Simulation and dynamic analysis of military marching using lower limbs anthropometric data

Background: Gait analysis is receiving increasing attention due to various applications in athletic performance, man-machine interfaces and especially in military services. This analysis involves the analysis of human locomotion augmented by body movements and biomechanics of joints. The kinematic motion of the body during a gait cycle capturing by cameras is then used as the desired target for modelling the motion of body segments. By taking advantage of gait analysis concept, this study aims to model the military marching, using anthropometric data with the focus on lower limbs while introducing top candidates with better healthy conditions in lower limb joints during a cycle of marching. Methods: Using 100 anthropometric data from military soldiers, equations of motion for the model are derived by applying Lagrangian methods in an inverse dynamic approach. In this model, the joints are simulated using springs and dampers while the actuators, simulated the muscles, acted like motors and applied enough torque on joints so that the model motion replicates normal military marching. Finally, all the springs and dampers coefficients are driven from optimization process. Results: Hip, knee and ankle torques were calculated after the optimization process for all 100 soldiers and then 5 candidates among them were established with less suffering forces and torques in their joints. Conclusions: In this study using biomechanics basics and anthropometry data at the same time, a standard could be evaluated to select the soldiers based on healthy condition of lower organs.


INTRODUCTION
Marching is a specific sample of daily human gait [1].That being said, in order to simulate marching, simulation of human gait cycle is required in which we should benefit from priori biomechanical knowledge on human motion and anthropometric data at the same time [2].The gait cycle consists of complex functional tasks requiring interaction between lower limb joints of the body [3].Motion of body segments during this interaction is studied as a part of kinematics without considering any forces.Typically, camera systems and electromagnetic devices are being used to record the motion of the body during a cycle [4].
While the body segments are considered as a mechanism, joints in hip, knee and ankle provide degrees-of-freedom (DOF) of the system.The joint reaction forces and muscle net torques convert this problem into a dynamic analysis in which inverse dynamic method is being used commonly.Then, equations of motion relate the kinematics of the body motion to the forces and torques that are causing those motions [5].Friction force calculation completes this set of equations as an external force.This force during a walking depends highly on the foot-ground contact simulation and modeling.Some of the studies used additional kinematic joint while other used temporal fixing to the support foot.But in some studies, realistic models for this section have been modeled which can simulate the reaction close to the real foot-ground contact [2,6].Some studies conduct the whole gait simulation by taking advantage of three dimensional models [7,9] while others consider two dimensional models [3,10,11] for simulation [4,8].Stability of such models is significantly important.Active feedback control systems [12] and simple PDI controller [10] are common methods used as a controller for the balance of gait models during a cycle.
Application of gait analysis in military services mainly limits to assessing the influence of boot [11,13] and carrying a backpack of soldiers during a gait cycle.Rare studies like [1] demonstrated one of the first researches on marching locomotion.As other applications of gait analysis, there have been some studies about modeling the gait analysis in order to analyze the dynamic of prosthesis during a gait cycle [3,10,14,18,19,20].
Towards simulation of marching, this paper conducts this analysis by computing the internal actuator forces and dynamic analysis of 100 soldiers.The major part of this simulation is to replicate normal kinematics of marching when subjected to normal muscle forces and joint torques during a cycle.In an inverse dynamic approach, using Lagrangian method the net joint torques is calculated from a sample military marching benchmark and external forces in a condition that muscles are the actuators of the musculoskeletal system.The optimization process then is needed to compute the forces and the torques in a way to generate the desired kinematic motion.

METHOD
From the engineering point of view, marching is in fact body segment movements in a mechanism way with couple of degrees of freedom (DOF).This mechanism consists of hip joint, knee joint and ankle joint.There is friction force between foot and ground which provides movement of the whole body.In this simulation, bones are considered as rigid elements while supporting the weight of the body and muscles act as elastic stimulators and displace the bones.
The appropriate method in studying the simulation of marching is by applying dynamic models.Using this method, the model was considered as a rigid body with several links.Considering exact and complete number of variables for dynamic simulation complicates the model and motion equations, therefore in this study some assumptions were employed to simplify the model.These assumptions include 1) the upper section of the body (head, trunk and hands) was considered as one link only, 2) friction was neglected in joints, 3) the weights of the elements were equally distributed, 4) modelling and solving equations of motion conducted in sagital plane (2D), 5) due to 2D modeling, joints were from hinge type and not spherical for hip joint.
The model for rigid body motion of a soldier contains seven links that include trunk, two thighs, two legs and two feet.The model was defined with nine generalized coordinates as consisting of seven links with vertical angles and hip coordinates in Cartesian coordinate system (Figure 1).In this model, the angles are trunk, right thigh, right leg, left thigh, left leg, right foot and left foot angles with vertical direction while the joints are hinge joints.One of the most important issues in this dynamic simulation is modeling the reaction force of the ground to the body.The modeling of this section was developed based on using a spring and a damper in contact points of foot to the ground so that the equivalent of damper and spring forces reflects the ground reaction force to the body [16].This estimation helps to calculate this reaction force during dynamic simulation and to simulate the viscoelastic contact of the foot.As it can be seen in Figure 2, a series of vertical springs and dampers were attached to the contact points to generate this vertical reaction force [10].In the case of no contact between foot and ground, the spring and damper contacts are disconnected and no force is reacted and whenever the contact between foot and ground occurs again, the spring and damper contacts are connected and so the force is reacted.In this model, the muscles are neglected and instead, actuators are employed in hip, knee and ankle joints.The torque being applied on the joints is a function of current angle in joint, desired angle and their derivatives.
Indeed, springs and dampers are used in joints so that the actuators act as motors which are linked with these springs and dampers.Therefore, appropriate modelling is needed for calculating the optimized variables for springs and dampers.The general equation for torque in the joints is In  In this study, anthropometric data of 100 military soldiers were analyzed through the dynamic simulation method.Table 1 shows the values of min, max, average and standard deviation parameters for all 100 soldiers.Using waist height and knee height, thigh length was calculated and based on the lengths of segments, the weights of segments were approached [15] (Table 2).

Table 2. Standards for mass calculation for body segments
While the anthropometric data varies from one soldier to another, the kinematic data is the same for all soldiers.The kinematic data must be considered the same for all since the ending result of joint torques for the model varies for various kinematic data but this study targets to analyze the joint torques only based on anthropometric data, so that as a result a standard of anthropometric data could be concluded.
Therefore kinematic data of a marching benchmark (Figure 3) was considered for all soldiers.As this figure depicts, right thigh and leg are raised to specific height in an extension movement and then in a felxion movement, right foot makes contact with the ground and then left segments follow the same path alternatively.Knee angle was considered the same as hip angle which means that the knee by itself does not have any flexion-extension movement.
Kinematic data includes, 1) angular speed and angle of hip joint 2) angular speed and angle of knee joint 3) angular speed and angle of ankle joint 4) angular speed and angle of trunk.Figure 4 shows the angle of hip and ankle during one cycle of marching.Fourier function was used in order to generate a curve passing through acquired kinematic data points.As it was mentioned in previous section, the equations of motion for this 2D model follow Lagrangian method.General formulation of the Lagrangian method is In which k is number of general coordinates which is 9 in this model, L is the difference between kinetic and potential energy of the whole system, Qk reflects non-conservative forces.In order to use these equations, initially kinetic and potential energy of the whole system were calculated and then after 9 equations of the order of 2 were produced.For the dynamic simulation using this method, initial conditions and angular speed of the links were provided.
Constant coefficients in all joint torques (discussed in previous sections) could be generated in an optimization process.The goal of optimization is to calculate these coefficients in the condition that the discrepancy between flexion-extension angles of joints in simulated and desired circumstances is minimal.The desired (normal) values for this simulation were obtained from [17].In order to conduct this optimization, genetic algorithm method was applied to decrease the difference between angles in desired and simulated modes.While in this optimization variables are the constant coefficients, the objective function is the summation squared of the difference between these angles.Below are the steps that the genetic algorithm takes to find the optimization coefficients:  Initial parameters are provided as the initial condition of the problem,  Initial guess of the spring and damper coefficients of the model is provided,  Based on the values from steps 1 and 2, the equation of the motion is formed,  From this equation of the motion, the angles function of the model is derived,  By having the angles of the model and desired angles (from the literature), the objective function is found,  The genetic algorithm comes to play and starts to minimize the objective function by finding the optimized values for spring and damper coefficients of the whole model  Once the optimized coefficients are found from the step 6, the values are replaced with coefficients of step 2 until final optimized converged coefficients are obtained. Return to step 3.

RESULTS
The numerical programming for dynamic simulation of a soldier was accomplished in MATLAB software.After providing all necessary parameters such as initial conditions, kinematic data, equations of motion and optimization, the whole program focused to generate the constant coefficients using the optimization method for a sample anthropometric data.Then, whole 100 anthropometric data was given to the program to calculate the joint torques in hip, knee and ankle.In fact this torque value is equivalent to maximum total torques acting on that specific joint.Table 3 shows the statistic about resultant torque values.In the initial analysis of this huge data, we focused on hip data.Since in this study having healthy joints are more desirable, we sorted the data from small to large values and then just consider the first top 10 as good candidates for healthy joints (Table 4).This narrowing down of the results helps us to analyze the data more consistently.
Moreover, Figure 5 shows how the waist height and knee height vary by the torque in the hip.It is interesting that both heights increase by gradually increasing of the hip torque.This fact is confirming that higher waist and knee heights can lead to larger torques in the hip.In the next step, knee torque was assessed by the numerical programming code in MATLAB and the results were again sorted from small to large and first 10 candidates were selected.This selection was done independently of previous selection for the hip.Indeed, this study approached the joint torques behaviour separately and then concludes all the resultant data together.The corresponding knee data can be found in Table 5. Comparing the torque values in knee joint to other two joints, it can be concluded that for this specific kinematic data for marching, the torque in knee is negligible.
Therefore, for the ending result of this study, knee did not play a major role and that is because flexion -extension movement was not considered for the knee.
Ankle torques is the next criteria considering as a result.Table 6 represents the torques in ankle joint during a full cycle of marching for the top 10 candidates who their ankle torques were minimal independent from other joint torques.Comparing the variation of the ankle torque versus waist and knee heights depicts the fact that ankle torque is highly dependent on the force being applied by the ground, since the waist and knee heights are almost constant with the variation of ankle torque (Figure 6).In fact the amount of ground force reaction plays the major role in the torque of the ankle.

DISCUSSION
In order to assess the healthy joints under marching condition, 2 factors were considered in this study: 1) hip torque and 2) ankle torque.Then based on the torque data of each joint, the whole data was sorted from smallest to largest value three times.Then, first 10 candidates in each section were highlighted and analyzed.In the first look, it seems like 20 different candidates with identical anthropometric data have been introduced, therefore it is needed to define another factor in order to select from those candidates, those with better healthy conditions in terms of minimum torque value in their joints.Due to less torque rate in knee comparing to hip and ankle, a new factor called "Minimal Summation Torque" is defined for this study which follows below equation, considering the fact that the maximum torque in hip is 22.39 N.m and in ankle is 4.35 N.m.
= ℎ   ℎ  +      By using the maximum values of torques we normalized our torque data.Then, we needed to calculate this factor beside other factors for all 100 data and sort again based on minimum value of torques.Table 7 shows this result.Analyzing Tables 4, 6, 7 reveals that there is a common point between all of them.There are some soldiers that exist in all those tables, showing that these candidates are common among all other candidates.This approves that this set of candidates contain a healthier body during a cycle of marching.We highlighted these candidates in all tables and allocated one colour to each of them.In fact, in this study using biomechanics basics and anthropometry data at the same time, a standard could be concluded to select the soldiers based on healthy condition of lower organs.
This study is one of the first studies being conducted in the area of specifying a standard for soldiers based on their lower organ healthy condition during a cycle of marching.

Figure 1 .
Figure 1.The model used in simulation with seven angles and hip coordinates

Figure 2 .
Figure 2. Foot-ground contact force simulated by spring and damper which Ki and Ci are constant coefficients, θ i m and θ ̇i m are angle and angular speed of links in each time frame,    and  ̇  are desired angle and angular speed of the links.Spring and damper are key factors in stability of the model during marching.If these factors are neglected, any small disturbance in the model at the contact time leads to instability of the whole system.On the other hand, calculating the optimized coefficients requires desired kinematic data and stability of whole model.

Figure 3 .
Figure 3. Marching benchmark used in kinematic section of this study

Figure 4 .
Figure 4. Hip and ankle angles during a gait cycle of the benchmark

Figure 5 .
Figure 5. Variation of waist and knee heights by hip torque increasing in top 10 candidates with lowest torque amount in hip joint

Figure 6 .
Figure 6.Variation of waist and knee heights by ankle torque increasing in top 10 candidates with lowest torque amount in ankle joint

Table 7 . Top 10 candidates with lowest Minimal Summation Torque Hip torque
Table 8 represents all of them in one separate table containing corresponding anthropometric data.