Model and measurement studies on stages of prosthetic gait.   

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Chapter 4

Gait termination
Controlling horizontal deceleration during gait termination in transfemoral amputees: Measurements and simulations
Helco G. van Keeken, Aline H. Vrieling, At L. Hof, Klaas Postema, Bert Otten,
In Med Eng Phys, 2011. [bib] [pdf] [doi]  
 
NOTICE: this is the author's version of a work that was accepted for publication in J Med Eng Phys. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication.
Narrow range of options for the dynamics variables to achieve a target center of mass velocity. The colored markers represent the velocity range. The black dots indicate the target velocity.

 

 

Controlling horizontal deceleration during gait termination in transfemoral amputees; measurements and simulations.

In this study we investigated how leading limb angles combined with active ankle moments of a sound ankle or passive stiffness of a prosthetic ankle, influence the center of mass velocity during the single limb support phase in gait termination. Also, we studied how the trailing limb velocity influences the center of mass velocity during this phase. We analyzed force plate data from a group of experienced transfermoral amputee subjects using a prosthetic limb, and the outcome from a two dimensional mathematical forward dynamics model. We found that when leading with the sound limb, the subjects came almost to a full stop in the single limb support phase, without the use of the prosthetic limb. When leading with the prosthetic limb, the center of mass deceleration was less in a relatively short single limb support phase, with a fast forward swing of the trailing sound limb. Slowing down the heavier trailing sound limb, compared to the prosthetic limb, results in a relatively larger braking force at the end of the swing phase. The simulations showed that only narrow ranges of leading limb angle and ankle moments could be used to achieve the same center of mass velocities with the mathematical model as the average start and end velocities of the prosthetic limb user. We conclude that users of prosthetic limbs have a narrow range of options for the dynamics variables to achieve a target center of mass velocity. The lack of active control in the passive prosthetic ankle prevents the transfermoral amputee subjects from producing sufficient braking force when terminating gait with the prosthetic limb leading, forcing the subjects to use both limbs as a functional unit, in which the sound limb is mostly responsible for the gait termination.
 
transfemoral prosthetic limb, gait termination, ground reaction force, inverse dynamics, forward dynamics

Introduction

Successful gait termination with a transfemoral (TF) prosthetic limb requires indirect control over a device with limited degrees of freedom. With fewer muscles compared to able-bodied individuals, TF amputees have to be able to control the prosthetic limb by making use of the limb properties and the environment in which the limb is used.
Gait termination studies in able-bodied subjects show that several strategies are used to reduce the forward motion of the center of mass (CoM). By placing the leading limb on the ground in front of the body, a center of pressure (CoP) under the foot is formed. The ground reaction force (GRF) originating from this CoP is used to decelerate the CoM. Also, by decreasing the push-off with the trailing limb the forward motion is reduced. 1; 2; 3; 4. During gait termination, the leading limb is for the most part responsible for the production of the necessary braking force 3.
Studies in prosthetic limb users show that the motion of the CoP is directly related to the stiffness of the prosthetic ankle, the orientation of the limb, the position of the foot and shaft and the type of foot that is used 5; 6. As a result of the absence of active control in the ankle joint, a prosthetic limb produces less braking ground reaction force under the leading prosthetic limb in anterior-posterior direction, compared to the force under the sound limb in a sound limb leading situation 7. To compensate for the limitations in the prosthetic foot and ankle, the leading prosthetic limb can be placed under a different angle with the vertical at initial contact, to change the position of the CoP, and therefore influence the CoM velocity. (The leading limb angle is defined as the angle between that limb and the vertical.) When making a larger leading limb angle, which results in a more forward positioned foot, the CoM decelerates more as a result of the more backward orientation of the GRF, compared to when making a smaller leading limb angle.

In the current study, we investigated how combinations of leading limb angles and internal active ankle moments of the sound ankle or passive stiffness of the prosthetic ankle influence the CoM velocity during the single limb support phase of the gait termination. Also, we considered if the trailing limb motion influences the CoM velocity during gait termination. Similar to gait initiation 8, we expect that when TF amputee subjects terminate gait with their prosthetic limb leading, the duration of the single limb support phase on the prosthetic limb is shorter compared to a sound limb leading condition. This strategy enables the TF amputee subjects to make use of the active muscle control possibilities in the trailing sound limb as quickly as possible. However, it may have its effect on the CoM velocity, as the fast forward acceleration and deceleration of the trailing sound limb toward the final stance position influences the direction of the GRF.

In our study, we used a two dimensional mathematical forward dynamics model based on Newton Euler and constraint equations and the gait termination data from a group of experienced TF amputee subjects using a prosthetic limb 7. We divided the gait termination process in a single and double limb support phase, in both a prosthetic limb leading condition and a sound limb leading condition. We compared the CoP position, the GRF and the CoM Velocity of the TF amputee subjects during the single limb support phase with the outcome of the forward dynamics model. The model consisted of a leading limb, either a sound limb or prosthetic limb, with an ankle joint, a trunk and a trailing limb. The two limbs were connected to the trunk via hip joints. The model enabled us to inspect systematically the whole range of possible combinations of leading limb angles, active ankle moments of a sound ankle or passive stiffness of a prosthetic ankle, and trailing limb accelerations. Testing for these parameters in human subjects, without the interference of compensation strategies was not feasible; therefore it was decided to approach this problem in a theoretical way. The outcome and insights we gained from this study can be used to understand how TF amputees can compensate for the limitations in the active control of the CoP position during gait termination by using different leading limb angles.

Methods

Subjects

For this study, TF amputee subjects were recruited by a prosthetics workshop with clients in the three northern provinces of the Netherlands. Inclusion criteria for the amputee group were having a TF amputation for at least one year, daily use of a prosthetic limb and the ability to walk with the prosthesis more than 50m without walking aids. Amputees were excluded if they had any medical condition affecting their mobility or balance, like neurological, orthopedic or rheumatic disorders, otitis media, cognitive problems, severely impaired vision, reduced sensibility of the sound limb, or the use of antipsychotic drugs, antidepressants or tranquilizers. Furthermore, amputee subjects with pain or wounds to the stump, and fitting problems of the prosthesis were excluded.
The study group consisted of seven TF amputee subjects (6/1 (m/f, absolute numbers); mean 44.0 y (SD 14.1); mean 81.4 kg (SD 12.4); mean 1.83 m (SD 0.06); time since amputation mean 210.7 months (SD 158.1); side of amputation 5/2 (l/r, absolute numbers)). The TF amputee subjects used different types of prosthetic knees, all supplied with a free movable knee: Teh Lin (3), computerized C-leg (1), Total knee (1), Otto Bock 3R60 (1) and Proteval (1). The following prosthetic feet were used by the subjects: C-walk (2), dynamic SACH (2) and Endolite (3). The subjects walked with their own shoes.

Apparatus

The measurements were performed at the Motion Analysis Laboratory of the Center for Rehabilitation of the University Medical Center Groningen. A force plate of 0.40 x 0.60 m was used to measure the three dimensional forces and moments, which were used to calculate the position of the CoP in the horizontal plane. The force plate data were sampled at 100 Hz. The gait termination was recorded with two video cameras: one recording the coronal plane, the other the sagittal plane. The video frame rate was 25 Hz. Recording, synchronization and analysis of the force measurements and video registration was done with a custom-developed Gait Analysis System (GAS) based on LabView software. Synchronization was done by the hardware, using the same clock pulse.

Procedure

The subjects were instructed to terminate walking by stepping with the leading limb on the force plate, followed by placing the trailing limb next to the leading limb. The subjects performed repeated runs, until the prosthetic limb and the sound limb were both used twice as leading limb in an adequate manner. The subjects performed at least three steps prior to the gait termination step, to achieve steady-state gait 9; 10; 11. Adjustment of the step length in order to hit the force plate was avoided by practicing the task in advance to select an appropriate distance from the starting point to the force plate. The subjects were instructed to look at the end of the walkway instead of at the force plate.
The data were obtained in vertical (GRFz; CoPz), anterior-posterior (GRFy (breaking force); CoPy) and mediolateral (GRFx; CoPx) direction. The CoM (CoMz; CoMy; CoMx) start velocity (CoM Velocity) and duration of the single limb support phase (leading limb foot on the ground) and the double limb support phase were calculated from the force plate data (figure 2).
The gait termination process was divided in two phases. The start of the single limb support phase was defined as the moment the leading limb was on the ground, the trailing foot was off the ground and the maximal GRFz was reached. The video images were used to determine if the trailing foot was off the ground. The moment of the transition from the single limb support phase to the double limb support phase was defined as the moment the CoPx reached the highest velocity when moving from under the single limb stance foot a point between the two feet. The end of the double limb support phase was defined as the moment the GRFy was reduced to zero.

Outcome Parameters

The force plate data of the group of TF amputee subjects were used to calculate the final outcome parameters. The CoM acceleration vector was directly calculated from Newton’s second law: m *⃗a = ⃗W + ⃗F, where m is the subject’s mass, ⃗a is the CoM acceleration, ⃗W is the subject’s gravity vector and ⃗F is the GRF vector. The instantaneous CoM velocity was obtained by integration of the acceleration.
The CoM velocity at the start of the single limb support phase (CoM Velocity1ft) and the double limb support phase (CoM Velocity2ft), the duration of gait termination process, the duration of the two phases (Duration1ft, Duration2ft) and the impulses used in the two phases (Impulse1ft, Impulse2ft) were used to determine the global differences in the sound limb and prosthetic limb leading conditions. The trajectories of the CoMy velocity, the GRF and the CoPy position during the one foot phase were compared to the outcome of simulations with the mathematical model.

Statistical Analysis

For each subject, individual means of the outcome parameters over the two trials for the leading and the two trials for the trailing (prosthetic and sound) limb condition were calculated. A paired two sample student’s t-test was used to determine the global differences between the two conditions. The level of significance was set to p 0.05.

Mathematical Model

A two dimensional forward dynamics model was used to inspect the whole range of possible combinations, that may enable TF amputee subjects to achieve the equal CoM velocities at the end of the single limb support phase during gait termination when varying the leading limb angle and the internal active ankle moment of the sound ankle or passive stiffness of the prosthetic ankle (figure 1). The four elements model is based on Newton Euler constrained equations with forward dynamics 12. We used Euler integration for the simulation steps (Δt = 0.0001 s). Some additions were made to simulate spring and damper limited joints and contact points with the external world.
The model consisted of a leading foot (length: 0.3 m, mass: 2kg; the proximal and distal end point can form contact points with the external world), with an ankle joint, and limb (length: 1.0 m, mass: 15 kg or 5 kg (respectively the sound limb and prosthetic limb)), a trunk (length: 1.0 m, mass: 48 kg) and a trailing limb (length: 1.0 m, mass: 15 kg or 5 kg) . The two limbs were connected to the trunk via hip joints. The limbs could be set with the inertial properties of a prosthetic limb or a sound limb. The elements were modeled as slender rods. Joints were simulated as frictionless pin point joints. Elements were constrained by joint forces based on equating the acceleration of element endpoints. Internal and external forces were passed on via the joint elements. Hip and ankle muscles were simulated as torque engines in the joints.
Input variables in the model were (1) the leading limb angle at initial contact, (2) the trailing limb motion during single limb support phase, and (3) the ankle stiffness or internal active moment of force of the ankle of respectively the leading prosthetic limb or the leading sound limb. The initial horizontal velocity of the CoM at the start of the gait termination was set to 0.7 m/s. Mixed dynamics, forward dynamics with some constraint or prescribed movement 12, were used to constrain the trunk element to the vertical of the external world by using hip moments. The known angular acceleration of the trunk was set to zero, keeping the trunk in the upright position. The velocity of the CoM was the result of the forces acting on the model, the inertial properties of the model and the internal moments produced by the model. A minimum jerk bell-shaped velocity curve 13 was defined for the angular velocity of the trailing limb. The angular acceleration derived from this angular velocity curve was scaled to the range the trailing limb was allowed to move forward. When the scale value was 1, the trailing limb ended hanging vertically next to the leading limb at the end of the single limb support phase. When it was 0, the trailing limb did not move forward. When the leading limb was set as a prosthetic limb, the passive prosthetic ankle was limited in its range of motion by a spring (Cspring) and damper (Cdamper) system, creating a counter torque (Tc) based on the joint angle (θ) and joint angle velocity (˙θ) when the ankle angle exceeds its limits (Equation 1).

                                   ˙
Tc = Cspring * (θlimit - θ)- Cdamper *θ
(1)

in which Cspring = 100 Nm and Cdamper = 25 Ns m.
When the leading limb was set as a sound limb, the internal active ankle moment (Tankle) was set to be the known in the external moments when the leading foot was in contact with the ground. The sound ankle was set to produce a constant internal plantar flexion moment of 54 Nm.
Ground reaction forces (⃗FGRF ) were formed when the leading foot made contact with the floor of the external world. Ground reaction forces were calculated based on continuously checking for the intrusion of the heel and toe (⃗
Xfooty,z) into the floor (⃗
Xfloory,z), which has a modeled stiffness (Cstiffness) and damping (Cdamping) (Equation 2).

                                                      ⃗         ⃗
⃗FGRF = Cstiffness *(⃗Xfloor  - (⃗Xfoot  ))- Cdamping * Δ-(Xfloory,z---Xfooty,z)
                        y,z        y,z                       Δt
(2)

in which Cstiffness = 1 * 105 N m-1 and Cdamping = 5 * 103 Ns m.


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Figure 1: Gait Termination Model consisting of a leading limb and foot with ankle joint, a trunk and a trailing limb.


The mathematical model was used to study the influence of the leading limb angle at initial contact, the ankle properties (passive stiffness or internal active ankle moments) of the leading limb, and the trailing limb motion during foot contact with the leading limb on the anterior-posterior CoM velocity. Combinations of settings were investigated to study the influence of the ankle and limb properties on the end velocity. Table 1 shows the ranges of the initial settings.
The mathematical model was validated by examining the conservation of energy and comparing the modeled external forces with measured ground reaction forces of the TF amputee subjects. The mixed dynamics moments of force constraining the trunk element were checked against unrealistic values by inspecting and comparing the calculated values with known maximal joint moments of force in healthy subjects 14.


Table 1: Ranges of the initial settings of the leading limb angle, ankle stiffness or aktive moment of force and trailing limb motion in the gait termination simulation.



Condition Sound limb leadingProsthetic limb leading



Leading limb angle 0.13 - 0.23 rad 0.18 - 0.28 rad
7.45 - 13.18 10.31 - 16.04



Ankle stiffness - 0 - 1 a)



Internal active ankle moment 19 - 69 Nm -



Trailing limb motion 0 - 1b) 0 - 1b)



a) 1: maximal stiffness (= 100 N m-1).
b) 1: trailing limb is next to the leading limb at the end of the simulation, hanging vertically.


Results

Subjects


Table 2: Gait termination: Averages and standard deviation of the velocity of the CoM (CoM Velocity) at the start (t = 0) of the single limb support phase (1ft) and double limb support phase (2ft), the gait termination duration (Duration; total = 1ft + 2ft) and the braking impulse (Impulse) in the sound limb leading and prosthetic limb leading conditions (n=6).



Condition Sound limb leadingProsthetic limb leading






CoM Velocity1ft(t=0)(m/s) 0.72(SD 0.12) 0.69(SD 0.15)
CoM Velocity2ft(t=0)(m/s) 0.14(SD 0.06)* 0.33(SD 0.08)



Durationtotal(s) 1.04(SD 0.41) 1.02(SD 0.31)
Duration1ft(s) 0.58(SD 0.15)* 0.32(SD 0.08)
Duration2ft(s) 0.46(SD 0.43) 0.70(SD 0.29)



Impulsetotal(N s) 56.1(SD 12.1) 54.5(SD 14.1)
Impulse1ft(N s) 48.7(SD 15.6)* 29.6(SD 8.1)
Impulse2ft(N s) 7.3(SD 6.6)* 24.9(SD 9.7)



Significant differences (p<0.05) between the two leading limb conditions are marked with *.


All subjects, except one, were able to come to a full stop in one gait termination cycle (both the leading and trailing limb placed on the ground). Although the CoMy velocity of the subject that could not come to a full stop, was heavily reduced to almost zero, one small step with his leading limb was made extra to come to the complete full stop in both conditions. We could not find a convincing biomechanical reason for this extra step. We excluded the data from this subject from the analyses.


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Figure 2: Exemplary gait termination by TF amputee subjects in sound limb and prosthetic limb leading conditions, divided in single limb support phase (1ft) and double limb support phase (2ft). Vertical lines represent the start and end of a phase.
A: Sound limb leading, with a short double limb support phase and a deviating decrease and increase in the GRF just after the single limb support phase start. 1 subject produced this GRF pattern.
B: Sound limb leading, with a long double limb support phase, although the CoMy velocity is already heavily reduced at the start of the double limb support phase. 5 subjects produced similar GRF patterns.
C: Prosthetic limb leading, with GRFy peak after start double limb support phase. 4 subjects produced similar GRF patterns.
D: Prosthetic limb leading, with GRFy peak before start double limb support phase. 2 subjects produced similar GRF patterns.
CoPy: 0 = center of force plate; positive value indicates anterior position of the CoP (toward toe of foot).


No significant differences were found in the velocity at the beginning of the gait termination process in the two conditions (table 2). Also, the overall duration and the impulse used to come to a full stop were almost equal. However, significant differences were found when the gait termination process was divided in two phases. When the gait was terminated with the sound limb leading, the gait termination was almost completely executed during the single limb support phase (figure 2A). 85 percent (SD 14) of the total impulse was generated in the first part of the process. The velocity at the end of the single limb support phase was reduced to 20 percent. On average, the duration of the single limb support phase and the double limb support phase were almost equal, but a large standard deviation was found for the double limb support phase (figure 2A,B). Although CoMy velocity was already heavily reduced for most subjects the moment they placed the trailing limb on the ground, the double limb support phase took quite long to come to a complete stop (figure 2B).
When terminating with the prosthetic limb, the average single limb support phase took 33 percent (SD 8) of the total process time. During this phase 55 percent (SD 10) of the total brake impulse was generated (figure 2D).
The data shows that our subjects use the CoP and the GRF to come to a complete stop with their CoM. Positive GRFy results in braking of the subjects.

CoP In the sound limb leading condition our subjects moved the CoP to anterior under the foot in the first part of the single limb support phase. During the double limb support phase, almost no CoP motion was found (figure 2A,B).
When leading with the prosthetic limb, the CoP moved slowly to anterior during the single limb support phase. When the trailing sound foot was placed on the ground the CoP continued moving forward, while transitioning the weight from the leading prosthetic limb toward the sound limb (figure 2C,D).

GRF In the sound limb leading condition the GRF decreased as presented in figure 2B in 5 subjects. In one subject the GRFy decreased and increased after the start of the single limb support phase (figure 2A).
In the prosthetic leading conditions the GRFy also decreased after the start of the single limb support phase, but was still higher at the end of the single limb support phase. Compared to the sound limb leading condition a clear increase was found around the start of the double limb support phase in the GRFy in four of the six subjects. This increase was found not only before but also after the weight shifting from leading limb to trailing limb (figure 2D).

CoM The curved trajectory of the CoMy velocity in single limb support phase of the sound limb leading condition differed from the relatively more linear trajectory in the prosthetic limb condition. The CoMy velocity showed an larger decrease around the start of the double limb support phase (figure 2C,D).

Mathematical Model

The CoMy velocity, the position of the CoP and the GRFy in the mathematical model showed similarities with the measurements (figure 3).

Initial Parameters

The average velocity of the TF amputee subjects at the beginning of the single limb support phase (0.7 m/s) was used as an initial parameter for the mathematical model in both the sound limb leading condition and prosthetic limb leading condition. The goal was to achieve end velocities with the model, which were similar to the average end velocity of the prosthetic limb group. In the sound limb leading condition, the average end velocity was 0.14 m/s. In the prosthetic limb leading condition the average end velocity was 0.33 m/s. These end velocities had to be reached in the same time as the subjects used to reach these end velocities. The average duration in the sound limb leading condition was 0.58 s and 0.32 s in the prosthetic limb leading condition. To achieve the end velocity in the sound limb leading condition, we set the leading limb angle of the model at initial contact to 0.18 rad (10.31). The sound ankle was set to produce a constant internal plantar flexion moment of 54 Nm. In the prosthetic limb leading condition, we set the leading limb angle of the model at initial contact to 0.21 rad (12.03). The prosthetic ankle of the model was set as a flexible ankle. In both conditions, the velocity of the trailing limb was set in such a manner that the limb would be next to the stance limb at the end of the single limb support phase.
When simulating with the initial parameters, a clear decrease in the CoMy velocity in the sound limb leading condition was found. In the prosthetic limb leading condition a plateau was found in the CoMy velocity halfway the single limb support phase. This plateau was accompanied by a clear increase in the GRFy. The CoPy in the sound limb condition showed a gradual translation toward the toe, while in the prosthetic limb leading condition the CoPy rapidly shifted from one position to another after the foot was flat on the ground.

Combinations of settings

Figure 4 shows that only small areas can be found in which the calculated end velocities of the model match the average end velocity of the group of TF amputee subjects. When chosen inadequate, the wrong combination of sound ankle moment and limb angle results in a CoM end velocity that differs substantially from the desired end velocity, which was achieved by the TF amputee subjects. The ankle stiffness appears to have limited influence on the CoM end velocity. The trailing limb motion seems only relevant for the CoM end velocity when the combination of the sound ankle moment or ankle stiffness and the limb angle result in a CoM end velocity which is close to the average end velocity of the TF amputee subjects. This can be seen around the small areas in which the exact end velocity is found.


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Figure 3: Results of the gait termination model during the single limb support phase, either leading with the sound limb (A) or with the prosthetic limb (B). Initial parameters were used to achieve initial velocities and end velocities that were similar to the average velocities of the TF amputee subjects.
Vend: end velocity (m/s); LA: leading limb angle (rad); AS: ankle stiffness (graduated scale from 0 to 1; 0: max. flexible, 1: max. stiff); AM: internal active ankle moment (Nm); VTL: Trailing limb motion (graduated scale from 0 to 1; 0: no motion, 1: inversely related to leading limb)
The sudden changes of the CoPy position and GRF at the transition of the heel contact to the foot flat on the ground, marked with , are artifacts of the mathematically modeled floor stiffness and elasticity, and the flat shoe sole, causing a quick alternation between heel and toe contact.
CoPy: 0 = center of modelled force plate; positive value indicates anterior position of the CoP (toward toe of foot).



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Figure 4: Results of a gait termination model with several different parameters; Relation between the influence of the leading limb angle (rad), sound ankle moment (Nm) or prosthetic ankle stiffness (graduated scale from 0 to 1; 0: max. flexible, 1: max. stiff) and trailing limb motion (graduated scale from 0 to 1; 0: no motion, 1: inversely related to leading limb) and the anterior-posterior CoM velocity at the end of the gait termination. The calculated end velocities of the model that are equal to the average end velocity of TF amputee subjects are visualized with black markers. The size of the markers indicates the magnitude of the correspondence with the average end velocity of the TF amputee subjects. The arrows show the direction of the change in velocity. The numbers at the end of the arrows show the range of the end velocity. Negative velocity values represent motion in the backward direction. The initial anterior-posterior CoM velocity is set to 0.7 m/s, the end velocity to 0.14 m/s in the sound limb leading condition and 0.33 m/s in the prosthetic limb leading condition, and the duration of the single limb support phase is set to 0.58 s in the sound limb leading condition and 0.32 s in the prosthetic limb leading condition.


Discussion and Conclusion

Although it seemed that no differences in the sound limb leading condition and prosthetic limb leading condition were found in the global outcome values of the overall process (CoM velocity, duration and impulse), clear differences were found when the process was divided in the single limb support phase and double limb support phase. When leading with the sound limb, the subjects came almost to a full stop in the single limb support phase, without the use of the prosthetic limb. When leading with the prosthetic limb, the CoM deceleration was less in a relatively short single limb support phase, with a fast forward swing of the trailing sound limb. One possible reason for limiting the single limb support phase can be given: the active control possibilites in the sound limb are necessary for the gait termination.
For succesful gait termination adequate CoP positioning and an associated backwards GRF are necessary. Anterior motion of the CoP during the single limb support phase was found in both leading limb conditions. When the leading sound limb was in contact with the ground, with the foot flat on the ground, the CoP motion was the result of an internal active ankle plantar flexion moment. When the leading prosthetic limb was in contact with the ground the limited CoP motion under the prosthetic foot was the result of a passive coupling between the foot and the changing limb angle during the gait termination. The lack of active control in the passive prosthetic ankle prevents the TF amputee subjects from producing sufficient braking force when terminating gait with the prosthetic limb leading. A more forward placed CoP produces a more backward GRF, but in the TF amputee, CoP can not be moved actively forward using muscular control. To compensate for this shortcoming, the leading prosthetic foot can be placed more forward, resulting in a larger GRFy, which contributes to the production of the braking force. However, the simulations show that CoM velocity is very susceptible to foot placement. This finding shows that using this strategy for accurate control is not very likely. Only narrow ranges of values could be used to achieve the same CoM velocities with the mathematical model as the average start and end velocities of the prosthetic limb user. These outcomes imply that a prosthetic limb user does not have much freedom choosing the leading limb angle, the trailing limb motion and the ankle moment. Based on the simulations, it seems that TF amputee subjects have to make use of sound limb depending strategies during gait termination. Instead of the leading limb being responsible for the necessary braking force as seen in healthy subjects, it is the sound limb, either as the leading limb or trailing limb, which is mostly responsible for the necessary braking force compensating for the shortcomings in the prosthetic limb. Also, it should be taken into account that not every limb angle is possible, as the CoP position and the resulting GRF are essential for passive knee extension. Small changes in CoP placement may result in buckling of the knee, depending on the position of the instantaneous knee axis.
This sound limb depending strategy has some consequences for the gait termination process. In the short single limb support phase when the prosthetic limb is leading, the relatively heavy sound trailing limb has to be moved forward. This influences the way the subjects can use the braking force to reduce the CoMy velocity. Slowing down the trailing sound limb velocity at the end of its swing phase by producing adequate moments of force with both hips results in a relatively larger braking force, compared to when slowing down the relatively light prosthetic limb in the sound limb leading condition. This relation explains the linear CoMy velocity decrease in the single limb support phase in the prosthetic limb leading condition (figure 2C,D).
It seems that the way the trailing sound limb is slowed down is critical, but only when the leading prosthetic limb angle is chosen correctly and in accordance with the stiffness of the prosthetic ankle as can be seen in figure 4. When the ankle is very stiff and the limb angle is large, the CoPy remains under the heel. Although the large limb angle would contribute to a higher braking force, the position of the CoP reduces this gain, since it remains under the heel for quite some time. When using a flexible ankle and the same limb angle, the CoPy moves forward at the beginning of the stance phase resulting in a larger braking force. For gait termination it seems that a flexible ankle or heel setting would be preferred as this setting increases braking force. In contrast to gait termination, during gait this setting is counter productive, because CoMy velocity is lost during early stance phase, hindering forward progression. The settings of the prosthetic ankle or heel should be different for gait termination than for gait. Therefore, a well-balanced choice should be made. The choice for the settings depends on the subject's needs.

Some remarks and considerations should be given when interpreting the outcome of the current study. In this study, we were merely interested in specific underlying principals of gait termination. Since no exact numerical correspondences were to be expected, we decided that the models anthropometrics had to match those of a human being roughly. For simplicity reasons we chose not to add moveable knee joints. Moveable knee joints have some benefits during gait termination. A moveable knee joint has its added value in the foot clearance process and can be used to decrease the impact of the GRF during heel strike. We realize that these knee joints influence the CoM velocity. Already we showed that the orientation of the GRF and the weight of the trailing limb are of importance for CoM velocity changes. Similar to this, the inertia of the limb, influenced by the orientation and position of the upper and lower part of the limb, has its effect on the CoM velocity changes. The absences of the knee joints might explain some of the minor differences between the results of our model and the measured data of the TF amputee subjects. Although we did not use moveable knee joints, it is our opinion that the current findings are relevant. Most of the prosthetic knees remain in full extension during the gait termination process, especially in the prosthetic leading limb condition, but also in the prosthetic limb trailing condition. In this last condition, many times the limb does not flex as a result of the low angular velocity and the minimal use of the GRF which is not sufficient to create a flexion moment around the knee joint.
Also for the sake of simplicity, we did not add a trailing foot. Adding a foot would probably enabled us to reproduce the quick braking force alteration around the beginning of the double limb support phase. Based on our observations during the measurements and the consequences of the definition of the single limb support phase to double limb support phase transition, we assume that the quick alteration is the result of the landing of the trailing foot and the moment in which the weight transition to the sound limb occurs.
In one subject the braking force decreased and increased at the beginning of the single limb support phase as a result of plantar flexion of the stance ankle joint of the sound limb. This plantar flexion was used to increase limb length to reach toe clearance of the trailing prosthetic limb at the beginning of its swing phase. Obviously, this phenomenon was not found in the mathematical model because the plantar flexion strategy was not necessary as we did not include a trailing foot.
Because of the absence of the moveable knees and the trailing foot, we were not able to study their influence on the CoM velocity. It is likely that TF amputees make use of these possibilities to influence their CoM velocity. Similar to this, it should be taken into account that also the trunk influences the CoM velocity, and that TF amputees probably use their trunk to influence the CoM velocity as well. We observed variation of the trunk position when analyzing the video images. Four out of the six subjects showed a more backward trunk position in the prosthetic limb leading condition. This strategy results is a more backward positioned CoM, relative to the CoP position, which contributes to the deceleration of the CoM. In our mathematical model, the trunk is controlled to stay in vertical orientation using adequate hip torques. A more advanced model should be used to study the influence of these items. The current model is used to study the relation between the leading limb angles, the internal active ankle moments of the sound ankle or passive stiffness of the prosthetic ankle, the trailing limb motion and the CoM velocity during the single limb support phase of the gait termination.

During gait termination TF amputee subjects can compensate for the absence of voluntary muscle activity and joints in the prosthetic limb by influencing the CoP position and the GRF under their prosthetic limb, and with their sound limb. Patients have to be trained in such a manner that they learn how to produce the optimum of braking force with both of their limbs used as a functional unit: using an adequate limb angle on the prosthetic side and using the capacities of the sound ankle. Because the duration of the total gait termination process and the produced braking impulse do not differ when leading with the prosthetic limb or with the sound limb, it seems that there is no reason to enforce a preferred leading limb during gait termination. However, when terminating gait with the sound limb leading, the TF amputee subjects are able to reduce their CoMy velocity more quickly during the first part of the process. This gait termination strategy is the preferred strategy in the experienced TF amputee subjects group.

Competing Interests

There are no competing interests related to this study.

Acknowledgement

The authors wish to acknowledge the OIM foundation, Beatrixoord foundation and Anna Foundation for their financial support.

Ethical Approval

The medical ethics committee of the University Medical Center Groningen approved the study protocol (reference number 2004.176). All subjects signed an informed consent before testing.

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