Finite element simulation of frontal plane adaptation using full-foot, split-toe and cam-linkage designs in prosthetic feet

INTRODUCTION

Frontal plane adaptability of the human foot occurs during many ambulatory activities including weight-shift in two-legged standing, side-step,¹ turning,² imprecise positioning of the foot,³ balance perturbations,^4,5 as well as uneven ground and cross-slope gait.^6,7 Studies of young adults during cross-slope walking have found kinematic adaptations through a series of foot elements: hallux relative to the forefoot, forefoot relative to the hindfoot, and hindfoot relative to the tibia.⁶ Frontal plane rotation of the foot improves contact area and friction with the ground. Damavandi et al. concluded that functional adaptations are required to maintain forward progression and dynamic stability in stance during cross-slope walking and running.⁸ Foot adaptability also changes the center of pressure at the foot, modifying torques acting on the body. Functional adaptations in the frontal plane are likely to result in comfortable locomotion on cross-slopes.⁶ In contrast, bracing the foot and ankle increased mediolateral torque at the knee joint.⁹

People with lower-limb amputation (LLA) have persistent problems with balance, falls, residual limb symptoms, functional mobility, the need for cognitive attention during gait, and satisfaction with participation in daily activities despite using prostheses.^10–14 Previous studies have suggested that lack of adaptability of currently available prosthetic feet in the sagittal plane can contribute to these issues and researchers proposed automatic adaptation to the walking surface as a solution.^15,16 Unexpected torques propagate through the kinetic chain with undesirable consequences, and can be attenuated with adaptable mechanisms. These problems may also be related to limited frontal plane (rotation and translation) adaptability of prosthetic feet because less than optimal alignment of the foot with the ground can cause abnormal frontal plane forces and torques at the socket-residual limb interface. Limited frontal plane adaptability of the prosthetic feet may cause mediolateral reaction forces and torques to be propagated up the lower-limb kinetic chain¹⁷ resulting in gait instability,¹⁸ residual limb discomfort,¹⁹ and skin breakdown.²⁰

Efficient gait requires smooth forward velocity of the body mass along a desired path regardless of perturbations at the foot. Maintaining forward velocity is less efficient when the effects of forces and moments acting external to the body cause speeding up, slowing down, or directional shifts of the center of mass that require extra energy to correct. For example, less than optimal tuning of the person’s mass to prosthetic foot stiffness causes slowing and elevation (dead spot), or speeding up and dropping (giving way) of the person’s mass.²¹ The work of walking has been shown to be increased with mediolateral displacement of the body mass.²² Theoretically, prosthetic foot design should allow smooth forward motion of the body mass with minimal correction by the individual despite initial ground-foot contact geometry.

Current methods to provide frontal plane adaptability in prosthetic feet include compliance mechanisms (stiffness) such as split toes. In our previous paper, using a finite element simulation approach, we investigated the effects of foot stiffness, foot width, and split toes on motion and foot contact during walking gait on cross-slopes compared to level ground.²³ That study found that one or more lengthwise splits through the toe of the prosthetic foot reduced performance differences on cross-slopes compared to flat ground. Compared to single cantilever spring variations, split-toe prosthetic feet had less difference in forward velocity, vertical displacement, and lateral deviation on −10° cross-slopes compared to level ground. Additional variations of prosthetic feet can be tested with the long-term goal of improving performance in a variety of daily tasks.

Despite numerous variations of prosthetic feet, Stevens et al. recently concluded that “…the scientific community has lagged behind in reporting empirical evidence….”²⁴ These authors reviewed the literature and found that “…supplements to prosthetic feet including multiple-axis ankle units, have received limited treatment.”²⁴ Computerized simulation of prosthesis design has advantages over human motion analysis studies for preliminary prosthetic foot studies because a large number of confounding variables are eliminated. The theoretical underpinnings of design changes can be established. Simulations of several variables can be accomplished without acclimatization phases that are necessary for human trials. Differences in outcomes between variations can be evaluated accurately compared to human tests. Efficacy of a design can be simulated and refined prior to expensive prototype fabrication and human motion analysis and community tests that are important for determining clinical significance.

The purpose of this study is to determine if a cam-linkage articulation, designed for frontal plane prosthetic foot adaptability, improved forward motion on cross-slopes compared to single or double cantilever spring designs (full-foot or split-toe, respectively) using finite element simulation.

METHODS

Finite Element Simulations

Model construction (material data, geometry, and mesh) and simulations were performed using Ansys LS-Dyna (ANSYS, Inc. Canonsburg, PA). Material properties were assigned from the Ansys materials library. Isotropic materials were used to avoid variations in anisotropic material properties. Default meshing was modified by varying element size to improve simulation efficiency while maintaining sufficient elements across thicknesses. Mesh density was increased in regions where greater mechanical changes were expected because of contact or geometry. Node position and software parameter modifications were used to improve surface-to-surface contact detection and eliminate gaps or penetration at surfaces. Changes in geometry, such as split toes, resulted in variations in mesh. Number of nodes and elements were approximately 34,000 and 72,500, respectively, for the basic models, and varied depending on automatic meshing of foot geometry (splits) and refinements to the mesh. Element volume in each part varied from 0.55 mm³ to 52.3 mm³ depending on part geometry, and expected surface-to-surface interactions (e.g., sliding surfaces) were more finely meshed.

Friction coefficients between the foot and ground surface were set at 0.9 in all simulations to avoid possible slippage on cross-slopes.

Simulated Foot Prosthesis

The Durawalk (WillowWood, Mt. Sterling, OH) prosthetic foot geometry was used as the base model, and simplified geometry was used for the anterior toepad, posterior elements, and pylon to reduce computational time. Footwear and the standard cosmetic cover were not incorporated into the model so that variations of prosthetic feet could be evaluated independently of other factors. An aluminum pylon, 0.41 m long, was bonded directly to the cam-linkage to simplify the model. Total mass of the prosthetic foot and pylon was 1.14 kg. The current linkage design increased the weight of the prosthetic foot by 0.17 kg. A concentrated mass (75 kg) was simulated as a cube (0.04 m in each dimension) bonded to the proximal pylon to represent the body mass. By bonding the simulated body mass to the proximal pylon, an inverted pendulum is created, in keeping with this theoretical model of energy conservation during gait.

Full-foot and split-toe variations were tested to determine the effect of a split, a common feature in prosthetic feet. The split was 0.5 mm in width and 137 mm in length centered on the distal toe and completely through the toe pad and spring plate.

Linkage Design

The linkage mechanism, to provide frontal plane prosthetic foot adaptability, is based on a theoretical model of the knee joint.²⁵ The specific geometry of this linkage mechanism allows knee sagittal plane angular rotation to occur with predictable surface-to-surface contact. The knee mechanism results in a simple, noncompliant, single degree of freedom linkage. Properties of a crossed 4-bar linkage articular model include predictable surface-to-surface contact, responsiveness to contact geometry and forces, improved joint contact, improved stability, and reduced joint wear.²⁵ Other modeling studies demonstrated that the conceptual approach could be extended to other joints including the ankle in the sagittal plane.²⁶ A simplified crossed 4-bar linkage coming into contact with a cross-slope is depicted in Figure 1. A previous study examined use of this linkage configuration in a hand prosthesis.²⁷

Figure 1.

A. The crossed 4-bar linkage system is fabricated similarly to the knee joint with concave and convex articulations and internal bars acting like the anterior and posterior cruciate ligaments. B. Rotation of the lower portion of the mechanism to become parallel to the slope, while the upper portion remains flat.

Several variations on the crossed 4-bar linkage theoretical system have been fabricated and used in prototype prosthetic feet, including the cam-linkage design used for the current simulation study (Figure 2). The cam-linkage follows the same path as previous 4-bar linkage designs with ±10° of rotation in the sagittal plane. This specific linkage design is passive and does not have springs or bumpers to simplify understanding of the mechanism during simulation.

Figure 2.

A. Prototype prosthesis at the initiation of the simulation. The toe is positioned on a 15° slope. A 75 kg mass is bonded to the top of the pylon, and has an initial velocity of 1 m/s. B. Close-up of the foot with a cam-linkage prototype connected between foot and pylon for frontal plane adaptability at the ankle region.

Initial conditions

The simulation was initiated at foot flat (mid-stance) with the foot stationary and in contact with fixed surfaces at the heel and toe (Figure 2). Standard earth gravity was applied vertically to all components of the simulation. An initial velocity of 1 m/s was applied to the body mass. After initiation of the simulation, inertia of the body mass and the force due to gravity in conjunction with friction at the foot ramp interface caused rotation of the prosthetic foot during the stance phase of gait, in keeping with the inverted pendulum theory of the stance phase of gait.^3,28

The duration of the simulation was 0.2 seconds and sagittal plane angular rotation of the pylon was measured to compare trials.

Variations

Full-foot (single cantilever spring) and split-toe (double cantilever spring) variations were tested with the cam-linkage locked (bonded) and unlocked (frictionless). In this way, all features of simulations were equivalent except for linkage motion. A cross-slope of 15° was chosen to be significant perturbation. A level surface (0°) was used to compare to the cross-slope condition.

Outcomes

Outcomes for the study were based on the inverted pendulum model of walking.²⁸ Forward progression of the simulated body mass over the foot on 15° cross-slope was compared to level ground. In biomechanical equations, force is equal to mass times acceleration (F=ma). Distance traveled is ½ average acceleration times elapsed time squared (d=0.5at²). Since mass and time are equivalent between trials, average forces acting on the mass is proportional to distance traveled. Therefore, less than optimal forces at the foot-surface interface act on proximal structures and on the body mass and alter displacements. Mediolateral deviation (mm), vertical displacement (mm), and change in forward velocity (% of initial) were obtained for a single node at the center of the superior-anterior edge of the proximal mass. Ideal mediolateral translation was assumed to be 0 mm because initial mediolateral velocity is 0 m/s. Ideal maximum vertical displacement of approximately 25 mm was estimated from previous studies because the simulation is started with the pylon in the vertical position – the highest point of the pendulum.²⁹ In addition, rotations of the pylon in the transverse and frontal planes were calculated as indicators of additional perturbation to prosthetic alignment. Forces sufficient to cause mediolateral displacement to the proximal simulated body mass and change forward velocity may be perceived by a person using a prosthesis and may contribute to gait instability. These outcomes also suggest that the individual would need to correct the direction and velocity of the body mass to maintain gait.

To verify that mediolateral displacements were correlated to forces acting at the pylon - body mass connection, those forces were calculated in the simulations using Ansys software algorithms. Calculated mediolateral forces were smoothed using a 12-Hz low-pass Butterworth filter (MATLAB, MathWorks, Natick, MA).

Foot contact with the simulated ground surface was measured as a percent of foot width because friction and base of support increases with increased foot contact.

RSULTS

Performance of prosthetic feet on level ground, as measured by mediolateral displacement, maintenance of forward velocity, pylon transverse angle, and frontal plane angle, were consistent, except for small but systematic differences in vertical displacement of the body mass (Table 1). On level ground, after 0.2 seconds simulation time, mediolateral displacements were, on average, 2.2 mm (standard deviation 1.0 mm). Final velocity was, on average, 98.6% (standard deviation 0.54%) of initial velocity for the 4 prosthesis variations. Average sagittal plane rotation was 24.3° (range 24.1 ° – 24.5 °). Average transverse plane rotation was 1.8 ° (range 0.8 ° – 3.1°). Average frontal plane rotation was 0.8 ° (range 0.3° – 1.7 °). Full-foot locked and unlocked vertical displacements on level ground were −16.2 mm and −18.4 mm, respectively. Split-toe locked and unlocked vertical displacements were −18.2 mm and −20.5 mm, respectively. Therefore, the frictionless cam-linkage introduced −2.3 mm of vertical displacement over the simulation.

Table 1.

Lateral and vertical displacement and change in velocity of the proximal mass at completion of stance phase simulation on 15° cross-slope.

	Lateral displacement (mm)		Vertical displacement (mm)		Change in velocity (% of original)
	0°	15°	0°	15°	0°	15°
Full-foot Locked	1.2	11.8	−16.2	−26.1	98.0	100.5
Full-foot Unlocked	4.6	4.4	−18.4	−33.7	98.5	102.0
Split-toe Locked	2.2	7.9	−18.2	−33.5	98.6	103.5
Split-toe Unlocked	0.7	4.0	−20.5	−36.0	99.5	104.0

On angled cross-slopes, lateral displacements of the proximal mass (Figure 3, Table 1) were decreased with split-toe locked compared to the full-foot locked variations by 33%. The frictionless cam-linkage decreased lateral displacement by 67% for the full-foot and 50% for the split-toe variations (Figure 3). Vertical displacements and resultant forward velocity were increased with the cam-linkage (7.6 mm and 1.5%, respectively) consistent with level ground results.

Figure 3.

Mediolateral deviation of simulated body mass for 4 prosthetic feet during toe contact on a 15° cross-slope. A single split creating two cantilever springs reduced lateral deviation by 33%. A frictionless cam-linkage with mediolateral adaptability decreased deviation for both the full-foot and split-toe variations.

Frontal plane angles of the pylon after 0.2 seconds of toe contact on the 15° cross-slope were small in all cases and were reduced with the cam-linkage (Table 2). Transverse plane rotations were also small at the end of the simulations, but the frictionless cam-linkage with full-foot introduced approximately 1.3° of transverse plane rotation whereas the split-toe with frictionless cam-linkage had the least transverse plane rotation (Table 2).

Table 2.

Frontal and transverse plane rotations and foot contact at the completion of stance phase simulation on 15° cross-slope.

	Frontal Plane Rotation (degrees)		Transverse Plane Rotation (degrees)		Foot Contact (% of foot width)
	0°	15°	0°	15°	0°	15°
Full-foot Locked	0.6	1.45	1.3	2.45	100	30
Full-foot Unlocked	1.7	0.01	3.1	3.70	100	46
Split-toe Locked	0.9	1.26	2.1	1.78	100	60
Split-toe Unlocked	0.3	0.44	0.8	0.54	100	61

Maximal mediolateral forces acting at the pylon-body mass connection were reduced with the split-toe compared to the full-foot and also were reduced with the frictionless linkage (Table 3, Figure 4). The maximal force was reduced from 83.0 N for the full-foot locked variation to 39.7 N for the split-toe unlocked variation, a reduction of 52%. The average reduction in maximal force due to the frictionless linkage (unlocked) was 27%.

Table 3.

Mediolateral contact forces acting at the proximal pylon – body mass connection during the stance phase of gait.

	Maximum force (N)	Percent of Full-foot Locked	Average force (N)	Percent of Full-foot Locked
Full-foot Locked	83.0	100%	57.4	100%
Full-foot Unlocked	64.1	77%	35.9	62%
Split-toe Locked	57.2	69%	37.1	65%
Split-toe Unlocked	39.7	40%	25.8	45%

Figure 4.

Contact forces acting at the pylon-body mass connection during the latter half of the stance phase of gait in contact with 15° cross-slope. Unlocked full-foot and split-toe linkage prosthetic foot variations decreased forces compared to locked variations, respectively.

Foot contact for the full-foot variation was increased with the frictionless cam-linkage. Split-toe variations also improved foot contact area (Table 2, Figure 5).

Figure 5.

Four prosthetic foot variations at terminal stance phase of gait in contact with a 15° cross-slope. Foot contact increases with the frictionless cam-linkage for the full-foot variation. Split-toe variations have increased foot contact. A. Full-foot locked. B. Full-foot unlocked. C. Split-toe locked. D. Split-toe unlocked.

DISCUSSION

This finite element simulation study found that a frictionless, frontal plane adaptable cam-linkage articulating at the foot-pylon interface can decrease lateral displacement of a 75 kg mass positioned at the proximal pylon as well as mediolateral contact forces at the proximal pylon during toe contact with 15° cross-slope compared to a bonded (locked) cam-linkage variation. Combinations of split-toe with the frictionless linkage resulted in the least frontal plane motion and forces. This suggests that this linkage system may reduce forces and moments acting on the socket-residual limb interface. Results of the study can be explained by instantaneous stiffness of the prosthetic foot during the stance phase of gait. The split-toe variation allows the foot to flex during contact, reducing torque acting on the proximal pylon. Whereas, the linkage allows frontal plane rotation of the foot, decreasing the moment arm of the reaction forces, also reducing torques acting on the proximal pylon. There do not seem to be similar studies to compare.

The current study simulated 75 kg of body mass directly atop the pylon. In traditional lower-limb prostheses, the socket would be positioned there. More recently, this would be the site of osseointegrated attachments. Forces that are sufficient to accelerate a 75 kg mass at this position may be significant because the actual local anatomical mass is much less. At the traditional residual limb – socket interface, forces have been implicated in skin conditions,³⁰ and the lateral aspect of the residual limb is relatively sensitive to pain with pressure.¹⁹ Mediolateral forces may be partly responsible for mediolateral stability challenges for some people with transtibial amputations during gait on uneven ground³¹ as well as step-width variability and increased work of walking.³² Future studies will need to determine if the reduction of forces and moments improves gait, balance, skin integrity, and pain.

Reduction of forces and moments also can have the positive effect of reducing correction forces required to maintain the desired path. Step-to-step ankle inversion/eversion torque modulation to offset body position errors over the foot has been recommended to reduce the work of walking.³³ Reducing corrective forces may also improve force and energy requirements for actuated prostheses or robotic feet.

The current finite element simulation study also confirmed that including a lengthwise split to change one cantilever spring into two (split toe) reduced lateral displacement. However, neither the split-toe nor the frictionless cam-linkage alone or in combination reduced lateral displacement to zero. Current results should be considered as a starting point because the current version of the linkage has not been optimized for kinematics or kinetics. It is also unknown the effect of series or parallel linkages.

Both the full-foot and split-toe frictionless cam-linkage variations reduced pylon frontal plane angle on cross-slopes. This may be advantageous in maintaining body center of mass over the foot area. However, the full-foot variation unexpectedly increased transverse plane rotation under these conditions compared to the split-toe variation. The split-toe also had the advantage of increasing mediolateral base of support. While these differences seem small, gait consists of many repetitions, and human studies will be required to determine if they are clinically significant.

Both the split-toe and the frictionless cam-linkage variations introduced a change in vertical displacement and resultant forward velocity. Vertical displacement is important for compression on the inferior socket as well as functional leg length on level ground and cross-slopes. Consequently, the mechanical properties of the entire configuration should be considered when tuning a prosthetic foot to someone’s needs.

The current linkage prototype is based on previous mathematical models of the knee and other joints.^25,34 However, there may be other appropriate mechanisms that may provide similar or improved performance.

The current study has several limitations. This finite element simulation study was intended to show performance trends for 4 variations under two conditions. It was assumed that adaptability of the foot would result in a benefit to physical performance for this narrow context. There are considerably more variations and conditions possible that might be tested. Personal preference or task-specific demands may change the optimal prosthetic foot design.

The material was modeled as an isotropic material. Carbon fiber and fiberglass reinforced materials would respond differently depending on the orientation of fibers. Finite element modeling was intended to illustrate trends in foot performance and the precise numbers generated in the current study are dependent on meshing and other factors within the model and may show some variation.

Many of the model characteristics are different compared to actual prostheses and their use. The current study did not include footwear and a cosmetic cover, although these would affect foot performance. The geometry and material properties of prosthetic feet are important to the performance qualities of the foot. The current study did not optimize these factors. There was only one base foot geometric configuration tested, and geometry of the prosthetic foot would likely change the numeric results. Orientation of the limb relative to the surface, velocity of walking and other factors would affect outcomes. A single linkage was positioned where the ankle might be, but other positions such as the forefoot or heel - the sites of anatomical adaptability - were not tested.

CONCLUSIONS

This finite element simulation study shows that a frictionless cam-linkage with frontal plane adaptability integrated into a prosthetic foot improved mediolateral displacement of the proximal pylon and frontal plane angular rotation of the pylon compared to a locked (bonded) linkage during the stance phase of gait in contact with a cross-slope. Proximal contact forces were also reduced by the linkage. These findings may have implications for people using lower-limb prostheses by improving gait, improving balance, reducing pain, reducing skin breakdown, and reducing the work of walking. This linkage, or another with similar outcomes, may also reduce the power needed to maintain forward motion in actuated feet. These findings should be tested in further studies.