Mechanical energy transfer by internal force during the swing phase of running        

Name: Nanako Yamazaki

Affiliation: Graduate School of Media and Governance


Running is a cyclic motion in which each leg has two phases, stance and swing. The energy generated by the muscular forces is dominant in the phase between initial contact and mid-stance. Previous researches have concentrated on the contribution of muscular activity during running. However, researches in the field of robotic proved that natural efficient gait could be achieved without using any actuators. The inexistence of the actuators is equivalent to using non-muscular internal forces in human. During the swing phase of running, non-muscular internal forces act to transfer mechanical energy between segments. Thus, in this research, we focused on how non-muscular (e.g. centrifugal and gravity) forces of each link generate, absorb, and transfer the mechanical energy. A mathematical analysis of the lower limb's movement in the swing phase of running at different speeds was carried out. A multi-body power analysis was performed using dynamical equations of a three-dimensional double pendulum to find the mechanism to behind efficient running form. Results show that the non-muscular forces acting on the hip joint is supplying energy. The internal force acting on the knee joint works throughout the swing phase to transfer the supplied energy. This energy is used to achieve the natural swing motion of the lower limb. It is also noted that the hip and knee joint muscular torque are very small in comparison to the internal force. However, the small muscular power achieves parametric excitation to maximize the energy needed at foot contact. The flow of the internal forces coordinates joint muscular activity to minimize its usage. Understanding the transfer mechanism using the internal forces could lead to economical running forms that prevent injuries and enhance the performance of the runners.


1. Introduction

Passive walking is one of the popular topics in robotics and biomechanical control of human locomotion. A mechanical system model of an inverted double pendulum with no actuator was developed to simulate the legs movement during walking and methods for generating dynamic gait using parametrically excited telescopic leg actuation were found [1]. These studies gave us more insight on human locomotion and motivated us to find efficient running forms that ensure economical energy transfer through the leg segments. This may possibly be achieved by finding the mechanism to propagate the energy without the use of muscular power. Therefore leading to better performance and avoid injuries such as hamstring injuries that occur mostly during the terminal swing phase [2]. Leg swing in running is a combination of active and passive mechanics [3]. Muscle actions are required for initiating and propagating leg swing and they are also needed for maintaining flexed knee and ankle joint positions during swing and arresting leg motion at the end of the swing phase [3]. However, non-muscular internal forces may play a role in transferring energy between segments. In this research, we focused on the internal forces which do not contribute to any mechanical work in the entire system but act as an energy pipeline between segments during the swing phase.We used inverse dynamics and multi-body power analysis [4] to prove how these internal forces are generated at the joints to transfer energy and the optimum position of the leg which maximizes this energy transfer from the proximal to the distal segments.    



  x0,  xg1,  x1,  xg2                                    Position vectors of the right hip joint, center of mass of the                                                     thigh link, Knee joint and center of mass of the shank link,                                                     respectively.

  m1, m2                          Mass of thigh and shank, respectively (mass of the foot                                                           included in m2).

  τ1, τ2                                       Torque vectors of hip joint and knee joint, respectively.

  lg1                                           Length from hip joint to the thigh's center of gravity.

  lg2                                          Length from the Knee joint to the shank's center of gravity.

  l1                                                        Length from the hip joint to the knee joint.

,                  Local unit vector of the thigh and the shank respectively.

 J1 , J2                                       Moment of inertia tensor of thigh and shank respectively.

 ω1, ω2                                      Angular velocity vector of the thigh and shank,                                                                     respectively.

 F1, F2                                       Force vector on hip joint and knee joint, respectively.

 g                                             Gravitational acceleration vector.


2. Coordinate frames

  We set a local moving coordinate frame on each segment of the leg. The local coordinate frames of the thigh and the shank are defined as (X',Y',Z') and (X”,Y”,Z”) respectively. As shown in Fig.1(a), the unit vectors j1, j2 of the Y',Y'' axes are defined as the longitudinal vector parallel to the coronal plane  and the unit vectors k1,k2 of the Z',Z'' axes are defined as the cross product of the horizontal vector that is parallel to the transverse plane and  j1, j2. Finally, the unit vectors i1, i2 of the X', X'' axes are defined as j1×k1  and  j2×k2.

3. The dynamic equation of three dimensional double pendulum

For the analysis, the leg was modelled as a two linked segments (thigh and shank) with two rotary joints each joint has three degrees of freedom (Fig.2.(b)). For the purpose of the analysis, the shank and the foot were considered as a one rigid link. Rigid body kinematical equations were used to obtain the velocities and accelerations.

By taking the second derivative of the position vectors, the accelerations can be written as follows:

Newton's equations were used to obtain the forces F1 and F2 as follows:   

by substituting (3) in (5) the internal force can be decomposed into:

Euler's equations of rotational dynamics were used to obtain the net torque vectors:


whereare angular accelerations in the local frames of the thigh and the shank, respectively.

The mechanical energy of the shank is defined as: 

where T2, U2 are the kinetic energy and the potential energy of the shank, respectively.

By differentiating (9) the power of the shank is obtained :


the inertial moment is zero [4] thus


thus Eq. (10) becomes:

The mechanical energy of the thigh is:


differentiating (12) to obtain the power of the thigh segment:

The total power of the entire system is :


Eq.(16) indicates that the external forces (,,) affect the production and absorption of the energy of the entire system while the internal force () is cancelled out. Therefore, this indicates that the internal force () transfers the energy and works as an energy pipeline along the subsequent segments. However, it does not affect the energy of the entire system.


4. Experimental methods

Four male subjects with the following average characteristics (Height= 175.7±8.1, Weight=69.7±11.4, Age=20.5±2.4) were asked to run along a 30 m path in a sports gymnasium at three self selected speeds at a jog, a comfortable run and a fast run . Twenty two reflective markers were placed on both of the lower limbs. Kinematic data was measured using motion capture system (Vicon Mx System) at a sampling rate of 500Hz.  

  a)                                   b)        

Fig. 1.a) Local coordinates systems of the thigh and shank links; Fig. 2.b) double pendulum model of the right leg with the forces and moments acting on both links.

5. Results and discussion

 The swing phase is defined from toe-off of the right foot until contact of the same foot (Fig.4.(c)). The results shown below are for the right leg of one of the subjects. The force (Fig.2.(a), (b)) and the moment (Fig.3.(a),(b)) generated at the knee joint are much smaller than those generated at the hip joint. Therefore this indicates that, in order to increase the swing speed efficiently it is better to increase the angular velocity of the thigh link to obtain a centrifugal force at the thigh that generates a pulling force to rotate the shank link. This way we can increase the swing speed with minimum amount of muscular force at the knee joint.

a)      b)           

Fig. 2.a) components of force F1 at the hip joint in (X',Y',Z'); Fig. 1.b) components of  internal force F2 at the Knee joint in (X'',Y'',Z'')

 As shown in Fig.4.(a) the power () is influential in the first half of the swing and it reaches a peak value at about 0.52s. At the same time the power component () reaches a maximum value. Its also noted from Eq. (16) that the power term () does not contribute in the total power of the system () but is included in the power of both links individually. Furthermore, the external forces (,,) contributes to the production and absorption of the energy of the entire system. Although the power () is a scalar quantity, it is the result of the inner product of two vectors. From Eq. (11) we write:



       a)   b)       

Fig. 3.a) components of the thigh link moment τ1 in local coordinate system (X'',Y'',Z''); Fig.3.b) components of the shank link moment τ2 in the local coordinate system (X',Y',Z')

By performing multi-body analysis for Eq. (17) the power can be broken down into its vector components. The Y''-axis components of the internal force () is influentially great (Fig.2.(a)) thus analysis of the power components in the Y''-axis was performed. From Eq. (6) and Eq.(17) the power from the internal force () can be split into five components.

where , pj21 is the power generated by the velocity of the hip and gravitational acceleration, Pj22 is the power generated by the angular acceleration of the thigh link,  pj23 is the power generated by the centrifugal acceleration of the thigh link, pj24 is the power generated by the angular acceleration of the shank and it is equal to zero, pj25 is the power generated by the centrifugal acceleration of the shank link. As shown in Fig.4.(d) the power pj25 has two local maximum peaks, the first at approximately around 0.054s which corresponds to the peak value of shank power (Fig.4.(a)) this indicates the internal force () is responsible for the propagation of the rotational power () that generates a centrifugal force at the shank link. In the second half of the swing at around 0.21s as the contralateral foot contacts the ground, the power () of the swinging leg decreases to reach a minimum. At approximately 0.25s the second maximum peak of shank's centrifugal force is generated from propagation of the pulling force caused by the angular acceleration of the thigh. It is noted here that the power generated by the moment () is small and has very little effect on the rotation of the shank. Thus the rotation of the shank may be solely dependent on the internal forces therefore it is possible to find  efficient running by transferring the energy through these internal forces.

6. Conclusion

  In order to increase the leg swing speed, it is important to control the angular velocity about the hip joint in the first half of the swing before the knee extension commences, which in turns increases the centrifugal force at the knee to swing the shank forward. In the second half of the swing after the contact of the contralateral leg, the power decreases to a minimum. However, there is a supply of power by the angular acceleration of the thigh which attempts to complete the leg swing propagate the energy through to the shank to generate more centrifugal force. This indicates the existence of a possible energy transfer caused by impact of the contact of the contralateral foot. It should be emphasized here that the rate of energy generated by moment in the knee joints were insignificant in comparison to the transferred energy through internal forces. In the future, this energy transfer in professional runners will be the focus of investigation to find  more efficient mechanism for running with minimum amount of muscular force.

a)    b)   

c)       d)     

Fig.4.a) Total power  and powers at the thigh  and at the shank  ; Fig.4.b) Power components  and  from equation(23), (26); Fig.4.c) stick figure of the swing phase; Fig.4.d) Components of power  in the Y''axis in the local coordinate system(X'',Y'',Z'') ; 


[1]Asano F, Luo Z.W. Energy-efficient and high-speed dynamic biped locomotion based on principle of parametric excitation. IEEE Transaction on robotics (2006): 24- 6:1289-1301

[2] Schache A.G,Wrigley T.V, Baker R, Pandy M.G. Biomechanical response to hamstring muscle strain injury. Gait & posture (2008): 29-2: 332-338.

[3] Modica J, Kram R. Metabolic energy and muscular activity required for leg swing in running. J Appl Physiol (2005) : 98 : 2126–2131

[4] Ohta K, Ohgi Y, Shibuya K. Multi-body power analysis of golf swing based on a double pendulum with moving pivot. 9th Conference of the International Sports Engineering Association (ISEA)(2012)