ4 (q0 ) + tφ˙ 4 (q0 ) − − − − − − − − − − − − − − −− φ∗1 (q0 )q˙ 0 − Vp 







      −At e    



em ym γ m (t)      ≡  −−  −  −−  −  −−   el yl γ l (t) where, A = diag {α1 , . . . , α5 }.

3

Control system design

The control system for the biped should be capable of implementing the constraints described above. The controller should account for the nonlinearities as well as parameter uncertainties and unmodeled dynamics. In this 7

section we first present the constraint equations that reflect trajectory tracking errors. Subsequently, we propose a variable structure control scheme that ensures minimization of the respective tracking errors.

3.1

Sliding Mode Equations

We formulate the sliding mode equations in terms of the modified error vector presented in the previous section (Eq.(10 )) as follows: 

˜ s=

Γm (q) Γl (q)

−1 

˜ e˙ m + Λ˜ em e˜l



(10)

where, Λ = diag(λ1 , . . . , λ5 ) is a positive definite matrix, the matrix Γm (q) is the Jacobian of φ(q) = {φ1 , φ2 , φ3 , φ4 }, and Γl (q) is given by Γl (q) = φ∗ (q)

(11)

The inverse in Eq. (10) exists everywhere except at the singular configurations of the biped. Using Eqs. (10) and (11) in (10) yields ˜ s = q˙ − q˙ r

(12)

where the reference velocity vector q˙ r is given by 

q˙ r =

Γm (q) Γl (q)

−1 

em y ¯˙ m (t) + γ˙ m (t) − Λ˜ y¯l (t) + γ l (t)



(13)

Differentiating Eq. (12) with respect to time yields ˜ s˙ = q ¨−q ¨r

(14)

where the reference acceleration q ¨r is given by 

q ¨r =

Γm (q) Γl (q)

−1 

¨ −Γ∗m (q, q) ˙ q˙ r + y ¯m (t) + γ ¨ m (t) − Λ˜ e˙ m ∗ −Γl (q, q) ˙ q˙ r + y¯˙ l (t) + γ˙ l (t)



(15)

and the components of the 4 × 5 and 1 × 5 dimensional matrices Γ∗m (q, q) ˙ ˙ can be written as and Γ∗l (q, q) Γ∗mi,j

n ∂ 2 φi ∂φ∗i ∗ = q˙ k and Γli = q˙ k k=1 ∂qj ∂qk k=1 ∂qk n

8

(16)

Having specified the sliding domain, we use a variable structure control algorithm that preserves the sliding motion about the discontinuity surface (see Chang and Hurmuzlu, 1993).

4

Gait Patterns

The main objective of the present article is to explore the potential of using the present model as a platform to analyze normal and pathological human gait. Human walking is a complex activity that involves coordination of the two lower limbs and the body mass. We accept the fact that the present model has significantly less complex structure compared to that of the human body. Nevertheless, we postulate that the inherent dynamical properties of the two systems should bear common characteristics that qualitatively and quantitatively link their gait patterns. We contrast the gait patterns of the model against human walk through the use of kinematic and kinetic joint profiles. First we outline the similarities and elaborate on the differences between the normal gait patterns of the two systems. In the case of pathological gait, functional effects of disability will be interpreted, for each system (human and model), based on the analysis of the gait patterns and their deviations from their respective norms. We particularly focus on documented abnormalities that occur during the specific phases of two gait pathologies. The gait cycle (interval between two sequential initial floor contacts by the same limb) is divided into two phases: stance and swing. Walking functionality is provided by the motion of the supporting limb during the stance with respect to the body and selective advancement of the other limb in swing. Therefore, hip, knee, and ankle of each limb are involved in a series of motion patterns. Each pattern of motion is associated with a particular phase of the gait cycle. The generic terminology developed by the Rancho Los Amigos gait analysis committee identifies total of eight gait phases: initial contact (IC), loading response (LR), mid stance (MSt), terminal stance (TSt), pre swing (PSw), initial swing (ISw), mid swing (MSw), and terminal swing (TSw) (Pathokinesiology Department, Rancho Los Amigos Medical Center, 1989). Functional significance of individual joint motions is analyzed in terms of the gait phases. Additionally, the gait phases are of great practical im9

portance for correlating individual joint motions into patterns of total limb function.

4.1

Normal Patterns

Normal gait patterns of the five link biped model were generated by running computer simulations with the following values of the gait parameters: σ = 0.2, step length SL = 0.8m, foot clearance Hm = 0.05m, and progression velocity of Vp = 1.4m/sec. The values of the parameters were chosen such that they are within the ranges observed for average normal human gait (Whittle, 1991). Averaged sagittal joint profiles of normal human gait were acquired using twenty normal subjects (see Hurmuzlu et al, 1994, Hurmuzlu and Basdogan, 1994). Figures (2a), (2b), and (2c) depict the average joint kinematics of normal human subjects as well as what can be considered as normal gait pattern of the five link biped model. As shown on Fig. (2a), the best agreement among the profiles was observed at the hip joint. The model response bears a strong resemblance to the human data both in trends and values for all phases of the gait cycle. Figure (2b) depicts joint kinematics of the knee joint. Although the basic trends of motion are similar, we observed a relative shift in the data in terms of the phases of the gait cycle. Joint kinematics of the knee is qualitatively the same during the initial contact, loading response, and mid stance. During the remaining phases of the gait cycle, however, a delay in knee flexion can be clearly observed. Specifically, the model does not flex the knee during the late phases of the stance. As a consequence, the knee flexion process during the swing is also delayed. This fact can be attributed to the combined effect of the fourth motion constraint (the “sigma” constraint) and the absence of the toe-foot complex in the model. The motion constraint dictates the control system to keep the knee flexed at a fixed angle throughout the stance. The humans, however, flex their knee during the terminal stance and pre swing due to heel rise and motion of the toe-foot complex. The ankle profiles of the two systems differed significantly, as one might expect due to the lack of toe-foot complex in our model. The motion of the toe-foot complex in stance in humans involves three phases: heel strike-foot flat, foot flat, and heel off-toe off. Therefore, the kinematics of the ankle joint of the model can only be compared with the kinematics of ankle of the humans during the foot-flat period, that is when the whole foot is in contact 10

10 0 -10 -20 60 40 20 0

Angular Position (rad)

-20

Angular Position (rad)

Angular Position (rad)

20

human (mean) human (deviation) five-link model

30 20 10 0 -10 -20 0 IC

20 LR

40

60

Mst

TSt

PSw

80 ISw

100 MSw TSw

Percent Gait Cycle

Figure 2: Normal Gait Patterns: a) Ankle, b) Knee, and c) Hip with the ground. This structural dissimilarity between the five-link model and the human foot is mainly responsible for the significant discrepancies in the joint kinematics of the respective ankle joints (see Fig. (2c)).

4.2

Pathological Patterns

The common causes of gait pathologies fall into four functional categories: deformity, muscle weakness, impaired control, and pain (Perry, 1992). Gait anomalies can be characterized in terms of one or more of the above causes and the affected muscle groups. In the present study we only consider patho-

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logical conditions that are caused by muscle weaknesses in the knee, and ankle joints. In human anatomy, a leg muscle can be characterized as a flexor or extensor, depending on its role in the limb motion. For example, quadriceps is an extensor muscle because it is responsible for the knee extension, while biceps femoris is a knee flexor. Accordingly, in the model, the control torque at each joint can be treated as if it is generated by either extensor or flexor muscles. This depends on the sign of the torque and the prevailing gait phase (stance or swing). Table (2) shows the association between the sign of the control torque and the action of extensor/flexor muscle depending on the gait cycle for each joint. Human movement requires coordination of many muscles that are sometimes redundant. The redundancy of the muscle system allows considerable adaptability at the motor control level (Winter et al., 1990b). It should be noted that our model does not incorporate models of muscles and tendons. Instead, we simply relate the computed control moments at a particular joint to the resultant muscular effort of the corresponding muscle or group of muscles that are responsible for the joint motion depending on the sign of the control torque and the gait phase (see Table (2)). Joint Stance Swing Torque hip − + flexion + − extension knee − + flexion + − extension Table 2 Association between the sign of the control torque and the action of extensor/flexor muscle One can model gait anomalies by simulating muscle weaknesses through the modification of control torques. We use two approaches to modify the control torques: The first approach introduces a weakness factor that scales down the torques by a certain percentage. The second approach simulates muscle weakness as a saturation function, where muscle torque is not allowed to exceed some prescribed saturation limit. This is equivalent to introducing a limit on the on the muscle maximum strength capacity.

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4.3

Modeling of Soleus and Hip Extensor Weakness (Myelodyspasia)

Myelodyspasia is characterized by the weakness of Soleus (an ankle extensor muscle) and hip extensors (Perry, 1992). To model the hip extensor weakness, we used 75% weakness factor for control torques of the hip extensors of both legs. Weakness of the Soleus muscle was modeled by assigning a 10% weakness factor to the control torques that correspond to the action of ankle extensors (also for both legs). The values of the four motion constraints, however, were unchanged. In this case, the effect of introducing pathology into dynamical model was observed by comparing the resulting deviations in the profiles of joint kinematics for normal and pathological cases. Next, we correlate our observations with the results of clinical studies (Perry, 1992). In clinical studies it is a common practice to address gait deviations by their effect on joint kinematics during individual phases of the gait cycle. We adopt this practice in evaluating the pathological model response. The results of our computer simulations indicate that during the IC and the LR phases (see Fig. (3a) and (3b)) both the knee and the hip joints are in excessive flexion compared to the normal case. Clinical studies also point out to the markedly flexed posture at the knee (Perry, 1992), with excessive hip flexion during the first two phases. As the motion advances into the next phases (mid stance and terminal stance), the model response exhibits excessive knee flexion and a reduced hip flexion. The clinical studies also report similar trends, but with greater increase in knee and hip flexions (pre swing and mid swing). They attribute this behavior to the necessity of lifting the foot clear of the floor in the presence of weak ankles. Thus, the absence of feet in the model leads to the discrepancies in joint profiles in the latter part of the swing phase. Nevertheless, the overall analysis shows that there is an agreement between the results of our simulation and clinically reported observations.

4.4

Quadriceps Insufficiency (Poliomyelitis)

Poliomyelitis is characterized by partial or complete paralysis of the quadriceps, which is the primary knee extensor muscle. We model the quadriceps weakness by the use of saturation function as described above. The saturation value for the knee extension torque was set at 25N m, which was 13

Angular Position (rad)

40 30 20 10

30

normal myelodysplasia

20 10 0 -10 0 IC

20 LR

40 Mst

60 TSt PSw

80 ISw

Angular Position (rad)

0

-20 100 MSwTSw

Percent Gait Cycle

Figure 3: Myelodyspasia Patterns: a) Knee and b) Hip approximately 16% of the maximum knee extension torque for the normal cycle. As in the previous case, the motion constraints were kept unaltered. The joint kinematics of the simulated gait for the pathological leg is presented in Fig. (4a) and (4b). One of the characteristic gait patterns for polio patients is knee hypertension during stance (Hurmuzlu et al., 1996). However, we did not observe this behavior in the model response. This fact can be explained by the presence of the fourth motion constraint (“sigma” constraint) that forces the control system to keep the knee flexed during the stance. During the swing phase, the joint kinematics of the model exhibits excessive knee and hip flexion. This is in direct agreement with the clinical data, where patients use excessive hip flexion during the swing phase to attain adequate knee flexion. In addition, in both modeled and clinical cases, the knee extension is incomplete during the terminal swing. Another frequent occurrence in Polio gait is the hyperextension of the 14

40 20 0 -20

40

normal poliomyelitis

30 20 10 0

0 IC

20 LR

40 Mst

60 TSt

PSw

80 ISw

Angular Position (rad)

Angular Position (rad)

60

-10 100 MSw TSw

Percent Gait Cycle

Figure 4: Polio Patterns: a) Knee and b) Hip knee joint during the stance phase. We can induce this behavior into the model by choosing a negative stance bias angle during the stance phase (i.e. σ < 0). We performed computer simulations with σ = −0.15. The results of the simulation showed that the five-link model was capable of stable walking for negative values of σ as well. However, there was a significant difference between the two gait patterns (with positive and negative σ values) in terms of control torque profiles for the pathological leg. There was a substantial control torque reduction at the knee joint for the case of negative sigma (see Fig. (5b)). Additionally, the simulation results depicted reduction of the torque at the hip joint as well in the middle of the gait cycle (see Fig. (5a)). Thus, the model requires significantly less control effort when the stance knee is hyperextended. This is an interesting conclusion that can perhaps explain the reason why polio patients hyperextend their knees during the stance phase. On the flip side, this behavior is known to cause damage in the knee joint of individuals who have suffered from Polio. 15

600 Torque (Nm)

400 200 0 -200 -400 -600

0 -100

normal poliomyelitis (σ=0.2) poliomyelitis (σ=−0.15)

0 IC

20 LR

40 Mst

60 TSt PSw

80 ISw

-200

Torque (Nm)

100

-300 100 MSwTSw

Percent Gait Cycle

Figure 5: Polio Torque Patterns: a) Knee and b) Hip

5

Discussion and Conclusion

The main objective of the present article was to demonstrate that one can use a relatively simple model to capture some of the fundamental aspects of normal and pathological human gait. We have demonstrated that, the normal gait patterns of the model agree with the human profiles of the hip joints, and to a great extent at the knee joints. On the flip side, discrepancies in the ankle joints and the knee joint did arise due to lack of toe-foot complex in the model. We further demonstrated that the model qualitatively duplicates the compensatory behavior prevalent in pathological human gait. We have also shown that hyperextending the swing knee in modeled polio gait does lead to significant gains in the control effort, which may provide a good explanation for the similar behavior in Polio gait. Finally we should emphasize that the present investigation is a prelimi-

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nary effort taken in the direction of generating mathematical models to simulate the dynamics of pathological human walk. Such models will be very useful in testing various therapies and evaluate their effect on gait dynamics. We are encouraged by the predictions of a simple, planar five-element model. We are in the process of developing more complex models that can better represent the dynamics of the human system.

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[8] Hurmuzlu, Y., Basdogan, C., and Carollo, J., 1994, “Presenting Joint Kinematics of Human Locomotion Using Phase Plane Portraits and Poincare Maps” , J. Biomechanics, V. 27, No. 12, pp.1495-1499. [9] Hurmuzlu, Y., Basdogan, C., and Stoianovici, D., 1996, “Kinematics and Dynamic Stability of the Locomotion of Post-Polio Patients”, Journal of Biomechanical Engineering, Vol. 118, pp.405-411. [10] Iqbal, K., Hemami, H., and Simon, S. (1993). “Stability and Control of a Frontal Four-Link Biped System”, IEEE Transactions on Biomedical Engineering, VOL. 40, NO. 10, pp.1007-1018. [11] Koopman, B. and Grootenboer, H. (1991). “Numerical Prediction of Walking with a Stiff Knee”, in proceedings of International Society of Biomechanics XIII Congress 1991. [12] Loeb, G. and Levine, W. (1990). “Linking Musculoskeletal Mechanics to Sensorimotor Neurophysiology”, in Multiple Muscle Systems: Biomechanics and Movement Organization, (Edited by Winters, J.M. and Woo, S. L-U.), Chapt. 10, pp. 165-181, Springer-Verlag, New York. [13] H. Minakata and Y. Hori (1998), ”Experimental Study of 3-D Speed variable Biped Walking,” Trans. on IEEJ. Vol. 118-D, No. 1, 10-15. [14] Pathokinesiology Department, Physical Therapy Department, “Observational Gait Analysis Handbook”, Downey, CA, The Professional Staff Association of Rancho Los Amigos Medical Center, 1989. [15] Perry, J. “Gait Analysis. Normal and Pathological Function”, SLACK Incorporated, 1992. [16] Slotine, J.J., and Lee, W., 1991, “Applied Nonlinear Control”, PrenticeHall, NJ. [17] Whittle, W., 1991, Gait Analysis: Heinemann, Oxford.

An Introduction. Butterworth-

[18] Winter, D., Ruder, G., and MacKinnon, C. (1990a). “Control of Balance of Upper Body During Gait”, in Multiple Muscle Systems: Biomechanics and Movement Organization, (Edited by Winters, J.M. and Woo, S. LU.), Chapt. 33, pp. 534-541, Springer-Verlag, New York. 18

[19] Winter, D., Olney, S., Conrad, J., White, S., Ounpuu, S., and Gage, G. (1990b). “Adaptability of Motor Patterns in Pathological Gait”, in Multiple Muscle Systems: Biomechanics and Movement Organization, (Edited by Winters, J.M. and Woo, S. L-U.), Chapt. 44, pp. 680-693, Springer-Verlag, New York. [20] Yamaguchi, G. (1990). “Performing Whole-Body Simulations of Gait with 3-D, Dynamic Musculoskeletal Models”, in Multiple Muscle Systems: Biomechanics and Movement Organization, (Edited by Winters, J.M. and Woo, S. L-U.), Chapt. 43, pp. 663-679, Springer-Verlag, New York. [21] Zajac, F. and Winters, J. (1990). “Modeling Musculoskeletal Movement Systems: Joint and Body Segmental Dynamics, Musculoskeletal Actuation, and Neuromuscular Control”, in Multiple Muscle Systems: Biomechanics and Movement Organization, (Edited by Winters, J.M. and Woo, S. L-U.), Chapt. 8, pp. 121-148, Springer-Verlag, New York.

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